HEAVY FLAVOUR PHYSICS THEORY AND EXPERIMENTAL RESULTS IN HEAVY QUARK PHYSICS AND CP VIOLATION HEAVY FLAVOUR PHYSICS THEORY A ND EXPERIMENTAL RESULTS IN HEAVY QUARK PHYSICS AND CP VIOLATION Proceedings of the Fifty-Fifth t Scottish tUniversities Summer School in Physics, St. Andrews, 7 August - 23 August 2001. A NATO Advanced Study Institute Edited by C Daavies-vies — Unive:p~ity nivefs of Glasgow S M Pllayfer-ayfer — University of Edinburgh Series Editor P Osbosborne-rne — University of Edinburgh Copublished by Scottish Universities Summer School in Physics & The Open Access version of this book, available at www.taylorfrancis.com, has been made available under a Creative Commons Attribution-Non Commercial 4.0 International.DOI: 10.1201/9780429187056Open Access funded by SCOAP3Copyright 2021 C Davies, S M Playfer, P OsborneThis eBook was converted to open access in 2021 through the sponsorship of SCOAP3 licensed under the terms of the creative commons Attribution-NonCommercial 4.0 International License (https://creativecommons.org/licenses/by-nc/4.0/) which permits use, sharing, adaptation distribution and reproduction in any medium or format, as long as you give appropriate credit to the author(s) and the source, provide a link to the creative commons license and indicate if changes were made, this license does not permit the Contribution to be used commercially.SUSSP Proceedings 1 1960 Dispersion Relations 2 1961 Fluctuation, Relaxation and Resonance in Magnetic Systems 3 1962 Polarons and Excitons 4 1963 Strong Interactions and High Energy Physics 5 1964 Nuclear Structure and Electromagnetic Interactions 6 1965 Phonons in Perfect and Imperfect Lattices 7 1966 Particle Interactions at High Energy 8 1967 Methods in Solid State and Superfluid Theory 9 1968 Physics of Hot Plasmas 10 1969 Quantum Optics 11 1970 Hadronic Interactions of Photons and Electrons 12 1971 Atoms and Molecules in Astrophysics 13 1972 Properties of Amorphous Semiconductors 14 1973 Phenomenology of Particles at High Energy 15 1974 The Helium Liquids 16 1975 Non-linear Optics 17 1976 Fundamentals of Quark Models 18 1977 Nuclear Structure Physics 19 1978 Metal Non-metal Transitions in Disordered Solids 20 1979 Laser-Plasma Interactions: 1 21 1980 Gauge Theories and Experiments at High Energy 22 1981 Magnetism in Solids 23 1982 Laser-Plasma Interactions: 2 24 1982 Lasers: Physics, Systems and Techniques 25 1983 Quantitative Electron Microscopy 26 1983 Statistical and Particle Physics 27 1984 Fundamental Forces 28 1985 Superstrings and Supergravity 29 1985 Laser-Plasma Interactions: 3 30 1985 Synchrotron Radiation 31 1986 Localisation and Interaction 32 1987 Computational Physics 33 1987 Astrophysical Plasma Spectroscopy 34 1988 Optical Computing 35 1988 Laser-Plasma Interactions: 4 /continued v SUSSP Proceedings (continued) 36 1989 Physics of the Early Universe 37 1990 Pattern Recognition and Image Processing 38 1991 Physics of Nanostructures 39 1991 High Temperature Superconductivity 40 1992 Quantitative Microbeam Analysis 41 1992 Spatial Complexity in Optical Systems 42 1993 High Energy Phenomenology 43 1994 Determination of Geophysical Parameters from Space 44 1994 Simple Quantum Systems 45 1994 Laser-Plasma Interactions 5: Inertial Confinement Fusion 46 1995 General Relativity 47 1995 Laser Sources and Applications 48 1996 Generation and Application of High Power Microwaves 49 1997 Physical Processes in the Coastal Zone 50 1998 Semiconductor Quantum Optoelectronics 51 1998 Muon Science 52 1998 Advances in Lasers and Applications 53 1999 Soft and Fragile Matter 54 2000 The Restless Universe 55 2001 Heavy Flavour Physics 56 2002 Ultrafast photonics 57 2003 LHC Phenomenology vi Lecturers Steve Abel Institute for Particle Physics Phenomenology, Durham, UK Gerhard Buchalla CERN, Geneva, Switzerland Christine Davies University of Glasgow, UK Konrad Kleinknecht University of Mainz, Germany Lawrence Krauss Case Western University, Cleveland, USA Peter Krizan University of Llubljana, Slovenia Tatsuya Nakada University of Lausanne, Switzerland Harry Nelson University of California, Santa Barbara, USA Yosef Nir Weizmann Institute, Rehovoth, Israel Ken Peach University of Edinburgh & Rutherford Appleton Laboratory, UK Jonathan Rosner University of Chicago, USA Klaus Schubert Technical University, Dresden, Germany Sheldon Stone Syracuse University, USA vii Organising Committee Professor K J Peach University of Edinburgh Co-Director Profssor Y Nir Weizmann Institute Co-Director Mr A Walker University of Edinburgh Secretary Dr C Froggatt University of Glasgow Treasurer Professor C Davies University of Glasgow Co-Editor Dr S M Playfer University of Edinburgh Co-Editor Dr A Khan University of Edinburgh Steward Dr P Soler University of Glasgow Steward Ms L O'Donnell University of Edinburgh Administration viii Preface The timing of a school on Heavy Flavour Physics in August 2001 was particularly fortunate. Both BaBar and Belle had reported in the previous month new results with clear evidence for CP-violation in Bd decays. The Standard Model had been subjected to a qualitatively and quantitatively new test, and passed with flying colours. The result on sin 20 ^ 0.7 ± 0.2 was in excellent agreement with the Standard Model expectations based on non-CP violating processes of sin 2,3 of about 0.7. The CP-violation observed more than 35 years ago in the neutral kaon system was confirmed as coming, at least predominantly, from the complex phase in the quark mixing matrix. Of course, there is more, much more, to Heavy Flavour Physics than CP violation and the b-quark systems, and the course of lectures reflected this—'heavy flavour' was interpreted generously to include the c-quark and as appropriate the s-quark. The subject was approached from a strongly phenomenological point of view. Jonathan Rosner gave an overview of the Standard Model and its development, Gerhard Buchalla introduced the topic of Heavy Quark Theory and Christine Davies explained how these ideas could be implemented on the lattice. Yosef Nir gave a thorough introduction to CP-violation, while Steve Abel introduced concepts beyond the current Standard Model. Sheldon Stone gave an overview of B-phenomenology, and the present and future exper-imental programme was reviewed by Klaus Schubert (B-factories), Peter Krizan (HERA and the Tevatron), and Tatsuya Nakada (LHCb and BTeV). Harry Nelson discussed the opportunities for observing physics beyond the Standard Model in the charm sector, and Konrad Kleinknecht discussed the status of CP-violation in the neutral kaon system. Two topics strictly outside the domain of heavy flavour, but with significant relevance were covered. Lawrence Krauss gave an introduction to cosmology, and Ken Peach discussed the possibility that CP-violation might one day be measurable in the neutrino sector at a neutrino factory. We believe that this series of lectures will provide a thorough introduction to the phenomenology of heavy flavour physics, not only those working on the B-factories, LHCb or BTeV, but also those working on HERA, the Tevatron and eventually the LHC general purpose detectors. While the wealth of data from the B-factories and the Tevatron will lead to new discoveries and, we hope, clues to the physics beyond the Standard Model, it is already clear that the model provides a very good description of the data. Meanwhile, these lectures will be a useful guide to these processes, and will enable the student, whether theorist or experimentalist, to judge the significance of these developments as they unfold. Summer schools are not just about science: they are also about dialogue, discussion, meeting people, and making friends. The school succeeded in this secondary aim, aided by full social programme and a friendly environment provided by the staff of the John Burnet Hall. Within this, the scientific discussions and personal interactions flourished. ix The Organizing Committee acknowledge the support of NATO, PPARC (the UK Particle Physics and Astronomy Research Council), the National Science Foundation. the Scottish Universities Summer Schools in Physics and the Physics and Astronomy Departments of the Universities of Edinburgh, Glasgow and St. Andrews, without which the school would not have been possible. We would also like to thank all of the lecturers and participants for their enthusiasm, for physics and for life, which helped make this a truly memorable school. Ken Peach and Yossi Nir Co-Directors, January 2002 Editors' Note We regret that the lectures presented by Harry Nelson, Lawrence Krauss and Ken Peach are not included in this volume. Christine Davies and Steve Plavfer Editors, April 2002 Contents Standard Model The Standard Model in 2001 1 Jonathan Rosner Heavy Quark Theory & CP Violation Heavy Quark Theory 57 Gerhard Buchalla Lattice QCD 105 Christine Davies CP Violation 147 Yosef Nir Supersymmetry 201 Steve Abel Experimental Results Phenomenology of B Decays 237 Sheldon Stone B Experiments at HERA and the Tevatron 303 Peter Krizan Kaon Decay Experiments 327 Konrad Kleinknecht Results from e+e— B-factories 345 Klaus Schubert B Physics at the LHC 373 Tatsuya Nakada Index 391 xi 1 The Standard Model in 2001 Jonathan L Rosner Enrico Fermi Institute and Department of Physics University of Chicago, USA 1 Introduction The "Standard Model" of elementary particle physics encompasses the progress that has been made in the past half-century in understanding the weak, electromagnetic, and strong interactions. The name was apparently bestowed by my doctoral thesis advisor, Sam Treiman, whose dedication to particle physics kindled the light for so many of his students during those times of experimental and theoretical discoveries. These lectures are dedicated to his memory. As graduate students at Princeton in the 1960s, my colleagues and I had no idea of the tremendous strides that would be made in bringing quantum field theory to bear upon such a wide variety of phenomena. At the time, its only domain of useful application seemed to be in the quantum electrodynamics (QED) of photons, electrons, and moons. Our arsenal of techniques for understanding the strong interactions included analyt-icity, unitarity, and crossing symmetry (principles still of great use), and the emerging SU(3) and SU(6) symmetries. The quark model (Gell-Mann 1964, Zweig 1964) was just beginning to emerge, and its successes at times seemed mysterious. The ensuing decade gave us a theory of the strong interactions, quantum chromo-dynamics (QCD), based on the exchange of self-interacting vector quanta. QCD has permitted quantitative cal-culations of a wide range of hitherto intractable properties of the hadrons (Lev Okun's name for the strongly interacting particles), and has been validated by the discovery of its force-carrier, the gluon. In the 1960s the weak interactions were represented by a phenomenological (and un-renormalisable) four-fermion theory which was of no use for higher-order calculations. Attempts to describe weak interactions in terms of heavy boson exchange eventually bore fruit when they were unified with electromagnetism and a suitable mechanism for generation of heavy boson mass was found. This electroweak theory has been spectacu-larly successful, leading to the prediction and observation of the W and Z bosons and to precision tests which have confirmed the applicability of the theory to higher-order calculations. DOI: 10.1201/9780429187056-1T S 2 Jonathan L Rosner In this introductory section we shall assemble the ingredients of the standard model —the quarks and leptons and their interactions. We shall discuss both the theory of the strong interactions, quantum chromo-dynamics (QCD), and the unified theory of weak and electromagnetic interactions based on the gauge group SU(2) 0 U(1). Since QCD is an unbroken gauge theory, we shall discuss it first, in the general context of gauge theories in Section 2. We then discuss the theory of charge-changing weak interactions (Section 3) and its unification with electromagnetism (Section 4). The unsolved part of the puzzle, the Higgs boson, is treated in Section 5, while Section 6 concludes. These lectures are based in part on courses that I have taught at the University of Minnesota and the University of Chicago, as well as at summer schools (e.g., Rosner 1988, 1997). They owe a significant debt to the fine book by Quigg (1983). Quarks Q=2/3 Q-1/3 t Le pt onS Q=-1 Q=0 2 cu CD —4 0 —6 an 0 —8 0 —2 Figure 1. Patterns of charge-changing weak transitions among quarks and leptons. The strongest inter-quark transitions correspond to the solid lines, with dashed, dot-dashed, and dotted lines corresponding to successively weaker transitions. 1.1 Quarks and leptons The fundamental building blocks of strongly interacting particles, the quarks, and the fundamental fermions lacking strong interactions, the leptons, are illustrated, along with their interactions, in Figure 1. Masses, as quoted by the Particle Data Group (2000), are summarised in Table 1. The relative strengths of the charge-current weak transitions between the quarks are summarised in Table 2. The quark masses quoted in Table 1 are those which emerge when quarks are probed at distances short compared with 1 fm, the characteristic size of strongly interacting particles and the scale at which QCD becomes too strong to utilise perturbation theory. When regarded as constituents of strongly interacting particles, however, the u and d quarks act as quasi-particles with masses of about 0.3GeV. The corresponding "constituent-quark" masses of s, c, and b are about 0.5, 1.5, and 4.9GeV, respectively. Standard Model 3 Quarks Leptons Charge 2/3 Charge —1/3 Charge —1 Charge 0 Mass Mass Mass Mass u 0.001-0.005 d 0.003-0.009 e 0.000511 ve < 3 eV c 1.15-1.35 s 0.075-0.175 it 0.106 lip < 190 keV t 174.3 ± 5.1 b 4.0-4.4 T 1.777 v,- < 18.2 MeV Table 1. The known quarks and leptons. Masses in GeV except where indicated other-wise. Here and elsewhere we take c = 1. Relative amplitude Transition Source of information (example) ,-, 1 u H d Nuclear 13-decay ,-- 1 c H s Charmed particle decays -, 0.22 u <4 s Strange particle decays ,--, 0.22 c H d Neutrino prod. of charm ,-- 0.04 c H b b decays ,-, 0.003-0.004 u H b Charmless b decays 1 t H b Dominance of t H 1/17b ,--, 0.04 t 44 s Only indirect evidence 0.01 t H d Only indirect evidence Table 2. Relative strengths of charge-changing weak transitions. 1.2 Colour and quantum chromo-dynamics The quarks are distinguished from the leptons by possessing a three-fold charge known as "colour" which enables them to interact strongly with one another. (A gauged colour symmetry was first proposed by Nambu 1966.) We shall also speak of quark and lepton "flavour" when distinguishing the particles in Table 1 from one another. The experimental evidence for colour comes from several quarters. 1. Quark statistics. One of the lowest-lying hadrons is a particle known as the A++, an excited state of the nucleon first produced in 7r+p collisions in the mid-1950s at the University of Chicago cyclotron. It can he represented in the quark model as uuu, so it is totally symmetric in flavour. It has spin J = 3/2, which is a totally symmetric combination of the three quark spins (each taken to be 1/2). Moreover, as a ground state, it is expected to contain no relative orbital angular momenta among the quarks. This leads to a paradox if there are no additional degrees of freedom. A state composed of fermions should be totally antisymmetric under the interchange of any two fermions, but what we have described so far is totally symmetric under flavour, spin, and space 4 Jonathan L Rosner interchanges, hence totally symmetric under their product. Colour introduces an addi-tional degree of freedom under which the interchange of two quarks can produce a minus sign, through the representation A++ cooPubu'. The totally antisymmetric product of three colour triplets is a colour singlet. 2. Electron-positron annihilation to hadrons. The charges of all quarks which can be produced in pairs below a given centre-of-mass energy is measured by the ratio o-(e+e- hadrons) R= E Q, • (1) (e+ e- --+ ft+ p-) For energies at which only 11{1, dd, and sS can be produced, i.e., below the charmed-pair threshold of about 3.7GeV, one expects R = Arc [G)2 (-31)2 ± (-31)2] = Are (2) for N, "colours" of quarks. Measurements first performed at the Frascati laboratory in Italy and most recently at the Beijing Electron-Positron Collider (Bai et al. 2001: see Figure 2) indicate R = 2 in this energy range (with a small positive correction associated with the strong interactions of the quarks), indicating N, = 3. 6 z cc> 5.5 5 4.5 4 3.5 3 2.5 2 1.5 2 3 4 5 Em (GeV) Figure 2. Values of R measured by the BES Collaboration. 3. Neutral pion decay. The 7r° decay rate is governed by_a quark loop diagram in which two photons are radiated by the quarks in 7r° = (uu — dd)/\. The predicted rate is S2 7n3 ( \ 2 r (7° 77) = 87rf, 27r) (3) Standard Model 5 where f„ = 131MeV and S = Ne(C4—Q3) = N,/3. The experimental rate is 7.8±0.6 eV, while (3) gives 7.652 eV, in accord with experiment if S = 1 and N, = 3. 4. Triality. Quark composites appear only in multiples of three. Baryons are composed of qqq, while mesons are qq (with total quark number zero). This is compatible with our current understanding of QCD, in which only colour-singlet states can appear in the spectrum. Thus, mesons Al and baryons B are represented by 1 1 = B (cake qlbe, ). 0,1 Val Direct evidence for the quanta of QCD, the gluons, was first presented in 1979 on the basis of extra "jets" of particles produced in electron-positron annihilation to hadrons. Normally one sees two clusters of energy associated with the fragmentation of each quark in e+e- into hadrons. However, in some fraction of events an extra jet was seen, corresponding to the radiation of a gluon by one of the quarks. The transformations which take one colour of quark into another are those of the group SU(3). We shall often refer to this group as SU(3),01our to distinguish it from the SU (3) flavour associated with the quarks u, d, and s. 1.3 Electroweak unification The electromagnetic interaction is described in terms of photon exchange, for which the Born approximation leads to a matrix element behaving as 1/q2. Here q is the four-momentum transfer, and q2 is its invariant square. The quantum electrodynamics of photons and charged point-like particles (such as electrons) initially encountered calcula-tional problems in the form of divergent quantities, but these had been tamed by the late 1940s through the procedure known as renormalisation, leading to successful estimates of such quantities as the anomalous magnetic moment of the electron and the Lamb shift in hydrogen. By contrast, the weak interactions as formulated up to the mid-1960s involved the point-like interactions of two currents, with interaction Hamiltonian = GF,/,./14 /Na where CF = 1.16637(1) x 10-5GeV-2 the current value for the Fermi coupling constant. This interaction is very singular and cannot be renormalised. The weak currents in this theory were purely charge-changing. As a result of work by Lee and Yang, Feynman and Gell-Mann, and Marshak and Sudarshan in 1956-7 they were identified as having (vector)-(axial) or "V—A" form. Hideki Yukawa (1935) and Oskar Klein (1938) proposed a boson-exchange model for the charge-changing weak interactions. Klein's model attempted a unification with elec-tromagnetism and was based on a local isotopic gauge symmetry, thus anticipating the theory of Yang and Mills (1954). Julian Schwinger and others studied such models in the 1950s, but Glashow (1961) was the first to realise that a new neutral heavy boson had to be introduced as well in order to successfully unify the weak and electromagnetic interactions. The breaking of the electroweak symmetry (Weinberg 1967, Salam 1968) via the Higgs (1964) mechanism converted this phenomenological theory into one which could be used for higher-order calculations, as was shown by 't Hooft and Veltman in the early 1970s. (4) 6 Jonathan L Rosner The boson-exchange model for charge-changing interactions replaces the Fermi in-teraction constant with a coupling constant g at each vertex and the low-q2 limit of a propagator, 1/(A, - q2) -> 1/Me,v, with factors of 2 chosen so that GF/Nfi = g2 /81.g, The q2 term in the propagator helps the theory to be more convergent, but it is not the only ingredient needed, as we shall see. The normalisation of the charge-changing weak currents Ju suggested well in advance of electroweak unification that one regard the corresponding integrals of their time com-ponents (the so-called weak charges) as members of an SU(2) algebra (Gell-Mann and Levy 1960, Cabibbo 1963). However, the identification of the neutral member of this multiplet as the electric charge was problematic. In the V-A theory the W's couple only to left-handed fermions (1 - 75 )0/2, while the photon couples to /./,/, + t/,R, where OR (1 + N)1112. Furthermore, the high-energy behaviour of the v/7 -+ W± W scattering amplitude based on charged lepton exchange leads to unacceptable divergences if we incorporate it into the one-loop contribution to vi) -# vv (Quigg 1983). A simple solution was to add a neutral boson Z coupling to W+W- and vii in such a way as to cancel the leading high-energy behaviour of the charged-lepton-exchange diagram. This relation between couplings occurs naturally in a theory based on the gauge group SU(2) U(1). The Z leads to neutral current interactions, in which (for example) an incident neutrino scatters inelastically on a hadronic target without changing its charge. The discovery of neutral-current interactions of neutrinos and many other manifestations of the Z proved to be striking confirmations of the new theory. Identifying the W+ and W- with raising and lowering operations in an SU(2), so that W± = (W i TiW2 )-1, then left-handed fermions may be assigned to doublets of this "weak isospin", with /3L (u, c, t)= (ye, vi„ = +1/2 and in(d, s,b)=I3L (e-, p-, r-) = -1/2. All the right-handed fermions have /L = I3L = 0. As mentioned, one cannot simply identify the photon with W3, which also couples only to left-handed fermions. Instead, one must introduce another boson B associated with a U(1) gauge group. It will mix with the W3 to form physical states consisting of the massless photon A and the massive neutral boson Z: A = B cos 0 + W 3 sin , Z = -B sin ,9 + W3 cos . (5) The mixing angle 0 appears in many electroweak processes. It has been measured to sufficiently great precision that one must specify the renormalisation scheme in which it is quoted. For present purposes we shall merely note that sine 0 0.23. The correspond-ing SU(2) and U(1) coupling constants g and g' are related to the electric charge e by e = g sin 0 = g' cos 0, so that (6) (7) where we show the approximate experimental values. The detailed check of these predic-tions has reached the precision that one can begin to look into the deeper structure of the theory. A key ingredient in this structure is the Higgs boson, the price that had to be paid for the breaking of the electroweak symmetry. 11 1 e2 = g2 2 • The electroweak theory successfully predicted the masses of the WI and Z: Mw 38.6 GeV/ sin 0 80.5 GeV , Mz Mw / cos 0 91.2 GeV , Standard Model 7 1.4 Higgs boson An unbroken SU(2) 0 U(1) theory involving the photon would require all fields to have zero mass, whereas the WI and Z are massive. The symmetry-breaking which generates W and Z masses must not destroy the renormalisability of the theory. However, a massive vector boson propagator is of the form Dpv(q) = q2 — M2 where M is the boson mass. The terms qov, when appearing in loop diagrams, will destroy the renormalisability of the theory. They are associated with longitudinal vector boson polarisations, which are only present for massive bosons. For massless bosons like the photon, there are only transverse polarisation states J. = ±J. The Higgs mechanism, to be discussed in detail later in these lectures, provides the degrees of freedom needed to add a longitudinal polarisation state for each of W+, W-, and W°. In the simplest model, this is achieved by introducing a doublet of complex Higgs fields: 0 = 60 I 0* = Here the charged Higgs fields 0± provide the longitudinal component of W± and the linear combination (0° — 0°)/iv provides the longitudinal component of the Z. The additional degree of freedom (oo corresponds to a physical particle, the Higgs particle, which is the subject of intense searches. Discovering the nature of the Higgs boson is a key to further progress in understanding what may lie beyond the Standard Model. There may exist one Higgs boson or more than one. There may exist other particles in the spectrum related to it. The Higgs boson may be elementary or composite. If composite, it points to a new level of substructure of the elementary particles. Much of our discussion will lead up to strategies for the next few years designed to address these questions. First, we introduce the necessary topic of gauge theories, which have been the platform for all the developments of the past thirty years. 2 Gauge theories 2.1 Abelian gauge theories The Lagrangian describing a free fermion of mass in is £free = 17(i — T12)2/'. It is invariant under the global phase change 0 —> exp(ia)0. (We shall always consider the fermion fields to depend on x.) Now consider independent phase changes at each point: 2/) —+ E exp[ia(x)10. (10) Because of the derivative, the Lagrangian then acquires an additional phase change at each point: SLf„, = The free Lagrangian is not invariant under such changes of phase, known as local gauge transformations. + god11/12 (8) 60 - (9) 8 Jonathan L Rosner Local gauge invariance can be restored by the replacement a„ D, ieA, in the free-fermion Lagrangian, which now becomes b(i IQ — m)v) = — rn)V, - eV) 4(x)0. (11) The effect of a local phase in 1/) can be compensated if we allow the vector potential Ap to change by a total divergence, which does not change the electromagnetic field strength (defined as in Peskin and Schroeder 1995; Quigg 1983 uses the opposite sign) F„ = apA, — 0,,A, . (12) Indeed, under the transformation zG -+ 1// and with A -> A' with A' yet to be determined, we have C m)'b' er) = 0(i 0 "2)0 - 0[0a(x)]0 - 4'4'0 • (13) This will be the same as r if A;,(x)= Ap(x)- -el 0,a(x) . (14) The derivative D, is known as the covariant derivative. One can check that under a local gauge transformation, D „V, ei'( x) D p r,b Another way to see the consequences of local gauge invariance suggested by Yang (1974) and discussed by Peskin and Schroeder (1995, pp 482-486) is to define -eA,(x) as the local change in phase undergone by a particle of charge e as it passes along an infinitesimal space-time increment between xt‘ and xi' + dxP. For a space-time trip from point A to point B, the phase change is then B 4) AB = exp (-ie A,(x)dx11) . (15) The phase in general will depend on the path in space-time taken from point A to point B. As a consequence, the phase (DAB is not uniquely defined. However, one can compare the result of a space-time trip along one path, leading to a phase (I)(, )Th with that along another, leading to a phase (1)(A13. The two-slit experiment in quantum mechanics involves such a comparison; so does the Bohm-Aharonov effect in which a particle beam traveling past a solenoid on one side interferes with a beam traveling on the other side. Thus, phase differences (1)TB(1)(AT = Oc = exp (-ie Aµ(x)de) , (16) associated with closed paths in space-time (represented by the circle around the integral sign), are the ones which correspond to physical experiments. The phase 4:1)c for a closed path C is independent of the phase convention for a charged particle at any space-time point xo , since any change in the contribution to 4:1)c, from the integral up to xo will be compensated by an equal and opposite contribution from the integral departing from xo. The closed path integral (16) can be expressed as a surface integral using Stokes' theorem: Ap(r)dxP = f Fpu(x)do-Pv , (17) Standard Model 9 where the electromagnetic field strength Fin, was defined previously and do-P' is an element of surface area. It is also clear that the closed path integral is invariant under changes (14) of Aµ (x) by a total divergence. Thus Fp, suffices to describe all physical experiments as long as one integrates over a suitable domain. In the Bohm-Aharonov effect, in which a charged particle passes on either side of a solenoid, the surface integral will include the solenoid (in which the magnetic field is non-zero). If one wishes to describe the energy and momentum of free electromagnetic fields, one must include a kinetic term LK = —(1/4)F,,,FP' in the Lagrangian. which now reads = --4F Fiw V)(z — 700 — 410. 1 (18) If the electromagnetic current is defined as Jpe m Tly-y,O, then this Lagrangian leads to Maxwell's equations. The local phase changes (10) form a U(1) group of transformations. Since such trans-formations commute with one another, the group is said to be Abelian. Electrodynamics, just constructed here, is an example of an Abelian gauge theory. 2.2 Non-Abelian gauge theories One can imagine that a particle traveling in space-time undergoes not only phase changes, but also changes of identity. Such transformations were first considered by Yang and Mills (1954). For example, a quark can change in colour (red to blue) or flavour (u to d). In that case we replace the coefficient eitp of the infinitesimal displacement dx, by an n x n matrix —g.A.„(x) (x)T, acting in the n-dimensional space of the particle's degrees of freedom. (The sign change follows the convention of Peskin and Schroeder 1995.) For colours, n = 3. The T, form a linearly independent basis set of matrices for such transformations, while the A are their coefficients. The phase transformation then must take account of the fact that the matrices Aµ (x) in general do not commute with one another for different space-time points, so that a path-ordering is needed: OAB = P [exp (ig fA At, (x)daP)] (19) When the basis matrices T, do not commute with one another, the theory is non-Abelian. We demand that changes in phase or identity conserve probability, i.e., that 4)AB be unitary: OtABO AB = 1. When 4)AB is a matrix, the corresponding matrices Aµ (x) in (19) must be Hermitian. If we wish to separate out pure phase changes, in which Aµ (x) is a multiple of the unit matrix, from the remaining transformations, one may consider only transformations such that det(4)AB) = 1, corresponding to traceless A„(x). The n x n basis matrices T, must then be Hermitian and traceless. There will be — 1 of them, corresponding to the number of independent SU(N) generators. (One can generalise this approach to other invariance groups.) The matrices will satisfy the commutation relations [Ti, T jj = iCiikTk (20) where the Cijk are structure constants characterising the group. For SU(2), cijk = Eijk (the Kronecker symbol), while for SU(3), cijk = fink , where the fink are defined in Gell-Mann 10 Jonathan L Rosner and Ne'eman (1964). A 3 x 3 representation in SU(3) is Ti = A1 /2, where A,/2 are the Gell-Mann matrices normalised such that Tr )03 = 26,3 . For this representation, then, Tr TiT3 = In order to define the field-strength tensor F„, = Fp'„T, for a non-Abelian transfor-mation, we may consider an infinitesimal closed-path transformation analogous to (16) for the case in which the matrices Aµ (x) do not commute with one another. The result (see, e.g., Peskin and Schroeder 1995, pp 486-491) is F„, = 8µA — — , Fp, = 0„24i, — + gc,,k Np Ak, . (21) An alternative way to introduce non-Abelian gauge fields is to demand that, by analogy with (10), a theory involving fermions zi) be invariant under local transformations 0(x) —> Oi(x) = U(x)0(x), UtU = 1 , (22) where for simplicity we consider unitary transformations. Under this replacement we find that G C, where .C1 (i — m)1,b1 = OU-1(i — m)U0 = 0(i 0 — 711)0 + i0U-17P (aliU)0 • (23) As in the Abelian case, an extra term is generated by the local transformation. It can be compensated by replacing a„ by —> D, a, — ig.k(r). (24) In this case r = tP(i D — m)0 and under the change (22) we find p/ - rn)0' = + g 4ti rn)uo = + ti,[g(t1-1 Aw— + iu-'(0u)]0• (25) This is equal to r if we take Aim = — (0,U)U-1 . (26) This reduces to our previous expressions if g = —e and U = e''( x ) . The covariant derivative acting on 0 transforms in the same way as 0 itself under a gauge transformation: 13,0 —> = UD,O. The field strength F„, transforms as Fµ —> = UF,,U-1. It may be computed via [Dµ, D"] = —igF,„; both sides transform as U( )(1-1 under a local gauge transformation. In order to obtain propagating gauge fields, as in electrodynamics, one must add a kinetic term L A- = —(114)Fp',Fi" to the Lagrangian. Recalling the representation Fp, = Foi,, in terms of gauge group generators normalised such that Tr(TiT3 ) = 6,3 /2, we can write the full Yang-Mills Lagrangian for gauge fields interacting with matter fields as = --2Tr(F mu FP'') + 1/)(i p - m),0 . (27) We shall use Lagrangians of this type to derive the strong, weak, and electromagnetic interactions of the "Standard Model." Standard Model 11 The interaction of a gauge field with fermions then corresponds to a term in the interaction Lagrangian AG = gi71)(x)-yit .A,(x)0(x). The [A,,Au] term in F,, leads to self-interactions of non-Abelian gauge fields, arising solely from the kinetic term. Thus, one has three- and four-field vertices arising from = )gcijk.4'`' Aek (12 kcim r, AmiA"A pm A . (28) These self-interactions are an important aspect of non-Abelian gauge theories and are responsible in particular for the remarkable asymptotic freedom of QCD which leads to its becoming weaker at short distances, permitting the application of perturbation theory. 2.3 Elementary divergent quantities In most quantum field theories, including quantum electrodynamics, divergences occurring in higher orders of perturbation theory must be removed using charge, mass, and wave function renormalisation. This is conventionally done at intermediate calculational stages by introducing a cutoff momentum scale A or analytically continuing the number of space-time dimensions away from four. Thus, a vacuum polarisation graph in QED associated with external photon momentum k and a fermion loop will involve an integral 11,,(k) f ( 14 " Tr ( 1 -y 1 (27)4 — m is+ y — m7v (29) a self-energy of a fermion with external momentum p will involve di g 1 1 (27)4 g2 'yµ _ mre and a fermion-photon vertex function with external fermion momenta p, p' will involve d4k 1 1 1 A i,(P P) f (27)4 k2 JP ± _ — 77? 7” The integral (29) appears to be quadratically divergent. However, the gauge invariance of the theory translates into the requirement kPflp, = 0, which requires 11,, to have the form 11,,,(k) = (k 2gp, — k,k,)11(k2 ) . (32) The corresponding integral for H(k2 ) then will be only logarithmically divergent. The inte-gral in (30) is superficially linearly divergent but in fact its divergence is only logarithmic, as is the integral in (31). Unrenormalised functions describing vertices and self-energies involving n B external boson lines and nF external fermion lines may be defined in terms of a momentum cutoff A and a bare coupling constant go (Coleman 1971, Ellis 1977, Ross 1978): Fljn B.n, (Pi , go, A) , (33) where pi denote external momenta. Renormalised functions FR may be defined in terms of a scale parameter p, a renormalised coupling constant g = g (go , A/ ii), and renormalisation constants ZB(A) and Zp-(A) for the external boson and fermion wave functions: F R(pi ,g, p41,1[ZE (A)]"B[ZF (A)]"FF„u 8,„,(pi , go , A). (34) (30) (31) 12 Jonathan L Rosner The scale p is typically utilised by demanding that FR be equal to some predetermined function at a Euclidean momentum p2 = —p2. Thus, for the one-boson, two-fermion vertex, we take 1 1,2 (0, p, — p) The unrenormalised function Fu is independent of p, while FR and the renormalisation constants ZB(A), ZF (A) will depend on y. For example, in QED, the photon wave function renormalisation constant (known as Z3 ) behaves as ao A2 Z3 = 1 — 71_ In —112 . (36) The bare charge e0 and renormalised charge e are related by e = e0 Z3/2 . To lowest order in perturbation theory, e < e0 . The vacuum behaves as a normal dielectric; charge is screened. It is the exception rather than the rule that in QED one can define the renormalised charge for q2 = 0; in QCD we shall see that this is not possible. 2.4 Scale changes and the beta function We differentiate both sides of (34) with respect to p and multiply by p. Since the functions Fu are independent of p, we find ( pap + Or where F"`pi B azB az,) zyzvr! , (37) (38) (39) , [I\ = (n zB p + a 0 p—ap + 0(g) 09 + nB-yB(g)+ nryB(g)iFR(pi,g,p) 0g p az8 zE = 0 , p DZF °(g)E ap -YF (g) = 7B(g)= ZB a zF a The behaviour of any generalised vertex function FR under a change of scale p is then governed by the universal functions (39). Here we shall be particularly concerned with the function OM . Let us imagine p Ay and introduce the variables t In A, 9(g, t) -a g(go,A/Ap), Then the relation for the beta-function may be written d9(g, t) = 13(9), 9(9, 0) = g(go, A/p) = g . (40) dt Let us compare the behaviour of 9 with increasing t (larger momentum scales or shorter distance scales) depending on the sign of /3(9). In general we will find 0(0) = 0. We take 0(9) to have zeroes at 9 = 0, gl , 92, .... Then: 1. Suppose /3(9) > 0. Then 9 increases from its t = 0 value 9 = g until a zero g, of /3(9) is encountered. Then 9 —> g, as t 00. 2. Suppose i3(9) < 0. Then 9 decreases from its t = 0 value 9 = g until a zero g of 0(9) is encountered. lim ZF2 ZB I 2T (0 p —p) p2=_ti2A o0 p2 = -p2 (35) Standard Model 13 In either case g approaches a point at which OM = 0, < 0 as t oc. Such points are called ultraviolet fixed points. Similarly, points for which /3(g) = 0, /3'(g) > 0 are infrared fixed points, and g will tend to them for t —Do (small momenta or large distances). The point e = 0 is an infrared fixed point for quantum electrodynamics, since /3'(e)> 0at e= 0. It may happen that ,31(0) < 0 for specific theories. In that case .0 = 0 is an ultraviolet fixed point, and the theory is said to be asymptotically free. We shall see that this property is particular to non-Abelian gauge theories (Gross and Wilczek 1973, Politzer 1974). 2.5 Beta function calculation In quantum electrodynamics a loop diagram involving a fermion of unit charge contributes the following expression to the relation between the bare charge eo and the renormalised charge e: e eo (1 — ln —µ , ao A as implied by (35) and (36), where ao E 4/47. We find e03 e3 /3(e) = 1272 127r2 where differences between e0 and e correspond to higher-order terms in e. Thus /3(e) > 0 for small e and the coupling constant becomes stronger at larger momentum scales (shorter distances). We shall show an extremely simple way to calculate (42) and the corresponding result for a charged scalar particle in a loop. From this we shall be able to first calculate the effect of a charged vector particle in a loop (a calculation first performed by Khriplovich 1969) and then generalise the result to Yang-Mills fields. The method follows that of Hughes (1980). When one takes account of vacuum polarisation, the electromagnetic interaction in momentum space may be written e2 e2 (43) 0,2 q2[1 + me)] Here the long-distance (q2-40) behaviour has been defined such that e is the charge measured at macroscopic distances, so n(o) = 0. Following Sakurai (1967), we shall reconstruct 11,(q2 ) for a loop involving the fermion species i from its imaginary part, which is measurable through the cross section for e+ —> i is Im = 47ra cr(e+ ii) , (44) where s is the square of the centre-of-mass energy and a e2/47r. For fermions f of charge e f and mass m f , Im Hf (s) = —ae2f (1 + 2m2f ) (1 47n.211/20(s — 4m.2f) 3 s s (41) (42) (45) 14 Jonathan L Rosner while for scalar particles of charge es and mass ms , cle2 4 m \ 3/2 Im Hs(s) = 12 s 1 ) B(s - 4m s2 ) . (46) The corresponding cross section for el- -> p+ , neglecting the muon mass, is o-(e+ e- µ+ µ-) = 47roz2 /3s, so one can define o-(e+ ii)/ (e+ p+ p-) , (47) in terms of which Im H2 (s) = aRi (s)13. For s 00 one has R f (s) of for a fermion and Rs(s) -> es/4 for a scalar. The full vacuum polarisation function Hi(s) cannot directly be reconstructed in terms of its imaginary part via the dispersion relation Hi(s) = -1 ro ds' Im 1-1(s') , (48) 4m2 S— S since the integral is logarithmically divergent. This divergence is exactly that encountered earlier in the discussion of renormalisation. For quantum electrodynamics we could deal with it by defining the charge at q2 = 0 and hence taking 1-1, (0) = 0. The once-subtracted dispersion relation for Hi (s) - 11,(0) would then converge: 1-1,(s) = -1 r ds' ,Im (s') . (49) 77 4m2 s'(s' - s ) However, in order to be able to consider cases such as Yang-Mills fields in which the theory is not well-behaved at q2 = 0, let us instead define H,(-p2 ) = 0 at some spacelike scale q2 = -µ2. The dispersion relation is then He(s) = 1 f ds' 4m2 For 1q21 >> ft' >> m2, we find 11,(q2) 37rR,(oo) and so, from (43), the "charge at scale e2 e2 1 1 Im Hi(s') as In 2 + const.1 , . (50) (51) (52) [ s' q" may 2 e - s [1 s' + p2 2 In be written + R,(0C) 37r — 1 + Hz (q2) q21 p2 The beta-function here is defined by )3(e) = p(Oe I a p) at fixed eq. This expression gives /3(e) = -/30e3/(167r2) + 0(e5 ) and one finds 00 = -(4/3)e, for spin-1/2 fermions and 00 = -(1/3)e2, for scalars. These results will now be used to find the value of /30 for a single charged massless vector field. We generalise the results for spin 0 and 1/2 to higher spins by splitting contributions to vacuum polarisation into "convective" and "magnetic" ones. Furthermore, we take into account the fact that a closed fermion loop corresponds to an extra minus sign in (s) Standard Model 15 (which is already included in our result for spin 1/2). The "magnetic" contribution of .9 a particle with spin projection , must be proportional to S. For a massless spin-S particle, S! = 52 . We may then write /30 = (-1)"F(aS2 +2b)(S 0) , where n F = 1 for a fermion, 0 for a boson. The factor of 2b for S 0 comes from the contribution of each polarisation state (S. = +S) to the convective term. Matching the results for spins 0 and 1/2, 1 a — 3 = b, —4 — (it + 2b) , (54) we find a = 8 and hence for S =1 = 8 —3 2 22 — = • 3 The magnetic contribution is by far the dominant one (by a factor of 12), and is of opposite sign to the convective one. A similar separation of contributions, though with different interpretations, was obtained in the original calculation of Khriplovich (1969). The reversal of sign with respect to the scalar and spin-1/2 results is notable. 2.6 Group-theoretic techniques The result (55) for a charged, massless vector field interacting with the photon is also the value of 00 for the Yang-Mills group SO(3) SU(2) if we identify the photon with Ap3 and the charged vector particles with At,± (At,1 ii1,,2 )/N.. We now generalise it to the contribution of gauge fields in an arbitrary group G. The value of 30 gauge fields depends on a sum over all possible self-interacting gauge fields that can contribute to the loop with external gauge field labels i and m: G G 00[G] , 'ijk Cm k /30[SU (2) ] _SU(2) Si1(2) ijk Crajk where ci9ik is the structure constant for G, introduced in (20). The sums in (56) are proportional to (Sim: ciik cmik = 8i,C2 (A) . (57) The quantity C2 (A) is the quadratic Casimir operator for the adjoint representation of the group G. Since the structure constants for SO(3) SU(2) are just c,sjIkj(2) = E zik , one finds C2 (A) = 2 for SU(2), so the generalisation of (55) is that /30 gauge fields = (11/3) C 2 ( A) . The contributions of arbitrary scalars and spin-1/2 fermions in representations R are proportional to T(R), where Tr (TiTi ) SiiT(R) (58) for matrices Ti in the representation R. For a single charged scalar particle (e.g., a pion) or fermion (e.g., an electron), T(R) = 1. Thus 00 spin 0 = —(1/3)T0 (R), while (-1)nF(aS2 + b)(S = 0) , (53) (55) (56) 16 Jonathan L Rosner N spin 1/2 = - (4/3)Tv2 (R), where the subscript on T(R) denotes the spin. Summarising the contributions of gauge bosons, spin 1/2 fermions, and scalars, we find 11 4 1 (30 = —3 C2 (A) - 3 - E To 3 (R f ) - E—To (R.9). (59) f s One often needs the beta-function to higher orders, notably in QCD where the per- turbative expansion coefficient is not particularly small. It is 7,3 g-5 /30) = N (167r2)2 , where the result for gauge bosons and spin 1/2 fermions (Caswell 1974) is di r = -3 {17[C2 (A)j2 - 10T(R)C2 (A) - 6T(R)C2 (R)} . (61) The first term involves loops exclusively of gauge bosons. The second involves single-gauge-boson loops with a fermion loop on one of the gauge boson lines. The third involves fermion loops with a fermion self-energy due to a gauge boson. The quantity C2(R) is defined such that [Ti (R)Tz (R)]ao = C2 (R)Sao (62) where o and Q are indices in the fermion representation. We now illustrate the calculation of C2 (A), T(R), and C2(R) for SU(N). More general techniques are given by Slansky (1981). Any SU(N) group contains an SU(2) subgroup, which we may take to be generated by T1 , T2, and T3. The isospin projection 13 may be identified with T3. Then the 13 value carried by each generator Ti (written for convenience in the fundamental N-dimensional representation) may be identified as shown below: i-- 2 -- 4- N - 2 -> 0 1 1/2 • • • 1/2 -1 0 -1/2 • • • -1/2 -1/2 1/2 0 • • • 0 • • • • • • • • • • • • • • • -1/2 1/2 0 • • • 0 Since C2 (A) may be calculated for any convenient value of the index i = in in (57), we chose i = In = 3. Then 1 2 C2 (A) = E (13)2,1+1+ 4(N - 2) () =N. adjoint (63) As an example, the octet (adjoint) representation of SU(3) has two members with 1/31= 1 (e.g., the charged pions) and four with I/31 = 1/2 (e.g., the kaons). (60) Standard Model 17 For members of the fundamental representation of SU(N), there will be one member with /3 = +1/2, another with /3 = —1/2, and all the rest with /3 = 0. Then again choosing i = in = 3 in (58), we find T(R)I fundamental = 1/2. The SU(N) result for fio in the presence of n1 spin 1/2 fermions and n, scalars in the fundamental representation then may be written /3o= 3N — —3n f - 6—n3 . 11 2 1 (64) The quantity C2 (A) in (63) is most easily calculated by averaging over all indices = 3. If all generators 7" are normalised in the same way, one may calculate the result for an individual generator (say, T3 ) and then multiply by the number of generators [N2 -1 for SU(N)]. For the fundamental representation one then finds C2(R) = —N1N — 1) 2) j — 2N 1 , 2 1 \ 2 ( \ 21 N2 — 1 (65) 2.7 The running coupling constant One may integrate (60) to obtain the coupling constant as a function of momentum scale Al and a scale-setting parameter A. In terms of d g2 /47r, one has dd 2 3 /4 ( M2 cIt' == 1•'4 i (47)2' t' 2t In A2 ) . (66) For large t' the result can be written as d(m2) 47r /31 ln t'l 0(e_2) Oot' (67) Suppose a process involves p powers of d to leading order and a correction of order dp+ 1 r = AdP[1 + Bc1+ (.9(d2 )]. (68) If A is rescaled to AA, then t' t' — 21n A = t'(1 — 21n A/e), so ap PO --+ tiP (1 + ° In A) . (69) 27r The coefficient B thus depends on the scale parameter used to define (1. Many prescriptions have been adopted for defining A. In one ('t Hooft 1973), the "minimal subtraction" or MS scheme, ultraviolet logarithmic divergences are parame-terised by continuing the space-time dimension d = 4 to d = 4 — e and subtracting pole terms f d4— E/p4 1/E. In another (Bardeen et al. 1978) (the "modified minimal subtraction" or MS scheme) a term 11 In 47r — 7E = E E 2 containing additional finite pieces is subtracted. Here 7E = 0.5772 is Euler's constant, and one can show that A r,, = Ams exp[(In 47r — 7E )/2[. Many 0(d) corrections are quoted in the MS scheme. Specification of A in any scheme is equivalent to specification of d(21.12). (70) 18 Jonathan L Rosner 0.30 0.25 ,----, - \ - c\ \ - - \ - 0.20 N\ N u) _ N _ _ N t _ N .N _ N. _ N.. _ --- 0.10 5 10 20 50 100 200 M (GeV) Figure 3. Scale-dependence of the strong-coupling constant as(M2) subject to the con-straint ds(MD = 0.118 ± 0.002. The solid line shows the central value; dashed lines indicate ±10 limits. 2.8 Applications to quantum chromodynamics A "golden application" of the running coupling constant to QCD is the effect of gluon radiation on the value of R in e 4 e- annihilations. Since R is related to the imaginary part of the photon vacuum polarisation function II(s) which we have calculated for fermions and scalar particles, one calculates the effects of gluon radiation by calculating the correc-tion to II(s) due to internal gluon lines. The leading-order result for colour-triplet quarks is R(s) —> R(s)[1 + d(s)17]. There are many values of s at which one can measure such effects. For example, at the mass of the Z, the partial decay rate of the Z to hadrons involves the same correction, and leads to the estimate Cks(MD = 0.118 ± 0.002. The dependence of as(M2) satisfying this constraint on M2 is shown in Figure 3. As we shall see in Section 5.1, the electromagnetic coupling constant also runs, but much more slowly, with a-1 changing from 137.036 at q2 = 0 to about 129 at q2 = M. A system which illustrates both perturbative and non-perturbative aspects of QCD is the bound state of a heavy quark and a heavy antiquark, known as guarkonium (in analogy with positronium, the bound state of a positron and an electron). We show in Figures 4 and 5 the spectrum of the cc and bb bound states (Rosner 1997). The charmonium (c0 system was an early laboratory of QCD (Appelquist and Politzer 1975). The S-wave (L = 0) levels have total angular momentum J, parity P, and charge-conjugation eigenvalue C equal to JP(-1 = 0+- and 1-- as one would expect for 1 S0 and 3S, states, respectively, of a quark and antiquark. The P-wave (L = 1) levels have J Pe = 1+-for the 1 P1 , 0++ for the 3P0, 1++ for the 3P1 , and 2++ for the 3P2. The J Pc = 1-- levels are identified as such by their copious production through single virtual photons in e+ annihilations. The 0-+ level lie is produced via single-photon emission from the J/V, (so its C is positive) and has been directly measured to have J P compatible with 0-. Numerous studies have been made of the electromagnetic (electric dipole) transitions between the S-wave and P-wave levels and they, too, support the assignments shown. 0.15 4.5 4/(4S/5S) DD IG(2D/43) DD ifr(3s) 4.0 — DD 4/(1D) 1//(2S) DD D) 3.5 OP( 1 S) 3.0 71c(1S)4 J PC 0 0- 2 M(D) BB BB T(6S) T(5S) BB T(4S) 2 M(B) 7T71 7T7T hb(2P) Xbo(2P) Xbt (2P) Xb2 ( 2P) - h b(1P) xbo(ip) xbi(113) xb2(1p) T(1S) 3b(1 S) V b(3S) n b(2s) Standard Model 19 Figure 4. Charmonium (c0 spectrum. Observed and predicted levels are denoted by solid and dashed horizontal lines, respectively. Arrows denote electromagnetic transitions (labeled by -y) and hadronic transitions (labeled by emitted hadrons). 11.0 10.5 aU 0 CC 10.0 ‘;c1 9.5 J PC 0-' 1 -- 0- 1.. Figure 5. Spectrum of bb states. Observed and predicted levels are denoted by solid and dashed horizontal lines, respectively. In addition to the transitions labeled by arrows, numerous electric dipole transitions and decays of states below BB threshold to hadrons containing light quarks have been seen. D. IY D. D / / D' Mesons Baryons. no - 6.2 6.0 5.8 5.6 B.0 5.4 — B. B B - V7 . B 5.2 Mesons Baryons Eh* = 01) - Ab 20 Jonathan L Rosner 2.8 2.6 cgc) 2.4 Ch 2.2 N 2.0 1.8 0 1 0 1 2 IStrangenessi Figure 6. Spectrum of lowest-lying states containing one charmed and one light quark. Observed and predicted levels are denoted by solid and broken horizontal lines, respectively. 0 1 0 1 2 IStrangenessl Figure 7. Spectrum of lowest-lying states containing one bottom and one light quark. Observed and predicted levels are denoted by solid and broken horizontal lines, respectively. N CJ Standard Model 21 The bb and cc levels have a very similar structure, aside from an overall shift. The similarity of the cc and bb spectra is in fact an accident of the fact that for the interquark distances in question (roughly 0.2 to 1 fm), the interquark potential interpolates between short-distance Coulomb-like and long-distance linear behavior. The Coulomb-like behav-ior is what one would expect from single-gluon exchange, while the linear behavior is a particular feature of non-perturbative QCD which follows from Gauss' law if chroino-electric flux lines are confined to a fixed area between two widely separated sources (Nambu 1974). It has been explicitly demonstrated by putting QCD on a space-time lattice, which permits it to be solved numerically in the non-perturbative regime. States consisting of a single charmed quark and light (u, d, or s) quarks or antiquarks are shown in Figure 6. Finally, the pattern of states containing a single b quark (Figure 7) is very similar to that for singly-charmed states, though not as well fleshed-out. In many cases the splittings between states containing a single b quark is less than that between the corresponding charmed states by roughly a factor of mc/rnb 1/3 as a result of the smaller chromomagnetic moment of the b quark. Pioneering work in understanding the spectra of such states using QCD was done by De RUjula et al. (1975), building on earlier observations on light-quark systems by Zel'dovich and Sakharov (1966), Dalitz (1967), and Lipkin (1973). 3 W bosons 3.1 Fermi theory of weak interactions The effective four-fermion Hamiltonian for the V—A theory of the weak interactions is GF f GF /— 7-1W = 1Yµ(1 — )02] [037P (1 — 'Y5)'4] = 4— 1'1/:Yit2L) (030P 04L) , (71) where GF and OL were defined in Section 1.3. We wish to write instead a Lagrangian for interaction of particles with charged W bosons which reproduces (71) when taken to second order at low momentum transfer. We shall anticipate a result of Section 4 by introducing the W through an SU(2) symmetry, in the form of a gauge coupling. In the kinetic term in the Lagrangian for fermions, LA-f = 0- 772)0 = O L(i 0)//q, + Ou(i 0)0u — (72) the 0 term does not mix OL and OR, so in the absence of the TO term one would have the freedom to introduce different covariant derivatives P acting on left-handed and right-handed fermions. We shall find that the same mechanism which allows us to give masses to the W and Z while keeping the photon massless will permit the generation of fermion masses even though ?L and OR will transform differently under our gauge group. We follow the conventions of Peskin and Schroeder (1995, p 700 ff). We now let the left-handed spinors be doublets of an SU(2), such as [11,1L , V e— - (73) L L 22 Jonathan L Rosner (We will postpone the question of neutrino mixing until the last section.) The W is introduced via the replacement D, a, — igTzWpi , T'/2 , (74) where T' are the Pauli matrices and W' are a triplet of massive vector bosons. Here we will be concerned only with the WI, defined by (WI! T iWtD/0. The field Wm+ annihilates a W+ and creates a W-, while WI; annihilates a W- and creates a W+. Then W P P+ = (W W-)/0 and W2 = i(W+ — W-)/0, so . 1 1/1/1 014/1 - Ti W' = P 2 014/t; —14/t3, The interaction arising from (72) for a lepton 1 = e, r is then .C(w,I = --- [Pio' Wp+ IL + 10'147,:vid where we temporarily neglect the Wi.3, terms. Taking this interaction to second order and replacing the W propagator (A — 9,2 ) — 1 y D its q2 = 0 value, we find an effective interaction of the form (71), with GF _ g2 814 • 3.2 Charged-current quark interactions The left-handed quark doublets may be written t d'l L s'l Lb' - L where d', s', and b' are related to the mass eigenstates d, s, b by a unitary transformation d' s' = V s VtV = 1 . (79) b' The rationale for the unitary matrix V of Kobayashi and Maskawa (1973) will be reviewed in the next section when we discuss the origin of fermion masses in the electroweak theory. The interaction Lagrangian for W's with quarks then is (75) (76) (77) (78) L LinWt Iquarks = g2 (Ucel/Vp+V DL ) + h.c. , U d U E D E (80) t b Standard Model 23 A convenient parameterisation of V (conventionally known as the Cabibbo-Kobayashi-Maskawa matrix, or CKM matrix) suggested by Wolfenstein (1983) is Vm 17,d Vd Vtd Vus V. Vls Vub Vcb Vtb _A2 —A AA3(1 — p — iy) A 1 _ A2 7 —AA2 AA3 (p — AA2 1 . (81) Experimentally A 0.22 and A 0.85. Present constraints on the parameters p and 71 are shown in Figure 8. The solid circles denote limits on 117„b/jVcd = 0.090 ± 0.025 from 0.6 0.4 1 0.2 0.0 -0.50 -0.25 0.00 0.25 0.50 Figure 8. Constraints on parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The plotted point at p = 0.21, ri = 0.38 lies in the middle of the allowed region. (See text.) charmless b decays. The dashed arcs are associated with limits on Vtd from Bo-B-0 mixing. The present lower limit on Be-Ps mixing leads to a lower bound on IVis /Vtd1 and the dot-dashed arc. The dotted hyperbolae arise from limits on CP-violating K°-.r.° mixing. The phases in the CKM matrix associated with 7/ # 0 lead to CP violation in neutral kaon decays (Christenson et al. 1964) and, as recently discovered, in neutral B meson decays (Aubert et al. 2001a, Abe et al. 2001). These last results lead to a result shown by the two rays, sin(20) = 0.79 ± 0.10, where 0 = Arg(—VedVI/Vtd Vtn. The small dashed lines represent lo- limits derived by Gronau and Rosner (2002) (see also Luo and Rosner 2001) on the basis of CP asymmetry data of Aubert et al. (2001b) for B° 7r+71--. Our range of parameters (confined by lo- limits) is 0.10 < p < 0.32, 0.33 < 71 < 0.43. Similar plots are presented in several other lectures at this Summer School (see, e.g., Buchalla 2001, Nir 2001, Schubert 2001, Stone 2001), which may be consulted for further details, and an ongoing analysis of CKM parameters by Wicker et al. (2001) is now incorporating several other pieces of data. 24 Jonathan L Rosner 3.3 Decays of the T lepton The r lepton (Perl et al. 1975) provides a good example of "standard model" charged-current physics. The r- decays to a yr and a virtual W which can then materialise into any kinematically allowed final state: e — Pe , p-r),„ or three colours of fa, where, in accord with (81), d' 0.975d + 0.22s. Neglecting strong interaction corrections and final fermion masses, the rate for T decay is expected to be 5 P(7-- -> all) = 5G2 19273 2 x 10-3 eV , (82) F corresponding to a lifetime of 7,- ^ 3 x 10-13 s as observed. The factor of 5 = 1 + 1 + 3 corresponds to equal rates into e-1/e, and each of the three colours of The branching ratios are predicted to be B(r- vre-Pe ) = 13(r- -> vr p-1/,) = 3 -1 13(r- -> = 20%. (83) Measured values for the purely leptonic branching ratios are slightly under 18%, as a result of the enhancement of the hadronic channels by a QCD correction whose leading-order behavior is 1 + as/7r, the same as for R in e+e- annihilation. The r decay is thus further evidence for the existence of three colours of quarks. 3.4 W decays We shall calculate the rate for the process W -> f f' and then generalise the result to obtain the total W decay rate. The interaction Lagrangian (76) implies that the covariant matrix element for the process W (k) f (p) (pi) is M (A) = 2v 2- (P)7P (1 - 75)v/ (P')€ (pA) (k) • Here A describes the polarisation state of the W. The partial width is 1 1 — (A) I2 r(W ft) = 2mw 3 pois I M 47rAfw where (2Mw )-1 is the initial-state normalisation, 1/3 corresponds to an average of W polarisations, the sum is over both W and lepton polarisations, and p* is the final centre-of-mass (c.m.) 3-momentum. We use the identity k k, E c(,)0(k)E(,,A).(k) + 2 (86) A Mw for sums over W polarisation states. The result is that 1 2 (1712 rnt2 )2 E IM(a)I2 = g2 [M124T — (77-12 ra' ) 2Mw2 pols (84) (85) (87) Standard Model 25 for any process W f .r, where m is the mass of f and m' is the mass of f'. Recalling the relation between GF and g2, this may be written in the simpler form _ F(W ft) = off' , 2p* p*2 + 3EE' (Dff' Mw ma, • (88) Here E = (p*2 + m2 ) 1 I 2 and E' = (p*2 + m'2 )1 /2 are the c.m. energies of f and f'. The factor 4).11, reduces to 1 as m, —> 0. The present experimental average for the W mass (Kim 2001) is Mw = 80.451 ± 0.033GeV. Using this value, we predict F(W e-0,)= 227.8 ± 2.3MeV. The widths to various channels are expected to be in the ratios CP, : : : : =1 : 1: 1 : 3 [1 + as(Mev)] : 3 [1 + as(Mw)1 , (89) 71 7r so as(Ma,) = 0.120 ± 0.002 leads to the prediction rtot (w) = 2.10 ± 0.02GeV. This is to be compared with a value (Drees 2001) obtained at LEP II by direct reconstruction of W's: rtot(w) = 2.150 ± 0.091GeV. Higher-order electroweak corrections, to be discussed in Section 5, are not expected to play a major role here. This agreement means, among other things, that we are not missing a significant channel to which the charged weak current can couple below the mass of the W. 3.5 W pair production We shall outline a calculation (Quigg 1983) which indicates that the weak interactions cannot possibly be complete if described only by charged-current interactions. We consider the process ve (q) + 1/e(V) 147±(k) +147 - (P) due to exchange of an electron e- with momentum p. The matrix element is M(A'A') = GF,./Mw2 1/(q1 ) PAA') (e)(1 — '75)p IA) (k)u(q) • (90) For a longitudinally polarised W+, this matrix element grows in an unacceptable fashion for high energy. In fact, an inelastic amplitude for any given partial wave has to be bounded, whereas .A.4(''A') will not be. The polarisation vector for a longitudinal W+ traveling along the z axis is E!,A)(k) = (iki• Mw) k,,/Mw, (91) with a correction which vanishes as Iki oo. Replacing e;,A) (k) by k„/Mw, using JE ='— X and 'u(q) = 0, we find A/t(A'A') --‘12GFMwEl(q1) fe) (k')u(q), (92) E (A 'A') 12 = 2ama,[8q, • E (A' ) q • E(A' ) — 4q • q' EP A' ) • f (A' ) ]. (93) lepton pol. This quantity contributes only to the lowest two partial waves, and grows without bound as the energy increases. Such behavior is not only unacceptable on general grounds 26 Jonathan L Rosner because of the boundedness of inelastic amplitudes, but it leads to divergences in higher-order perturbation contributions, e.g., to elastic vv scattering. Two possible contenders for a solution of the problem in the early 1970s were (1) a neutral gauge boson Z° coupling to vv and W+W- (Glashow 1961, Weinberg 1967, Salam 1968), or (2) a left-handed heavy lepton E+ (Georgi and Glashow 1972a) coupling to veW±. Either can reduce the unacceptable high-energy behavior to a constant. The Z° alternative seems to be the one selected in nature. In what follows we will retrace the steps of the standard electroweak theory, which led to the prediction of the W and Z and all the phenomena associated with them. 4 Electroweak unification 4.1 Guidelines for symmetry We now return to the question of what to do with the "neutral W" (the particle we called 14/3 in the previous section), a puzzle since the time of Oskar Klein in the 1930s. The time component of the charged weak current j(+) = NopL L UL-yi,VDL , (94) where NL and L L are neutral and charged lepton column vectors defined in analogy with UL and DL, may be used to define operators Q(+) f d3x4+) Q(-) Q(-1-1f (95) which are charge-raising and charge-lowering members of an SU(2) triplet. If we define Q3 = (1/2)[Q(+) ,Q"], the algebra closes: [Q3 , Q(±)] = +Q(±) . This serves to normalise the weak currents, as mentioned in the Introduction. The form (94) (with unitary V) guarantees that the corresponding neutral current will be J(3) = 2 [SLry,NL - Lop L L + Unp.UL - Di:4LN • (96) which is diagonal in neutral currents. This can only succeed, of course, if there are equal numbers of charged and neutral leptons, and equal numbers of quarks of charge 2/3 and and quarks of charge -1/3. It would have been possible to define an SU(2) algebra making use only of a doublet (Gell-Mann and Levy 1960) ; L 1 = Vudd + Vuss which was the basis of the Cabibbo (1963) theory of the charge-changing weak interactions of strange and non-strange particles. If one takes Vud = cos Oc, I/;„ = sin 0c, as is assumed in the Cabibbo theory, the u, d, s contribution to the neutral current Ji(;') is 1 Jµ3) u,d,s —2 [acyoL - cos2 Oc joiAL - sin2 t9c,§0,,SL - sin Oc cos Oc(d L-y ps L + -.§L-Y /AL )] • (98) (97) L Standard Model 27 This expression contains strangeness-changing neutral currents, leading to the expectation of many processes like K+ 7r+vP, K2 -> , . . ., at levels far above those observed. It was the desire to banish strangeness-changing neutral currents that led Glashow et al. (1970) to introduce the charmed quark c (proposed earlier by several authors on the basis of a quark-lepton analogy) and the doublet [ c c = [ (99) s' x Vcdd + 17,5.9 L • In this four-quark theory, one assumes the corresponding matrix V is unitary. By suitable phase changes of the quarks, all elements can be made real, making V an orthogonal matrix with Vud=17cs = cos 3c,17.., = -Vm = sin 9c. Instead of (98) one then has 1 J (3) iud = - u -y c d -yd — g-ys] , 1.11, LILL - LIILLµL ''s' e 2 which contains no flavour-changing neutral currents. The charmed quark also plays a key role in higher-order charged-current interactions. Let us consider 10-Ti° mixing. The CP-conserving limit in which the eigenstates are K1 (even CP) and K2 (odd CP) can be illustrated using a degenerate two-state system such as the first excitations of a drum head. There is no way to distinguish between the basis states illustrated in Figure 9(a), in which the nodal lines are at angles of ±45° with respect to the horizontal, and those in Figure 9(b), in which they are horizontal and vertical. (100) (a) (b) Figure 9. Basis states for first excitations of a drum head. (a) Nodal lines at ±45° with respect to horizontal; (b) horizontal and vertical nodal lines. If a fly lands on the drum-head at the point marked "x", the basis (b) corresponds to eigenstates. One of the modes couples to the fly; the other doesn't. The basis in (a) is like that of (K0, Te), while that in (b) is like that of (K1 , K2 ). Neutral kaons are produced as in (a), while they decay as in (b), with the fly analogous to the 71-7 state. The short-lived state (K 1, in this CP-conserving approximation) has a lifetime of 0.089ns, while the long-lived state K2 ) lives 600 times as long, for 52ns. Classical illustration of CP-violating mixing is more subtle but can be achieved as well, for instance in a rotating reference frame (Rosner and Slezak 2001, Kosteleck3'T and Roberts 2001). 28 Jonathan L Rosner The shared xx intermediate state and other low-energy states like 7r°, u, and rj are chiefly responsible for CP-conserving K° -K° mixing. However, one must ensure that large short-distance contributions do not arise from diagrams such as those illustrated in Figure 10. Figure 10. Higher-order weak contributions to K°.--K° mixing due to loops with internal u, c,t quarks. If the only charge 2/3 quark contributing to this process were the u quark, one would expect a contribution to Arnic of order f h2 m mw2 GF A2,771K (g2 /1672 ArriK I. 4 sin2 Oc cos2 Oc/1672 (101) where fic is the amplitude for ds to be found in a K°, and the factor of 1672 is character-istic of loop diagrams. This is far too large, since AmK FKs G2Ffkm3K. However, the introduction of the charmed quark, coupling to —d sin 9c + s cos Oc, cancels the leading contribution, leading to an additional factor of [(mc2 "4,)/MM ln(Varn2 ) in the above expression. Using such arguments Glashow et al. (1970) and Gaillard and Lee (1974) estimated the mass of the charmed quark to be less than several GeV. (Indeed, early candidates for charmed particles had been seen by Niu, Mikumo, and Maeda 1971.) The discovery of the ,110 (Aubert et al. 1974, Augustin et al. 1974) confirmed this prediction; charmed hadrons produced in neutrino interactions (Cazzoli et al. 1975) and in e+e-annihilation (Goldhaber et al. 1976, Peruzzi et al. 1976) followed soon after. An early motivation for charm relied on an analogy between quarks and leptons. Hara (1964), Maki and Ohnuki (1964), and Bjorken and Glashow (1964) inferred the existence of a charmed quark coupling mainly to the strange quark from the existence of the µ—doublet: : leptons c : quarks . (102) Further motivation for the quark-lepton analogy was noted by Bouchiat et al. (1972), Georgi and Glashow (1972b), and Gross and Jackiw (1972). In a gauge theory of the electroweak interactions, triangle anomalies associated with graphs of the type shown in Figure 11 have to be avoided. This cancellation requires the fermions f in the theory to contribute a total of zero to the sum over f of QyalL. Such a cancellation can be achieved by requiring quarks and leptons to occur in complete families so that the terms Leptons : (0)2 (2) + (_1)2 ( 2) = 21 (103) Quarks : 3 [(3)2 (21) (_ 13)2 (_i1)] = (104) Standard Model 29 Figure 11. Example of triangle diagram for which leading behaviour must cancel in a renormalisable electroweak theory. sum to zero for each family. We are then left with a flavour-preserving neutral current ./f,3) , given by (100), whose interpretation must still be given. It cannot correspond to the photon, since the photon couples to both left-handed and right-handed fermions. At the same time, the photon is somehow involved in the weak interactions associated with W exchange. In particular, the WI themselves are charged, so any theory in which electromagnetic current is con-served must involve a -y147±W- coupling. Moreover, the charge is sensitive to the third component of the SU(2) algebra we have just introduced. We shall refer to this as SU(2)L, recognising that only left-handed fermions 0/, transform non-trivially under it. Then we can define a weak hypercharge Y in terms of the difference between the electric charge Q and the third component I3L of SU(2)L (weak isospin): Q = 1-3L ± -2 . (105) Values of Y for quarks and leptons are summarised in Table 3. Particle(s) Q 131, Y vex 0 1/2 -1 ei -1 -1/2 -1 uL 2/3 1/2 1/3 dL -1/3 -1/2 1/3 e R -1 0 -2 uR 2/3 0 4/3 dR -1/3 0 -2/3 Table 3. Values of charge, In, and weak hypercharge Y for quarks and leptons. If you find these weak hypercharge assignments mysterious, you are not alone. They follow naturally in unified theories (grand unified theories) of the electroweak and strong interactions. A "secret formula" for Y, which may have deeper significance (Pali and Salam 1973), is Y = 2138 + (B - L), where I3R is the third component of "right-handed" isospin, B is baryon number (1/3 for quarks), and L is lepton number (1 for leptons such as e- and ve ). The orthogonal component of I3R and B - L may correspond to a higher-mass, as-yet-unseen vector boson, an example of what is called a Z'. The search 30 Jonathan L Rosner for Z' bosons with various properties is an ongoing topic of interest; current limits are quoted by the Particle Data Group (2000). The gauge theory of charged and neutral W's thus must involve the photon in some way. It will then be necessary, in order to respect the formula (105), to introduce an additional U(1) symmetry associated with weak hypercharge. The combined electroweak gauge group will have the form SU(2)L 0 U(1)y. 4.2 Symmetry breaking Any unified theory of the weak and electromagnetic interactions must be broken, since the photon is massless while the W bosons (at least) are not. An explicit mass term in a gauge theory of the form m'Ait,APi violates gauge invariance. It is not invariant under the replacement (26). Another means must be found to introduce a mass. The symmetry must be broken in such a way as to preserve gauge invariance. A further manifestation of symmetry breaking is the presence of fermion mass terms. Any product 00 may be written as = (h+ OR)(1PL +OR) = obR + ORO', , (106) using the fact that iL = ( 1 + -y5 )/2 and IT R = T1, (1 — -y5 )/2. Since 0/, transforms as an SU(2) /, doublet but OR as an SU(2) L singlet, a mass term proportional to T/)0 transforms as an overall SU(2)L doublet. Moreover, the weak hypercharges of left-handed fermions and their right-handed counterparts are different. Hence one cannot even have explicit fermion mass terms in the Lagrangian and hope to preserve local gauge invariance. One way to generate a fermion mass without explicitly violating gauge invariance is to assume the existence of a complex scalar SU(2)L doublet 0 coupled to fermions via a Yukawa interaction: LY = ±gY(OLOOR ±h.C.) ( E [ C:01 • (107) Thus, for example, with TPL = (Pe, e) L and OR = eR, we have = —9,(PeL40+,, + eL(YPeR +h.c.). (108) If 0° acquires a vacuum expectation value, (0°) # 0, this quantity will automatically break SU(2) L and U(1)y, and will give rise to a non-zero electron mass. A neutrino mass is not generated, simply because no right-handed neutrino has been assumed to exist. (We shall see in the last section how to generate the tiny neutrino masses that appear to be present in nature.) The gauge symmetry is not broken in the Lagrangian, but only in the solution. This is similar to the way in which rotational invariance is broken in a ferromagnet, where the fundamental interactions are rotationally invariant but the ground-state solution has a preferred direction along which the spins are aligned. The d quark masses are generated by similar couplings involving z/,y, = (U, j)i, and OR = dR, so that = ±gYd(f11,95+dR d Led R h.C.) . (109) Standard Model 31 To generate u quark masses one must either use the multiplet (:/3 _z_. 0 = iT20*, [ —0- (110) which also transforms as an SU(2) doublet, or a separate doublet of scalar fields With 0L = d)L and = uR, we then find Ly,„ = — 91,u(lirj°aL — GIL0-i/L + h.c.) (112) if we make use of 0, or Ly,„ = -gyu (fiLeUL C1L01-uL + h.c.) (113) if we use 0'. For present purposes we shall assume the existence of a single complex doublet, though many theories (notably, some grand unified theories or supersymmetry) require more than one. 4.3 Scalar fields and the Higgs mechanism Suppose a complex scalar field of the form (107) is described by a Lagrangian = (apo)f(aPo)— 4 (of )2 + 22 0to (114) Note the "wrong" sign of the mass term. This Lagrangian is invariant under the symmetry SU(2) /, 0 U(1)y. The field 0 will acquire a constant vacuum expectation value which we calculate by asking for the stationary value of aro it2 = 0 (Of 0) = (115) 5(00) We still have not specified which component of 0 acquires the vacuum expectation value. At this point only 01- 0 = 10+ 1 2 +10°12 is fixed, and (Re 6+, Im , Re 0°, Tin 0°) can range over the surface of a four-dimensional sphere. The Lagrangian (114) is, in fact, invariant under rotations of this four-dimensional sphere, a group SO(4) isomorphic to SU(2) 0 U(1). A lower-dimensional analogue of this surface would be the bottom of a wine bottle along which a marble rolls freely in an orbit a fixed distance from the centre. Let us define the vacuum expectation value of 0 to be a real parameter in the 0° direction: 0 vi‘h The factor of 1/ is introduced for later convenience. We then find, from the discussion in the previous section, that Yukawa couplings of 0 to fermions 0i generate mass terms (0) = (116) 32 Jonathan L Rosner m, = gy,v/N5. We must now see what such vacuum expectation values do to gauge boson masses. (For numerous illustrations of this phenomenon in simple field-theoretical models see Abers and Lee 1973, Quigg 1983, and Peskin and Schroeder 1995.) In order to introduce gauge interactions with the scalar field 0, one must replace 0, by D„ in the kinetic term of the Lagrangian (114). Here Ti = ig 2pzg , (117) where the U(1)y interaction is characterised by a coupling constant g' and a gauge field 13„, and we have written g for the SU(2) coupling discussed earlier. It will be convenient to write 0 in terms of four independent real fields (e2 , zi) in a slightly different form: 0 = exp ( je'71 v 77 1 . 2v ) We then perform an SU(2) L gauge transformation to remove the e dependence of 0, and rewrite it as 0 = v + (119) Nfi The fermion and gauge fields are transformed accordingly; we rewrite the Lagrangian for them in the new gauge. The resulting kinetic term for the scalar fields, taking account that If = 1 for the Higgs field (107), is G,4, = (D,O)t(DPO) ap g Wm' i1471,2 gl . (120) 2 BP v 2 + —W 3 N.5 This term contains several contributions. 1. There is a kinetic term ((907)(01177) for the scalar field g. 2. A term vao is a total divergence and can be neglected. 3. There are WW77, BBy, WW7/2 , and B1372 interactions. 4. The v2 term leads to a mass term for the Yang-Mills fields: 2 Lni,ym = 8 ig2R w 1 )2 + (W2)2] + (gW3 giB)2} (121) The spontaneous breaking of the SU(2) 0 U(1) symmetry thus has led to the appearance of a mass term for the gauge fields. This is an example of the Higgs mechanism (Higgs 1964). An unavoidable consequence is the appearance of the scalar field 17, the Higgs field. We shall discuss it further in Section 5. The masses of the charged W bosons may be identified by comparing (121) and (75): (gv)2 /8 = M4,/2, or Mw = gv/2. (122) (118) 2 Standard Model 33 Since the Fermi constant is related to g/Mw, one finds GF g2 1 8Ma, 2v2 The combination gW„3 - g1 B, also acquires a mass. We must normalise this com-bination suitably so that it contributes properly in the kinetic term for the Yang-Mills fields: 11 LK,ym = 4I' Wis" - 4 -BI` ,B" , (124) where 141w apvv, - , B„ -= 8,13# . (125) Or v = 2-114G-F112 = 246 GeV . (123) Setting cos 0 = (g2 gig ) 1/2 sin 0 = (g2 + y2)1/2 g, (126) we may write the normalised combination gWii3 - g'Bp which acquires a mass as Z = W3 cos - B sin 6? . (127) - The orthogonal combination does not acquire a mass. It may then be identified as the photon: A, = B, cos 0 + 147,,3 sin 0 . (128) The mass of the Z is given by (g2 8 gl2 ) v2 M. mw (g2 g/ 2 )1/2 mw Or MZ cos e' (129) 2 using (126) in the last relation. The W's and Z's have acquired masses, but they are not equal unless g' were to vanish. We shall see in the next subsection that both g and g' are nonzero, so one expects the Z to be heavier than the W. It is interesting to stop for a moment to consider what has taken place. We started with four scalar fields 0+, 0-, 0°, and 00. Three of them [0+, 0-, and the combination ((b° - .b°)/i-V2] could be absorbed in the gauge transformation in passing from (118) to (119), which made sense only as long as (0° + cl;.)/ had a vacuum expectation value v. The net result was the generation of mass for three gauge bosons W+, W-, and Z. If we had not transformed away the three components ei of 0 in (118), the term ,C1,-,0 in the presence of gauge fields would have contained contributions W,OP0 which mixed gauge fields and derivatives of (b. These can be expressed as wo,0 = aP(i47,0)- (aow,)0 (130) and the total divergence (the first term) discarded. One thus sees that such terms mix longitudinal components of gauge fields (proportional to 811/1c) with scalar fields. It is necessary to redefine the gauge fields by means of a gauge transformation to get rid of such mixing terms. It is just this transformation that was anticipated in passing from (118) to (119). The three "unphysical" scalar fields provide the necessary longitudinal degrees of free-dom in order to convert the massless WI and Z to massive fields. Each massless field possesses only two polarisation states (J, = ±J), while a massive vector field has three (J, = 0 as well). Such counting rules are extremely useful when more than one Higgs field is present, to keep track of how many scalar fields survive being "eaten" by gauge fields. 34 Jonathan L Rosner 4.4 Interactions in the SU(2) U(1) theory. By introducing gauge boson masses via the Higgs mechanism, and letting the simplest non-trivial representation of scalar fields acquire a vacuum expectation value v, we have related the Fermi coupling constant to v, and the gauge boson masses to gt, or 02+9,2 )1/2 v. We still have two arbitrary couplings g and g' in the theory, however. We shall show how to relate the electromagnetic coupling to them, and how to measure them separately. The interaction of the fermions with the gauge fields is described by the kinetic term Ch = PO. Here, as usual, VI YV ig ig —2 /3 . 2 i (131) The charged-W interactions have already been discussed. They are described by the terms (76) for leptons and (80) for quarks. The interactions of W3 and B may be re-expressed in terms of A and Z via the inverse of (127) and (128): W3 = Z cos 0 A, sin , B„ = —Z„ sin° + A, cos . (132) Then Z) the covariant derivative for neutral gauge bosons is i 'neutral -= 0 — iglu( 7 cos e+ sina) — ig (Q — 13L )(— 7 sin e+ cos e) . (133) Here we have substituted Y/2 = (Q — In). We identify the electromagnetic contribution to the right-hand side of (133) with the familiar one —ieQ 4, so that e = g' cos = g sin r9 . (134) The second equality, stemming from the demand that 13L A terms cancel one another in (133), is automatically satisfied as a result of the definition (126). Combining (126) and (134), we find e= Vg2 the result advertised in the Introduction. The interaction of the Z with fermions may be determined from Eq. (133) with the help of (126), noting that g cos + g' sin = g2 g/2 )1/2 and g' sin = (g2 + g' 2) 1 /2 sin2 0. We find 'neutral= 0 _ ieQ _ i(g2 gi2) 1/2 ( in sin2 0) 7. (136) Knowledge of the weak mixing angle 8 will allow us to predict the I4 and Z masses. Using GF/12- = g2 /8M4, and g sin B = e, we can write M [ ira 1 37.3 GeV (137) w N. 1/2 G1-] sin = sin if we were to use a' = 137.036. However, we shall see in the next section that it is more appropriate to use a value of a" 129 at momentum transfers characteristic of the W mass. With this and other electroweak radiative corrections, the correct estimate is raised to Mw 38.6 GeV/ sin 8, leading to the successful predictions (7). The Z mass is expressed in terms of the W mass by Mz = Mw/ cos O. gg' 1 1 1 Or e2 = g2 ± (135) Standard Model 35 4.5 Neutral current processes The interactions of Z's with matter, rint,Z = (g2 + g12 )1 /2 0(I3L - Q sin2 0) , (138) may be taken to second order in perturbation theory, leading to an effective four-fermion theory for momentum transfers much smaller than the Z mass. In analogy with the rela-tion between the W boson interaction terms (76) and (80) and the four-fermion charged-current interaction (71), we may write wivvc = 4GF ‘/[TP ( /3L - Q sin2 0)71-'02][7'3 ( /3L - Qsin2 0)-y01)41, (139) where we have used the identity (g2 + g'2 )/8/1/4 = GF// following from relations in the previous subsection. Many processes are sensitive to the neutral-current interaction (139), but no evidence for this interaction had been demonstrated until the discovery in 1973 of neutral-current interactions on hadronic targets of deeply inelastically scattered neutrinos (Hasert et al. 1973; Benvenuti et al. 1974). For many years these processes provided the most sensitive measurement of neutral-current parameters. Other crucial experiments (see, e.g., reviews by Amaldi et al. 1987 and Langacker et al. 1992) included polarised electron or muon scattering on nucleons, asymmetries and total cross sections in e+e- -> µ± or 7+ parity violation in atomic transitions, neutrino-electron scattering, coherent 77° production on nuclei by neutrinos, and detailed measurements of W and Z properties. Let us take as an example the scattering of leptons on quarks to see how they provide a value of sin2 O. In the next subsection we shall turn to the properties of the Z bosons, which are now the source of the most precise information. One measures quantities o-(v,4 v ...) a(0A -> + ...) R o-(vA ft- + ...) R u(0A µ± + ...). (140) These ratios may be calculated in terms of the weak Hamiltonians (71) and (139). It is helpful to note that for states of the same helicity (L or R, standing for left-handed or right-handed) scattering on one another, the differential cross section is a constant: do- do- -> RR) = dS2(LL LL) = —uo , (141) 47r where o-0 is some reference cross section, while for states of opposite helicity, a() ( 2 1 ± cos 8, ,„ 2 du d (RL RL) = —du 47r (LR LR)= (142) cl52 Thus o-(RR -> RR) = a(LL -> LL) = 3o (RL RL) = 3a(LR LR). (143) We first simplify the calculation by assuming the numbers of protons and neutrons are equal in the target nucleus, and neglecting the effect of antiquarks in the nucleon. (We shall use the shorthand v = vµ and 7) = i"),.) Then o-(vu vu) + o (vd vd) o- (7)u -3 7)u) + o (7)d 0d) R, = R, = (144) o-(vd au- u) o- (0u µ±d) 36 Jonathan L Rosner One can write the effective Hamiltonian (139) in the form ANC G F [1)-Y ( 1 - 75)V] [Ere (1 - -y5)uEL(u)+ fre(1+75)uER(u) + d7µ(1 - 75 )d€L (d)+ d-y"(1 + 'Ys)clER(d)] (145) where E L (u) = 1 3 - sine 9, E R(u) = -3 sine 0 , (146) 2 E L (d) = 2 3 3 + sin2 , E R(d) = 1 sin2 9. (147) Taking account of the relations (143), one finds = [EL (u)]2 + 31 [ER(u)]2 + [et, (02 + 31 [ER (d)]2 Rv = [61,(u)]2 3[6 8(u)]2 [EL(d)]2 + 3[€ R(0 2 , where we have used the fact that ct(vd µ-d) = 3o (vu p+d). The results are 1 = - sin 20 + 20 sin, 1 0 , R„ - sin200 sin, 0 . (150) 2 27 2 9 If we consider also the antiquark content of nucleons, this result may be generalised (Llewellyn Smith 1983) by defining o-(ON p+ X) r o-(vN p- X) . Instead of (150) one then finds 1 5 R, = sin2 + 5(1 r) sin4 , 1 5 - sin2 + (1 + ) sin4 . (152) Some experimental values of flu, R„, and r are shown in Table 4 (Conrad et al. 1998). The relation between R, and R, as a function of sin2 B is plotted in Figure 12. This result has a couple of interesting features. Experiment R, fl, r CHARM 0.3091 ± 0.0031 0.390 ± 0.014 0.456 + 0.011 CDHS 0.3135 + 0.0033 0.376 + 0.016 0.409 ± 0.014 Average 0.3113 ± 0.0023 0.384 + 0.011 0.429 ± 0.011 Table 4. Neutrino neutral-current parameters. The observed R„ is very close to its minimum possible value of less than 0.4. Initially this made the observation of neutral currents quite challenging. Note that R„ is even smaller. Its value provides the greatest sensitivity to sin2 O. It is also more precisely measured than R, (in part, because neutrino beams are easier to achieve than antineutrino beams). The effect of 1' on the determination of sin2 0 is relatively mild. (148) (149) (151) Standard Model 37 R, 0.6 0.5 0.4 0.3 01 0.2 0.3 0.4 0.5 06 Rv Figure 12. The Weinberg-Salam "nose" depicting the relation between 11,, and R,. The solid line corresponds to r = 0.429, close to the actual situation; the dashed line corresponds to the idealised case r = 1/3 in which antiquarks in the nucleon are neglected. The plotted point with error bars corresponds to the average of measured values. A recent determination of sin2 8 (Zeller et al. 1999), based on a method proposed by Paschos and Wolfenstein (1973), makes use of the ratio o-(v N -> v X ) - o-(0 N -> 1).A") R„ - r 1 R- sin B. (153) o-(vNµX) - o-(0 N -> X) 1 - r 2 In these differences of neutrino and antineutrino cross sections, effects of virtual quark-antiquark pairs in the nucleon ("sea quarks," as opposed to "valence quarks") cancel one another, and an important systematic error associated with heavy quark production (as in vs c) is greatly reduced. The result is sin2 0 (on—shell) = 0.2253 ± 0.0019(stat.) f 0.0010(syst.) , (154) which implies a W mass May = Mz cos 0 (on—shell) = 80.21 + 0.11 GeV . (155) The "on-shell" designation for sin2 8 is necessary when discussing higher-order electroweak radiative corrections, which we shall do in the next section. A more recent analysis by Zeller et al. (2001) finds sin2 0 (on—shell) = 0.2277 + 0.0014(stat.) f 0.0008(syst.) , (156) equivalent to Mw = 80.136 ± 0.084GeV. Incorporation of this result into the electroweak fits described in the next section is likely to somewhat relax constraints on the Higgs boson mass: See Rosner (2001). 38 Jonathan L Rosner 4.6 Z and top quark properties We have already noted the prediction and measurement of the W mass and width. The Z mass and width are very precisely determined by studying the shape of the cross section for electron-positron annihilation as one varies the energy across the Z pole. The results (LEP Electroweak Working Group [LEP EWWG] 2001) are Mz = 91.1875 + 0.0021 GeV , Fz = 2.4952 ± 0.0023 GeV . (157) In much of the subsequent discussion we shall make use of the very precise value of Mz as one of our inputs to the electroweak theory; the two others, which will suffice to specify all parameters at lowest order of perturbation theory, will be the Fermi coupling constant Gp = 1.16637(1) x 10-5 GeV-2 and the electromagnetic fine-structure constant, evolved to a scale M3: a- (ms)(A,) = 128.933 + 0.021 (Davier and Hocker 1998). This last quantity depends for its determination upon a precise evaluation of hadronic contributions to vacuum polarisation, and is very much the subject of current discussion. The relative branching fractions of the Z to various final states may be calculated on the basis of (138). One may write this expression as rz ff = (g2 + r g'2 )1 /2 / -y5 )a L + (1 + -y5)coiii • (158) The values of a L and a R for each fermion are shown in Table 5. Channel aL a R Partial width (MeV) Number of channels Subtotal (MeV) vv ll vu, cc dd, s.S bb 1 4 2 (-- + sin2 a) 2 (21 — 32 sin2 8) 2 (— + s sin2 (9) 2 (— 2 + s sin2 a) 0 sine 9 1 2 -2 21 (_ 32 sin2 0) -- (3 sin2 0) 2 (3 sin2 0) 165.9 83.4 296.5 382.1 372.8 3 3 2 2 1 498 250 593 764 373 Total 2478 Table 5. Contributions to rz predicted in lowest-order electroweak theory (including leading-order QCD corrections to hadronic channels). Here we have taken sin2 9 = 0.231 and as(MD = 0.12. The partial width of Z into f f is 4G F 1'(Z f f) = 371-‘7 A.d(a2L + a R)nc , where n, is the number of colours of fermions f: 1 for leptons, 3 for quarks. The predicted partial width for each Z vv channel is independent of sin2 8: GF MZ F(Z vv) =127r = 165.9 MeV (159) (160) Standard Model 39 using the observed value of Mz. The partial decay rates to other channels are expected to be in the ratios v&: e+ e- : : dd= 1: 1- 4 sin2 + 8 sin4 : 3- 8 sin2 32 ± sin4 : 3- 4 sin2 + -8 sin4 0, (161) 3 3 or 1: 0.503: 1.721: 2.218 for sin2 9 = 0.231. A small kinematic correction for the non-zero b quark mass leads to a suppression factor 4m,,2 ) 112 9,,2 (DU, = Cl " [ f v (1 ± fA (1 4mb2 )] M2 /U m2 where fv and IA =1- f v are the relative fractions of the partial decay width proceeding via the vector a L-Fa R) and axial-vector a L-aR) couplings. For sin2 = 0.23, fv 1/3, IA 2/3, and 'bb 0.988. A further correction to F(Z bb), important for the precise determinations in the next section, is associated with loop graphs associated with top quark exchange (see the review by Chivukula 1995), and is of the same size, about 0.988. Taking a correction factor (1 + as/7r) with as(111) = 0.12 for the hadronic partial widths of the Z, we then predict the contributions to Fz listed in Table 5. (The ti channel is, of course, kinematically forbidden.) The measured Z width (157) is in qualitative agreement with the prediction, but above it by about 0.7%. This effect is a signal of higher-order electroweak radiative corrections such as loop diagrams involving the top quark and the Higgs boson. Similarly, the observed value of F(Z —> e+ el, assuming lepton universality, is 83.984 + 0.086 MeV, again higher by 0.7% than the predicted value of 83.4MeV. We shall return to these effects in the next section. The width of the Z is sensitive to additional vi) pairs. Clearly there is no room for an additional light pair coupling with full strength. Taking account of all precision data and electroweak corrections, the latest determination of the "invisible" width of the Z (see the compilations by the LEP EWWG 2001 and by Langacker 2001) fixes the number of "light" neutrino species as = 2.984 ± 0.008. The Z is produced copiously in e+ e- annihilation when the center-of-mass energy /7 is tuned to Mz. The Stanford Linear Collider (SLC) and the Large Electron-Positron Collider at CERN (LEP) exploited this feature. The cross section of production of a final state f near the resonance, ignoring the effect of the virtual photon in the direct channel, should be u(e+ e- _4 Z° _> f) = 127rF ( Z e+e-)F(Z f) (163) (s MD2 ± Mg2z • Setting 8,- +e- F(Z° el-e—)/rz 3.37% we see that at resonance the peak total cross section should be apeak = 127rL3e+ e-/MZ 59.4 nb, corresponding to o-(e+e- Z° -> all) = 913,+,- 5000. (164) cr(e+e- it+ µ--) a2 This is a spectacular value of R, which is only a few units in the range of lower-energy e+e- colliders. Of course, not all of the cross section at the Z peak is visible: Nearly 12 nb goes into neutrinos! Another 6 nb goes into charged lepton pairs, leaving apeak, hadrons = 41.541 ± 0.037 nb. (162) 40 Jonathan L Rosner We close this subsection with a brief discussion of spin-dependent asymmetries at the Z. These are some of the most powerful sources of information on sin2 O. They have been measured both at LEP (through forward-backward asymmetries) and at SLC (through the use of polarised electron beams). The discussion makes use of an elementary feature of vector- and axial-vector cou-plings. Processes involving such couplings to a real or virtual particle (such as the Z) always conserve chirality. In the direct-channel reactions CO- 4 Z -4 f f this means that a left- (right-)handed electron only interacts with a right- (left-) handed positron, and if the final fermion f is left- (right-)handed then the final anti-fermion f will be right-(left-) handed. Moreover, such reactions have characteristic angular distributions, with do-(q IL ) (1 + cos 9, ) 2 = ao(aeL)2 (a{02 (165) c/12 2 do-(eRc/S2 fR) (1 + cos 0, ) 2 = ogaeR)2(afR)2 2 (166) do-(e 52 2 f R) (1 — cos 0 c ) 2 = ao(a1)2 (a1a)2 (167) d dale dct 2 —> = 0-0(aea)2laii2 (1 — cos Oc.m.) 2 (168) where o-0 is some common factor, and the a L, R are given in Table 5. Several asymmetries can be formed using these results. The polarised electron left-right asymmetry compares the cross sections for producing fermions using right-handed and left-handed polarised electrons, as can he produced and monitored at the SLC. The cross section asymmetry is given by o-(e el- hadrons) — o-(eRe+ —> hadrons) ALR(hadrons) o-(eie+ hadrons) + a(e-Re+ hadrons) = 0'1)2 (aea)2 1 — 4 sine -- (6E1)2 + (aeR)2 1 — 4 sin2 0 + 8 sin4 (169) The measured value (LEP EWWG 2001) ileLR(hadrons) = 0.1514 ± 0.0022 corresponds to sin2 9 = 0.23105 ± 0.00028 using this formula. (We shall discuss small corrections in the next section.) The forward-backward asymmetry in e+e- —> f I uses the fact that f °I) d cos 0(1 ± cos 0)2 = ±2 , o so that (f 1 o \\ + f d cos 0(1 ± cos 0)2 = , (170) cr(ee±—>f t)f,,d — o-(ee±—>f f)back o-(e-e±—> f + o-(e-e+—>f f )back 3 ((6)2 — (aeR)2 ) ((afL )2 — (afR)2 ) — ./Ac Af 4 (aL )2 + (aa)2 ) (afr,)2 + (crfa)2 — 4 LR LR AFB (171) Standard Model 41 These quantities can be measured not only for charged leptons, but also for quarks such as the b, whose decays allow for a distinction to be made (at least on a statistical basis) between b and b. The discovery of the top quark by the CDF (1994) and DO (1995) Collaborations cul-minated nearly two decades of detector and machine work at the Fermilab Tevatron. A ring of superconducting magnets was added to the 400GeV Fermilab accelerator, more than doubling its energy. Low-energy rings were added to accumulate and store antipro-tons, which were then injected into the superconducting ring and made to collide with oppositely-directed protons at a centre-of-mass energy of 1.8TeV. The top quarks were produced in the reaction pp -4 tt The top quark's mass is currently measured to be mt = 174.3 ± 5.1 GeV. It cou-ples mainly to b, as expected in the pattern of couplings discussed in Section 3. One determination (see Gilman, Kleinknecht, and Renk 2000 for details) is that iVt612 = 0.99 ± 0.29 . (172) Vtdi2 + iVisi2 + This result makes use of the measured fraction of the decays t be+v, in top semileptonic decays. The top quark is the only quark heavy enough to decay directly to another quark (mainly b) and a real W. Its decay width is G Frq [( Alev ) 2 ( Mi24, F(t W±b) = 2 + 2-2- 1.53 GeV . (173) 871:2- mt This is larger than the typical spacing between quarkonium levels (see Figures 4 and 5), and so there is not expected to be a rich spectroscopy of tt levels, but only a mild enhancement near threshold of the reaction e+e- tt, associated with the production of the 1S level (Kwong 1991, Strassler and Peskin 1991). A good review of present and anticipated top quark physics is given by Willenbrock (2000). 5 Higgs boson and beyond 5.1 Searches for a standard Higgs boson Let us assume that all quark and lepton masses and all W and Z masses arise from the vacuum expectation value of a single Higgs boson: (0°) = where the strength of the Fermi coupling requires v = 246GeV. The Yukawa coupling gy f (107) for a fermion f is related to the fermion's mass: gy f = /v. (It is a curious feature of the top quark's mass that, within present errors, gyt = 1. Since fermion masses "run" with scale p., it is not clear how fundamental this relation is.) Those quarks with the greatest mass then are expected to have the greatest coupling to the physical Higgs boson H = - (Here we use H to denote the field represented by it in the previous section.) The Higgs boson has a well-defined coupling to W's and Z's as a result of the discussion in the previous section. The term (D,O)t(DPO) in the Lagrangian leads to 02 + 9,2) 2 I/2HMz Gyww = (Wm-WP+) , LIIZZ = (4Zµ) (174) 42 Jonathan L Rosner To lowest order, one find £HZ7 = Lyn = 0. Processes involving the couplings (174) include qq W virtual —) W +H and especially e+e Zreal or virtual —4 Zvirtual or real + H , (175) where the final Z° can be detected (for example) via its decay to e+ ,a+11-, or even its existence inferred from its vv decay. For a virtual intermediate and real final Z, the cross section (Quigg 1983) is ircv2 (p.2 + 3m3) 2p* a(e+ —) ZH) = (1 4 sine + 8 sin4 -1- 24 sin4 cos4 at 114-2 _ vkiviZ 81 A.7:9 where p* is the final c.m. 3-momentum. This cross section behaves as 1/s for large s (as does any cross section for production of qq, it+ ...), so that as s —> oc, cr(e+e- —) Z11) 1— 4 sine + 8 sin4 9 1 --+ u(e+e- 128 sin4 8 cos4 8 — 8 (177) At very high energies, the Higgs boson can be produced by means of W+W- and ZZ fusion; the (virtual) W's and Z's can be produced in either hadron-hadron or lepton-lepton collisions. A further proposal for producing Higgs bosons is by means of muon-muon collisions. For Higgs bosons far above WW and ZZ threshold, one expects (Eichten et al. 1984) 3GF MGeV )3 F H =F —> (H W±W-) +F167r-(H —> ZZ) = (500 60 GeV (178) as one can show with the help of (174). The longitudinal degrees of freedom of the W and Z provide the dominant contribution to the decay width in this limit. For = 1TeV, this relation implies that the Higgs boson's width will be nearly 500GeV. Such a broad object will be difficult to separate from background. However, mixed signals for a a much lighter Higgs boson have already been received at LEP. At the very highest LEP energies attained, \rs < 209 GeV, the four LEP collabo-rations ALEPH, DELPHI, L3, and OPAL have presented combined results (LEP Higgs Working Group 2001) which may be interpreted either as a lower limit on the Higgs bo- son mass of 114.1GeV, or as a weak signal for a Higgs boson of mass illy 115.6GeV produced by the above process. This latter interpretation is driven in large part by the ALEPH data sample (Barate et al. 2001). The main decay mode of a Higgs boson in this mass range is expected to be bb, with T+T- taking second place. LEP now has ceased operation in order to make way for the Large Hadron Collider (LHC), which will collide 7TeV protons with 7TeV protons and should have no problem producing such a boson. The LHC is scheduled to begin operation in 2006. In the meantime, the Fermilab Tevatron has resumed pp collider operation after a hiatus of 5 years. Its scheduled "Run II" is initially envisioned to provide an integrated luminosity of 2 fb-1, which is thought to be sufficient to rival the sensitivity of the LEP search (Carena et al. 2000), making use of the subprocess qq W — virtual W + H. With 10 fb-1 per detector, a benchmark goal for several years of running with luminosity improvements, it should be possible to exclude a Higgs boson with standard couplings nearly up to the ZZ (176) Standard Model 43 threshold of 182GeV, and to see a 3a signal if My < 125 GeV. Other scenarios, including the potential for discovering the Higgs boson(s) of the Minimal Supersyminetric Standard Model (MSSM) are given by Carena et al. (2000). Meanwhile, we shall turn to the wealth of precise measurements of electroweak properties of the Z, W, top quark, and lighter fermions as indirect sources of information about the Higgs boson and other new physics. 5.2 Precision electroweak tests We have calculated processes to lowest electroweak order in the previous section, with the exception that we took account of vacuum polarisation in the photon propagator, which leads to a value of cr+1 closer to 129 than to 137.037 at the mass scale of the Z. The lowest-order description was found to be adequate at the percent level, but many electroweak measurements are now an order of magnitude more precise. As one example, we found that the predicted total and leptonic Z widths both fell short of the corresponding experimental values by about 0.7%. Higher-order electroweak corrections are needed to match the precision of the new data. These corrections can shed fascinating light on new physics, as well as validating the original motivation for the electroweak theory (which was to be able to perform higher-order calculations). We shall describe a language introduced by Peskin and Takeuchi (1990) for precise electroweak tests which allows the constraints associated with nearly every observable to be displayed on a two-dimensional plot. The Standard Model implies a particular locus on this plot for every value of rat and My, so one can see how observables can vary with mi (not much, now that Mt is so well measured) and A/H. Moreover, one can spot at a glance if a particular measurement is at variance with others; this can either signify physics outside the purview of the two-dimensional plot, or systematic experimental error. The corrections which fall naturally into the two-dimensional description are those known as oblique corrections. The name stems from the fact that they do not directly affect the fermions participating in the processes of interest, but appear as vacuum polarisation corrections in gauge boson propagators. In that sense processes which are sensitive to oblique corrections have a broad reach for discovering new physics, since they do not rely on a new particle's having to couple directly to the external fermion in question. The oblique correction first identified by Veltman (1977), still the most important, is that due to top quarks in W and Z boson propagators. The large splitting between the top and bottom quarks' masses violates a custodial SU(2) symmetry (Sikivie et al. 1980) responsible for preserving the tree-level relation Mw = Mz cos B mentioned in the previous section. As a result, an effect is generated which is equivalent to having a Higgs triplet vacuum expectation value. For the photon, gauge invariance prohibits contributions quadratic in fermion masses, but for the W and Z, no such prohibition applies. The vacuum polarisation diagrams with W+ —> W+ and Z -+ (tt, bb) Z lead to a modification of the relation between G F, coupling constants, and Mz for neutral-current exchanges: GF g2 + g' 2 8Af" GF__g2 .9' 2 3G p P P 1 n4 • ' 8A/4 872 (179) 44 Jonathan L Rosner The Z mass is now related to the weak mixing angle by 71IX (180) v-2-GFpsin2 B cost where we have omitted some small terms logarithmic in nit . A precise measurement of Mz now specifies 8 only if mt is known, so 8 = O(mt ) and hence Mil, = 7ra/(V GF sine 0) is also a function of mi. The factor of p in (179) will multiply every neutral-current four-fermion interaction in the electroweak theory. Thus, for, example, cross sections for charge-preserving interac-tions of neutrinos with matter will be proportional to p2 , while parity-violating neutral-current amplitudes (to be discussed below) will be proportional to p. Partial decay widths of the Z, since they involve the combination 22 + g'2 , will be proportional to p. A large part of the 0.7% correction mentioned previously is due to p > 1. The observed values of Mw/Mz = p cos 8 and sin2 8 also are much more compatible with each other for a value of p exceeding 1 by about a percent. The W and Z propagators are also affected by virtual Higgs-boson states due to the couplings (174). Small corrections, logarithmic in MH , affect all the observables, but notably p. In order to display dependence of electroweak observables on such quantities as the top quark and Higgs boson masses mt and MH, we choose to expand the observables about "nominal" values calculated by Marciano (2000) for specific mt and MH . We thereby bypass a discussion of "direct" radiative corrections which are independent of mt , MH , and new particles. We isolate the dependence on Mt , MH, and new physics arising from "oblique" corrections associated with loops in the W and Z propagators. For mt = 174.3GeV, MH = 100GeV, the measured value of Mz leads to a nominal expected value of sin2 8eff = 0.2314. In what follows we shall interpret the effective value of sin2 O as that measured via leptonic vector and axial-vector couplings, namely sin2 Oeft. E (1/4)(1 — [gf,I Defining the parameter T by .gyp aT, we find T 3 I ir4 — (174.3 GeV)2 ] 3 In MH (181) A 4-2 1v1W cost 0 100 GeV 1671- sine The weak mixing angle 0, the W mass, and other electroweak observables now depend on mt and MH. The weak charge-changing and neutral-current interactions are probed under a num-ber of different conditions, corresponding to different values of momentum transfer. For example, muon decay occurs at momentum transfers small with respect to Mw, while the decay of a Z into fermion-antifermion pairs imparts a momentum of nearly Mz/2 to each member of the pair. Small "oblique" corrections, logarithmic in nit and MH, arise from contributions of new particles to the photon, W, and Z propagators. Other (smaller) "direct" radiative corrections are important in calculating actual values of observables. We may then replace the lowest-order relations between GF, couplings, and masses by GF g2 (i+ crSw GFp — g2 g' 2 _L. CVSZ 4 sin2 0) N/ 8/4 4 sin2 cost 8) (182) Standard Model 45 where Sw and Sz are coefficients representing variation with momentum transfer. To-gether with T, they express a wide variety of electroweak observables in terms of quantities sensitive to new physics. (The presence of such corrections was noted quite early by Velt-man 1977.) The Peskin and Takeuchi (1990) variable U is equal to Sw - Sz, while S Sz. Expressing the "new physics" effects in terms of deviations from nominal values of top quark and Higgs boson masses, we have the expression (181) for T, while contributions of Higgs bosons and of possible new fermions U and D with electromagnetic charges Qu and Qy to Sw and 5z, in a leading-logarithm approximation, are (Kennedy and Langacker 1990) sz =6r [In 100 GeV/c2 mil + E Arc (1 ni 4Q In U D SW = 1 r [In MH 100 GeV/c2 in E — 4QD In )] . 67 n The expressions for 5w and Sz are written for doublets of fermions with NC colours and mu > mD >> mz, while Q (Qu +QD)/2. The sums are taken over all doublets of new fermions. In the limit mu = mD, one has equal contributions to Sw and Sz. For a single Higgs boson and a single heavy top quark, (183) and (184) become 1[ in MH nit SZ 21n 6ir 100 GeV/c2 174.3 GeV/c2 1 [ln MH mt Spy 41n + 67r 100 GeV/c2 174.3 GeV/c2 where the leading-logarithm expressions are of limited validity for MH and nit far from their nominal values. (We shall plot contours of S and T for fixed mt and MH values without making these approximations.) A degenerate heavy fermion doublet with N, colours thus contributes ASz = ASW = Nc/67r. For example, in a minimal dynamical symmetry-breaking ("technicolour") scheme, with a single doublet of N, = 4 fermions, one will have AS = 2/37r 0.2. This will turn out to be marginally acceptable, while many non-minimal schemes, with large numbers of doublets, will be seen to be ruled out. Many analyses of present electroweak data within the S, T rubric are available (e.g., Swartz 2001). We shall present a "cartoon" version after discussing possible extensions of the Higgs system. Meanwhile we note briefly a topic which will not enter that discussion. The anomalous magnetic moment of the electron and muon have been measured ever more precisely. The latest measurement of the µ± (Brown et al. 2001), performed in a special storage ring at Brookhaven National Laboratory, gives ap+,exp _ (9 -2 2),+ = 11 659 202(14)(6) x 10-1° (1.3 ppm) , (186) The theoretical value (CPT invariance implies awF = ) is a,,th 11 659 177(7) x 10-b0 (0.6 ppm) , (187) (183) (184) (185) 46 Jonathan L Rosner being the sum of three terms: aµ,QED ap,weak a had = = 11 658 470.56(0.29) x 10-10 15.1(0.4) x 10' 691(7) x 10-10 (0.025 ppm) , (0.03 ppm) , (0.6 ppm) , where we have incorporated a recently-discovered sign change in the hadronic light-by-light scattering contribution (Knecht and Nyffeler 2001; Hayakawa and Kinoshita 2001). The difference, ap+,th = 25(17) x 10-1° , (188) is not yet known precisely enough to test the expected weak contribution. Results of analysing a larger data sample are expected shortly. 5.3 Multiple Higgs doublets and Higgs triplets There are several reasons for introducing a more complicated Higgs boson spectrum. Rea-sons for introducing separate Higgs doublets for u-type and d-type quarks include higher symmetries following from attempts to unify the strong and electroweak interactions, and supersymmetry. We examine the simplest model with more than one Higgs doublet, in which a single doublet couples to d-type quarks and charged leptons, and a different, dou-blet couples to u-type quarks. This model turns out to naturally avoid flavour-changing neutral currents associated with Higgs exchange (Glashow and Weinberg 1977). Let us denote by 0„ the Higgs boson coupling to u-type quarks and by Od the boson coupling to d-type quarks and charged leptons. We let ((u) = vu/N, (0d) = yd/5 • (189) The contribution of 0„ and Od to W and Z masses comes from CK (DpPu)i (DP c5u) (DµOd)t (DP0d) • (190) We find the same 1/1/,, — Bp mixing pattern as before, and in fact this pattern would remain the same no matter how many Higgs doublets were introduced. The parameters v„ and tad may be related to the quantity v = 246GeV introduced earlier by vu+vd = v2 , whereupon all previous expressions for Mw and Mz remain valid. One would have v2 = Ei /,z for any number of doublets. The quark and lepton couplings to Higgs doublets are enhanced if there are multiple doublets. Since rn, = gyv,/ (q = u or d) and v„ < v, one has larger Yukawa couplings than in the standard single-Higgs model. A more radical consequence, however, of multiple doublets in the SU(2) /, gauge theory is that there are not enough gauge bosons to "eat" all the scalar fields. In a two-doublet model, five "uneaten" scalars remain: two charged and three neutral. The phenomenology of these is well-described by Gunion et al. (1990). The prediction Mz = Mw/ cos 0 is specific to the assumption that only Higgs doublets of SU(2)L exist. [SU(2)L singlets which are neutral also have Y = 0, and do not affect 147 and Z masses.] If triplets or higher representations of SU(2) exist, the situation is changed. We shall examine two cases of triplets: a complex triplet with charges (++.+,0) and one with charges (+,0,-). Standard Model 47 Consider first a complex triplet of the form +1 '3L = 0 • (191) —1 Since Q = 131 + (Y/2), one must have 1' = 2 for this triplet. In calculating ID,FI2 we will need the triplet representation for weak isospin: I3= The result, 1 0 —1 if (40) = (IDA12 ) 1 II = V1,_1 /‘, is that V2 g2 { 0 1 0 1 0 1 0 1 0 (1472 )21+ = (_gw3 0 —1 0 1 0 —1 1 0 0)21 . (192) (193) = 2 2 Rw1 ) 2 The same combination of W3 and B gets a mass as in the case of one or more Higgs doublets, simply because we assumed that it was a neutral Higgs field which acquired a vacuum expectation value. Electromagnetic gauge invariance remains valid; the photon does not acquire a mass. However, the ratio of W and Z masses is altered. In the presence of doublets and this type of triplet, we find , (2 m2 = 3 21,7 1,2 -1\ Mw 4 \ / IL1Z = 9,2 \ 2 4_ / 41 12 v2 ± (194) (92 so the ratio p = (Mw / Mz cos 8) 2 is no longer 1, but becomes v2 2V12 1 (195) P = v2 + 41712_ 1 This type of Higgs boson thus leads to p < 1. A complex triplet 4>+ (196) is characterised by Y = 0. If we let (4)(5 = 1/1,0 / \F).,, we find by an entirely similar calculation, that 4>+ 4,0 „2 m2w = 4v12,0) m2 (g2 gi2 )v2 4 (197) Here we predict 4V120 p = 1 + 212 ' , (198) so this type of Higgs boson leads to p > 1. 48 Jonathan L Rosner We now examine a simple set of electroweak data (Rosner 2001), updating an earlier analysis (Rosner 1999) which may be consulted for further references. (See also Peskin and Wells 2001.) We omit some data which provide similar information but, are less constraining. Thus, we take only the observed values of Mw as measured at the Fermilab Tevatron and LEP-II, the leptonic width of the Z, and the value of sin2 Oeff as measured in various asymmetry experiments at the Z pole in e+e- collisions. We also include parity violation in atoms, stemming from the interference of Z and photon exchanges between the electrons and the nucleus. The most precise constraint at present arises from the measurement of the weak charge (the coherent vector coupling of the Z to the nucleus), Qw = p(Z - N - 4Z sin2 0), in atomic cesium. The prediction Qw (Cs) = -73.19 + 0.13 is insensitive to standard-model parameters once Mz is specified; discrepancies are good indications of new physics. The inputs, their nominal values for int = 174.3GeV and MH = 100GeV, and their dependences on S and T are shown in Table 6. We do not constrain the top quark mass; we display its effect on S and T explicitly. Each observable specifies a pair of parallel lines in the S-T plane. The leptonic width mainly constrains T; sin2 Oaff provides a good constraint on S with some T-dependence; and Mw lies in between. Atomic parity violation experiments constrain S with almost no T dependence. Although the errors on S they entail are too large to have much impact, we include them for illustrative purposes. Since the slopes associated with constraints are very different, the resulting allowed region is an ellipse, shown in Figure 13. [Note added: Milstein and Sushkov (2001) have noted that a correction due to the strong nuclear field changes the central value of Qw (Cs) in Table 6 to -72.2, while Dzuba et al. (2001) include this and further corrections to obtain Qw = -72.39 ± 0.58.] Quantity Experiment Theory Mw ( GeV/c2 ) 80.451 + 0.033 a) 80.385 b) — 0.29S + 0.45T Fee(Z) (MeV) 83.984 ± 0.086 a) 84.011 b) — 0.18S + 0.78T sin2 Bair 0.23152 ± 0.00017 `) 0.23140 b) + 0.00362S - 0.00258T Qw(Cs) -72.5 1 0.8 -73.19 - 0.800S - 0.007T Qw (T1) -115.0 ± 4.5 -116.8 - 1.17S - 0.06T Table 6. Electroweak observables described in fit. The superscripts refer to (a) Charlton (2001), (b) Marciano (2000) and (c) LEP EWWG (2001). References for atomic physics experiment and theory are given by Rosner (2001). Figure 13 also shows predictions by Peskin and Wells (2001) of the standard elec-troweak theory. Nearly vertical lines correspond, from left to right, to Higgs boson masses MH = 100, 200, 300, 500, 1000 GeV; drooping curves correspond, from top to bottom, to +la-, central, and -lo values of rni = 174.3 ± 5.1 GeV. In the standard model, the combined constraints of electroweak observables such as those in Table 6 and the top quark mass favour a very light Higgs boson, with most analyses favouring a value of MH so low that the Higgs boson should already have been discovered. The efficacy of a small amount of triplet symmetry breaking has recently been 0.4 0.2 0.0T -0.2 0.4 -0.2 0.0 03 S Standard Model 49 Figure 13. Regions of 68% (inner ellipse) and 90% (outer ellipse) confidence level values of S and T based on the comparison of the theoretical and experimental electroweak observables shown in Table 6. Details are given in the text. stressed in a nice paper by Forshaw et al. (2001). It is also implied in the discussions of Dobrescu and Hill (1998), Collins et al. (2000), He et al. (2001), and Peskin (2001). The standard model prediction for S and T curves down quite sharply in T as MH is increased, quickly departing from the region allowed by the fit to electroweak data. (Useful analytic expressions for the contribution of a Higgs boson to S and T are given by Forshaw et al. 2001.) However, if a small amount of triplet symmetry breaking is permitted, the agreement with the electroweak fit can be restored. As an example, a value of V1,o /v = 0.03 permits satisfactory agreement even for MH = 1TeV, as shown by the vertical line in the Figure. 5.4 Supersymmetry, technicolour, and alternatives What could lie beyond the standard model? The odds-on favourite among most theorists is supersymmetry, an extremely beautiful idea which may or may not be realised at the electroweak scale, but which almost certainly plays a role at the Planck scale at which space and time first acquire their meaning. The simplest illustration of supersymmetry (in one time and no space dimensions!) does back to Darboux in 1882, who factored second-order differential operators into the product of two first-order operators. Dirac's famous treatment of the harmonic oscillator, writing its Hamiltonian as H = hw(at a + is an example of this procedure, which was generalised by SchrOdinger in 1941 and Infeld and Hull in 1951. Some of this literature is reviewed by Kwong and Rosner (1986). The Hamiltonian is the generator of time translations, so this form of supersymmetry essentially amounts to saying that a time translation can be expressed as a composite of more fundamental operations. 50 Jonathan L Rosner Modern supersymmetry envisions both spatial and time translations as belonging to a super-algebra. The Lorentz group is isomorphic to SU(2) 0 SU(2) (with factors of i thrown in to account for the Minkowski metric); under this group space and time translations transform as (1/2,1/2). The supercharges transform as (1/2,0) and (0,1/2), clearly more fundamental objects. Electroweak-scale supersymmetry is motivated by several main points. You will hear further details in this lecture series from Abel (2001). 1. In any gauge theory beyond the standard SU(3)cotpur® SU(2) L, if the scale A of new physics is very high, this scale tends to make its way into the Higgs sector through loop diagrams, leading to quadratic contributions g2 A2 to the Higgs boson mass. Unless something cancels these contributions, one has to fine-tune counter-terms in the Lagrangian to exquisite accuracy, at each order of perturbation theory. This is known as the "hierarchy problem." 2. The very nature of a A(00)2 term in the Lagrangian is problematic when considered from the standpoint of scale changes. This is known as the "triviality problem." 3. In the simplest theory by Georgi and Glashow (1974) unifying SU(3)coiourO SU(2) L, based on the gauge group SU(5), the coupling constants approach one another at high scale, but there is some "astigmatism." In a non-supersymmetric model, they do not all come together at the same scale. This is known as the "unification problem." It is cured in the simplest supersymmetric model, as a result of the dif-ferent particle content in loop diagrams contributing to the running of the coupling constants. The model has a problem, however, in predicting too large a rate for p -+ K+0 (Murayama and Pierce 2001, Peskin 2001). An alternative scheme for solving these problems, which has had a much poorer time constructing any sort of self-consistent theory, is technicolour, the notion that the Higgs boson is a bound state of more fundamental constituents in the same way that the pion is really a bound state of quarks. This mechanism works beautifully when applied to the generation of gauge boson masses, but fails spectacularly (and requires epicyclic patches!) when one attempts to describe fermion masses. The basic idea of technicolour is that there is no hierarchy problem because there is no hierarchy; a wealth of TeV-scale new physics awaits to be discovered in the simplest version (applied to gauge bosons) of the theory. A further, even more radical notion, is that both Higgs bosons and fermions are composite. This scheme so far has run aground on the difficulty of constructing quarks and leptons, keeping their masses light by nearly preserving a chiral symmetry ('t Hooft 1980). One can make guesses as to quantum numbers of constituents (Rosner and Soper 1992), but a sensible dynamics remains completely elusive. 5.5 Fermion masses We finessed the question of the origin of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It comes about in the following way. The electroweak Lagrangian, before electroweak symmetry breaking, may be written Standard Model 51 in flavour-diagonal form as — Lint = - 9 —[U' Lry"WiY-) + h.c.] , (199) where U' (u', c', t') and D' (d', s', b') are column vectors describing weak eigenstates. Here g is the weak SU(2) L coupling constant, and iP L E (1 - -y5)-0/2 is the left-handed projection of the fermion field = U or D. Quark mixing arise because mass terms in the Lagrangian are permitted to connect weak eigenstates with one another. Thus, the matrices Mu, D in = -[U' RMuU' + D' RM D h.c.] (200) may contain off-diagonal terms. One may diagonalise these matrices by separate unitary transformations on left-handed and right-handed quark fields: FOQMQLQ = L-16M-IQRQ = AQ . (201) where Q' L = LQQL ; QT R = RQQR (Q = U, D) . (202) Using the relation between weak eigenstates and mass eigenstates: LP L = LU UL and D' L = LDDL, we find Ant = --g 4 [/L-e14,7,VDL + h.c.] , (203) where U (u, c, t) and D 1--_- (d, s, b) are the mass eigenstates, and V a Lti L D. The matrix V is just the Cabibbo-Kobayashi-Maskawa matrix. By construction, it is unitary: V+V = VV± = 1. It carries no information about Ru or RD. More information would be forthcoming from interactions sensitive to right-handed quarks or from a genuine theory of quark masses. Quark mass matrices can yield the observed hierarchy in CKM matrix elements. As an example (Rosenfeld and Rosner 2001), the regularities of quark masses evolved to a common high mass scale can be reproduced by the choice 0 6 3 ei`k 0 MQ = m3 E3e-'° E2 ( 62 (204) 0 €2 1 where m3 denotes the mass eigenvalue of the third-family quark (t or b), and E 0.07 for u quarks, 0.21 for d quarks. Hierarchical descriptions of this type were first introduced by Froggatt and Nielsen (1979). The present ansatz is closely related to one described by Fritzsch and Xing (1995). This type of mass matrix leads to acceptable values and phases of CKM elements. The question of neutrino masses and mixings has entered a whole new phase with spectacular results from neutrino observatories such as super-Kamiokande ("Super-K") in Japan and the Sudbury Neutrino Observatory (SNO) in Canada. These indicate that: 1. Atmospheric muon neutrinos oscillate in vacuum, probably to T neutrinos, with near-maximal mixing and a difference in squared mass Am,' 3 x 10-3 eV2. 52 Jonathan L Rosner 2. Solar electron neutrinos oscillate, most likely in matter, to some combination of muon and T neutrinos. All possible Amt values are at most about 10-4 eV2; several ranges of parameters are permitted, with large mixing favoured by some analyses. In addition one experiment, the Liquid Scintillator Neutrino Detector (LSND) at Los Alamos National Laboratory, suggests 0, --+ fie oscillations with Am' 0.1 to 1 eV, with small mixing. This possibility is difficult to reconcile with the previous two, and a forthcoming experiment at Fermilab (Mini-BooNE) is scheduled to check the result. For late news on neutrinos see the Web page maintained by Goodman (2001). A possible explanation of small neutrino masses (Gell-Mann, Ramond, and Slansky 1979, Yanagida 1979) is that they are Majorana masses of order mm = m2D /A/m, where Trip is a typical Dirac mass and Alm is a large Majorana mass acquired by right-handed neutrinos. Such a mass term is invariant under SU(2)L, and hence is completely acceptable in the electroweak theory. The pattern of neutrino Majorana and Dirac masses, and the mixing pattern, is likely to provide us with fascinating clues over the coming years as to the fundamental origin and nature of mass. 6 Summary The Standard Model of electroweak and strong interactions has been in place for nearly thirty years, but precise tests have entered a phase that permits glimpses of physics beyond this impressive structure, most likely associated with the yet-to-be discovered Higgs boson. Studies of mixing between neutral kaons or neutral B mesons, covered by Stone (2001) in these lectures, are attaining impressive accuracy as well, and could yield cracks in the Standard Model at any time. It is time to ask what lies behind the pattern of fermion masses and mixings. This is an input to the Standard Model, characterised by many free parameters all of which await explanation. 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Zweig G, 1964, CERN Reports 8182/TH 401 and 8419/TH 412 (unpublished), second paper reprinted in Developments in the Quark Theory of Hadrons, 1980, editors Lichtenberg D B and Rosen S P (Hadronic Press), v 1, pp 22-101. 57 Heavy quark theory Gerhard Buchalla Theory Division, CERN, Switzerland 1 Introduction The dedicated study of b-flavoured hadrons has developed into one of the most active and promising areas of experimental high-energy physics. The detailed investigation of b decays at the B-meson factories and at hadron colliders will probe the flavour physics of quarks with unprecedented precision. To fully exploit this rich source of data a systematic theoretical approach is necessary. The required field theoretical tools are the subject of these lectures. We shall discuss the construction of effective weak Hamiltonians, introducing the op-erator product expansion (OPE) and the renormalisation group (RG) and presenting the effective Hamiltonians for nonleptonic AB = 1 and AB = 2 transitions as examples. The subsequent chapter is devoted to heavy-quark effective theory (HQET). It explains the basic formalism as well as the application to heavy-light and heavy-heavy currents, dis-cussing the B-meson decay constant f B and the exclusive semileptonic decay B -+ D*1v, respectively. Inclusive b decays and the heavy-quark expansion (HQE) are treated next, in particular the general formalism, the issue of quark-hadron duality, the theory of b-hadron lifetimes and of inclusive semileptonic decays. Finally, we discuss QCD factorisation for exclusive hadronic B decays, focusing on B D7r and B -4 71-7r. We conclude with a short summary. The effective-Hamiltonian framework is the oldest and most general of the methods we shall discuss. It dates back, more or less, to the beginnings of the standard model itself. HQET and HQE are later developments that have been established in the second half of the eighties and at the beginning of the nineties. The last topic, QCD factorisation for exclusive B decays is the most recent. It is the least well established among these methods and it continues to be studied and developed in further detail. DOI: 10.1201/9780429187056-258 Gerhard Buchalla We would like to mention a very short selection of literature, which we hope will be helpful to obtain further information on the various subjects related to the contents of these lectures. Very useful resources are the BaBar Physics Book (Harrison Si Quinn 1998) and the Fermilab B Physics Report (Anikeev et al. 2002). They collect nice reviews on both theoretical and experimental topics in B physics. A textbook more specifically directed towards theoretical heavy quark physics is the work of Manohar Si Wise (2000). Review articles on particular subjects are (Buchalla, Buras & Lautenbacher 1996) on effective weak Hamiltonians, (Neubert 1994) on HQET and (Bigi, Shifman and Uraltsev 1997) on HQE. 2 Introduction and overview 2.1 Motivation In the following chapters we will study the theoretical tools to compute weak decay properties of heavy hadrons. To put the formalism into perspective, we start by recalling the main motivation for this subject. The central goal is the investigation of flavour physics, the most complicated sector in our understanding of fundamental interactions. A good example is give by particle-antiparticle mixing, as first studied with neutral kaons. The strong interaction eigenstates K° (M) and K° (ds) can be transformed into each other through second order weak interactions, which leads to a tiny off-diagonal entry M12 in the mass matrix (Figure 1). d C K° + K° K = M M12 K° M12 M ) K° Figure 1. - K° mixing. The physical eigenstates are K L,s = (K° +170)/ -V-2 with a mass difference AmK =mL -ms given by 20412 1 so that AniK L7"K"K I VCSVCd12 MC2 = 7 • 10-15 (1) mK 672 where the number on the right-hand side is the experimental value. The theoretical expression is derived neglecting the third generation of quarks and CP violation, which is a valid approximation for AmK. The factors fkBK (BK 1 for the purpose of a first estimate) account for the binding of the quarks into mesons. A crucial feature of (1) is the Glashow-Iliopoulos-Maiani (GIM) cancellation mechanism: the orthogonality of the quark mixing matrix 4;, (i=u, c; j=d, s) leads to a cancellation among the various contributions with intermediate up and charm quarks (symbolically the amplitude has the form (nu) - (uc) - (cu) + (cc)). For may = me this cancellation would be complete, giving AmK = 0. Still, even for mu mc, the contributions from virtual momenta k > me, Wu Heavy quark theory 59 cancel since both mu and me are negligible in this case. What is left is a characteristic effect proportional to mc2, the up-quark contribution being subleading for 17/u < AQCD < Mc. This circumstance allowed Gaillard and Lee in 1974 to correctly estimate the charm-quark mass me 1.5 GeV, before charm was eventually discovered in the Fall of the same year. In a similar way, the discovery of Bd - Bd mixing by the ARGUS collaboration (Al-brecht et al. 1987) proved to be another milestone in flavour physics. In full analogy to K - K mixing we have m2 AMB G2Ffi3BB ivovtd 12 t = 6 • 10-14 , 172B 672 7Dr2 Iv7141 (2) where now the top-quark contribution is completely dominant. The unexpectedly large value observed by ARGUS provided the first evidence that the top-quark mass (Mt,pole = 176 GeV) was comparable to the weak scale and in any case much heavier than anticipated at the time. These examples show very nicely the "flavour" of flavour physics: precision observables are sensitive probes of high-energy scales and yield crucial insights into the fundamental structure of weak interactions. At the same time we see that hadronic effects manifest in quantities such as f B, BB, and strong interactions of the participating quarks in general, play an important role. Their understanding is necessary to reveal the underlying flavour dynamics and is the main subject of heavy quark theory. B - B mixing, CP violation in B decays and other loop-induced reactions of b-flavoured hadrons are of great interest and continue to be pursued by numerous experiments. A central target is the Cabibbo-Kobayashi-Maskawa (CKM) matrix 2 Vud Vus vub 1 - -- A A.\ 3(g - 2 V = Lcd Ves Vcb 1 _ -- AA2 Lid Vts Vtb AA3(1 - g - 11) - A A2 1 which parametrises charged-current weak interactions in the standard model. The second equality in (3) introduces the approximate form of the Wolfenstein parametrisation, where the four independent CKM quantities are .\, A, g and I/. The unitarity triangle, as defined in terms of g and n is shown in Figure 2, indicating the CP violating angles a, /3 and Figure 2. Definition of the unitarity triangle. (3) Gerhard Buchalla 60 1 0.8 0.6 0.4 0.2 Figure 3. Global fit of the unitarity triangle (darker shaded region), without sin 20 from CP violation in B -+ Ks. The constraint from the world average sin 20 (hatched areas) is overlaid (Hocker et al. 2001). The status of the unitarity triangle (in terms of 0 = g(1 — A2/2) and fi = 71(1 — A2 /2)) is displayed in Figure 3, with input from CP violation in the kaon sector (E K ) and from B decays (IVo/Vebl, Amu, Arris and sin 20). 2.2 B decays — overview A vast number of different B-decay observables is available to further test the standard scenario of flavour dynamics. One may distinguish the following broad classes. • Dominant decays: b —> cud, b —> ccs, b —> cIP B 137 B tlf K , B -4 D(*)//) • Rare decays: b -# ufid, b —> uus, b ulP B 77, B B --> rrW, B IP . Rare and radiative (loop induced) decays: b —> s(dh, b —> s(d)l+l- , b —> s(d)vv B —> K(*)-y, B —> p-y, . AB = 2 oscillations: Bd Bd, B, - B, mixing Heavy quark theory 61 Other obvious classifications are between inclusive and exclusive processes or between hadronic and (semi)leptonic decays. A few key properties of b-hadrons enhance considerably the possibilities in B physics both experimentally and theoretically. First, the smallness of Vd, = 0.04 leads to a long lifetime of TB •'>-.1 1.6 ps. In addition the b-quark mass is large compared with the QCD scale mb >> AQCD = 0.3GeV. (4) The exact value of mb depends on the definition. In particular, the running MS mass filb(filb) = 4.2+0.1GeV, the pole mass mkpoie 4.8GeV, whereas the mass of the lightest b-hadron is mB = 5.28GeV. The smallness of AQCD/mb provides us with a useful expansion parameter. Together with the property of asymptotic freedom of QCD and as (mb) < 1, this opens the possibility of systematic approximations, which are exploited in the various applications of heavy quark theory. 3 Effective weak Hamiltonians Computing weak decays of hadrons is a complicated problem in quantum field theory. Two typical cases, the first-order nonleptonic process B° --+ 7+7-, and the loop-induced, second-order weak transition B- K- in) are illustrated in Figure 4. The dynamics Figure 4. QCD effects in weak decays. of the decays is determined by a nontrivial interplay of strong and electroweak forces, which is characterised by several energy scales of very different magnitude, the W mass, the various quark masses and the QCD scale: nit , Mw >> mb , me >> AQcD >> m„, md, (me). While it is usually sufficient to treat electroweak interactions to lowest non-vanishing order in perturbation theory, it is necessary to consider all orders in QCD. Asymptotic freedom still allows us to compute the effect of strong interactions at short distances perturbatively. However, since the participating hadrons are bound states with light quarks, confined inside the hadron by long-distance dynamics, it is clear that also non-perturbative QCD interactions enter the decay process in an essential way. To deal with this situation, we need a method to disentangle long- and short-distance contributions to the decay amplitude in a systematic fashion. A basic tool for this purpose is provided by the operator product expansion (OPE). 62 Gerhard Buchalla 3.1 Operator product expansion We will now discuss the basic concepts of the OPE for B meson decay amplitudes. These concepts are of crucial importance for the theory of weak decay processes, not only in the case of B mesons, but also for kaons, mesons with charm, light or heavy baryons and weakly decaying hadrons in general. Consider, for instance, the basic W-boson exchange process shown on the left-hand side of Figure 5. This diagram mediates the decay of d u QCD C as) • Figure 5. OPE for weak decays. a b quark and triggers the nonleptonic decay of a B meson such as B° -4 7+71--. The quark-level transition shown is understood to be dressed with QCD interactions of all kinds, including the binding of the quarks into the mesons. To simplify this problem, we may look for a suitable expansion parameter, as we are used to do in theoretical physics. Here, a key feature is provided by the fact that the W mass Mw is very much heavier than the other momentum scales p in the problem (mb, ryas ). We can \ Mb, AQCDI Mu, rild, therefore expand the full amplitude A, schematically, as follows (p2 A = C as (Q) + 0 Mai) (5) which is sketched in Figure 5. Up to negligible power corrections of C9(p2/111;24,), the full amplitude on the left-hand side is written as the matrix element of a local four-quark operator Q, multiplied by a Wilson coefficient C. This expansion in 1/Mw is called a (short-distance) operator product expansion because the nonlocal product of two bilinear quark-current operators (du) and (ftb) that interact via W exchange, is expanded into a series of local operators. Physically, the expansion in Figure 5 means that the exchange of the very heavy W boson can be approximated by a point-like four-quark interaction. With this picture the formal terminology of the OPE can be expressed in a more intuitive language by interpreting the local four-quark operator as a four-quark interaction vertex and the Wilson coefficient as the corresponding coupling constant. Together they define an effective Hamiltonian Ref( = C-Q, describing weak interactions of light quarks at low energies. Ignoring QCD, the OPE reads explicitly (in momentum space) ,2 Yth' r* - 8 vud vub k2 _i mev (du)v-A (141))V -A . * 2 G V VtibC (Q) k2 N lid w ) with C = 1, Q = (du)v-A(ftb)t._A and GF 7u* , 11eff = V d Vub (aU)V-A ( UU)V-A • N/ As we will demonstrate in more detail below after including QCD effects, the most important property of the OPE in (5) is the factorisation of long- and short-distance A= (6) (7) Heavy quark theory 63 contributions: all effects of QCD interactions above some factorisation scale (short distances) are contained in the Wilson coefficient C. All the low-energy contributions below (long distances) are collected into the matrix elements of local operators (Q). In this way the short-distance part of the amplitude can be systematically extracted and calculated in perturbation theory. The problem to evaluate the matrix elements of local operators between hadron states remains. This task requires in general nonperturbative techniques, as for example lattice QCD or QCD sum rules, but it is considerably simpler than the original problem of the full standard-model amplitude. In some cases also the approximate flavour symmetries of QCD (isospin, SU(3)) can help to determine the non-perturbative input. This is true in general for hadronic weak decays. A decisive advantage of heavy hadrons is the fact that the heavy-quark mass itself is still large in comparison to Awl:). The limit AQCD /Tnb << 1 can then be exploited, which is achieved, depending on the application, by using heavy quark effective theory, heavy quark expansion or QCD factorisation for exclusive nonleptonic decays. The short-distance OPE that we have described, the resulting effective Hamiltonian, and the factorisation property are fundamental for the theory of B decays. However, the concept of factorisation of long- and short-distance contributions reaches far beyond these applications. In fact, the idea of factorisation, in various forms and generalisations, is the key to essentially all applications of perturbative QCD, including the important areas of deep-inelastic scattering and jet or lepton pair production in hadron-hadron collisions. The reason is the same in all cases: Perturbative QCD is a theory of quarks and gluons, but those never appear in isolation and are always bound inside hadrons. Nonperturbative dynamics is therefore always relevant to some extent in hadronic reactions, even if these occur at very high energy or with a large intrinsic mass scale. Thus, before perturbation theory can be applied, nonperturbative input has to be isolated in a systematic way, and this is achieved by establishing the property of factorisation. It turns out that the weak effective Hamiltonian for nonleptonic B decays provides a nice example to demonstrate the general idea of factorisation in simple and explicit terms. We would therefore like to discuss the OPE for B decays in more detail, including the effects of QCD, and illustrate the calculation of the Wilson coefficients. A diagrammatic representation for the OPE is shown in Figure 6. The key to calculating the coefficients = ci. Figure 6. Calculation of Wilson coefficients of the OPE. Ci is again the property of factorisation. Since factorisation implies the separation of all long-distance sensitive features of the amplitude into the matrix elements of (Q,), the short-distance quantities Ci are, in particular, independent of the external states. This means that the Ci are always the same, no matter whether we consider the actual physical amplitude where the quarks are bound inside mesons, or any other, unphysical amplitude with on-shell or even off-shell external quark lines. Thus, even though we are ultimately interested in, e.g., B -4 771- amplitudes, for the perturbative evaluation of Ci we are free to choose any treatment of the external quarks according to our calculational convenience. A convenient choice that we will use below is to take all external quarks massless and Tai 3 64 Gerhard Buchalla with the same off-shell momentum p (p2 0). The computation of the C, in perturbation theory then proceeds in the following steps: . Compute the amplitude A in the full theory (with W propagator) for arbitrary external states. . Compute the matrix elements (Qz) with the same treatment of external states. . Extract the C, from A = C, (Q2). We remark that with the off-shell momenta p for the quark lines the amplitude is even gauge dependent and clearly unphysical. However, this dependence is identical for A and (Q,) and drops out in the coefficients. The actual calculation is most easily performed in Feynman gauge. To O(&) there are four relevant diagrams, the one shown in Figure 6 together with the remaining three possibilities to connect the two quark lines with a gluon. Gluon corrections to either of these quark currents need not be considered, they are the same on both sides of the OPE and drop out in the Ci. The operators that appear on the right-hand side follow from the actual calculations. Without QCD corrections there is only one operator of dimension 6 Q1 = (fljbj) V-A (8) where the colour indices have been made explicit. To (.9(cts ) QCD generates another operator Q2 = (di1ii)v_A V-A (9) which has the same Dirac and flavour structure, but a different colour form. Its origin is illustrated in Figure 7, where we recall the useful identity for SU(N) Gell-Mann matrices (diTiakuk )(ftiT;b1 ) = —21N (cl jui )(ftiki ) + 2(cl jui)(uibi ). (10) Tiak Figure 7. QCD correction with colour assignment. It is convenient to employ a different operator basis, defining Q± = Qi ± Q2 2 The corresponding coefficients are then given by C± = ± C2. (12) Heavy quark theory 65 If we denote by S± the spinor expressions that correspond to the operators Q± (in other words: the tree-level matrix elements of Q± ), thee full amplitude can be written as A= (1+ -y_Fas ln M1 S+ + 1 + -y_a, ln 1'v S_. (13) p2 p2 Here we have focused on the logarithmic terms and dropped a constant contribution (of order as, but nonlogarithmic). Further, p2 is the virtuality of the quarks and -y+ are numbers that we will specify later on. We next compute the matrix elements of the operators in the effective theory, using the same approximations, and find n2 (Q± = -y+ a, ( —1 _p2 + In ) S. (14) The divergence that appears in this case has been regulated in dimensional regularisation (D = 4 — 2E. dimensions). Requiring A = C+ (Q+) + C_(Q_), (15) M2 C± = ± 7+a, In 2 , (16) where the divergence has been subtracted in the minimal subtraction scheme. The effec-tive Hamiltonian we have been looking for then reads GF Neff = 7/ - V„dVo (C± (µ)Q+ + C_ (11)Q _) , (17) with the coefficients C± determined in (16) to (9(as log) in perturbation theory. The following points are worth noting: . The 1/E. (ultraviolet) divergence in the effective theory (14) reflects the Mw -f oo limit. This can be seen from the amplitude in the full theory (13), which is finite, but develops a logarithmic singularity in this limit. Consequently, the renormalisation in the effective theory is directly linked to the In Mw dependence of the decay amplitude. . We observe that although A and (Q± ) both depend on the long-distance properties of the external states (through p2 ), this dependence has dropped out in C+. Here we see explicitly how factorisation is realised. Technically, to 0(a, log), factorisation is equivalent to splitting the logarithm of the full amplitude according to 2 2 P2n2 • In Mp2 w = In M w + In (18) 2 Ultimately the logarithms stem from loop momentum integrations and the range of large momenta, between Mw and the factorisation scale tt, is indeed separated into the Wilson coefficients. . To obtain a decay amplitude from Jeff in (17), the matrix elements (fIQ±IB)(P) have to be taken, normalised at a scale tt. An appropriate value for /..t is close to the (-quark mass scale in order not to introduce an unnaturally large scale into the calculation of (Q). we obtain 66 Gerhard Buchalla . The factorisation scale /..t is unphysical. It cancels between Wilson coefficient and hadronic matrix element, to a given order in as, to yield a scale independent decay amplitude. The mechanism of this cancellation to 0(as) is clear from the above example (13-16). . In the construction of fleff the W-boson is said to be "integrated out" , that, is, removed from the effective theory as an explicit degree of freedom. Its effect is still implicitly contained in the Wilson coefficients. The extraction of these coeffcients is often called a "matching calculation", matching the full to the effective theory by "adjusting" the couplings C. . If we go beyond the leading logarithmic approximation 0(as log) and include the finite corrections of 0(as ) in (13), (14), an ambiguity arises when renormalising the divergence in (14) (or, equivalently, in the Wilson coefficients C± ). This ambi-guity consists in what part of the full (non-logarithmic) 0(as ) term is attributed to the matrix elements, and what part to the Wilson coefficients. In other words, coefficients and matrix elements become scheme dependent, that is, dependent on the renormalisation scheme, beyond the leading logarithmic approximation. The scheme dependence is unphysical and cancels in the product of coefficients and ma-trix elements. Of course, both quantities have to be evaluated in the same scheme to obtain a consistent result. The renormalisation scheme is determined in particular by the subtraction constants (minimal or non-minimal subtraction of 1/e poles), and also by the definition of -y5 used in D 4 dimensions in the context of dimensional regularisation. . Finally, the effective Hamiltonian (17) can be considered as a modern version of the old Fermi theory for weak interactions. It is a systematic low-energy approximation to the standard model for b-hadron decays and provides the basis for any further analysis. 3.2 Renormalisation group Let us have a closer look at the Wilson coefficients, which read explicitly C+ = 1 + 0,5(P) 'YV) In (0) _ 4 - 47r 2 /114, -8 (19) where we have now specified the exact form of the 0(as log) correction. Numerically the factor as (mb)7T) /(87) is about +3.5% (-7%), a reasonable size for a perturbative correction (we used as (tt = 4.2 GeV) = 0.22). However, this term comes with a large logarithmic factor of In(µ2 //11124,) -6, for an appropriate scale of /7, = 4.2 GeV. The total correction to C+ = 1 in (19) is then -21% (42%)! The presence of the large logarithm spoils the validity of a straightforward perturbative expansion, despite the fact that the coupling constant itself is still reasonably small. This situation is quite common in renormalisable quantum field theories. Logarithms appear naturally and can become very large when the problem involves very different scales. The general situation is indicated in the following table, where we display the form of the correction terms in higher orders, Heavy quark theory 67 denoting P = ln(p1Mw): LL NLL as f a, a'sr asP as cie3 a8e2 as3e as 4. 0(1) 0(as ) In ordinary perturbation theory the expansion is organised according to powers of as alone, corresponding to the rows in the above scheme. This approach is invalidated by the large logarithms since as f, in contrast to as, is no longer a small parameter, but a quantity of order 1. The problem can be resolved by re-summing the terms (as f)" to all orders n. The expansion is then reorganised in terms of columns of the above table. The first column is of 0(1) and yields the leading logarithmic approximation, the second column gives a correction of relative order as , and so forth. Technically the reorganisation is achieved by solving the renormalisation group equation (RGE) for the Wilson coefficients. The RGE is a differential equation describing the change of C± (ii) under a change of scale. To leading order this equation can be read off from (19) ( dln itC±(P)= 47'Y±o) .C±(P)• (21) The factors (as /47)4) are called the anomalous dimensions of C. To understand the term "dimension", compare with the following relation for the quantity fin, which has (energy) dimension n: d = (22) d ln The analogy is obvious. Of course, the C± (µ) are dimensionless numbers in the usual sense; they can depend on the energy scale pc only because there is another scale, Mw, present under the logarithm in (19). Their "dimension" is therefore more precisely called a scaling dimension, measuring the rate of change of C± with a changing scale att. The nontrivial scaling dimension derives from 0(as ) loop corrections and is thus a genuine quantum effect. Classically the coefficients are scale invariant, C± 1. Whenever a symmetry that holds at the classical level is broken by quantum effects, we speak of an "anomaly". Hence, the 4) represent the anomalous (scaling) dimensions of the Wilson coefficients. We can solve (21), using das a2 33 — 2f d In = 200 s47, /30 3 , o =C± ( Mw ) = 1, (23) µ and find (o) /,L,,0 °)/2/30 C± (ii) = [ 8 a (Mw)r± P 11 0 as(P) 1n 4247 —')/± = [ + o (24) as(A) J 47 µ2 I This is the solution for the Wilson coefficients C± in leading logarithmic approximation, that is to leading order in RG improved perturbation theory. The all-orders re-summation of as log terms is apparent in the final expression in (24). (20) q 0 q 68 Gerhard Buchalla 3.3 The AB = 1 effective Hamiltonian In this section we will complete the discussion of the AB = 1 effective Hamiltonian. So far we have considered the operators — (diPi)V -A(Pjbj)V - A) (25) (212 = (diPj)V-A(Pjbi)V-117 (26) which come from the simple W-exchange graph and the corresponding QCD corrections (Figure 8). We have slightly generalised our previous notation, allowing for the cases p = c. Figure 8. Correction to W exchange. Figure 9. QCD-penguin diagram. In addition, there is a further type of diagram at 0(as ), which we have omitted until now: the QCD-penguin diagram shown in Figure 9. It gives rise to the four new operators Q3 = (dibi)V-AE(Mi)V-Al (27) Q4 = (dibi)v_ A E(M4)v -A, (28) Q5 = (dibi)V-AE(qjqj)V-FAI (29) Q6 — (dibi)V-A E(gjqi)V-FA. q (30) Two structures appear when the light-quark current (qq)v from the bottom end of the diagram is split into V — A and V + A parts. In turn, each of those comes in two colour forms in a way similar to Qi and Q2. Finally, one further gauge-invariant operator of dimension six appears in the matching procedure, the chromomagnetic penguin operator Chg = — 87-r2 mb dzo-iw (1 + 75)T4b, Gaw,. (31) This operator corresponds to the diagrams in Figure 9 with the lower quark line omit-ted. The gluon is thus an external field, represented in (31) by the field-strength tensor Note that the characteristic tensor current necessitates a helicity flip in the b —> d transition, which is accompanied by a factor of the quark mass mb (the effect of and is neglected). The contribution of Q89 would be very small for the Hamiltonian of K decays, which only involves light external quarks, but it is unsuppressed for b decays. The operators Qi , , Q6, Q89 mix under renormalisation, that is the RGE for their Wilson coefficients is governed by a matrix of anomalous dimensions, generalising (21). In this way the RG evolution of C1,2 affects the evolution of C3, , C6, C89. On the other Heavy quark theory 69 hand C1,2 remain unchanged in the presence of the penguin operators Q3, ... , Q6, Q8g , so that the results for C1,2 derived above are still valid. The construction of the effective Hamiltonian follows the principles we have discussed in the previous sections. First the Wilson coefficients C,(uw), i = 1, , 6, 8g, are de- termined at a large scale ,uw = O(/Ilvv , m,) to a given order in perturbation theory. In this step both the W boson and the heavy top quark are integrated out. Since the renormalisation scale is chosen to be pw = O(Mw, m1 ), no large logarithms appear and straightforward perturbation theory can be used for the matching calculation. The anoma-lous dimensions are computed from the divergent parts of the operator matrix elements, which correspond to the UV-renormalisation of the Wilson coefficients. Solving the RGE the C, are evolved from pcw to a scale µ = O(mb ) in a theory with f = 5 active flavours q = a, d, s, c, b. The terms taken into account in the RG improved perturbative evaluation of C,(µ) are, schematically: LO: (a, In )n, NLO: a, (a, In )n, at leading and next-to-leading order, respectively. The final result for the AB = 1 effective Hamiltonian can be written as B = 1 G F eff = ciV+c2(g+ > C,Q, + h.c., (32) 1=3,...,6,8g where Ai, EZ Vp*dVpb . In principle there are three different CKM factors, Au, A and A1, cor-responding to the different flavours of up-type quarks that can participate in the charged-current weak interaction. Using CKM unitarity, one of them can be eliminated. If we eliminate A,, we arrive at the CKM structure of (32). The Hamiltonian in (32) is the basis for computing nonleptonic b-hadron decays within the standard model (to lowest order in electroweak interactions) with AB = 1 and AS, AC = 0. The Hamiltonian for b-decays with different flavour quantum numbers of the light quarks has a completely analogous form. For instance, AB = 1 transitions with a simultaneous change in strangeness, AS = 1, are simply described by (32) after the replacement d -4 s. When new physics is present at some higher energy scale, the effective Hamiltonian can be derived in an analogous way. The matching calculation at the high scale auw will give new contributions to the coefficients Ci (iii,v), the initial conditions for the RG evolution. In general, new operators may also be induced. The Wilson coefficients C", are known in the standard model at NLO. A more detailed account of 111B=1 and information on the technical aspects of the necessary calculations can be found in (Buchalla, Buras & Lautenbacher 1996) and (Buras 1998). 3.4 B — B mixing at NLO In the following section we present the effective Hamiltonian for a AB = 2 transition, which is relevant for B - B mixing. In this case only a single operator contributes. The form of the Hamiltonian is therefore particularly simple. We use this example to illustrate the structure of Wilson coefficients at next-to-leading order. The mass difference AmB in the B - B system is related to the effective Hamiltonian /3-2 by 1 Arna = 21/1/121B = ms (B17de=21B) (33) 70 Gerhard Buchalla In order to construct 7-a/3-2, the full standard model amplitude for AB = 2 transitions is matched onto the effective theory amplitude at the matching scale tit = 0(mt ) = 0(114w ). This is sketched in Figure 10. u, c, t d W + QCD ti C(µt ) • b Figure 10. OPE for B - B mixing. There is only one local operator Q = (bd)e_A(bd)v-A• (34) The Wilson coefficient, up to next-to-leading order, can be written as C(ut ) = C(°) (p,t ) + :f71C(1) (p,t ). (35) where C(') is the lowest order result and C(1) comes from the corrections with one-gluon exchange. The RG evolution from the high scale ut down to a scale p, = O(mb) has the form Op) = [1 + as as(Pt) J51 Fas(iLt) 6/23 47r L a,(µ) • C(ut ). (36) The second factor on the right-hand side is fami iar from the leading logarithmic approx-imation (the only difference is that at NLO the two-loop expression for ois (µ) has to be used). The first factor represents the next-to-leading order correction. Here J5 is a scheme-dependent constant, which in the usual, so-called NDR, scheme reads J5 = 5165/3174. We now have the ingredients to write the effective Hamiltonian up to NLO precision eff 2w ( 1EbVid)2 . C(A)Q 167 G2F /11,2 (Vtb Vtd)2So(xt)77B[cts(ii)] —6/23 [1 + (II) 16JF,1Q 72 471 (37) The result is entirely dominated by the top-quark contribution. It is common practice to separate the coefficient C(I) into the function So (xt ) (with act = m2t /M121,), which would be the coefficient in the absence of QCD effects, into the terms that depend on as (µ), and the remainder, which is defined as the QCD-correction factor Tut . This has been done in the second equation in (37). Taking the matrix element of 703-2 between the B and the B state and using (33) gives (-22 11/ 2 AmB = F'"V 11,7* 17 12S D 1.2 tb td -0,X071B 1-tBJ B 111, B• 67r 2 One encounters the hadronic matrix element of Q, which is written as (PIQ113)(11) = Inm2uBB(11), b (38) (39) Heavy quark theory 71 defining the (scale and scheme dependent) hadronic parameter BB (p). The combination BB =-L- BB(p)[cts(p)]-6123 [1. as47rP() 51 (40) is formally scale and scheme independent and has been used in (38). The parameter BB(//) is a nonperturbative quantity and has to be determined e.g. by lattice calculations. At present the value of BB is still very uncertain, in contrast to the short-distance QCD corrections, which are precisely known. A numerical illustration is given in Table 1, where we have put BB(µ) = 0.9 as an example. Table 1. 0.846 0.9 = 0.761 T1B [ces(P)]-6/23 4-/Ir'l J5] {1 + Be(li) 0.551 1.38 = 0.761 Two different definitions of a short-distance QCD factor can be considered, depending on where the terms with as(µ) are included. One possibility is to include them with 703 into a Wilson coefficient (=0.846), which is to be multiplied by the hadronic matrix element BB(µ) = 0.9. The other possibility is the formally scheme independent separation into tiB = 0.551 and BB = 1.38 (for 7)B this is the precise result; BB = 1.38 is only true in our example). The purpose of this exercise is to remind us that different definitions are sometimes employed for the parameter BB and care has to be taken which one is being used, in order to combine it with the appropriate short-distance corrections. We can also see that the large deviation of the QCD correction factor 7h3 from 1 is merely a consequence of pulling out the large factor [as(µ)]-6123. It is somewhat artificial and does certainly not indicate a problem for perturbation theory. In fact, the coefficient 0.846 is the one that has the proper limit, approaching 1 as a, --+ 0. It is indeed much closer to unity in accordance with the expectation for a perturbative correction factor. Still, the use of 7/B = 0.551 is often adopted due to its formally scheme invariant definition. An important application is the ratio of the mass differences for Bd and Bs mesons, for which (38) implies ArnedVed 2 MBd f/3d BBd = (41) Am Bs Ves rnB,, f This quantity is a very useful measure of 11/td/14,1. All other short-distance physics (top-dependence, 7)B) has dropped out. Hadronic uncertainties are reduced in the ratio of matrix elements, which is 1 in the limit of unbroken SU(3) flavour symmetry. The can-cellation of the short-distance contribution is a direct consequence of the factorisation property of the OPE. Lattice calculations give for the ratio of matrix elements (Hocker et al. 2001, and references therein) f Bs 3B Bs = 1.16 + 0.06. f Bd 3B Bd (42) The ratio AmBJA77t Bs is a very powerful constraint for the unitarity triangle, as can be seen in Figure 3. 72 Gerhard Buchalla 4 Heavy quark effective theory 4.1 Basic formalism Heavy quark effective theory (HQET) is an effective field theory designed to systematically exploit the simplifications of QCD interactions in the heavy-quark limit for the case of hadrons containing a single heavy quark. The HQET Lagrangian can be derived as follows. We start from the usual QCD Lagrangian for a heavy-quark field with mass m r = - mtp,p, (43) with the covariant derivative = a, - (44) The heavy-quark momentum can be decomposed as p = my + k, (45) where v is the 4-velocity of the heavy hadron. Once my, the large kinematical part of the momentum is singled out, the remaining component k is determined by soft QCD bound state interactions, and thus k = 0(AQcD) << m. We next decompose the quark field qf into ht,(x) = etiriV• x 1 + W(x) 2 Hv(x) = /3 W(x), 2 which implies (x) = (h„(x) + H,(x)) . (48) The expressions (1 +0)/2 are projection operators. Their action represents the covariant generalisation of decomposing tY into upper and lower components. Using the standard representation for 'y-matrices, this is evident in the rest frame where 0 = . Note also that the equation of motion with respect to the large momentum components, m(0 —1)h„ = 0, is manifest for hv. The exponential factor exp(imv-x) in (46), (47) removes the large-frequency part of the x-dependence in W(x) resulting from the large momentum my. Consequently, the x-dependence of hv, Hv is only governed by the small residual momentum and derivatives acting on hv and H„ count as 0(AwD). (Our sign conventions are appropriate for a heavy quark. To describe the case of a heavy anti-quark, similar definitions are valid with the sign of v reversed.) Multiplying the QCD equation of motion (4) — m)kli = 0 with the projectors (1 and (1 + 0)/2, and using (46) - (48) and the definition = DP — v"v•D, (49) we obtain the coupled system of equations iy•Dhu = (50) (iv.D + 2m)110 = tgliftv• (51) (46) (47) Heavy quark theory 73 They represent the equation of motion in terms of ht, and Hy. The second equation implies that kit, = (9(AwD/m)h, by power counting. Hence H„ is suppressed with respect to h„ in the heavy-quark limit. In other words, ft, contains the large components, H„ the small components of The HQET Lagrangian is obtained starting from (43), expressing in terms of hv, H„ and eliminating fit, using (51). We find = hviv•Dh, + hviDi 1 iv•D + 2 IJI m (52) Alternatively, Hi, as obtained from (51) in terms of h„ can be inserted into (50) to yield the equation of motion for h„. This equation is just the equation of motion implied by (52) (upon variation with respect to h„, i.e. 6r/8h, = 0). The Lagrangian may thus be written down immediately given the equation of motion for the field h„. The second term in (52) contains the nonlocal operator (iv.D + 2m)-1. It can be expanded in powers of AQcD /m to yield a series of local operators. Keeping only the leading-power correction we can simply replace (iv•D + 2m)-1 by (2m)-1 and get = hviv•Dh„ + 1hy(iD1) 2h„ + —h aPvG h v ILV V • (53) 2m 4m We now discuss some important aspects of this result. The first term on the right hand side of (53) is the basic, lowest-order Lagrangian of HQET. It describes the "residual" QCD dynamics of the heavy quark once the kinematic dependence on m is separated out. Since there is no longer any reference to the mass in, the only parameter to distinguish quark flavours, this term is flavour symmetric: The dynamics is the same for b and c quarks in the static limit. Since the operator v-D contains no 7-matrices, which would act on the spin degrees of freedom, the leading HQET Lagrangian also exhibits a spin symmetry. This corresponds to the decoupling of the heavy-quark spin in the in oo limit. Together, we have the famous spin-flavour symmetries of HQET (Isgur & Wise 1989). They lead to relations among different heavy-hadron form factors. From the Lagrangian kiv•Dh, the Feynman rules for HQET can be read off. The propagator is igv"Ta. (55) These Feynman rules enter in the computation of QCD quantum corrections. The remaining terms in (53) are the leading power corrections. They have an intu-itive interpretation. In the first term one recognises the operator for the nonrelativistic kinetic energy p2/(2m), which describes the residual motion of the heavy quark recoiling against the light degrees of freedom inside the heavy hadron. The last term represents the chromomagnetic interaction of the heavy-quark spin with the gluon cloud. Both effects violate flavour symmetry, the chromomagnetic term also spin symmetry, but they are power suppressed. i 1 + 0 (54) v•ic 2 where the projector (1 + 0)/2 appears since is a constrained spinor (see Equation 46). The interaction of the heavy-quark field hi, with gluons is given by the vertex 74 Gerhard Buchalla So far we have only considered QCD interactions. Weak interactions introduce exter-nal currents, which can also be incorporated in HQET. A generic heavy-light transition current 4FT, arising for instance in semileptonic decays, can be represented as qfkli =6, Fh,+0(-1 m), replacing the heavy-quark field by the HQET field hp using (48). (56) 4.2 Theory of heavy-hadron masses Before considering HQET in the context of weak decays, let us discuss a first application of the basic HQET Lagrangian (53) in the spectroscopy of heavy hadrons. To be specific, we shall analyse the masses of the ground-state mesons B and B*. These mesons constitute a doublet that arises because the spin 1/2 of the heavy quark couples with the total spin 1/2 of the light degrees of freedom in their ground state to form a spin-0 and a spin-1 meson, the pseudo-scalar B and the vector B*, respectively. Because the b-quark spin decouples in the heavy-quark limit, the state of the light cloud is identical for B and B* to leading order, and the angular-momentum coupling described above is the appropriate scheme. If we neglect the power corrections in (53), we can immediately write down the composition of the meson masses (0) (0) M B = T11 = mb + A. 13— Evidently the meson mass has a component mb from the heavy quark. In addition it has a term A = O(AQcD) from the energy of the light constituents. The latter is determined only by the interactions among the light degrees of freedom and their interaction with the static b-quark (hp) through the first term in (53). It is therefore independent of mb. The sum of mb and A is a physical quantity, however, separately both parameters are dependent on the scheme used to define them. In order to include the first power corrections, we treat the 1/m terms in (53) as perturbations to the lowest-order HQET dynamics. To first order in perturbation theory the corrections to (57) are then simply given by the expectation values of the 1/m terms. The proper normalisation is obtained as follows. If = —Glin, is the Hamiltonian (density) corresponding to the correction term Li m, in (53), and H = f d3xli is the Hamilton operator, the mass correction due to IL is just omB = (B111-11B1), (58) where 1B1 ) is the B-meson state normalised to one, (B1 1B1 ) = 1. Using the conventionally normalised states with (BIB) = 2mB V, we can write 1 1 (BH(0)113) Ornu = (BI f d3x1i(x)1B) f d3x (B17-1(0)IB) = , (59) 2mBV 2n2BV 2mB where we have used the translation invariance of 9-t and f d3x = V. Defining (gh(iD)2hIB) 1 (Blhgo-•Gh113) A2 (60) 2mB 6 2mB (57) = mb + A A1 = inc + A 2me 2774' Al (70) (71) Heavy quark theory 75 we obtain A l + 3A2 8mB — 2m (61) b Note that we may replace D2 by Di in the definition of A1 , up to higher order corrections (see (49), (50)). The parameter Al corresponds to (minus) the expectation value of the momentum squared of the heavy quark, Al = —(ph) = 0(A cD). This gives a positive correction in (61) representing the (small) kinetic energy of the heavy-quark. The A2-correction to the mass reflects the interaction energy of the heavy-quark spin with its hadronic environment, as already discussed in the previous section. While the A1 -term is independent of the heavy-quark spin and identical for B and B*, the chromomagnetic correction — A2 = 0(A,2wD) is different for B*. We have — A2 with the same A, Al and A2 as before. These results have a few interesting consequences. First, A2 parametrises the spin-splitting between the pseudo-scalar and the vector mesons: M D* — M D =— = 141 MeV. (68) me HQET predicts that the spin-splitting scales inversely proportional to the heavy-quark mass. This is seen to be quite well fulfilled given that mb 3me. Relation (67) call be used to determine the nonperturbative quantity A2 from experiment A2 = 4 (m2B* — B 7122B) = 0.12 GeV2 • (69) On the other hand, the quantity A l has to be estimated theoretically. Finally one may introduce the spin-averaged masses 70 B +3mB* 4 InD +3mD* 4 omB* = 2 (62) 7-rib Including (57) we arrive at the following expansion for the meson masses - A l + 3A2 mB mb + A 2m (63) b A1 — mB* = Mb (64) 2m A2b where the dependence on mb is explicit order by order. If we apply the heavy-quark limit to D mesons, we obtain analogous relations - A l + 3A2 mD me + A (65) 2m, A1 — A2 M D* 771,c + A (66) 2me mB* — MB = 2A2 = 46 MeV, (67) mb 2A2 mB fhu 76 Gerhard Buchalla This eliminates A2 and yields the useful result — = (the — ri/D) (1 _ _ 2MBMD (72) Since the ),1-correction is fairly small, the quark-mass difference is rather well determined, much better than individual quark masses. Exercise Derive the relative factor between the chromomagnetic correction to the mass of the B and the B* meson. Solution: Denote the heavy-quark spin by s, the total spin of the light degrees of freedom by j and the total spin of the meson by J = s + j. The chromomagnetic field of the light cloud has to be proportional to j. Hence the energy of the interaction between this field and s is proportional to (s-j). The angular momentum algebra then gives (2s.j) = J(J + 1) — s(s + 1) — j(j + 1), which is (-3/2) for B and (1/2) for B*, hence the relative factor (-1/3) of the A2-term in (62) with respect to (61). 4.3 Heavy-light currents and f B The B-meson decay constant f B is defined by the matrix element (01A,IB(P)) = —iismBv,, (73) of the heavy-light axial vector current A, = gry„-y5k1f. (74) Here q is the light-quark, DIY the heavy-quark field in full QCD, with 'If = b in the present case. The B-meson momentum is p = mBv. Let us analyse A, in HQET, including QCD corrections. The expansion of A, in HQET to leading order in 1/m, but allowing for QCD effects, has the form A = C1(p)A1 + C2(A)A2 + 0 (I) (75) A1 = 4'T/17511v A2 = qVA75hv (76) The matching conditions at the b-quark mass scale µ = mb are Ci (mb) = 1 + 0(as ) C2(mb) = 0(a5 ). (77) To leading order in QCD only Al is present in HQET, with coefficient one. Radiative corrections at 0(a5) modify C1 and generate a new operator A2. Note that the matching calculation of the full-QCD current A onto HQET, leading to (75), is completely anal-ogous to the OPE procedure of constructing the effective weak Hamiltonian from the W-exchange amplitude in the full standard model, which we have discussed in Section 3. The difference is only that a 1/Mw expansion is performed in the latter case, and a 1/mb Heavy quark theory 77 expansion in the case of HQET. The basic philosophy is essentially the same. In partic-ular, a factorisation of long and short-distance contributions is obtained: contributions from large scales greater than including the mb-dependence, are again contained in the coefficient functions C1,2. Soft scales less than µ, are factorised into the hadronic matrix elements of A1,2. In contrast to the full-QCD current A, the HQET currents do have an anomalous dimension, reflecting a logarithmic dependence of f B on the heavy-quark mass at 0(a8 ). The logarithms can be re-summed by renormalisation group methods, again in full analogy to the procedure in Section 3. In leading logarithmic approximation (LLA) C2 can be neglected and C1 acquires the familiar form [cfs(mbn -200 Ci (µ) = (78) as(i-t) Here the LLA assumes the hierarchy as (mb) < 1, a8 1n(mbh.c) = 0(1), which holds in the heavy-quark limit (mb large, tt = 0(1 GeV)). To express f B in HQET via (73), (75) and (78), we need the matrix element of Al (olAilB(P)) = (79) Since the dynamics of HQET is independent of mb, the reduced decay constant f (p) is mb-independent. The only mb-dependence in (79) enters through a trivial factor Vine from the normalisation of the B-meson state, which in the usual convention is given by (BIB) = 2mBV. (80) Collecting the ingredients, (75) yields I(µ) as (mb )1 -2A fa — (81) VmB as (p) This expression for f a is true to leading order in the HQET expansion in Acx,p/Mb and in leading logarithmic approximation in QCD. The factor f (µ) in (81) is still a nonperturbative quantity to be determined by other methods. However, the dependence of f B on the heavy-quark mass is now explicit. Equation (81) implies the scaling behaviour f B ti 1/VmB, up to a calculable logarithmic dependence on mb. In principle such a relation can be used to relate f B to the analogous quantity fa for heavy mesons with charm. In practice, it turns out that the leading order scaling result for f B is not very well fulfilled even for the b-mass scale and that subleading power corrections are important in this case. Nevertheless the result in (81) is of conceptual interest and can serve as a simple example of an application of HQET. 4.4 Heavy-heavy currents: B—>D(*)1P and Vcb One of the most important results of HQET is the extraction of 17,1, from exclusive semilep-tonic B —› D*10 decay. We will here give a short outline of the main steps in this analysis. The starting point is the differential decay rate dT(B > D*1P) = Veb2 k(w)F2(w) (82) dw 78 Gerhard Buchalla in the kinematical variable w = v•v', where v and v' are the 4-velocities of B and D*, respectively. The dependence of (82) on I Vcbl, the quantity of interest, is obvious, and K(w) is a known kinematical function. Finally, F(w) contains the nontrivial QCD dynamics encoded in the B D* transition form factors. The corresponding matrix elements of the weak currents can be written in the heavy-quark limit as 1 OnD*771B 1 vrnp*TriB (D*(vt ,E)1&-y„-y5b1B(v)) = e(w) [(1 + w)c„ - (c-v)vil . (84) In the heavy-quark limit, that is to lowest order in HQET, all hadronic dynamics is expressed in a single function 1 (w), the Isgur-Wise function (Isgur & Wise 1989). In this limit we further have -7(w) = (w). (85) Moreover, is absolutely normalised at the no-recoil point -(1) = 1. (86) The no-recoil point w = 1 corresponds to the kinematical situation where the D* meson stays at rest in the rest frame of the decaying B (v' = v = w = 1). Measuring dF/dw at w = 1, 117c1,1 can then be determined from (82) since all ingredients are known. Because w = 1 is at the edge of phase space, an extrapolation is necessary to find dF ldw l w_ i from the measured spectrum. For a realistic analysis corrections to the heavy-quark limit need to be considered. An important property of B D*10 is that linear power corrections in HQET are absent, = 0, where m can be either me or mb. Consequently the leading corrections enter only at second order and are thus greatly reduced. This result is known as Luke's theorem. The absence of linear corrections does not hold for B -+ DIP decays, hence the particular importance of B D*IP. Including corrections, the lowest order approximation T(1) = (1) = 1 is modified to .F(1) = 77,4 (1 + (87) where a1/ni2 are the second order power corrections and riA is a correction from perturbative QCD. To first order in a, it reads as (mt, + mb 8) 7 A = 1 + In 7r - me me 3J b (88) detailed numerical analysis yields (Harrison & Quinn 1998) F(1) = 0.913 ± 0.042, (89) which gives (Hocker et al. 2001) V reb = 0.0409 ± 0.0014,„p f 0.0019th• (90) To summarise the crucial points for the extraction of lid) from B D*I0 decay: (D*(v' ,c)1C7,101(v)) = e(w) E(U, E, , V), (83) Heavy quark theory 79 . Heavy-quark symmetry relates the various semileptonic form factors (four different, functions V, Ao, A1 , A2 in full QCD) to a single quantity e(w), the Isgur Wise function. . The function e is absolutely normalised, e(1) = 1. This property has an intuitive reason: At the kinematical point w = 1 the decaying b-quark at rest is transformed into a c-quark, also at rest. Since both quarks are treated in the static approximation (mb, oo, mbltn, fixed), the light hadronic cloud doesn't notice the flavour change b —> c and is transfered from the B to a D meson with probability one. The function e is identical for B D and B D* transitions, because these are related by heavy-quark spin symmetry. . HQET provides a framework for systematic corrections to the strict heavy-quark limit governed by e(w). Luke's theorem guarantees the absence of first-order cor-rections in 1/m for B —> D*ln. 4.5 HQET: conclusions We would finally like to summarise the basic ideas and virtues of HQET, and to re-emphasise the salient points. . HQET describes the static approximation for a heavy quark, covariantly formu-lated as an effective field theory and allowing for a systematic inclusion of power corrections. . Order by order in the expansion in AQcD /m HQET achieves a factorisation of hard, perturbative contributions (momentum scales between in and a factorisation scale p) and soft, nonperturbative contributions (scales below p). The former are contained in Wilson coefficients, the latter in the matrix elements of HQET operators. . The procedure of matching full QCD onto HQET is analogous to the construction of the effective weak Hamiltonian 'Leff . The difference lies in the massive degrees of freedom that are being integrated out: the W boson (mass Mw ) for 9-leff , the lower-component spinor field H, (mass 2m) for HQET. The perturbative matching can be supplemented by RG resummation of logarithms, ln(Mw /p) in the former case, ln(m/p) in the latter. . The usefulness of HQET is based on two important features: The spin-flavour sym-metry of HQET relates form factors in the heavy-quark limit and thus reduces the number of unknown hadronic quantities. The dependence on the heavy-quark masses is made explicit (scaling, power corrections). We conclude with briefly mentioning another field, called large energy effective theory (LEET), which has some similarities with HQET. LEET is needed for matrix elements of the form (MIqFb1B) at large recoil of the light meson M = p, K (*), etc. Then HQET is not sufficient in this situation because not only soft but also collinear infrared singularities need to be factorised. The latter occur due to the light-like kinematics of the fast and energetic light quark emitted from the weak current. To define LEET the 80 Gerhard Buchalla usual heavy-quark limit can be considered for the B meson with velocity v. The large-energy limit is taken for the light meson M with light-like momentum vector En. Here E = 0(mb) is the energy of M and n is a light-like 4-vector with n2 = 0 and v•n = 1. The momentum of the energetic light quark q is written as 7,, = En + k, with a residual momentum k = 0(Ac2cD ). In formal analogy to the fields by and 1-1, in HQET, the new light-quark fields qn(x) = eiEn.x2 q(x) Qn(x) = eiE 2 ?3'q(x), (91) can be defined and used in the construction of LEET (Dugan & Grinstein 1991, Charles et al. 1999, Beneke & Feldmann 2001, Bauer et al. 2001a). As a consequence of the LEET limit the ten form factors needed to describe all matrix elements (Mla F bir3) of bilinear heavy-light currents can be reduced to only three independent functions. LEET has received increasing interest quite recently and is still under active development. 5 Inclusive decays and the heavy quark expansion 5.1 Basic formalism and theory of lifetimes The heavy-quark limit, rn >> Awl), proves to be extremely useful also for the computation of inclusive decay rates of heavy hadrons (Chay et al. 1990, Bigi et al. 1992, 1997). The specific technique appropriate for this application is distinct from HQET and goes by the name of heavy quark expansion (HQE). Consider the total decay rate FH of a heavy hadron H. The starting point for the HQE is the following representation of FH FH 1(HITIH) = (7), (92) where the transition operator T is defined as Im i f d4 x T 7-leff (x)fleff (0), (93) with lieff the effective weak Hamiltonian. Equations (92), (93) express the total decay rate as the absorptive part of the forward scattering amplitude H —> H under the action of ?Jeff. This expression is referred to as the optical theorem by analogy to a similar relation in optics. One may rewrite (92), (93) in a more directly understandable form by inserting a complete set of states IX)(X I between the two factors of 7-ieff in (93) and removing the T-product by explicitly taking the absorptive part. This yields FH (1-117-leff IX)(XineffiH), (94) where one immediately recognises the decay rate as the modulus squared of the decay amplitude (summed over all final states X). The reason to introduce (93) is that the T-product, by means of Wick's theorem, allows for a direct evaluation in terms of Feynman diagrams. In order to compute Ft/ an operator product expansion is applied to (93), resulting in a series of local operators of increasing dimension. The coefficients of these operators are Heavy quark theory 81 correspondingly suppressed by increasing powers of 1/mb. The series has the form T = Fb bb + —zG2 bgo-•Gb + E 3 bF q q Fib + (95) mb mbwhere we have written the first few operators of dimension three (bb), five (bga•Gb) and six (bF2q 0, Fib). The matrix elements of the operators contain the soft, nonperturbative physics, their Wilson coefficients Fb, Zk the hard contributions, which are calculable in perturbation theory. Again, the coefficients are determined by an appropriate matching calculation between (93) and the r.h.s. of (95). The Feynman diagrams for the three terms in (95) are shown in Figure 11 The two weak-interaction vertices in these diagrams b W A d c b (B d) b 4 tt p 6N2__\c b (BO Figure 11. Heavy quark expansion for the total decay rate of b-hadrons. correspond to the two factors of 'Ha' in the definition of T in (93) (the absorptive part of the diagrams is understood). Obviously, the heavy quark expansion is different from HQET. However, we may still use HQET in conjunction with (95) in order to further analyse the hadronic matrix elements. An important example is the leading dimension-three operator bb. Its matrix element between heavy-hadron states H can be expanded in HQET as (bb) =1+ 1 (h(iD)2h) + (hgaGh), (96) 2mb2 4Mb where (. .) (HI 1H)/(2mH ). Equations (92), (95) and (96) imply that to leading order in the HQE F H = Fb, that, is the total decay rate of all b-flavoured hadrons is equal to the rate of free b-quark decay. Pictorially this can be seen from the first diagram in Figure 11, which represents essentially the amplitude squared for the partonic decay of a b-quark. Note also that perturbative QCD corrections to Fb can consistently be taken into account. The gluonic corrections to inclusive b-quark decay are infrared safe, as required for Fb in its role as a Wilson coefficient of the HQE. Also, corrections proportional to powers of crs (mb) ti 1/ ln(mb/A) are only suppressed by inverse powers of In mb in the heavy-quark limit, and hence formally leading in comparison to higher corrections in the HQE, which are suppressed by powers of A/mb. The calculation of heavy-quark decay in the parton picture has been used since the beginnings of heavy-quark physics as an approximation for inclusive decays of the corresponding heavy hadrons. As we have seen, the HQE gives a formal justification for this approach and provides us with a theoretical framework to compute nonperturbative corrections. The first correction term in (96) depends on the expectation value of the momentum squared (p2) of the heavy quark inside the hadron. This matrix element is non-zero 82 Gerhard Buchalla because the heavy quark is recoiling against the light degrees of freedom through gluonic interactions in the hadronic bound state. This term has a very intuitive interpretation. It corresponds to a correction factor 1 — (p2 )/(2mg) = 1 — (vg)/2, which is just the reduction of the free decay rate from time dilatation due to the recoil motion of the heavy quark. The second correction comes from interactions of the light hadronic cloud with the heavy-quark spin. We have 1 (hgaGh) = 32 (m26* — m6) H = B 0 H = Ab (97) The result is zero for the Ab baryon since the light degrees of freedom are in a state of zero total angular momentum. Note that the spin interaction enters twice in (95), explicitly with coefficient zG and via the expansion of (bb). The leading nonperturbative corrections start only at second order. There is no cor-rection linear in 1/mb. This is because there is no gauge-invariant operator of dimension four that could appear in the HQE. At order 1/mg contributions appear where the spectator quark participates directly in the weak interactions. For b-mesons they can be interpreted as the effect of weak annihilation (WA) of the b-quark with the valence d-quark (for Bd ) and as the effect of Pauli interference (PI)(for BO. The latter phenomenon occurs because in the nonleptonic decay of a B„, b(u) cfid(u), two identical u-quarks are present in the final state. These corrections distinguish, in particular, between Bd and B„ mesons and are responsible for their lifetime difference. Despite the suppression by three powers of rnb these effects can be relatively important due to their two-body kinematics, which brings a phase-space enhancement factor of 1672 in comparison to the leading three-body decay. As one of the possible applications, the HQE provides us with a theory of heavy-hadron lifetimes. The deviations of lifetime ratios from unity probes the power corrections. At present_ there are still sizeable theoretical uncertainties due to the hadronic matrix elements (b Fq q Fb). They can in principle be computed with the help of lattice gauge theory. Table 2 shows a comparison of theoretical predictions and experimental results (see for instance (Ligeti 2001)). Table 2. Experiment Theory T(B+)/T(B3) 1.068 ± 0.016 1 — 1.1 T (Bs)/ T(Bd) 0.947 + 0.038 0.99 — 1.01 7-(Ab)/7-(Bd ) 0.795 ± 0.053 0.9 — 1.0 5.2 Local quark-hadron duality A systematic uncertainty within the HQE framework, which is often debated in the lit-erature, arises from the issue of quark-hadron duality. In this paragraph we give a brief and heuristic discussion of the basic idea behind this topic. The theoretical prediction for an inclusive decay rate obtained from the HQE has the Heavy quark theory 83 2 1 . 5 F/ro 0.5 mb Figure 12. F/Fo as function of mb (arbitrary units). form A ) n r/ro = 1 ± > zn - , n=2 (98) where we have denoted the leading, free-quark result by Fo. Let us consider the decay rate as a function of mb, keeping A = Awl') constant. Then the quantity r/ro, to any finite order in (A/m), is a simple polynomial expression in this variable. This is sketched as the monotonic curve in Figure 12 showing F/F0 as function of mb (in arbitrary units). Now since, by construction, the HQE for r/ro yields a power expansion in (A/m), any term of the form exp (— ( b )k) sin ( b ) \ A / (99) for example, present in the true result for F/F0 would be missed by the HQE. This is due to the exponential suppression in the expansion parameter. In fact, the function exp(-1/x) is non-analytic. Its power expansion around x = 0 gives identically zero. However, such (or similar) terms are expected to be part of the true r/ro on general grounds. The corresponding complete result for r/ro, including such a term, is sketched as the oscillating graph in Figure 12. This true curve represents the physical result for the decay rate F/Fo, which consists of the inclusive sum over all the different exclusive decay channels. It is intuitively understandable that the true mb-dependence will have such a damped oscillating behaviour: if we imagine continually increasing mb, F/F0 will undergo a small jump whenever it reaches a value at which the presence of a further higher hadronic resonance in the final state becomes kinematically allowed. Since the excited hadrons have finite widths, the threshold behaviour will be smoothed out, resulting in the pattern of damped oscillations. The term quark-hadron duality refers to the idea that the inclusive rate as the sum over all exclusive hadronic decay channels and the inclusive rate as predicted by the heavy quark expansion are dual to each other. This means they are both valid representations of the same quantity using different descriptions, the hadron level or the quark level. The 84 Gerhard Buchalla term local refers to the fact that the energy scale mb is a fixed quantity, as opposed to e.g. the centre-of-mass energy in e+e- annihilation, which can be averaged to obtain suitably defined "global" quantities. In principle, the hadronic description gives the true result, measured in experiment. The problem is, however, that we would have to compute all exclusive rates first, which is far beyond our current control of nonperturbative QCD. On the other hand, the HQE calculation can be performed, within some uncertainties, but it is clear that the result need not be identical to the true answer. A deviation between the latter and the HQE (including power corrections) is referred to as a violation of quark-hadron duality. Indeed, contributions violating quark-hadron duality are expected (see(99)), but the numerical size of these terms cannot be strictly computed at present. Conceptually this is no problem because they are formally subleading in comparison to power corrections, so that the HQE still makes sense even at higher orders. The remaining question is how large can violations of quark-hadron duality be numerically. While there are at the moment, within the uncertainties intrinsic to HQE, no established cases in inclusive B decays where duality is violated, the issue clearly needs further investigation, both theoretically and phenomenologically. A more detailed account of the status of quark-hadron duality can be found in the papers by Blok, Shifman & Zhang (1998), Shifman (2000) and Bigi & Uraltsev (2001). 5.3 Inclusive semileptonic decays: 11-„b and Veb The HQE cannot only be applied to the total decay rates, but also to inclusive rates with specific flavour quantum numbers in the final state, such as semileptonic processes. Furthermore one can analyse differential decay rates. An example of special interest is the inclusive decay B —> X,lv, which can be used to extract Vet, The HQE for the integrated rate has the form 2 veb 92F27:3 , 4rn3A2 Z3 -I- 0+ 2 ) 6A2 + z5 —2- rnb (Tri2B* - rn2B). 0(as), (100) (101) (102) (103) F(B Xelv) = with 1 (bb) = + I ± 3A2 b (baGb) = 6A2 = - x4 - 12x2 In x + The Wilson coefficients read z3 = z5 = 1 - 8x + 8x3 -(1 - x)4 , where x = (me/mb)2. A major source of theoretical uncertainty for the determination of I Vcd using (100) is the b-quark mass. This appears to be especially problematic since rro, comes with the fifth power in (100). Fortunately, however, the actual situation is not as bad. Taking into account the phase-space function z3, one finds that the combined dependence on mb and me shows the approximate behaviour F(B Xclv) ti mb2.3 (nib - no2.7 (104) mb — Mc = (m6mD) (1 _ _ 2mBrnD = 3.40 ± 0.03 ± 0.03GeV, (105) Heavy quark theory 85 Since the difference mb - me is better known than the individual quark masses, the corresponding uncertainty is reduced. The quark-mass difference is in fact constrained by HQET, which gives (72) where 7'11 (Tit 3mB*)/4. The QCD corrections to z3 are known to (.9(ces ) and partly at 0(cts2 ). The special class of corrections 0(13Vi crns ) has been calculated to all orders n. Numerically the inclusive method gives (HOcker et al. 2001) V = 0.04076 ± 0.00050ex, ± 0.00204th , (106) which can be compared with the result from the exclusive determination via B D*10 (90). One can also try to extract IV„b1 from B Xulv decays. This is more difficult since the very large background from semileptonic b -+ c transitions requires kinematical cuts (in the lepton energy, the hadronic or the dilepton invariant mass), which renders the HQE less reliable and introduces larger uncertainties. A recent discussion has been given by Bauer et al. (2001b). The HQE has further useful applications, for instance in the case of the inclusive rare decays B Xs,d7, B Xs,dl+l-, or B Xs4vP. Exercise Show that quark-hadron duality is exactly fulfilled for the semileptonic b -+ c transition rate in the Shifman-Voloshin (small-velocity, or SV) limit mb, me >> mb - Mc >> AQCD• This holds with only two exclusive channels on the hadronic side of the duality relation, that is the inclusive rate is saturated as F(B -> xclo F(B Dlv)+F(B -4 D* Iv) in this limit. Solution: We start from the exclusive differential decay rates in the heavy-quark limit. They read (see e.g. Harrison & Quinn 1998): dl (B Dlv) dw dF(B D*lv) dw In the strict SV limit we have G4872F1Vc3 b12 (m6 Dip )2m3D ( w2 - 1)312 2(w) (107) G2F Vet, 12 (TDB - DID* )2 m3D* vw2 - 1(w 1)2 4873 4w m2B - 2wmBrnD* + m2D* x (1 + w + 1 (mB - mD* )2 . (108) ms = mb, mD* = mD = Inc, me = mb(1 - f), (109) where e (mb - me)/mb is a small parameter. The variable w is related to the dilepton invariant mass q2 through 2 rs q2 = MB + "I2 D - zmemDw. (110) 86 Gerhard Buchalla The kinematic limits of q2 are easily identified as „2 „,2 gmax 1"'b — "'T))2 Vmm = 0. The corresponding limits of w are Wmin = 1, Mb2 4- Inc Wmax = • 2171bM, (112) Defining s w — 1 we have 0 < s < e2 /2, where the upper limit is valid to leading order in e. Expanded to leading order in e, (107) gives F(B —> Dlv) = GF6171/3'eb12 7/2b5 V/72 fo(2 /2 S3/2 ds = G2F1V,b12 6073 (mb me)5' (113) which is the decay rate in the SV limit. In this derivation we have made use of the fact that -(w) = (1) + 0(e2), which can be approximated by (1) = 1. In this way any dependence on nontrivial hadronic input has disappeared. Similarly we can expand the integral over (108) in e to extract the leading contribution in the SV limit. We obtain iV b 2 F(B —> D*lv) = G2F0 c (Mb — mer. (114) 73 We also observe that higher D-meson resonances and hadronic multiparticle states have wave functions of the light degrees of freedom that are orthogonal to the ground state wave function of the light cloud (identical for D, D* and B) in the SV limit. There is therefore no overlap of those higher excitations with the initial B and the corresponding rates vanish. Finally, we need to take the SV limit of the inclusive rate as obtained from the heavy quark expansion in (100). In this limit the second-order power corrections and perturba-tive QCD corrections disappear, and we only have to expand the phase space function z3 in the small-f limit. We find z3 = 64e5/ 5 + 0(e6) and G2 iV1573 12 eb F(B Xclv) = (Mb — nic)5. (115) We see that indeed the inclusive HQE result (115) is saturated by the sum of just the two exclusive rates (113) and (114). Clearly, the SV limit is a very special situation. Nevertheless, it is an interesting example of exact (local) quark-hadron duality. Moreover, the semileptonic rates into D and D* measured in experiment account for roughly two thirds of the inclusive rate, indicating that the SV limit is not even entirely unrealistic. 6 QCD factorisation in exclusive hadronic B decays 6.1 Introduction Decay amplitudes for exclusive nonleptonic B decays, such as B 77i, can be computed starting from the effective weak Hamiltonian discussed in Section 3.3. Whereas the Wilson coefficients Ci are well understood, the main problem is posed by the hadronic matrix elements of the operators Qi. In some cases this problem can be circumvented (CP asymmetry in B J/kli Ks), or at least reduced using SU(2) or SU(3) flavour symmetries Heavy quark theory 87 and an appropriate combination of various channels. However, an improved understanding of the QCD dynamics in exclusive hadronic B decays would greatly enhance our capability to extract from these processes the underlying flavour physics. Indeed, it turns out that the heavy-quark limit leads to substantial simplifications also in the problem of hadronic two-body decays of heavy hadrons. Again the main feature is the factorisation of short-distance and long-distance contributions. In the case of the matrix elements of four-quark operators Q, the factorisation takes the form , (7 (11)7 (4) 1(2z1 = fa (42) ri du 7;1 (u)(13.„(u) + f ckdudv (e,u,u)( )13(e)(1)7,(u)(1)„ (v) . (116) This factorisation formula is valid up to corrections of relative order AQCD/rnb. Here f 13-'7 (q2) is a B 7r form factor evaluated at q2 = m2 0, and I (1.13 ) are leading- twist light-cone distribution amplitudes "wave functions") of the pion (B meson). These objects contain the long-distance dynamics. The short-distance physics, dominated by scales of order mb, is described by the hard-scattering kernels which are calculable in perturbation theory. starts at 0(a°), Ti" at 0(asi) (see Figure 13). In (116) long- Figure 13. Order as corrections to the hard scattering kernels TI (first two rows) and Ti" (last row). In the case of TZI , the spectator quark does not participate in the hard interaction and is not drawn. The two lines directed upwards represent the two quarks forming the emitted pion. and short-distance contributions are thus systematically disentangled, that is factorised. The long-distance sensitive quantities (form factors and wave functions) still need to be determined by other means, but they are universal quantities and much simpler than the original full B 7r7r matrix elements we started with. They could in principle be cal- culated by nonperturbative methods or extracted experimentally from other observables. In any case (116) represents a substantial simplification of our problem. The general expression (116) further simplifies when we neglect perturbative as-corrections. The T" term is then absent and the kernel V becomes a constant in u, such that the pion distribution amplitude integrates to the pion decay constant. The matrix element of operator Qi, for instance, reduces to (7+7-1(a)v _ A (du)v_A1/3) —> (7+ (t7b) V-A113) • (Tr- RdU)v_A 1°) = .771213.i B-"r (0) f7r. (117) This procedure, termed "naive factorisation" has long been used in phenomenological application, but the justification had been less clear. An obvious issue is the scheme 88 Gerhard Buchalla and scale dependence of the matrix elements of four-quark operators, which is needed to cancel the corresponding dependence in the Wilson coefficients. This dependence is lost in naive factorisation as the factorised currents are scheme independent objects. In QCD factorisation (116) the proper scale and scheme dependence is recovered by the inclusion of 0(as) corrections as we will see explicitly below. A qualitative justification for (117) had been given by Bjorken (1989). It is based on the colour transparency of the hadronic environment for the highly energetic pion emitted in B decay (the 7r- in the above example, which is being created from the vacuum). This is related to the decoupling of soft gluons from the small-size colour-singlet object that the emitted pion represents. The QCD factorisation approach as encoded in (116) may be viewed as a consistent formalisation and generalisation of Bjorken's colour transparency argument. This treatment of hadronic B decays is based on the analysis of Feynman dia-grams in the heavy quark limit, utilising consistent power counting to identify the leading contributions. The framework is very similar in spirit to more conventional applications of perturbative QCD in exclusive hadronic processes with a large momentum transfer, as the pion electromagnetic form factor (see the article by Sterman and Stoler (1997) for a recent review). It justifies and extends the ansatz of naive factorisation. In particular the method includes, for B 771, the hard nonfactorisable spectator interactions, penguin contributions and re-scattering effects (Figure 13). As a corollary, one finds that strong re-scattering phases are either of C9(as ), and calculable, or power suppressed. In any case they vanish therefore in the heavy quark limit. QCD factorisation is valid for cases where the emitted particle (the meson created from the vacuum in the weak process, as opposed to the one that absorbs the b-quark spectator) is a small size colour-singlet object, e.g. either a fast light meson (7r, g, K, K*) or a J/kIi. Note that the term factorisation is used here for two a priori entirely different things. In the case of QCD factorisation (116), it refers to the factorisation of short-distance and long-distance contributions. In the sense of the phenomenological approach of naive factorisation (117), it simply denotes the separation of the hadronic matrix element, of a four-quark operator into two factors of matrix elements of bilinear currents. It is a nontrivial result that the latter, naive factorisation is obtained as the lowest order ap-proximation of QCD factorisation. To avoid confusion, it is useful to keep the distinction in mind. For example, the hard gluon exchange corrections between the two quark cur-rents in Figure 13 are "nonfactorisable" in the sense of naive factorisation, although they are a consistent ingredient of (116), hence "factorisable" in the sense of QCD. In the following we shall discuss QCD factorisation in some detail using the example of B DIT decays. In this case the b u transition current is replaced by a heavy-heavy b c current. This case is somewhat simpler than B irir since the spectator interaction (the T H term, bottom line of Figure 13) does not contribute to leading power. This is because for a heavy-to-heavy transition the spectator quark, and hence the gluon attached to it, is always soft. This leads to a suppression, according to the colour transparency argument, when this gluon couples to the emitted pion. Also penguin contributions are absent for B --+ Dir. We shall illustrate explicitly how factorisation emerges at the one-loop order in this specific case, and in the heavy-quark limit, defined as inb, 174 >> AQcD with Tri,c /mb fixed. Further details on QCD factorisation in B decays and additional literature can be found in the articles by Beneke et al. (1999, 2000 & 2001). Heavy quark theory 89 6.2 B Dir: factorisation to one-loop order 6.2.1 Preliminaries The effective Hamiltonian relevant for B Mr can be written as GF * Neff = 7V d V, cb (C000 + C808) (118) with the operators 0 = E7"(1 - 75)b d7µ(1 - 75)u, (119) 08 = E7P(1 — 75)Tab d-y,(1 — 75)Tau. (120) Here we have chosen to write the two independent operators in the singlet-octet basis, which is most convenient for our purposes, rather than in the more conventional bases of Qi , Q2 or Q+, Q_ (see the discussion in Section 3; because all four quark flavours are different in (118), penguin operators are absent). The Wilson coefficients Co, C8 have been calculated at next-to-leading order in renormalisation-group improved perturbation theory (Altarelli et al. 1981, Buras & Weisz 1990) and are given by N, +1 N, —1 Co = C+ + C, C8 = C+ — C_, (121) 2N, 2N, where C± (µ)= (1 + a4(P) B± ) C± (µ), (122) 7r Ltas s(M(P) uld± [1+4 as(Mw) 7r as(A) C±(P) = (B± — J± )]. (123) (The coefficients Co, C8 are related to the ones of the standard basis by Co = C1 + C2/3 and C8 = 2C'2.) We employ the next-to-leading order expression for the running coupling, oes (µ)= 47r /31 In ln(p2/AcD)-1 Ooln(112/ Ai-4cD) i3 In(p2 /AcD) J 11N, — 2f 342 10 N2 — 1 3 0o = 01 = —3 Arc — —3 NJ — 2CF.i, CF - e (125) ' 2N ' where N is the number of colours, and f the number of light flavours. AQCD E A& is the QCD scale in the MS scheme with f flavours. Next we have (0) 7± (o) ey± = ±12N, + 1 B+ = ±N, 1 B. cl+ = 2i30 , (126) 2N, The general definition of 4 may be found in (Buchalla, Buras & Lautenbacher 1996). Numerically, for N, = 3 and f = 5 6 6473 23, 374' d+ = B+ — 4 = 12 9371 — 23' —1587' (124) (127) b c b d q q q q 90 Gerhard Buchalla (a) (b) (c) Figure 14. Basic quark-level topologies for B B71- decays (q = a, d): (a) class-I, (b) class- (c) weak annihilation. Bd D4- 7- receives contributions from (a) and (c), Bd D°71-° from (b) and (c), and B- D°71-- from (a) and (b). Only (a) contributes in the heavy-quark limit. The quantities (30 , X31 , d± , B± —,I+ are scheme independent. The scheme dependence of the coefficients at next-to-leading order is parametrised by B± in (122). In the naive dimen-sional regularisation (NDR) and 't Hooft-Veltman (HV) schemes, this scheme dependence is expressed in a single number B with BNDR = 11 and BFD/ = 7. The dependence of the Wilson coefficients on the renormalisation scheme and scale is cancelled by a corre-sponding scale and scheme dependence of the hadronic matrix elements of the operators 00 and 08. Before continuing with a discussion of these matrix elements, it is useful to consider the flavour structure for the various contributions to B Thr decays. The possible quark-level topologies are depicted in Figure 14. In the terminology generally adopted for two-body non-leptonic decays, the decays Bd -+ D+7-, Bd 13°70 and B- D°7r- are referred to as class-I, class-II and class-III, respectively. In both Bd D+71- and B- D°7- decays the pion can be directly created from the weak current. We may call this a class-I contribution, following the above terminology. In addition, in the case of Bd D+7T- there is a contribution from weak annihilation and a class-II amplitude contributes to B- —> D°7- , see Figure 14. The important point is that the spectator quark goes into the light meson in the case of the class-II amplitude. This amplitude is suppressed in the heavy-quark limit, as is the annihilation amplitude. The amplitude for Bd D°7°, receiving only class-II and annihilation contributions, is therefore subleading compared with Bd D+71"- and B-- —> D°7r- , which are dominated by the class-I topology. The treatment of this leading class-I mechanism will be the main subject of the following sections. We shall use the one-loop analysis for Bd D+71-- as a concrete example on which we will illustrate explicitly how the factorisation formula can be derived. 6.2.2 Soft and collinear cancellations at one-loop In order to demonstrate the property of factorisation for Bd D+7r- we have to anal- yse the "non-factorisable" one-gluon exchange contributions (Figure 15) to the b cud transition. We consider the leading, valence Fock state of the emitted pion. This is justi-fied since higher Fock components only give power-suppressed contributions to the decay amplitude in the heavy-quark limit. The valence Fock state of the pion can be written as kr(q)) = du d211 1 f Ni fut 1673 V2N, " " t (/ )bt (/-) — at (1 )bt (l q-)) 0)1 W(u, 11), (128) Heavy quark theory 91 (a) (b) (c) (d) Figure 15. "Non-factorisable" vertex corrections. where at, (bI) denotes the creation operator for a quark (antiquark) in a state with spin s =1- or s =1,, and we have suppressed colour indices. This representation of the pion state is adequate for a leading-power analysis. The wave function kIi(u, II) is defined as the amplitude for the pion to be composed of two on-shell quarks, characterised by longitudinal momentum fraction u and transverse momentum 11. The on-shell momenta (424 = 0) of the quark (4) and the antiquark (1,) are given by 12, 12 19 = uq +11+4uE n_ 1, = - 11+4f.tE n_. (129) In these expressions q = E(1, 0, 0, 1) is the pion momentum, E = pB • q/m B the pion energy and n_ = (1, 0, 0, -1). Furthermore l_L.q = 11.n_ = 0. For the purpose of power counting L AQCD < E mb. Note that the invariant mass of the valence state is (4+4)2 = 11/(uu), which is of order A cip and hence negligible in the heavy-quark limit, unless u is in the vicinity of the endpoints (0 or 1). In this case the invariant mass of the quark-antiquark pair becomes large and the valence Fock state is no longer a valid representation of the pion. However, in the heavy-quark limit the dominant contributions to the decay amplitude come from configurations where both partons are hard (u and both of order 1) and the two-particle Fock state yields a consistent description. The suppression of the soft regions (u or u << 1) is related to the endpoint behaviour of the pion wave function. We will provide an explicit consistency check of this important feature later on. As a next step we write down the amplitude 2 (7(q)1u(0).A(Y)010) = f du 6x373 v21 N tlf *(u,11)(')/5 4)0 expli4.0, (130) which appears as an ingredient of the B Dir matrix element. The right-hand side of (130) follows directly from (128). Using (130) it is straightforward to write down the one-gluon exchange contribution to the B Dir matrix element of the operator 08 (Figure 15). We have (D+7- Pad) 1-gluon = (131) - 1 d24_ kIJ*(u,11) ; (d2741;4 (.13+124 1 (k)b1Bd } f du tr[-y54,42 (/,, 14, k)], k2 o 1673 V2N, where Ai (k) = 7A (/3, - IE + mc)r r(i3b + + Tr/07 A 2pc.k - k2 2pb.k + k2 ' A2 (/q lq k) = r(j4 + 0)-yA ?'a(/, + 16)F , , 219.k + k2 219.k + k 2 • (132) (133) qA q • k (135) 92 Gerhard Buchalla Here F = (1-75) and pb, pc are the momenta of the b quark and the c quark, respectively. Note that this expression holds in an arbitrary covariant gauge. The gauge-parameter dependent part of the gluon propagator gives no contribution to (131), as can be seen from (132) and (133). There is no correction to the matrix element of 00 at order as, because in this case the (c/ft) pair is necessarily in a colour-octet configuration and cannot form a pion. In (131) the pion wave function 4/(tt, II) appears separated from the B D transition. This is merely a reflection of the fact that we have represented the pion state by (128). It does not, by itself, imply factorisation, since the right-hand side of (131) involves still nontrivial integrations over 11 and gluon momentum k, and long- and short-distance contributions are not yet disentangled. In order for (131) to make sense we need to show that the integral over k receives only subdominant contributions from the region of small k2. This is equivalent to showing that the integral over k does not contain infrared divergences at leading power in 1/mb. To demonstrate infrared finiteness of the one-loop integral f dik k2 Ai (k) ® A2(4,14, k) (134) at leading power, the heavy-quark limit and the corresponding large light-cone momentum of the pion are again essential. First note that when k is of order rub, 1 for dimensional reasons. Potential infrared divergences could arise when k is soft or when k is collinear to the pion momentum q. We need to show that the contributions from these regions are power suppressed in mb. (Note that we do not need to show that J is infrared finite. It is enough that logarithmic divergences have coefficients that are power suppressed.) We treat the soft region of integration first. Here all components of k become small simultaneously, which we describe by scaling k A. Counting powers of A (d4 k --, 1/k2 A-2, 1/p • k .\-') reveals that each of the four diagrams (corresponding to the four terms in the product in (134)) is logarithmically divergent. However, because k is small, the integrand can be simplified. For instance, the second term in 42 can be approximated as 2 \ ')/A(fq + A (U4 " 01/E) + 14) F 2uq • k + 211 • k + (2121 •k E k where we used the fact that 4 to the extreme left or right of an expression gives zero due to the on-shell condition for the external quark lines. We get exactly the same expres-sion but with an opposite sign from the other term in A2 and hence the soft divergence cancels out when adding the two terms in A2. More precisely, we find that the integral is infrared finite in the soft region when / I is neglected. When 11 is not neglected, there is a divergence from soft k which is proportional to 121/q AcD/Ing• In either case, the soft contribution to J is of order Awamb or smaller and hence suppressed relative to the hard contribution. This corresponds to the standard soft cancellation mechanism, which is a technical manifestation of colour transparency. Each of the four terms in (134) is also divergent when k becomes collinear with the light-cone momentum q. It is convenient to introduce, for any four-vector v, the light-cone 21q • k + k2 Heavy quark theory 93 components V0 ± v3 V+ = The collinear region is then described by the scaling k+ A°, k1 A, A, IC A2. (137) Then d4 k dk+ dk- c12 k A4 and q • k = q+k A2, k2 = 2k+ + k1 A2. The divergence is again logarithmic and it is thus sufficient to consider the leading behaviour in the collinear limit. Writing k = aq + . . . we can now simplify the second term of A2 as rya(l q + 2(u + oz)F 21, • k + k2 qA 21, • k + k2. (138) No simplification occurs in the denominator (in particular 11 cannot be neglected), but the important point is that the leading-power contribution is proportional to qA. Therefore, substituting k = aq into A l and using q2 = 0, we obtain qA241 0(0, + mc)r r(fib +mb)ra = 0, (139) 2ap, • q 2apb • q employing the equations of motion for the heavy quarks. Hence the collinearly divergent region is seen to cancel out via the standard collinear Ward identity. This completes the proof of the absence of infrared divergences at leading power in the hard-scattering kernel for Bd D+7r- to one-loop order. In other words, we have shown that the "non- factorisable" diagrams of Figure 15 are dominated by hard gluon exchange. The proof at two loops has been given by Beneke et al. (2000) and a proof to all orders by Bauer et al. (2001c). Since we have now established that the leading contribution to J arises from k of order mb ("hard" k), and since 1111 << E, we may expand A2 in 1111/E. To leading power the expansion simply reduces to neglecting /I altogether, which implies 19 = uq and 14 = fiq in (133). As a consequence, we may perform the 11 integration in (131) over the pion wave function. Defining f d211 (u,1±) _ (13. 7, (u), J 1673 v2N, - the matrix element of 08 in (131) becomes (D+71 108 1B41—gluon 2 eF r d4k + _ - gs —81V, j (2)4 (D IcA1(k)b1.13d) —1 f Jo 1 du (1),,.(u) tr[-y50,42 (uq, k)]. k2 Putting y on the light-cone in (130), y+ = y1 = 0, hence /, • y = /q+y- = uqy, we see that the 11-integrated wave function (I),(u) in (140) is precisely the light-cone distribu-tion amplitude of the pion in the usual definition. Indeed, the leading-twist light-cone distribution amplitude for pseudoscalar mesons (P) reads (136) (140) (141) (P(q)1q(y)cd (x),9 1 0) f _ = 4p (0y5)0, I du ez(uqs+ugY ) p (U, 11). 2= 0 (= y)' (142) 94 Gerhard Buchalla Here it is understood that the operator on the left-hand side is a colour singlet. We use the "bar"-notation, i.e. V 1 — v for any longitudinal momentum fraction variable. The parameter p, is the renormalisation scale of the light-cone operators on the left-hand side. The distribution amplitude is normalised as fc; du (I) p(U, 1,1) = 1. One defines the asymptotic distribution amplitude as the limit in which the renormalisation scale is sent, to infinity. The asymptotic form is (1) p(u, pi) 11—=°° 6uu. (143) The decay constant appearing in (142) refers to the normalisation in which f„ = 131 MeV. There is a path-ordered exponential that connects the two quark fields at different posi-tions and makes the light-cone operators gauge invariant. In (142) we have suppressed this standard factor. This derivation demonstrates the relevance of the light-cone wave function for the factorisation formula. Note that the collinear approximation for the quark-antiquark momenta emerges automatically in the heavy-quark limit. After the k integral is performed, the expression (141) can be cast into the form 1 (D+71" 1081-B41—gluon ^ FB=1,(0) du T8 (u, z) D+7- amplitude factorises in the heavy-quark limit into a matrix element of the form (D+1[...]b1Bd ) for the B D transition and a matrix element (7- Id(x)[. .]u(0)10) with x2 = 0 that gives rise to the pion light-cone distribution amplitude. Leaving aside power-suppressed contributions, the essential requirement, for this conclusion was the absence of both soft and collinear infrared divergences in the gluon exchange between the (CI)) and (du) currents. This gluon exchange is therefore calculable in QCD perturbation theory. We now present these corrections explicitly to order a,. The effective Hamiltonian (118) can be written as eff G F - N — 1 as(t1) CF Va* 2N Ne + 1 dVeb {[ C± (ft) + C_(p) + BC801.1 00 , 4ir 2N , + [C8(P)] 08}, (145) where the scheme-dependent terms in the coefficient of the operator 00, proportional to the constant B defined after (127), have been written explicitly. Schematically, the matrix elements of both 00 and 08 can be expressed in a form analogous to (144). Because the light-quark pair has to be in a colour singlet to produce the pion in the leading Fock state, only 00 gives a contribution to zeroth order in a,. Similarly, to first order in as only 08 can contribute. The result of computing the diagrams Heavy quark theory 95 in Figure 15 with an insertion of 08 can be written in a form that holds simultaneously for H = D, D* and L = 7r, p, using only that the (ftd) pair is a colour singlet and that the external quarks can be taken on-shell. One obtains (z = rn,/mb ) CF f 1 (H (p')L(q)108IBd(p)) - 47r 2N, 1j f L du DL (u) (146) X {- (61n +13) ((iv)-( JA))+F(u, z) (4)- F (u, -z) (141, mb where (Jv) = (H (1,1)14 blBd(p)), (JA) = (H (014-y5b1Bd(p)). (147) In obtaining (146) we have used the equations of motion for the quarks to reduce the operator basis to Jv and JA. The form of (146) is identical for pions and longitudinally polarised p mesons. The production of transversely polarised p mesons is power suppressed in Ac4cD /rrtb. In the case of a distribution amplitude (I)L (u) that is symmetric under u U, which is relevant for L = 7, p, the function F(u, z) appearing in (146) can be compactly written as F(u, z) = 31n z2 - 7 + f (u, z) + f (u, 1/z), (148) with u(1 - z2)[3(1 - u(12 z2 )) + f (u, z) = ln[u(1 z2 )] (149) [1 - u(1 - z)]2 1 - u(1 - z2 ) . The contribution of f (u, z) in (148) comes from the first two diagrams in Figure 15 with the gluon coupling to the b quark, whereas f (u, 1/z) arises from the last two diagrams with the gluon coupling to the charm quark. The absorptive part of the amplitude, which is responsible for the occurrence of strong rescattering phases, arises from f (u, 1/z) and can be obtained by recalling that z2 is z2 - if with F > 0 infinitesimal. We then have 1 Im F(u, z) _(1 - u)(1 - z2 )[3(1 - u(1 - z2 )) + [1 - u(1 - z2 )]2 (150) As mentioned above, (146) is applicable to all decays of the type Bd -4 D(*)+L-, where L is a light hadron such as a pion or a (longitudinally polarised) p meson. Only the operator Jv contributes to Bd ---+ D+L, and only in contributes to Bd D*+ . (Due to helicity conservation the vector current B D* matrix element contributes only in conjunction with a transversely polarised p meson and hence is power suppressed in the heavy-quark limit.) Our final result can therefore be written as (D+ L where L = 100,00 =- (D+ 0'1(1 p, and the hard-scattering To (u, z) = 1 + 0(aD, C — 75)biBd) . iftgµ Jo du To,8(u, z) <12. L (u), kernels are [ IL2 61n m-2- - B + F(u, z)1 + 0(01,2). b (151) (152) (153) T8(u, z) = 47r 2NF, 96 Gerhard Buchalla When the D meson is replaced by a D* meson, the result is identical except that F(u, z) in (153) must be replaced by F(u, -z). Since no order a, corrections exist for 00, the matrix element retains its leading-order factorised form (D+L-100 IBd ) = if Lq, (13+ IC-211 - 1/5)b1Bd ) (154) to this accuracy. From (149) it follows that T8(u, z) tends to a constant as u approaches the endpoints (u 0, 1). Therefore the contribution to (151) from the endpoint region is suppressed, both by phase space and by the endpoint suppression intrinsic to (1)[,(u), which can be represented as (I)L (u) = 6u(1 - u) [1 + a,/,'(//) C,3/2 (2u - 1)] , (155) n=-1 vanishing u (1 - u)) at the endpoints. Here Cr (x)=3x, (x)=-(5x2-1), etc. are Gegenbauer polynomials. The Gegenbauer moments an (it) are nonperturbative quan- tities. They are multiplicatively renormalised and approach zero as µ co. (The scale dependence of these quantities enters the results for the coefficients only at order oz:2,, which is beyond the accuracy of a NLO calculation.) As a consequence of the endpoint suppression the emitted light meson is indeed dom-inated by energetic constituents, as required for the self-consistency of the factorisation formula (151). Combining (145), (151), (152) and (153), we obtain our final result for the class-I, non-leptonic Bd D(*)+L- decay amplitudes in the heavy-quark limit, and at next- to-leading order it as. The results can be compactly expressed in terms of the matrix elements of a "transition operator" * T = NV dV, — b [a l (DL) Qv - at (D* L) QA] , (156) u where Qv = e-yttb d-y,(1 - -y5)u, QA = &Y"'Y5b c6,(1 - 75)u, (157) and hadronic matrix elements of Qyyt are understood to be evaluated in factorised form, i.e. (DLIJ I j21/3) = (JAMB) (Lij210). (158) Equation (156) defines the quantities a l (D(*)L), including the leading "non-factorisable" corrections, in a renormalisation-scale and -scheme independent way. To leading power in AQCD/mb these quantities should not be interpreted as phenomenological parameters (as is usually done), because they are dominated by hard gluon exchange and thus calculable in QCD. At next-to-leading order we get ai (DL) (D* L) = + - - - f 1 du F(u, z) 413. !Jul , (159) f du F(u, -z)(13.L(u) (160) ] N2N,1 C-1- (/-1) + a, CF \ 2N,1 mb 1 + -47 —2N, u8(p)[-D + 1 /\,/, C+(P) ± 2N, a, CF 2N, "P) tt2 [-61n m b C8(µ)) + 2N, le) = F(Bd D(*)± L) dF(Bd D(*)±l -P) dq2 = 6721v.d12fi la' (D(*) L)I 2 x(*) , = m (162) Heavy quark theory 97 These expressions generalise the well-known leading-order formula aL0 = Ne + 1 cLo (it) + Ne —1 cLo 2N, + (161) We observe that the scheme dependence, parametrised by B, is cancelled between the co-efficient of 0 in (145) and the matrix element of 08 in (151). Likewise, the it dependence of the terms in brackets in (159) and (160) cancels against the scale dependence of the coefficients CAA), ensuring a consistent physical result at next-to-leading order in QCD. The coefficients a i (DL) and ai (D*L) are seen to be non-universal, i.e. they are ex-plicitly dependent on the nature of the final-state mesons. This dependence enters via the light-cone distribution amplitude (1)L (u) of the light emission meson and via the analytic form of the hard-scattering kernel (F(u, z) vs. F(u, —z)). However, the non-universality enters only at next-to-leading order. Exercise Verify that the coefficients al (D(*)L) in (159) and (160) are independent of the unphysical renorrnalisation scale ft up to terms of 0(ol2). 6.2.4 Phenomenological applications for B —> Dir An important test of QCD factorisation can be performed by comparing the hadronic decays Bd D(*)± I,- with the semileptonic processes B d D(*)+ 1- V . It is useful to introduce the ratios where X,. = XP = 1 for a vector meson (because the production of the lepton pair via a V — A current in semi-leptonic decays is kinematically equivalent to that of a vector meson with momentum q), whereas X, and X: deviate from 1 only by (calculable) terms of order m27,./m2B, which numerically are below 1%. The main virtue of (162) is that the B D(*) form factors cancel in the ratio. The theoretical prediction for the QCD coefficients, based on QCD factorisation to leading power and at NLO in perturbative QCD, is (DO') L) I = 1.05. The uncertainty of this leading-power result is small, about ±0.01. The prediction is to be compared with the experimental results, extracted from (162), which read (D*7)1 = 1.08 ± 0.07, lai (D* p)1 = 1.09 ± 0.10 and la' (D*ai )1 = 1.08 + 0.11. These values show fair agreement with the theoretical number, albeit within experimental uncertainties that are still large. Another interesting consideration concerns the comparison of class-I modes with those of class II and III. For B D(*)7, all three decay modes, which are related by isospin, have been measured. A nice discussion of the present experimental status and its in-terpretation in the context of QCD factorisation has been given by Neubert & Petrov (2001). Let us briefly discuss here a few important aspects. The experimental status is 98 Gerhard Buchalla B —> D7T B —> D* 71- B° —) D(*)+71-- 3.0 + 0.4 2.76 + 0.21 B° -4 D(*)°70 0.27 + 0.05 0.17 ± 0.05 B- -+ D(*)°R-- 5.3 ± 0.5 4.6 + 0.4 IC— AVIT + AI 0.42 ± 0.05 0.35 ± 0.05 8Tc (56 + 20)° (51 + 20)° Table 3. Experimental data for B D(*)7r branching ratios (in units of 10') and extracted parameters IC — + Al, 67,c, (see Neubert 6 Petrov (2001)). summarised in Table 3. Denoting the basic topologies from Figures 14(a,b,c) by T, C and A, respectively (where the notation refers to "tree", "colour-suppressed tree" and "annihilation"), we have A(Bd D+ 7-) = T + A, (163) \A(Bd D°7°) = C — A, (164) A(B- —> D°7-) = T + C. (165) For later use we may further define the spectator-interaction contribution to T, Tspec (see bottom row of Figure 13). A similar decomposition holds for the B —> D*7 modes. Isospin symmetry is reflected in the amplitude relation A(Bd DH-7-) + A(Bd -+ D°7°) = A(B- —> D°7r-), which is manifest in the parametrisation of (163) - (165). This means that there are only two independent amplitudes, which we can take to be (T + A) and (C— A). These amplitudes are complex due to strong phases from final-state interactions. Only the relative phase of the two independent amplitudes is an observable and we define 6Tc to be the relative phase of (T + A) and (C — A). The phase can then be extracted from the data through the relation r(13°)) B(B- —>D°7-) — B(B° —>D+R--) — 2B(B°—>D°70) cos Orc = \3B(B° —>D-E7r-) B(B°—>D°7°) (166) In the usual heavy-quark limit, mb ti in, > Awl), only T is calculable. Twec, C and A are not, but they are power suppressed. It is instructive to consider the alternative limit mb >> me >> Awl), which allows us to distinguish between rub and me (Beneke et al. 2000). In this case, due to me << mb , also the D becomes a "light" meson and in that respect the process is similar to B —> 77, where both Tspec and C are calculable. However, since me >> Awl) A, the D-meson wave function TD(u) is highly asymmetric and strongly peaks at ii (1 — u) Atatc , where u is the c-quark momentum fraction. These properties can be used to derive the scaling rules A A Tspec A T rub T me ' C A T me . (167) Heavy quark theory 99 The amplitude A is still not calculable in this scheme, while T, Ts pec and C are. Note that from (167) we can recover the two standard scenarios we have been discussing: In the heavy-quark limit me rub (167) reduces to a simple power suppression A/mb for A, Tspec and C compared to T. On the other hand, if m, becomes a truly light quark, corresponding to the case of B 77, we count m, ti A and see that both Tspec arid C are of the same order as T, while A is still power suppressed. The general scenario mb > me > Awl) allows us to interpret the experimental results in Table 3. We can even make numerical estimates for T, Tspec and C based on the factorisation formula for light-light final states (116). These are somewhat model dependent because (1 D is not known at present. It is not too difficult to accommodate substantial values IC - AI/IT + 0.2 - 0.3 and 87-c ti 40°, in qualitative agreement with Table 3. Given the special role of the charm quark (not light, but also not too heavy), the current experimental situation is not in contradiction with QCD factorisation in the large-mb limit. For a comparison with experiment it is useful to keep in mind that, according to (167), the suppression of C over T is only A/m, (not A/mb) and that 5T(: is naturally expected to be of order one. 6.3 CP violation in B -4 7r+7r- decay Hadronic B decays into a pair of light mesons, such as B 7K or B 77, have a very rich phenomenology. Their main interest lies in their sensitivity to short-distance flavour physics, including CKM parameters, CP violation and the search for new physics. By way of an outlook we mention here the important example of CP violation in B 7+7- decay. The starting point for computing the required decay amplitudes is the effective Hamiltonian in (32). The needed matrix elements of the operators Qi can be analysed within QCD factorisation using (116). We will not go into the technical details of such an analysis and the discussion of limitations of the approach, in particular from power corrections in AQCD/mb. These can be found in (Beneke et al. 2001). Here we just want to present the phenomenological motivation and to illustrate that a theoretical approach towards a direct dynamical calculation of hadronic matrix elements will be very valuable, even if it necessitates some approximations. The observable of interest is the time-dependent CP asymmetry in the decays of B° and B° to 7+7-- ; it is sensitive to the Bd--Bd mixing phase c.-2'/3 . We define B(B°(t) - B(B°(t) 71+71- ) AU,r, (t) = B(B°(t) 7F+ 7F- ) B(B°(t) 71+71- ) = -S„ sin(AmB t) + C„cos(Am B t) (168) where 2 Im A„ 1 - 1A7r7 = e12 _20 e-il" P„/T„ C A S7't = 1 + 1A„12 "Tr 1 + A?T'712 , „ e17 + P„/T7,7, (169) The amplitudes T„ ("tree") and P, ("penguin") are the components of the B 7+7- amplitude corresponding to the terms in (32) involving Au and Ac, respectively. In the standard phase conventions A, is real and Au has a weak phase -7, which has been factored out above. The coefficient C„, which is a function of 7, represents direct CP violation 100 Gerhard Buchalla 1 - 0.5 0 0.5 1 sin 2a Figure 16. Relation between sin 2a and the mixing-induced CP asymmetry S„, assuming sin 2/3 = 0.48. The dark band reflects parameter variations of the first kind, the light band shows the total theoretical uncertainty. The lower portion of the band refers to values 45° < a < 135°, the upper one to 0 < a < 45° (right branch) or 135° < a < 180° - (left branch). and is expected to be small. We shall not discuss it further here. The mixing-induced asymmetry S„ depends on -y and /3. In fact, in the limit where P,„/T„ is set to zero it follows that A„„ = e'i(19+7) = Oa, and hence S„ = sin 2a. In this limit A„ is just the relative weak phase between the direct amplitude B -> 7+71-- and the one with mixing B B 7+7-. All dependence on hadronic input has cancelled in this situation. In practice, however, P„„/T„„ is not fully negligible. It is here that information on hadronic dynamics becomes crucial. QCD factorisation predicts that P„„/T„„ is suppressed (either by a, or by powers of AQCD/mb), because T„ can arise at tree level, P„ only through loops. Estimates within this framework give values of about 0.25-0.3. To illustrate the effect of penguin amplitudes, we first assume that 1V,,,b/17,b1 and the weak phase /3 have been determined accurately. Then using 7 = 180° -a- 0 the expression for A„ in (169) becomes a function of a and our prediction for the penguin-to-tree ratio P„„/T„„. If we further assume that the unitarity triangle lies in the upper half of the (fi, 7)) plane, then a measurement of S„ determines sin 2a with at most a two-fold discrete ambiguity. Figure 16 shows the relation between the two quantities for the particular case where IV„6 /17,1,1 = 0.085 and /3 = 14.3°, corresponding to sin 2/3 = 0.48. The dark band shows the theoretical uncertainty due to input parameter variations, whereas the light band indicates the total theoretical uncertainty including estimates of the effect of power corrections. We observe that for negative values sin 2a as preferred by the global analysis of the unitarity triangle, a measurement of the coefficient S„ could be used to determine sin 2a with a theoretical uncertainty of about ±0.1. Interestingly, for such values of sin 2a the "penguin pollution” effect enhances the value of the mixing-induced CP asymmetry, yielding values of S„ between -0.5 and -1. Such a large asymmetry should be relatively easy to observe experimentally. 1 0.5 0 - 0.5 Heavy quark theory 101 0.6 0.4 0.2 0 - 0.2 - 0.4 - 0.6 - 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 p Figure 17. Allowed regions in the (fi,fi) plane corresponding to constant values of the mixing-induced asymmetry S„ assuming the Standard Model. The widths of the bands reflect the total theoretical uncertainty. The corresponding bands for positive values of S„ are obtained by a reflection about the p axis. The light circled area in the upper-right quadrant shows the allowed region obtained from the standard global fit of the unitarity triangle (Hacker et al. 2001). Although it illustrates nicely the effect of "penguin pollution" on the determination of sin 2a, Figure 16 is not the most appropriate way to display the constraint on the unitarity triangle implied by a measurement of S„. In general, there is a four-fold discrete ambiguity in the determination of sin 2a, which we have reduced to a two-fold ambiguity by assuming that the triangle lies in the upper half-plane. Next, and more importantly, we have assumed that I Vub/Veb I and /3 are known with precision, whereas a is undetermined. However, in the Standard Model I V„b /17,b1 and the angles a, i(3, -y are all functions of the Wolfenstein parameters p and It is thus more appropriate to represent the constraint implied by a measurement of S„ as a band in the (p, fi) plane. To this end, we write P. 3152 ± 7)2 e (1-p)2 -7? -2ifi(i-p) P„, r„ ( 1 — 15) 2 + Tirir V192 + F12 ' (170) where r„ elo—r is independent of p and p. We now insert these relations into (169) and draw contours of constant S„ in the (p, p) plane. The result is shown by the bands in Figure 17. The widths of the bands reflect the total theoretical uncertainty (including power corrections). For clarity we show only bands for negative values of S„; those corresponding to positive S, values can be obtained by a reflection about the p axis (i.e., —fl). Note that even a rough measurement of S„ would translate into a rather narrow band in the (p, f)) plane, which intersects the ring representing the Vub Veb constraint at almost a right angle. In a similar way, the constraint is also quite robust against hadronic uncertainties. Even the approximate knowledge of hadronic matrix elements, as provided by QCD factorisation, will therefore be very valuable and can lead to powerful constraints on the Wolfenstein parameters. 102 Gerhard Buchalla 7 Summary In these lectures we have discussed the theory of heavy quarks, focusing on the important case of B physics. (All methods relying on the heavy-quark limit could in principle he applied to charmed hadrons as well, but they are in most cases much less reliable due to the smaller value of the charm mass.) We shall conclude by summarising the key points. . A crucial and general idea for dealing with QCD effects is the factorisation of short-distance and long-distance dynamics. We have encountered this principle in many different forms and applications: — The OPE to construct the effective weak Hamiltonians (7leAffB=1,2) factorises the short-distance scales of order Mw, mt from the scales of order mb. — The heavy-quark scale in treated as a short-distance scale can be factorised further from the intrinsic long-distance scale of QCD, Awl:). This leads to a systematic expansion of observables simultaneously in 1/m and a, (m) with often very important simplifications. The precise formulation of this class of factorisation depends on the physical situation and can take the form of HQET, LEET, HQE or QCD factorisation in exclusive hadronic B decays. . HQET exhibits the spin-flavour symmetry of QCD in the heavy-quark limit, which allows us to relate different form factors, and makes the mQ dependence explicit. Examples of typical applications are B -+ D(*)1v or f B. . HQE is a theory for inclusive B decays. It justifies the "parton model" and allows us to study the nonperturbative power corrections. This is of great use for processes as B —r X,,,i1v, B -+ Xs-y, B s1+1-, and the lifetimes of b-flavoured hadrons. . QCD factorisation, finally, refers to a framework for analysing exclusive hadronic B decays with a fast light meson as for instance B -+ Drr, B —4 rrrr, B —> 7rIt. and B —r With these tools at hand we are in a good position to make full use of the rich experi-mental results in the physics of heavy flavours. We can determine fundamental parameters of the flavour sector, such as Vub, Vcb, Vtd I Vts, 71 and sin 2a, and probe electroweak dy-namics at the quantum level through b -+ sey or B - B mixing. This will enable us to thoroughly test the standard model and to learn about new structures and phenomena that are yet to be discovered. 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Sterman G and Stoler P, 1997, Hadronic form factors and perturbative QCD, hep-ph/9708370. Lattice QCD Christine Davies University of Glasgow, UK 1 Introduction Lattice QCD was invented, way ahead of its time, in 1974. It really became a useful technique in the 1990s when a huge amount of progress was made in the understanding and reduction of systematic errors. Now, we are poised to start a second lattice revolution with the onset of Teraflop supercomputing around the world and further improvements in methodology. This will enable calculations using lattice QCD to reach errors of a few percent, over the next five years. At this level, lattice results, where they exist, will be the theoretical calculations of choice for the experimental community. It seems, then, a good time to review the fundamentals of lattice QCD, for an audience of experimental particle physicists. As 'consumers' of lattice calculations, it is important to be aware of how these calculations are done so that a critical assessment, of different results can be made. I have tried to keep technical details to a minimum in what follows but it is necessary to understand some of them, to appreciate the significance and the limitations of the lattice results that you might want to use. For a more detailed discussion see, for example (Gupta, 1998) or (Di Pierro, 2001). This school is largely concerned with CP violation and heavy quark physics, so in Section 4 I concentrate on lattice results relevant to these areas. 2 Lattice QCD formalism and methods 2.1 The path integral approach Lattice QCD is just QCD, no more and no less. We take the theory, express it in Feynman Path Integral language, and calculate the integral as well as we can. We would like to be able to do this in the continuous space-time of the real world, but this is not possible. Instead, we must break space-time up into a 4-d grid of points, i.e. a lattice (Figure 1), and evaluate the Feynman Path Integral by Monte Carlo methods on a computer. It turns out to be a calculation that requires a huge amount of computing power and tests the fastest supercomputers that we have. 105 DOI: 10.1201/9780429187056-3106 Christine Davies • • • • • • • • • • • • • • • • Figure 1. A 2-dimensional rendition of a 3-dimensional cubic lattice. Lattice QCD calculations use a 4-dimensional grid. In the Feynman Path Integral approach, we first express the quantity that we want to calculate as the matrix element in the vacuum of an operator, 0, which will be a product of quark and gluon fields so that, for example, 0 = (00),(1/*),,. creates a hadron at a point x and destroys it at a point y. We will discuss later other forms that 0 might take to calculate useful quantities. Then: f [dO] PO] [d An] Okl) , d), s MOP = (1) f PO] [dVd[dAp]e-s where S is the action, the integral of the Lagrangian: S = f G d4 x. (2) We are using Euclidean space here (imaginary time) so that the integrand doesn't contain the oscillatory els, but the more easily integrated e-s. The integral of Equation 1 can then be evaluated numerically if we can convert it to a finite-dimensional problem. Currently the integral runs over all values of the quark and gluon fields '45 and A at ev-ery point in space-time. We need to make the number of space-time points (and therefore field variables) finite and we do this by taking a 4-d box of space-time and discretising it into a cubic grid, or lattice. It is then a relatively simple matter to transcribe the contin-uous theory onto the lattice, and we use the standard methods used for discretising e.g. differential equations for numerical solution. Continuous space-time (x, t) becomes a grid of labelled points, (xi, ti ) or (nta, nta) where a is the spacing between the points, called the lattice spacing. The fields are then associated only with the sites, 0(x ,t) —> (rt, , nt ). The action must also be discretised, but this is also straightforward. The Lagrangian typically contains fields and derivatives of fields. The fields are replaced with fields at the lattice sites and the derivatives replaced with finite differences of these fields. The integral over space-time of the Lagrangian becomes a sum over all lattice sites: (f a a4 ). There are inevitably discretisation errors associated with this procedure (just as there are for differential equations) because the lattice Lagrangian only matches the continuum Lagrangian at a = 0. At non-zero a there are effectively additional unwanted terms in the lattice Lagrangian that are proportional to powers of a. We will discuss this further later. Another view of the lattice is that it provides an ultra-violet cut-off on the theory Lattice QCD 107 in momentum space, since no momenta larger than 7r/a make sense (the wavelength is then smaller than a). In this way it is an alternative regularisation of QCD. As an illustration of the simplicity of the discretisation procedure, let us consider a scalar field theory with Lagrangian 1 + 2_m202 + A04. (3) The lattice action, S is then S z a4 (2 i i0(n +1,) — — 1„)] 2 lin2 02(n) ± A4 (n)) 2 (4) 2a The point n +1, is one lattice point up from the point n in the µ direction. We are always free to rescale parameters and fields and we do this on the lattice, rescaling by powers of the lattice spacing, so that the parameters and fields we work with are dimensionless. Everything is then said to be in 'lattice units'. In the scalar theory above we rescale to primed quantities where q' = Oa, m' = ma, A' = A. Then S = (012 (n) [2 + ma] + 4 — 41 CY(n + 1µ) '(n — in)) • (5) The rescaling has the effect of removing the lattice spacing explicitly from the action. A lattice calculation is done then without input of any value for the lattice spacing, or even knowing what it is. We will discuss later converting results back from lattice units to physical units, so that we can compare results to the real world. Equation 5 has in addition been rearranged to collect similar lattice terms together, using En to move the space-time indices. It now looks very like a spin model, revealing a deep connection between lattice field theory and the statistical mechanics of spin systems. 2.2 Lattice gauge theories for gluons To discretise gauge theories such as QCD onto a lattice requires a little additional thought because of the paramount importance of local gauge invariance. The role of the gluon (gauge) field in QCD is to transport colour from one place to another so that we can rotate our colour basis locally. It should then seem natural for the gluon fields to 'live' on the links connecting lattice points, if the quark fields 'live' on the sites. The gluon field is also expressed somewhat differently on the lattice to the continuum. The continuum Ai, is an 8-dimensional vector, understood as a product of coefficients Al;i times the 8 matrices, Tb, which are generators of the SU(3) gauge group for QCD. On the lattice it is more useful to take the gluon field on each link to be a member of the gauge group itself i.e. a special (determinant = 1) unitary 3 x 3 matrix. The lattice gluon field is denoted Up(ni,nt), where µ denotes the direction of the link, ni refer to the lattice point at the beginning of the link, and the color indices are suppressed. We will often just revert to continuum notation for space-time, as in U5(x). The lattice and continuum fields are then related exponentially, U = (6) X 108 Christine Davies X x+1 x x+1 041--• U,(x) U_,(x +1) = Uµ 1 = Uµ(x) Figure 2. The gluon field on the lattice. where the a in the exponent makes it dimensionless, and we include the coupling, g, for convenience. If U,(x) is the gluon field connecting the points x and x + 1. (see Figure 2), then the gluon field connecting these same points but in the downwards direction must be the inverse of this matrix, Uli-1(x). Since the U fields are unitary matrices, satisfying UtU = 1, this is then U?,(x). This form for the gluon field makes it possible to maintain exact local gauge invariance on a lattice. To apply a gauge transformation to a set of gluon fields we must specify an SU(3) gauge transformation matrix at each point. Call this G(x). Then the gluon field U„,(x) simply gauge transforms by the (matrix) multiplication of the appropriate G at both ends of its link. The quark field (a 3-dimensional colour vector) transforms by multiplication by G at its site. 0,9) (x) = G(x)U„(x)G1(x + 1„) 1/)(9) (x) = G(x)0(x) 0(9) (x) = 1/)(x)GI(x). (7) To understand how this relates to continuum gauge transformations try the exercise of set-ting G(x) to a simple U(1) transformation, ega(x), and show that Equation 7 is equivalent to the QED-like gauge transformation in the continuum, ilgp = Aµ — a„oz. M X2 4 Figure 3. A string of gluon fields connecting quark and antiquark fields (left) and a closed loop of gluon fields (right). Gauge-invariant objects can easily be made on the lattice out of closed loops of gluon fields or strings of gluon fields (Figure 3) with a quark field at one end and an anti- quark field at the other, e.g. 0(x1 )Um (x1 )U,,(x 1 + UE(x2 — 1,)0(x2). Under a gauge transformation the G matrix at the beginning of one link 'eats' the GI at the end of the previous link, since GIG = 1. The G matrices at x1 and x2 are 'eaten' by those transforming the quark and anti-quark fields, if we sum over quark and antiquark colors. Lattice QCD 109 The same thing happens for any closed loop of Us, provided that we take a trace over color indices. Then the G at the beginning of the loop and the Gt at the end of the loop, the same point for a closed loop, can 'eat' each other. (Try this as an exercise, re-membering that U fields going in the downward direction are Uts and, from Equation 7, Utt/g) (x) = G(x +1,)Utti (x)Gt(x).) The purely gluonic piece of the continuum QCD action is Scoot = f dix-1 Tr F ,,FM" (8) 4g2 and the simplest lattice discretisation of this is the so-called Wilson plaquette action: Slat. = E (1 - 3Re {Tr Up}) ; Q = 6 . (9) Up is the closed 1 x 1 loop called the plaquette, an SU(3) matrix formed by multiplying V x Figure 4. A plaquette on the lattice. 4 gluon links together in a sequence. For the plaquette with corner x in the i, j plane we have (Figure 4): Up(x) = U1(x)U3 (x + 1,)Uit(x + 1;)U; (x) (10) Tr in Siatt denotes taking the trace of Up i.e. the sum of the 3 diagonal elements. Sian, sums over all plaquettes of all orientations on the lattice. /3 is a more convenient version for the lattice of the QCD bare coupling constant, g2. This is the single input parameter for a QCD calculation (whether on the lattice or not) involving only gluon fields. Notice that the lattice spacing is not explicit anywhere, and we do not know its value until after the calculation. The value of the lattice spacing depends on the bare coupling constant. Typical values of for current lattice calculations using the Wilson plaquette action are 6. This corresponds to a 0.lfm. Smaller values of /3 give coarser lattices, larger ones, finer lattices. Other improved discretisations of the gluon action are also used. In these the bare coupling constant appears in a different way and so comparison of the bare coupling constant between different gluon lattice actions is meaningless. The only comparison which makes sense is that of the resulting values for the lattice spacing. That Siatt of Equation 9 is a discretisation of Scant is not obvious, and we will not demonstrate it here. It should be clear, however, from Equations 6 and 10 that Slat does contain terms of the form aw4p. Siatt is gauge-invariant, as will be clear from our earlier discussion. Thus lattice QCD calculations do not require gauge fixing or any discussion of different gauges or ghost terms. We simply calculate the appropriate Feynman Path Integral using Siatt . Since we are only describing calculations for gluons at this stage, 0 will be some gauge-invariant product of U fields, for example the closed loop of Figure 3. Such a calculation is fully non-perturbative since the Feynman Path Integral includes all possible interactions in 110 Christine Davies the matrix element that we are evaluating. In contrast to the real world, however, the calculations are done with a non-zero value of the lattice spacing and a non-infinite volume. In principle we must take a 0 and V -4 ao by extrapolation. In practice it suffices to demonstrate, with calculations at several values of a and V, that the a and V dependence of our results is small, and understood, and include a systematic error for this in our result. 2.3 Algorithms The Feynman Path Integral (Equation 1) for gluons only becomes f [dU] 0e-s ROM = f [dU] e-s • To evaluate this integral we can generate random sets of U fields on the lattice and work out the result: E czte s„ mop = (12) sass {v},, is a set of U matrices, one for each link of the lattice, and is called a configuration. ,„ is the value of 0 on that configuration (e.g. the trace of a closed loop of Us). A set of configurations is an ensemble. This is a very inefficient way of working. If Sa is large for a particular configuration it contributes very little to the result. Instead it is better to generate the configurations with probability e-s. This is called 'importance sampling' since we preferentially choose configurations with a large contribution to the integral. If we have a set of configurations so distributed then N (01010) = (0) = 1 E da, (13) — a=1 i.e. the result simply becomes the ensemble average of the value of the operator 0 evalu-ated on each configuration. The calculation then has a statistical uncertainty associated with it, which varies with the ensemble size, N, as 1/VTV. Several algorithms exist to generate an ensemble of configurations with distribution e-s. The Metropolis algorithm is the earliest and simplest, but shares several features with later more sophisticated algorithms. The first step is to generate a starting configuration, {U}1 , e.g. by setting all the U matrices to the unit 3 x 3 matrix or by generating random SU(3) matrices. The algorithm then sweeps round the configuration, one U matrix at a time. For each U matrix a small change is proposed, i.e. a random matrix close to the unit matrix is generated which could multiply U. The change in S is calculated if this change to U were to happen. If S is reduced, the change is accepted; if not, it is accepted with probability e-°s (by comparing e-°s to a random number between 0 and 1). Once this has been done for every U E {U}1 we have a new configuration, {U}2. We then repeat to obtain {U}3 etc. Once we have an ensemble we can do any number of different calculations (often called 'measurements') on it for different operators 0. Ensembles are the equivalent of experimental data sets created by collaborations of theorists. They are Lattice QCD 111 often stored for years and re-used many times. Some ensembles are publicly available -see http://qcd.nersc.gov/ and http://www.ph.ed.ac.uk/ukqcd/. An important point to note is that each member of an ensemble is generated from a previous member. The ensemble therefore has a (computer) time history. We have to worry about the 'equilibration time' and the `decorrelation' (autocorrelation) time of the ensemble. The equilibration time is the number of sweeps required to reach a configuration typical of the distribution e-s that we are trying to create, i.e a configuration which has `forgotten' the starting configuration. The autocorrelation time is the number of sweeps it takes to generate a sufficiently different configuration that results can be considered statistically independent. The autocorrelation time can be determined from the sequence of results for 0 and will depend on 0. In general if 0 is an operator with large extent, e.g. a closed loop of U fields over many lattice sites, it will have a longer autocorrelation time than if 0 is a small loop. This is because the changes to a configuration spread out randomly from a point, one step per sweep. As we try to reach smaller values of a, closer to the continuous space-time of the real world, we expect a phenomenon called `critical slowing-down'. This is because a given physical distance, say the size of a hadron, takes up many more lattice sites as a gets smaller. For an ensemble to decorrelate on this physical distance scale then requires more sweeps. This makes the numerical cost of reducing the lattice spacing at fixed physical volume far worse than the naïve al (see Figure 5). Figure 5. A given physical distance requires more lattice points to cover it as a is reduced. 2.4 Quarks on the lattice 2.4.1 The fermion doubling problem The inclusion of quarks in the lattice QCD action causes several difficulties related to their fermionic nature and makes lattice QCD calculations very costly in computer time. The so-called `fermion doubling' problem is apparent even for free quarks, in the absence of any interaction with the gluon field. The continuum action for a single flavor of free fermions is S1 = f dl x 27te ap m0. (14) The obvious (so-called naïve) lattice discretisation gives 4 clatt,naive a4 E •-y,21)x±11` + m•Ox0x • 2a x A=1 (15) 112 Christine Davies 0 Ida Figure 6. The doubling problem for lattice fermions. The sine curve shows the lattice quark inverse propagator in 1-d. The straight lines through p = 0 (solid) and through p = 7r/a (dotted) are those for a continuum quark. The problems become evident when we Fourier transform this and compare the lattice inverse propagator: i sin p„a Glatt,naive (P) = i'712 M a to that obtained in the continuum from Equation 14, Gcolnt (P) = 17µPµ M• The two are plotted for a massless quark in one-dimension in Figure 6 over one lattice Brillouin zone (momenta beyond ±ir/a are equivalent to those in this range). The lattice result looks continuum-like around p • " 0, where the inverse propagator is close to zero. The lattice inverse propagator is also close to zero around p 7r/a, however. Since 7r/a and —7r/a are periodically connected on the lattice, another continuum-like line can be drawn at this point (with opposite slope to the one at the origin). Thus in one-dimension, our lattice fermion contains two continuum-like fermions rather than one! On a 4-dimensional lattice we have 24 fermions instead of one. The 15 excess fermions are called doublers. The doubling problem is clearly a consequence of the fact that the sine function appears in Equation 16 and this is because of the single derivatives in the Dirac action for a relativistic fermion, Equation 14. For a scalar particle (Equation 5) we would have a cosine instead, and no difficulty. 2.4.2 Wilson quarks There are several approaches to the doubling problem. The most severe in terms of its effects, but currently the most popular for a lot of applications, is the Wilson quark action. In this the doublers are entirely removed, by adding a 'Wilson' term to the action which gives them a much larger mass than ma, so that they drop out of the physics. The term added is a double derivative so appears with an extra power of a (a5) in order to have the (16) (17) Lattice QCD 113 same dimensions as the other terms in Sir (Equation 15): Sw = srive _ r a5 7/3x 2 (18) 1=10x = 4 Ox+1 — 20x + i„ a2 µ=1 (19) where r is the Wilson parameter (almost always set to 1). The extra power of a in Equation 18 means that the correspondence between the lattice and continuum actions as a 0 is not changed. However, if we look at the inverse propagator again, there is a difference. 2 4 Gt70 (P) = Gnaive E sin2(Na/2). (20) a /2,1 If we substitute for G.alive from Equation 16 and expand out the sin function around p 0 we get ra 4 (P) = + m + 2 E p2v. (21) Comparing this to the continuum form (Equation 17) as a —+ 0, the r term will disappear and a fermion of mass 171 will have the right form. If instead we look at the doublers, we must expand around p 7r/a. If we call /5 the momentum difference between p and iria and consider the case where p has only one component close to 7r/a, and the others are close to zero, then G-1(p) = YµPµ + m + + • • • • (22) Now as a 0, the mass of the doubler, m + 2r/a —> oo. The doublers at other corners of the Brillouin zone pick up masses of 4r/a, 6r/a, 8r/a : check this as an exercise. Thus we are assured that our quark action describes only the one quark that we intended, but there is a price for this, as we shall see below. The Wilson quark action is converted to dimensionless units by a rescaling 0120 0, leaving the quark mass parameter as a mass in lattice units, ma (previously called Tre). sr = E {ox E [(7„ - r)ox+1,, - (-y,i + + (ma + 4r)7,bx0x } . (23) s A It is conventional to define a 'hopping parameter' called ic which is 1/(2ma + 8r) and so 1/k plays the role of the quark mass. 0 is conventionally rescaled by A/2Tc so that i moves to multiply the terms connecting the '0 field on different sites (thus allowing 'hops'). If we now couple in a gluon field, the 0 field will become a 3(color)x4(spin) dimensional vector on each site. The gluon field must be included in such a way as to keep the action gauge-invariant. From our earlier discussion it is then obvious that U matrices must be inserted as a link between the 0 and 0 fields when they are on neighbouring sites. The Wilson quark action is then conventionally written: Sf = E {k [E - ou,(x)ox+,„ - Ox-F1µ + rwit(x)oxl + OxtPx (24) 114 Christine Davies The price we pay for using the Wilson quark action is that we break explicitly the chiral symmetry of continuum QCD. This is a symmetry of the derivative terms in S (Equa-tion 14) which allows us to rotate separately right- and left-handed components of the quark field. The spontaneous breaking of this symmetry gives us a massless pseudoscalar meson called the pion as a Goldstone boson and has other important consequences for particle physics. Chiral symmetry is broken explicitly by a quark mass (so that the real pion is not actually massless) but also, more seriously for the lattice, by the Wilson term. As a --> 0, chiral symmetry will be recovered, but for real lattice calculations at non-zero a, the lack of chiral symmetry can cause difficulties for some calculations. One surprising feature of Wilson quarks is that it is still possible to get a massless pion even at non-zero a, when chiral symmetry is broken. However, we have to search for the value of 1/ic at which it occurs it is not simply the point 1/ic = 8r, as it would be in the free theory, above. Lattice calculations of the mass of the pseudoscalar meson (Mps) must be done at various input values of K (see Section 3) for a given ensemble. A plot of MA against 1/K is then extrapolated to the point where Mps is zero. The value of K at this point is called Kaitical and is the point at which the bare quark mass in the interacting theory is zero (but matrix elements will not necessarily show chirally symmetric behaviour). The bare quark mass in lattice units, ma, at other values of ic can then be taken to be (1/2ic - 1/21ccritical)• Another problem for the Wilson quark action is the presence of large discretisation errors. The naive quark action has discretisation errors proportional (at lowest power) to a2 because (see Equations 16 and 17) sin(pa)/a = p(1 -2p a2/6+ ). In the measurement of a hadron mass, the terms proportional to p2a2 in the action will induce an error pro-portional to A2a2 where A is some typical momentum scale inside the hadron in question, say 300MeV. For lattice spacing values we can reach, around 0.lfm (= (2GeV)-1 when It = c = 1), this gives an expected error of order 2%. The Wilson term (Equation 18) that we added, however, is proportional to a, so that Sy = + 0(a). Now hadron masses will have an error of typical size Aa, which could be 15% at a = 0.lfm. One can extrapolate this error away by doing calculations at several values of a but the size of the extrapolation adds uncertainty. Instead, we can 'improve' the quark action, by adding additional terms to counteract the errors at any order in a. This is equivalent to a higher order discretisation scheme for differential equations. For the Wilson quark action we can add the so-called clover term, making the clover, or Sheikholeslami-Wohlerti, action: sclover iacs,nr v, Q W (25) = 4 x ap, Fp, O • The standard discretisation of aFp, is as a set of 4 plaquettes arranged in a clover-leaf shape. If the clover coefficient, csw, is chosen correctly then the clover action has leading order errors proportional to a2 again. It is in the correct choice of this coefficient that the difficulties of discretising a field theory, as opposed to a standard differential equation, appear. We are trying to match QCD with an ultraviolet momentum cut-off of 7r/a to QCD with an infinite momentum cut-off. Gluonic interactions with gluon momenta between 7r/a and co in the continuum must be accounted for on the lattice by a renormalisation of coefficients in the action. Thus the naive (tree-level) value of 1 for csw is renormalised by an amount which depends on the QCD coupling constant at some momentum scale around 7/a. This momentum scale is typically quite large (for a = 0.lfm Lattice QCD 115 it is 6GeV) so that a perturbative calculation of cs,, can work well. csw = 1 + ci as (v/a) + c2as2 (7r/a) + • • • . In fact it has been shown that a lot of the perturbative correction can be absorbed into a renormalisation of the U field by a factor called Ito, and this is called tadpole-improvement (Lepage, 1993). Alternatively cs, can be determined within the lattice calculation itself (i.e. non-perturbatively) by insisting that some continuum relationship, broken by the discretisation errors, works on the lattice (Sommer, 1998). For cs, we can impose Ward identities from chiral symmetry, for example. This improvement programme for the lattice action can be carried further at the cost of introducing more coefficients that have to be determined by a match to continuum QCD. However, this must be compared to the cost of not improving the action, which requires calculations on very fine lattices to achieve small enough discretisation errors for the accuracy we require and is generally prohibitive. 2.4.3 Staggered quarks Here we return to the naïve quark action and ask, what was so bad about having 16 quarks instead of 1? If we had 16 flavors of quarks of the same mass in Nature, the naive action might be fine. In fact we only have two quarks that might be considered degenerate, u and d. They both have masses of a few MeV. Although we do not believe that their masses are the same, the difference is much smaller than any other mass, and they are treated as degenerate in most lattice calculations at present. We can 'thin' the degrees of freedom of the naïve lattice quark action by removing the 4 spin degrees of freedom (which can be shown to be multiple copies of the same thing). The quark field, X, then becomes a 3(colors) xl(spin) component object on a site and the staggered (Kogut-Susskind) fermion action is: S'fs = E xs E 7,s,„ (up(x)xx+,„ — u,f2(x — 1,)xx„) + maxx} (26) 71x is ±1 according to the formula 7ix,, = (-1)k where k xu. This action describes 16/4 = 4 quarks, now much closer to the real world, if we want to interpret the doublers as flavors. We might hope that if the 4 flavors do behave as 4 copies of the same thing we can reduce their effect by a factor of two or four (depending on how many degenerate flavors we want to simulate) by multiplication with the required factor at appropriate points (as we could in QCD perturbation theory). The 4 spin degrees of freedom for the 4 flavors are made from the 16 components of the X field on a 24 hypercube, which is a complication if we need to separate out the flavors. The staggered action, however, has a remnant of chiral symmetry which ensures the very desirable feature that the quark mass (and the associated Goldstone boson pion mass) vanish at ma = 0. This behaviour gives the added benefit of making staggered quarks rather better behaved and computationally much faster to work with than Wilson-type quarks. The down-side of staggered quarks is again the discretisation errors. These are formally 0(a2), just as for naive quarks, but some of the errors induce flavor-changing interactions and so are rather dangerous. In practice they produce a larger than expected effect for simple a2 errors. A quark with momentum around 0 can be scattered to one with momentum around 7r/a i.e a doubler, and therefore a different flavor, by the interaction of Figure 7. One of the results of this is that the 16 different pions (for 4 flavors) no 116 Christine Davies p.-1E/a Figure 7. A flavor-changing interaction for staggered quarks on the lattice. longer have the same mass and only one of them has a mass which vanishes as ma 0. Improvement terms have recently been developed which can be added to the action to reduce these interactions to a much lower level, and the masses of the different pions are then much closer together (Bernard, 2001, MILC collaboration). This makes the prospects for working with staggered quarks in lattice QCD calculations much better, and a lot more work with these quarks will certainly be done. 2.4.4 Ginsparg-Wilson quarks A recent development has been a set of quark actions which maintain chiral symmetry of the action while still describing only one quark flavor, but at the cost of a very complicated lattice discretisation of the continuum derivative. This is then costly to implement. For example, in the domain-wall formulation an additional 5th dimension is required whose length, in principle, must go to infinity. A lot of work is being done to develop algorithms for these quark actions which may make them feasible in the long-term. In the meanwhile, they are already being used for calculations that really need chiral symmetry at finite lattice spacing, such as that of the CP-violating parameter in the K system, ei. 2.5 Algorithms for quarks Another problem with handling quarks in lattice QCD is that they are fermions, obeying the Pauli Exclusion Principle, and therefore cannot be represented by ordinary numbers in a computer. We must do the quark functional integral by hand: f [dU] [d] [d] = f [dU] detM e-s9 (27) where the form for the matrix M depends on the quark formulation and can be derived from the forms given above for the quark action (Equations 24, 25 and 26). The QCD action then becomes s E(1- N,Re Tr (Up)) — ln (det M) . (28) We now generate ensembles of gluon fields (only) with importance sampling based on this action. The standard algorithm for doing this is called Hybrid Monte Carlo . The second term is a very expensive one to include, because it requires frequent calculations of M-1 Lattice QCD 117 (various algorithms, such as Conjugate Gradient exist to do this) and M is a large matrix (4(forWilson) x 3 x V 2 x 106 on a side). If this term is missed out for expediency (so that the action is just S9) then we talk of using the 'quenched approximation'. Most calculations in the past have been quenched (and most of the results I discuss later will be in the quenched approximation) but recently calculations using the full QCD action (`unquenched' or 'with dynamical/sea quarks') have been attempted and in the future we hope that the quenched approximation will become redundant. We can think of the ln(det M) term as giving rise to a sea of quark/anti-quark pairs appearing and disappearing in the vacuum. For every quark flavor for which we have a separate matrix M we should in principle include a term of the form ln(det M) in the dynamical quark action. However, it is only the production of light (u, d, s) quark/anti-quark pairs that we envisage having a significant effect for most of the quantities that we calculate. Dynamical lattice calculations are then done with Nf = 2 for u, d dynamical quarks or 2+1 if s is included. Quarks must also be integrated out of the operators, 0. For 0 =- (00),(00)s, the form mentioned earlier, which creates a meson at the point x and destroys it at the point y, then f [dU][d0][4],14,.. 74,a 114,6 74,19 e — S f [dU](ivc,,,,[u-])ab(myr,b. ,1,d j) detMe-s9 . (29) is the quark propagator from x to y on a given gluon configuration, obtained by solving Mx = b where b is a vector with a 1 at x (and a certain color and spin index) and 0 everywhere else. We have been explicit here about the flavor indices, which we have taken as u and d, although lattice calculations usually then assume that u and d are degenerate and therefore the two M 1 factors are the same. However, if the hadron actually does contain two quarks of the same flavor then 'disconnected' pieces containing Mz1 will appear, as well as the 'connected' pieces above. The color indices, a and b, are also explicit (and summed over) and make 0 gauge-invariant. The sums over spin indices have not been made explicit because in this case they follow the color indices (but see Section 2.6). On an importance-sampled ensemble (either quenched or unquenched) for 1-, this example we then have to calculate Trcolor,son(Mz y,u)(my 2'd) on every configuration and average over configurations. Calculating M-1- is computationally expensive and gets harder as M develops small eigenvalues, which happens as ma -+ 0 (for staggered quarks) or tc„it, (for Wilson or clover quarks). Thus, even in the quenched approximation, we cannot actually calculate with quark masses close to those of real u and d quarks. Instead we work with heavier quarks and perform so-called chiral extrapolations to the chiral limit where u and d quarks would be (almost) massless. 2.6 Relating lattice results to physics Above we have given an example for 0, which includes the creation of a valence quark and anti-quark at the point x and their destruction at the point y. This is a so-called hadron correlator or 2-point function on the lattice since it simply has a source and a sink, and is one of the simplest quantities to calculate. It is shown pictorially at the left of Figure 8, where the solid lines indicate the valence quark propagators, and the blobs 118 Christine Davies 0 T 0 T Figure 8. A graphical representation of two types of 2-point functions calculated on the lattice. Left, for a hadron mass calculation; right, for a decay constant. at the two ends indicate the creation and annihilation of the meson. A baryon would of course have 3 valence quark propagator lines. Usually we project onto specific values of p for the hadron, so in Figure 8 we have suppressed spatial indices at the source and sink and just refer to the time index, 0 at the source and T at the sink. The figure shows how, as the valence quarks propagate, they interact any number of times by exchange of gluons. This is a pictorial representation of the fully non-perturbative nature of a lattice QCD calculation. The interactions include the production of dynamical quark/anti-quark pairs if a dynamical calculation is being done. The calculation of this 2-point function will enable the extraction of the hadron mass (see below), for the hadron corresponding to the J'e quantum numbers of 0. We make different quantum numbers by inserting -y matrices between the 0 and 'V) fields in each piece of 0. For example, (Ttry,50)x creates a pseudoscalar meson (such as x) and 7,;-). i 0 a vector (such as p). When the quark functional integral is done, as in Equation 29, -y matrices will appear between the two M' factors and appropriate sums over spin indices will have to be done. The blobs in Figure 8 indicate that we can use more complicated forms for 0 for a given hadron, e.g. the 0 and TP fields do not both need to taken at the point x. We can separate them spatially, either by inserting U fields to keep 0 gauge-invariant, or by fixing a gauge to allow spatial separation without including U fields. This enables us to feed in information, or prejudice, about the relative spatial distribution of the quarks in the hadron, i.e. its `wavefunction'. Each piece of 0 takes the form Tpx,,,0(r)ox (suppressing the U fields) where 0 is some function of the separation between V) and td: it, is known as the 'smearing' function and 0 is then a smeared operator. When the quark functional integral is done, factors of 0 will appear between the Al-' factors. The factor of 0 is absorbed at the source by solving Mx = 0 for, say, the quark (making a `smeared quark propagator') and Mx = 6 for the anti-quark (a 'local quark propagator'). The two propagators are then put together with an explicit insertion of 0 at, the sink. Often calculations measure separately hadron correlators with several different smearing functions at both source and sink, enabling a more precise determination of the hadron mass. Another type of 2-point function is shown on the right of Figure 8. In this case we create the hadron with a smeared operator and destroy it with a local operator. This is a `smeared-local' or 'smeared-current' correlator, since the quantity that we can extract from this is the matrix element of the appropriate current operator, J, between the vacuum and the hadron. For example, this is used to calculate the decay constant, f r , related to the vacuum to 77 matrix element of the axial vector current (denoted by its time component, Lattice QCD 119 Ao, in Figure 8). This couples to the W particle and mediates the purely leptonic decay of a 7r meson. See the Lagrangian for the weak interactions in (Rosner, 2002), but note that the W particle is not included explicitly in lattice QCD calculations. 0 in this case then takes the form (Ox+r-y5cb(r)zi)s)(0-y0750),, where the first factor creates the pion with a smeared operator at x and the second destroys it with the time component of the local axial vector current. The quark functional integral converts this to the same type of quantity, with two factors of M-1, that we discussed above. 0 T Figure 9. A graphical representation of a 3-point function (for sernileptonic decay) calculated on the lattice. Figure 9 shows a lattice 3-point function appropriate to the semi-leptonic decay of a hadron. One of the valence quark lines emits a W and changes to a different, flavor. A new hadron is then formed with the spectator quark. The emission of the W can he represented by the insertion of a current on one of the valence quark lines. The Figure shows a vector current (with temporal component Vo and spatial component 1;) which contributes to the decay of a pseudoscalar meson to a pseudoscalar meson (e.g. B —> D). We then have a (smeared) source and sink at 0 and T, and a (local) current insertion at, t, i.e. 3 points. When the quark functional integral is done there will be 3 factors of M-' , one for the original valence quark which decays (from 0 to t), one for the final valence quark (from t to T) and one for the spectator (from 0 to T). In fact the roost, efficient way to do this calculation is to solve for the final valence quark propagator from T to t, taking as a source the spectator quark propagator from 0 to T. 3 Lattice QCD calculations 3.1 The steps of a typical lattice calculation Step 1 A volume and a rough lattice spacing are chosen. A volume of (3fm)3 is considered to be large enough not to 'squeeze', and therefore distort, typical hadrons placed on it. The time extent is usually taken as twice the spatial size since masses etc are extracted from the time dependence of hadron correlators (see below). The selection of the lattice spacing is a trade-off between getting close to the continuum limit (and therefore small discretisation errors) and the cost of the calculation, which grows as some large power of a-1. Improvement of the action, discussed above, helps here by giving small discretisation errors on coarser lattices. Lattice spacings around 0.lfm are reasonable on both counts. From experience we know roughly what value of the bare QCD coupling constant to take 120 Christine Davies in the gluon part of the QCD action to achieve various values of a (determined after the calculation, see below). However, the quark contribution to the action affects this also, and we have much less experience with this. A (3fm)3 x 6fm lattice with a ti 0.lfm requires (30)3 x 60 sites. Step 2 A quark formulation, number of quark flavors, and masses in lattice units, ma, are chosen for the quark part of the QCD action. Again we have a trade-off between trying to take realistically small masses for the u and d quarks, and the cost. Again we do not know what the quark mass actually is until after the calculation, when we have calculated the masses of hadrons containing that quark. Recent calculations have been able to take dynamical quark masses down to the s quark mass and some have gone further; future calculations need to reach much smaller masses than this. Extrapolations to u and d quark masses will continue to be necessary, however (see step 8). Some interpolation will always be necessary too since the masses chosen will inevitably not be exactly correct, e.g. for the physical strange quark mass. Step 3 An ensemble of gluon configurations must then be generated using importance sampling with e'. As discussed above, dynamical quarks appear implicitly through the quark determinant. Step 4 Quark propagators are calculated on each gluon configuration of the ensemble by inverting the quark matrix, M, to make the 'valence' quarks inside the hadron. Where they are supposed to have the same flavor as the dynamical quarks, they should have the same mass in lattice units, ma. However, we can also calculate valence quark propagators for quarks with different mass from the dynamical quarks, and perform separate extrapolations in valence and dynamical quark masses. This is sometimes useful and particularly so if there is a very limited set of dynamical quark masses. It is known as the partially quenched approximation (PQA). Step 5 The quark propagators are then put together in various combinations to form hadron correlators (see the discussion of the form taken for operators, 0, above) which are then averaged over all the configurations in the ensemble. We are concentrating here on operators 0 which are related to quark-based hadrons but gluonic operators can also be measured on the ensemble and averaged in the same way. Step 6 The hadron correlators are fitted to their expected theoretical form to extract hadron masses and matrix elements. For the 2-point function described for the spectrum, the ensemble average of the product of smeared quark propagators described above gives us the vacuum expectation value of a hadron correlator, (011/f (T)H(0)10) (see Equation 1). The hadron creation(destruction) operator can create(destroy) from the vacuum all the hadron states which have the same JPC quantum numbers as the operator. For example, if the operator has the quantum numbers of a pseudoscalar meson containing u and d quarks, the 7r and all its radial excitations can be created(destroyed). The amplitude, A, with which a particular state is created or destroyed depends on the overlap with that Lattice QCD 121 1'1 '1'111'i 10 20 30 2E 0.50 0.48 0.46 0.44 0.42 0.40 0.38 T Figure 10. The effective mass of a 1-- bb (T) correlator, calculated from a lattice 2-point function with local source and sink. state of the operator used, i.e. the smearing function, at source or sink. Thus we obtain (01Ht (T)H (0)10) = E Agrc2Ensnk,n e_E„T (30) n where the factor e-E"T arises because the two hadron operators are offset by a time dis-tance T in Euclidean space, and En is the energy of the nth state. The states which dominate the fit, especially at large values of T, are those with lowest energy; if a projec-tion on zero momentum has been done, these will be the states with lowest mass. Often we are interested in the one state with lowest mass, the ground state (the 7r in the example above), and then try to design a good smearing function to have large overlap with that state, and very small overlap with its radial excitations. In that case fits can be done in which data at small values of T are thrown away and only a single exponential is used in the fit. The extent to which this works can be gauged by plotting the 'effective mass', the log of the correlator at time t divided by t. If one state completely dominates the fit, a constant result is obtained as a function of t - the effective mass is said to 'plateau'. The plateau value is the ground state mass. Figure 10 shows the result for the effective mass of the correlator for an T particle (see Section 4) calculated on the lattice. A clear plateau is seen but only for t > 15. For smaller t the correlator clearly contains excitations of higher mass, because no smearing was used in this case. The best calculations use several different smearing functions at source and sink and perform simultaneous multi-exponential fits of the type in Equation 30. If the masses of several states can be obtained from the fit the reliability of the ground state mass is increased. It should also be pointed out that correlated fitting techniques must be used since the correlators at adjacent times are not statistically independent of each other. For the 2-point function used to calculate decay constants, the amplitude with which the hadron is destroyed at the sink is the vacuum to hadron matrix element of the current. (01J1n) e-E„T (01J (T)11(0)10) = E Asrc 2E„ (31) 122 Christine Davies and (00017r(p = 0)) = f,rni,. To isolate the part proportional to the decay constant requires dividing the total amplitude of the ground state exponential by This can be obtained from a fit of the type in Equation 30, if the same smearing function is used at source and sink so that As„,„ = Asno,. For the 3-point function we have two sets of hadrons with different flavor quarks, separated by a current insertion. fin)Asnk,nt (rn1 — (01Hrt(T)J(t)H(0)10) = EE Asi-c,n 2E'n2Ern e E„t e (32) re m where n runs over hadrons with the quantum numbers of the operator at 0 and trt, those of the operator at T. Again the matrix element of interest, that of the current between two hadrons (usually for the ground states in the two cases), can be obtained by dividing out the amplitudes at source and sink from two separate 2-point fits for the two different hadrons of the kind in Equation 30. Step 7 It is now possible to determine what the lattice spacing was in the simulation. This then sets the single dimensionful scale so that everything can be converted to physical units (GeV) from lattice units. The lattice spacing is determined by requiring one dimensionful quantity to take its real world value. Usually a hadron mass is chosen, because these are easiest to determine on the lattice, but it should not be one whose mass depends strongly on valence quark masses to be determined in the next step (see below) otherwise a complicated iterative tuning procedure will result. The most popular quantity to use at present is known as To, a parameter associated with the potential between two infinitely heavy quarks. It is extracted from the energy exponent of a gluonic operator (the closed loop of Figure 3), so can be precisely determined and does not contain any valence quark masses. The only problem is that it is not an experimentally accessible quantity, and we rely on potential model results to give a phenomenological value, estimated to be 0.5fm. Another quantity frequently used is the mass of the p meson, obtained by chiral extrapolation to the point where the 7F meson mass, and therefore the u,d quark mass, is (almost) zero. The chiral extrapolation, however, can produce large errors. A better quantity is the orbital excitation energy, i.e the splitting between P states and S states, in bb or cc systems, since these don't contain light quarks and this splitting is even insensitive to the heavy quark mass. (The treatment of heavy quarks on the lattice will be discussed in Section 4.) Step 8 The step above yields all hadron masses in GeV. However, before we can compare to experiment we must tune the quark masses. This requires calculations at several different values of the bare quark masses in an appropriate region. For each quark mass we then select a hadron whose mass will be used for tuning (and is therefore not predicted). For that hadron we interpolate/extrapolate the results to find the bare quark mass at which that hadron mass is correct. The masses of other hadrons containing that quark are then predicted if we interpolate/extrapolate those masses to the same quark mass, or combination of quark masses. In the process we learn about the dependence of hadron masses on the quark mass and this can be useful theoretical information. The hadrons used for tuning should be low-lying states with accurate experimental masses which can be calculated precisely on the lattice. The 7r mass is usually used to fix the u, d mass Lattice QCD 123 (taken to be the same), although sometimes the approximation in„. = 0 is used. The mass of the K, K*, or can be used to fix the s quark mass. The K or K* obviously require the u and d masses to have been fixed. The dimensionless ratio of the K* to the K mass can also be used, and this is then less dependent on the quantity used to fix the lattice spacing. For the c(b) quarks, the D(B), D8 (138 ) or 0(T) systems are convenient ones. The interpolation/extrapolation of hadron masses as a function of bare quark masses is a relatively simple procedure in the quenched approximation. Then there is no feedback from the quark sector into the gluon sector. We can create gluon field configurations at a fixed value of the lattice spacing (as determined, for example, from a purely gluonic quantity such as ro ) and measure hadron masses at many different quark masses on those configurations. The issues are then the correlations between results at different quark masses that must be taken into account and the spurious non-analytic behaviour in quark mass that can arise in the quenched approximation in extrapolations to u and d masses (`quenched chiral logarithms'). When we include dynamical quarks in the calculation, the effects of the quark deter-minant at a particular quark mass feed into the gluon field configurations. Results at different dynamical quark masses then represent a completely new calculation, generating a new ensemble of gluon configurations with statistically independent results. The inter-polations/extrapolations in quark mass take on a new dimension and there are subtleties associated with how to do this. Some groups have chosen to generate configurations at fixed bare coupling constant and various dynamical bare quark masses. Then the lattice spacing will vary with quark mass and extrapolations in quark mass must be done in lattice units, before fixing the lattice spacing at the end. I believe a more satisfactory ap-proach from a physical perspective is to adjust the bare coupling constant at different bare quark masses so that the lattice spacing remains approximately the same (as determined from ro, for example). This then allows interpolations/extrapolations for physical hadron masses, and a better picture of the physical dependence of quantities on the presence of dynamical quarks. Several groups have also carried out this procedure. In all of these approaches we must extrapolate to reach the physical u/d mass region, and so we need to know the appropriate functional form for this extrapolation. This can be derived for light enough u/d mass using an effective theory of Goldstone pions called chiral perturbation theory. This shows that logarithmic behaviour of quantities as a function of the 7T mass (the variable representing the u/d quark mass) should be present in general as well as simple power-law behaviour. These 'chiral logarithms' will only show up at rather small quark masses (mu,d m3/4) and so it is important for dynamical simulations to reach quark masses low enough to be able to match on to this behaviour and extrapolate down. Step 9 The calculation needs to be repeated at several values of the lattice spacing to check that the dependence of physical results on the lattice spacing is at an acceptable level and/or to extrapolate to the continuum limit a = 0. Extrapolations again obviously require knowledge of an appropriate functional form. Step 10 Compare to experiment or give a prediction for experiment! 124 Christine Davies Concluding remark Above we have described an ideal situation. Lack of computer power has meant compro-mising on one or more aspects in existing calculations. A lot of calculations have used the quenched approximation. More recent dynamical calculations have used heavy dynamical masses on rather coarse and sometimes rather small lattices. These difficulties should be overcome in the next few years and this will represent a huge improvement in the reliability of lattice results. 3.2 Control of lattice systematic errors We aim for errors of a few percent from future lattice calculations. This requires both improved statistical errors in general and good control of systematic errors. Improved sta-tistical accuracy is obtained by generating larger ensembles of configurations with a cost proportional to the square of the improvement. Improved systematic accuracy requires theoretical understanding of the sources of error and how to remove them. It is this under-standing, described below, that has been responsible for the development of good lattice techniques and the convergence of lattice results in the quenched approximation through the late 1990s. This must be carried further in the next phase of dynamical simulations to reach the goal of providing quantitative tests of QCD and input to experiment. 3.2.1 Discretisation errors As discussed earlier, these arise from errors in the lattice form of the Lagrangian, and operators 0, compared to the continuum versions. Lattice results, even when converted to physical units, have some dependence on a. This will be as a power series in a, starting at an. As discussed earlier, n = 1 if the Wilson quark action is used, 2 for the clover quark action and 2 for the staggered quark action. n is also 2 for the Wilson plaquette gluon action of Equation 9. We expect the size of the a dependence to be controlled by a typical momentum scale relevant to the quantity being calculated. Quantities sensitive to shorter distances than others will be more susceptible to discretisation errors, even though the value of Ti depends only on the action used. Improved gluonic and quark actions are available in which higher order terms are added to G to increase n, and therefore reduce the a dependence, and these can be tested for their efficacy in the quenched approximation. The systematic improvement method is known as Symanzik improvement (Gupta, 1998). Figure 11 shows a scaling plot of the vector meson mass (the p, except that the quark mass is heavier than the real u, d mass) in GeV versus the lattice spacing for various quark actions (Toussaint, 2002). Some of the calculations use an improved gluon action, with discretisation errors reduced beyond 0(a2), but others use the Wilson plaquette action. There is very little difference between these (compare fancy diamonds and squares) so that most of the difference arises from the quark action used. A variant of r0, called r1 , is used to set the lattice spacing so the vector mass and scale are given in units of r1. The plot shows results for clover quarks (improved Wilson quarks), staggered quarks, improved staggered quarks and Ginsparg-Wilson (domain wall) quarks. The last two formulations, which are both improved to remove 0(a2) errors show an impressively flat line, i.e. very little a dependence for this quantity. The clover quarks shown here have a clover improvement coefficient (see Section 2.4.1) chosen using tadpole-improvement. I I Lattice QCD 125 2.0 1.8 1.6 1.4 - a plaqiconyentional KS — 0 Sym_lloop/Asq_tad t iid - o Wilson/Dr 1.2 0 0 0.2 0.4 06 (a/r,)z Figure 11. A scaling plot in the quenched approximation for the vector meson at a quark mass such that the pseudoscalar meson has mass marl = 0.807. The vector meson mass is given in units of r1 , a variant of r0 , where r1 0.35fm = 0.57GeV-1. It is plotted versus the square of the lattice spacing, also given in units of r1 . The squares and fancy diamonds use an improved gluon action; the others use the Wilson plaquette action. The quark actions used are: circles, staggered (Kogut-Susskind); squares, improved stag-gered; fancy squares and fancy diamonds, tadpole-improved Wilson (clover); diamonds, Ginsparg-Wilson (domain wall). (Toussaint, 2002) This reduces the a dependence of Wilson quarks to ctsa but it is clearly still visible. A non-perturbative determination of the clover improvement coefficient can reduce the a dependence further to 0(a2), and then this formulation looks rather better. Notice the large discretisation errors visible for unimproved staggered quarks, despite the fact that the errors are 0(a2) (and results therefore lie on a straight line in the Figure). Provided that all the different quark formulations have been fixed to the same physical quark mass, all the results for the vector meson mass should agree in the a —> 0 limit. This does seem to be true, within the statistical errors shown. 3.2.2 Finite volume Lattice results will be distorted if the space-time box in which the calculation is done is too small to adequately represent the infinite space-time volume of the real world. For large enough volumes the error should be exponential in the lattice size, oc for a lattice of size L in physical units. This means that it is possible to reduce finite volume errors rapidly to zero by taking large enough volumes. The lightest particle is the 7, so this sets the volume required as we reduce the u, d quark masses to their physical values. For u, d quark masses of m8 /4, Tri„L > 5 for L > 3fm, giving a finite volume error of less than 1%. Most recent lattice calculations have used volumes of this size, although there has been little systematic dependence of the volume dependence of results. ti 126 Christine Davies 3.2.3 Matching hadronic matrix elements to the continuum The calculation of hadronic matrix elements of various currents, J, on the lattice is dis-cussed for 2- and 3-point functions in Section 2.6. An important point is that these depend in general on how QCD has been regularised and a finite renormalisation is then required to convert lattice results to those appropriate to a continuum scheme (such as MS). Since lattice QCD and continuum QCD differ in the ultra-violet (for momenta greater than 7r/a), this renormalisation can be calculated in perturbation theory, by matching the matrix elements of J between quark states. We usually need several lattice currents to make up the continuum current and a mixing and matching calculation must be done. Jcont = Zo Ji(aTt + a Zi Ji(at), + • • • Zi = 1 + c,(1) a,,(2/a) + c,(2)as2 (2/a) + (33) Lattice perturbation theory is done in the same way as continuum perturbation the-ory, in terms of the field A, and including gauge-fixing and ghost terms, if necessary. Relatively little lattice perturbation theory has been done up to now and few results exist beyond 0(a8). This leaves errors of 0(as2), 5-10% if we take a scale for a, of 2/a at a=0.lfm. Higher order calculations will be required to reduce this to the required level of 2-3%, and techniques are being developed to do this. It is also sometimes possible to fix the normalisation of lattice currents non-perturbatively using symmetry arguments or to match numerically between lattice and continuum MOM-type schemes. In whatever way it is done, the matching of lattice matrix elements to the continuum is a lot of work and an area where improvements are still necessary. 3.2.4 Unquenching The neglect of dynamical quarks in the quenched approximation is obviously wrong, but how wrong? For many years systematic errors from the quenched approximation were obscured by the size of the statistical and discretisation errors. Now improved quenched calculations are showing internal inconsistencies and disagreement with experiment winch we believe will be removed once realistic dynamical calculations can be done. One effect expected in the quenched approximation is the incorrect (too fast) running of the coupling constant from one scale to another because of the absence of g —> —> g pieces in the vacuum polarisation to give quark screening of the color charge. From this we might expect that the determination of the lattice spacing would depend on the quantity used to fix it, since different quantities will be sensitive to different distance/momentum scales and these will not be connected correctly by the running of a, in the quenched approximation. (Using a quantity to fix a is equivalent to fixing the QCD coupling constant at the momentum scale relevant to that quantity). This is indeed found and illustrated by the quenched point in Figure 12. Likewise hadron masses depend on the hadron used to fix the quark mass. Then if a set of hadron masses is studied, sensitive to a range of scales and containing different combinations of quarks, errors will show up (see Figure 13 (Aoki, 2000, CP-PACS collaboration)). The quenched approximation also does not allow the decay of particles where this requires the production of a -0 pair from the vacuum, e.g p 77. Once dynamical quarks are light enough for this to happen, it will in fact be difficult to determine pip 1.4 Lattice QCD 127 -quenched dynamical a rn a., 1.2 Ta 1 00 0.5 I I 1 . 0 1.5 20 m„2 GeV2 Figure 12. The ratio of inverse lattice spacings, a-1, obtained from the orbital excitation energy, the splitting between 1P and 1S states, in the T system and from r0 . Results are given for quenched simulations and for dynamical simulations using two ,flavors of dynamical quarks at three different values of the quark mass, all heavier than Ins , indicated by the square of the corresponding pion mass along the x axis. (Marcantonio, 2001, UKQCD collaboration) 1.0 0.8 0.6 0.4 0 ......= I I Q _ A 7" H —. —rr 4) II E* N —-. I K* —e. I A mg _ • K input o input — experiment Figure 13. The spectrum of light mesons and baryons obtained in the quenched approxi-mation after extrapolation to u, d quark masses and to the continuum limit. The p and n masses are missing since they were used to fix the lattice spacing and u, d masses. Results are compared using the K or the cb to fix the strange quark mass and disagreement between the two is seen. The size of the discrepancy with experiment depends on this and varies between hadrons, but is at the level of 10%. (Aoki, 2000, CP-PACS collaboration) 128 Christine Davies 10 8 Cost in Tflops year 6 4 0J" ' I ' 0 0 0.2 0.4 0.6 0.8 1 0 Figure 14. The computer cost in teraflop-years of generating 500 303 x 60 configurations with a = 0.1fm and a dynamical quark mass which gives the ratio of pseudoscalar to vector meson masses along the x axis. Clover and improved staggered quarks are compared, assuming the same scaling behaviour as (m, I Tri p)3 . The straight line shows what is possible in 6 months on a 5Tflops computer. since we will obtain instead the lighter mass of the two-pion system. It is then important in dynamical simulations to use hadrons which are stable in QCD, or have very narrow widths, to fix the quark masses in the QCD action. It has been stressed that the numerical cost of unquenched calculations is very high. It increases very rapidly as a is reduced at fixed physical volume and as m,„,d is reduced, although the exact scaling behaviour is not completely clear. Figure 14 estimates the cost of generating an ensemble of 500 gluon configurations on an L3 x T lattice with L = 3fm and T = 2L at a lattice spacing, a = 0.lfm, as a function of the u, d dynamical quark mass. The x axis is plotted as the ratio m,/m, where the 7r and p are the pseudoscalar and vector mesons made with valence quarks of the same mass as the dynamical quarks. The real world has m„/m, = 0.2. For mt,,d = ms the ratio is 0.7, for mu,d = m8 /2, 0.55 and for m8 /4, 0.4. For m8 /2 the ratio is obtained from the K and K* masses. For m, and m3/4 some arguments must be made about the scaling of hadron masses with quark masses because, for example, no pure .5.3 pseudoscalar meson exists. The cost varies here as (m,/m,)3, which is based on estimates from simulations. Figure 14 compares the cost for clover quarks and improved staggered quarks, again based on simulations at one quark mass, and using the same scaling formula. The cost advantage of improved staggered quarks is clear on this plot. One disadvantage is that the algorithm generally used for two flavors of dynamical staggered quarks is not exact, unlike that for clover. This means that there are systematic errors, rather like discretisation errors, which increase with the computer time step, €, which is used to generate one gluon configuration from the previous one. Checks must to be done to make sure that this systematic error is at an acceptable level and/or an extrapolation to e = 0 must be done. Recent unquenched calculations, albeit with rather heavy dynamical quark masses, clover - improved — staggered 1.0 1.2 1.05 • Nf.2 o N1=0 Improved q N1.0 Standard 1.00 0 0.95 E 0.90 0.85 0.0 0.2 0.4 0.6 0.8 a [Gee] Lattice QCD 129 Figure 15. The masses of the 0 and K* mesons as a function of lattice spacing, a, for quenched simulations and those using two flavors of dynamical quarks. The K meson is used to fix the strange quark mass. The experimental results are indicated by diamonds at a .= 0. The Iwasaki improved gluon action was used with a clover quark action. (Ali Khan, 2002, CP-PACS collaboration) have shown encouraging signs that systematic errors from the quenched approximation are being overcome. Figure 12 shows that the ratio of a' values obtained from two different quantities is closer to 1 on dynamical configurations (using two flavors of dynamical quarks with a mass around 7728) than it was on quenched configurations (Marcantonio, 2001, UKQCD collaboration). From this we can hope that with 2 dynamical light quarks and a dynamical strange quark there will be only one value of the lattice spacing, corresponding to the one dimensionful scale of QCD in the continuum. Figure 15 compares results for the masses of 0 and K* mesons on quenched and unquenched configurations as a function of a. The K meson is used to fix ms and gives poor results for the K* and the 0 in the quenched approximation, as described earlier. For two flavors of dynamical quarks, the K* and 0 masses are much closer to experiment, at least after a continuum extrapolation (Ali Khan, 2002, CP-PACS collaboration). One worrying feature of this plot is the size of discretisation errors in the unquenched case, implying that the improved action used does not work very well unquenched. Figure 16 shows another quantity from light hadron physics that gives a problem in the quenched approximation. This is the difference of the squared vector and pseudoscalar masses for given quark combinations. Experimentally the result is very flat as a function of quark mass, being 0.55GeV2 from the 7, p to the D, D*. In the quenched approx-imation this quantity has a pronounced downward slope as the quark mass is increased. Recent results from the MILC collaboration with 2 (m8 /4) + 1 (ms) flavors of dynamical improved staggered quarks show qualitatively different behaviour, much closer to that of experiment (Bernard, 2001, MILC collaboration). This is the strongest indication yet that calculations with dynamical quarks will overcome the disagreements between the quenched approximation and experiment. 20 0.5 0 0 0.5 1.0 1.5 inn' in GeV' 0.8 q quenched x unquenched expt a, 0.7 0 " C E 0.6 130 Christine Davies Figure 16. The difference of the squares of the vector and pseudoscalar masses for various light hadrons, obtained with quenched and dynamical lattice QCD. The dynamical results have 2+1 flavors of dynamical quarks with masses m5/4 and ms . The dynamical results are given only for valence quark masses equal to the dynamical ones. Experimental results are given by the bursts, using an estimated mass for the pseudoscalar s-s- meson. The lattice spacing has been obtained using ro = 0.5fm. Errors do not include errors from, fixing the lattice spacing (Bernard, 2001, MILC collaboration). 4 Lattice QCD results The Proceedings of each year's lattice conference provide a useful summary of current results and world averages. See (LAT2000, LAT2001). Almost all lattice papers can be found on the hep-lat archive, http://arXiv.org/hep-lat/ . I have deliberately chosen to refer to reviews where possible and these should be consulted for fuller access to the literature. 4.1 Methods for heavy quarks Bottom and charm quarks are known as heavy quarks since they have masses much greater than the typical QCD scale, AQcD, of a few hundred MeV. Top quarks are also heavy, of course, but do not have interesting bound states so are not studied by lattice QCD. b and c quarks could be treated in the same way as u, d, or s quarks on the lattice except that, with current lattice spacings of about 0.lfm, we have mba in the interval 2--3 and Tr/cc/ in 0.5-1. If ma is not small then discretisation errors of the form ma, (ma)2 etc. will not be small either and such an approach will not give accurate results. Relativistic momenta, p m, can also not be well simulated if pa is not small: pa of 0(1) corresponds to wavelengths which are in danger of being small enough to 'fall through' the holes in the lattices. To reach the very fine lattices that would be required to give mba << 1 and accurate simulations for b quarks would require an amount of computing power way beyond our current hardware even in the quenched approximation. Luckily the physics of heavy Lattice QCD 131 quark systems in the real world means that we do not have to do this; indeed, it, would be largely a waste of computer power. b and c quarks are non-relativistic in their bound states, so that m and p m are irrelevant dynamical scales. The non-relativistic nature is evident from the hadron spectrum. There are heavy-heavy bound states in which both the valence quark and anti-quark are heavy (T, 'tb and Bc ) and heavy-light bound states in which the heavy (anti-)quark is bound to a light partner (B, B,, D, D,) or partners, in the case of baryons (A1, Ac). In all cases the mass difference (splitting) between excitations of these quark systems is much less than the mass of the hadrons. For example m(T')—m(T) = 560MeV, m(T) = 9.46GeV. The internal dynamics, which controls these splittings, operates with scales much smaller than the quark mass. Instead the important scales are the typical momentum carried by the quark inside the hound state, mv, and the typical kinetic energy, 2mv2. That these scales are small compared to m implies that v/c < 1. The use of non-relativistic techniques on the lattice is then a good match to the physics of b and c systems as well as providing an efficient way to handle them numerically on the lattice. There are several ways to proceed, and it is important when reading the lattice lit-erature to understand which method has been used. In the remainder of this section we consider three methods in particular: (a) static quarks, (b) NRQCD (a non-relativistic version of QCD) and (c) heavy relativistic quarks. Static quarks This is the m = oo limit of heavy quarks. In this limit Heavy Quark Symmetry holds and quarks become static sources of colour charge with no spin or flavor. This is evident on the lattice as the quark propagator becomes simply a string of gluon fields along the time direction (Eichten, 1990). Obviously no real quarks have infinite mass but this is a useful limit for studying heavy-light systems. Corrections away from the infinite mass limit are the subject of Heavy Quark Effective Theory (Buchalla, 2002). NRQCD NRQCD is a non-relativistic version of QCD (Lepage, 1992). The Lagrangian for heavy quarks is the non-relativistic expansion of the Dirac Lagrangian: D2 cr-B rQ = TP(D1 2m c2mQa + )V) (34) Qa where additional terms can be added to go to higher order in v/c. V.) is now a 2-component spinor since the quark and anti-quark fields of the Dirac fields decouple from each other. D is a covariant derivative, including coupling to the gluon field. B is the chromomagnetic field, related to space-space components of the field strength tensor, Bi = TriQ is the quark mass; heavy quarks are frequently generically denoted Q in contrast to the q used for light quarks. Notice that the quark mass term OmQazP has been dropped. This simply redefines the zero of energy so that the energies of all hadrons determined on the lattice are less than 1. The NRQCD Lagrangian can be discretised onto a lattice and leads to much simpler and faster numerical algorithms for calculating the quark propagator than for light quarks. 132 Christine Davies Instead of having to explicitly invert a matrix using an expensive iterative procedure such as Conjugate Gradient, the propagator is simply calculated by stepping through the lattice in time and calculating the propagator at time t from that at time t — 1. This is simply illustrated if we look at the Lagrangian in the infinite mass limit, where it becomes the Lagrangian for static quarks. Only the first term above contributes and we have: SQ = E zp(x) (ut(x)1p(x + - o(x)) (35) M is then an upper triangular matrix, using the notation of Equation 27, and the quark propagator is given by: (Al,t1 (?) = Utt (Wti ' (2 ). (36) The general start and end points, x and y, are simply denoted here by their t co-ordinates, 0 for the origin and t for the end point. To move from end point t to t +1 just requires multiplication by the appropriate U field in the time direction, so M' does not change spatially and becomes a string of U fields as described for static quarks above. For NRQCD with non-infinite masses, the evolution equation in t for the propagator is not as simple and does contain spatial variations (e.g. from the spatial covariant derivatives in Equation 34) but the same principles apply. A smearing function, 0, is chosen at the time origin and then the propagator calculated from 0 to later times by an evolution equation from one t to the next. This makes NRQCD numerically very attractive. Heavy quark propagators, once calculated, can be combined together or with a light quark propagator to make 2- and 3-point functions for heavy hadrons as described for light hadrons earlier. As described there also, the value for the bare heavy quark mass in lattice units, mo, is adjusted, given a value for a, until a heavy hadron mass is correct in GeV. The energies of heavy hadrons calculated on the lattice do not in fact equate directly to their masses because the mass term was removed from the Lagrangian. Instead, for one heavy hadron we have to calculate an energy-momentum dispersion relation and derive its mass from the momentum dependence (E cx p 2 / 2 /11). NRQCD is an effective theory, containing the right physics for low momentum heavy quarks. Adding more relativistic corrections to the Lagrangian can make this more ac-curate. These higher order terms appear with coefficients (such as c in equation 34) which must be determined by matching to relativistic QCD. These coefficients represent the effect of relativistic momenta missing from NRQCD and they are governed by a, at this high momentum scale and so are perturbative. High momenta for both quarks and gluons are missing anyway on the lattice because of the discretisation of space-time. We described earlier how a better match between lattice QCD and QCD is made by adding terms to the lattice QCD Lagrangian which are higher order in a, with a coefficient which depends on the strong coupling constant at the lattice cut-off scale. That the two pro-cedures are very similar is not an accident; indeed, the same higher dimension operators appear in both cases. In this case NRQCD is simply making a virtue of the existence of the lattice cut-off. The difference is, however, that in the NRQCD case the operators appear with inverse powers of incja (in a dimensionless lattice notation) and so mo, and therefore a, cannot be taken to zero in this approach. NRQCD has no continuum limit, but this does not prevent physical results being obtained at finite lattice spacing. It is just necessary to show that the results are sufficiently independent of a over a range of values of a. Lattice QCD 133 Heavy relativistic quarks This method looks very different from NRQCD, but has a lot of features in common. The use of a relativistic action, such as the Wilson/clover action, for heavy quarks on a lattice does not have to be incorrect if the results are interpreted carefully (El-Khadra, 1997). The main point to realise is that the existence of a large value for ma breaks the symmetry between space and time. The inverse quark propagator in momentum space has an energy at zero momentum very different from its mass (e.g. for a free Wilson quark, E(p = 0) = ln(1 + ma)) but its momentum dependence for small momenta is correct (i.e. as p2 /ma). Thus, we can ignore the ma errors in the energy if we fix masses from the energy-momentum relation as for NRQCD. For more precision we must add higher order discretisation/relativistic corrections. These will appear with coefficients chosen to match continuum relativistic QCD. As we have seen the coefficients are a power series in a, at the cut-off scale and they will depend on ma. For small ma the coefficients will be those of a discretisation correction to the action; for large ma they will go over to the NRQCD coefficients. For example, the clover term corrects for an 0(a) error in the Wilson action for light quarks; for heavy quarks, it becomes the relativistic correction which couples the quark spin and the chromomagnetic field. In this way an action can be developed that smoothly interpolates between heavy and light quark physics, at the numerical cost of having to handle heavy quarks in the same way as light ones. This method is sometimes known as the Fermilab method, since it was pioneered there. The charm quark mass is not very heavy on the finest of current quenched lattices, and some groups have taken the standard relativistic approach in this case. To reach the b quarks then requires an extrapolation jointly in the heavy quark mass and the lattice spacing (Maynard, 2002, UKQCD collaboration) to avoid confusing discretisation and relativistic corrections. Such an extrapolation inevitably has rather large errors. A better approach is to consider a formalism which explicitly breaks space-time symmetry in order to restore the relativistic energy-momentum relation for heavy quarks. For example, you can take an anisotropic lattice which has a much finer spacing in the time direction than in the space directions. mat is then small and the heavy quark looks like a light one, at the cost of having many more timeslices on the lattice, and having to determine the lattice spacing in both directions (Chen, 2001). 4.2 The heavy hadron spectrum The spectrum of heavy-heavy states has largely been the province of NRQCD (Davies, 1998). Figure 17(a) shows the radial and orbital excitations of the bb T system, obtained both on quenched gluon configurations and those with two flavors of dynamical quarks (Marcantonio, 2001, UKQCD collaboration). For these results the lattice spacing has been fixed by demanding that the splitting between the T(1S) and the spin-average of the P-wave (Xb ) states is correct. The b quark mass has been fixed by requiring that the T(1S) mass be correct. It is only the 2S (T'), 3S (T") and 2P (Xib) states that are predicted from this calculation, and they have rather large statistical errors at present. It is a general feature of lattice calculations that ground state masses are more precise than excited state masses. For both excited and ground states the noise is controlled by the ground state mass. For excited states the signal/noise ratio is then much worse and becomes exponentially bad at large T. xbo 3S 2P (.1) 2S - 1P 20 — 0— -Oak T T/ (2* hb —20-- C) -,J)7.1 .-! - X62 X61 3,51 1P1 - 40- CD rib I 0 - 60— 10.5- 10.0- 9.5- 134 Christine Davies (a) (h) GeV MeV Figure 17. (a) The radial and orbital excitations in the bb system, as calculated in lattice QCD using NRQCD for the b quarks (Marcantonio, 2001, UKQCD collaboration). (b) The fine structure of low-lying bb states. Legend: the horizontal dashed lines are experimental values; open circles show the quenched approximation; solid circles correspond to 2 flavors of clover dynamcal quarks with mass rns . (The lowest cluster of points on the right show an extrapolation to lighter dynamical mass and to N1=3. Of more immediate interest is the fine structure of the low-lying S and P states, shown in Figure 17(b). These can be determined very precisely on the lattice, particularly the `hyperfine' splitting between the spin-parallel vector T state and the not-yet-seen spin- antiparallel pseudoscalar A comparison with experiment, when it exists, for this splitting will provide a very good test of lattice QCD and our b quark action, which will be important for the lattice predictions of B matrix elements described in Section 4.3. The accuracy of the NRQCD, or other lattice action, for heavy-heavy bound states can be estimated by working out what order in an expansion in powers of v/c is represented by each term. e.g. the first two terms in the NRQCD action of Equation 34, i.e. the time derivative and the kinetic energy term, are both 0(v2 /c2). This is because the `potential energy' and kinetic energy terms are roughly equal for two heavy particles. These terms give rise to the radial and orbital splittings, and the ratio of these 500MeV) to half the T mass gives an estimate of v2 /c2 0.1 for b quarks in an T. Higher relativistic corrections, such as the D4 /8mQ term, are 0(v4/0) and should give roughly a 10% correction to these splittings. These terms were included here, but not the v6 I c6 corrections, so an error of roughly 1% remains. The a • B term of Equation 34 is the first spin-dependent term and is 0(0/c4 ). It gives rise to the hyperfine splitting and a similar term of the same order, proportional to a • D x E, gives rise to the P fine structure. The fine structure is indeed roughly 10% of the radial and orbital splittings. Including only these terms in the NRQCD action, as was done here, implies an error of roughly 10% in these splittings. A more precise calculation, necessary to test this action against Lattice QCD 135 = 0 a a 01 Eti 11] B -a- -1E1- .H.• A A A * * exotic gluon excitations 0 1 0— 1— 1' 2- 2'2 3— 3' 3 0— 1— 2— Figure 18. The spectrum of a states, as calculated in lattice QCD using anisotropic quenched configurations (Chen, 2001). experiment, will require the v6 /c6 spin-dependent terms and the as v4 /c4 terms implied by calculating the coefficient c in front of the o • B term in equation 34. This is now being done. Figure 17(b) does show, however, that the hyperfine splitting increases when two flavors of dynamical quarks are included, and continues to increase as the dynamical quark mass is reduced towards real u and d quark masses. We expect the T to see also s quarks in the vacuum and extrapolating the number of dynamical flavors to three increases the splitting further. The charmonium, lb, system is more relativistic that the T system and correspondingly less well-suited to NRQCD. Estimates as above give v2 /c2 c•-.. 0.3. Figure 18 shows the charmonium spectrum obtained from anisotropic relativistic clover quarks in the quenched approximation (Chen, 2001). The lattice spacing and charm quark mass were fixed in the analogous way to that described above, except that the spin average of the vector JO and pseudoscalar masses was used to fix in,. Since the Tie mass is known experimentally this gives improved precision since the spin-average is not sensitive to any inaccuracies in spill-dependent terms. The spectrum given in Figure 18 includes some gluonic excitations of the cc system, i.e. cg states, called hybrids. Their existence is expected simply from the non-Abelian nature of QCD which allows gluons themselves to carry color charge. Some of these hadrons have exotic quantum numbers not available to mesons made purely of valence quarks, and the prediction of their masses will be important for their experimental discovery. Figure 19 shows the spectrum of mesons made from one b quark and one light (u/d or s) anti-quark in the quenched approximation (Hein, 2000). NRQCD was used for the b quark, and the clover action for the light quark. In this case the lattice spacing was fixed using a quantity from the light hadron spectrum, ni p, because heavy-light systems are more similar in terms of internal momentum scales to light hadrons than heavy-heavy ones. See the comments in Section 3.2 on how the lattice spacing in the quenched approximation depends on the quantity used to fix it. The ul d and s quark masses were fixed using the 7 and K masses. The b quark mass was fixed from the spin-average of the B and B* meson masses. Taking a spin-average, as above for charmonium, avoids 4.8 4.4 4 <0 E 3.6 3.2 2.8 136 Christine Davies 6.5 GeV 6.0 -1-1- 5.5 _0- B B. B. B: B; B; Lr., Figure 19. The spectrum of bound states of a b quark with a light anti-quark as calculated in lattice QCD in the quenched approximation using NRQCD for the b quark (Hein, 2000). any errors from spin-dependent terms in the action. The b quark mass obtained this way differs from that obtained above from the T system, and is another feature of the quenched approximation. In the 'real world' there is only one lattice spacing and one set of quark masses and parameters fixed from the T system will be used to predict the entire B spectrum. The power counting in v/c for terms in the Lagrangian works rather differently in heavy-light systems compared to heavy-heavy ones. Now there is one quark that carries almost all the mass of the heavy-light system and it sits in the centre surrounded by the swirling light quark cloud. This picture makes sense even in the limit in which the heavy quark has infinite mass when the Lagrangian would contain only the covariant temporal derivative Dt (static quarks). The higher order terms in the Lagrangian can then be ordered in terms of the inverse powers of the heavy quark mass that they contain. This is equivalent to an expansion in powers of v/c. The typical momentum of a heavy quark in a heavy-light system is O(AQCD) (as is that of the light quark) and so v/c AwD/n),(2. This gives v/c P.-- 10% for the B and 30% for the D. Again the power counting exercise enables us to understand the approximate relative sizes of different mass splittings in the spectrum and the accuracy of our lattice QCD calculation to a given order in v/c. The leading spin-independent term in the action is Dt giving rise to the orbital and radial excitations of 500MeV. The kinetic energy term, D2 /2mQ gives a AQcD /mQ correction to this, which depends on the quark mass and, therefore flavor. This explains why these excitation energies are so similar for B and D systems; the similarity between '0 and T is more accidental. The leading spin-dependent term is 0-B/2mQ, which gives rise to fine structure such as the splitting between the pseudoscalar B and vector B*. This splitting should then be smaller by a factor of AQCD /mQ compared to the spin-independent splittings and this is indeed observed. To calculate this splitting precisely on the lattice requires the inclusion of higher order terms in the Lagrangian, as well as a better matched coefficient c for the cr • B term and this I " I AVerade ' I 74'" liadronic Jets -0,- e.e- rates 'e+e- event shapes 0 ; Fragmentation Z width ep event shapes Polarized DIS Deep Inelastic Scattel-in4(DIS) ° T decays Lattice. Y decay • —o-- ; Lattice QCD 137 0.1 0.12 0.14 cts(Mz) Figure 20. A comparison of determinations of the strong coupling constant, expressed as ar(Mz) (PDG, 2001). will be done in future calculations. We have stressed that lattice QCD is simply a way of handling QCD. It has the same a priori unknown parameters as QCD, the overall scale (equivalent to the coupling constant) and the quark masses. These parameters come from a deeper theory and must simply be fixed in the QCD Lagrangian using experiment and the results from a calculation in QCD. As described in Section 3, Lattice QCD provides the most direct way of doing this. The values for the parameters obtained are then useful input to other theoretical techniques. Determination of the lattice spacing at a given lattice bare coupling constant, is equiva-lent to (and can be converted into) a determination of the renormalised coupling constant, a, at a physical scale in GeV. To compare to other determinations of as, this can be converted to the MS scheme and run to /Viz. Figure 20 shows a comparison of different determinations of a, from the Particle Data Group (PDG, 2001). It is clear that the lattice result is one of the most precise. All methods for determining a, have three components: 1. Theoretical input: a perturbative expansion in as, for some quantity. 2. A value for that quantity. 3. An energy scale. Most methods use an experimental result for stage 2, where the lattice uses a non-perturbative evaluation on the lattice of the vacuum expectation value of a simple short-distance gluonic operator (such as the plaquette). This avoids the problems of hadronisa-tion etc which reduce the precision of methods based on the experimental determination 138 Christine Davies of jet shapes or cross-sections. All methods use experiment for stage 3, and here the lattice-based determination needs an experimental result to fix the lattice spacing. A good quantity to use here is the orbital excitation energy (1P-1S) in, say, the T system since this is well-determined on the lattice and directly measured experimentally. Quark masses are also well determined on the lattice. Since quarks are not, freely available to be weighed, as an electron would be, care must be taken in defining what exactly is meant by the quark mass. The bare mass in the lattice QCD Lagrangian for a particular action, determined by the requirement to get a particular hadron mass correct and converted to physical units, is a well-defined quantity but not very convenient. We can convert it perturbatively into, say, the running quark mass in the MS scheme. The best current determination of the b quark mass is in fact from the static approximation in which b quarks have infinite mass. There is no bare b quark mass in that case; instead the binding energy mB — mb is calculated, and from that, mb is determined. The binding energy is small compared to m B and has only weak dependence on the b quark mass, so for this quantity the static approximation is a good one. The b quark mass obtained in this way is 4.30(10)GeV in the quenched approximation, with some indications that it is slightly lighter when dynamical quarks are included (Lubicz, 2001). 4.3 Heavy hadron matrix elements Precise lattice calculations of matrix elements for B decay are essential to the experimental B factory programme (Stone, 2002). This aims to test the internal consistency of the Standard Model in which CP violation occurs through the Cabibbi-Kobayashi-Maskawa matrix. The weak decays of the b quark are particularly useful in giving us access to poorly known elements of this matrix. The unitarity of the CKM matrix can be represented by a triangle; the position of the upper vertex being constrained by the determination of angles and sides, see Figure 21. The angles are determined directly by measurement of asymmetries. The determination of the sides requires both the experimental measurement -0.5 0 0.5 Figure 21. The unitarity triangle with constraints on the upper vertex obtained from different quantities (Hockey, 2001). The lower vertices are at Ti = 0, 7)- = 0 and 1. 500 4001 300:: ()/1"V) 2001 - • 100- Lattice QCD 139 of a decay rate and its theoretical calculation.This allows the magnitude of one of the CKM elements to be extracted. Below we describe the lattice calculation of the matrix elements most important for this programme. The extent to which the unitarity triangle can be tested depends on both the experimental and the theoretical errors. It is critical to reduce the errors from lattice calculations to a few percent, otherwise they will dominate the uncertainties from experiment. The simplest 2-point matrix element that can be calculated on the lattice is that for the decay constant of the charged pseudoscalar heavy-light mesons (see Figure 8). For the B this is known as f B and it is obtained from the vacuum to B matrix element of the axial vector current which couples to the W. (0 I API B) = B' (37) The purely leptonic decay rate of the B meson is then proportional to n times kinematic factors times the square of the CKM element which multiplies the appropriate axial vector current in the Lagrangian, in this case ftvy5b (Rosner, 2002). In principle an experimental determination of the leptonic decay rate could be combined with the lattice calculation to yield Vub, but in practice the experiment is very hard to do because the rate is so low. For other heavy-light mesons, it may be possible. fos has been measured experimentally, but not very precisely as yet. It can be used, with lattice calculations, to give V. It is important to realise that, although we are discussing the weak decay of a b or c quark, the calculations are done in lattice QCD. The quark cannot decay in isolation, but must be bound into a hadron by the confinement property of QCD. The determination of the decay matrix element must then take into account all the QCD interactions inside the hadron (see Figure 8) and this requires lattice QCD. We do not put the W boson on the lattice. As far as QCD is concerned the B meson annihilates into the vacuum. The virtual W boson decay to leptons is put in by hand when we calculate the decay rate. Lattice calculations of f B improved markedly through the 1990s (this has been true of most lattice calculations) as we got to grips with the systematic errors. Figure 22 shows a timeline of results in the quenched approximation. It shows both that lattice calculations have markedly improved and that early calculations had very unreliable estimates of their Figure 22. A timeline of results for the B meson decay constant, f B, calculated in lattice QCD in the quenched approximation. 140 Christine Davies errors. The most recent and best calculations do a careful job of matching the lattice representation of the axial vector current to the continuum. For heavy-light mesons we have to be careful both about relativistic (Ac4cD/mQ) corrections and discretisation corrections to the leading order lattice current. Since mQa is a dimensionless number, these two corrections in fact appear together and can be considered simultaneously. The matching between lattice and continuum is currently done only to 0(as) and this is the major source of error in the quenched approximation. Table 1 shows a typical 'error budget' for such a calculation. We need a more precise matching, either to as or non-perturbatively (both of which can be done with a lot of hard work), to improve the errors beyond the 10% level. Source percent error statistical + fitting 3 discretisation O((aA)2 ) 4 perturbative 0(4 a's I (a M)) 7 NRQCD 0((A/M)2, asA/M) 2 light quark mass a 1 (,,,,, \ ki"Pl 4 4 Total 10 Table 1. Source of error in a typical lattice calculation of f B using NRQCD for the heavy quark in the quenched approximation. a 0.1fm, M is the b quark mass, A a typical QCD scale of a few hundred MeV and a, is evaluated at 2/a. Recent reviews of lattice results (Ryan, 2002), (Bernard, 2001) have given the following `world averages' for lattice results in the quenched approximation: • f B = 173 ± 23MeV • fp, = 230 ± 14MeV • fed fe = 1.15(3); fpjfp = 1.12(2). (Note that the B, does not decay purely leptonically but the calculation of the appropri-ate matrix element can still be done in lattice QCD and yields useful information on its dependence on the light quark mass.) Large-scale calculations on dynamical configura-tions are only just beginning, so unquenched results are still unclear. It seems likely that decay constants will be 10-20% larger unquenched. A more important quantity from the point of view of the B factory programme is the mixing amplitude for neutral B mesons, B° and B,. This mixing gives rise to a difference in mass between the CP-eigenstates, Am, which can be measured experimentally through oscillations between particle and anti-particle (Stone, 2002). The mixing amplitude is given by the 'box diagram' (see Figure 23) in which the b quark and light anti-quark convert to a b anti-quark and light quark through the mediation of virtual Ws and (pref-erentially) t quarks. The mixing amplitude is then proportional to the matrix element of the box between, say, a B° and a T3° multiplied by the product of CKM elements 14617td. The current determination of 1Vtd12 from experiment and theory gives a curve Lattice QCD 141 Figure 23. The B box diagram, related to that of a 4-quark operator. on the unitarity triangle plot (marked Arrid on Figure 21). Future experiments will be able to see oscillations of the B, and then ratios of AmB,I AmB will allow a more precise determination of 114.5/Vtd12, since some of the systematic errors will cancel out. As explained earlier, W bosons do not appear in lattice QCD calculations. The matrix element of the box diagram is calculated in lattice QCD by replacing it with the equivalent 4-quark operator which appears in the effective (low-energy) weak Hamiltonian (Rosner, 2002). Conventionally this matrix element for the B is set equal to (8/3)JIMP3B, giving a definition of the parameter confusingly called BB. BB is the amount by which the matrix element differs from the result that would be obtained by saturating the Hw vertex of Figure 23 with the vacuum (comparing this to the right hand picture of Figure 8 we can see that this would be n). BB is generally expected to be roughly 1, and this explains why lattice calculations originally concentrated on calculating f B. To calculate BB is harder, but is now being done. It requires, as for f B, a careful matching between the lattice and the continuum, and this has again been done to 0(cts ) so far. Recent world averages for the renormalisation-group-invariant definition of BB in the quenched approximation have been given as (Ryan, 2002), (Bernard, 2001): . BBd = 1.30(12)(13) • fkAMB, =230(40)MeV • EBJEBs, = 1.00(4) • fB, VEBs /f ki \MB, = 1.15(6) A lot of the matching errors cancel out in the ratios between B, and Bd so that the errors in these ratios are less than 10%. The ratio may also not be significantly affected by unquenching. Heavy-light mesons decay semi-leptonically through a diagram in which the heavy quark changes flavor, emitting a virtual W, and the other (spectator) quark in the meson combines with the new quark flavor to make a new meson. In this way B mesons can decay to D or D* mesons if b c and to 7 or p mesons if b u. In each case the appropriate CKM element appears at the current vertex in the three-point diagram (see Figure 9) and can therefore be determined by a comparison of the experimental exclusive rate to the theoretical one. The ratio Vub/Veb gives an important circular constraint in the unitarity triangle (see Figure 21). The calculation of the matrix element for B semi-leptonic decay on the lattice re-quires the calculation and simultaneous fitting of the 3-point function of Figure 9 and the appropriate 2-point functions necessary to isolate the matrix element. It is therefore 142 Christine Davies significantly harder than a simple 2-point calculation. In addition the matrix element depends on q2, the squared difference of 4-momenta between the initial and final meson. This can take a range of values, because the decay is a three-body one. The matrix el-ement can then be written as a combination of form-factors which are q2 dependent, in contrast to the two-body leptonic decay which was parameterised by a single number, f B. For example the pseudoscalar to pseudoscalar transition (e.g. B to D or 71-) proceeds only through the vector current and has two form factors, f+ and fo: - t 1 (1111)1Vp1P(P)) = f+(q2){(P + p1 ), q2Mi ql1.1 Mq2)m2 P 2 The differential decay rate is proportional to the square of f+ because the leptonic current coupling to the W has qµ L, = 0 for massless leptons. The pseudoscalar to vector transition proceeds through both the vector and axial currents and has 5 form factors, 3 of which appear in the decay rate. To explore different values of q2 for semi-leptonic decay on the lattice it is easiest to insert different 3-momenta at the final meson and at the current, and then work out the resulting 3-momentum of the initial meson. We are restricted to values of 3-momentum allowed on the lattice, i.e. the components of p have the form pi a = ti,3 27r/L, where = 0,1, 2 ... and L is the number of lattice sites in the j direction. The smallest non-zero value of p, is then 27r/(La) where La is the physical size of the lattice in a spatial direction. A big physical volume is then required to achieve a fine discretisation of momentum space and avoid a large jump from one momentum to the next. In general results at higher momenta are much noisier than those at small momenta (this is for the same reason that excited states are noisier than ground states, discussed above) and calculations tend to be restricted to a few of the smallest possible momenta. Discretisation errors will also be larger at larger values of pa, so systematic errors will be higher. For the matrix element for B to D(*) semi-leptonic decay it is useful to consider both the b quark and the c quark in the heavy quark limit. In that limit, as discussed above, the Lagrangian for heavy quarks becomes insensitive to the heavy quark spin or flavour (Buchalla, 2002). The light quark cloud in the meson cannot tell whether it is surrounding a b or a c quark or one whose spin is pointing parallel or anti-parallel to its spin. Thus the form factors for B D and B —> D* will become identical (or vanish) and the same as the B —> B elastic form factor, provided they are viewed as a function of the right variable. This is not q2 but v•v' where v is the 4-velocity (pp /rn) of the initial meson and v' is the 4-velocity of the final meson. v-v' is often given the symbol w. In the notation of Equation 38 w = + MP, — q2)/(2MpMp'). The limit w = 1 is known as the `zero-recoil' limit because this corresponds to the kinematic point where the B meson at rest decays to, say, a D meson at rest and the decay products of the W come out back-to-back. This point has maximum q2 = (Alp — M02. The B B elastic form factor takes the form (B(v')117,IB(v)) = Mg(w)(v + v') (39) in the limit of infinite b quark mass, where (w) = f+ (q2), = 0. (w) is known as the Isgur-Wise function. - (1) = 1 is an absolute normalisation in the continuum because 67,b is a conserved current. The lattice current is not a conserved one (except for the NRQCD/static actions) but if we are interested only in the shape of (w) we can (38) mo=1.8, x=0.153 0.95 LP 0.90 0.85 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 vv' Figure 24. The B -4 B elastic form factor calculated in lattice QCD using NRQCD for the b quark and plotted versus w (Hashimoto, 1996). 1.00 Lattice QCD 143 renormalise to match 1 at w =- 1. There have been several calculations of the B B form factor on the lattice, for various heavy quark masses. Figure 24 shows such a calculation using NRQCD with a mass close to that for the b quark (Hashimoto, 1996). The interest in calculating the Isgur-Wise function is that, in the Heavy Quark Symme- try picture described above, it is also applicable to B D and B D* decays. In these cases, however, there is an additional overall perturbative renormalisation because b-y,c is not a conserved current, and there are corrections which appear as differences of inverse powers of the b and c quark masses. For kinematic reasons, B D* is experimentally easier to measure in the w —> 1 region. The differential decay rate is dF dTv = iv eb 2K(w)F2(w) (40) where Va is the CKM element that we want to determine, K(w) is a kinematic factor and is the form factor for the decay. Figure 25 shows results from the CLEO collaboration (CLEO, 2000) for .T(w)1Ved. The lighter hashed curve is the result from the lattice shown in Figure 24 rescaled by a constant to match at w = 1. Given lattice results for B D* rather than B B, the constant required for rescaling would be Ilied which would then be determined. In fact, a number of simplifications can be made to the lattice calculation at the w = 1 point and so it is currently better to perform a phenomenological extrapolation of the experimental data to w = 1 and divide the extrapolated result by the lattice result for T(1). The Fermilab group, using heavy relativistic (Fermilab) quarks and 0(crs) matching • D*+lv — Fit 144 Christine Davies 1 . 1 1.2 1.3 1.4 I.5 w Figure 25. I Vcb1F(w) extracted from the experimental B D* decay rate plotted as a function of w (CLEO, 2000). The shorter curve on 1 D*ln decay width and 117,4 heP-ex/ 0007052. Davies C, 1998, The heavy hadron spectrum in Computing Particle Properties, Springer Lec- tureNotes in Physics, Eds Gausterer, Lang; hep-ph/9710394. Di Pierro M, 2001, From Monte Carlo Integration to Lattice Quantum Chromodynamics, hep- lat/0009001 Eichten E and Hill B, An effective field theory for the calculation of matrix elements involving heavy quarks, Phys. Lett. B243, 511. 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Lepage P et al, 1992, Improved nonrelativistic QCD for heavy quark physics, Phys. Rev. D46, 4052; hep-lat/9205007. Maynard C et al, 2002, UKQCD collaboration, Heavy-light decay constants on the lattice, Nucl. Phys. B (Proc. Suppl. 106), 388; hep-lat/0109026. Luhicz V, 2001, Quark masses on the lattice, light and heavy, Nucl. Phys. B (Proc. Suppl. 94) 116; hep-lat/0012003. Marcantonio L et al, 2001, UKQCD collaboration, The unquenched T spectrum, Nucl. Phys. B (Proc. Suppl. 94), 363; hep-lat/0011053. PDG, 2001, http://pdg.lbl.gov/. Rosner .J, 2002, The Standard Model in 2001, this volume; hep-ph/0108195. Ryan S, 2002, Heavy quark physics from lattice QCD, Nucl. Phys. B (Proc. Suppl. 106) 86; hep-lat/0111010. Sommer R, 1998, Non-perturbative renormalisation of QCD in Computing Particle Properties, Springer Lecture Notes in Physics, Eds Gausterer, Lang; hep-ph/9711243. Stone S, 2002, B Phenomenology, this volume; hep-ph/0112008. Toussaint D, 2002, Spectrum results with Kogut-Susskind quarks, Nucl. Phys. B (Proc. Suppl. 106) 111; hep-lat/0110010. 147 CP violation — a new era Yosef Nir Weizmann Institute of Science, Israel 1 Introduction The Standard Model predicts that the only way that CP is violated is through the Kobayashi-Maskawa mechanism [1]. Specifically, the source of CP violation is a sin-gle phase in the mixing matrix that describes the charged current weak interactions of quarks. In the introductory chapter, we briefly review the present evidence that supports the Kobayashi-Maskawa picture of CP violation, as well as the various arguments against this picture. 1.1 Why believe the Kobayashi-Maskawa mechanism? Experiments have measured to date three independent CP violating observables: . Indirect CP violation in K 71-7 decays [2] and in K Tel, decays is given by: E K = (2.28 + 0.02) x 10-3 u7 l4. (1) . Direct CP violation in K 71-7 decays [3, 4, 5, 6, 7] is given by = (1.72 + 0.18) x 10-3. (2) . CP violation in B 'bKs decay and other, related modes has been measured [8, 9, 10, 11, 12]: av,K, = 0.79 + 0.10. (3) All three measurements are consistent with the Kobayashi-Maskawa picture of CP violation. In particular, the two recent measurements of CP violation in B decays [11, 12] have provided the first precision test of CP violation in the Standard Model. Since the model has passed this test successfully, we are able, for the first time, to make the following statement: The Kobayashi-Maskawa phase is, very likely, the dominant source of CP violation in low-energy flavor-changing processes. DOI: 10.1201/9780429187056-4148 Yosef Nir In contrast, various alternative scenarios of CP violation that have been phenomeno-logically viable for many years are now unambiguously excluded. Two important examples are the following: . The superweak framework [13], that is, the idea that CP violation is purely indirect, is excluded by the evidence that EVE 0 0. . Approximate CP, that is, the idea that all CP violating phases are small, is excluded by the evidence that a,pK, = 0(1). The experimental result (3) and its implications for theory signify a new era in the study of CP violation. In this series of lectures we will explain these recent developments and their significance. 1.2 Why doubt the Kobayashi-Maskawa mechanism? 1.2.1 The baryon asymmetry of the Universe Cosmology shows that the Kobayashi-Maskawa phase cannot be the only source of CP violation: baryogenesis, that is, the history of matter and antimatter in the Universe, cannot be accounted for by the Kobayashi-Maskawa mechanism. To understand this statement, let us provisionally switch off all CP violation. Then, for every process that occurs in Nature, the corresponding CP conjugate process proceeds with a precisely equal rate. Let us further assume that the initial conditions are such that the number density of quarks and the number density of the matching antiquarks are equal. Then, CP invariance guarantees that the number densities remain equal to each other along the history of the Universe. In other words, the baryon asymmetry, (nB — nB)/n,, is guaranteed to remain zero. Two particularly significant processes are proton-antiproton annihilation and production. While the first would happen at any temperature, the latter is allowed only if the energy of the photons is large enough to produce a proton-antiproton pair. At high enough temperatures, T 27n,,, annihilation and production will keep the protons and antiprotons in equilibrium and their number densities would be (precisely equal to each other and) similar to the photon number den-sity, nB = nB n.y. But at temperatures well below GeV, proton-antiproton production slows down until it practically stops. Since annihilation continues to take place, the num-ber densities of protons and antiprotons (remain equal to each other but) decrease, and at present there would be practically neither matter nor antimatter. This is, of course, inconsistent with observations. Now let us switch on CP violation. That allows a different rate for a process and its CP conjugate. Such a situation would have relevant consequences if two more conditions are met [14]: there is a departure from thermal equilibrium and baryon number can be vi-olated. When all three conditions are satisfied, a difference between the number densities of quarks and of antiquarks can be induced. We assume that the number of quarks be-comes slightly larger than the number of antiquarks. This scenario is called baryogenesis. At the electroweak phase transition (temperatures of order a few hundred GeV, t 10-'1 seconds) baryon number violating processes become highly suppressed, and the baryon number cannot change any longer. The history of matter and antimatter in the Universe CP Violation 149 proceeds along the same lines as described in the previous paragraph. In particular, at temperatures well below GeV the number densities of protons and antiprotons decrease. There is however an important difference: at some time, practically all antiprotons would disappear. But the small surplus of protons have no matching antiprotons to annihilate with. It remains there forever. The resulting picture of the present Universe is then as follows: there is no antimatter. There is a small amount of matter, with the present ratio (nB/n,),:, reflecting the baryon asymmetry, RN — n4)/nd BG, induced by baryogenesis. This picture is qualitatively consistent with observations. Thus we have good reasons to think that we understand the general mechanism of baryogenesis. The important point for our purposes is that baryogenesis is a consequence of CP violating processes. Therefore the present baryon number, which is accurately deduced from nucleosynthesis constraints (for a recent analysis, see [15]), nB = (5.5 ± 0.5) x 10-10, (4) n-r is essentially a CP violating observable! It can be added to the list of known CP violating observables, Equations (1), (2) and (3). Within a given model of CP violation, one can check for consistency between the data from cosmology, Equation (4), and those from laboratory experiments. The surprising point is that the Kobayashi-Maskawa mechanism for CP violation fails to account for (4). It predicts present baryon number density that is many orders of magnitude below the observed value [16, 17, 18]. This failure is independent of other aspects of the Standard Model: the suppression of nB In, from CP violation is much too strong, even if the departure from thermal equilibrium is induced by mechanisms beyond the Standard Model. This situation allows us to make the following statement: There must exist sources of CP violation beyond the Kobayashi-Maskawa phase. Three important examples of viable models of baryogenesis are the following: 1. GUT baryogenesis (for a recent review see [19]): the source of the baryon asym-metry is in CP violating decays of heavy bosons related to grand unified theories. In general, baryon number is not a conserved quantity in GUTs. Departure from thermal equilibrium is provided if the lifetime of the heavy boson is long enough that it decays when the temperature is well below its mass. The relevant CP violating parameters are not expected to affect low energy observables. 2. Leptogenesis (for a recent review see [20]): lepton asymmetry is induced by CP violating decays of heavy fermions that are singlets of the Standard Model gauge group (sterile neutrinos). Departure from thermal equilibrium is provided if the lifetime of the heavy neutrino is long enough that it decays when the temperature is below its mass. B + L-violating processes are fast before the electroweak phase transition and convert the lepton asymmetry into a baryon asymmetry. The CP violating parameters may be related to CP violation in the mixing matrix for the light neutrinos (but this is a model dependent issue [21]). 3. Electroweak baryogenesis (for a review see [22]): the source of baryon asymme-try is the interactions of top (anti)quarks with the Higgs field during the electroweak phase transition. CP violation is induced, for example, by supersymmetric inter-actions. Sphaleron configurations provide baryon number violating interactions. 150 Yosef Nir Departure from thermal equilibrium is provided by the wall between the false vac-uum ((q) = 0) and the expanding bubble with the true vacuum, where electroweak symmetry is broken. 1.2.2 The strong CP problem Non-perturbative QCD effects induce an additional term in the SM Lagrangian, — OQCD vpoF4wa FP" • 3272 (5) This term violates CP. In particular, it induces a neutron electric dipole moment (EDM). The leading contribution in the chiral limit is given by [23] dN = ln(MN/m7r) 97TNN.97NN 4Ir2MN 5 x 10-16 0QcD ecm, (6) where MN is the nucleon mass, and g7NN (9 NN ) is the pseudoscalar coupling (CP-violating scalar coupling) of the pion to the nucleon. (The leading contribution in the large N, limit was calculated in the Skyrme model [24] and leads to a similar estimate.) The experimental bound on d N is given by d N < 6.3 x 10-26 ecm [25]. ( 7) It leads to the following bound on 0QcD: OQCD <10-in. (8) Since eQcD arises from non-perturbative QCD effects, it is impossible to calculate it. Yet, there are good reasons to expect that these effects should yield 0QcD = 0(1) (for a clear review of this subject, see [26]). Within the SM, a value as small as (8) is unnatural, since setting 0QcD to zero does not add symmetry to the model. [In particular, as we will see below, CP is violated by 6KM = 0(1).] Understanding why CP is so small in the strong interactions is the strong CP problem. It seems then that the strong CP problem is a clue to new physics. Among the solutions that have been proposed are a massless u-quark (for a review, see [27]), the Peccei-Quinn mechanism [28, 29] and spontaneous CP violation. As concerns the latter, it is interesting to note that in various string theory compactifications, CP is an exact gauge symmetry and must be spontaneously broken [30, 31]. 1.2.3 New physics Another motivation to measure CP violating processes is that almost any extension of the Standard Model provides new sources of CP violation. These sources often allow for significant deviations from the Standard Model predictions. Moreover, various CP violat-ing observables can be calculated with very small hadronic uncertainties. Consequently, CP violation provides an excellent probe of new physics. CP Violation 151 1.3 Will new CP violation be observed in experiments? The SM picture of CP violation is testable because the Kobayashi-Maskawa mechanism is unique and predictive. These features are mainly related to the fact that there is a single phase that is responsible to all CP violation. As a consequence of this situation, one finds two classes of tests: . Correlations: many independent CP violating observables are correlated within the SM. For example, the SM predicts that the CP asymmetries in B OKs and in B OKs, which proceed through different quark decay processes, are equal to each other. Another important example is the strong SM correlation between CP violation in B OKs and in K —> TvP. . Zeros: since the KM phase appears in flavor-changing, weak-interaction couplings of quarks, and only if all three generations are involved, many CP violating observables are predicted to be negligibly small. For example, the SM predicts no CP violation in the lepton sector, practically no CP violation in flavor-diagonal processes (i.e. a tiny electric dipole moment for the neutron) and very small CP violation in tree level D decays. In addition, some of the CP violating observables can be calculated with very small hadronic uncertainties. The strongest argument that new sources of CP violation must exist in Nature comes from baryogenesis. Whether the CP violation that is responsible for baryogenesis would be manifest in measurements of CP asymmetries in B decays depends on two issues: . The scale of the new CP violation: if the relevant scale is very high, such as in GUT baryogenesis or leptogenesis, the effects cannot be signalled in these measurements. To estimate the limit on the scale, the following three facts are relevant: First, the Standard Model contributions to CP asymmetries in B decays are 0(1). Second, the expected experimental accuracy would reach in some cases the few percent level. Third, the contributions from new physics are expected to be suppressed by (AEw /ANp)2. The conclusion is that, if the new source of CP violation is related to physics at ANp > 1 TeV, it cannot be signalled in B decays. Only if the true mechanism is electroweak baryogenesis, can it potentially affect B decays. . The flavor dependence of the new CP violation: if it is flavor diagonal, its effects on B decays would be highly suppressed. It can still manifest itself in other, flavor diagonal CP violating observables, such as electric dipole moments. We conclude that new measurements of CP asymmetries in meson decays are particu-larly sensitive to new sources of CP violation that come from physics at (or below) the few TeV scale and that are related to flavor changing couplings. This is, for example, the case in certain supersymmetric models of baryogenesis [32, 33]. The search for electric dipole moments can reveal the existence of new flavor diagonal CP violation. Of course, there could be new flavor physics at the TeV scale that is not related to the baryon asymmetry and may give signals in B decays. The best motivated extension of the SM where this situation is likely is that of supersymmetry. We will discuss supersymmetric CP violation in the last chapter. 152 Yosef Nir 2 The Kobayashi-Maskawa mechanism 2.1 Yukawa interactions are the source of CP violation A model of elementary particles and their interactions is defined by three ingredients: (1) the symmetries of the Lagrangian, (2) the representations of fermions and scalars and (3) the pattern of spontaneous symmetry breaking. For the Standard Model (SM) the corresponding ingredients are as follows: • The gauge symmetry is GSM = SU(3)c x SU(2)L x U(1)y. (9) • There are three fermion generations, each consisting of five representations of Gsm: Q/L, (3, 2)+1/6, Ulfh (3, 1)+2/3, Dim (3, 0_1/3, LL(1,2)_ 112, EIR,(1, 1)-1. (10) Our notation means that, for example, left-handed quarks, CYL, are triplets of SU(3)c, doublets of SU(2)L and carry hypercharge Y = +1/6. The super-index I denotes interaction eigenstates. The sub-index i = 1, 2, 3 is the flavor (or generation) index. There is also a single scalar representation, 0(1, 2)+1/2. (11) . The scalar assumes a vacuum expectation value (VEV), (12) so that the gauge group is spontaneously broken, GSM 4 SU(3)c x U(1)Em. (13) The Standard Model Lagrangian, .Csm, is the most general renormalisable Lagrangian that is consistent with the gauge symmetry (9). It can be divided to three parts: .CSM = Ckinetic .CHiggs LYukawa• (14) We consider each of these terms in turn, Kinetic terms For the kinetic terms, to maintain gauge invariance, one has to replace the derivative with a covariant derivative: = aµ + igsq:La + igiVrTb + ig'13PY. (15) CP Violation 153 Here Gf: are the eight gluon fields, n the three weak interaction bosons and BP the single hypercharge boson. The La 's are SU(3)c generators (the 3 x 3 Gell-Mann matrices Aa /2 for triplets, 0 for singlets), the Tb's are SU(2)L generators (the 2 x 2 Pauli matrices erb /2 for doublets, 0 for singlets), and Y are the U(1)y charges. For example, for the left-handed quarks Q1L, we have £kinetic(QL) = (aP + gsGna + gWri-b + 6i OP) QL, (16) while for the left-handed leptons L IL, we have Lkinetic(LL) = 2LLiyµ (aP + 2i g1417-1, — ig BP) Lim. (17) These parts of the interaction Lagrangian are always CP conserving. The Higgs potential The Higgs potential, which describes the scalar self interactions, is given by: £H iggs = p2of — A (0t02. (18) For the Standard Model scalar sector, where there is a single doublet, this part of the Lagrangian is also CP conserving. For an extended scalar sector, such as that of a two Higgs doublet model, Llliggs can be CP violating. Even in the case that it is CP symmetric, it may lead to spontaneous CP violation. Quark Yukawa interactions The quark Yukawa interactions are given by —4u tt = &u,/'?; + h.c.. (19) This part of the Lagrangian is, in general, CP violating. More precisely, CP is violated if and only if [34] Im det[YdYdt, YuYnf]} # 0. (20) An intuitive explanation of why CP violation is related to complex Yukawa couplings goes as follows. The hermiticity of the Lagrangian implies that Lyukawa has its terms in pairs of the form Yz30 m01:13 + RjOt /'LI• (21) A CP transformation exchanges the operators OLikRj <4 ORj Othi (22) but leaves their coefficients, Yii and unchanged. This means that CP is a symmetry of LYukavva if Yij = Yi;!`. 154 Yosef Nir Lepton Yukawa interactions The lepton Yukawa interactions are given by _eyupkatonwa -1,4LL0EiR j h.c.. (23) It leads, as we will see in the next section, to charged lepton masses but predicts massless neutrinos. Recent measurements of the fluxes of atmospheric and solar neutrinos provide evidence for neutrino masses. That means that GSM cannot be a complete description of Nature. The simplest way to allow for neutrino masses is to add dimension-five (and, therefore, non-renormalisable) terms, consistent with the SM symmetry and particle con-tent: —.Ctw = m dim-5 —Li L + h.c.. a Yiv (24) The parameter M has dimension of mass. The dimensionless couplings Yip are symmetric .(Yiv = Yu) We will refer to the SM extended to include the terms 4,,mkaw5a of Equation (24) .7 , as the "extended SM" (ESM): LESM = £kinetic £Higgs LYukawa Ltaw5a. (25) The inclusion of nonrenormalisable terms is equivalent to postulating that the SM is only a low energy effective theory, and that new physics appears at the scale M. How many independent CP violating parameters are there in ,Criakarkwsa? Each of the two Yukawa matrices Yq (q = u, d) is 3 x 3 and complex. Consequently, there are 18 real and 18 imaginary parameters in these matrices. Not all of them are, however, physical. One can think of the quark Yukawa couplings as spurions that break a global symmetry, U(3)Q x U(3) D x U(3)u U(1) B. (26) This means that there is freedom to remove 9 real and 17 imaginary parameters [the number of parameters in three 3 x 3 unitary matrices minus the phase related to U(1)B]. We conclude that there are 10 quark flavor parameters: 9 real ones and a single phase. This single phase is the source of CP violation in the quark sector. How many independent CP violating parameters are there in the lepton Yukawa in-teractions? The matrix Ye is a general complex 3 x 3 matrix and depends, therefore, on 9 real and 9 imaginary parameters. The matrix Yv is symmetric and depends on 6 real and 6 imaginary parameters. Not all of these 15 real and 15 imaginary parameters are physi-cal. One can think of the lepton Yukawa couplings as spurions that break (completely) a global symmetry, U(3)L x U(3)E. (27) This means that 6 real and 12 imaginary parameters are not physical. We conclude that, there are 12 lepton flavor parameters: 9 real ones and three phases. These three phases induce CP violation in the lepton sector. CP Violation 155 2.2 CKM mixing is the (only!) source of CP violation in the quark sector Upon the replacement Re(0) (v+H°)/ [see Equation (12)], the Yukawa interactions (19) give rise to mass terms: 1 DI + ( 1 U I U1 h = - Li Ri sMu,ij Li Rj + *C•, where V Mq = — and we decomposed the SU(2)L quark doublets into their components: Uji QL - = Li) • (30) The mass basis corresponds, by definition, to diagonal mass matrices. We can always find unitary matrices VqL and Vq R such that L mg vqtR Mdiag (q = d), (31) with Mgdiag diagonal and real. The quark mass eigenstates are then identified as qLi = qRi = (V, R)iigiRi (q = u, d). (32) The charged current interactions for quarks [that is the interactions of the charged SU(2)L gauge bosons W± = v2 ,--(1/1/1P ± iW2)1 which in the interaction basis are described tt by (16), have a complicated form in the mass basis: ,Cg = — uLCY A (Vu L )OdLjW ti+ h.c.. W± The unitary 3 x 3 matrix, -17cKm = 1/uLVdtLI (VCKM VCKM — 1), is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix for quarks [35, 1]. A unitary 3 x 3 matrix depends on nine parameters: three real angles and six phases. The form of this matrix is not unique and in fact there are two distinct freedoms which must be fixed. 1. There is freedom in defining V'cKm in that we can permute between the various generations. This freedom is fixed by ordering the up quarks and the down quarks by their masses, i.e. (u1 , u2, u3) (u, c, t) and (d1 , d2, d3) —> (d, s, b). The elements of VcK m are written as follows: Vud Vus 17ub VCKM = Vcd Vcs Vcb) • (35) Vtd Vts Vtb (28) (29) (33) (34) 156 Yosef Nir 2. There is further freedom in the phase structure of Vc K m . Let us define 13, (q = u, d) to be diagonal unitary (phase) matrices. Then, if instead of using 1/qL and VqR for the rotation (32) to the mass basis, we use VqL and VqR, defined by VqL = Pq 1/4 and VqR = PqVqR, we still maintain a legitimate mass basis since Mid'ag remains unchanged by such transformations. However, VcK m does change: VC K M Pu VC K M Pd* • (36) This freedom is fixed by demanding that Vc K m has the minimal number of phases. In the three generation case Vc K m has a single phase. (There are five phase differences between the elements of 131, and Pd and, therefore, five of the six phases in the CKM matrix can be removed.) This is the Kobayashi-Maskawa phase 6KM which is the single source of CP violation in the quark sector of the Standard Model [1]. As a result of the fact that Vc K m is not diagonal, the IV gauge bosons couple to quark (mass eigenstates) of different generations. Within the Standard Model, this is the only source of flavor changing quark interactions. 2.3 The three phases in the MNS mixing matrix The leptonic Yukawa interactions (23) and (24) give rise to mass terms: = (11,),34,43 + (111,),, + h.c., where vM2 = Mv , 2 r v, and we decomposed the SU(2)L lepton doublets into their components: vi L = Li . Li We can always find unitary matrices VeL and V, such that VeL Me g' = diag(m 2e 77,121, ), xtmixj = diag(n4, (40) The charged current interactions for leptons, which in the interaction basis are described by (17), have the following form in the mass basis: = + h.c.. (41) The unitary 3 x 3 matrix, VmNs = VeLVt , (42) is the Maki-Nakagawa-Sakata (MNS) mixing matrix for leptons [36]. As for the CKM matrix, the form of the MNS matrix is not unique: there are the same two freedoms but they are fixed by different conventions: (37) (38) (39) sucia s13e-th C12C23 s12s23.913ez S23C13 ,oi45 — C12823 — Si2C23,313c, L-23.-13 (46) CP Violation 157 1. We can permute between the various generations. This freedom is usually fixed in the following way. We order the charged leptons by their masses, i.e. (el , e2, e3) (e, For the neutrinos, one takes into account that the interpretation of atmo- spheric and solar neutrino data in terms of two-neutrino oscillations implies that OmATM >> Am6. It follows that one of the neutrino mass eigenstates is separated in its mass from the other two, which have a smaller mass difference. The conven-tion is to denote this separated state by v3. For the remaining two neutrinos, vi and v2, the convention is to call the heavier state v2. In other words, the three mass eigenstates are defined by the following conventions: >> Om21 > 0. (43) Note in particular that v3 can be either heavier or lighter than v1,2. The elements of VMNS are written as follows: VMNS = (Vel Vet Vp1 Vp2 VT1 V72 Ve3 Vp3 VT3 (44) 2. There is further freedom in the phase structure of VMNS. (In the MNS paper [36] there is no reference to CP violation.) One can change the charged lepton mass basis by the transformation e(L, R), e( L,R)z = (P,)„e(L, R),, where Pe is a phase matrix. There is, however, no similar freedom to redefine the neutrino mass eigenstates: From Equation (37) one learns that a transformation Pvvi, will introduce phases into the diagonal mass matrix. This is related to the Majorana nature of neutrino masses, assumed in Equation (24). The allowed transformation modifies VMNS: VMNS PeVMNS• (45) This freedom is fixed by demanding that VMNS will have the minimal number of phases. Out of six phases of a unitary 3 x 3 matrix, the multiplication by Pe can be used to remove three. We conclude that the three generation VmNs matrix has three phases. One is the analog of the Kobayashi-Maskawa phase: it is the only source of CP violation in processes that conserve lepton number, such as neutrino flavor oscillations. The other two phases can affect lepton number changing processes. With VMNS 1, the W± gauge bosons couple to lepton (mass eigenstates) of different generations. Within the ESM, this is the only source of flavor changing lepton interactions. 2.4 The flavor parameters Examining the quark mass basis, one can easily identify the flavor parameters. In the quark sector, we have six quark masses and four mixing parameters: three mixing angles and a single phase. The fact that there are only three real and one imaginary physical parameters in VcKm can be made manifest by choosing an explicit parameterisation. For example, the standard parameterisation [37], used by the Particle Data Group, is given by Ci2C13 VCKM = ( — 812C23 — c12823s13e S12823 — ci2c23.913eth 158 Yosef Nir where C,3 E- cos 0,3 and sii E sin O. The three sin 0,3 are the three real mixing parameters while S is the Kobayashi-Maskawa phase. Another, very useful, example is the Wolfenstein parameterisation, where the four mixing parameters are (A, A, p, q) with A = = 0.22 playing the role of an expansion parameter and ij representing the CP violating phase [38]: VCKM = —A 1 — A2 T A'2 + 0(A4 ). (47) \ AA3(1 — p — in) —AA2 1 / Various parameterisations differ in the way that the freedom of phase rotation, Equa-tion (36), is used to leave a single phase in VcKm. One can define, however, a CP violating quantity in VcKm that is independent of the parameterisation [34]. This quantity, JCKM, is defined through 3 = JCKM E fikrriejln, (i, j,k,1 = 1,2,3). (48) m,n=1 In terms of the explicit parameterisations given above, we have JCKM = Cl2C23C13812,923,913 sin (5 A6A2i1. (49) It is interesting to translate the condition (20) to the language of the flavor parameters in the mass basis. One finds that the following is a necessary and sufficient condition for CP violation in the quark sector of the SM: Anizt,Am2tuAnic2uAinLA74, d'-1 A mseiCKM 0. Here AM?. -= 770 — zi — 3 Equation (50) puts the following requirements on the SM in order that it violates CP: . Within each quark sector, there should be no mass degeneracy; . None of the three mixing angles should he zero or 7r/2; . The phase should be neither 0 nor 7r. For the lepton sector of the ESM, the flavor parameters are the six lepton masses, and six mixing parameters: three mixing angles and three phases. One can parameterise VmNs in a convenient way by factorising it into VIANS = VP. Here P is a diagonal unitary matrix that depends on two phases, e.g. P = diag(ez°',0'2 , 1), while V can be parameterised in the same way as (46). The advantage of this parameterisation is that for the purpose of analyzing lepton number conserving processes and, in particular, neutrino flavor oscillations, the parameters of P are usually irrelevant and one can use the same Chau-Keung parameterisation as is being used for VCKM. (An alternative way to understand these statements is to use a single-phase mixing matrix and put the extra two phases in the neutrino mass matrix. Then it is obvious that the effects of these `Majorana-phases' always appear in conjunction with a factor of the Majorana mass that is a lepton 1— A2 A AA3(p — i7)\ -2- (50) (51) VtdVtb = V(1 — p)2 + 7/2 . (56) VudVub Rai Vp2 n2, Rt VedVcb VcdVcb CP Violation 159 number violating parameter.) On the other hand, the Wolfenstein parameterisation (47) is inappropriate for the lepton sector: it assumes 117231 << V121 << 1, which does not hold here. In order that the CP violating phase S in V would be physically meaningful, i.e. there would be CP violation that is not related to lepton number violation, a condition similar to (50) should hold: A ,,,,,,2 A „,,,2 A A 2 A ,r.r.,2 A 7 `31".M22 1 MNS 0. (52) 2.5 The unitarity triangles A very useful concept is that of the unitarity triangles. We will focus on the quark sector, but analogous triangles can be defined in the lepton sector. The unitarity of the CKM matrix leads to various relations among the matrix elements, e.g. lildVu*s + 17cd 17c4.: + Vtd 17t's — (53) s V:b Velt, = 0, (54) Vu d Vu*b + d cb + Vtd Vt': = 0. (55) Each of these three relations requires the sum of three complex quantities to vanish and so can be geometrically represented in the complex plane as a triangle. These are "the unitarity triangles", although the term "unitarity triangle" is usually reserved for the relation (55) only. It is a surprising feature of the CKM matrix that all unitarity triangles are equal in area: the area of each unitarity triangle equals IJcKMI/2 while the sign of Jcicm gives the direction of the complex vectors around the triangles. The resealed unitarity triangle is derived from (55) by (a) choosing a phase convention such that (Vcd Vcn is real, and (b) dividing the lengths of all sides by 1Vcd Ve*,1. Step (a) aligns one side of the triangle with the real axis, and step (b) makes the length of this side 1. The form of the triangle is unchanged. Two vertices of the resealed unitarity triangle are thus fixed at (0,0) and (1,0). The coordinates of the remaining vertex correspond to the Wolfenstein parameters (p, ii). The area of the resealed unitarity triangle is Ii /2. Depicting the resealed unitarity triangle in the (p, n) plane, the lengths of the two complex sides are The three angles of the unitarity triangle are defined as follows [39, 40]: a _= ar f VtdVt'6 [ VcdVc: [ VudVu*b 1 7 Tr* , arg 7 arg (57) v ud v ub V 17 d ' Id tb I7cdVe't] They are physical quantities and can be independently measured by CP asymmetries in B decays [41, 42, 43, 44, 45]. It is also useful to define the two small angles of the unitarity triangles (54) and (53): Os arg [Vts Vtbl oK = arg Vcs vcd VcsVcb VnsVud • (58) 160 Yosef Nir 11 2 Figure 1. Present Standard Model constraints and the result from the global CKM fit. To make predictions for future measurements of CP violating observables, we need to find the allowed ranges for the CKM phases. There are three ways to determine the CKM parameters (see e.g. [46]): . Direct measurements are related to SM tree level processes. At present, we have direct measurements of Yuctl, IVu81, 114Lbi, 117ccii, 1Vebi and • CKM Unitarity (QKm VcKm = 1) relates the various matrix elements. At present, these relations are useful to constrain IVtdi, 114,1, IV and I'Ved. . Indirect measurements are related to SM loop processes. At present, we con-strain in this way I14bVdI (from AmB and AmB,) and 5KM or, equivalently, i or (from eK and aoKs )• When all available data are taken into account, one finds [47]: A p sin 2/3 = = = 0.2221 ± 0.0021, A -= 0.827 ± 0.058, 0.23 ± 0.11, i = 0.37 ± 0.08, 0.77 ± 0.08, sin 2a = —0.21 ± 0.56, 0.43 < sine 'y < 0.91. (59) (60) (61) Of course, there are correlations between the various parameters. The full information in the (p, 77) plane is given in Figure 1 [47]. CP Violation 161 2.6 Uniqueness of the standard model picture of CP violation In the previous subsections, we have learnt several features of CP violation as explained by the Standard Model. It is important to understand that various reasonable (and often well-motivated) extensions of the SM provide examples where some or all of these features do not hold. Furthermore, until a few years ago, none of the special features of the Kobayashi-Maskawa mechanism of CP violation was experimentally tested. This situation has dramatically changed recently. Let us survey some of the SM features, how they can be modified with new physics, and whether experiment has shed light on these questions. 1. 6KM is the only source of CP violation in meson decays. This is arguably the most unique feature of the SM and gives the model a strong predictive power. It is violated in almost any low-energy extension. For example, in the supersymmetric extension of the SM there are 44 physical CP violating phases, many of which affect meson decays. The measured value of aoKs is consistent with the correlation between K and B decays that is predicted by the SM. It is therefore very likely that SR 1,4 is indeed the dominant source of CP violation in meson decays. 2. CP violation is small in K 77 decays because of flavor suppression and not because CP is an approximate symmetry. In many (though certainly not all) super-symmetric models, the flavor suppression is too mild, or entirely ineffective, requiring approximate CP to hold. The measurement of av,K, = 0(1) confirms that not all CP violating phases are small. 3. CP violation appears in both OF = 1 (decay) and OF = 2 (mixing) amplitudes. Superweak models suggest that CP is violated only in mixing amplitudes. The measurement of EYE confirms that there is CP violation in AS = 1 processes. 4. CP is not violated in the lepton sector. Models that allow for neutrino masses, such as the ESM framework presented above, predict CP violation in leptonic charged current interactions. The data from neutrino oscillation experiments makes it very likely that charged current weak interactions violate CP also in the lepton sector. 5. CP violation appears only in the charged current weak interactions and in conjunc-tion with flavor changing processes. Here both various extensions of the SM (such as supersymmetry) and non-perturbative effects within the SM (0QcD) allow for CP violation in other types of interactions and in flavor-diagonal processes. In particu-lar, it is difficult to avoid flavor-diagonal phases in the supersymmetric framework. The fact that no electric dipole moment has been measured yet poses difficulties to many models with diagonal CP violation (and, of course, is responsible to the strong CP problem within the SM). 6. CP is explicitly broken. In various extensions of the scalar sector, it is possible to achieve spontaneous CP violation. It will be very difficult to test this question experimentally. This situation, where the Standard Model has a very unique and predictive description of CP violation and the number of experimentally measured CP violating observables is 162 Yosef Nir very limited (E K , E7E and aoKs ), is the basis for the strong interest, experimental and theoretical, in CP violation. There are two types of unambiguous tests concerning CP violation in the Standard Model: First, since there is a single source of CP violation, all observables are correlated with each other. For example, the CP asymmetries in B OKs and in K R-v0 are strongly correlated [48, 49, 50]. Second, since CP violation is restricted to flavor changing fermion processes, it is predicted to be highly suppressed in the lepton sector and practically vanish in flavor diagonal processes. For example, the transverse lepton polarisation in semileptonic meson decays, CP violation in tt production, and (assuming eQcD = 0) the electric dipole moment of the neutron are all predicted to be orders of magnitude below the (present and near future) experimental sensitivity. We conclude that it is very important to search for CP violation in many different systems. 3 Meson decays In the previous section, we explained how CP violation arises in the Standard Model. In the next three sections, we would like to understand the implications of this theory for the phenomenology of CP violation in K, D and B decays. To do so, we first present a model-independent analysis of CP violation in meson decays. We distinguish between three different types of CP violation in meson decays: . CP violation in mixing, which occurs when the two neutral mass eigenstate admix-tures cannot be chosen to be CP-eigenstates; . CP violation in decay, which occurs in both charged and neutral decays, when the amplitude for a decay and its CP-conjugate process have different magnitudes; . CP violation in the interference of decays with and without mixing, which occurs in decays into final states that are common to B° and B°. 3.1 Notations and formalism To define these three types and to discuss their theoretical calculation and experimental measurement, we first introduce some notation and formalism. We refer specifically to B meson mixing and decays, but most of our discussion applies equally well to K, B, and D mesons. A B° meson is made from a b-type antiquark and an d-type quark, while the B° meson is made from a b-type quark and an d-type antiquark. Our phase convention for the CP transformation law of the neutral B mesons is defined by CPIB°)= welB°), C131 13°) = wLIB°), (Pei = 1)- (62) Physical observables do not depend on the phase factor WB. The light, BL, and heavy, BH, mass eigenstates can be written as linear combinations of B° and B°: IBL) = PIB°) + qIB°), 1BH) = PI B°) — 41B°), (63) 163 (64) (65) (66) (67) CP Violation with lq12 1 p12 = 1. The mass difference Ame and the width difference ArB are defined as follows: MI/ — OP-PH —PL. The average mass and width are given by MH FH +FL 772 B -7- 2 Fe = 2 It is useful to define dimensionless ratios x and y: The time evolution of the mass eigenstates is simple: BH(t)) e-imH te-r H011311), IBL(t)) e-imoe-ri,e121BL). (68) The time evolution of the strong interaction eigenstates is complicated and obeys a Schrodinger-like equation, (m ii2r) (g)' (69) where M and r are 2 x 2 Hermitian matrices. The off-diagonal terms in these matrices, M12 and F12, are particularly important in the discussion of mixing and CP violation. M12 is the dispersive part of the transition amplitude from B° to B°, while F12 is the absorptive part of that amplitude. Solving the eigenvalue equation gives (Am)2 — 41 (AF)2 = (41M1212 — IF1212), AmAF =- 4Re(M121- 2), (70) q2V2 — Am— AF = — (71) P Om— 2m1.2 —iF12 In the B system, 11'121 << 1M121 (see discussion below), and then, to leading order in ir12/M121, Equations (70) and (71) can be written as AMB = 21M121, 2Re(M12112) AFB = (72) 1 11/121 MI; P IM121 . To discuss CP violation in mixing, it is useful to write Equation (71) to first order in Ir12/m121 [rather than to zeroth order as in (73)]: q P 1W2 [ [1 1 T ( r12 )1 (74) IM121 2illi Ut/112) AP Y 2r • (73) 164 Yosef Nir To discuss CP violation in decay, we need to consider decay amplitudes. The CP transformation law for a final state f is CP1f) = wilf), CiDif) = 1). (75) For a final CP eigenstate f = f = fcp, the phase factor wJ is replaced by i1cP = ±1, the CP eigenvalue of the final state. We define the decay amplitudes Af and A f according to AJ = Af = (f IndiB°), (76) where lid is the decay Hamiltonian. CP relates A f and AI. There are two types of phases that may appear in Af and Af. Complex parameters in any Lagrangian term that contributes to the amplitude will appear in complex conjugate form in the CP-conjugate amplitude. Thus their phases appear in Ai. and Af with opposite signs. In the SM these phases occur only in the mixing matrices that parameterise the charged current weak interactions, hence these are often called "weak phases". The weak phase of any single term is convention dependent. However the difference between the weak phases in two different terms in A f is convention independent because the phase rotations of the initial and final states are the same for every term. A second type of phase can appear in scattering or decay amplitudes even when the Lagrangian is real. Such phases do not violate CP and they appear in Af and Al with the same sign. Their origin is the possible contribution from intermediate on-shell states in the decay process, that is an absorptive part of an amplitude that has contributions from coupled channels. Usually the dominant re-scattering is due to strong interactions and hence the designation "strong phases" for the phase shifts so induced. Again only the relative strong phases of different terms in a scattering amplitude have physical content, an overall phase rotation of the entire amplitude has no physical consequences. Thus it is useful to write each contribution to A in three parts: its magnitude Az; its weak phase term eul; and its strong phase term ez . Then, if several amplitudes contribute to B f, we have A f A f Ei Aiej(5,-0,) (77) To discuss CP violation in the interference of decays with and without mixing, we introduce a complex quantity Af defined by q A f A f = p - —A • (78) .11 f We further define the CP transformation law for the quark fields in the Hamiltonian (a careful treatment of CP conventions can be found in [51]): q wqq, 9 -4 w9 4, (Voq = 1)- (79) The effective Hamiltonian that is relevant to M12 is of the form 1 -2i°B [fry/1(1 -Y5 )d]2 Hy-2 a e+2i08 [d,yi, (1 - ,y5 )1)] 2 + e where 20B is a CP violating (weak) phase. (We use the SM V-A amplitude, but the results can be generalised to any Dirac structure.) For the B system, where 1F121 << 1M1211 this leads to Q = -2i0B = wBwb wde • (80) (81) CP Violation 165 (We implicitly assumed that the vacuum insertion approximation gives the correct sign for M12. In general, there is a sign(BB) factor on the right hand side of Equation (81) [52].) To understand the phase structure of decay amplitudes, we take as an example the b qgcl decay (q =- u or c). The decay Hamiltonian is of the form Hd OC e±i°f [04(1 - 75)d] [b7,(1 - -y5)q] + e-'47 [q7P(1 - 75)1)] [d7,(1 - 75)q] , (82) where O f is the appropriate weak phase. (Again, for simplicity we use a V-A structure, but the results hold for any Dirac structure.) Then A - f = * „* _ -220 A —w f w Bwbuid ' • Equations (81) and (83) together imply that for a final CP eigenstate, = nfepe2i(OBA-0j). 3.2 The three types of CP violation in meson decays 3.2.1 CP violation in mixing lq/p1 1. (85) This type of CP violation results from the mass eigenstates being different from the CP eigenstates, and requires a relative phase between M12 and F12 . For the neutral B system, this effect could be observed through the asymmetries in semileptonic decays: r(13°,hys(t) f+ vX) -F(Bp°hys (t) t-vX) asL = F(Bghys(t) e-EvX)+F(g;hys (t) f-vX) . (86) In terms of q and p, 1 - 14/P14 asL = (87) 1 + 14/Pie CP violation in mixing has been observed in the neutral K system (Re EK 0). In the neutral B system, the effect is expected to be small, < 0(10-2). The reason is that, model independently, the effect cannot be larger than 0(AFB/AmB). The difference in width is produced by decay channels common to B° and B°. The branching ratios for such channels are at or below the level of 10-3. Since various channels contribute with differing signs, one expects that their sum does not exceed the individual level. Hence, we can safely assume that AFB/I B = 0(10-2). On the other hand, it is experimentally known that AmB/F B 0.7. To calculate asL, we use (87) and (74), and get: an = IM(r12/M12)• (88) To predict it in a given model, one needs to calculate M12 and F12 . This involves large hadronic uncertainties, in particular in the hadronisation models for F12. (83) (84) 166 Yosef Nir 3.2.2 CP violation in decay (89) This appears as a result of interference among various terms in the decay amplitude, and will not occur unless at least two terms have different weak phases and different strong phases. CP asymmetries in charged B decays, r(B+ f+) — r(B- f -) aft F(B+ f+)+F(B- f -)' are purely an effect of CP violation in decay. In terms of the decay amplitudes, 1— lAf - A1-1-1 2 CP violation in decay has been observed in the neutral K system (Re EiK # 0). To calculate a ft, we use (91) and (77). For simplicity, we consider decays with con-tributions from two weak phases and with A2 << A1 . We get: af t = —2(A2 /A1 ) sin(62 — (51) sin(02 — 01)• (92) The magnitude and strong phase of any amplitude involve long distance strong interaction physics, and our ability to calculate these from first principles is limited. Thus quantities that depend only on the weak phases are much cleaner than those that require knowledge of the relative magnitudes or strong phases of various amplitude contributions, such as CP violation in decay. 3.2.3 CP violation in interference between decays with and without mixing Im Af, 0. (93) This effect is the result of interference between a direct decay amplitude and a first-mix-then-decay path to the same final state. For the neutral B system, the effect can be observed by comparing decays into final CP eigenstates of a time-evolving neutral B state that begins at time zero as B° to those of the state that begins as B°: F(flthys (t) fcp) — F(Bi°hys(t) fcp) afcp (t) = (94) F(Yr)thys (t) fcp) + F(g;hys (t) —> fcp) This time dependent asymmetry is given, in general, by 1 — lAk.p 1 2 2ImAf„ a fep(t) = 2 COS(AMBt) NCP1 1 ± Afcp12 sin(AmBt). I In decays with Afcp = 1, (93) is the only contributing effect: a f „(t) = ImAf sin(Am Bt). We often use _ 2ImAf„ af(P — 1 + 1A /CP aft = 1 + I Af- I A f+1 2. (90) (91) (95) (96) (97) CP Violation 167 CP violation in the interference of decays with and without mixing has been observed for the neutral K system (Im EK 0) and for the neutral B system (aibK„, 0 0). In the latter, it is an effect of 0(1). For such cases, the contribution from CP violation in mixing is clearly negligible. For decays that are dominated by a single CP violating phase (for example, B MKS and KL 71-°v0), so that the contribution from CP violation in decay is also negligible, af„ is cleanly interpreted in terms of purely electroweak parameters. Explicitly, ImAf„ gives the relative phase between the B — B mixing amplitude and the relevant decay amplitudes [see Equation (84)]: Im 10 = -11k,p Si11[2(0B + Of)]. (98) 3.2.4 Direct and indirect CP violation The terms indirect CP violation and direct CP violation are commonly used in the lit-erature. While various authors use these terms with different meanings, the most useful definition is the following: . Indirect CP violation refers to CP violation in meson decays where the CP violating phases can all be chosen to appear in OF = 2 (mixing) amplitudes. . Direct CP violation refers to CP violation in meson decays where some CP violating phases necessarily appear in OF = 1 (decay) amplitudes. Examining Equations (85) and (71), we learn that CP violation in mixing is a manifes-tation of indirect CP violation. Examining Equations (89) and (76), we learn that CP violation in decay is a manifestation of direct CP violation. Examining Equations (93) and (78), we learn that the situation concerning CP violation in the interference of de-cays with and without mixing is more subtle. For any single measurement of ImAf 0 0, the relevant CP violating phase can be chosen by convention to reside in the OF = 2 amplitude [O f = 0, OB 0 in the notation of Equation (84)], and then we would call it indirect CP violation. Consider, however, the CP asymmetries for two different final CP eigenstates (for the same decaying meson), fa and fb. Then, a non-zero difference between ImAf“ and ImAfi, requires that there exists CP violation in AF = 1 processes (0 f,, — O f, 0 0), namely direct CP violation. Experimentally, both direct and indirect CP violation have been established. Below we will see that E K signifies indirect CP violation while E', signifies direct CP violation. Theoretically, most models of CP violation (including the Standard Model) have pre-dicted that both types of CP violation exist. There is, however, one class of models, that is superweak models, that predict only indirect CP violation. The measurement of E'K $ 0 has excluded this class of models. 4 K decays Measurements of CP violation have played an enormous role in particle physics. First, the measurement of E K in 1964 provided the first evidence that CP is not a symmetry of Nature. This discovery revolutionised the thinking of particle physicists and was essential (1ro7ro In KL (99) (7+71-- INIKL) (70-71-- NIKs) . noo = (7o7ro Define, for (ij) = (00) or (+-), 168 Yosef Nir for understanding baryogenesis. Second, the measurement in 1988 of e provided the first evidence for direct CP violation and excluded the superweak scenario. In the future, the search for CP violation in K -4 rrvi) decays will add significantly to our understanding of CP violation. 4.1 eK and eiK Historically, a different language from the one used by us has been employed to describe CP violation in K 7r71- and K rrev decays. In this section we 'translate' the language of E K and E1K to our notation. Doing so will make it easy to understand which type of CP violation is related to each quantity. The two CP violating quantities measured in neutral K decays are = ( Rj1 (110), A. - q z 7jINIK°), = (-) K Ai (100) Then 1 - A00 1 - A+_ noo 71+- = (101) 1 + Aoo 1 + Ad • The r100 and ri_F _ parameters get contributions from CP violation in mixing (1(q/p)1K 1) and from the interference of decays with and without mixing (ImA23 0) at 0(10-3) and from CP violation in decay (124„/A23 1 1) at 0(10-6). There are two isospin channels in K ->71-7 leading to final (27r) /=0 and (27)1-2 states: (7°7°I = 11/3((77)/=o1 V2/3((77)/=21, (7+7-1 = V2/3((77)/=o1+1/1/3((77)/=21. (102) The fact that there are two strong phases allows for CP violation in decay. The possible effects are, however, small (on top of the smallness of the relevant CP violating phases) because the final I = 0 state is dominant (this is the A/ = 1/2 rule). Define = ((rrr)/17-11K°), Al = ((r7r)/17illf°), = )p (103) Experimentally, I A2 /A0 I 1/20. Instead of ri00 and rid__ we may define two combinations, E K and Ex , in such a way that the possible effects of direct (indirect) CP violation are isolated into Eli( (ex). The experimental definition of the ex parameter is 1 E K3(77oo + 2n+-). (104) CP Violation 169 The experimental value is given by Equation (1). To zeroth order in A2 /A0, we have No = 77+- = E K . However, the specific combination (104) is chosen in such a way that the following relation holds to first order in A2/Ao: 1 — Ao eK 1 + Ao • Since, by definition, only one strong channel contributes to Ao, there is indeed no CP violation in decay in (105). It is simple to show that Re eK 0 is a manifestation of CP violation in mixing while Im E K 0 0 is a manifestation of CP violation in the interference between decays with and without mixing. Since experimentally arg E K 7r/4, the two contributions are comparable. It is also clear that EK 0 0 is a manifestation of indirect CP violation: it could be described entirely in terms of a CP violating phase in the M12 amplitude. The experimental definition of the Eli( parameter is E K The quantity that is actually measured 1 — The world average is given in Equation , EK , in No 1/00)• experiment is 2 = 6Re(E'/E). The theoretical expression is oo — A+-)• (2). 6(A (106) (107) (108) 71+- 1 Obviously, any type of CP violation which is independent of the final state does not contribute to eK. Consequently, there is no contribution from CP violation in mixing to (108). It is simple to show that Re EK $ 0 is a manifestation of CP violation in decay while Im 0 is a manifestation of CP violation in the interference between decays with and without mixing. Following our explanations in the previous section, we learn that E'K 0 0 is a manifestation of direct CP violation: it requires 02 — 00 0 0 [where 0/ is the CP violating phase in the Al amplitude defined in (103)]. 4.1.1 The eK parameter in the Standard Model An approximate expression for E K , that is convenient for calculating it, is given by ezr/4 EK = AMK (109) A few points concerning this expression are worth emphasising: • Equation (109) is given in a specific phase convention, where A2 is real. Within the SM, this is a phase convention where 17,,„V,4*, is real, a condition fulfilled in both the standard parameterisation of Equation (46) and the Wolfenstein parameterisation of Equation (47). (105) 170 Yosef Nir . The phase of 714 is approximate. It is determined by hadronic parameters and therefore is independent of the electroweak model. Specifically, arg(E K ) arctan(-2AmK /AFK) 7r/4. (110) • A term of order 2 Im A Re M12 < 0.02 is neglected when (109) is derived. tLeHp Im M12 . There is a large hadronic uncertainty in the calculation of M12 coming from long distance contributions. There are, however, good reasons to believe that the long distance contributions are important in Re M12 (where they could be even com-parable to the short distance contributions), but negligible in Im M12. To avoid this uncertainty, one uses ImM12 /AmK with the experimentally measured value of AmK , instead of ImM12 /2Re M12 with the theoretically calculated value of Re M12. • The matrix element (k° I (M)v_A(gcl)V-A1-10) is yet another source of hadronic un-certainty. If both Im M12 and Re M12 were dominated by short distance contribu-tions, one would use the ratio ImM12 /Re M12 where the matrix element cancels out. However, as explained above, this is not the case. Within the Standard Model, Im M12 is accounted for by box diagrams. We follow here the notations of reference [53], where precise definitions, numerical values and appropriate references are given. One obtains: E K = e"CEBK InI(Vts* Vtd) Re(Vc*, Ved)[771S0(xe) — xt )] — Re(1.71*, 17,d )772So (xt) (111) r2 where CE G is a well known parameter, the 7i, are QCD correction factors, 6.7r2ArnK So is a kinematic factor, and BK is the ratio between the matrix element of the four quark operator and its value in the vacuum insertion approximation. We would like to emphasise the following points: . CP violation was discovered through the measurement of EK. Hence this measure-ment played a significant role in the history of particle physics. . For a long time, EK has been the only measured CP violating parameter. Roughly speaking, this measurement set the value of 8KM (and, by requiring 5KM = 0(1), made the KM mechanism plausible) but could not serve as a test of the KM mech- anism. (More precisely, a value of 1EK 10-3 would have invalidated the KM mechanism, but any value IEK1< 10-3 was acceptable.) It is only the combination of the new measurement of aoKs with E K that provides the first precision test of the KM mechanism. . Within the SM, the smallness of E K is not related to suppression of CP violation but rather to suppression of flavor violation. Specifically, it is the smallness of the ratio 1(VtdVts)/(VudV..01,-,, that explains 1EK I ti 10-3. . Until recently, the measured value of EK provided a unique type of information on the CKM phase. For example, the measurement of sign(Re E K ) > 0 tells us that > 0 and excludes the lower half of the p — 71 plane. Such information cannot be obtained from any CP conserving observable. CP Violation 171 . The E K constraint gives hyperbolae in the p — ri plane. It is shown in fig. 1. The measured value is consistent with all other CKM-related measurements and further narrows the allowed region. . The main sources of uncertainty are in the BK parameter, BK = 0.85 + 0.15, and in the 117,b 4 dependence. . E K is an extremely powerful probe of new physics. Its small value poses a problem to any model of new physics where the flavor suppression is less efficient than the GIM mechanism [54] of the SM. For example, the construction of viable supersymmetric models is highly constrained by the requirement that they do not give contributions that are orders of magnitude higher than the experimental value. 4.1.2 The ex- parameter in the Standard Model Direct CP violation in K 7r7r decays was first measured in 1988 [3]. Two recent measurements achieved impressive accuracy: E' (20.7 ± 2.8) x 10' KTeV [6], (112) E (15.3 ± 2.6) x 10-4 NA48 [7]. In combination with previous results [4, 5], the present world average has an accuracy of order 10% [see Equation (2)]. A convenient approximate expression for E'K is given by: A2 T, EK = v 2 el(62-6°) sin(02 — 0o). (113) Note that: . The approximations used in (113) are lq/p1 = 1 and 1A2 /A01 << 1. . The phase of EiK is determined by hadronic parameters and is, therefore, model independent: arg(EK) = 7r/2+ 62 — 60 7r/4. The fact that, accidentally, arg(EK ) arg(6',), means that Re(E'/E) e' /e. (114) . Re eK 0 0 requires 62 — 60 0, consistent with our statement that it is a manifes- tation of CP violation in decay. eK 0 0 requires 02 — 00 0, consistent with our statement that it is a manifestation of direct CP violation. The calculation of EYE within the Standard Model suffers from large hadronic uncertain- ties. A very naive order of magnitude estimate gives e' /e (A212-10)(Ar,enguin /Atoree) 10-3. Note that E'/E is not small because of small CP violating parameters but because of hadronic parameters. The value of the phase 13K cancels in the ratio E' /E and therefore did not affect our estimate. In actual calculations, one usually uses the experimental value of E h and the theoretical expression for EK. Then the expression for E'/E depends on the CP violating phase. 172 Yosef Nir The detailed calculation of EyE is complicated. There are several comparable contri-butions with differing signs. The final result can be written in the form (for details and references, see [53]): = in(Vtdvts*) [p(I/2) p(3/2)] 13 ImAMs (Vid V„*) ( 7-?2,(2 GeV))) 340 MeV 110 MeV 2 [ki/2)( i_ Q„„,) — 0.4/43/21 ( n it 2.5 165 GeV) } • We have omitted here a phase factor using the approximation arg(EK) = arg(E'K). P(1 /2) , which is dominated by QCD penguins, gives the contributions from A/ = 1/2 transitions, while P(3/2) , which is dominated by electroweak penguins, gives the con-tributions from A/ = 3/2 transitions. The E3 112) and /4312) factors parameterise the corresponding hadronic matrix elements. The QCD penguin contributions are suppressed by isospin breaking effects (mu # md), parameterised by The resulting estimates vary in the range [53]: Re(E'/E)sm = (0.5 — 4) x 10-3. (116) We would like to emphasise the following points: . Direct CP violation was discovered through the measurement of e'. . The SM range (116) is consistent with the experimental result (2). . The main sources of uncertainties lie then in the parameters ms, f41/2) , .k312) , S-2,±„, and A. The importance of these uncertainties is increased because of the cancellation between the two contributions in (115). . The large hadronic uncertainties make it difficult to use the experimental value of E'/E to constrain the CKM parameters. Still, a negative value or a value much smaller than 10' would have been very puzzling in the context of the SM. . The experimental result is useful in probing and constraining new physics. 4.2 CP violation in K 71-vv Observing CP violation in the rare K 7v0 decays would be experimentally very chal- lenging and theoretically very rewarding. It is very different from the CP violation that has been observed in K 717 decays which is small and involves theoretical uncertainties. Similar to the CP asymmetry in B z/iK5, it is predicted to be large and can be cleanly interpreted. Furthermore, observation of the KL ev0 decay at the rate predicted by the Standard Model will provide further evidence that CP violation cannot be attributed to mixing (AS = 2) processes only, as in superweak models. Define A,CW (7°P1) 14 IK°), Aicov, = (7r°vP17i Ar„ = (q ) AevI7. (117) p K A,o„ (115) CP Violation 173 The ratio between the neutral K decay rates is then F(KL 7r°v0) 1 — 2ReA„,, F(Ks — 1 + IA„p12 + 2ReA„p We learn that the KL 71-°vP decay rate vanishes in the CP limit (Airup = 1), as expected on general grounds [55]. (The CP conserving contributions were explicitly calculated within the Standard Model [56] and within its extensions with massive neutrinos [57] and with extra scalars [58] and found to be negligible.) CP violation in decay and in mixing are expected to be negligibly small, of order 10-5 and 10-3, respectively. Consequently, A„,7 is, to an excellent approximation, a pure phase. Defining 20K to be the relative phase between the K—K mixing amplitude and twice the s e2zOK, dvP decay amplitude, namely A„,? = we get from (118): evP) = tan2 OK. (119) F(Ks 7°i/0 Using the isospin relation A(K° 71-°v0)/A(K+ 7r+vP) = 1/V2, we get F(K L —> air., -7- = -sin BK . F(K+ -> ir+vi7) (120) The present experimental searches give B(K+ ir+v0) = (1.5tn x 10-10 [59], 13(KL --÷7°v0) < 5.9 x 10-7 [60]. (121) Equation (120) implies that a„, < 1. This inequality is based on isospin considera- tions only. Consequently a measurement of F(K+ 71-+vP) can be used to set a model independent upper limit on F(KL 7r°v0) [61]: 13(KL -4 7TO VP) < 4.4 B(K± R-+vo). (122) From the range given in (121) for the K± decay, the isospin bound on the K L decay is 13(KL 71-°vP) < 2.6 x 10-9, which is more than two orders of magnitude below the direct bound. Within the Standard Model, the K 71-v0 decays are dominated by short distance Z-penguins and box diagrams and can be expressed in terms of p and n (see [53] for details and references) B(K± 7r+vO) = 4.11 x 10-11[x(xt)]2A4 [772 + (P0 10)21 13(KL 71-°v0) = 1.80 x 10-10[x(x0]2A4,02. (123) P0(h) Here Po = 1+ A2 Px xt) , and X(xt ) and P0(X) represent the electroweak loop contributions in NLO for the top quark and for the charm quark, respectively. We would like to emphasise the following points: (118) 174 Yosef Nir . The K 7v0 decays are theoretically clean. The main theoretical uncertainty in the K+ decay is related to the strong dependence of the charm contribution on the renormalisation scale and the QCD scale, Po(X) = 0.42 + 0.06. The K L decay has hadronic uncertainties smaller than a percent. . In the future, these decays will provide excellent p- ri constraints. . Present constraints on the CKM parameters give the SM predictions [47]: B(K+->7r± v0) = (7.0 ± 1.9) x 10-11 B(K L->7° N7) = (2.9 + 1.1) x 10-11. (124) The experimental range for the K+ decay (121) is then consistent with the SM but not yet accurate enough to constrain it, while the experimental bound on the K1, decay is still four orders of magnitude above the SM range. . The CP violations in K Tv!) and in B OKs are strongly correlated and can provide the most stringent test of the Kobayashi-Maskawa mechanism. . The K 7rvt) decays are interesting probes of CP violation related to new physics. 5 D decays Within the Standard Model, D-D mixing is expected to be well below the experimental bound. Furthermore, effects related to CP violation in D-D mixing are expected to be negligibly small since this mixing is described to an excellent approximation by physics of the first two generations. An experimental observation of D-D mixing close to the present bound or, more strongly, of related CP violation, will then be evidence for New Physics. To explain how D-D mixing is searched for and how CP violation can be signalled, we use notation similar to that of the B system. We thus use Equation (63) to define the two mass eigenstates 1/31,2), Equation (66) to define the average width r, Equation (67) to define the width and mass differences y and x, Equation (76) to define the decay amplitudes Af and Af and Equation (78) to define Af . 5.1 D K 7r and D -+ K K decays Processes that are relevant to the most sensitive measurements at present are the doubly-Cabibbo-suppressed D°->K+R-- decay, the singly-Cabibbo-suppressed D°-+K+K- decay, the Cabibbo-favored D°-4(-7r+ decay, and the three CP-conjugate decay processes. We follow here the analysis presented in reference [62]. We write down approximate expres-sions for the time-dependent decay rates that are valid for times t < 1/F. We take into account the experimental information that x, y and tan 0, are small. In particular, the smallness of tan 0, implies that A;(+,- I « I; IAK-,+I « I. (125) CP Violation 175 We expand each of the rates only to the order that is relevant to present measurements: r[D°(t)-K+71--] = Crt1AK+7,-1214/P12 x 11AK1+,-12+[Re(CK1+,-)Y + Im(Ak1+7,._)x]Ft+ 71 (y2+x2 )(Ft)2} , r[D°(t)-#K-ir+] = e-rt IAK 7,+121p/412 x {1AK-,+12 +[Re(AK-7,+)y+Im(AK„+)x]Ft+ 41 (y2+x2)(D) 126) F[D° (t)-4(+ K = e rt IAK+K-12 {1 + [ReNc+K-)y Im(AK+1,--)x]Ft} , F [D° ( t ) -4( K- ] = e rt IAK+K- 1 2 {1 + [Re(AKl+K _ )y - Im(Akl+K _ )x]Ft} , (127) F[D° (0-> e-rtiAK 70_ 1 2 , F [D° (t)-#K+71-1 rt IAK+,-12. (128) Within the Standard Model, the physics of D°-D° mixing and of the tree level decays is dominated by the first two generations and, consequently, CP violation can be safely neglected. In almost all 'reasonable' extensions of the SM, the six decay modes of Equa-tions (126), (127) and (128) are still dominated by the SM CP conserving contributions [63, 64]. On the other hand, there could be new short distance, possibly CP violating, contributions to the mixing amplitude M12. Allowing for only such effects of new physics, the picture of CP violation is simplified since there is no direct CP violation. The effects of indirect CP violation can be parameterised in the following way [65]: - R;,] e-i(6+4'D ) , =Rm e-i(6-°D), = V-R ei°D. Here R and Rm, are real and positive dimensionless numbers. CP violation in mixing is related to Rm 1 while CP violation in the interference of decays with and without mixing is related to sin O D 0. The choice of phases and signs in (129) is consistent with having O D = 0 in the SM and (5 = 0 in the SU (3) limit. We further define x' x cos 6 + y sin 6, y' y cos 6 - x sin 6. (130) With our assumption that there is no direct CP violation in the processes that we study, and using the parameterisations (129) and (67), we can rewrite Equations (126), (127) and (128) as follows: F[D°(t)-4(+7-] = e rcOx r+' 2 x[R+ NrkIlm(y1 cos (i5 D-X1 sin OD)rt+—R 4 (y2±x2)(Ft)21 F[D° (t) e-rti R-2 x [R+VRR,,,l (y' cos 0D+x' sin 0D)rt+ 4 (r2+x2)(Ft)2](131) (129) 176 Yosef Nir F[D°(t)—Y1C+If -] = e-rtiAK+K_12 [1 — R„,(y cos OD — x sin q5 D)Ft] , r[Do(t)—>K+K — ] = e-rt 114K+ K-12 [1 — Rm-1(y cos OD X sin OD)rt] , (132) F[D°(t)-4K-R-+] -= F[D°(t) K+7-] = (133) Of particular interest is the linear term in Equation (131) which is potentially CP violating [66, 67]. It is useful to define a CP violating quantity aD,K, which depends on the six measurable coefficients in (131): Re(AK-+)y + Im(AK-+)x Re(AK1+,,_)y + Im(AK-1+7,_)x + Y2 21AK1+,,_ I Vx2 + y2 X1 =- VX 2 ± y2 sin OD. (134) Observing aD,K„ 0 would be the most convincing evidence for new physics in D—D mixing. The CLEO measurement [68] gives the coefficient of each of the three terms [1, Ft and (ri)2] in the doubly-Cabibbo suppressed decays (131). Such measurements allow a fit to the parameters R, R,„, x' sin 0, yi cos 0, and x2 + y2. Fit A of reference following one sigma ranges: R = (0.48 + 0.13) x 10-2, [68] quotes the y' cos JD x' = = (-2.51U) x 10-2, (0.0 + 1.5) x 10-2, (135) Am sin OD = = 0.2311, 0.0 ± 0.6. It is assumed here that Rm is not very different from one and can be parameterised by a small parameter Am, R7t2 = 1 ± Am. (136) We would like to make two further comments in this regard: . The experimental results in Equation (135) do not show any signal of CP violation, that is, both sin OD and Am are consistent with zero. Consequently, there is no hint of new physics in the present results. . To test models of new physics, it would be useful to know the value of the strong phase b. Such an estimate is a difficult theoretical task [69, 70, 71] but experimental data on related channels would be useful [72, 73]. For the singly-Cabibbo suppressed modes of Equation (132), several experiments fit the time dependent decay rates to pure exponentials. We define F to be the parameter that is extracted in this way. More explicitly, for a time dependent decay rate with F[D(t) -+ f] cc e-rt(1 — zft + • • .), where IzI << 1, we have F(D ---> f) = r(1 + z). The above equations imply the following relations: f(D° K+K-) = r [1 + Rm(ycos OD — X sin OD)], (D° K+K-) = r [1 +Rm.-1(y cos OD ± X sin OD)], (137) f(Do K— e) = ND° -4 K+71--) = F. CP Violation 177 Note that deviations of f (D K±K -) from F do not require that y 0. They can be accounted for by x 0 and sin OD 0, but then they have a different sign in the D° and D° decays. Combining the two D-4 K+K- modes, one obtains the CP conserving quantity ycp: f (D K+K-) YCP = 1 = y cos OD - X sin 0, (138) (D° K-7r+) 2 where we made the approximations of zero production asymmetry and small Am [62]. The one sigma ranges measured by various experiments are given by (3.4 ± 1.6) x 10 -2 FOCUS [74] (0.8 + 3.1) x 10-2 E791 [75] (-1.1 ± 2.9) x 10-2 CLEO [76] (0.5 ± 1.3) x 10-2 BELLE [77] giving a world average of YCP = (1.3 ± 0.9) x 10-2. (140) Finally, we note that direct CP violation has been searched for in the Cabibbo-favored [78], singly-Cabibbo-suppressed [79, 80, 81] and doubly-Cabibbo-suppressed [68] decays with all results consistent with zero. We conclude that at present there is no evidence for mixing and certainly not for CP violation in the neutral D system. These results are consistent with the SM and constrain models of new physics. If evidence is found in the future, the D K7r and D KK decays will provide rich enough information that we will be able to point out the origin of the signals in much detail. 6 B decays 6.1 CP violation in mixing CP violation in mixing is related to a non-zero value for the following quantity [see (74)]: ( M r , 2 2 The effect can be isolated by measuring the asymmetry in semileptonic decays [see This has been searched for in _ asL — giving a world average of 2(1 — I9/M) Im(F12/M12). (142) several experiments, with sensitivity at the level of 10-2: (1.4 + 4.2) x 10-2 CLEO [82] (0.4 ± 5.7) x 10-2 OPAL [83] (143) (-1.2 ± 2.8) x 10-2 ALEPH [84] (0.48 ± 1.85) x 10-2 BABAR [85] asp = (0.2 + 1.4) x 10-2. (144) YCP = (139) 1 — q p (141) (87)1: 178 Yosef Nir As explained above, in the Bd system we expect, in a model independent way, that IF12/11/121‹ 1. Within any given model we can actually calculate the two quantities from quark diagrams. Within the SM, M12 is given by box diagrams. For both the Bd and B, systems, the long distance contributions are expected to be negligible and the calculation of these diagrams with a high loop momentum is a very good approximation. F12 is calculated from a cut of box diagrams [86]. Since the cut of a diagram always involves on-shell particles and thus long distance physics, the calculation is, at best, a reasonable approximation to F12. (For F12(Bs) it has been shown that local quark-hadron duality holds exactly in the simultaneous limit of small velocity and large number of colors. We thus expect an uncertainty of 0(1/Nc) 30% [87, 88]. For F12 (Bd) the small velocity limit is not as good an approximation but an uncertainty of order 50% still seems a reasonable estimate [89].) Within the Standard Model, M12 is dominated by top-mediated box diagrams (see [53] for details and references): G. F 2 n 4.2 1 TT*)2 \ M12 = 1272 rni3Mw77/3DBJ Bl vtb vtd) '30kXt), where So(xt ) is a kinematic factor, 77 8 is a QCD correction, and Bgn parameterises the hadronic matrix element. For F12 , we have [91, 92, 93] G2F F12 = 247rTrisrq,BBfi3(VtaVe'n2 [5 m2B Bs , (K2 Ki)+ 4 (2Ki +K2)+8 (K1+K2) T4 Veb lrd , (146) x , 3 lmb+md)2 108 3 Ing 1' rtb l Irl where K1 = —0.39 and K2 = 1.25 [93] are combinations of Wilson coefficients and Bs parameterises the (S — P)2 matrix element. New physics usually takes place at a high energy scale and is relevant to the short distance part only. Therefore, the SM estimate in Equation (146) remains valid model independently. Combining (145) and (146), we learn that Iri2/m121 = 0(ing/74), which confirms our model independent order of magnitude estimate,IF12/m121 , < 10-2. For the imaginary part of this ratio, we have asi, = Im ,,, Fi2 ,-;., 1.4 x 10-3 I/ (147) ivii2 (1 — P)2 + 772* The suppression by a factor of 0(10) of asL compared to Ir12/m121 comes from the fact that the leading contribution to F12 has the same phase as M12 . Consequently we have asL = 0(me2/n-q). The CKM factor does not give any further significant suppression, Im ysiLy4 ( Vtb Vtd = 0(1). In contrast, for the B, system, where the same expression holds except that Vcd/Vtd is replaced by V„/Vt,,, there is an additional CKM suppression from Im (Vct. VC5 = O(t 2) Via Vts In the SM and in most of its reasonable extensions, both F12 and b cis transitions are dominated by SM tree level decays. Consequently, new physics affects asL and av,K, only through its contributions to M12. This leads to interesting correlations between asL and av,K, that can be used to probe flavor parameters [94, 95]. Conversely, one can use the measured value of a,1 K5 to give model independent predictions for asL [96, 97]. (145) CP Violation 179 6.2 Penguin pollution In purely hadronic B decays, CP violation in decay and in the interference of decays with and without mixing is > 0(10-2). We can therefore safely neglect CP violation in mixing in the following discussion and use Q Ub Vld * B • P Vt6 Vtd (From here on we omit the convention-dependent quark phases b.), defined in Equa-tion (79). Our final expressions for physical quantities are of course unaffected by such omission.) A crucial aspect of our discussion is the number of relevant weak phases for a given decay process: . If there is a single weak phase that dominates the decay, CP violation in decay will be small and difficult to observe. On the other hand, CP asymmetries in neutral B decays into final CP eigenstates are subject to clean theoretical interpretation: we will either have precise measurements of CKM parameters or be provided with unambiguous evidence for new physics. . If there are two (or more) weak phases that contribute comparably, hadronic uncer-tainties will appear in the theoretical interpretation of CP violation in the interfer-ence of decays with and without mixing. On the other hand, if there are also large strong phase differences, CP violation in decay can be observed in the corresponding charged and neutral B decays. In many cases of interest, different weak phases are carried by tree and penguin con-tributions. The difficulties arising from hadronic uncertainties related to comparable tree and penguin contributions have become known as "penguin pollution." To illustrate the problem, we will consider two relevant CP asymmetries. First. the CP asymmetry in B OKs is an example of a case where the penguin pollution is negligibly small and a theoretically very clean interpretation of the experimental measurement is possible. Second, the CP asymmetry in B 7r 7r is an example of a case where penguin pollution cannot be a-priori ignored. We also list various ways in which the problem might he overcome. 6.3 B MKS The first evidence for CP violation outside K decays has been provided by the recent BaBar and Belle measurements of the CP asymmetry in B 01-(5, I 0.59 f 0.15 Babar [11] a'Pics = l 0.99 ± 0.15 Belle [12] These results in combination with previous ones [8, 9, 10] give the world average quoted in Equation (3). The process B OKs is one where the penguin contribution is harmless and the CP asymmetry is subject to an impressively clean theoretical interpretation. (148) (149) 180 Yosef Nir The decay is mediated by the quark transition 6 — ccs. It gets contributions from a tree level diagram and from penguin diagrams with intermediate u, c and t quarks. Using the unitarity relation (54), we can write the various contributions in terms of two CKM combinations: A(b C;c.§) = (Tes + P: — P:)17,':Vcs + (P: — Pst )V.IVus. (150) The second term is suppressed by two factors. First, there is the ratio between penguin and tree contributions, P — P: bryPTasEry,TacIB°) PT [ In (151) — To( — Tees + P; — "4 (01(s I 127r mg (7r± 7r- lboAcih.,-yoL IB°) The term in the square brackets is 0(0.03) but the ratio of matrix elements may partially compensate for this suppression. Secondly, there is the ratio of CKM elements, I (17:aVu Ve K., )1 — V. We conclude that the second term is suppressed by a factor of 6/,',T < 10-2 and we can safely neglect PoKs. Thus the B OK decay is dominated by a single weak phase, that is, arg(KI Vcs ). Neglecting PoK, means that, to a very good approximation, we have I A,kbK = 1, alp Ks = IMAL,Ks , (152) and that the experimental value of av,Ks [Equation (3)] can be cleanly interpreted in terms of a CP violating phase. A new ingredient in the analysis is the effect of K—K mixing. For decays with a single Ks in the final state, K —K mixing is essential because B° K° and B° —> k°, and interference is possible only due to K — K mixing. This adds a factor of (72) _ ves vc*d u,* K Kt" K (153) into (A/A): AoK (Kb V c*5 (Vc s V c*d) w* AoKs s 4> A2, then B Kir would be dominated by the penguin process. Let us provisionally make the following assumptions: (i) flavor SU(3) 0.03+0.54 182 Yosef Nir symmetry in the QCD matrix elements; (ii) electroweak penguins and "color suppressed" processes are negligible; (iii) penguins dominate B K7r, so TK, may be ignored in BR(B° -+ K+r-); (iv) penguins make a small enough contribution to B -4 71-71- that P,,, may be ignored in BR(B° 7r 4 7-). Then P„ Pnn PlOr VubVud (BR(B° K+7r-)V2 BR(B° 71+71 ) • (163) T„ PK7r T„, vtsVtb Recent measurements [100, 101, 102] give world averages BR(B° —r 7 4- 7-) = (4.4 + 0.9) x 10-6 and BR(B° 1-(+7r-) = (17.3 + 1.5) x 10'. We thus find BR(B° K+7-)/BR(B° —r 7+7- ) 3.9 and obtain the rough estimate 1r 7,77,1 —0.2-0.3. (164) It is clear that penguin effects are unlikely to be negligible in B 7r7r . Combining Equations (148) and (160), we find A (B 7r7r) Vub Vu*d 1 ± rirp7-(Vt& Vt'fi)/(VubVu*d) Vtb VutVud 1 + r7P7T(Vt't Vtd)/(VutVud) If the last factor could be approximated by unity, that is r7,14. = 0, we would obtain = 1 and a„ = sin 2a. (166) This approximation is however unjustified. To get an idea of the effects of P„ 0, we give the leading corrections due to a small IrpTI: IA77,1 = 1 — 2(Rt /RiL )Im(714.) sin a, ImA = sin 2a + 2(Rt /Ru)Re(rWT) cos 2a sin a. (167) (For a more detailed discussion, see [98].) Note that if strong phases can be neglected, rpT is real and IA„1=- 1 would be a good approximation. But it is not clear whether the strong phases are indeed small. In any case, one needs to know 7-7,7r, to extract a from a„„(t). This is the problem of penguin pollution. A variety of solutions to this problem have been proposed, falling roughly into two classes. The first type of approach is to convert the estimate given above into an actual measurement of IPK,1. (The list of papers on this subject is long. Early works include [103, 104, 105]. For a much more comprehensive list of references, see [98].) Once I Pit' I is known, flavor SU(3) is used to relate I PK,1 to I I • One must then include a number of additional effects: . Electroweak penguins. The effects are calculable [106]. . Color suppressed and re-scattering processes. These must be bounded or estimated using data and some further assumptions. . SU(3) corrections. Some, such as f K /f,, can be included, but SU(3) corrections generally remain a source of irreducible uncertainty. (165) CP Violation 183 The second type of approach is to exploit the fact that the penguin contribution to P,„ is pure A/ = 1/2, while the tree contribution to T„ contains a piece which is A/ = 3/2. (This is not true of the electroweak penguins [107], but these are expected to be small.) Isospin symmetry allows one to form a relation among the amplitudes B° 7+7r-, 707° and B+ 4 7+7°, A(B°-71-+71-- ) + A(B°—>7r°7°) = A(B+—>7r+7°). (168) There is also a relation for the charge conjugate processes. A simple geometric construc-tion then allows one to disentangle the unpolluted A/ = 2amplitudes, from which sin 2a may be extracted cleanly [108]. The key experimental difficulty is that one must measure accurately the flavor-tagged rate for B° zr°7°. Since the final state consists of only four photons, and the branching fraction is expected to be of 0(10-6), this is very hard. It has been noted that an upper bound on this rate, if sufficiently strong, would also allow one to bound P„ usefully [109, 98, 110]. An alternative is to perform an isospin analysis of the process B° [JR- 7+7r-moo [111, 112, 113, 114]. Here one must study the time-dependent asymmetry over the entire Dalitz plot, probing variously the intermediate states p±71-+ and p°7°. The advantage here is that the final states with two 7°'s need not be considered. On the other hand, thousands of cleanly reconstructed events would be needed. Finally, one can attempt to calculate the penguin matrix elements. Model-dependent analyses are not really adequate for this purpose, since the goal is the extraction of fundamental parameters. Precise calculations of such matrix elements from lattice QCD are far in the future, given the large energies of the 71'S and the need for an unquenched treatment. Recently, a new QCD-based analysis of the B 71-7 matrix elements has been proposed [115, 116, 117, 118]. For details, see [119]. 7 CP violation in supersymmetry 7.1 CP violation as a probe of new physics We have argued that the Standard Model picture of CP violation is rather unique and highly predictive. We have also stated that reasonable extensions of the Standard Model have a very different picture of CP violation. Experimental results are now starting to decide between the various possibilities. Our discussion of CP violation in the presence of new physics aims to demonstrate that, indeed, models of new physics can significantly modify the Standard Model predictions and that measurements in the near future will therefore have a strong impact on the theoretical understanding of CP violation. To understand how the Standard Model predictions could be modified by new physics, we focus on CP violation in the interference between decays with and without mixing. As explained above, this type of CP violation may give, due to its theoretical cleanliness, unambiguous evidence for new physics most easily. We now list some of the questions can be answered when many such observables are measured. 184 Yosef Nir • Is there new physics in B — B mixing? Consider a, Ks , the CP asymmetry in B-4OKs. This measurement will cleanly determine the relative phase between the B — B mixing amplitude and the b ces decay amplitude (sin 2/3 in the SM). The b ces decay has Standard Model tree contributions and therefore is very unlikely to be significantly affected by new physics. On the other hand, the mixing amplitude can be easily modified by new physics. We parameterise such a modification by a phase 9d: 29d = arg ( Mi2 /M,s2m ) (169) so that the the SM prediction for av,K, becomes aibK, -= sin[2(13 + Od )]. (170) Od 0 0 would be clear evidence of new physics (see e.g. [120]). It is interesting to note that already now the measured value of aoKs (3), which is consistent with the SM range, excludes many models that require a modification of CP violation in B — B mixing due to new physics. Among these are various models of soft CP violation [121, 122] aimed at solving the strong CP problem, models of geometric CP violation due to extra dimensions [123], models of spontaneous CP violation in the left-right symmetric framework [124, 125], and several models that aim to solve the supersymmetric CP problems [126, 127, 128]. . Is the new physics related to AB = 1 or AB = 2 processes, or both? Consider aoKs., the CP asymmetry in B cblCs. This measurement will cleanly determine the relative phase between the B — B mixing amplitude and the b decay amplitude (sin 20 in the SM). The b s.§.9 decay has only Standard Model penguin contributions and therefore is sensitive to new physics. We parameterise the modification of the decay amplitude by a phase 6 A [129]: OA = arg(AoKs /ilr,4,,). (171) Then aohs = sin[2(/3 0d + GA)]. (172) Comparing aoKs to aoKs , that is, examining whether 0A 0 0, will tell us if the new physics is related to AB = 1 or AB = 2 processes. . Is the new physics related to the third generation, or all generations? Consider the CP violating ratio of K irvz) decays, defined in (120). This measurement will cleanly determine the relative phase between the K — K mixing amplitude and the s —> dvi decay amplitude (of order sine /3 in the SM). The experimentally measured small value of E K requires that the phase of the K —K mixing amplitude is not modified from the Standard Model prediction. (More precisely, it requires that the phase of the mixing amplitude is very close to twice the phase of the s -+ dun decay amplitude [130].) On the other hand, the decay, which in the SM is a loop process with small mixing angles, can be easily modified by new physics. Examining whether the SM correlation between Cl7,0 and aoKs is fulfilled will give us information on the generations involved. CP Violation 185 . Is the new physics related to the down sector or the up sector or both? Consider aD,K,, the CP violating quantity in D -4 I(± 7r decays defined in (134). It depends on OD, the relative phase between the D—D mixing amplitude and the c dgu and c —> sdu decay amplitudes. Within the Standard Model, the two decay channels are tree level. It is unlikely that they are affected by new physics. On the other hand, the mixing amplitude can be easily modified by new physics. Examining whether aD,K, = 0, that is, whether O D (and/or Od ) 0, will provide information on the sectors involved. . Are the new sources of CP violation flavor changing, flavor diagonal or both? Consider d N , the electric dipole moment of the neutron. We did not discuss this quantity so far because, unlike CP violation in meson decays, flavor changing cou-plings are not necessary for d N. In other words, the CP violation that induces dN is flavor diagonal. It does in general get contributions from flavor changing physics, but it could be induced by sectors that are flavor blind. Within the SM (and ignor-ing OQCD), the contribution from (5K m arises at the three loop level and is at least six orders of magnitude below the experimental bound (7). If the bound is further improved (or a signal observed), we may elucidate the flavor dependence. It is no wonder then that with such rich information, flavor and CP violation provide an excellent probe of new physics. We will now demonstrate this situation more concretely by discussing CP violation in supersymmetry. 7.2 The supersymmetric framework Supersymmetry solves the fine-tuning problem of the Standard Model and has many other virtues. But at the same time, it leads to new problems: baryon number violation, lepton number violation, large flavor changing neutral current processes and large CP violation. The first two problems can be solved by imposing R-parity on supersymmetric models. There is no such simple, symmetry-related solution to the problems of flavor and CP vio-lation. Instead, suppression of the relevant couplings can be achieved by demanding very constrained structures of the soft supersymmetry breaking terms. There are two impor-tant questions here. First, can theories of dynamical supersymmetry breaking naturally induce such structures? (For an excellent review of dynamical supersymmetry breaking, see [134) Second, can measurements of flavor changing and/or CP violating processes shed light on the structure of the soft supersymmetry breaking terms? Since the answer to both questions is in the affirmative, we conclude that flavor changing neutral current processes and, in particular, CP violating observables will provide clues to the crucial question of how supersymmetry breaks. 7.2.1 CP violating parameters A generic supersymmetric extension of the Standard Model contains a host of new fla-vor and CP violating parameters. (For a review of CP violation in supersymmetry see [132, 133].) It is an amusing exercise to count the number of parameters [134]. The su-persymmetric part of the Lagrangian depends, in addition to the three gauge couplings 186 Yosef Nir of Gsm, on the parameters of the superpotential W: W = E (Yi7HuchiuL + + Y4H,LL,ELJ) + pHuHd. (173) 1,3 In addition, we have to add soft supersymmetry breaking terms: £soft = — (A iu31-1uQ ii LitLi + Ad + AierildLLitLi + BHu Hd +h.c.) 1 3 — E (7.2s),,A,A1 — —2 E (fii(a) (AA)(,) + h.c.) . (174) all scalars (a)= I where S = QL,DL,U L,LL,EL. The three Yukawa matrices Yf depend on 27 real and 27 imaginary parameters. Similarly, the three A1-matrices depend on 27 real and 27 imaginary parameters. The five ms hermitian 3 x 3 mass-squared matrices for sfermions have 30 real parameters and 15 phases. The gauge and Higgs sectors depend on Owl:), in' (1), Th(2)1 75'1(3)1 91,92, 93, /1, B, mho , mhd , (175) that is 11 real and 5 imaginary parameters. Summing over all sectors, we get 95 real and 74 imaginary parameters. The various couplings (other than the gauge couplings) can be thought of as spurions that break a global symmetry, U(3)5 x U(1)pQ x U(1) R U(1) R x U(1) L. (176) The U(1)pQ x U(1) R charge assignments are: Hu Hd QU QD LE U(1)pQ 1 1 —1 —1 —1 . (177) U(1) R 1 1 1 1 1 Consequently, we can remove 15 real and 30 imaginary parameters, which leaves 124 = { 80 real physical parameters. (178) 44 imaginary physical parameters. In particular, there are 43 new CP violating phases! In addition to the single Kobayashi-Maskawa of the SM, we can put 3 phases in MI , M2, it (we used the U(1)pQ and U(1) R to remove the phases from itB* and M3, respectively) and the other 40 phases appear in the mixing matrices of the fermion-sfermion-gaugino couplings. (Of the 80 real parameters, there are 11 absolute values of the parameters in (175), 9 fermion masses, 21 sfermion masses, 3 CKM angles and 36 SCKM angles.) Supersymmetry provides a nice example to our statement that reasonable extensions of the Standard Model may have more than one source of CP violation. The requirement of consistency with experimental data provides strong constraints on many of these parameters. For this reason, the physics of flavor and CP violation has had a profound impact on supersymmetric model building. A discussion of CP violation in this context can hardly avoid addressing the flavor problem itself. Indeed, many of the supersymmetric models that we analyze below were originally aimed at solving flavor problems. For details on the supersymmetric flavor problem, see [135]. For CP violation, one can distinguish two classes of experimental constraints. First, bounds on nuclear and atomic electric dipole moments determine what is usually called the .supersymmetric CP problem. Second, the physics of neutral mesons and, most impor-tantly, the small experimental value of eK pose the supersymmetric E R problem. In the next two subsections we describe the two problems. CP Violation 187 7.2.2 The supersymmetric CP problem One aspect of supersymmetric CP violation involves effects that are flavor preserving. Then, for simplicity, we describe this aspect in a supersymmetric model without additional flavor mixings, i.e. the minimal supersymmetric standard model (MSSM) with universal sfermion masses and with the trilinear SUSY-breaking scalar couplings proportional to the corresponding Yukawa couplings. (The generalisation to the case of non-universal soft terms is straightforward.) In such a constrained framework, there are four new phases beyond the two phases of the SM (8KM and OQcD). One arises in the bilinear ii-term of the superpotential (173), while the other three arise in the soft supersymmetry breaking parameters of (174): (the gaugino mass), A (the trilinear scalar coupling) and B (the bilinear scalar coupling). Only two combinations of the four phases are physical [136, 137]: = arg(A*rii), OB = arg(riv,B*). (179) In the more general case of non-universal soft terms there is one independent phase OA, for each quark and lepton flavor. Moreover, complex off-diagonal entries in the sfermion mass-squared matrices represent additional sources of CP violation. The most significant effect of OA and Og is their contribution to electric dipole moments (EDMs). For example, the contribution from one-loop gluino diagrams to the down quark EDM is given by [138, 139]: dd = and ea3 (IA sin OA + tan 187rm' MOIsin OB) (180) where we have taken rri2Q ti m2D ti rn rit 2, for left- and right-handed squark and gluino masses. We define, as usual, tan 0 = (11,i)/(Ild). Similar one-loop diagrams give rise to chromoelectric dipole moments. The electric and chromoelectric dipole moments of the light quarks (u, d, s) are the main source of d N (the EDM of the neutron), giving [140] dN 2 (100 GeV )2 sin cb A, B X 10-23 e cm, rh where, as above, rn represents the overall SUSY scale. In a generic supersymmetric framework, we expect riz = O(mz) and sin (bA,B = 0(1). Then the constraint (7) is generically violated by about two orders of magnitude. This is the Supersymmetric CP Problem. Equation (181) shows two possible ways to solve the supersymmetric CP problem: . Heavy squarks: fir > 1 TeV . . Approximate CP: sin cbA,B < 1. Recently, a third way has been investigated, that is cancellations between various contri-butions to the electric dipole moments. However, there seems to be no symmetry that can guarantee such a cancellation. This is in contrast to the other two mechanisms mentioned above that were shown to arise naturally in specific models. We therefore do not discuss any further this third mechanism. (181) 188 Yosef Nir The electric dipole moment of the electron is also a sensitive probe of flavor diagonal CP phases. The present experimental bound, de l < 4 x 10-27 ecm[141], (182) is also violated by about two orders of magnitude for 'natural' values of supersymmetric parameters. A new experiment [142] has been proposed to search for the electric dipole moment of the muon at a level smaller by five orders of magnitude than present bounds; such improvement will make d1, another sensitive probe of supersymmetry [143]. 7.2.3 The supersymmetric eK problem The supersymmetric contribution to the eK parameter is dominated by diagrams involving Q and d squarks in the same loop. For fit = 774. ^ mQ ^ mD (our results depend only weakly on this assumption) and focusing on the contribution from the first two squark families, one gets (see, for example, [144]): Here 5 ot3 fkm, = -= m ic \ 2 3 —25] Im [(42)m,(42)R7d K1d2L I i{dRI " 12 , • (183) (184) EK = 1620 ril2 A7nK (6d12)LL (6d12) RR ms + md) + — (771 2 2 ) 77/Q- m2 m2 — b2 —151) M— 2 ( D where Kg' (IqP) are the mixing angles in the gluino couplings to left-handed (right-handed) down quarks and their scalar partners. Note that CP would be violated even if there were two families only [145]. Using the experimental value of 6K, we get panKEK rusy 107 ( 7 nc2- 300 GeV )2 rnc. —2 (AmKEK )EXP rIl ) 2 2 (rnii2 - m 131 IKff'Kg1 ) rii2_ D sin 0, (185) where 0 is the CP violating phase. In a generic supersymmetric framework, we expect fit = 0(7714, 6771617742, D = 0(1), 4' D = 0(1) and sin 0 = 0(1). Then the constraint (185) is generically violated by about seven orders of magnitude. The AmK constraint on Re [(612)/./.(42)RR] is about two orders of magnitude weaker. One can distinguish then three interesting regions for (b12) = VAILL(02)RR 0.003 << (02) excluded, 0.0002 << (02) < 0.003 viable with small phases, (186) (42) << 0.0002 viable with 0(1) phases. The first bound comes from the ArrI K constraint (assuming that the relevant phase is not particularly close to 7r/2). The bounds here apply to squark masses of order 500 GeV and scale like /h. There is also dependence on m-drh, which is here taken to be one. Equation (185) shows other possible ways of solving the supersymmetric EK problem: CP Violation 189 . Heavy squarks: rit >> 300 GeV; . Universality: bm2Q,D < m2Q,D; • Alignment: 11-q12 I << 1; • Approximate CP: sin q << 1. 7.2.4 A supersymmetric e' / e? In this section we discuss the question of whether supersymmetric contributions to Ei/E can be dominant. A typical supersymmetric contribution to E'/E is given by [146] 1 3 (158 MeV (500 GeV) = 58BG (15(m4) [(6(1,R)12 WR)21'11 las(500 GeV)] ms-Emd mg el , (187) where BG parameterises the matrix element of the relevant four-quark operator. Conse-quently, the supersymmetric contribution saturates E'/E for Im [(61,R)12 (-1,R)* 7 500 GeV where, motivated by flavor symmetries (see below), we parameterise the suppression by powers of A 0.2. Without proportionality, a naive guess would give ti ms1Vusl ti A5-6 1714 7 TT/ and A5-6 Mt - • This is not far from the value required to account for e' /e [147]. Thus, it is certainly possible that supersymmetry accounts for, at least, a large part of E'/E (see, for example, the models of references [148]-[154]). Yet, it has been argued that such a situation is not generic [155]. The problem is that Equation (189) gives an overestimate of the supersymmetric contribution in most viable models of supersymmetry breaking that have appeared in the literature. We will encounter concrete examples of this statement when we survey the various supersymmetric flavor models. 7.3 Supersymmetry breaking and flavor models Before turning to a detailed discussion, we define two scales that play an important role in supersymmetry: As, where the soft supersymmetry breaking terms are generated, and AF, where flavor dynamics takes place. When AF >> As, it is possible that there are no genuinely new sources of flavor and CP violation. This leads to models with exact universality. When AF < As, we do not expect, in general, that flavor and CP violation are limited to the Yukawa matrices. One way to suppress CP violation would be to assume that, similarly to the Standard Model, CP violating phases are large, but their effects are screened, possibly by the same physics that explains the various flavor puzzles, such as models with Abelian or non-Abelian horizontal symmetries. It is also possible that CP violating effects are suppressed because squarks are heavy. Another option, which is now excluded, was to assume that CP is an approximate symmetry of the full theory (namely, CP violating phases are all small). (188) (8dLR)12 (°dLR)21 (189) 190 Yosef Nir 7.3.1 Gauge mediated supersymmetry breaking If at some high energy scale squarks are exactly degenerate and the A terms proportional to the Yukawa couplings, then the contribution to E K comes from Renormalisation Group Evolution (RGE) and is GIM suppressed, that is log(As/mw ) 2 (190) [Wed *2] Yt4 Elf Im 1672 This contribution is negligibly small [136]. The contribution from genuinely supersym-metric phases (i.e. the phases in At and /1) is also negligible [156, 157]. (This does not necessarily mean that there is no supersymmetric effect on E K . In some small corner of parameter space the supersymmetric contribution from stop-chargino diagrams can give up to 20% of E K [158, 159].) In Gauge Mediated Supersymmetry Breaking (GMSB) [160, 161], superpartner masses are generated by the SM gauge interactions. These masses are then exactly universal at the scale As at which they are generated (up to tiny high order effects associated with Yukawa couplings). Furthermore, A terms are suppressed by loop factors. The only contribution to E K is then from the running, and since As is low it is highly suppressed. These models can also readily satisfy the EDM constraints. In most models, the A terms and gaugino masses arise from the same supersymmetry breaking auxiliary field, that is, they are generated by the same SUSY and U(1) R breaking source. They therefore carry the same phase (up to corrections from the Standard Model Yukawa couplings), and OA vanishes to a very good approximation: A „ v2-v2 T YA`-'`--‘t.cibdCKM [log(As/mw)]4 1672 (191) The resulting EDM is dN < 10-31 e cm. This maximum can be reached only for very large tan) — 60 while, for small tan ,Q 1, d N is about 5 orders of magnitude smaller. This range of values for d N is much below the present 10' e cm) and foreseen 10-28e cm) experimental sensitivities (see e.g. [162]). The value of OB in general depends on the mechanism for generating the p term. However, running effects can generate an adequate B term at low scales in these models even if B(As) = 0. One then finds [163] —B = At (As) + M2(As) (-0.12 + 0.171Yt 1 2 ) , (192) where M2 is the SU(2) gaugino mass. Since OA ^ 0, the resulting g5 B vanishes, again up to corrections involving the Standard Model Yukawa couplings [164]. There is therefore no CP problem in simple models of gauge mediation, even with phases of order one. The supersymmetric contribution to D—D mixing is similarly small and we expect no observable effects. For the Bd system, GMSB models predict then a large CP asymmetry in B y Ks, with small deviations (at most 20%) from the SM. More generally, in any supersymmetric model where there are no new flavor violating sources beyond the Yukawa couplings, CP violation in meson decays is hardly modified from the SM predictions [165]. CP Violation 191 7.3.2 Gravity, anomaly and gaugino mediation If different moduli of string theory obtain supersymmetry breaking F terms, they would typically induce flavor-dependent soft terms through their tree-level couplings to Standard Model fields. There are however various scenarios in which the leading contribution to the soft terms is flavor independent. The three most intensively studied frameworks are dilaton dominance, anomaly mediation and gaugino mediation. Dilaton dominance assumes that the dilaton F term is the dominant one. Then, at tree level, the resulting soft masses are universal and the A terms proportional to the Yukawa couplings. Both universality and proportionality are, however, violated by string loop effects. These induce corrections to squark masses of order -Q`.7 -42, where ax = [27r(S + S*)]-1 is the string coupling. There is no reason why these corrections would be flavor blind. However, RGE effects enhance the universal part of the squark masses by roughly a factor of five, leaving the off-diagonal entries essentially unchanged. The flavor suppression factor is then [166] 2 one—loop 1 M12 aX 4x 10-4 m2 7r 25 (193) Dilaton dominance relies on the assumption that loop corrections are small. This probably presents the most serious theoretical difficulty for this idea, because it is hard to see how non-perturbative effects, which are probably required to stabilise the dilaton, could do so in a region of weak coupling. In the strong coupling regime, these corrections could be much larger. However, this idea at least gives some plausible theoretical explanation for how universal masses might emerge in hidden sector models. Given that dilaton stabilisation might require that non-perturbative effects are important, the estimate of flavor suppression (193) might well turn out to be an underestimate. We now turn to the flavor diagonal phases that enter in various EDMs. The phase OA vanishes at tree-level, so that [166, 167] OA = 0 (ax /7r). [The smallness of OA implies that, there is a suppression of 0(ax /7r) ti 10' compared to (189) and the supersymmetric contribution to EVE is small.] However, OB is unsuppressed, even when and through it B, are generated by Kahler potential effects through supersymmetry breaking, in which case B = 2m3*12µ [168]. While the size of m312 is determined from the requirement that the cosmological constant vanishes, its phase remains arbitrary, and in fact depends on the phase of the constant term that is added to the superpotential in order to cancel the cosmological constant. We conclude that the supersymmetric EK problem is solved in these models but the EDM problem, in general, is not. For EDM contributions to be small in these models, the gravitino mass had better give a small physical phase. Anomaly mediation (AMSB) provides another approach to solving the flavor prob-lems of supersymmetric theories, as well as to obtaining a predictive spectrum. In the presence of some truly 'hidden' supersymmetry breaking sector, with no couplings to the SM fields (apart from indirect couplings through the supergravity multiplet) the confor-mal anomaly of the Standard Model gives rise to soft supersymmetry breaking terms for the Standard Model fields [169, 170]. These terms are generated purely by gravitational 192 Yosef Nir effects and are given by 1 a'Y(P) 2 l3(µ) 1 77712)(1-1) = 4 1nµ m312, mv2(P) = 9(p)m3/2, A(ii) 2 -Y(P)rn3/2 where and 7 are the appropriate beta function and anomalous dimension. Thus, apart from the Standard Model gauge and Yukawa couplings, the soft terms only involve the parameter m312. In general, naturalness considerations suggest that couplings of hidden and visible sectors should appear in the Kahler potential, leading to soft masses for scalars already at tree level, and certainly by one loop. As a result, one would expect the contributions (194) to be irrelevant. However, in "sequestered sector models" [169], in which the visible sector fields and supersymmetry breaking fields live on different branes separated by some distance, the anomaly mediated contribution (194) could be the dominant effect. This leads to a predictive picture for scalar masses. Since the soft terms (194) are generated by the Standard Model gauge and Yukawa couplings, they are universal, up to corrections involving the third generation Yukawa couplings. However, the resulting slepton masses-squared are negative, so this model requires some modification. We will not attempt a complete review of this subject here. Our principal concerns are the sources of CP violation, and the extent to which the AMSB formulae receive corrections, leading to non-degeneracy of the squark masses. For Equation (194) to correctly give the leading order soft terms, it is necessary that all moduli obtain large masses before supersymmetry breaking, and that there be no Planck scale VEVs in the supersymmetry breaking sector [171]. A possible scenario for this to happen is if all moduli but the fifth dimensional radius, R, sit at an enhanced symmetry point, and that R obtains a large mass compared to the supersymmetry-breaking scale (say, by a racetrack mechanism). Even in this case, however, there is a difficulty. One might expect that some of the moduli have masses well below the fundamental scale. If there are light moduli in the bulk, there are typically one-loop contributions to scalar masses-squared from exchanges of bulk fields, proportional to 74/2 /R3 times a loop factor [169]. If these contributions are non-universal, they may easily violate the Arni< and EK constraints [133]. If there are no light moduli, and if the contributions described above are adequately suppressed, some modification of the visible sector is required in order to generate ac-ceptable slepton masses. Different such solutions have been suggested. In some of these models, there are no new contributions to CP violation simply because there are few enough new parameters in the theory that they can all be chosen real by field redefini-tions [172, 173, 174]. Furthermore, it is possible to generate the it term in these models from AMSB, so that bB vanishes. These models are then similar to GMSB models from the point of view of CP violation. We conclude then, that in generic sequestered sector models it is difficult to obtain strong degeneracy and a special phase structure is required. It is conceivable that there might be theories with a high degree of degeneracy, or with no new sources of CP violation. In such theories, the SM predictions for CP violation are approximately maintained. Gaugino mediation [175, 176] is in many ways similar to anomaly mediation, and poses similar issues. These models also suppress dangerous tree level contact terms by invoking extra dimensions, with the Standard Model matter fields localised on one brane (194) CP Violation 193 and the supersymmetry breaking sector on another brane. In this case, however, the Standard Model gauge fields are in the bulk, so gauginos get masses at tree level, and as a result scalar masses are generated by running. Scalar masses are therefore universal. Furthermore, the soft terms typically involve only one new parameter, namely, the singlet F VEV that gives rise to gaugino masses. Therefore, they do not induce any new CP violation. Again, however, if there are non-universal tree and one loop contributions to scalar masses, significant violations of degeneracy and proportionality can be expected, and a special structure of CP violating phases is required. 7.3.3 Supersymmetric flavor models Various frameworks have been suggested in which flavor symmetries, designed to explain the hierarchy of the Yukawa couplings, impose at the same time a special flavor structure on the soft supersymmetry breaking terms that helps to alleviate the flavor and CP problems. In the framework of alignment, one does not assume any squark degeneracy. In-stead, flavor violation is suppressed because the squark mass matrices are approximately diagonal in the quark mass basis. This is the case in models of Abelian flavor symme-tries, in which the off-diagonal entries in both the quark mass matrices and in the squark mass matrices are suppressed by some power of a small parameter, A, that quantifies the breaking of some Abelian flavor symmetry. A natural choice for the value of A is sin Oc, so we will take A 0.2. One would naively expect the first two generation squark mixing to be of the order of A. However, the AmK constraint is not satisfied with the 'naive alignment', K(12 A, and one has to construct more complicated models to achieve the required suppression [177, 178]. One can solve the supersymmetric E K problem by flavor suppression, that is, models with (b;2) A5 [179]. These models are highly constrained and almost unique. It is simpler to construct models where (02) A3 but the CP vi- olating phases are also suppressed [128]. Such models predict that aoKs < 1 and are therefore now excluded. (Models with (82) A3 could still be viable with phases of order one if the RGE contributions enhance squark degeneracy.) For flavor diagonal phases, the question is more model dependent. There is however a way to suppress these phases without assuming approximate CP [179]. The mecha-nism requires that CP is spontaneously broken by the same fields that break the flavor symmetry ("flavons"). It is based on the observation that a Yukawa coupling and the corresponding A term carry the same horizontal charge and therefore their dependence on the flavon fields is similar. In particular, if a single flavon dominates a certain coupling, the CP phase is the same for the Yukawa coupling and for the corresponding A term and the resulting OA vanishes. Similarly, if the /1 term and the B term depend on one (and the same) flavon, OB is suppressed. For E'/E, the E K constraint requires that the relevant terms are suppressed by at least a factor of A2 compared to (189) [155] and the contribution is therefore small. We conclude that one can construct models in which an Abelian horizontal symmetry solves both the E K and the d N problems. These models are however not the generic ones in this framework. 194 Yosef Nir Non-Abelian horizontal symmetries can induce approximate degeneracy between the first two squark generations, thus relaxing the flavor and CP problems [180]. A review of E K in this class of models can be found in [132]. Quite generically, the supersymmetric contributions to E K are too large and require small phases (see, for example, the models of reference [181]). There are however specific models where the E K problem is solved without the need for small phases [182, 183]. Furthermore, universal contributions from RGE running might further relax the problem. For flavor diagonal phases, it is difficult (though not entirely impossible) to avoid OA > .\ 2 ^, 0.04 [132]. This, however, might be just enough to satisfy the d N constraint. With a horizontal U(2) symmetry, the two contributions to EVE in (187) cancel each other. (More generally, this happens for a symmetric A matrix with A11 = 0 [184].) We conclude that, similar to models of Abelian flavor symmetries, one can construct models of non-Abelian symmetries in which the symmetry solves both the E K and the dN problems. These models are however not the generic ones in this framework. Finally, one can construct models of heavy first two generation squarks. Here, the basic mechanism to suppress flavor changing processes is actually flavor diagonal so that m412 20 TeV. Naturalness does not allow higher masses, but this mass scale is not enough to satisfy even the AmK constraint [185], and one has to invoke alignment, A. This is still not enough to satisfy the E K constraint of Equation (185), and a somewhat small phase is required. Three more comments are in order: First, in this framework, gauginos are significantly lighter than the first two generation squarks, and so RGE cannot induce degeneracy. Second, the large mass of the squarks is enough to solve the EDM related problems, and so it is only the E K constraint that motivates a special phase structure. Finally, the contribution to E'/E is negligibly small. Instead of (189), a more likely estimate is [155] (61,R),,, Inz(Md)il which suppresses the relevant matrix elements by a factor (10 TeV)2 ' of order 104. 7.4 (S)Conclusions We would like to emphasise the following points: • For supersymmetry to be established, a direct observation of supersymmetric parti-cles is necessary. Once it is discovered, then measurements of CP violating observ-ables will be a very sensitive probe of its flavor structure and, consequently, of the mechanism of dynamical supersymmetry breaking. • It seems possible to distinguish between models of exact universality and models with genuine supersymmetric flavor and CP violation. The former tend to give d N < 10-31 e cm while the latter usually predict d N > 10-28 e cm. • The proximity of ati,K, to the SM predictions is obviously consistent with models of exact universality. It disfavors models of heavy squarks such as that of refer-ence [185]. Models of flavor symmetries allow deviations of order 20% (or smaller) from the SM predictions. To be convincingly signalled, an improvement in the CP Violation 195 theoretical calculations that lead to the SM predictions for aoK, will be required [186]. . 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B 423, 319 (1998) [hep-ph/9711215]. [188] A. J. Buras, A. Romanino and L. Silvestrini, Nucl. Phys. B 520, 3 (1998) [hep-ph/9712398]. [189] G. Colangelo and G. Isidori, JHEP 9809, 009 (1998) [hep-ph/9808487]. CP and flavour in supersymmetry Steven Abel Institute for Particle Physics Phenomenology Durham University, U.K. 1 Introduction The brief for these lectures is to introduce some of the ideas and challenges for super-symmetry coming from flavour and CP physics, aimed at an audience that includes 'non-experts'. It is usual for courses on this subject to jump in at the superpotential for the Minimal Supersymmetric Standard Model, and leave the reader to sort out for him or herself what the superpotential actually means. In this course I have instead decided to begin at square one, assuming no prior knowledge other than field theory of scalars and fermions. While it might seem a challenge to get from this point to the MSSM in a just page or two of working, this is in fact possible (and only slightly painful) if we do not spend too long worrying about things like `superspace', which although quite fundamen-tal, do not really have much bearing on the question of phenomenology. Thus the first section will develop the idea of a superpotential by giving a quick and dirty demonstration of how to `supersymmetrise' a fundamental scalar. By following the (very explicit) steps in this part, those readers who are unfamiliar with supersymmetry can get a grasp of the various concepts involved. In particular we will see at first hand the famous cancellation of divergences that makes supersymmetry such an important idea. This will lead natu-rally to the MSSM superpotential and from there we can go on to analyse the particular problems that flavour and CP throw up for supersymmetry. The interests of a SUSSP audience are typically rather broad, so the course is organised as follows. The reader who requires a brief overview of the subject will be able to follow the main body of the text which is self contained. However, since some of the audience may want to develop a deeper knowledge of the subject, I have made copious use of appendices to include more detailed discussions of various aspects as we go along. From these it should be possible for the reader to follow some of the more formal texts in [1]. 201 DOI: 10.1201/9780429187056-5202 Steven Abel Why supersymmetry? Although there are several responses to this question, the most important is undoubtedly cancellation of divergences. This is a remarkable feature of supersymmetric models that gives stability to any theory, specifically stability of mass scales. We shall shortly derive it in detail, but in brief it goes as follows. Consider a theory involving scalars with self coupling Ab, the Higgs particles, and also fermions which couple to the Higgs particles in Yukawa couplings with strength Af . •-e-• eee Figure 1. Contributions to the Higgs potential. At one loop order, the Higgs potential will get a divergent contribution from the diagrams shown in Figure 1, which is of the form 61/Higgs (IA/12 2)H2A2. (1) The fermionic loop gets an extra minus sign. This type of contribution is a disaster for electroweak symmetry breaking, since we will have to tune our Ab and Al couplings very precisely (to at least one part in 10" if we take A > MGuT 1016 GeV) if the physical Higgs mass and hence weak gauge bosons are to end up around the 100GeV mark. Moreover, the A couplings will themselves have logarithmic radiative corrections from one-loop diagrams, and in fact we need to re-tune at each order in perturbation theory. Finally the low energy theory will be sensitive to unknown high energy physics -for example an unknown GUT theory or string theory. The problem introduced by the existence of vastly different mass scales is referred to as the gauge-hierarchy problem. If you believe that the Higgs fields are the result of some non-perturbative process (a top quark condensate perhaps) happening just above the weak scale, or that there are no new scales above Mw, or more generally, that there are no fundamental scalars, then this will not be a problem for you. Supersymmetry solves these technical problems by ensuring that Af = Ab, and in fact all contributions to the effective potential then vanish. The only renormalisation of the bare couplings which takes place comes from field renormalisations. The price we have to pay for such a useful cancellation is high; we have to introduce a fermionic(bosonic) degree of freedom for every boson(fermion) (known collectively as superpartners). The general form of the couplings will be A(1,2) — A(,i)log (2) P2 Now couplings which start off small at the GUT scale remain small. (This holds to all orders in perturbation theory.) CP and flavour in supersymmetry 203 This then is the main reason for taking supersymmetry seriously. Other reasons (in descending order of compulsion) are that the gauge couplings of the supersymmetric standard model (MSSM) appear to exactly unify at some (presumably GUT) scale, or that the only known theory of quantum gravity (i.e. string theory) strongly suggests it, or that the cosmological constant is zero. This last reason is probably invalid, since supersymmetry must be broken at some scale, reintroducing a cosmological constant (albeit one which is only out by 52 orders of magnitude instead of 120). One additional interesting hint for extra scalars comes from the fact that for relatively low mass Higgs particles (around 115GeV say) the Standard Model vacuum is unstable to decay, as shown in [2] and references therein, because the top mass is large. This instability can be removed if there are more scalars than just the Higgs fields. 2 Express supersymmetry Our aim in this section is to derive (or at least explain the form of) supersymmetric inter-actions from their non-supersymmetric counterparts (SM). We begin in this subsection by supersymmetrising a simple Lagrangian describing a massless, free boson. We will then discuss how to write down interactions using the superpotential and examine the top quark Yukawa coupling in detail, in particular verifying the cancellation discussed above. Readers already familiar with supersymmetry can skip to Section 3. Readers who would like to see the more detailed conventional approach can consult the standard texts [1]. Let us begin by considering the Lagrangian for a free boson: = — (apO*)(aP0). (3) Since we wish to describe both particles and anti-particles, 0 is complex with two degrees of freedom. Also 0 has spin 0 and dimension 1. This Lagrangian resembles that of the Higgs which is already present in the SM. Eventually, each chirality (left or right) of matter fields will aquire a scalar superpartner of this type, together forming chiral supermultiplets. Now let us introduce our fermion superpartner which must also have two degrees of freedom for a sensible transformation. Before continuing the reader should familiarise themselves with the Weyl notation for fermions which has become standard for this subject and is outlined in Appendix A. The supersymmetry transformation should by definition turn the scalars into fermions so, noting that a two component spinor has the same number of degrees of freedom as the scalar, let us introduce an infinitesimal (linear) supersymmetry transformation (5 on the complex boson above, involving such a spinor: (50 = aCX„. (4) The fermionic parameter describes the magnitude of the transformation, and a is some as yet undetermined constant. We will often drop the fermionic indices a when they are contracted. The spin and dimension on both sides must match, so that 1 must have spin 1/2 (it anticommutes with itself and X) and dimension 1/2. Next we wish to have an equivalent transformation for the spinor itself. Matching spin and dimensions again, this can be of the form, (VC, = bo-c,P6 60,0 + c„F. (5) 204 Steven Abel where F is some field of spin 0 and dimension 2 whose existence will be explained shortly. Then for F we can have; StF = dee,&6-8,2x. (6) In order to determine the constants a, b, c, d we need to consider the commutation of two transformations. For example, [8e' (5,]0 = ab(no-PA — Xo"77)000. (7) This commutator involves only one field on each side, and is therefore related to the underlying symmetry (supersymmetry). It describes a relation between the 4 and a, operators, and should not depend on which field (0, X or F) we choose to express it. (The equivalent statement in gauge theory, is that the Lie-algebra [Ta = i fa,,Tc is formed by successive gauge transformations on any of the fields in the theory, irrespective of their representation.) We need to ensure that the algebra is the same for the field X. and this is why the field F is required; in fact [50 5„]X = cd(no-"X — xo-Ava„x, (8) giving ab = cd. The extra field F is known as an auxiliary field since it is there simply to make the algebra close. It does not propagate, and so it is easily removed from the Lagrangian using its equation of motion. The full chiral supermultiplet consists of the three fields 1 = {0, X, F} (and the anti-chiral supermultiplet, = {0*, X, F*}). Having determined the multiplet structure, we are able to write down the simplest invariant Lagrangian. Using the usual normalisations, our first guess is, £KE = —iX0-"aA X 0*820+ F* F (9) In order for this to be invariant (i.e. to transform into itself plus total derivatives under the operations (4)—(6)), we find that we require b = ia, d = ic, (10) which leaves only one degree of freedom a which normalises the operators 4. Conven-tionally we take a = f. So far we have only the kinetic terms of our Lagrangian. In order to be able to include Yukawa couplings, for example, we need to consider additional interaction terms. As a starting point, notice that the auxiliary field F transforms into a total derivative under a supersymmetry transformation. Thus LP = F (11) is a suitable invariant. By itself this is of course not much use, since we also wish to include mass terms and gauge symmetries. In order to do this we must first generalise our infinitesimal transformations to make them finite; a transformation through a finite spinor parameter 0„, is given by the operator exp 69. Arbitrary transformations through 0,, of the scalar field cb(x) will change its value in a one-to one manner. We can now construct a scalar quantity, (I)(x, 0, 0), which gives 0(x) for arbitrary values of Taking 0(x) as the starting value when 0,, = 0, 4)(x , 0, 0) = (1)(x), (12) CP and flavour in supersymmetry 205 we can find the general form of (I) by transforming 0; (x, 0, 19) = (1 + 69 + 1 42 ± 693 ± 694 ) 0(x) = 0(y) + f2-0X(y) + 00F(y) (13) where yP = e + ie0-49. We have used the identities 020a = 0 (because 00, has only two components which anticommute), and 013 00 = —20'190. These scalar functions are referred to as chiral-superfields. The Taylor expansion in Os of any function terminates at the 00 term, and consequently any general function of y and 0 is a chiral superfield, and its components must transform as the components of a chiral supermultiplet. Beginning with 0* results instead in anti-chiral superfields (denoted (I)) which are functions of y* and 0 only. Chiral superfields enable us to build up more general Lagrangians, once we notice that (trivially) the product of two or more chiral superfields is also of this form and therefore is also a chiral superfield. So, for example, the F-term of the product of two chiral fields gives mass terms, 1 a., £Yuk = Aifievel'fFas k + h.c. = AiikFi0j0k — Awe ze0ja0k + h.c. (15) F The generic indices i, j, k stand for summation over gauge representation and/or field labels (e.g. Higgs, matter etc.). Terms of order higher than three make the theory non-renormalisable and will not be considered further. The most general interaction Lagrangian is therefore of the form Lint = W(CFZ)I F h.c. (16) where W(4)i) is a third order polynomial of the chiral superfields, known as the super-potential. In terms of the superpotential, the Lagrangian is ,aw 1 a2w Lint — Pi h.c. (17) aoi 2 °' 3 aoiao, where only the scalar part of the potential is taken, and summation over indices is implied. Finally, by using the equations of motion of Ft = aw (18) we are able to eliminate it, to give 2 OOP . Lint —r1 102147 + h.c. o 2 3 00i00j In particular we can see that the supersymmetry potential is positive definite, and has a minimum at aW/aoi = 0 where it vanishes. .C,,, = (13.2 F + h.c. = OF — 2 1O'Oa + h.c. (14) It is easy to check that this is invariant under the infinitesimal supersymmetry transfor-mations. For the Yukawa interactions we need the product of three superfields, (19) 206 Steven Abel So we have derived a Lagrangian which is able to describe scalars and their fermionic superpartners. For this introduction we have been avoiding the notion of superspace; the interested reader can consult Appendix B which explains the connection. Now let us progress to the simplest non-trivial example, the top quark Yukawa, in order to see the promised cancellation of one loop divergences. The top quark Yukawa To write down a top-quark Yukawa we naturally define the superfields to have the exactly the same charges as they would in the Standard Model: Hu = hu flu Fit,. Q= -4 q FQ te = I', te Ft, Thus we have Wtop—Yukawa = AtQffutc (20) leading to a Lagrangian of the form Ltop—Yukawa = -.tghutc -.t4(hutc) - At (qhu)i, + 41/it:TX + 41hu-412 414T,12- (21) It is customary to write the superpartner of any field as that field with a tilde and we shall follow that convention here. Now consider the one loop contributions to the hu mass squareds. The internal fermions can only be the pair q and to , whereas the scalar 'bubble' diagram gets two contributions separately from -4- and t. Including the symmetry factor of 1/2 in the scalar diagram, and -1 for the fermion loop, we find exact cancellation as promised. As yet we have not included the gauge fields (we have no spin one fields), however we can guess what their Lagrangian might look like from the above. Supersymmetry makes fermions and bosons indistinguishable, so not surprisingly the net effect on the Yukawa coupling was to introduce duplicate `super'-Yukawa couplings with the fermions and bosons permuted (the first three terms in Equation 21). The same duplication hap-pens in the gauge sector as well. Here the physical components of the vector supermul- tiplets are {AA (where A is a group index) which are respectively the gauginos, gauge bosons (of the Standard Model) and auxiliary fields. The matter-gauge-matter couplings become _ vA = gXto-iivVi + i-V2g0i* At.t X + h.c. (22) Where we use the usual notation that vri = v',4Ti7. As with the Yukawa, the introduction of supersymmetry has simply added supersymmetric counterpart couplings with the roles of boson and fermion reversed. The inclusion of gauge symmetry and the supersymmetrisation of the Yang-Mills terni involves so-called vector supermultiplets. It is possible (although much more arduous) to build these supermultiplets beginning with the lowest component, as we did above for the chiral supermultiplets. However, for the discussion of CP and flavour phenomenology we now have everything we need; the reader interested in these more technical aspects can turn to Appendix C where a discussion of the total gauge invariant Lagrangian is included for completeness. CP and flavour in supersymmetry 207 3 The softly broken MSSM We now have a general framework for building supersymmetric gauge invariant La-grangians. Our aim is to build a model which includes this, and yet mimics the Standard Model at low enough energies. A consensus seems to have been established as to the most likely way this might happen in Nature. This is referred to as the softly broken Minimal Supersymmetric Standard Model (MSSM). There are a number of influences guiding the form of the MSSM, one of the most overriding being 'naturalness'. A case in point is the scale of supersymmetry breaking itself (clearly a requirement). Most would agree that this should not be too high, otherwise fine-tuning is re-introduced. Because of this, a ceiling of a few TeV is usually placed on this parameter. One should bear in mind however that there are no other upper limits on the scale of supersymmetry breaking. This problem is compounded by the fact that there is still a large number of additional parameters left in the model (supersymmetry breaking scalar masses for example). Often these are reduced by considering a particular kind of supersymmetry breaking mediation - for example mediation by supergravity (local supersymmetry) or gauge mediation, but also they are reduced by considering experiment itself. (For example, the absence of large FCNCs is often cited as the reason for taking scalar masses to be degenerate at the GUT scale.) Because of this, getting genuine predictions out of supersymmetry is difficult, and usually the 'prediction' is merely that supersymmetry is consistent with a large (g —2), or electric dipole moment of the neutron or b s-y or whatever. In fact, usually supersymmetry is also consistent with values of (g — 2), etc. which totally violate current experimental bounds, thus lending an air of `just around the corner'-ness to the whole business. This difficulty really arises out of our ignorance about the nature of supersymmetry breaking and leads to what has become known as the supersymmetric flavour and CP problems. These general problems will be illustrated in this section, and expanded upon later. First however, let us present the superpotential of the MSSM. 3.1 The MSSM superpotential Since we wish to stay as close to the SM as possible, we assume that the same gauge groups and representations will appear. Thus, the group will be SU(3), x SU(2) L x U(1)y. The gauge bosons (g, W1,2,3 and B) are accompanied by fermionic superpartners, the gluinos, winos and bino. The matter and Higgs fields become chiral supermultiplets. These are usually represented by capital letters, so that we have Q = L = {l L, I L }, Uc fu R* , utR1 and so on for each of the quarks and leptons, and H = h} for the Higgs doublet. L, Q and H are SU(2) doublets with ("L) L) Ch+) (23) The superpartners are the squarks, sleptons and Higgsinos respectively. We now encounter a minor problem. In the Standard Model we were able to make do with only one Higgs field, to generate two masses. The Yukawa couplings which did this 208 Steven Abel were of the form £Yuk = AelLheR Ad4Lhdft AOLTIUR h.c.. (24) The h scalar has hypercharge Y = 1, and the h scalar is given by h = iT2h* with I = —1. (The Ti are the Dirac matrix generators of SU(2), acting on isospin indices, not to he confused with the as.) Clearly, if we turn these fields into chiral supermultiplets, the two quark mass terms cannot both be derived from a superpotential since if h corresponds to a chiral superfield then h* corresponds to an anti chiral one. An additional problem comes from triangle anomalies. These are a potential problem in any theory with chiral fermions, and their disappearance requires that E, Y = EZ I? = 0. In the Standard Model, these cancel exactly, but now this is no longer the case since we have introduced an extra fermion, the Higgsino, which also couples to isospin. Happily, both these problems are solved by the introduction of a new Higgs superfield with hypercharge —1, which replaces h. Our total particle content has the following assignments and transformations under the Standard Model gauge group, ° L = (vL H ) (1,2,-1) H, = 1 H- (1,2,-1) , ut, (3, 2, 1/3) H2 d L H2° (1, 2, 1) (25) EC (1,1,2) DC (3,1,2/3) Ue (3,1, —4/3) where H, and H2 are two entirely separate superfields. (There is, by definition, no right handed neutrino in the MSSM although clearly experimental data now suggests one. Neutrino physics requires a review in itself and we shall not attempt to cover it here.) The Yukawa piece of the superpotential (taking a little more care with our isospin indices for a moment) may now taken to be WYuk = AEEcHTE'L ADDcHireC2 AuUc HTE1Q h.c., (26) where we have suppressed colour and generation indices. The e' = iT2 is the same matrix as f, the prime being there simply to remind us that its indices act on isospin. SU(2) gauge invariance follows from the fact that (IT' = — (Ti )TE, so that (eiA.T)T (1 ei' = e'. The extra Higgs superfield has introduced another parameter; tan = (14)/(H?). The fact that this ratio can be large is seen as a possible explanation for the large value of the top quark mass. In addition to the Yukawa terms, we will require some Higgs self-couplings in order to generate the correct electro-weak breaking; WHiggs = PHITE'H21 (27) where it has dimensions of mass. In order to get the correct scale for breaking (i.e. the correct values of Mw and Mz) it turns out that we need to have µ, = O(Mw). The fact that we have an additional dimensional parameter (besides the scale of SUSY breaking) that needs to be set by hand, is known as the ̀ µ-problem,'. Q = CP and flavour in supersymmetry 209 3.2 Soft SUSY breaking If supersymmetry is the correct extension of the Standard Model, then it is expected to be softly broken by dimensionful parameters (such as squark mass terms), in order for the broken supersymmetry to still remove quadratic divergences and provide a solution to the problem of maintaining a hierarchy between the electroweak scale and the Planck scale. The physics of the breaking is not yet well understood and there are several mech-anisms that can achieve it. To take account of our ignorance one can write a general, gauge invariant, Lorentz invariant, effective Lagrangian; r = rSUSY rsoft (28) where the first term preserves supersymmetry invariance and the second violates super-symmetry and is given by (dropping the E') —rsoft = (1113-# M2WW M:13:6 h.c.) + Alzrej*Cj + mzile* ric + AViy*15c + + A/Ake* k AuAurle012 ADADTice2111+ AEAE:EcLili h.c. + rrq/AK H2 + m2H1 Hi Hl + (BH2H1 + h.c.) where all generation indices have been suppressed. The most general softly broken MSSM has a total of a 124 physical parameters. The count goes as follows; 6 of the parame- ters come from gaugino mass terms of the form A = I M2 I . The squark and slepton masses are in principle 3 x 3 Hermitian matrices with complex matrix elements, contribut-ing 5 x6 x 2 = 60 parameters. Trilinear couplings between the sfermions and Higgs bosons are arbitrary 3 x3 complex matrices which constitute 2 x 9 x 2 = 36 parameters. Addi-tional parameters arise from the µ term, resulting in 33 mass eigenstates, 43 phases and the CKM angle. 3.3 Flavour and CP problems: mass insertion approximation Given the most general form of supersymmetry breaking in (29) we see immediately that there are likely to be phenomenological problems with both flavour changing and CP violating processes. For example, using the new supersymmetric contributions to the Lagrangian in equations 21 and 22, we can construct a diagram (Figure 2) with an internal wino and squark—one of many possible supersymmetric contributions to b .5-y. h orW ),, g or x S t, E Figure 2. A supersymmetric contribution to b .5-y. (29) 210 Steven Abel The diagram includes a flavour changing mass insertion on the squark line and our lack of knowledge about supersymmetry breaking means that in principle this insertion could be as large as any of the other terms. Likewise, the supersymmetry breaking terms can have arbitrary CP violating phases and it is not difficult to construct diagrams that contribute to electric dipole moments (of which more later). The appropriate way to treat these problem generally is to diagonalise all the mass matrices and calculate the diagrams with the mass-eigenstates. However there is a useful approximation, the mass-insertion approximation, that can be used when the supersym-metry breaking is approximately generation independent. This type of situation is usually motivated by particular kinds of supersymmetry breaking mediation, for example media-tion by gravity resulting in mSUGRA models. In such models the canonical assumption is that the supersymmetry breaking terms are flavour universal at the GUT scale. Thus the most commonly considered scenario has rn-Z, = dig mo '177 MA = M112 = Apii = AEi = (30) Together with the parameters bt and B we then have 5 independent parameters. 11112 can be determined by electroweak symmetry breaking, and traditionally we trade µB for tan /3, the ratio of Higgs VEVs, leaving us with 4 independent parameters, mo, A/1/2 , A, tan /1. In addition there are 2 physical phases OA = arg(AM,*f2) and Oa = arg(a/111f2 ). All other phases can be rotated away. Typically these phases must be small in order to satisfy all electric dipole moment (EDM) requirements (including that of the mercury ED1\1 [6]). This model, sometimes referred to as the Constrained MSSM (CMSSM), is an example of a Minimal Flavour Violating model (MFV model), defined as models where all flavour and CP violation is attributed to the Yukawa couplings. Typically, running the renormalisation group equation from the GUT to the elec-troweak scales, will result in flavour non-universality thanks to the contribution from the Yukawa couplings, however they will generally be small. (Note that splittings between the squared masses of left and right-handed scalars, and also between squarks and sleptons will generally be large because of their different gauge couplings.) Thus, when considering a particular process, it is useful to consider a situation where the squark masses matrices are flavour universal and real, and to find the experimental limits on perturbations. This approach, pioneered by Gabbiani et al in [3], allows one to state the flavour and CP prob-lems more quantitatively since it leads to model independent bounds on mass insertions. One begins by defining perturbations from universal scalar masses, f m2-, as follows: (/ma-2)Liti3 (TIqA)LL (n qA)RR (31) 61,n m = Or 2 " (5q 73 2 772- 772- f It is then fairly straightforward to calculate the leading contributions to various processes by considering the diagrams with a single insertion. The approximation is represented pictorially in Figure 3. The fully diagonalised quark propagator would have diagonalisa-tion matrices for the masses, Via where i labels flavour and a labels one the 6 squark mass eigenstates, on either end. However, this is equivalent to treating the flavour changing g q CP and flavour in supersymmetry 211 Ma V. qa Figure 3. The insertion approximation for the quark propagator. as an interaction vertex (denoted X). The experimental bounds can then be expressed as limits on the 8's, given in Table 1 and Table 2. For b s'y decays with mi.-----500GeV the main contributions are give in Figure 4 and the limits in Table 3. CP violating processes constrain imaginary insertions. e.g. for M3 = mi=500GeV we have the limits in Table 4. Limits on- VIRe(51, ti~)2 1 VIRe(81Rii ) 2 1 0.11e(OLL )Re(61,Ri )1 ii = sd 0.04 0.004 0.003 ij = bd 0.1 0.03 0.02 ij = cu 0.1 0.03 0.02 Table 1. Limits from OF = 2 with 11,13(Mw ) = m5=500GeV. Limits on- 1 6i,rij 1 1 8er,ftij I ij = ,ue 0.008 2.10-6 ij = re 29 0.1 ij = 77..i 5 0.02 Table 2. Limits from li 1.0 decays with M1 = mi=100GeV. Figure 4. Main contributions to b s-y decays with mi=500GeV. 212 Steven Abel x = I t4117715 141,23 1 14%3 1 0.3 4.4 0.013 1 8 0.016 4 26 0.03 Table 3. Limits from b s-y decays with rn7=500GeV. E \I IIM(4L,2)21 < 0.002 VI/n2(OL12 )2 < 0.0006 EVE 1/m(61,L12)1 < 0.5 I/m(4,R12 )1 < 2.10-5 dn, < 6.10-26 lim(61,R,31 < 1.10-6 lirn(6'1,R11 ) < 3.10-6 Table 4. CP violating processes constrain imaginary insertions. Limits are for M3 = m-=500GeV. Satisfying experimental bounds/results on flavour changing and CP violating processes obviously prohibits significant perturbations from universality for many of these param-eters (although other reasons besides universality have been considered in the literature, and will be mentioned later). This apparent fine tuning is the quantitative form of the flavour and CP problems. Incidentally, in the case of the mSUGRA, all of these constraints are satisfied. Note that, had this not been the case (i.e. if running the RGE's in mSUGRA had resulted in supersymmetry breaking terms that violated these bounds) there would have been very little hope of ever explaining electroweak phenomena (e.g. symmetry breaking and masses) in terms of Planck or GUT scale physics. 4 The mass-spectrum and generic SUSY breaking The mass insertion approximation outlined in the previous section is useful for addressing the question: how far can we deviate from universality and still satisfy experimental bounds? However, there may be (and often are) other wildly different patterns of soft-supersymmetry breaking which can have a significant degree of non-universality and yet have cancellation of contributions to various processes. (One example is models with Hermitian flavour structure, in which the EDMs are suppressed). In order to calculate in more general scenarios, it is necessary to diagonalise all mass matrices and find the physical mass-eigenstates. The particle content and diagonalisations proceed in the following manner. First we collect the components with the same e.m. charges together into squarks, charginos, neutralinos: iiRi=1 .3 - 6'a= 1..6 ®W+ - x+ it,- ED W- e e fv0 e -* —• ',a= 1..4 (32) CP and flavour in supersymmetry 213 The sfermion mass-squared terms are given by f—mass—squared ( fn)iM- f2ii ( f RL) where the MZ are 6x6 matrices hi rn f (A f - p*R f ) MZ-Fm2f +M,2( -Q f sine Ow) cos 20 ( h.c. + + 11/1,2Q f sin2 Ow cos 2/3 with R1 = cot Q ( or tan 0) for /3 = 1/2 ( or -1/2) and tan 0 = (14)/(h7). We diagonalise this mass matrix by making a unitary transformation on the squarks Ma 6 - (VI Ng V--) dr, — d d d at) (35) The charginos have a mass term Mj = (33) (34) Lcharginos = (CV +, hd (36) where 1112 ‘,/-2-Mw sin 0 Mc = V//w cos 0 and we require a bi-unitary transformation to diagonalise the mass matrix: m±= vC* 17-1 x C c For the neutralinos we have: Lneutralinos = (k, WO, hOd, Tit) i MN.. 130 / !To hd where MN is symmetric: MI 0 -Mz sin 0w cos 0 Mz sin 0w sin 0 T 1112 Mz cos Ow cos -ltiz cos Ow sin T T 0 T T T 0 with VNMN V N (41) MX° The quantities which then enter calculations of FCNC and CP violating processes are then of course the diagonalisation matrices, VQ, Vc, Uc VN which appear on the vertices. MN = (37) (38) (39) (40) g oc —+ —0 ?)226el fR 214 Steven Abel 4.1 An example calculation and the SuperGIM mechanism All calculations follow more or less the same format, so let us examine just one in detail, the supersymmetric contribution to the EDMs. EDMs are quite complicated beasts, and get contributions from electromagnetic, chromomagnetic as well as purely gluonic contributions shown in Figure 5. See references [4, 5, 6]. Figure 5. Leading SUSY contributions to the EDMs. The photon and gluon lines are to be attached to the loop in all possible ways. For simplicity I will consider only the electromagnetic contributions to the fermion EDMs. These are given by [5] c/ (E) —2as 6 M--- 9 ` T,„,, (r1k) g n_ p ( g 37r .'=1 ''''' q ) M2 ‘q '-' M2 ) ' = e qk qk 2 2 d X± (E) 6 2 MX+ Mx+ Mx+ —aem = E im(r.ik) ii12 [Qii-B( 14-2' ) + (Q.- (22,-) A Mi )1 , e 47r sin2 Ow k=1 i=1 d A dk dA u Xd .4 + (E) 47r sin2 O —aem 6 2 M + (M2x+ \ 7n2 + e E E im(rdik) r B\111 ) + (Qd — $9;1) A c 11172! )1 , w k-1 i-1 d ° (E) I e = aem Ei Imo? f a ) M 2 ) nim2; ) , \ mX° n_ 13 ( X, (42) 47r sin Ow ' k=-1 i=1 fk fk where f qik = e-i4)3 V-q4k17:ik , (43) with 03 being the gluino phase. The chargino factor rfik comes from products of vertices with attached charged Higgsino or wino: and analogously for the electron. The Af are the Yukawa couplings Au = ruik = Aul'A2Viii k (UL Vilk r dik = AdUA2 mu rnd,e Villk(V il l7:1k — Aul A21/:4k) — AdULV;4k ) , Ad,e = (45) Oinw sin 0 Ornw cos 0 . uk ilk 2 dX+ (E) aem 2 MX+ Im(Fet) A Mrci- 2 ) ' e 47r sin2 Ow m- (44) CP and flavour in supersymmetry 215 The neutralino vertex rif ik is given by rifik = [— \/ {tan Ow (Q f — I31)Viv1i + 1.3 f N 2i} 11 k 4k ] X [f tan OwQtVN1iVj4k — A117 N bi Vi 1 k] (46) where /3 is the third component of the isospin, b = 3 (or 4) for /3 = —1/2 (or 1/2). The loop functions A(r), B(r), and C(r) are defined by 3 r+ 21nr A(r) = 1 ( 2(1 — r)2 1 — r) B(r) = 2(r 1 1)2 (1 + r + 2r In r 181n r C(r) = 6(r _1 1)2 (10r 26 + 1 — r 1 — r) (47) Note that the indices '1' and '4' appear in the above because for the EDMs of the down, up and electron we pick out the left and right chirality fermions of the first generation. This completes the electromagnetic contribution to the quark and electron EDMs. We can now see in a more general way why flavour universality gives cancellation of contributions. Consider for example the gluino contribution to EDMs. In the event that, the squark masses are universal and equal to A/2g this can be written as d9 (E) _ Q B g —2a, m- m-- 6 rlk e 37r M? M2 i g _( q q ) • q k=1 The last factor in this expression is im(rgik ) =e-i43 (1/0,141) 41 = (), (49) and consequently the gluino contribution exactly vanishes because of the unitarity of the diagonalisation matrices, Vq-. With exact universality the leading contribution is from the chargino diagram, however these are proportional to Im ELI (Fdik ). Again unitarity means that the main contribution here is from the phase on the µ term (unfortunately the least well understood of all the parameters in the MSSM). Note that even in the CMSSM, unitarity is broken between left and right chiralities at the weak scale because of the different gauge coupling contributions to the mass-squared RGEs. Because of this there is also a small dependence on the phase of the A-term (OA ) even in the CMSSM. A similar sort of cancellation occurs for flavour changing processes, where it is referred to as the superGlM mechanism. For example consider the chargino contributions to,MK shown in Figure 6. The Ks — K L mass difference is given by where K = 2Re(M12 ) (50) 2 6 2 D(I)14, M12 = fK E E A2ik A1ile A2je A*1 jk 24 kk' ij 2 ) (51) 1172 2r1rir 1 — r (48) 216 Steven Abel SL dL dL SL Ub Figure 6. Chargino contribution to AMK . and 3 Afik = Vc*i l Vii*fk — E A, ff, _z_ u(f , +3)k (52) and where f , f' label flavours, and D is a dimensionless one-loop integral which is of order one if the chargino and squark masses are similar. Here the first term in A comes from the fldW term and the second from the a/it term in the Lagrangian. Again we see that degenerate squark masses leads to suppression because of the uni-versality of the diagonalisation matrices. Indeed because only left handed external quarks are involved in this particular diagram, the contribution to AMK from the first term in the As is proportional to the mass splitting in the first two generations of left handed up-squarks which in the CMSSM is driven by the renormalisation group effects of the Yukawas: 2 AMK oc Mc • (53) rn- In addition to this contribution there are contributions involving cross-terms in the As, but because we need two cross terms (there are 4 As) they will be of the same order of magnitude. This overall suppression is the super-GIM mechanism. 5 Aspects of CMSSM phenomenology The universality of supersymmetry breaking is found not to hold in most string derived models and in general supergravity models, and we expect at least some non-universality in the flavour structure of the A terms. However because of the super-GIM mechanism, the CMSSM is the minimal model from the point of view of avoiding nasty FCNC and CP violating contributions. It is important therefore to estimate what the new super-symmetric contributions to these processes are likely to be in the CMSSM, in order to provide a kind of lower bound on the expectations for new physics. These effects come from the running of the renormalisation group equations, whence the flavour dependence of the Yukawas enters the supersymmetry breaking. These effects are small and so the mass insertion approximation is valid. However, there is no general way to estimate the resulting size of the (5s induced by the renormalisation group, because the RGEs are non-linear and in any case some of the parameters depend on the minimisation of the one-loop electroweak potential. Thus a useful approach is to concentrate on the domi-nant contributions in the CMSSM (and indeed Minimal Flavour models). As we have CP and flavour in supersymmetry 217 seen, contributions from diagrams involving gluinos are suppressed in minimal flavour models (this is a general result) and the neutralino contributions are also suppressed in the CMSSM. Hence the dominant contributions are those involving charginos and charged Higgs particles. This greatly reduces the number of parameters and means that one can survey the parameter space rather generally without having to perform the minimisation of the one-loop potential. A study along these lines was carried out in [7] and recently improved and updated in [8] where many more rare decays were also considered. (This reference provides a useful compendium of expressions for calculating various rare processes in SUSY, and summarises the new operators that are introduced into the basis for Heff. . Many of the expressions for the supersymmetric contributions are also found in [9].) Here the dis-cussion will concentrate on the supersymmetric contributions to the Wilson coefficients calculated at Mw, so the reader is referred to that reference for more information on how to turn these into the measured parameters.) The later study includes NLO-QCD and QED corrections as well as constraints from electroweak precision studies, exclusion bounds on Mh, and the B -+ Xs-y decay rate. The input parameters required for such a study are only those that effect the chargino and Higgs masses and mixings: m- m- , t, x m- • tan 13, met,, p, ml t2i (54) where.12 are the heavy and light stop masses, is the stop mixing angle, 111,-X the lightest chargino mass, and mi the common slepton mass (for the rare decays). Note that there are more parameters here than in the CMSSM (including the breaking of electroweak symmetry), but in performing a study using these parameters one is addressing a wider class of models, including all MSSM models that have Minimal Flavour Violation (for example those in which the scalar masses are flavour independent but not necessarily degenerate). One then scans over reasonable values of these parameters. Again what constitutes 'reasonable' is open to interpretation. However, imposing masses less than 1TeV seems appropriate in order to avoid reintroducing fine tuning. The parameter 1/11 can typically be larger than the range 1µ1 < 500GeV chosen in this study, although larger values reduce the mixing in the chargino and therefore its contributions. In addition values of large tan13 > 6 are excluded in this study. Although the results tend to asymptote for medium tan 13 there is a region of very high tan 13 which is interesting from a theoretical point of view; it allows Yukawa unification at the GUT scale. Unfortunately this implies large b-quark Yukawas and hence the top-quark dominance approximation made for the low-medium tan 13 studies is no longer valid. Before summarising the results, let us briefly discuss the main effects of supersyin-metry. For the kaons, the most interesting effect is that one tends to find a depletion of EVE, unfortunately contrasting with what we would perhaps like given the fact that the measured value is slightly larger than one might expect from the Standard Model (modulo uncertainties in hadronic matrix elements). The dominant contributions are from Z° and photo-penguin diagrams involving charginos and charged Higgs. The two contributions come with opposite sign (the chargino being positive), so that there are indeed regions where an enhancement over the SM value can occur, although this is only of the order of 10%. On the other hand the depletion can be up to 40%. A summary of the most interesting processes is given in Table 5. Again, the result for e'/e is plagued by hadronic uncertainties, whereas the rare decays are theoretically much cleaner places to search for 218 Steven Abel Process \ Ratio w.r.t SM max min EYE 0.53 1.07 K+ -4 7+07 0.65 1.02 K L -+ 7r°vP 0.41 1.03 K L —> 7r°e+ e- 0.73 1.10 B —> Xsiiii 0.73 1.34 B -4 /LA 0.68 1.53 Table 5. Ratios of CMSSM to SM results for various processes. signals of supersymmetry. One additional interesting feature of Minimal Flavour models is that they predict an absolute lower bound of sin 2/3 > 0.42 [9, 8]. Such a lower bound is possible because one can generalise the expression for the E hyperbola to include supersymmetric contributions. where 0.31 is we get [( 1 P)A2712Ftt + 0.31] the charm contribution. From 1.6 = A2nA R2 ° = 0.266 0.31n(Ftt )] (55) (56) (57) 27g - p) 1.6 A2Fte 0.226 2/3 sin = Ri RO772 1.242.13K This function may now be plotted as a function of the parameter Ftt which is a collection of all SM and SUSY contributions and is typically of order 1 in the CMSSM. (Note that it is not monotonic because the variables are not independent of Ftt .) A value of sin 2,3 less than this value would have been the first indication of flavour structure in soft supersymmetry breaking and arguably an indirect detection of supersymmetry. Unfortunately, at the time of writing, the value of sin 2/3 coming from B —> K5 measurements at Belle and BaBar indicate higher values. 6 Beyond the CMSSM and MFV Up to this point we have been looking at the minimal case where all flavour and CP vio-lation in the MSSM is attributed to the Yukawa sector. However, an interesting question to ask is whether there might be other, non-universal scenarios, in which some of these processes are consistent with experimental limits. This is an important question because string models do not generally predict the favourable CMSSM scenario very naturally. There are a number of ideas for suppressing FCNC and CP processes apart from uni-versality. They are: cancellations of SUSY contributions amongst themselves, flavour alignment, particular flavour structures (e.g. Hermitian or off-diagonal phases), decou-pling and dynamical (i.e. derived from minimising a string derived potential). In the CP and flavour in supersymmetry 219 next subsection I will discuss cancellations with respect to EDMs and make some general comments. Then I will mention decoupling and in the following section discuss some recent developments in the last of these, FCNC and CP suppression in dynamical string derived patterns of supersymmetry breaking. 6.1 Non-universality and cancellations When the flavour structure is non-universal at the GUT scale there can be cancellations to various processes. Intuitively it seems unlikely that they will occur simultaneously for different processes, and this indeed turns out to be the case when numerical studies are done, as we shall now see. Non-universality has been studied as a general idea in [10] but more recently there has been a resurgence of interest thanks to 'new' string inspired scenarios, motivated by explicit constructions of SM-like models in for example type I models. These models can lead to flavour non-universality which is relatively 'well-controlled', so naturally there has been some speculation as to whether a particular type of flavour non-universality might lead to cancellations 'across the board' as effectively as universality [11, 12]. The answer to this, at least for those cases that have been studied, unfortunately appears to be negative as we shall now see. The measurables that first received attention in this context were again the EDM's of the neutron and electron, both of which have rather strict experimental bounds. In [11] it was noted that for particular patterns of supersymmetry breaking there can be a simul-taneous cancellation of these parameters, even if the CP violating phases on the it term is large, much larger than would be allowed in the naive CMSSM with phases. This idea has since attracted some attention and has become known as the 'cancellation mechanism'. The analysis was based on supersymmetry breaking pattern inspired by so-called D-brane models in which the SU(2) gauge group was associated with a different brave from the SU(3) and U(1)y groups of the SM. The result is that the supersymmetry breaking gauginos masses can have different values at the string scale (assumed to he near the GUT scale). On the other hand the A-terms can have a good deal of universality. The overall pattern of supersymmetry breaking can therefore be assumed to be identical to that of the CMSSM with the exception of the gaugino masses which are of the form: = 1113 = idm3/2 (cos 0)01 .7112 = Om3/2 (cos 19)82 (58) where m312 Mw is the gravitino mass and the angles here are a convenient parameteri-sation of the distribution of supersymmetry breaking amongst the different fields. If we in addition allow an arbitrary phase, Ow for the µ term, then one finds large values of phases where the neutron and electron EDMs cancel simultaneously as shown in Figure 7(a). Unfortunately, when additional experiments are considered, the cancellation region begins to be excluded. In this case the experiment in question is the bound on the mercury EDM [6]. This parameter also has a cancellation region, but unfortunately it tends to cut across the previous cancellation region, and all three EDMs are only consistent with bounds in the trivial small phase region [6]. Steven Abel 2. 3x12 (PA X x.12 ov 0 0 x110 9p 220 3x12 Figure 7. The cancellation regions for EDMS of the electron (dark vertical curves), neutron (lighter vertical curves overlapping those of the electron) and mercury atom (di-agonal lines). The parameters are tan /3 = 2, of simultaneous n,e,Hg EDM cancellation in mSUGRA for the same tan /3 and m312. Note that in contrast, mSUGRA (CMSSM with phases) has a region of cancellation for all three EDMs as shown in Figure 7b which allows any value of OA but restricts < 7r/100. (The weak constraint on OA is a result of the small contribution of OA to the EDMs, as explained earlier). Adding a small phase to any of the gauginos separates the regions of small n, e and Hg EDMs and destroys this cancellation. In both cases the cancellation regions decrease in size at more realistic, larger, tan /3. Thus for this example the cancellations are not 'across the board'. Indeed, given the fact that the cancellation regions shift considerably depending on the adopted model of the neutron (valence-quark or parton), finding a (non-symmetry based) stringy underlying reason for cancellation would in any case be difficult. Although here I have concentrated on EDMs I see no reason why the conclusion would be different with any other CP or FCNC process unless there was some underlying flavour symmetry. 6.2 Decoupling This possibility is based on the fact that as the supersymmetric particles become heavier their contribution to FCNC and CP processes begins to decouple. For example, even if one allows 0(1) CP violating phases, the contribution to the EDMs will be negligible if the SUSY spectrum is sufficiently heavy [13]. Generally, SUSY fermions are required to be lighter than the SUSY scalars by, for example, cosmological considerations. So the decoupling scenario can be implemented with heavy sfermions only. Note that for FCNCs for example, the contribution is still proportional to (59) where i and j label different flavours. Thus if, for example, we put down the suppression of AiliKsus, to large first and second generation masses, we still require a proportionate universality in their values. So this scenario is in a sense 'universality where we need it'. CP and flavour in supersymmetry 221 Here the SUSY contributions to the EDMs are suppressed even with maximal SUSY phases because the squarks in the loop are heavy and the mixing angles are small. In Figure 8 we display the EDMs as functions of the universal scalar mass parameter mo for the mSUGRA model with maximal CP-phases ¢µ = OA = 7r/2 and m1/2 = A = 200GeV. We observe that all EDM constraints except for that of the electron require mo to he around 5TeV or more. The mercury constraint is the strongest one and requires (mo)decoupI. 10 TeV . (60) This leads to a serious fine-tuning problem. Recall that one of the primary motivations for supersymmetry was a solution to the naturalness problem. Certainly this motivation will be entirely lost if a SUSY model reintroduces the same problem in a different sector, i.e. for example a large hierarchy between the scalar mass and the electroweak scale. 2000 4000 6000 8000 10000 m0 Figure 8. EDMs as a function of the universal mass parameter mo . 1: electron. 2: neutron (chiral model). 3: neutron (parton model). A: mercury. The experimental limit is given by the horizontal line. Here tan Q = 3, m1/2 = A = 200GeV and 0, ,-- = 7r/2. The degree of fine-tuning can be quantified as follows. The Z boson mass is determined at tree level by 1 2 2 Tri2H, - Tr/ 2H tan213 2 µ • (61) = tan2 13 — 1 One can define the sensitivity coefficients [14],[15] 0 In Tn2z ln aF where ai are the high energy scale input parameters such as m1/2, mo, etc. Note that it is treated here as an independent input parameter. A value of ci much greater than 1 would (62) 222 Steven Abel indicate a large degree of fine-tuning. The Higgs mass parameters are quite sensitive to mo so, for mo=10TeV, we find cnio 5000 (63) for the parameters of Figure 8. For the universal scalar mass of 5 (or 3) TeV the sensitivity coefficient reduces to 1300 (or 500). This clearly indicates an unacceptable degree of fine-tuning. This problem can be mitigated in models with focus point supersymmetry, that is when mH, is insensitive to mo [15]. However, this mechanism works for mo no greater than 2-3TeV which is not sufficient to suppress the EDMs. Another interesting possibility is presented by models with a radiatively driven inverted mass hierarchy, i.e. the models in which a large hierarchy between the Higgs and the first two generations scalar masses is created radiatively [16]. However, a successful implementation of this idea is far from trivial [17]. One can also break the scalar mass universality at the high energy scale [18]. In this case, either a mass hierarchy appears already in the soft breaking terms or certain relations among the soft parameters must be imposed (for a review see [19]). These significant complications disfavour the decoupling scenario as a way to solve the SUSY CP problem, yet it remains a possibility. Concerning the other phenomenological consequences, we remark that the SUSY con-tributions to the CP-observables involving the first two generations (such as E, E' ) are negligible, although those involving the third generation may be considerable. The corre-sponding CP-phases are constrained through the Weinberg operator contribution to the neutron EDM, which typically prohibits the maximal phase 0A,, GUT= 7r/2 while still allowing for smaller 0(1) phases. 6.3 New contributions to FCNC processes As well as the question of how SUSY contribution can be generally suppressed with non-universality, some attention has been paid to the question of whether non-universality might be able to explain excesses of, for example, E'/E. One of the reasons for the em-phasis on non-universality is that there are rather general arguments that, show that, in the absence of a CKM phase, there are no new contributions to E, EVE or hadronic B° asymmetries even with arbitrary soft phases. Thus supersymmetric CP violation is intimately connected with flavour non-universality. CP in the presence of new flavour structures was reviewed recently in [20], mainly concentrating on enhance-ments to E'/e. Contributions conic from the chargino diagrams in Figure 9 as well as the usual gluino diagrams, and these were recently analysed in [21]. Thus flavour and CP constrain low energy supersymmetry but at the same time provide interesting new areas to search for supersymmetry indirectly. Generally we do expect new CP violating phases in the SUSY sector; however the conclusion of these studies is that, they will not produce sizable effects in E and E'/E unless there is also some flavour structure in the SUSY breaking. q q q q g,y,Z SL CP and flavour in supersymmetry 223 Figure 9. Chargino contributions to EVE'. 7 The future Supersymmetry breaking is clearly the biggest factor in flavour and CP phenomenology and yet remains the biggest unknown. Ideally one would like to have a picture of CP and flavour from first principles but at the time of writing this seems some way off. Nevertheless there are two hints to do with CP violation that some progress might be possible. The first hint is that in the SM the CP violation is insufficient to account for the observed baryon asymmetry. Secondly, general arguments show that the four dimensional CP is a discrete gauge symmetry in string theory. Thus any viable string scenario must break CP spontaneously. (See [23] for the original observation, and some more recent references on the general connection between supersymmetry breaking and CP.) If the CP violation responsible for baryogenesis is associated with the supersymmetry breaking sector then at some point in the evolution of the Universe these phases must, have been quite large. On the other hand the SUSY CP problem seems to indicate that today they are small. An interesting idea that connects with the stringy picture of CP violation is therefore that the spontaneous breaking of CP happens at the same time as supersymmetry breaking, and that at the minimum of the potential the CP phases are zero. Thus at the time of supersymmetry breaking there was sufficient time dependent CP violation to account for baryogenesis, but today these phases have been set dynamically to zero. Some progress in this direction has been made in recent investigations into the nature of CP in string theory and the precise way in which it can be violated, either by string theory moduli (describing the compactification) or by the Froggatt- Nielsen fields which are responsible for flavour structure in effective string models [24, 25]. There it was found quite generally that it is quite difficult to generate a flavour and CP structure in the Yukawa couplings (by whatever means) without at the same time generating large EDMs [25]. This problem, which is at once more specific and more serious than the supersymmetric CP problem seems to indicate that there is more to be learnt by adopting more of a 'top-down' approach. Eventually such approaches might shed some light on the origin of CP violation in string theory and even provide a direct connection between string theory and experiment. Certainly CP and flavour remain two of the most important aspects of supersymmetry phenomenology, and as we have seen may allow detection of supersymrnetry indirectly before the LHC. Beyond that, should supersymmetry be found, experiments directed at flavour and CP violating phenomenology will help us to pin down the parameters of the 224 Steven Abel supersymmetry breaking sector and allow us to piece together the type of mechanism responsible for breaking and mediating supersymmetry, providing a window onto string theory. In these lectures I have given a purely personal overview of the subject and have omitted many topics, my main aim being to equip the reader with the techniques rather than to be comprehensive. To finish, I will try to make some amends by summarising some of these developments and directing the reader to relevant works. Muon dipole moments have been discussed in [26] where it has been shown that a muon EDM may in the future be at the 10-22e.cm level and hence measurable. The relations between dipole moment operators have been examined in [27] where it was shown that the muon anomalous magnetic moment is correlated with the electron EDMs. The question of (g — 2), in mSUGRA with and without phases and universality has been analysed in citearnl. In mSUGRA there are correlations between this quantity and B. p-p+ (see Dedes et al in the same reference). Alternative approaches to spontaneous CP violation that concentrate instead on CP breaking in extensions of the MSSM with gauge singlet have been considered in [29]. Other possibilities for CP violation are reviewed in [22]. The neutrino sector (to which I have paid scandalously little attention) deserves a review of its own. As well as the direct interest in the flavour structure of the neutrino sector, various patterns of supersymmetry breaking in the neutrino sector may lead to observable effects in rare decays such as p ey and p eee, as well as interesting possibilities for CP and leptogenesis. The reader is referred to [30] and references therein. I have also been unable to cover R-parity violating models which obviously have interesting implication for CP and flavour. For a review see [33]. The masses of the Higgs sector are strongly dependent on the CP violating phase in the p term (something that was somewhat skated over in the text). This is discussed in [32]. Finally there are a host of other interesting ideas and I cannot even begin to summarise them. One of many intriguing examples is the idea that all fermion masses are actually the result of radiative corrections and therefore soft. This is discussed in [34]. Acknowledgments It is a pleasure to thank the organisers for hosting and organising this School and for giving me the opportunity to contribute. The first part of this course was developed with the help of Jeff Forshaw whilst we were both in RAL, to whom I am indebted. I would also like to thank my collaborators Shaaban Khalil and Oleg Lebedev for their contributions to some of the topics I discussed here. 11 0 (68) CP and flavour in supersymmetry 225 Appendix A: Conventions We here briefly recall the essential features of the \Vey' notation for the discussion of Section 1. In the Weyl representation, a four component, Dirac fermion looks like where X and X, are two-component spinors (a, ,3 = 1, 2). The gamma matrices are ( 0 a-P irtt 0 (64) where az' = (1,o'), .TylL = (1, —a-1), and ai are the Dirac matrices. The -y5 matrix looks like . 1 0 'Y5 — z 0 —1 so that the chiral projections (given by R/L (1 ± i75)/2) on the fermions are, 0 0/, = I X 0 ) — Xc) The suffix 'c' refers to charge conjugate, since = 2 a 2 7(4 i0.2X so that Xca = —ia„,20)-(,*°. The matrix 0 Co -= -i(7,20 = acts like a metric tensor for the unbarred spinors. For example the invariant mass terms are 'OP = X, Xp + h.c. ci3aX„Xo + h.c. = ( 14)a(0.0)+ h.c. (69) Because of this we identify the complex conjugate of a field, X*/3 with V. (The last line we included to make the point that we may identify the two component spinors in terms of the original Dirac ones, so that (Pt)a 02,. The dagger we shall reserve to indicate an operation on the Dirac spinors.) This means that (nX)f = Tn. Similarly, the complex conjugates form their own representation of the Lorentz algebra, and have their (65) (66) (67) 226 Steven Abel own Lorentz invariant quantities. To avoid confusion the indices of the latter are written with a dot, so that —Ea. /3. = a2 . ao' and, in general, Lorentz invariant quantities are written as (70) xana, x- 0 at . (71) and so on. Appendix B: Superspace In the text we saw that the transformations of the {0, X, F} supermultiplet, could be expressed as the behaviour of a single scalar quantity, the superfield. The eight 'coor-dinates' (x, 0, 0) are known collectively as superspace. Supersymmetry transformations are equivalent to transformations in 0. Transformations in Minkowski space are gener-ated by the momentum P„, which is realised over space-time coordinates as a differential operator P, —ia,, so that if x x + a, then exp(ia-P) induces the transformation —> 0(x) + a4a,0(x). In fact Pp is the conserved charge associated with the transfor- mation x x + a, which, by Noether's theorem is Pp= fdx3 —a, f — (5p° L (72) Of where f represents summation over all fields (fermions and bosons) in the Lagrangian. In a similar way, transformations in B can be written in terms of differential operators acting in the superspace; = e + (73) where Q„ = a a aoa P Qa = a Jo-a/ea,. (74) a0A Acting (infinitesimally) with these operators on the coordinates gives the following trans-formations; bo x = ioae — ieu0 (50 = (50 = e (5. y = 2i0a-e. (75) It is easy to check that, firstly, (I)(y, 0) + kl)(y, 0) = (I)(y + 2i0oC0 + e) = (I)(y, 0) + 6“b + -\/206X + 00V (76) CP and flavour in supersymmetry 227 so that the differential operators above do indeed produce the infinitesimal transforma-tions, and secondly, that the momentum is related to the supersymmetry generators as, {Qa,Q6 } = (77) all other anti-commutators vanishing. This is one piece of the supersymmetry algebra and leads to Equation 7. The remaining pieces are provided by the Poincare algebra of normal special relativity (i.e. the algebra of commutations of the momentum P, and six Mpu generators of the Lorentz group - three boosts Ao and three angular momentum - which represent currents conserved under a Poincare transformation xP Aix" + 0). The generator CL has a current associated with it too, the super-current. This is simply the No-ether current for the infinitesimal supersymmetry transformations given above; s„, (2xia a, aw = o (o-,X j),„) (78) with Qc, = f dx3S.o• The algebra that we have described is called N = 1 supersymmetry, because there is one generator Q. Models with N generators are referred to as having N-extended supersymmetry, and their multiplets become increasingly complicated. N > 8 is not normally considered because it requires helicities of 5/2 and higher. Besides, we also have more than one graviton for these cases. N = 4 is the largest value of N that we see corning from heterotic strings. However, N = 1 is favoured because it allows chiral generations to exist (in the sense that we may have different representations of left and right handed fields). Appendix C: Gauge invariant supersymmetry Using the technology presented in the main text, we are already able to construct gauge invariant interactions such as Yukawa couplings using the superpotential. However gauge invariance also requires some modification of the kinetic terms, as well as a Yang-Mills term. This appendix is devoted to the task of determining gauge invariant expressions for these and is included for completeness. I should point out that the calculations required to produce the results of this appendix become more tedious as we progress. Because of this it is not possible (and would not be very useful) to give their derivation in the same detail that was possible for the superpotential, and the main aim will be to present the flow of the argument.. Gauge invariant kinetic terms Our first task is to introduce vector-superfields. Vector superfields are the other convenient way of deriving supersymmetry invariant Lagrangians. Like the chiral-superfields, they are dimensionless, but obey V(x, 0, 0) = V(x, 0, 0). (79) In fact (in a roundabout way) we already know how to construct a supersymmetry in-variant Lagrangian from such a field; the kinetic energy terms may be derived from the K(x, 0, 0) = c14. (80) Clearly this obeys (79), and Taylor expanded in Os, it becomes K(z, 0, 0) = O*0 +'(0X + --#X) + 000* F + 000F* — OcrAB (Xo + 2cba,0* - Jo* 8,0) + eee6 (i/Na,,,(Troax,„0*)) - (2fia„(0-P x 0)) + 8888 (2 (R0-x+axax)--1218012+ 4 -1 020* +-14 o*a2 0+F F*) . (81) N otice that the series terminates at the 0000 term, which is the highest possible term. Also notice that, apart from some total derivatives, the final term is the kinetic energy piece of the Lagrangian. Traditionally, the 0000 term of a vector field is referred to as the D-term, so that we may now write the kinetic energy more succinctly as LKE = (82) D As we have seen this term transforms into a total derivative under a supersymmetry transformation. In fact this is a general property of D-terms, which is perhaps not so surprising when we realise that D-terms, like F-terms, are auxiliary fields which do not propagate. In order to show this we need to find the supersymmetry transformations for the components of an arbitrary vector superfield, which we can write in its Taylor expanded form as, 228 Steven Abel vector superfield V (x, 0, 0) = C(x) + 0((x) + 0((x) + 00G(x) + 00G* (x) — OuP0vp(x) + i000 (A(x)6 + — i000' (Ac(x) — a:68,1 1 + 0000 1 2D(x) + —4 a2C(x)) . The spin one component vu lends its name to the entire vector superfield. In gauge superfields, the components A, A and v# will be the gaugino, anti-gaugino and gauge boson respectively. There are a number of ways to determine the supersymmetry transformations of the components in this multiplet. We could perform the same operation as in the text for the chiral supermultliplet (i.e. begin with (5. -C(x) = + and continue up through the multiplet, checking at each stage that the supersymmetry algebra operates correctly on each component), or we could deduce it by comparison with K(x, 0, 0), or most simply, we could extract the components of V (x+i0cre—iecrO, 0+e, 0+e) = V (x, 0,0) + (50/ = V (x, 9,0) + (eQ + eQ)V (84) where Q,Q are the differential operators defined in the Introduction. The result for the last three components is 6AI (x) = iGtDA _ 21 (ape )usFiit viiA(x) = (eo-PX A + AA a" xi) — aµ (eo-P (A + 8-DA (x) = ( C/Pap A A ali AAe) . (85) (83) CP and flavour in supersymmetry 229 The D-term transforms into a total derivative as required. Having determined this fact, let us now try to introduce gauge invariance into our kinetic term. Under a gauge trans-formation, we wish our supermultiplets to retain all their properties, so that the natural extension of the transformation 0: = (el 03 (86) (different A, with )lii = AAT21, and A, i, j as group indices) is (I) -4 = (e1A),i ci (87) where now A is a chiral superfield (remember that the product of two chiral superfields is another chiral superfield). From now on we shall drop the explicit gauge indices. The first thing to notice is that our kinetic term K,(x, 0, 0) _ 4,e -inein4, (88) is not gauge invariant, since the two exponentials do not cancel. It can be made gauge invariant by sandwiching another superfield with the correct transformation properties, between the chiral superfields. This field is the gauge-superfield, and since the kinetic term must transform as a vector superfield, it is also a vector superfield. Conventionally the gauge superfield is defined as g17,3 = gAVATill, such that the kinetic term is K(x,e,#)= (Te29v.:1). (89) The constant g will turn out to be the conventional coupling constant. For gauge invari-ance, under a gauge transformation V must transform so that it cancels the troublesome phases in Equation 88. That is, we must have e29 7 ezKogy e-in. (90) We now have nearly all the ingredients we need for our Lagrangian. The kinetic energy term is found, as before, by extracting the D-term which transforms as a total derivative, but is now gauge invariant too, (Te (DI £KE = 2gV D. (91) In order to extract the D-term, we need to expand the exponential. To do so, we first make a gauge transformation such that C, G = 0. As we shall see below, by examining infinitesimal gauge transformations, one finds that this gauge, the Wess-Zumino gauge, is always possible. In the Wess-Zumino gauge, things are particularly simple since, VA (S, a, 0) = — 00-P4VA (X) + i000A(x) — MO.\ + 10000D(x), VAVB = 2000froA,, VA VB Vc = 0 (92) so that e2917 = 1 + V + V2/2. After some lengthy manipulations and Fierz-rearranging, the kinetic terms in the Lagrangian are found to be £KE = —iXCYPT,p1 — (DpOt DPO F* F + (0* AX — XAAO) + gODO*. (93) 230 Steven Abel Note that there are suppressed indices floating around here. For example we have g0.0.0* = E Aii gA0i DAT4(/):7 and gcb* AX = >Aij giv/Til,\,1 Xic, where gA allows for group factors and U(1) charges. The derivatives are the usual covariant ones; Dpi, = oi + igAT Av 23 AP gAfBAcvCA. DµAB = 6ABaµ (94) In order to maintain the Wess-Zumino gauge, supersymmetry transformations must be accompanied by the required gauge transformation. That is, one performs an infinitesi- mal supersymmetry transformation on the gauge superfield. This generates new C, G terms which are found to cancel for a certain additional infinitesimal gauge transforma-tion on V. Thus the purpose of Wess-Zumino gauge is to make explicit all the physi-cal degrees of freedom. (Explicitly, the supersymmetry transformations give 5 C = 0, (V. = = 20, and the gauge transformation which cancels it is given by A = {0 N/ —0 }.) The combined transformation we shall refer to as a supergauge transformation 6r. Roughly speaking, this changes the derivatives in the supersymmetry transformations into covariant ones. That is, for the chiral superfields we have fix(/' SSGX = + -\/eF 6'r F = if2- .54-LDA X + 2g0Ae, and for the gauge boson and gauginos and D-field we have 6iscActa (x) jeaDA _ 2 _1 (0./ivaiFlia, \ bisGvprA (x) (aPNA — A Ao-Pxi) ss.GDA (x) (ecrADABon. Dp/30.1 ) The Yang Mills terms This completes the kinetic terms for the matter fields. The final terms that we need are the Yang-Mills terms. These can be found as in the text by forming an additional chiral superfield Wa(y, 0) out of the three physical fields, Xn , FiAi„, and D A. Since the gaugino is the component with lowest dimension, we begin by defining; W,,, (x, 0, 0) = (x). (97) This chiral superfield is a function with dimension 3/2 and helicity 1/2, and is the field-strength superfield. Its general form may be found by repeating what we did in the main text but this time for finite supergauge transformations. Making an arbitrary supergauge transformation, and using the transformation rules for the components above we find, VV01(y,O) = (559'GVV;),(x,0) zAc4 ( y) ± 00 (61,3j)A + 2 \ Ac ( 0 1.1511,3,FpAu) ± 090.A . DAC act P (98) where = `-'11' v av — YAJBeu4 fA ,,kt uv,B (95) (96) CP and flavour in supersymmetry 231 The required, supergauge invariant, Lagrangian can be formed in the usual way by taking the F-component of a combination of these superfields; it is given by LYM 4Tr A WaA ;F + Oa W— A F * ) Fµ\ A 6Ai„fTµA^CL2 DA DA — A (99) in the Wess-Zumino gauge. The full N = 1 supersymmetric, gauge invariant Lagrangian is given by the sum of equations 91,16,99: = £KE + Lint + LYM (100) This is the main result of this Appendix. Those who would rather avoid any discussion of superfields at all, could simply verify that this Lagrangian is indeed invariant under the general supergauge transformations given above. Notice again that the auxiliary D field may be easily removed by using its equations of motion, DA = 9A(bi*Til Oi • (101) The remainder of this Appendix expands a little on the role played by the chiral 1/1)„ field; it may be skipped at the reader's discretion. In standard gauge theory, the gauge field is the connection required for covariant, derivatives. These are quantities which, under a gauge transformation Joi -4 = (e'A) , (102) transform covariantly: DO, —> = (eiA),3 (103) The connection is everything other than the derivative in Dµ = a,— gvµ, and the trans-formation of v,. is therefore defined so that Di = (eiA)231)0, (e- . (104) Quite generally, there is only one non-trivial tensor that can be made from the the con-nection, by taking its covariant derivative. This is the curvature, which is given by F„ = DpA, — (105) So much for standard Yang-Mills theory, but what about its supersymmetric counterpart'? The field strength superfield Wes , may also be related to the connection in a super/gauge covariant derivative (just as the usual field strength Fp, is related to the gauge bosons). We begin with the derivatives which anti-commute with Q„, and C26 which we shall attempt to make gauge covariant; these are Dc, = aaa + io-".0a a o aa = --a — aea (106) 232 Steven Abel We can easily verify that {D„, Qc,} = {Da, Q6,} = {Da, (1} = {Da, Q.} = O. In addition Day = 0. This means that an alternative (and in fact the usual) definition of a chiral superfield is Da(1)(y,0) = 0. The fact that 156D2 = 0 gives us a useful chiral projection operator, P = -4-D2' (107) which, when applied any function of turns it into a chiral superfield (that is a function of y and 0 only). Now consider the derivative Da4). (108) This derivative is not gauge covariant. A covariant derivative is one which transforms as follows under gauge transformations: = Vic,e2^ 4) = elA (VoA.). (109) Thus via ezAvae-iA. (110) By examining the gauge transformations of V in Equation 90, we see that va e-29V Dae2gv transforms in the correct way, where Da acts on everything to its right. The connection superfield is defined as the difference between this term and Da. i.e. Fa = e-29v(Dae29v). (112) Finally the chiral field 14).„ which we derived earlier is the chiral projection of this: Wa = pra. (113) The gauge invariance of the Yang-Mills term is now manifest. 8 Appendix D: R-symmetry and R-parity As well as the terms in the superpotential of the MSSM there are many other Yukawa couplings that we could have included but didn't. For example, Wproblem = PiLTE'112 A'ELTeL+AuDLT N+ Ameab,UaDbDe, (114) where a, b, c are colour indices, are all invariant under SU(3), x SU(2) L x U(1)y. Including all of the these terms would lead to disastrously fast proton decay (suppressed by only the inverse squark mass squared). Thus, counting over generation indices, we have to explain the absence of 30 possible couplings in the MSSM superpotential. This can be done by the introduction of a global U(1) R symmetry known as R-symmetry. An R-transformation may be defined as a transformation of the spinor coordinates, ez'0 (115) CP and flavour in supersymmetry 233 Under this transformation we may assign a charge, the R-charge, to each of the chiral superfields: 0', 0') = eza (x, 0, 0). (116) For the elements of the supermultiplet this means Xi, e'aXi, {(4.:', Fi*1 e-ja {4, e2ja F! }. (117) Since the elements transform differently this also means that R transformations do not commute with supersymmetry transformations (i.e. in order to be consistent the su- persymmetry generators in Appendix A must obey Q e'Q which may easily be checked). Now let us examine which of the terms in the Lagrangian is invariant under R- transformations. The 0 kinetic term is invariant only if we assign R„ = 0 to the gauge vector, since R4, = —117. Examining the kinetic terms in, for example, Equation 22, we find A etaA. The supersymmetric Yang-Mills Lagrangian (shown in Appendix C) is also invariant under this transformation. This leaves the interaction term which looks like •Cint = W(4))1F + h.c. h.c. (118) By looking at the transformations in (117), the interaction Lagrangian transforms as Aijk(Di€1)AkIF R,-2) IF (119) and therefore only the terms with E R, = 2 (120) survive in the R-invariant superpotential. A choice of charges which results in the MSSM is Ri = 3B, + L, (121) where B, and L, are the baryon number and lepton number of (I),. Clearly Equation 120 is not satisfied by any of the 30 couplings in Equation 114. When supersymmetry is broken (for example by a Majorana mass for the gauginos), a residual discrete symmetry remains in the Lagrangian, given by fi )2Si -1-313; +Li fi (122) where f, is an arbitrary field of spin Si. This particular choice of discrete symmetry is called R-parity or matter-parity. Other (equally valid) choices are said to be R -parity breaking. For example, one particularly interesting choice, called baryon-parity is [35], f . (_02Si+3Bi fi. 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D 63, 035009 (2001) [hep-ph/0008237]; M. Carena, J. Ellis, A. Pilaftsis and C. E. Wagner, Nucl. Phys. B 586, 92 (2000) [hep-ph/0003180]; Phys. Lett. B 495, 155 (2000) [hep-ph/0009212]. [33] B. Allanach et al., arXiv:hep-ph/9906224. [34] F. Borzumati, G. R. Farrar, N. Polonsky and S. Thomas, Nucl. Phys. B 555 (1999) 53 [arXiv:hep-ph/9902443]. [35] L. E. Ibanez and G. G. Ross, Nucl. Phys. B 368 (1992) 3. [36] H. Dreiner and G. G. Ross, Nucl. Phys. B 410 (1993) 188 [arXiv:hep-ph/9207221]. B phenomenology Sheldon Stone Syracuse University, USA 1 The Standard Model and B decays Studies of b and c physics are focused on two main goals. The first is to look for new phenomena beyond the Standard Model and the second is to measure Standard Model parameters including CKM elements and decay constants. These lectures concern "B Phenomenology," a topic so broad that it can include almost anything concerning b quark decay or production. I will cover an eclectic ensemble of topics that I find interesting and hope will be educational. 1.1 Theoretical basis The physical states of the "Standard Model" are comprised of left-handed doublets con-taining leptons and quarks and right handed singlets (Rosner 2001) (u) (:) 1 :)L ,L RR, dR, CR, SR, tR, bR (1) te- ) 7-- eR, TR— , VeR, R, 117 R• (2) ve I. ( vjL The gauge bosons, W±, -y ( zir )L and Z° couple to mixtures of the physical d, s and b states. This mixing is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix (Kobayashi 1973). The Lagrangian for charged current weak decays is 9 t Lc„ -= --- /JtcWp h.c., where eL dL 4 = (1 e, Ow Or) -Yil liMNS PL + (UL, CL, tL) 7'µ/CKM SL TL 1)1, 237 (3) (4) DOI: 10.1201/9780429187056-6238 Sheldon Stone and VCKM = (Vud V. Vcd Ves Vtd Vts Vub Vcb Vtb • (5) Multiplying the mass eigenstates (c/L, s L, b L ) by the CKM matrix leads to the weak eigenstates. VmNs is the analogous matrix required for massive neutrinos (we will not dis-cuss this matrix any further). There are nine complex CKM elements. These 18 numbers can be reduced to four independent quantities by applying unitarity constraints and the fact that the phases of the quark wave functions are arbitrary. These four remaining num-bers are fundamental constants of nature that need to be determined from experiment, like any other fundamental constant such as a or G. In the Wolfenstein approximation the matrix is written to order A3 for the real part and A4 for the imaginary part as (Wolfenstein 1983) 1 — A2 /2 A A.\ 3(p — i71)(1 — )2/2) VCKM = —A ( 1 — A2 /2 — i71A2 A4 A)\ 2 (1 + ir1A2 ) (6) AA3(1 — p — ) —AA2 1 The constants A and A are determined from charged-current weak decays. The measured values are A = 0.2205 ± 0.0018 and A=0.784±0.043. There are constraints on p and n from other measurements that we will discuss. Usually the matrix is viewed only up to order A3. To explain CP violation in the K° system the term of order A4 in % is necessary. Figure 1. Diagram for muon decay. 1.1.1 Determination of GF Muons, being lighter than the lightest hadrons, must decay purely into leptons. The process is it- e-Fev, as shown in Figure 1. The total width for this decay process is given by G2 F = 192 7r F3 M5 x (phase space) x (radiative corrections). Since r„ • Tt, = h, measuring the muon lifetime gives a direct measure of GF- (7) 1.1.2 Determination of IVIL81 A charged current decay diagram for strange quark decay is shown in Figure 2. Here the CKM element Vus is present. The decay rate is given by a formula similar to Equa-tion (7), with the muon mass replaced by the s-quark mass and an additional factor B phenomenology 239 K Figure 2. Semileptonic K- decay diagram. of 117.b,12. Two complications arise since we are now measuring a decay process involving hadrons, K- 70c-P rather than elementary constituents. One is that the s-quark mass is not well defined and the other is that we must make corrections for the probability that the u-spectator-quark indeed forms a 7r° with the u-quark from the s-quark decay. These considerations will be discussed in greater detail in the semileptonic B decays section. Fortunately there are theoretical calculations that allow for a relatively precise measure-ment because they deal with hadron rather than quark masses and have good constraints on the form-factors. Using these models we have A = Vas = 0.2205 ± 0.0018. 1.1.3 Using semileptonic B decays to determine Kb and Vub Semileptonic b decays arise from a similar diagram to Figure 2, where the b quark replaces the s quark. In this case the b quark can decay into either the c quark or the the u quark, so we can use these decays to determine Veb and Vab providing we have three ingredients: • B lifetimes, • relevant B branching fractions, . a theory or model to take care of hadronic physics. 1.2 Lifetime measurements The "b-lifetime" was first measured at the 30GeV e+e- colliders PEP and PETRA, where b-quarks were produced via e±e- annihilation into bb pairs. Experiments at these colliders measured the average lifetime of all b-hadron species using the distribution of the "im-pact parameter", which is the minimum distance of approach of a track to the primary production vertex. This distance is related to the lifetime (Atwood 1994). A more direct measurement would be to measure the actual decay distance L and the momentum of the b hadron. Since L = ryikt, where t is the decay time of the individual particle (also called the proper time), and /3 = pl E and 'y = E/mb, the distribution of decay times t can be determined. Events will be distributed exponentially in t as with T being the lifetime. Uncertainties come from errors on L, the b hadron momentum, and contributions from non b-quark backgrounds. Precision lifetimes of individual b-flavored hadrons have been measured at LEP where the production process is e+ bb and at CDF in 1.8TeV pp collisions. Large samples of semileptonic decays have been used to determined the B° and B- lifetimes. Note that the CPT theorem guarantees that the lifetime of the anti-particles is the same Bo-•12."1 v • Data O B ▪ BkP, 4 6 a 16 t (pa) 111 t (us) 240 Sheldon Stone ALEPH Figure 3. Proper time distributions of exclusive semi-leptonic decays from ALEPH. as the particles. The decay distributions for two semileptonic B decay channels are shown in Figure 3. The B° --+ D*-P±v channel has mostly signal with some background from B- decays and other backgrounds as indicated in the figure. It takes a great deal of careful work to accurately estimate these background contributions. The clear exponential lifetime shapes can be seen in these plots. Some data has also been obtained using purely hadronic final states (Sharma 1994). B B° B5 Ab H H 1.653±0.028 ps 1.548±0.032 ps 1.493±0.062 ps 1.23±0.08 ps 1.39+:34.28 ps I • I 11 1.2 1.3 1.4 1.5 1.6 1.7 18 Figure 4. Measured lifetimes of different B species. A summary of the lifetimes of specific 6-flavored hadrons is given in Figure 4 (Groom 2001). Note that the ratio of B+ to B° lifetimes is 1.074+0.028, a 2.6a difference from unity, which is quite significant. Also, the Ab lifetime is much shorter than the B° lifetime. According to proponents of the Heavy Quark Expansion model, there should be at most a 10% difference between them (Bigi 1997). To understand lifetime differences we must first analyze hadronic decays. c or u E , d s 17, b E d s q b) hadronic: color suppressed b d) W exchange c or u q a) simple spectator c) annihilation ► c or u 4 if B phenomenology 241 1.3 B decay mechanisms Figure 5 shows sample diagrams for B decays. Semileptonic decays are depicted in Fig-ure 5(a), when the virtual W- materialises as a lepton-antineutrino pair. The name "semileptonic" is given, since there are both hadrons and leptons in the final state. The leptons arise from the virtual while the hadrons come from the coupling of the spec- tator anti-quark with either the c or u quark from the b quark decay. Note that the B is massive enough that all three lepton species can be produced. The simple spectator diagram for hadronic decays (Figure 5(a)) occurs when the virtual W- materialises as a quark-antiquark pair, rather than a lepton pair. The terminology simple spectator comes u,c, b t d W _ d 4 4 b u,c,t e) box: mixing Figure 5. Various mechanisms for B meson decay. from viewing the decay of the b quark, while ignoring the presence of the spectator an-tiquark. If the colors of the quarks from the virtual W- are the same as the initial b quark, then the color suppressed diagram, Figure 5(c), can occur. While the amount of color suppression is not well understood, a good first order guess is that these modes are suppressed in amplitude by the color factor 1/3 and thus in rate by 1/9, with respect to the non-color suppressed spectator diagram. The annihilation diagram shown in Figure 5(c) occurs when the b quark and spectator anti-quark find themselves in the same space-time region and annihilate by coupling to a virtual W. The probability of such a wave function overlap between the b and u-quarks is proportional to a numerical factor called f B. The decay amplitude is also proportional to the coupling Vub. The mixing and penguin diagrams will be discussed later. Each diagram contributes differently to the decay width of the individual species, although the simple spectator diagram is always expected to be the dominant one. Note that for b baryons there are even more diagrams than those shown in Figure 5. Currently there is no direct evidence for diagrams (c) and (d), although (c) is expected to be responsible for the purely leptonic decay B- The semileptonic decay width, Tm, is defined as the decay rate in units of inverse seconds into a hadron (or hadrons) plus a lepton-antineutrino pair. Decay rates can also 242 Sheldon Stone be expressed in units of MeV by multiplying by h. The width, rsl , is related to the semileptonic branching ratio Bsi and the lifetime T by: rst = Bs' • rtotal = BsI/T• (8) The semileptonic width should be equal for all b species. This is true for D° and D+ mesons, even though their lifetimes differ by more than a factor of two. Thus, it is differences in the hadronic widths among the different b hadrons that drive the lifetime differences. Let us now consider the case of B° and B- lifetime differences. There is some indi-cation that the B has a shorter lifetime, which would imply that there are more decay channels available. Figure 6 shows the color allowed and color suppressed decay diagrams for two-body decays into a ground-state charmed meson and a The color suppressed diagram only exists for the B. . The relative rate F(B- —> D°7-) F(B° = 1.8 + 0.3, (9) and the same is true for all other similar two-body channels, such as D* p- . Thus we would expect, if most B decays are given by these diagrams that the B- would have a shorter lifetime than the re, opposite to what the data suggests. An explanation is that this ratio reverses for higher multiplicity decays, but this is an interesting discrepancy that needs to be kept in mind. d n (q=d) cji (q_d) 41. D" (q=u) qi Figure 6. (left) Spectator diagram for TY —> 17;1+ 7r — and B- D°7--. (right) Color suppressed spectator diagram for B- D°71-- only. 2 Semileptonic B decays 2.1 Formalism of exclusive semileptonic B decays The same type of semileptonic charged current decays used to find Vus are used to find I 'cb and Vub. The basic diagram is shown in Figure 5(a). We can use either inclusive decays, where we look only at the lepton and ignore the hadronic system at the lower vertex. or exclusive decays where we focus on a particular single hadron. Theory currently can predict either the inclusive decay rate, or the exclusive decay rate when there is only a single hadron in the final state. Now let us briefly go through the mathematical formalism of semileptonic decays. We start with pseudoscalar B to pseudoscalar m transitions. The decay amplitude is given by Grinstein (1986) and Gilman (1990): A(B —> me-v) = —G 1,/ (10) B phenomenology 243 where = fie7" — 1,5) vv, (11) and H1= tad (0)113) = f+(q2)(P +p)µ + f-(q2)(P — p)„, (12) where q2 is the four-momentum transfer squared between the B and the m, and P, p are four-vectors of the B, m respectively. H1 is the most general form of the hadronic matrix element. It is written in terms of the unknown f functions that are called "form-factors." It turns out that the term multiplying the f_(q2) form-factor is the mass of lepton squared. Thus for electrons and muons (but not r's), the decay width is given by dq2 2472 If+(q2)i2, dFsi G2F IV,i121C3 (13) where K= 1 2LB [( 11128 2 q 2)2 — 4 Arn 'us 2 ,r2 ]1/2 (14) is the momentum of the particle m (with mass m) in the B rest frame. In principle, dradq2 can be measured over all q2. Thus the shape of f4 (q2 ) can be determined experimentally. However, the normalisation, MO) must be obtained from theory, for to be measured. In other words, FSL oc 1V1:1121M0)121 f K 3g(g2)dq2, TB (15) where g(q2) = f± (q2)/ MO). Measurements of semileptonic B decays give the integral term, while the lifetimes are measured separately, allowing the product I Vz.712 1.i-1-(0)12 to be experimentally determined. For pseudoscalar to vector transitions there are three independent form-factors whose shapes and normalisations must be determined (Richman 1995). 2.2 Measurement of IVebl 2.2.1 Heavy quark effective theory and ft We can use measurements of exclusive B decays, coupled with "Heavy Quark Effective Theory" (HQET) to find Veb (Isgur 1994). We start with a quick introduction to HQET. It, is difficult to solve QCD at long distances, but it is possible at short distances. Asymptotic freedom, the fact that the strong coupling constant a, becomes weak in processes with large q2, allows perturbative calculations. Large distances are of the order 1/AQou fm, since AQCD is about 0.2GeV. Short distances, on the other hand, are of the order of the quark Compton wavelength; .\Q ti 1/mQ equals 0.04 fm for the b quark and 0.13fm for the c quark. For hadrons, on the order of lfm, the light quarks are sensitive only to the heavy quark's color electric field, not the flavor or spin direction. Thus, as mQ oo, hadronic systems which differ only in flavor or heavy quark spin have the same configuration of 244 Sheldon Stone their light degrees of freedom. The following two predictions follow immediately (the actual experimental values are shown below): mB. - mBd = mD, — ?no+ (16) (90.2 ± 2.5) MeV (99.2 ± 0.5) MeV , 2 2 2 2 (17) 0.49 GeV2 0.55 GeV2. The agreement is quite good but not exceptional. Since the charmed quark is not that heavy, there is some heavy quark symmetry breaking. This must be accounted for in quantitative predictions, and can probably explain the discrepancies above. The basic idea is that if you replace a b quark with a c quark moving at the same velocity, there should only be small and calculable changes. In lowest order HQET there is onlymoi2 m e±m Bm2for -factorD*_ function F which is a function of the Lorentz invariant four-velocity transfer w, where B D* CV = 2 (18) The point w equals one corresponds to the situation where the B decays to a D* which is at rest in the B frame. Here the "universal" form-factor function F(w) has the value, F(1) = 1, in lowest order. This is the point in phase space where the b quark changes to a c quark with zero velocity transfer. The idea is to measure the decay rate at this point, since we know the value of the form-factor, namely unity, and then apply the hopefully small and hopefully well understood corrections. In HQET, the form factor F can also be applied to the analysis of B Dt-17, but the overall decay rate for the pseudoscalar final state is only 40% of D*.e- ,and the decay rate vanishes at w = 1 much faster, making the measurement more difficult. 2.2.2 Detection of B D* iv Since this is a semileptonic final state containing a missing neutrino, the decay cannot be identified or reconstructed by merely measuring the 4-vectors of the final state particles. One technique used in the past relies on evaluating the missing mass (MM) where MM2 = (EB — ED* — Ee)2 — (PB — PD* — pf )2 (19) = + MD2 * + — 2EB • (ED* + Et) + 2ED* Et —2pD*.pe + 2pB•(pD* + Pe). For experiments using e+e- T(4S) -+ BB, the B energy, EB, is set equal to the beam energy, Ebeam, and all quantities are known except the angle between the B direction and the sum of the D* and lepton 3-vectors (the last term). A reasonable estimate of MM2 is obtained by setting this term to zero. The signal for the D* tv final state should appear at the neutrino mass, namely at MM2 = 0. In an alternative technique MM2 is set to zero and the angle between the B momentum and the sum of the D* and lepton 3-vectors is evaluated as (20) \ 2E/3 (ED* + Et) - - MD *e cos (0B.D*1) = 2 1PB1 (PD* + PP) -10 -4 -2 0 2 • D*1 v • D**1 v 1111 combinatoric continuum • uncorrelated • correlatT1 2 4 -8 -6 -4 -2 0 Cole„ D•1 1.306a6c1.357 • D*1 v • D**I v • combinatoric continuum • uncorrelated • correlated eo 60 40 20 0 g• 22.5 ti; 20 17.5 15 12.5 10 7.5 5 2.5 0 B phenomenology 245 where MD*i indicates the invariant mass of the D*-lepton combination. A Monte-Carlo simulation of cos (0B.D*t) is given for the final state of interest and for the main background reaction in Figure 7. For the correct final state only a few events are outside the "legal" region of ±1, while when there are extra pions in the final state the shape changes and many events are below -1. 16 14 12 10 8 6 4 2 -10 -8 -6 -4 -2 0 2 4 COSOB_Dfl , 1 35 D*Iv 30 25 20 J 15 10 5 -10..8 ..-6 -4 -2 0 2 4 COSOB.D.1 Figure 7. The cosine of the angle between the B momentum vector and the sum of D* and lepton momentum vectors for (left) the final state D*ev and (right) D* X fv, where X refers to an additional pion. Recent CLEO data has been analyzed with such a technique. The data in two specific w bins is shown in Figure 8. The final result for all w bins is shown in Figure 9. The result is characterised by both a value for F(1)117,b1 and a shape parameter p2. Figure 8. The signal and background contributions in two different w bins for the final state D*.ev. A D.*/ v V eels — Fit . 1.0 1.1 1.2 1.3 1.4 1 5 Figure 9. The CLEO results for D* ev for both B and B-. The curve is a fit to a shape suggested in (Caprivi 1998). 0.05 0.04 0.03 0.02 0.01 246 Sheldon Stone 2 (Nra. 1 • 13°-> D*-t+ v Belle :DELP PAL ALEPH 30 40 F(1)1Vo1x10-3 Figure 10. The correlation between the slope parameter p2 and F(1)117,1,1. The contours are for a change in the fit X2 of one unit. 2.2.3 Evaluation of Vet, using B D* tv Figure 10 and Table 1 give recent experimental results on exclusive B D* ev decays. The CLEO results are not in particularly good agreement with the rest of the world including the BELLE results. Experiment F (1)117,b l ( x 10-3) P2 ALEPH (Buskulic 1997) 33.0 + 2.1 ± 1.6 0.74 + 0.3 ± 0.4 BELLE (Tajima 2001) 35.4 ± 1.9 ± 1.9 1.35 0.17 + 0.18 CLEO (Heltsley 2001) 42.2 + 1.3 ± 1.8 1.61 ± 0.09 ± 0.21 DELPHI (Abreu 2001) 34.5 ± 1.4 ± 2.5 1.2 ± 0.1 ± 0.4 OPAL(7e) (Abbiendi 2000) 37.9 + + 2.4 1.2 + 0.2 ± 0.4 OPAL (Abbiendi 2000) 37.5 ± 1.7 ± 1.8 1.4 ± 0.2 ± 0.2 Average 37.8 + 1.4 1.37+ 0.13 Table 1. Modern determinations of F(1)1V,b 1 using B D* VP decays. To extract the value of 11.7,b1 we have to determine the corrections to F(1) that lower its value from unity. The corrections are of two types: quark mass, characterised as some coefficient times AQCD/mQ, and hard gluon, characterised as 77A. The value of the form-factor can then be expressed as (Neubert 1996) F(1) = (1 + 0 • AwD/ing + c2 • (AQCD /MQ) 2 •• • •) = (1 + 6). (21) The zero coefficient in front of the 1/mQ term reflects the fact that the first order correc-tion in quark mass vanishes at w equals one. This is called Luke's Theorem (Luke 1990). Recent estimates for 71,4 and 6 are 0.967+0.007 and -0.55+0.025, respectively. The value predicted for F(1) is then 0.91+0.05. This is the conclusion of the PDG review done by B phenomenology 247 Artuso and Barberio (Artuso 2001). There has been much controversy surrounding the theoretical prediction of this number. In the future Lattice-Gauge Theory calculations will presumably become more accurate when unquenched calculations become possible. Cur-rent lattice calculations are done in the quenched approximation, without including light quark loops. They give F(1) = 0.913Ir 0.01612341M12,64 , where the uncertain-ties come respectively, from statistics and fitting, matching lattice gauge theory to QCD, lattice spacing dependence, light quark mass effects and the quenching approximation. (Hashimoto 2001). Using the Artuso-Barberio value for F(1) we have Vcb = (41.5 + 1.5 ± 2.3) x 10-3, (22) where the first error is experimental and the second the error on the calculation of F(1). 2.2.4 117,b l from inclusive semileptonic decays The inclusive semileptonic branching ratio b Xtv has been measured by both CLEO and LEP to reasonable accuracy. CLEO finds (10.49 + 0.17 ± 0.43)%, while LEP has (10.56 + 0.11 ± 0.18)%. These are not quite the same quantities as the CLEO number is an average over 13- and B° only, while the LEP number is a weighted average over all b hadron species produced in Z° decays. Thus using the LEP number one should use the average "b quark" lifetime of 1.560±0.014ps. Using the Heavy Quark Expansion (HQE) model (Bigi 1997) we can relate the total semileptonic decay rate at the quark level to I Vcb 1 17c612 = h(Ai, A2, A) x F(b cev) (23) = h(A1, A2, A) x 8(b cev) x 1 Tb where B(b -4 cev) is the inclusive semileptonic branching ratio minus a small b —> utv component. To be precise, it is the decay of a B meson to a lepton-antineutrino pair plus any charmed hadron. Tb is the lifetime of that particular meson or average lifetime of the combination of the b-flavored hadrons used in the analysis, suitably weighted. In the HQE model the semileptonic rate is described to order (AQCD /mb)2 by three parameters: . Al is the kinetic energy of the residual motion of the b quark in the hadron: = —2 MB ( 1) B(v) ,(113)2 hdB(v)), (24) • A2 is the chromo-magnetic coupling of the b quark spin to the gluon field: A2= MB (B( v) 2 9 hv - a' -2 Gpatv B(v)) , (25) . A is the strong interaction coupling, given in terms of MB = (MB + 3MB* )/4, the spin averaged B meson mass, as Al A = MB — mb + • 2MB (26) 248 Sheldon Stone These parameters are further related by = mb ± A (Al2+m3b A2) = mb +A (A1 — Az) 2mb • These relations allow us to determine A2 from the MB* — MB mass splitting as 0.12GeV2. The function h(Ai , A2, A) can be calculated from the Heavy Quark Expansion (HQE) model. This involves both perturbative and non-perturbative pieces. Although we will go through this derivation of Kb, there is a disturbing assumption of quark-hadron duality; the idea of duality is that if you integrate over enough exclusive charm bound states and enough phase space, the inclusive hadronic result will match the quark level calculation. However, we do not know what size is the uncertainty associ-ated with the duality assumption. In fact, Isgur said "I identify a source of AQcD/mQ corrections to the assumption of quark-hadron duality in the application of heavy quark methods to inclusive heavy quark decays. These corrections could substantially affect the accuracy of such methods in practical applications and in particular compromise their utility for the extraction of the CKM matrix element Kb" (Isgur 1999). Let us move to the details of the calculation. In one implementation Al and A are derived; the relationship between the inclusive b du branching fraction (about 99% of b X hi) is given as 0.0411\ 8(6 —r X,Pv) /1.55 ps (1 0.02 — 0.5 GeV2) (28) iVebi = 0.105 V rb 0.1 GeV2 ) x (1 + 0.015pert. 0.010m6 ± 0.0121/m0 , (29) where ii2 is the negative of Al modulo QCD corrections and is taken as (0.5 + 0.1) GeV2 (Bigi 1997). This leads to a value of IVeb l = (40.7 ± 0.5 ± 2.4) x 10-3 (30) from the LEP data alone, with a similar value from CLEO. The first error contains the statistical and systematic error from the experiments while the second error contains an estimate of the theory error from sources other than duality. In another implementation the parameters Al and A are obtained from data. Here we use the HQE formula G2FiVebl2 MB x 0.369 x 1927r3 [ A A2 AI A2 1 1-1.54—as-1.65 (1-1.087 c ) —0.95-A-4-3.18—Ag+0.02—m123 7r MB ir Determining Al and A can be accomplished by measuring, for example, the "moments" of the hadronic mass produced in b du decays. The first moment is defined as the de-viation from the D mass (MD) as (M2 x — MD) and the second moment as2 H. 2 ((MX D) )* MB MB* (27) rsl (31) 1500 - Fit Div D*tv =1 XHiv 2000 di 500- -4 -2 0 2 4 2 M x(GeV2 ) 10 B phenomenology 249 It is also possible to use the first and second moments of the lepton energy distribution in these decays, or moments of the photon energy in the process b —> s-y. In fact any two distributions can be used; in practice it will be critical to use all of them to try and ascertain if any violations of quark-hadron duality appear and to check that terms of order (AQcD/mb)3 are not important. CLEO has used the first and second moments of the hadron mass in b cev decays. They find the Mx distributions by using missing energy and momentum in the event to define the v four-vector. Then detecting only the lepton and requiring it to have a momentum above 1.5GeV/c, they calculate: MX = (En — Et — Eu)2 — (pB — pi — p02 (32) = — 2ER (Ee + Ev) + 2PB• (Pe Pv) MB + Mev — 2EBEtv, (33) where Mt, is the invariant mass of the lepton-neutrino pair. The measured MI- distri-bution is shown in Figure 11. We do not see distinct peaks at the mass of the D and D* mesons because ignoring the last term in Equation (32) causes poor resolution. This term must be ignored, because using this technique, we do not know the direction of the B meson. Figure 11. The mass distribution for b du events from CLEO. XH indicates an additional pion plus a D or D*. CLEO finds values of the first and second moments of (0.287±0.023±0.061) GeV2 and (0.712±0.056+0.176)Ger4, respectively. These lead to the determination of A1, A and Veb shown in Figure 12. Later we will see a different determination using b s-y (Section 5.3.1). In summary the exclusive measurements of Veb are to be trusted while the inclusive determination, though consistent, has an unknown source of systematic error and should not be used now. 250 Sheldon Stone 0 A l ( -0. Figure 12. Constraints in Al versus A from CLEO measurements of first and second hadronic moments in semileptonic decay. The darker band gives experimental uncertain-ties alone, while the lighter outer band also includes uncertainties from unknown 3rd order theoretical parameters. Bands of constant Kb and b quark mass, mb, are also shown, where the band widths represents theoretical uncertainties due to unknown 3rd order parameters. 2.3 Measurement of I Vub This is a heavy to light quark transition where HQET cannot be used directly as in finding Vcb. Unfortunately the theoretical models that can be used to extract a value from the data do not currently give precise predictions. Three techniques have been used. The first measurement of 17,d, done by CLEO (Fulton 1990) and subsequently confirmed by ARGUS (Albrecht 1990), used only leptons which were more energetic than those that could come from b -4 cP-P decays. These "endpoint leptons" can occur b -4 c background free at the 7(4S), because the B's are almost at rest. The CLEO data are shown in Figure 13. Since the lepton momentum for B —> Db, decays is cut off by phase space, this data provides incontrovertible evidence for b —> uev decays. Unfortunately, there is only a small fraction of the b —> uf-P lepton spectrum that can be seen this way, leading to model dependent errors. The models used are either inclusive predictions, sums of exclusive channels, or both: see Isgur (1995), Bauer (1989), Korner (1989), Melikhov (1996), Altarelli (1982) and Ramirez (1990). The average among the models is Yub/Vcal = 0.079 ± 0.006, without a model dependent error. These models differ by at most 11%, making it tempting to assign a ±6% error. However, there is no quantitative way of estimating the error. ALEPH (Barate 1999), L3 (Acciarri 1998) and DELPHI (Abreu 2000) isolate a class of events where the hadron system associated with the lepton is enriched in b -4 u and thus depleted in b —> c. They define a likelihood that hadron tracks come from b decay by using a large number of variables including, vertex information, transverse momentum, not being a kaon etc.. Then they require the hadronic mass to be less than 1.6GeV, which greatly reduces b —> c, since a completely reconstructed b -+ c decay has a mass greater 120 tr) 80— --. 0401 .tm 2 --- 0 " ' B phenomenology 251 2.00 2.25 2.50 2.75 3.00 Lenton Momentum (GeV/el Figure 13. Sum of inclusive electron and muon distributions from CLEO. The solid points are data taken on the peak of the T(4S) while the open circles are data taken on the continuum 30MeV below the resonance(suitably normalised). The dashed line is a fit to the continuum data and the solid line is the predicted curve from b -+ cEv dominated by B -+ Rev near the end of the allowed lepton spectrum. 03 Figure 14. The lepton energy distribution in the B rest frame from DELPHI. The data have been enriched in b —> u events, and the mass of the recoiling hadronic system is required to be below 1.6G eV. The points indicate data, the light shaded region, the fitted background and the dark shaded region, the fitted b —> uPti signal. than that of the D (1.83GeV). They then examine the lepton energy distribution, shown in Figure 14 for DELPHI. The average of all three results is I Vubl = (4.131§1133124510.20) x 10-3, resulting in a value for I Vub /17,b1 = 0.102 ± 0.018, using 11761 = 0.0405 ± 0.0025. I have several misgivings about this result. First of all the experiments have to understand the systematic errors very well. To understand semileptonic b and c decays and thus find their b -f ?Lev efficiency, they employ different models and Monte Carlo manifestations of these models. To find the error they take half the spread that different models give. This alone may be a serious underestimate. Secondly they use one model, the HQE model, to translate their measured rate to a value for 1Vubl. This model assumes duality, and there are no successful experimental checks: The model fails on the Ab lifetime prediction. Furthermore, the quoted theoretical error, even in the context of the model, has been estimated by Neubert 35 30 2 ir) 20 r 15 c a) 10 5 0 50 10 0 a) 25 252 Sheldon Stone to be much larger at 10% (Neubert 2000). Others have questioned the effect of the hadron mass cut and estimate 10-20% errors due to this alone (Bauer 2001). It may be possible to use the spectrum of photons in b 57 to reduce the theoretical error in the endpoint lepton method or to make judicious cuts in q2 instead of hadronic mass to help reduce the theoretical errors: see Wise (2001). The third method uses exclusive decays. CLEO has measured the decay rates for the exclusive final states 7rev and phi (Alexander 1996). The data are shown in Figure 15. The model of Korner and Schuler (KS) is ruled out by the measured ratio of p/7r. Other models include those of Isgur (1995, 1989), Wirbel (1985), Bauer (1989), Korner (1988), Melikhov (1996), Altarelli (1982) and Ramierz (1990). CLEO has presented an updated analysis for pev where they have used several different models to evaluate their efficiencies and extract Vub. These theoretical approaches include quark models, light cone sum rules (LCRS), and lattice QCD. The CLEO values are shown in Table 2. 5.12 5.14 5.16 5.18 5.20 5.22 5.24 5.26 5.28 5 30 Mcand (GeV) Figure 15. The B candidate mass distributions and the signal bin lepton momentum spectra (insets) for the pion modes (top) and the sum of p and w (vector) modes (bottom). The points are the data after continuum and fake background subtractions; the dark- shaded, cross-hatched and unshaded histograms are b cX , b uev feed-down and signal, respectively. For the ir (vector) modes, the light-shaded and hatched histograms are ir -4 7r (vector—) vector) and vector -+ 7r (ir -+ vector) cross-feed, respectively (charge final states can feed neutral and vice-versa). The histogram normalisations are from the nominal fit. The arrows indicate the momentum cuts. B phenomenology 253 Model Vub (x10-3) ISGW2 (Isgur 1989) Beyer/Melikhov (Beyer 1998) Ligeti/Wise (Ligeti 1996) LCSR (Ball 1998) UKQCD (Debbio 1998) 3.23 3.32 2.92 3.45 3.32 ± 0• -0.29 14+0-22 ± 0.15+8:30 0• -0 13+.1 ± 0.269 0• -0 15+.2 ± 0.321 ± 0 14+0.21 • -030 Table 2. Values of IVubi using B -+ pi-Ti and some theoretical models. The uncertainties in the quark model calculations (first three in the table) are guessed to be 25-50% in the rate. The Ligeti/Wise model uses charm data and SU(3) symmetry to reduce the model dependent errors. The other models estimate their errors at about 30-50% in the rate, leading conservatively to a 25% error in I Vubl. Note that the models differ by 18%, but it would be incorrect to assume that this spread allows us to take a smaller error. It may be that the models share common assumptions, e.g. the shape of the form-factors. At this time it is prudent to assign a 25% model dependent error realising that the errors in the models cannot be averaged. The fact that the models do not differ much allows us to comfortably assign a central value I Vubl = (3.25±0.141:33±0.80) x 10-3, and a derived value IVub/Kbl = 0.08 ± 0.02 . Lattice QCD has predicted form-factors and resulting rates for the exclusive semilep-tonic final states zrev and pPv (Sachrajda 1999) in the quenched approximation. These calculations require the momentum of the final-state light meson to be small in order to avoid discretisation errors. This means we only obtain results at large values of the invari- ant four-momentum transfer squared, q2. Figure 16 shows the predictions of the B pEv width as a function of q2. Note that the horizontal scale is highly zero suppressed. The region marked "phase space only" is not calculated but estimated using a phase space extrapolation from the last lattice point. 3 S 2 CD a I 14 16 18 22 q2/GeV2 Figure 16. The UKQCD lattice calculation for B pEv shown as circles. The line is an estimate. The integral over the region for q2 > 14GeV2 gives a rate of DI' = 8.3 Yubi2 ps-1-GeV2. CLEO measurements in the same interval give (7.1±2.4) x 10-5ps-1-GeV, yielding a value for Vub = (2.9 ± 0.5) x 10-3 (Sachrajda 1999). Ultimately unquenched lattice calculations when coupled with more precise data will yield a much better value for Vub. 254 Sheldon Stone 3 Facilities for b studies 3.1 b production mechanisms Although most of what is known about b physics has been obtained from e+e- colliders operating either at the T(4S) or at the Z° pole, interesting information is now appearing from the hadron collider experiments, CDF and DO, which were designed to look for considerably higher energy phenomena. The appeal of hadron colliders arises mainly from the large measured b production cross-sections. At the T(4S) the total b production cross-section is only 1.05 nb, whereas at the FNAL Tevatron collider, with 1.8TeV in the pi) center-of-mass, the b cross-section has been measured as --,100p,b. It is expected to he about five times higher at the LHC (Artuso 1994). The different production mechanisms of b quarks at various accelerators leads to dif-ferent methods of measurement. Figure 17 shows the production of B- and B° mesons at the T(4S), while Figure 18 shows the production mechanism of the different b species at a higher energy e+ e- collider such as LEP. Figure 19 shows the production mechanisms for a b or c quark at a hadron collider. The third order diagrams appear to be as important as the second order diagrams and the overall theoretical calculation gives about 1/2 of the measured value. word B+13or B°11° or d 6 Figure 17. B production at the T (4S). b} B° Bs, Ab, 61 Figure 18. b production in the continuum at e+ e- colliders. Figure 19. Heavy quark, Q, production at hadron colliders. Left four diagrams are second order in as, while right four are third order. B phenomenology 255 3.2 Accelerators for b physics Experiments on b decays started with CLEO and ARGUS using e+ colliders operating at the T(4S). They were quickly joined by the PEP and PETRA machines operating around 30GeV. Table 3 lists some of the machines used to study b quarks in the last century (Artuso 1994). Machine Beams Energy (GeV) o-(b) b fraction G cm-2s-1 Total # b-pairs CESR e+e- 10.8 1.05 nb 0.25 1.3x 1033 9.8 x 106 DORIS e+e- 10.8 1.05 nb 0.23 1031 0.4 x 106 PEP e+e- 29 0.4 nb 0.09 3.2 x 1031 PETRA e+e- 35 0.3 nb 0.09 1.7x 1031 LEP e+ e- 91 9.2 nb 0.22 2.4x 1031 1.8 x 106 SLC e+e- 91 9.2 nb 0.22 3.0x 103° 8.8 x 104 TEVATRON pp 1800 100 itb 0.002 3 x 1031 Table 3. Machines used for b physics in the 20th century. The total number of bb pairs accumulated per experiment is also listed when known. In the year 2000 the PEP II and KEK-B storage rings began operation. These ma-chines have separate e- and e+ magnet rings so they can operate at asymmetric energies; PEP II has beam energies of 9.0GeV and 3.1GeV, while KEK-B has energies of 8.0GeV and 3.5GeV. This allows the B mesons produced at the T(4S) to move with a velocity 13 0.6, which turns out to be very important in measurements of CP violation, since time integrated CP violation via mixing must be exactly zero due to the C odd nature of the T(4S). These machines are operating at very high luminosities. Current and future machines for B physics are listed in Table 4. The CDF and DO experiments will continue at the Tevatron with higher luminosities. CDF has already made significant contributions including studies of b production, lifetimes and the discovery of the B, meson (Abe 1998). Machine Exp. Beam Energy (GeV) a (b) b fraction £(Design) cm's-1 Interactions per crossing PEP II BABAR e+e- 10.8 1.05 nb 1/4 3x1033(1) < 1 KEK-B BELLE e+e- 10.8 1.05 nb 1/4 1034 < 1 HERA HERA-b pN 800 10 nb 2 • 10-6 4 Tevatron BTeV pp 2000 100 /../b 1/500 2 x 1032 2 LHC LHCb pp 14000 500 tib 1/160 2 x 103' 0.6 t Machine has already exceeded design luminosity. Table 4. Machines in use or approved for dedicated b physics experiments. Magnetic Coil Barrel Crystals Time of Flight Beam Pipe Silicon Vertex Detector 256 Sheldon Stone BTeV and LHCb will go into operation around 2007 with much larger event rates. The CMS and ATLAS experiments at the LHC will also contribute to b physics especially in the early stages when the luminosity will be relatively low; at design luminosity these ex-periments have an average of 23 interactions per crossing making the finding of detached vertices difficult. 3.3 e+e- detectors Most experiments at e+ e- storage rings look quite similar. CLEO II, shown in Figure 20, was the first detector to have both an excellent tracking system and an excellent elec-tromagnetic calorimeter. Starting from the inside there is a thin beryllium beam pipe Figure 20. Electrons view of the CLEO II detector. surrounded by a silicon vertex detector; this detector measures positions with an accu-racy of 10/1m. Then there is a wire drift chamber whose main function is to measure the curving trajectories of particles in the 1.5T solenoidal magnetic field. The next device radially outward is time-of-flight system to distinguish pions, kaons and protons. This system only works for lower momenta. The next device is an electromagnetic calorimeter that uses Thallium doped CsI crystals; indeed this was the most important new technical implementation done in CLEO II and has also been adopted by BABAR and BELLE. Afterwards there is segmented iron that serves as both a magnetic flux return and a filter for muon identification. The most important advance in the new CLEO III, BELLE and BABAR detectors is much better charged hadron identification. Each experiment uses different techniques based on Cherenkov radiation to extend irIK separation up to the limit from B decays. Figure 21 shows a view of the BELLE detector parallel to the beam. 11111Frit:f C.I KIM sat I rd J LI I I L L_1 1 ( M ) CS ACC ACC B phenomenology 257 Figure 21. The BELLE detector. 3.4 b production characteristics at hadron colliders To make precision measurements, large samples of b's are necessary. Fortunately, these are available. With the Fermilab Main Injector, the Tevatron collider will produce 4 x 1011 b hadrons/107 s at a luminosity of 2 x 1032cm's-I. These rates compare very favorably to e+e- machines operating at the T (4S). At a luminosity of 1034cm's-1 they would produce 2 x 108 B's/107 s. Furthermore B8 , Ab and other b-flavored hadrons are accessible for study at hadron colliders. The LHC has about a five times larger b production cross-section than the Tevatron. Also important are the large charm rates, ,-10 times larger than the b rate. In order to understand the detector design it is useful to examine the characteristics of b quark production at pp collider. It is often customary to characterise heavy quark production in hadron collisions with the two variables pt and 7/, where i = — In (tan (012)) , and 0 is the angle of the particle with respect to the beam direction. According to QCD based calculations of b quark production, the B's are produced "uniformly" in ?I and have a truncated transverse momentum spectrum, pt, characterised by a mean value approximately equal to the B mass (Artuso 1994). The distribution in is shown in Figure 22(a). The flat 7/ distribution hides an important correlation of bb production at hadronic colliders. In Figure 22(c) the production angles of the hadron containing the b quark is plotted versus the production angle of the hadron containing the b quark according to the Pythia generator. Many important measurements require the reconstruction of a b decay and the determination of the flavor of the other b, thus requiring both b's to be 10 75 -2.3 2.3 258 Sheldon Stone Figure 22. (a) The B yield versus n. (b) dry of the B versus (c) The production angle (in degrees) for the hadron containing a b quark plotted versus the production angle for a hadron containing b quark. observed in the detector. There is a very strong correlation in the forward (and backward) direction: when the B is forward the B is also forward. This correlation is not present in the central region (near 900). By instrumenting a relative small region of angular phase space, a large number of bb pairs can be detected. Furthermore the B's populating the forward and backward regions have large values of ,87 Figure 22(b). BTeV is a dedicated heavy flavor experiment approved to run at the Fermilab Tevatron collider, uses two forward spectrometers (along both the p and p directions) that utilise the boost of the B's at large rapidities. This is of crucial importance because the main way to distinguish b decays is by the separation of decay vertices from the main interaction. LHCb, approved for operation at the LHC, needs a larger detector to analyze the higher momentum decay products, and thus has only one arm. 3.4.1 The BTeV detector description I will describe BTeV here though LHCb shares many of the same features. There are difficulties that heavy quark experiments at hadron colliders must overcome. First of all, the huge b rate is accompanied by an even larger rate of uninteresting interactions. At the Tevatron the b-fraction is only 1/500. In addition all b experiments have problems searching for rare decay processes, at the level of parts per million, where the background from other b decays is dominant. To observe such decays the large data rate of b's must be handled. For example, BTeV, has 1 kHz of b's into the detector, and these events must be selected and written out. The electromagnetic calorimeter must be robust enough to deal with the particles from the underlying event and still have useful efficiency. Furthermore, radiation damage can destroy detector elements. The BTeV Detector is shown in Figure 23 (Skwarnicki 2001) and the LHCb detector in Figure 24 (Muheim 2001). The central part of the BTeV detector has a silicon pixel detector inside a 1.5T dipole magnet. The LHCb experiment uses silicon strips. The BTeV pixel detector provides precision space points for use in both the offline analysis and the trigger. The pixel geometry is sketched in Figure 25(a). Pulse heights are measured on each pixel. Prototype detectors were tested in a beam at Fermilab, and excellent resolutions were obtained (Figure 25(b)), especially when reading out pulse heights (Appel 2001). The final design uses a 3-bit ADC for this. B phenomenology 259 Pixels 1I1'\iagnet II RICH EM RICH beam line Muon wire chambers Muon Doublet position along beam (a) 50m Pixel Orientation 4.25 ± 6mm x±6nun Beam hole Beams Figure 23. Schematic of the BTeV detector. Figure 24. A schematic diagram of the LHCb detector . Key: 1, vertex detectors; 2,aerogel and gas RICH's; 3, magnet yoke; 4, coils; 5, magnetic shielding; 6, tracking chambers; 7, gas RICH; 8, EM-calorimeter; 9, hadron calorimeter; 10, muon system. 0.1 0.2 0.3 0.4 0.5 0.6 Track Angle (rad) Figure 25. (a) Pixel detector geometry in BTeV. The detector is inside the beam pipe. (b) The spatial resolution as a function of the incident track angle for both 2-bit and 8-bit ADC's as measured in an 800GeV/c pion beam. The straight lines are piecewise fits to the data used in the Monte Carlo simulation. The dashed line near the top indicates the resolution obtainable without using pulse height information. b d 260 Sheldon Stone The pixel tracker provides excellent vertex resolution which is good enough to trigger on events with detached vertices characteristic of b or c decays. BTeV shows a rejection of 100:1 for minimum bias events in the first trigger level while accepting about 50% of the usable b decays. A good explanation of the trigger algorithm can be found at the website www-btevinal.gov/public_documents/animations/Animated_Trigger/. Further trigger levels reduce the background by about a factor of twenty while decreasing the b sample by only 10%. The trigger system stores data in a pipeline that is long enough to ensure no deadtime. The data acquisition system has sufficient throughput to accommodate an output of 1kHz of b's, 1kHz of c's and 2kHz of junk. Tracking is accomplished using straw tube wire chambers with silicon strip chambers in the high track density region near the beam. Charged particle identification is done using a Ring Imaging CHerenkov detector. A gaseous C4F10 radiator is used with a large mirror that focuses light onto a plane of photon detectors; these are currently Hybrid Photo-Diodes. They have a photocathode and a 20KV potential difference between the photocathode and a silicon diode that is segmented into 163 individual pads. The photoelectron is accelerated and focused onto the diode yielding position information for the initial photon. The system will provide four standard-deviation kaon/pion separation between 3-70GeV/c, electron/pion separation up to 22GeV/c and pion/muon separation up to 15GeV/c. Because protons don't radiate until 9GeV/c they can't be distinguished from kaons below this momenta. BTeV is considering an additional liquid C6F14 radiator, lcm thick, in front of the gas along with a proximity focused phototube array adjacent to the sides of the gas volume, to resolve this ambiguity. BTeV also has an excellent Electromagnetic calorimeter made from PbWO4 crystals, based on the design of CMS. Finally, the Muon system is used to both identify muons and provide an independent trigger on dimuons (BTeV 2000). 4 B0-71I° mixing 4.1 Introduction Neutral B mesons can transform to their anti-particles before they decay. The diagrams for this process are shown in Figure 26 for the Bd. There is a similar diagram for the B,. Although u, c and t quark exchanges are all shown, the t quark plays a dominant role mainly due to its mass, as the amplitude of this process is proportional to the mass of the exchanged fermion. Figure 26. The two diagrams for Bd mixing. Under the weak interactions the eigenstates of flavor, which are degenerate in pure QCD, can mix. Let the quantum mechanical basis vectors be {I1),12)) {IB°), IT3°)}. B phenomenology Then the Hamiltonian is = m _ 11/ m12) 2 M12 M 2 Diagonalising we have Am = mBH — mBL = 2 IM121• r ri2 r12 r 261 (34) (35) Here H refers to the heavier and L to the lighter of the two weak eigenstates. Bd mixing was first discovered by the ARGUS experiment (Albrecht 1983), although there was a previous measurement by UA1 indicating mixing for a mixture of /33 and /38° (Albajar 1987). At the time it was quite a surprise, since mt was thought to be in the 30GeV range. It is usual to define R as probability for a B° to materialise as a r divided by the probability it decays as a B°. The OPAL data for R (Akers 1995) are shown in Figure 27. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 R 0 2 4 6 8 10 t (ps) Figure 27. The ratio, R, of like-sign to total events as a function of proper decay time, for selected B D*+ t-i) events. The jet charge in the opposite hemisphere is used to determine the sign correlation. The curve is the result of a fit to the mixing parameter. Data from many experiments has been combined by a LEP Working Group to ob-tain an average value of Arnd = (0.489 ± 0.0008) x 1012 1-is-1. Values from individual experiments are listed in Figure 28. 0 -2 262 Sheldon Stone ALEPH * (3 + 1 prel) DELPHI * (4 + 1 prel) L3 (3) OPAL (5) SLD * (5 prel) CDF * (4 + 2 prel) BABAR * (2 prel) BELLE (1) average of above after adjustments ARGUS+CLEO (xd measurements) 0.446±0.020±0.018 ps-1 0.504±0.019±0.012 ps-1 0.444±0.028±0.028 ps-1 0.479±0.018±0.015 ps-1 0.507±0.023±0.019 ps-1 0.495±0.026±0.025 0.503±0.009±0.011 ps-1 0.463±0.008±0.016 ps-1 0.489±0.009 ps-1 0.489±0.032 ps-1 world average 0.489±0.008 ps-1 working group average without adjustments 0.4 0.45 0.5 0.55 Amd (PS 1) Figure 28. Values of the Bd mixing parameter Amd for each experiment. The probability of mixing is given by (Gaillard 1974) * (Bigi 2000) m G2 m2 x A B f 2 MB-1-13 tb td12m2 t F INCD7 IV V Or = 67 F2 B Mw (36) where BB is a parameter related to the probability of the d and 6 quarks forming a hadron which must be estimated theoretically, F is a known function which increases approximately as 74, and wcp is a QCD correction, with a value of about 0.8. By far the largest uncertainty arises from the unknown decay constant, f B. In principle f B can be measured, since the decay rate of the annihilation process B- f- 17 is proportional to the product of Li IV„b1 2. In practice, even if Vub were well known, this is a very difficult process to measure. Our current best hope of determining fs is to rely on unquenched lattice QCD which can use the measurements of the analogous D+ -> v decay as check. Measurements of these D decays requires the construction of a "y-charm factory." Since the Bd mixing measurement determines: 1VNd12 cx 1(1 - "/ 77)12 = (P -1)2 + 7/2, (37) B phenomenology 263 it can be related to the CKM parameters; giving a circle centered at (1,0) in the p-n plane. This could in principle be a very powerful constraint. Unfortunately, the parameter BB is not experimentally accessible, and f B, although in principle measurable, has not been and may not be for a very long time, so it too must be calculated. The errors on these calculations are still quite large. 4.2 B3 mixing in the Standard Model 135° mesons can mix in a similar fashion to /33 mesons. The diagrams in Figure 26 are modified by substituting s quarks for d quarks, thereby changing the relevant CKM matrix element from Vtd to Vt„. The time dependent mixing fraction( is Am, G2F in2 = rn„ ivbK vs12774F is E i/QC131 (38) 672 bas ' t I Mw rs which differs from Equation (36) by parameters relevant for the B, rather than the Bd. Measuring x, allows us to use ratio of xd /x, to provide constraints on the CKM parameters p and 71. We still obtain a circle in the (p, n) plane centered at (1,0): Ivtd12 A2A4 [(1 p)2 (39) IVtd 2 = P)2 772. 1Vt.512 However we must calculate only the SU(3) broken ratios BBd /BB., and f Bd /f 13 . B° mixing has been searched for at LEP, the Tevatron, and the SLC. The probability, PM for a B, to oscillate into a B, is given by: (B, -+ B8) = [1 + cos (Amst)] , (40) where t is the proper time. To combine different experiments a framework has been established where each ex-periment finds amplitudes A for a spectrum of test oscillation frequencies, (.4.), defined as 79(t)= 2 -1 Fse-r'i [1 + A cos (wt)] . (41) Figure 29 shows the world average measured amplitude A as a function of the test frequency w = Am, (Leroy 2001). For each frequency the expected result is either zero for no mixing or one for mixing. No other value is physical, although allowing for measure-ment errors other values are possible. The data do indeed cross one at a Oats of 16ps-1, however here the error on A is about 0.6, precluding a statistically significant discovery. The quoted upper limit at 95% confidence level is 14.6 ps-1. This is the point where the value of A plus 1.645 times the error on A reaches one. Also indicated on the figure is the point where the error bar is small enough that a 4a discovery would be possible. This is at Am, = 11ps-1. One should also be aware that all the points are strongly correlated. The upper limit on Am, translates into an upper limit x, <21.6, also at 95% confidence level. CDF plans to probe higher sensitivity and eventually LHCb and BTeV can reach values of 264 Sheldon Stone 1 I I I "I' ••I I World average (prel.) data ± 1 a • 95% CL limit 14.6 ps4 --- 1.645 a -0 sensitivity 18.3 ps"' MI data ± 1.645 a D data ± 1.645 a (stet only) 4a Discovery Limit / .......... 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 m = Am (ns-11 Figure 29. Combined experimental values of the amplitude A versus the test frequency w = Arnd as defined in Equation 41. The inner (outer) envelopes give the 95% confi-dence levels using statistical (statistical and systematic) errors. The "sensitivity" shown at 18.3ps-1 is the likely place a 95% c.l. upper limit could be set. Also indicated is the maximum value, 1 1ps-1, where a 4o- discovery would be possible. 5 Rare b decays 5.1 Introduction These processes proceed through higher order weak interactions involving loops, which are often called "penguin" processes, for unscientific reasons (Lingel 1998). A Feynman loop diagram is shown in Figure 30 that describes the transition of a b quark into a charged -1/3 s or d quark, which is effectively a neutral current transition. The dominant charged current decays change the b quark into a charged +2/3 quark, either c or u. b s,do Figure 30. Loop or "penguin" diagram for a b s or b d transition. The intermediate quark inside the loop can be any charge +2/3 quark. The relative size of the different contributions arises from different quark masses and CKM elements. In terms of the Cabibbo angle (A=0.22), we have for the b s case t:c:u .\2:A2:A4. The mass dependence favours the t loop, but the amplitude for c processes can be quite large Moreover, as pointed out by Bander, Silverman and Soni (1979), interference can 2.5 2 1.5 A 0.5 -0.5 Hell = 07 = r(b s-y) = 4GF (VoVt312 C7 (rnb)07 1671-2 e rrit, Lcrii„bRFAP G2 Cart5 VtbVt::2 b 1C7I 2 327r4 (45) (46) (47) B phenomenology 265 occur between t, c and u diagrams and lead to CP violation. In the Standard Model it is not expected to occur for b s, due to the lack of a CKM phase difference, but could occur for b d. It is always worth looking for such CP violation effects; all that needs to be done, for example, is to compare the number of 13- K*--y events with the number of B+ 3 K*+-y events. There are other ways for physics beyond the Standard Model to appear. For example, the W- in the loop can be replaced by some other charged object such as a Higgs; it is also possible for a new object to replace the t. 5.2 Standard Model theory In the Standard Model the effective Hamiltonian for the intermediate t quark is given by (Deshpande 1994) 4GF 10 Heff = — Vtblit* E Ct (P)02 (A). (42) v 2 8 i=1 Two of the more important operators are 01 = .91^/AbiTi-eci, 07 = 1672 mbgiLo-,„63RFA'. (43) The matrix elements are evaluated at the scale au = Mw and then evolved to the b mass scale using renormalisation group equations, which mix the operators: Ca (µ) = E (ft, (p, mw)c,(mw ). (44) 5.3 b s y This process occurs when any of the charged particles in Figure 30 emits a photon. The only operator which enters into the calculation is CAA). We have for the inclusive decay It is far more difficult to calculate the exclusive radiative decay rates, but they are much easier to measure. Note that the reaction B Kry would violate angular momentum conservation, so the simplest exclusive final state is B K*-y. Different techniques are used for reconstructing exclusive and inclusive decays and unique methods are invoked for exclusive decays on the T(4S). At other machines the decay products, i, from an exclusive B decays are used to reconstruct an "invariant mass" via M2 = Ei Ez - E, pit. At the T(4S) the decay products are first tested to see if the sum of their energies is close to the beam energy, Ebeam. If this is true then the "beam constrained" invariant mass is calculated as M2 = Ebeam - Et p,2. These methods can be used for all exclusive B decays. Figure 31 shows the BELLE data for the reaction 524 5.26 5r.Nt M (GeV/c2) -0.2 -a15 -al -0.05 0 0.05 at 0.15 0 2 eE (GeV) 5.2 5.22 5.24 5.26 5.28 5 3 M [GeV/c2] "k)160 140 X120 ,,▪ 100 co 80 60 Q) 40 20 5.3 100 BO — at 60 F. 40 20 (b) 0.15 0.1 ."' 0.05 o -0.05 -0.1 -a15 5.22 • Data — Fit • BG • • (c) 266 Sheldon Stone Figure 31. BELLE data for the reaction TY —> D*1- . (a) The correlation between AE and M. The box shows the signal region. (b) The projection in AE for events in the M signal region; the line shows a fit to the background. (c) The M distribution for AE in the signal region; the line shows a fit to the background. —o B —> D*+71-- , where the D*+ 71-+D°. BELLE first selects events with candidate D°'s. Then they require an additional 7+ where the measured mass difference between the D°7+ minus the D° candidate is consistent with the known mass difference. Selecting the D*+ candidates they combine them with candidate 7r — . Figure 31(a) shows the the correlation between the difference in measured energy AE = Et Ez — Ebeam and the beam-constrained invariant mass. In Figure 31(b) AE is shown after selecting the signal region in M, and in Figure 31(c) M is shown after selecting on E. These plots show how clean signals can be selected. CLEO first measured the exclusive rate into K*(892)7 (Ammar 1993) shown in Fig-ure 32. Here several different decay modes of the K*(892) are used. The current world average value for 13(B —> K*0) = (4.2 ± 0.8 ± 0.6) x 10-5. 6 a) 4 En a▪ ) 2 5.200 5.220 5.240 5.260 5.280 5.300 M (GeV) Figure 32. First published CLEO data for the reaction B —> K*.y showing the 111 distribution for AE in the signal region. To find inclusive decays two techniques are used. The first one, which provides the cleanest signals, is to sum the exclusive decays for the final states Kn7ry, where n < 4 and only one of the pions is a 70. These requirements are necessary or the backgrounds become extremely large. Both charged and neutral kaons are used. Of course, imposing these restrictions leads to a model dependence of the result that must be carefully evaluated. This is why having an independent technique is useful. That is provided by detecting only Total Contin uum 1SR Continuum x° J— BB b—wr 1071 5, w 10 10 to Conti !mum Continuum + TSR B phenomenology 267 0 0.5 1 1.5 2 2.5 3 3.5 4 E7(GeV Figure 33. Levels of inclusive photons from various background processes at the 7(4S) labeled largest to smallest at 2.5GeV/c. Also shown is the expected signal from b s-y. the high energy photon. The technique used is to form a neutral network to discriminate between continuum and T(4S) data using shape variables. The momentum spectrum of the 7 peaks close to its maximum value at half the B mass. If we had data with only B mesons, it would be easy to pick out b —> s-y. We have, however, a large background from other processes. At the T(4S), the 7 spectrum from the different background processes is shown. The largest is 70 production from continuum e+e- collisions, but another large source is initial state radiation (ISR), where one of the beam electrons radiates a hard photon before annihilation. The backgrounds and the expected signal are illustrated in Figure 33. Similar backgrounds exist at LEP. Signal Figure 34. Examples of idealised event shapes. The straight lines indicate hadrons and the wavy lines photons. CLEO also made the first measurement of the inclusive rate for b s-y (Alam 1995). To remove background CLEO used two techniques, one based on "event shapes" and the other on summing exclusively reconstructed B samples. Examples of idealised events are shown in Figure 34. CLEO uses eight different shape variables and defines a variable r using a neural network to distinguish signal from background. The idea of the B re-construction analysis is to find the inclusive branching ratio by summing over exclusive 268 Sheldon Stone modes. The allowed hadronic system is comprised of either a K, —> 7+71-- candidate or a KU combined with 1-4 pions, only one of which can be neutral. The restriction on the number and kind of pions maximises efficiency while minimising background. It does however lead to a model dependent error. For all combinations CLEO evaluates ,2.13 (MB — 5.279)2 + (EB Ebeam) 2 , '` 0- A4 UE JJ where MB is the measured B mass for that hypothesis and EB is its energy. X 2B is required to be < 20. If any particular event has more than one hypothesis, the solution which minimises X2B is chosen. For events with a reconstructed B candidate CLEO also considers the angle between the direction of the B and the thrust axis of event with the B candidate removed, cos(Ot). This is highly effective in removing continuum background. A neural network is used to combine r, XB, cos(0t) into a new variable re and events are then weighted according to their value of rc. This method maximises the statistical potential of the data. Figure 35 shows the photon energy spectrum of the inclusive signal from CLEO combining both reconstruction techniques. The signal is compared to a theoretical prediction based on the model of Ali and Greub (Ali 1991). A fit to the model over the photon energy range from 2.0 to 2.7GeV/c gives the branching ratio result shown in Table 5, where the first error is statistical, the second systematic and the third shows the dependence on the theoretical model (Chen 2001). (48) 1 ON401.0717 Data — Spectator Model >40 - 8 ci) 0 1 5 2.5 3.5 4.5 Er (GeV) Figure 35. The background subtracted photon energy spectrum from CLEO. The spectrum is not corrected for resolution or efficiency. The solid lines show the spectrum from a simulation of the Ali-Greub spectator model with the b quark mass set to 4.690GeV and the Fermi momentum set to 410MeV/c. ALEPH reduces the backgrounds by weighting candidate decay tracks in a b —> sry event by a combination of their momentum, impact parameter with respect to the main vertex and rapidity with respect to the b-hadron direction (Barate 1998). Current results are shown in Table 5. The data are in agreement with the Standard Model theoretical prediction to next to leading order, including quark mass effects of B phenomenology 269 Experiment B x 10-4 CLEO 3.21 ± 0.43 ± 0.271:18 ALEPH 3.11 ± 0.80 ± 0.72 BELLE 3.36 ± 0.53 ± 0.441:N Average 3.23 ± 0.42 Table 5. B(b s7). (3.73 ± 0.30) x 10-4 (Hurth 2001). A deviation here would show physics beyond the Standard Model. More precise data and better theory are needed to further limit the parameter space of new physics models, or show an effect. 5.3.1 using moments of the photon energy spectrum In Section(2.2.4) we found a value of Kb using the first and second hadronic mass moments. Here we use the first moment of the photon energy distribution in b s-y. The values found for the moments and A which is directly proportional to (E7) are (Chen 2001) (E.,) = 2.346 ± 0.032 ± 0.011 GeV (49) - (E.02 = 0.0226 ± 0.0066 ± 0.0020 GeV2 (50) A = 0.35 ± 0.08 ± 0.10 GeV. (51) In Figure 36 we show the combination of first moments from photon energy in b .97 and hadron moments in b -+ cif,. This implies a value of Vd, around 0.0406. 0.1 X1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A Figure 36. Correlation between Ai, X. and Vet, derived from (E.,) and (Mk - MD). The lighter bounds include both experimental and theoretical errors. 270 Sheldon Stone 5.4 Rare hadronic decays The decays T3° -+ 7r+7r- and r K 71+ do not contain any charm quarks in the final state, so they must proceed either via the tree level Vub process shown in Figure 37(left), or via the penguin process shown in Figure 37(right). 7C Figure 37. Decay diagrams for 7-3° 7+7-. (left) Via tree level Vub process. (right) Via a penguin process. These diagrams can interfere with each other, and they can also interfere through B° mixing, thus complicating any weak phase extraction. The same diagrams are applicable for T3° 71+ by replacing W- fid in the tree level diagram by W- us and replacing the td coupling in the penguin by a is coupling. Figure 38. Diagrams for B- -+ K-7°, (a) and (b) are tree level diagrams where (b) is color suppressed; (c) is a penguin diagram. (d) shows B- K°7- , which cannot be produced via a tree diagram. Other diagrams for producing K7r final states are shown in Figure 38. In Section 7.1 it will be shown that CP violation can result from the interference between two distinct decay amplitudes leading to the same final state. Consider the possibility of observing o 5 2 5.225 5.25 5.275 5 3 M (GeV/c2) 0 40 30 20 10 0 6 c) ic,A* 0 • ' -0.25 -0.125 0 0.125 0.25 AE (GeV) r c Ken° 2 A nil.. 0 5 2 5.22 5.24 5.26 5.28 M (GeV/c2) 8 ° C.) b) K'ne C O (.3 Nu 10 > 8 2 6 O C7 4 cn 2 C > 8 2 6 = a) en 0 - 4 Ir 2 0 0 0 10 = b) 7.5 = Jl 5 2.5 0 e) Ken° 6 4 2 -0.3 -0.15 0 0.15 .6,E (GeV) -0 45 10 4 a) en° - L a) en' B phenomenology 271 Figure 39. Signals in M and AE for two-body decay modes from BELLE. The data result from projections of a Likelihood fit that takes into account event shape and particle identification information. The dashed lines are the signal projections. The dotted lines in the AE distributions are projections of the background component from the ir <=> K substitution. CP violation by measuring a rate difference between B- K-7° and B+ -4 K+7°. The K-7° final state can be reached either by tree or penguin diagrams. The tree diagram has an imaginary part coming from the Vut, coupling, while the penguin term does not, thus ensuring a weak phase difference. This type of CP violation is called "direct." Note also that the process B-->K°71-- can only be produced by the penguin diagram in Figure 38(d). Therefore, we do not expect a rate difference between B--3C°7- and B+-407+. Measurements of these rates have been made by several groups. Recent data from BELLE are shown in Figure 39 (Abe 2001a). Table 6 lists the currently measured branch-ing ratios. Mode CLEO BABAR BELLE Average R-+7-- 4.7+1:g + 0.6 4.1 ± 1.0 ± 0.7 5.61:0 + 0.4 4.5+N 7+ 7,0 <12 <9.6 <13.4 K±7I-+ 18.8±2:6 + ± 1.3 16.7 ± 1.6 ± 1.3 19.3t Ilt .0 17 7+1.6 ' ' • ' -1.5 K+7° K°7- K°7° 12.1-+23:80-+21:41 18.21:,1 + 1.6 14.8+5'9+2 '4 -5.1-3.3 10.8+N 1 1.6 18.2+3 0 ± 2.0 8.2+2:7 ± 1.2 16.3+13+14 13.71:78-±1: 16.0+_57:92±22..75 12.1t1:76 17.3+_2:a 10• -2 4+27 .5 Table 6. Branching Ratios for B K7r and B -> 77 in units of 10-6. 272 Sheldon Stone 6 Hadronic decays 6.1 Introduction Mark Wise in his talk at the 2001 Lepton Photon conference (Wise 2001) gave some advice to theorists: "If you drink the nonleptonic your physics career will be ruined and you will end up face down and in the gutter." Presumably Mark's statement was inspired by the difficulty in predicting hadronic decays. Here we have lots of gluon exchange with low energy gluons, while perturbation theory works well when the energies are large compared with Awl:, ",200MeV. Furthermore, multibody decays are currently impossible to predict, so we will consider only two-body decays. 6.2 Two-body decays into a charm or charmonium We start by considering two-body decays into a charmed hadron (Neubert 1997). Figure 6 shows the processes for both B- and 7-30 decays into a D and a 7r-. There is only one possible process for the TY, the simple spectator process (left), while the color suppressed spectator (right) is also allowed for the B. We call decays with only the simple spectator diagram allowed "class I" and decays where both the simple and color suppressed diagrams are allowed "class III". Note, that because the colors of all the outgoing quarks must be the same in the color suppressed case, naively the amplitude is only "4/3 that of the simple spectator case where the 147- can transform into quarks of all three colors. "Class II" decays are processes than can only be reached by the color suppressed spectator diagram, for example the B° J/OKs decay shown in Figure 40. it J/ d Ks Figure 40. Color suppressed spectator decay diagram for B° J/11)Ks• The effective Hamiltonian consists of local 4-quark operators renormalised at the scale µ and the Wilson coefficients from the Operator Product Expansion, ci (n). We have Heff = G F iVcb [C1 (.1)(2V) + C2 iN2b1 Vub • • •} N7 (52) Qlb = [(ctu)v—A+ c)v —Al (Tb)v_A = (Tu)v—A(clb)v—A+ (c)v—A(74)v —A, where the notation (qi q2)v.... A — -y5)q2. Without QCD corrections c1 (µ) = 1 and c2(p) = 0. From the non-leading order correction using the renormalisation group equations, we have ci (p) = 1.132 and c2(p) = —0.249. B phenomenology 273 We can factorise the amplitude by assuming that the current producing the 7r- is independent of the one producing the charmed hadron. Let us consider a class I case, —o B D±r - . The amplitude can be written as IT* I Afact = yd. v udal \ v (au) A0) (D+ l(a)vir) (53) The part of the amplitude dealing with the 1r- is known from pion decay. We have (7r-Idvy5u10) = i f„pi„, where the axial vector structure is made explicit, p, is the pion four-vector and f, is given by measuring the decay width for 71- -4 icy. The term al is defined as al = ci (p f )+ cz(pf ), (54) where is equal to the number of colors and the scale ti f is on the order of the b quark mass. Then Afact = GFV* —Vcbaifr (Mg m2D)F0/3-",(m20. ud The F0 form factor can either be calculated or measured in semileptonic decays. For a class II process B JIWK: GF Afact = ----Valles* a2(.1/0 I (Z0v10)(K 1(3b)vlB), where a2 = cz(it.f) + eci(ii.f)• A class III decay has a term al + xa2 in the amplitude, where x equals one from flavor symmetry. The actual values of al , x, and a2 are not well predicted from theory, but we can obtain them from the data. One method is to use the class I decays to obtain al = 1.08 ± 0.04. It is possible to calculate az/ai as shown in Figure 41 (Neubert 1997). Using these values the measured branching ratios are compared with the predicted ones in Table 7. Here x = +1 is used, taken from the data. This is opposite to the interference in D decays but is expected from the calculation shown in Figure 41. 0.6 0.4 0.2 B -0-1) -0.2 -0.4 D-P-1{ n -0.6 , ............. ..... 0 0.2 0.4 0.6 0.8 1 as(ilf) Figure 41. Calculated values of a2/al versus as . The arrows indicate the ratios chosen for B and D decays. (55) (56) 274 Sheldon Stone Class I Class II Class III Mode Model Data Mode Model Data Mode Model Data D+7- 30 30+4 J/OK- 11 10+1 D°7r- 48 53+5 D*-1- 7- 30 28+2 J/OK* 17 15+2 D*°7- 49 46+4 D±p- 70 79+14 D°7r° 0.7 2.8+0.4 D'p- 110 134+18 Dt+p- 85 73+15 D*°7° 1.0 2.1+0.9 D"p- 119 155+31 Table 7. Predicted Branching Ratios (10-4) (Neubert 1997) Compared to Measurement For Two-Body Hadronic Decays. The agreement is rather good except for the newly measured D°(*)7° modes where it is rather miserable. CLEO and BELLE both have rates for D°70 of (2.7 + 0.3 + 0.5) x 10' and (2.911 + 0.6) x 10', respectively, while CLEO alone has measured D*°r° as (2.1 ± 0.5 ± 0.8) x 6.2.1 Isospin analysis of the B ./37r system All the decay rates for B -+ Thr have now been measured. The four-quark operator (du)(-eb) has isospin I=1 and I3=+1. It transforms the r into final states D4 7- and D°7° with 1=1/2 or 1=3/2. The B- decays into D°71-- with 1=3/2 only. It is thought that the isospin amplitudes cannot be modified by final state interactions, so we can look for evidence of final state phase shifts by doing an "isospin analysis." The decay amplitudes form a triangle as shown in Figure 42. Figure 42. The isospin amplitude triangle for B D7- decays. The relationship among B decay amplitudes and isospin amplitudes is given by 1 2 Acre D+7-) = -3A3/2 +3Aii2 V) --+ D ° 71- 3 ° 1 A ( ) = -2 A312 - 3A172 A(B- VF-) = NA3/2. (57) These equations may be solved for the isospin amplitudes and the relative phase shift between the two amplitudes. The solution is A112 2 = A(T30 D+7-) 2 + A(76° D°7°) 2 1 3 A(B- D°7-) 1 2 (58) 31A(B- )1 2 - 2 2 2 A112 A312 Tests of this equation for D''± and a 7r-, p- or aT are satisfied at about 15% accuracy (Bortoletto 1990) (Browder 1996). Another test compares the polarisation of the D* in both hadronic and semileptonic cases: FL, FL - (B D*0 = - (B D*0 (60) q2 =m2h where FL denotes the longitudinally polarised fraction of the decay width. Comparisons with data will be shown in the next section. There are more modern approaches to factorisation (Beneke 2001, and Bauer 2001). However these approaches predict F(B- D°7 = 1 (9(AQcD/mb) F(./'€) -> D+7r-) (61) B phenomenology 275 A1/2 2 1 3 A(B D° 7r- ) 2 cos cos (83/2 - 81/2) - 2 N/2 A112 Solving these equations for the D7r final states gives cos S = 0.88±0.05 which indicates a phase shift of about 28 ± 8 degrees, which is not statistically significant enough to rule out zero. 6.2.2 Factorisation tests using semileptonic decays The factorised amplitude for D7r- decays in Equation 56 is the product of two hadronic currents, one for W- 7- and the other for B D. In semileptonic decay (Figure 5) we have the product of the known lepton current and the pion current. At q2 = m2, the B D should be the same in both decays, at least for class I. The comparison for the general case of any hadron h- is dr F (B D(*)h) = 67r2c4fi, 1V.,,d1" —dq2 (B D(* ) 17) A312 (59) ,2 =7,22h which seems to contradict current observations. 6.3 Observation of the p' in B decays CLEO made the first statistically significant observations of six hadronic B decays shown in Table 8 that result from studying the reactions B D(*)7±7r-7r-70(Alexander 2001). The signal in one of these final states B D(*+)7T+1- 7-e , where D*-F 7r+ D° and D° K-7T+ is shown in Figure 43. In examining the substructure of the four-pions, a clear w signal was observed in the 7+71-7° mass as can be seen in Figure 44, leading to a significant amount of D(*) aur-Furthermore, there is a low-mass resonant substructure in the unr- mass (Figure 45). The spin and parity of the air- resonance (denoted by A temporarily) is determined by considering the decay sequence B A D; A bEr and w 71-±7-7°. The angular 100 200 1, 0 200 100 ( b ) 0 520 5.24 5.26 IA, (GeV) 5.22 5.26 5.30 276 Sheldon Stone Mode B (%) 5 20 0 05 —o B D*-1-7r+r-71--7° —o B —> D*+ unr- -o B D+ unr- B- D"unr- B- 1.72±0.14±0.24 0.29±0.03±0.04 0.28±0.05±0.03 1.80±0.24±0.25 0.45±0.10±0.07 0.41±0.07±0.04 Table 8. Measured Branching Ratios Figure 43. The B candidate mass spectra for the final state D*±7r+7r-71--7r°, with D° K-71-+ (a) for AE sidebands and (b) for AE consistent with zero. The curve in (a) is a fit to the background distribution described in the text, while in (b) the shape from (a) is used with the normalisation allowed to float and a signal Gaussian of width 2.7MeV is added. ' I I ' • Signal MB Sideband _ AE Sideband _ + + 0.6 0.7 0.8 0.9 1.0 11 M Jura (GeV) Figure 44. The invariant mass spectra of 7r+7r-r° for the final state D*+7r+71-7-7r° for all three D° decay modes K-71+, 1C-7r+7r+7r- and K-7r+70. The solid histogram is the background estimate from the MB lower sideband and the dashed histogram is from the AE sidebands; both are normalised to the fitted number of background events. There is an additional cut selecting the centre of the Dalitz plot. 0 e 1 2 2- ....... - -1 0 -0.5 0 0.5 cos O. -0.5 0 0.5 10 cos O., 20 — B phenomenology 277 Alt11111 1111111111111.11.111 1.5 1.0 1.5 2.0 2.5 3.0 1.1.,... (GeV) Figure 45. (left) The background subtracted efficiency corrected W7T — mass spectrum from —o B (D*+ + D° + Dlunr- decays fit to a Breit-Wigner shape. distributions are shown in Figure 46. Here OA is the angle between the w direction in the A rest frame and the A direction in the B rest frame; 0,, is the orientation of the w decay plane in the w rest frame; and X is the angle between the A and w decay planes. Figure 46. The angular distribution of OA (top-left),0„, (top-right) and X (bottom). The curves show the best fits to the data for different J P assignments. The 0- and 1+ are almost indistinguishable in cos 0A, while the 1- and 2+ are indistinguishable in cos 0„, and X. The vertical axis gives e f iciency corrected events (104 data events are used). The data are fit to the expectations for the various JP assignments. The w polarisation is very clearly transverse (sine 0,) and that infers a 1- or 2+ assignment. The cos BA distribution prefers 1-, as does the fit to all three projections. This structure is identified with the p' because it has the correct JP and is at approx-imately the right mass. To determine the mass and width parameters, which are not well known, we write the decay width as a function of unr- mass as dr (B Dunr-) = 2MB A(B -4 D p') • 811 )(m„) • A(p' unr )2 (62) dm2, x dP(B Dp') • dP(p' unr-) 2; , thin 3.5 4 0 = NIP 278 Sheldon Stone where dP indicates phase space and the Breit-Wigner amplitude is given by BW(m") = — mp2,) — im,,,,rtot(mw7r) • 1 (63) The Breit-Wigner fit assuming a single resonance and no background gives a mass of 1349±251°MeV with an intrinsic width of 547±86+VeV. The fit shows that the unr-mass spectrum is consistent with being entirely one resonance. This state is likely to be the elusive p' resonance (Clegg 1994). These are by far the most accurate and least model dependent measurements of the p' parameters. Since the p' dominates the final state, the branching ratios for D(*)unr- apply also to D(*) sures the relative rates to be r (B D* (B Dp'-) = 1.06+0.1710.04, consistent Heavy quark symmetry predicts equal partial widths for D* p' andDp'. CLEO mea- within the relatively large errors. Factorisation predicts that the fraction of longitudinal polarisation of the D*+ is the same as in the related semileptonic decay B D* CO at four-momentum transfer q2 equal to the mass-squared of the p' FL (B D*+ FL (B D* (64) F (B D*+ p'-) F (B q2=m2, CLEO's measurement of the D*+ polarisation is (63±9)%. The model predictions in semileptonic decays for a q2 of 2GeV2, are between 66.9 and 72.6% (Isgur 1990, Wirbel 1985, Neubert 1991). Figure 47 shows the measured polarisations for the D*+ , the D*+ , (Artuso 1999) and the Dt+ D:- final states (Stone 2000). The latter is based on a new measurement using partial reconstruction of the D*+ (Ahmed 2000). Thus this prediction of factorisation is satisfied. 1 I I I I f t 1 1 I I 1 I I i I 1 I q2 0 Factorization Prediction (la) Region — 2 4 6 10 q2 (Gev2) Figure 47. Measured D*+ polarisation versus semileptonic model predictions. X 1.0 0.8 -a 0.6 3 0.4 o 0.2 0 B phenomenology 279 7 CP violation 7.1 Introduction Recall that the operation of Charge Conjugation (C) takes particle to anti-particle and Parity (P) takes a vector r to —r. CP violation can occur because of the imaginary term in the CKM matrix, proportional to 77 in the Wolfenstein representation (Bigi 2000). Consider the case of a process B —> f that goes via two amplitudes A and B each of which has a strong phase, sA, and a weak phase, W A. Then we have F(B —> f) = (1A1 ( sA±wA ) +181ei("+") ) 2 (65) F(B f) = (1A1e' ( sA-wA) + 181ez("-"))2 (66) F(B f) — r(B f) = 2 IABI sin(sA — ss) sin(wA — wB). (67) Any two amplitudes will do, though it is better that they be of approximately equal size. Thus charged B decays can exhibit CP violation as well as neutral B decays. In some cases, we will see that it is possible to guarantee that Isin(sA — sB)I is unity, so we can get information on the weak phases. In the case of neutral B decays, mixing can be the second amplitude. 7.2 Unitarity triangles The unitarity of the CKM matrix, namely Vud Vus Vub Vud 17c*d 171',‘ 1 0 o Vcd ves Kb K*8 VCs vt,* ) = o 1 o Vtd Vt, Vtb Vul Vc'6 17,1; o 0 1 allows us to construct six relationships of which the most useful turns out to be To a good approximation Vud •=-%Vtb•=,' so that Equation 68 becomes Now 1 1 A 17.d17t,; 1, 1 Vt* AK() Vtd Vts Vub Kb + 17us17t: + 17ubl7t Vub —1±--1 = A Kb = O. Vus = A, (68) (69) (70) (72) (73) 1 Vi i1)2 U + 772 vP2 ± 712 , , 280 Sheldon Stone Figure 48. The CKM unitarity triangle in the p-71 plane. The left side is determined by measurements of b u/b c and the right side can be determined using mixing mea- surements in the B, and Bd systems. The angles can be found by making measurements of CP violating asymmetries in hadronic B decays. so that Equation 70 defines a triangle in the p-ri plane. This is the CKM triangle depicted in Figure 48. We know the length of two sides already: the base is defined as unity and the left side is determined by the measurements of 111„b/Vcb I, but the error is still quite substantial. The right side can be determined using mixing measurements in the neutral B systems. Figure 48 also shows the angles labelled a, /3, and 'y. These angles can be determined by measuring CP violation in the B system. Another constraint on p and n is given by the measurement of E in CP violation in IC2 decays (Buras 1995): BK [(1 — p)A2(1.4 ± 0.2) + 0.35] A217—.75 --- (0.30 ± 0.06), (74) where BK is parameter that cannot be measured and thus must be calculated. A reason-able range is 0.6 < BK < 0.9, given by an assortment of theoretical calculations (Buras 1995). This number is one of the largest sources of uncertainty on the e constraint. Other constraints come from current measurements on VublV,b, Bd mixing and B3 mixing. The current status of these constraints on p and n is shown in Figure 49 (Hocker 2001). The width of these bands are also dominated by theoretical errors. This shows that the data are consistent with the standard model. It is crucial to check if measurements of the sides and angles are consistent, i.e., whether or not they actually form a triangle. If they do not the standard model is incomplete. The standard model has many parameters including the four CKM numbers, six quark masses, gauge boson masses and coupling constants. Perhaps measurements of the angles and sides of the unitarity triangle will bring us beyond the standard model and show us how these parameters are related, or what is missing. New physics can also be observed by measuring branching ratios which violate standard model predictions. Especially important are rare decay processes such as B K it+ µ- or D µ±µ-. These processes occur only through loops, and are an important class of penguin decays. B phenomenology 281 1 0.8 0.6 0.4 0.2 -0.5 0 0.5 5 Figure 49. The regions in p-ri space (shaded), where -0 = p(1—A2 /2) and fl =77(1— A212), consistent with measurements of CP violation in .K2 decay (e), VublVcb in semileptonic B decay, Bd mixing, and the excluded region from limits on B° mixing. The allowed region is defined by a fit from (Hocker 2001). The large width of the Bd mixing band is dominated by the uncertainty in BB M. The lines that are not specified are at 5% confidence level. 7.3 Formalism of CP violation in neutral B decays Consider the operations of Charge Conjugation, C, and Parity, P: CIB(P)) = IMP)), CIMP)) = IMP)) (75) PIMP)) = —1/(-13)), PIB(P)) = — 1 13(-13)) (76) CPIB(P)) = — IB( —P)), CPIB(P)) = — 1 13(-13)). (77) For neutral mesons we can construct the CP eigenstates IB?) = (I13°) ir)) , (78) where IBZ) = (IB°) + 0)) , (79) cP113°) = 113?), (80) cp piA)) = (81) Since B° and B can mix, the mass eigenstates are a superposition of al/3°) +1/.74°) which obey the Schrodinger equation 2 . a 7( -1t b) = (ab) = (114 (ab) (82) 282 Sheldon Stone If CP is not conserved then the eigenvectors, the mass eigenstates IBL ) and IBH ), are not the CP eigenstates but are 1BL) = plB°)+ 4'0 ), 1BH) = plB°) — (83) 1 1 + € B 1 1 — E B P = v2 /- — /- V1 +10312' v2 V1 +10312 CP is violated if 1B 0 0, which occurs if lq/p1 0 1. The time dependence of the mass eigenstates is BL(t)) = e-F012elmi,t12 1BL (0)) e_rol2eini x ti2 IBH (0))7 1BH(t)) leading to the time evolution of the flavor eigenstates as i Bo(t)) = e-(irn+rmt A2nd cos IB°(0)) sin Anil IT3° p 2 (13)) (85) (86) (87) IB (t)) = e-(im+r/2)t (ip sin A2rnt 0(0)) + cos A2rnt 1B(0))), g where m = (mL + Tr/ H )/2, Am = my — mL, r = FL FM and t is the decay time in the B° rest frame, the so called proper time. Note that the probability of a B° decay as a function of t is given by (B°(t)IB°(t))*, and is a pure exponential, e-"i2 , in the absence of CP violation. 7.3.1 CP violation for B via interference of mixing and decays Here we choose a final state f which is accessible to both B° and T3° decays (Bigi 2000). The second amplitude necessary for interference is provided by mixing. Figure 50 shows the decay into f either directly or indirectly via mixing. It is necessary only that f be Figure 50. Two interfering ways for a B° to decay into a final state f. accessible directly from either state; however if f is a CP eigenstate the situation is far simpler. For CP eigenstates CPIfce) = ±IfcP)• (89) It is useful to define the amplitudes A= A= (fcPriiir)- (90) where (84) (88) B phenomenology 283 If A/A 1, then we have "direct" CP violation in the decay amplitude, which we will discuss in detail later. In general CP can be violated by having A — —q • —A 1 (91) p which is also the case if A acquires a non-zero phase, i.e AlI could be unity and CP violation can occur. A comment on neutral B production at e+ colliders is in order. At the T(4S) resonance there is coherent production of B°B° pairs. This puts the B's in a C = —1 state. In hadron colliders, or at e+ e- machines operating at the Z°, the B's are produced incoherently. For the rest of this article I will assume incoherent production except where explicitly noted. The asymmetry, in this case, is defined as F (B°(t) fcp) — r (Tem —> Ice) which for lq/p1 = 1 gives (1 — 1Al 2) cos (Amt) — 2ImA sin(Amt) afcp — 1 + 1Al 2 For the cases where there is only one decay amplitude A, 1Al equals 1, and we have afcp = —ImA sin(Amt). (94) Only the amplitude, —ImA, contains information about the level of CP violation, the sine term is determined only by Bd mixing. In fact, the time integrated asymmetry is given by arei, = 1 + x2Im A = —0.48Im A. (95) This is quite lucky as the maximum size of the coefficient for any x is —0.5. Let us now find out how ImA relates to the CKM parameters. Recall A = (q/p)(A/A). The first term is the part that comes from mixing: q (V V t d) 2 (1 — p — in)2 e-2z0 P 11407td12 ( 1 p+ in) (i — and Im p = q (12(102 p) 7i2 = sin (20) . (97) — + 7/ To evaluate the decay part we need to consider specific final states. For example, consider f 7+7r- . The simple spectator decay diagram is shown in Figure 37 (left). For the moment we will assume that this is the only diagram which contributes, although we know this is not true. For this b —> 77,7-zd process we have A (17u*dvuby (p — i17)2 A 1Vudilubl 2 (P -221)(P+ 271) aloe F (B°(t) fcp) + F (17 0 (t) fcp)' (92) (93) (96) (98) 284 Sheldon Stone and Im (.A) = Im (e-2eSe-gory) = Im (e2za) = sin(2a). (99) For our next example let us consider the final state Jiff,. The decay diagram is shown in Figure 40. In this case we do not get a phase from the decay part because A (VcaVe':) 2 A ivcbv,s 1 2 is real to order 1/A4. In this case the final state is a state of negative CP, i.e. CPWOKs) = -1,//01(s). This introduces an additional minus sign in the result for Im A. Before finishing discussion of this final state we need to consider in more detail the presence of the K, in the final state. Since neutral kaons can mix, we pick up another mixing phase (similar diagrams as for B°, see Figure 26). This term creates a phase given by 1\ = (vzvcs)2 P)If IVedVcs12 which is real to order A4. It necessary to include this term, however, since there are other formulations of the CKM matrix than Wolfenstein, which have the phase in a different location. It is important that the physics predictions not depend on the CKM convention. (Here we don't include CP violation in the neutral kaon since it is much smaller than what is expected in the B decay. The term of order Azi in 17,5 is necessary to explain K° CP violation.) In summary, for the case of f = Jiff s, Im .A = - sin(20). 7.3.2 Comment on the penguin amplitude In principle all processes can have penguin components. One such diagram is shown in Figure 37(right). The 7±7r- final state is expected to have a rather large penguin ampli-tude ,--,10% of the tree amplitude. Then IAI 0 1 and a, (t) develops a cos(Arnt) term. It turns out that sin(2a) can be extracted using isospin considerations and measurements of the branching ratios for B+ 7+7° and B° eir°, or other methods, the easiest of which appears to be the study of B° In the J/OK, case, the penguin amplitude is expected to be small since a cc pair must be "popped" from the vacuum. Even if the penguin decay amplitude were of significant size, the decay phase is the same as the tree level process, namely zero. 7.4 Results on sin 20 For years observation of large CP violation in the B system was considered to be one of the corner stone predictions of the Standard Model. Yet it took a very long time to come up with definitive evidence. The first statistically significant measurements of CP violation in the B system were made recently by BABAR and BELLE. This enormous achievement was accomplished using an asymmetric e+ e- collider on the T(4S) which (100) (101) B phenomenology 285 Experiment sin 20 BABAR 0.59 ± 0.14 ± 0.05 BELLE 0.99 + 0.14 ± 0.06 Average 0.79+0.11 CDF 0.79'0.44 ALEPH 0.84±0.04 ± 0.16 OPAL 3.2±A ± 0.5 Table 9. Statistically significant measurements of sin 20 compared with previous results. was first suggested by Pierre Oddone. The measurements are listed in Table 9, along with other previous indications (Groom 2001). The average value of 0.79 + 0.11 is taken from BABAR and BELLE only. These two measurements do differ by a sizeable amount, but the confidence level that they correctly represent the same value is 6%. The value of 0.79 is consistent with what is expected from the other known constraints on p and 7/. We have T7 = (1 — P) sin 20 There is a four fold ambiguity in the translation between sin 2/3 and the linear constraints in the p-71 plane. These occur at /3, 7r/2 — 0, 7r + 0 and 37x/2— ,0. Two of these constraints are shown in Figure 51. The other two can be viewed by extending these to negative We think n > 0 based only on the measurement of e in the neutral kaon system. -0.5 0 0.5 1 Figure 51. Constraints from sin 20 measurement overlaid with other constraints (Hocker 2001). The inner band is at la while the outer band, shown on one band only, is at 2o-. 1 + 3 — sin2 20 (102) 286 Sheldon Stone 7.5 Remarks on global fits to CKM parameters The fits shown in this paper (Hocker 2001) for p and n have been done by others with a somewhat different statistical framework (Ciuchini 2001, Mele 1999). The latter group uses "Bayesian" statistics which means that they use a priori knowledge of the probability distribution functions. The former are termed "frequentists" (Groom 2001), almost for the lack of a better term. The frequentists are more conservative than the Bayesians. The crux of the issue is how to treat theoretically predicted parameters that are used to translate measurements into quantities such as Viib or E K that form constraints in the p — n plane. This is of course a problem because it is difficult to estimate the uncertain-ties in the theoretical predictions. Both groups treat the experimental measurements as Gaussian distributions with the la width derived from combined statistical and system-atic errors. Note, that experimental systematic errors are also difficult to quantify and are not necessarily Gaussian, but they are judged to be sufficiently well known to not cause a problem. Hocker et al. (2001) have decided to use a method which restricts the theoretical quantities to a 95% confidence interval where the parameter in question is just as likely to lie anywhere in the interval. They call this the "Rfit" method. Of course assigning the 95% confidence interval is a matter of judgment which they fully admit. Ciuchini et al. (2001) treat the theoretical errors in the same manner as the experimental errors. They call theirs "the standard method" with just a bit of hubris. They argue that QCD is mature enough to trust its predictions, that they know the sign and rough magnitude of corrections and they can assign reasonable errors, so it would be wrong to throw away information. Hocker et al. point out an interesting but generally unknown danger with the Bayesian approach, which is that in multi-dimension problems the Bayesian treatment unfairly predicts a narrowing of possible results. The following discussion will demonstrate this. Let xi denote N theoretical parameters over the identical ranges [—A, +A]; then the theoretical prediction is TJ N~ 11 xi. (103) In the 95% scan scheme [71,N)] = [—AN, +AN], while in the Bayesian approach the con-voluted Probability Density Function (PDF) is +co N p(T) =- f f dx,G(x j)(S(T — 71,N) ), (104) where the G(xi ) are the PDF's for the individual variables which taken to be equal here. This function has a singularity in p(T) that goes as (— ln T) '̂-1, so it increases as N increases. Now suppose G(xi ) is flat, then for N = 1 both approaches are the same, but for N > 2, the Bayesian approach gets a p(T) that peaks at zero. In effect, when the number of theoretical predictions entering the computation increases, and hence our knowledge of the corresponding observable decrease the Bayesian approach claims the converse. Lets look at a specific example where N = 3 and A = 0. Consider both the sum Ts = xi + x2 + x3 and product Tp = xi x2x3 distributions. For Rfit the allowed ranges are -8 -6 -4 -2 0 2 4 6 8 1.5 I: I 0.5 1 F" 10.2 e'31 10 -3 10 -4 10 -8 -6 -4 -2 0 2 4 6 8 0.3 0.2 0 - — Bayesian (Gauss) - Bayesian (flat) 1 Rfit (error ranges) - Confidence Level 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 B phenomenology 287 TS = x l +x2+X3 TP = X I X2 X3 Figure 52. Convolution of the sum Ts = xi+ 12+13 for both Rfit and Bayesian methods (left) and the product Tp = 111213 (right) of 3 parameters for the Bayesian method only. Plotted is the PDF p(T) obtained using for G(x) a uniform (solid lines, A = 0) or a Gaussian (dashed lines, a = 1) distribution. Both PDF's of Tp have a singularity at the origin which is not shown. The Rfit ranges of Ts and Tp are indicated by the arrows located in both instances at +30. From (Rocker 2001). identical being [-30, 30]. The left side of Figure 52 shows the probability density p(T) for Ts, while the right side shows p(T) for Tp with G(x) in the Bayesian case being either a Gaussian with a = 1 (solid lines) or a uniform distribution over the range [— 0, + (dashed lines). The later distribution is closest to the Rfit method where the allowed range for either T is [-30, +3/] indicated by the arrows. In the Rfit scheme the two predictions for Tp and Ts are identical, while in the Bayesian scheme there is a large difference in the PDF's and it really doesn't matter if a Gaussian or uniform distribution is chosen. There is a clear distinction between the Rfit and Bayesian predictions for Tp in this case, and the Bayesian one is unreasonable because it produces a very narrow PDF peaked very close to zero. An example of how this can effect the results is shown on Figure 53 where predictions of sin 20 from the indirect measurements are shown for the Rfit technique and either uniform or Gaussian Bayesian PDF's. The predictions are quite different. — Rfit - • - • - Bayesian (Gauss) Bayesian (uniform) =, Figure 53. Comparison between Rfit (broad solid curve) and Bayesian fits for the indirect CKM constraints on sin 20. For the Bayesian fits: Gaussian (uniform) theoretical PDF's are depicted as dashed-dotted (dotted) curves. (This example does not include the newer data.) 288 Sheldon Stone 8 New Physics studies 8.1 Introduction There are many reasons why we believe that the Standard Model is incomplete and there must be physics beyond. One is the plethora of "fundamental parameters," for example quark masses and mixing angles. The Standard Model cannot explain the smallness of the weak scale compared to the GUT or Planck scales; this is often called "the hierarchy problem." In the Standard Model it is believed that the CKM source of CP violation extensively discussed here is not large enough to explain the baryon asymmetry of the Universe (Gavela 1993). Finally, gravity is not incorporated. John Ellis said "My personal interest in CP violation is driven by the search for physics beyond the Standard Model" (Ellis 2000). We can also take the view that we will discover additional large unexpected effects in b and/or c decays. We must realise that all our current measurements are a combination of Standard Model and New Physics; any proposed models must satisfy current constraints. Since the Standard Model tree level diagrams are probably large, let us consider them as a background to New Physics. Therefore loop diagrams and CP violation are the best places to see New Physics. The most important current constraints on New Physics models are . The neutron electric dipole moment, d N < 6.3 x 10-26e-cm. . B(b sry) = (3.23 ± 0.42) x 10-4 and B(b ge+C) < 4.2 x 10'. . CP violation in KL decay, f K = (2.271 ± 0.017) x 10-3. . B° mixing parameter Arnd = (0.487 + 0.014) ps-1. 8.2 Generic tests for new physics We can look for New Physics either in the context of specific models or more generically, looking for any deviations from the Standard Model expectation. One example is to examine the rare decays B -4 Kite— and B —> K*.e+P- for branching ratios and polari-sations. According to Greub et al., "Especially the decay into K* yields a wealth of new information on the form of the new interactions since the Dalitz plot is sensitive to subtle interference effects" (Greub 1995). Another important tactic is to test for inconsistencies in Standard Model predictions independent of specific non-standard models. The unitarity of the CKM matrix allows us to construct six relationships. These may be thought of as triangles in the complex plane shown in Figure 54. (The bd triangle is the one depicted in Figure 48.) All six of these triangles can be constructed knowing four and only four independent angles (Silva 1997) (Aleksan 1994). These are defined as: = arg 146144(;) Vcb17:1) ry = arg 17:b Vi d VIVed) (105) Kb) = arg 17,7dVus Vt'.:17tb 17:Yes — X = arg (106) ds vcdvc: VtdVt: Vu dVut uc * sb VudVcci Vus-V:s vt.vti3' Vc.11,t, vusvut ct Vedift*d V„sVt, Vcblqb Vt,Vu: to B phenomenology 289 Figure 54. The six CKM triangles. The bold labels, e.g. ds refer to the rows or columns used in the unitarity relationship. The angles defined in equation (105) are also shown. (a can be used instead of 'y or /3.) Two of the phases /3 and y are probably large while X is estimated to be small Pz-.:0.02, but measurable, while X' is likely to be much smaller. It has been pointed out by Silva and Wolfenstein (Silva 1997) that measuring only angles may not be sufficient to detect new physics. For example, suppose there is new physics that arises in B° - T3° mixing. Let us assign a phase 0 to this new physics. If we then measure CP violation in B° J/V)K, and eliminate any penguin pollution problems in using B° 71-±7r-, then we actually measure 2/3' = 2/3 + 0 and 2a' = 2a - O. So while there is new physics, we miss it, because 2/3' + 2a1 = 2a + 2/3 and a' + + y = 180°. 8.2.1 A critical check using X The angle X, defined in Equation 105, can be extracted by measuring the time dependent CP violating asymmetry in the reaction B, -+ J 101(0 , or if one's detector is incapable of high quality photon detection, the J/1/4 final state can be used. However, in this case there are two vector particles in the final state, making this a state of mixed CP, which requires a large statistics sample to do a time-dependent angular analysis to extract X. Measurements of the magnitudes of CKM matrix elements all come with theoretical errors. Some of these are hard to estimate. The best measured magnitude is that of A = II/us /V.(11 = 0.2205 + 0.0018. Silva and Wolfenstein (Silva 1997) (Aleksan 1994) show that the Standard Model can be checked in a profound manner by seeing if: Vud sin(/3 + -Y) • Vus 2 sin 0 sin y (107) Here the precision of the check will be limited initially by the measurement of sin X, not of A. This check can reveal new physics, even if other measurements have not shown any anomalies. Other relationships to check include: Vu6 2 sin -y sin(/3 + '7) Vcb sin sin X - sin X - (108) Vtd 2 sin /3 sin(/3 + Ves sin 'y sin X = (109) 290 Sheldon Stone The astute reader will have noticed that these two equations lead to the non-trivial relationship: 2 (J sine Vid — Vcs 2 • 2 = S111 Vub V eb 2 (110) This constrains these two magnitudes in terms of two of the angles. Note, that it is in principle possible to determine the magnitudes of Vub I Vebl and 1Vtd/Vts1 without model dependent errors by measuring 3, -y and X accurately. Alternatively, 0, 'y and A can be used to give a much more precise value than is possible at present with direct methods. For example, once 0 and y are known I 7 2 v 7,b sin2 ,2 (111) Vcb sin2 (/3 + ey) Table 10 lists the most important physics quantities and the decay modes that, can be used to measure them. The necessary detector capabilities include the ability to collect purely hadronic final states labeled here as "Hadron Trigger," the ability to identify charged hadrons labeled as "Kx sep," the ability to detect photons with good efficiency and resolution and excellent time resolution required to analyze rapid B, oscillations. Measurements of cos(2q) can eliminate 2 of the 4 ambiguities in ch that are present when sin(20) is measured. Physics Quantity Decay Mode Hadron Trigger K7r sep 7 det Decay time a sin(2a) B° -> Air -> 71-±71--70 V V V cos(2a) B° -4 pir -4 7+7-7° V V V sign(sin(2a)) B° -# pir & B° -4 7r+7— V V V sin(y) B, -> _,OK V V V sin(7) B- -> 0 K- V V sin(y) B° -> 7+7r- & B, -4 K+ If - V V V sin(2X) sin(20) cos(20) cos(20) Bs -4 J1071', J/Vni B° -* JNK, B° JIOK°, K° -> irtv B° J17,bK*° & B, - JI00 V V V xs B, -> D3 f-ir- V V Or for B, Bs -4 ,11071', D;Fir-, K±K - V V V V Table 10. Required CKM Measurements for b's 8.2.2 Finding inconsistencies Another interesting way of viewing the physics was given by Peskin (2000). Non-Standard Model physics would show up as discrepancies among the values of (p, n) derived from in-dependent determinations using CKM magnitudes (1V,ib /17,1,1 and Vid/Vts1), or B3 mixing (0 and a), or B, mixing (X and -y). B phenomenology 291 8.2.3 Required measurements involving Besides a more precise measurement of sin 20 we need to resolve the ambiguities. There are two suggestions on how this may be accomplished. Kayser (1997) showed that time dependent measurements of the final state J/010, where K° —> 7r.ev, give a direct mea-surement of cos(20) and can also be used for CPT tests. Another suggestion is to use the final state J/OK", Kt° —> Ks7°, and to compare with B, J/00 to extract the sign of the strong interaction phase shift assuming SU(3) symmetry, and thus determine cos(20) (Dighe 1998). 8.2.4 Required measurements involving a and y It is well known that sin(20) can be measured without problems caused by penguin processes using the reaction B° J/01Cs. The simplest reaction that can be used to measure sin(2a) is B° -+ 7r+7-. This reaction can proceed via both the tree and penguin diagrams shown in Figure 37. Current measurements (Table 6) show a large penguin component. The ratio of the penguin amplitude to the tree amplitude in the 7+7- channel is about 15% in magnitude. Thus the effect of the penguin must be determined in order to extract a. The only model independent way of doing this was suggested by Gronau and London, but requires the measurement of B+ 7+7° and B° 707°, the latter being rather daunting. There is however, a theoretically clean method to determine a. The interference between tree and penguin diagrams is exploited by measuring the time dependent CP violating effects in the decays B° -+ pir 7+7-7° as shown by Snyder and Quinn (1993). The pit final state has many advantages. First of all, it has been seen with a relatively large rate. The branching ratio for the p°7r+ final state as measured by CLEO is (1.5+0.5+ 0.4) x 10-5, and the rate for the neutral B final state p±7rT is (3.5+1:0 ± 0.5) x 10-5, while the p°7° final state is limited at 90% confidence level to < 5.1 x 10-6 (Gao 1999). BABAR (Bona 2001) measures B (B° p±7±) as (28.9 ± 5.4 + 4.3) x 10-6. These measurements are consistent with some theoretical expectations (Ali 1999). The associated vector-pseudoscalar 107r decay modes have conquerable or smaller branching ratios. Since the p is spin-1, the it spin-0 and the initial B is spinless, the p is fully polarised in the (1,0) configuration, so it decays as cost 0, where 0 is the angle of one of the p decay products with respect to the other 7r from the B, determined in the p rest frame. This causes the periphery of the Dalitz plot to be heavily populated, especially the corners. A sample Dalitz plot is shown in Figure 55. This kind of distribution is good for maximising the interferences, which helps minimise the error. Furthermore, little information is lost by excluding the Dalitz plot interior, a good way to reduce backgrounds. To estimate the required number of events Snyder and Quinn preformed an idealised analysis that showed that a background-free, flavor-tagged sample of 1000 to 2000 events was sufficient. The 1000 event sample usually yields good results for a, but sometimes does not resolve the ambiguity. With the 2000 event sample, however, they always succeeded. This technique not only finds sin(2a), it also determines cos(2a), thereby removing two of the remaining ambiguities. The final ambiguity can be removed using the CP asymmetry in B° 7r±7r- and a theoretical assumption (Grossman 1997). IS 1 03 00 25 50 75 10E1 7 (ag) 125 150 175 K°) ts. (s*K"Yets .... +. =rmi ..00111 23 2 13 23 5 so 125 150 175 75 1CO Y (Op) 0.5 0 3 23 2 292 Sheldon Stone nu+0 Figure 55. The Dalitz plot for B° p71" 7+7r-7° from Snyder and Quinn (1993). Several model dependent methods using the light two-body pseudoscalar decay rates have been suggested for measuring -y. The basic idea in all these methods can be sum- marised as follows: the tree diagram for B° 71-±7- has the weak decay phase 7. In order to reproduce the observed suppression of the decay rate for 7+7i — relative to KI7rT we require a large negative interference between the tree and penguin amplitudes. This puts ,y in the range of 90°. A great deal of theoretical work is required to understand the effects of re-scattering, form-factors etc. We are left with several ways of obtaining model dependent limits, due to Fleischer and Mannel (1998), Neubert and Rosner (1998), Fleis-cher and Buras (2000), and Beneke et al. (2001). The latter use a sophisticated model of QCD factorisation and apply corrections. Figure 56 shows values of 'y that can be found Figure 56. Model predictions from (Beneke 2001) as a function of the indicated rate ratios. The dotted curve shows the predictions from naive factorisation. The curved bands show the total model uncertainties where the inner band comes from theoretical input uncertainties, while the outer band allows for errors to corrections on the theory. The specific sensitivity tolV„b 1 is showed as the dashed curves. The gray bands show the current data with a la error while the lighter bands are at 2a. V,0, D's- B phenomenology 293 91(- B u(D",-- { I;s 1,11K+ Figure 57. (left) The two diagram diagrams for B, .133K that interfere via B, mixing. (right) The two interfering decay diagrams for B- —> 15°K- where one is a b --+ u transition and the other a doubly-Cabibbo suppressed decay. in their framework, once better data are obtainable. In fact, it may be easier to measure 7 than a in a model independent manner. There have been two methods suggested. (1) Time dependent flavor tagged analysis of B, D.5±.K. This is a direct model independent measurement (Du 1986) (Aleksan 1992) (Aleksan 1995). Here the Cabibbo suppressed Vub decay interferes with a somewhat less suppressed Vb decay via B, mixing as illustrated in Figure 57 (left). Even though we are not dealing with CP eigenstates here there are no hadronic uncertainties, though there are ambiguities. (2) Measure the rate differences between B- ri°K— and B± D0 K+ in two different D° decay modes such as K-7± and K+IC-. This method makes use of the interference between the tree and doubly-Cabibbo suppressed decays of the D°, and does not depend on any theoretical modeling (Atwood 1997, Gronau 1991). See Figure 57 (right). 8.3 New Physics tests in specific models 8.3.1 Supersymmetry Supersymmetry is a kind of super-model. The basic idea is that for every fundamental fermion there is a companion boson and for every boson there is a companion fermion. There are many different implementations of couplings in this framework (Masiero 2000). In the most general case we pick up 80 new constants and 43 new phases. This is clearly too many to handle so we can try to see things in terms of simpler implementations. In the minimum model (MSSM) we have only two new fundamental phases. One 8D would arise in B° mixing and the other, BA, would appear in B° decay. A combination would generate CP violation in D° mixing, call it OK, when the decay is D° (Nir 1999). Table 11 shows the CP asymmetry in three different processes in the Standard Model and the MSSM. Two direct effects of New Physics are clear here. First of all, the difference in CP asymmetries between B° —> J/Olf, and B° -+ OK, would show the phase OA. Secondly, 294 Sheldon Stone Process Standard Model New Physics B° —> J/OK., sin 2/3 sin 2(0 + OD) B° -÷ OK, sin 20 sin 2(0 + OD + OA) D° -+ K-71-+ 0 ,.., sin OK7, Table 11. CP Violating Asymmetries in the Standard Model and the MSSM. there would be finite CP violation in D° IC-71+ where none is expected in the Standard Model. Manifestations of specific SUSY models lead to different patterns. Table 12 shows the expectations for some of these models in terms of these variables and the neutron electric dipole moment d N ; see (Nir 1999) for details. Note, that "Approximate CP" has already Model d N x 10-25 8D BA sin OK, Standard Model < 10-6 Approx. Universality > 10-2 0(0.2) 0(1) 0 Alignment > 10-3 0(0.2) 0(1) 0(1) Heavy squarks 10-i 0(1) 0(1) 0(10-2) Approx. CP 10-i -/3 0 0(10-3) Table 12. Some SUSY Predictions. been ruled out by the measurements of sin 20. In the context of the MSSM there will be significant contributions to B, mixing, and the CP asymmetry in the charged decay BT OK+. The contribution to B, mixing significantly enhances the CP violating asymmetry in modes such as B, —> (Recall the CP asymmetry in this mode is proportional to sin 2X in the Standard Model.) The Standard Model and MSSM diagrams are shown in Figure 58. The expected CP asymmetry in the MSSM is sin 0, cos OA sin(Am,t), which is approximately 10 times the expected value in the Standard Model (Hinchliff 2001a). b s fe 5 4 '_x ' 4 W+ Figure 58. The Standard Model (left) and MSSM (right) contributions to 13° mixing. We observed that a difference between CP asymmetries in B° JIOK, and OK, arises in the MSSM due to a CP asymmetry in the decay phase. It is possible to observe this directly by looking for a CP asymmetry in 13 OKT . The Standard Model and MSSM B phenomenology 295 u us K Figure 59. The Standard Model (left) and MSSM (right) contributions to B- --+ OK-. diagrams are shown in Figure 59. Here the interference of the two diagrams provides the CP asymmetry. The predicted asymmetry is equal to (Mw/msquark)2 Sin Op in the MSSM, where msquark is the relevant squark mass (Hinchliff 2001a). The OK and OK* final states have been observed, first by CLEO (Briere 2001) and subsequently by BABAR (Aubert 2001). The BABAR data is shown in Figure 60. The average branching ratio is 13(B- —> OK-) = (6.8 ± 1.3) x 10-6 showing that in principle large samples can be acquired especially at hadronic machines. 12 - (a) iff - (b) - (d) IL J 7 0 5.20 5.25 0 5.30 5.20 mES (GeV/c2) 5.25 5.30 Figure 60. Projections of the maximum likelihood fit onto the B candidate mass 171E5 from BABAR. (a) B+ OK+; (b) OK°; (c) B+ qK*+; (d) B° -4 OK*°. In (c) the histogram is the sum of the two contributing K* channels while the shaded area is K°71-+ alone (the other channel is K+7°). The solid line shows the projection of the signal plus background fit while the dashed line shows the projection of the background only. 296 Sheldon Stone 8.3.2 Other new physics models There are many other specific models that predict New Physics in b decays. I list here a few of these with a woefully incomplete list of references, to give a flavor of what these models predict. . Two Higgs and Multi-Higgs Doublet Models predict large effects in f K and CP vio- lation in D° r+ with only a few percent effect in B° (Nir 1999). Expect to see 1-10% CP violating effects in b -÷ sry (Wolfenstein 1994). . Left-Right Symmetric Model have contributions that compete with or even dominate over Standard Model contributions to Bd and B, mixing. This means that CP asymmetries into CP eigenstates could be substantially different from the Standard Model prediction (Nir 1999). . Extra Down Singlet Quarks are likely to give dramatic deviations from Standard Model predictions for CP asymmetries in b decays (Nir 1999). . FCNC Couplings of the Z boson in the decay B K* 2+P- mean that both the sign and magnitude of the decay leptons carry sensitive information on new physics. Potential effects are on the of 10% compared to an entirely negligible Standard Model asymmetry of ,--, 10-3 (Buchalla 2000). These models also predict a factor of 20 enhancement of b de+ f- and could explain a low value of sin 2/3 (Barenboim 2001a). . Non-commutative Geometry. If the geometry of space time is non-commutative, i.e. [x xy] = then CP violating effects may be manifest a low energy. For a scale <2TeV there are comparable effects to the Standard Model (Hinchliffe 2001b). . MSSM without new flavor structure can lead to CP violation in b s-y of up to 5% (Bartl 2001). Ali and London propose (Ali 1999) that the Standard Model formulas are modified by Supersymmetry as Amd = Amd(SM) 11 + f (inx ,mi-R ,rni f±, tan (1 (112) Am, = Arns (SM) [1 + f (mx , miR, m 1.1± , tan 0)] (113) G2F f 2 M K M47 1E1(1 = 6 72,6,MK B K (A2 A671) {Ye (71cth(Ye, Yt) — 7/cc) 7/ttYth(Yt) [1 f rrt H± , tan /3)] A2 A4 (1 — ,5)} , (114) where Am(SM) refers to the Standard Model formula and the expression for 1EK would be the Standard Model expression if f were set equal to zero. Ali and London show that it is reasonable to expect that 0.8 > f > 0.2. Since the CP violating angles will not change from the Standard Model, determining the value of (p, using the magnitudes Ams /Amd and loc i will show an inconsistency with values obtained using other magnitudes and angles. . Extra Dimensions are beginning appear in papers predicting b decay phenomena. See Agashe (2001), Barenboim (2001b), Branco (2001), Chang (2001) and Papayas-siliou (2000). B phenomenology 297 I close this section with a quote from Masiero and Vives (2001): "The relevance of SUSY searches in rare processes is not confined to the usually quoted possibility that indirect searches can arrive 'first' in signaling the presence of SUSY. Even after the pos-sible direct observation of SUSY particles, the importance of FCNC and CP violation in testing SUSY remains of utmost relevance. They are and will be complementary to the Tevatron and LHC establishing low energy supersymmetry as the response to the electroweak breaking puzzle." I agree, except that I would replace "SUSY" with "New Physics." 8.4 Expected data samples It is clear that precision studies of b decays can bring a wealth of information to bear on new physics, that probably will be crucial in sorting out anything seen at the LHC. This is possible because we do expect to have data samples large enough to test these ideas from existing and approved experiments. In Table 13 I show the expected rates in BTeV for one year of running (107 s) and an e+e- B-factory operating at the T(4S) with a total accumulated sample of 500 fb-1, about what is expected around 2006. (LHCb numbers are the same order of magnitude as the BTeV numbers for many of the modes.) Mode BTeV (107)s Yield Taggedt S/B B-factory (500 fb') Yield Taggedt S/B B, —r J/Vni(') 22000 2200 >15 B- —r 11000 11000 >10 700 700 4 B° —> OK, 2000 200 5.2 250 75 4 B° K*°p+p- 4400 4400 11 --,50 ? D*+ 7r+D°; K-71-+ 108 108 large 8 x 105 8 x 105 large Tagged here means that the initial flavor of the B is determined. Table 13. Comparisons of BTeV and B-factory Yields on Different Time Scales. 9 Conclusions I have attempted to cover the length and breadth of b physics, and have only scratched the surface. There is much more to be said and much more to learn. Why are there three families? What is the connection with neutrinos and that mixing matrix? How do we explain the mystery of flavor? These and many more unanswered questions I leave to the students. 298 Sheldon Stone Acknowledgments This research was supported by the US National Science Foundation. I thank Ken Peach and Steve Playfer for a well organised and interesting school that was very enjoyable to attend. 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Lett. 74, p.2809 [hep-ph/9410253]. 303 Heavy quarks at HERA-B and the Tevatron Peter Kraan University of Ljubljana, Slovenia 1 Introduction The two lectures which are written up in this paper, try to answer the questions why hadron machines look very promising for B physics studies, what are the problems in b hadron detection, and how these are solved in three experiments, CDF, an example of a general purpose collider experiment, HERA-B, a fixed target experiment, and BTeV, a specially designed collider experiment in the forward direction. Some of the advantages of hadron machines are evident from Table 1. The bb production cross section is typi-cally much higher in a hadron machine, as compared to an e+e- machine. At HER .A-B, where the cross-section is comparable to e+c- production, the bb production rate can Production Mechanism e+ e- -r T(4s) --- BB e+ e- -r Z° -+ bb pA bbX pp -> bbX pp(p) -> bbX forward Accelerators CESR, DORIS LEP, SLD HERA (P) Tevatron Tevatron PEPII, KEKB Detectors CLEO, ARGUS ALEPH, DELPHI HERA-B CDF, DO BTeV BABAR, BELLE L3, OPAL, SLD a (bb) a (bb):a (had) r.-, lnb :-..,-• 0.26 6nb 0.22 r-:-... 12nb 10-6 .--:-.' 50µ1) 10-3 •-:-.', 100µ13 2.10-3 B°, B+ yes yes yes yes yes Bs°, .13E , AI° no yes yes yes yes boost (07) 0.06 (0.5) 6 :::: 20 .--:_i 2-4 :::: 4-20 multiple events no no yes, 4 yes yes, 2 main trigger inclusive inclusive lepton pairs leptons displaced vertex Table 1. Comparison of bb production at e+e- and hadron machines DOI: 10.1201/9780429187056-7 ) 5.9 Phi ”1.7 304 Peter Kriian be improved by increasing the luminosity. Another advantage of b hadrons produced in hadronic collisions, compared to the B-factories at the T(4S), is the large boost which results in an average decay length, 44 = 480 prn. Finally, we note that the T(4S) ma-chines only produce B° and B± mesons, whereas the hadron machines also produce other b hadrons, including B,, Be and Ab. When designing a detector for b hadrons at a hadron machine, we have to consider several issues. High interaction rates are needed because of the rather low ratio of the signal cross section to the total cross section. In most cases this means we are dealing with multiple interactions per event, so the detector has to be able to tolerate high track densities (see Figure 1). The detector components that are used have to be radiation hard, and have a sufficiently fine granularity to separate out the track hits. Figure 1. Comparison of bb events: CDF(top), HERA-B(bottom) 00 (1 = 2.0 ;-,-;;;F?- 2.0 - LAYER 00 SVX II 77= 3.0 3 .0 2.0 3.0 m INTERMEDIATE SILICON LAYERS (SOLENOID HERA-B and Tevatron 305 The low ratio of signal to total cross section also requires an early selection of in-teresting events. Two approaches are used in developing selective triggers with good background suppression. The first approach relies on the fact that the decay products of a B meson have rather high transverse momenta in few-body final states, such as B twX, B JIWK, tip7r7r, B r+71--. The final states with a single lepton or lepton pairs are particularly attractive because lepton identification can be done rather easily. A second approach takes advantage of the displacement of the decay vertices of b hadrons from the interaction point. In the case of HERA-B this distance is typically lcm. Both approaches are used in the experiments presented in this paper. The detection of high PT tracks only takes a few hundred nanoseconds, while the reconstruction of the vertex takes more time, and is a part of the higher trigger levels, although the BTeV experiment plans to reverse this order. The next section reviews the excellent results from the CDF detector. These serve as a proof of principle for doing B physics at a hadron machine. Then follows a brief discussion of how to measure B, mixing. The remaining sections describe the HERA-B and BTeV experiments. 2 B physics with CDF 2.1 CDF spectrometer The Tevatron collider provides pp collisions at a center-of-mass energy of 2TeV in two interaction regions, for the two experiments, CDF and DO. The CDF detector (Abe 1988) is a general purpose magnetic spectrometer with a silicon microstrip vertex detector, a vertex TPC, a main tracking chamber, an electromagnetic and a hadronic calorimeter, and a system of muon chambers (Figure 2). Figure 2. CDF spectrometer: isometric view (left) with a vertex detector system, a central tracking chamber, electromagnetic and hadronic calorimeters, a superconducting solenoid, and muon chambers; side view after the recent upgrade (right) with a new sili-con vertex detector including the additional layer (L00), silicon tracker, central tracking chamber (COT), and additional calorimeters. E 0 Z.10 4io 3 10 2 10 -0.2 0 0.2 0.4 (cm) 306 Peter Kriian C:DF Run 1 25000- 110 pb-' -1- 15000 1- ,;',10000 - 0 2 9 2.95 3 3.05 3.1 3.15 3.2 3.25 3 3 in., (GeV/0 Figure 3. Examples of CDF performance: distribution of di-moon invariant mass (left), decay length distribution for di-moon events from the J 10 mass band (right). For b physics studies there are two interesting figures of merit illustrated in Figure 3. The di-muon invariant mass resolution in the J/1/) region is 16MeV/c2, and the resolution on the position of the JO vertex is 40-50 pm. The performance of the vertex detector is essential for improving the signal to background ratio, as can be seen for the Bd —> J10K+ decay channel in Figure 4. 0 • • •* I¤ • •• ¤ # • • ....* • • • 0 • d's 5.2 .3 , Mass (J/vr)5 [GeV/el 5.4 5.2 5.3 5.4 Mass (J/iiiK+) [GeV/e] Figure 4. Cleaning up the J MK+ signal: a cut on decay length (>1000m) significantly improves the signal to background ratio (bottom). ;••". 300 200 . 100 fr4 0 Z. 150 100 .4 50 as W 0 HERA-B and Tevatron 307 2.2 CDF results This is a brief summary of the results from CDF. The interested reader should consult overview papers (Paulini 1999, Pitts 2000) for more details. The excellent performance of the vertex detector allowed measurements of the time evolution, and hence of b lifetimes and BB mixing. The results of the lifetime measure-ments of b hadrons are summarized in Figure 5. The precision of these results, and of the Bd mixing measurements (Gay 2000), match the precision of the LEP experiments (Figure 5). A search for a B, mixing signal resulted in a lower limit Arn, > 5.8ps-' at 95% confidence level. Among the contributions to the spectroscopy of b hadrons the most important were the first observation of the B, meson and the determination of the masses of the B5, Ab and B, hadrons. CDF also measured the production cross section o-(pp bX) which turns out to be larger than theoretical predictions by about a factor of 2 (Bedeschi 2001). CDF studied time-dependent CP violation in Bd J101( 3, obtaining the result sin 2,3 = 0.791:114 (Affolder 2000). For the next generation of precision b physics ex-periments a measurement of the polarization in the B5 -+ J/00 decay is important as an input for the measurement of SF/F (Affolder 2000a). CDF has also shown that rare decays, such as Bd K*µµ, can be studied at hadron machines with sensitivities com- parable to e+e- machines (Affolder 1999). t(13°) t(13+) T(BD T(Ab) c(13+c) inc. i(l)) CDF B Lifetimes Arnd 1.51 ± 0.05 ps ALEPH 1.66 ± 0.05 Ps DELPHI 1.36 ± 0.10 Ps 1.32 I 0.17 ps OPAL 0.46 ± 0.17 ps SLD 0.0.1 ps CDF 1.09 ± 0.05 Average Results 0.446 ± 0.020 ± 0.018 ps-1 0.497 ± 0.027 ± 0.023 ps-1 14b 0.028 0.466 ± 0.022 ± 0.016 ps-1 0.526 ± 0.043 I" 0.031 ps 0.495 ± 0.026 ± 0.025 ps-1 0.480 ± 0.010 ± 0.013 ps' I " I H H H - H - 11111 I/11. 11f111111111 H H t(13+)/T(B°) 1111111111111111 0.5 1 1.5 0.4 0.6 Amd [P$-11 Figure 5. CDF results: lifetimes of b hadrons (left), Bd mixing parameter Aind compared to LEP and SLC results(right). 308 Peter Kriian 2.3 CDF plans for future For Run II the Tevatron collider has been upgraded with a new injector and antiproton source. The bunch spacing has been changed from 3.5ps to 396ns, and will be further reduced to 132ns in the second stage. It is expected that 2f13-1 of integrated luminosity will be accumulated in the years 2001-2004, and as much as 15fb-1 by 2007. Both numbers should be compared to the 0.lfb-' of data collected in Run I. Major upgrades of the CDF and DO detectors have been carried out to take advantage of the machine improvements. The upgrades to the CDF detector, shown in Figure 2, in-clude increased muon system coverage, a larger vertex detector, and a new central tracker consisting of a drift chamber and additional silicon layers between the vertex detector and the drift chamber. A time-of-flight counter has been added for 7r/K separation up to 1.6GeV/c. An additional vertexing layer(L00) close to the beam significantly improves the reach of the experiment for Bs mixing (see next section). The trigger system now includes a fast track search in the first level, and a silicon vertex tracking algorithm in the second level. This will lower the Pt threshold for muons, as well as providing a two track trigger for hadronic decays With an integrated luminosity of 2fla-1 in the first period of Run II, it is expected that the CP violation parameter sin 20 will be measured with an error of 0.043 (Tanaka 2001). A Bs mixing observation will be possible up to xs = 60 at 5a significance. Ar/r for Bs mesons will be measured with an error of 0.05 using the decay channel Bs J/00. It is also expected that the forward-backward asymmetry AFB in the rare decay Bd —* K* pp will be measured. The bb production cross section will be studied further, as well as exclusive decays of the B, meson, B, -> JitAev and Jitinr. 3 How to measure B, mixing To observe mixing in the B, system requires a measurement of the probability that a B3 at time t = 0 turns into B, when it decays at a later time t. The B, has to decay to a flavour specific final state with a precisely measured decay vertex and Bs momentum. Several decay channels have been considered, including the semileptonic channel B, pDsv, where Ds -> Or+, K+Ks, K+K*, the hadronic channel Bs -k J/OK*, where J/7/7 -+ pp, K* Kr, as well as more conventional hadronic decays such as Bs A*)7+ or D3*) 71-±7r+7-. A flavour tag is needed to determine the initial Bs flavour at t = 0. This is best provided by the charge of the kaon or lepton from the associated decay of the other b quark. In what follows, a simplified picture is given of the maximal mixing parameter x„ an experiment can measure, as a function of the resolution and available statistics. The interested reader should consult Moser & Roussarie 1997 for a full discussion. For a perfectly tagged B, at t=0, the probability that this Bs meson decays as a B, depends on the proper time t according to {1 - cos(xst/T)} exp (-t/r). Imperfect tagging and backgrounds reduce the amplitude of the cosine term by a "dilution" factor (D): r(B, B„) = e-t/T{i- D cos(xst/T)}. For lepton tagged events the value of D is about 0.19 (Figure 6). HERA-B and Tevatron 309 tit 2 1 0 1 2 tit 0.4 - 0.2 - 0 2 tic 2 1.25 0.75 0.5 0.25 0 D = 1 Figure 6. Example of B, mixing with x, = 30 (Am, = 22ps-1): (a) perfect resolution and no dilution, (b) with a dilution factor D = 0.19, (c) with dilution factor D = 0.19 and a finite resolution at = 45f s The amplitude of the mixing signal is further reduced by the finite resolution in proper time t. In the limit of very good resolution (at << xsr), a simple quantitative estimate of the effect can be made by observing that the amplitude is reduced by events moving from the "up" side of the wave to the "down" side, the effect being most important in an interval of order at around the node of the oscillating term (Figure 7). In this region the function is approximately linear, and the fraction of events in this interval can be roughly estimated to be olx/27-2. A full derivation involves a convolution of the time evolution curve and the detector response. The amplitude turns out to be reduced by a factor exp(-44/27-2), in agreement with the rough estimate to first order in al /-r2. time (t/x, ) Figure 7. Dilution of the oscillatory term due to finite resolution. signal b quark p lepton tag K kaon tag 310 Peter Kriian The dilution D is replaced by D' = D exp(—cqxs2 /2-r2 ) , giving the distribution shown in Figure 6(c). A simple estimate of the statistics required for a significant measurement can be made by fixing x,, and dividing the events into two classes, those from the "up" part of the wave, and those from the "down" part (Figure 7). The measured oscillation amplitude for a given x, differs from zero if the two classes are found to be differently populated. The distribution over the two classes is binomial, with probability for the "up" part equal to p = 0.5 + cD', where c is a constant of order 1. The error on p, o-(p) = \Ip(1 — p)/N, for a measurement with N reconstructed and tagged events. This expression can be simplified for p 0.5 to a(p) 1/(2VTV). The resulting error on the amplitude D' is aD 1/(2c", and the significance of the measurement is: ,.2,2 Di /Or D, 2c D exp 27-2 For a given significance, the number of events needed, and thus the duration of the measurement, is proportional to exp(44/27-2). This is a very steep function of the proper time resolution at and the mixing parameter xs, particularly in the region where the quadratic term in the exponent is greater than 1. In the upgraded CDF detector, where crt =45fs is expected, this value is reached at x, = 30. 4 HERA-B HERA-B is a fixed target experiment at the storage ring HERA at DESY (Lohse et al. 1994). The experiment was originally designed to measure CP violation in the decays of neutral B mesons to J///}4 (Figure 8). Bottom hadrons at HERA-B are produced with a beam of 920GeV protons from the storage ring which hits an internal wire target. The average momentum of the B mesons is 120GeV, so they travel about lcm before they decay. The J/0 from the decay B° J1010 can be triggered and reconstructed in its tagging b quark Figure 8. A B Jli,bKcs' event showing the typical momenta and decay distances expected in the HERA-B experiment. Tap Wow flIng Imaging CAICAIT.iet MO Cherenkov County,. 90 tiluaft CiolKlar 15 Itetrox Vesset Om Side View figmaost n Beam lectnan Boom HERA-B and Tevatron 311 dileptonic decay mode. From two existing measurements of cr(bb) in the relevant kinematic region (Jansen et al. 1995, Alexopoulos et al. 1999), the fraction of events containing b quarks is of order 10-6. Since the branching ratio for the decay B° J /OK° is of order 10-3, and J/0 1+1- is 6%, the ratio of the signal channel to the total inelastic cross section is of order 10-11. The HERA-B design requires a very high interaction rate of 40MHz, with an average of four proton-nucleon interactions occurring for each hunch crossing (every 96ns). This interaction rate leads to fluxes of charged particles through the detector components which are comparable to the fluxes expected at the LHC. Figure 9. The HERA-B spectrometer. 4.1 The HERA-B spectrometer HERA-B is a large magnetic forward spectrometer (Figure 9) with outer dimension of 20m x9m x 7m. It covers a polar angle between 10mrad and 200mrad with respect to the beam direction corresponding to a 90% coverage in the center of mass system. The main detector components are wire targets, a vertex detector system, a tracking system, a ring imaging Cerenkov counter, an electromagnetic calorimeter, muon chambers and a trigger system. An important requirement for HERA-B was to not interfere with HERA operation for ep collider experiments. The solution was to put several target wires in the halo of the proton beam (Ehret 2000). Protons which leave the core of the beam, and are therefore SL 1 4 stereo views / SL t2.5, 87.5, 92.5 deg IP 312 Peter Kriian internam Figure 10. HERA-B wire target concept (left), reconstructed vertex positions (right). lost for collisions, interact in 8 wires which are arranged in 2 stations (Figure 10). This arrangement allows for an efficient separation of primary vertices for events with multiple interactions. The interaction rate can be changed by moving the wire targets within the beam halo, and the number of wires in operation can be varied by retracting some of them. Operation of the wire target is very stable, and meets the criteria for small interference with collider operation. Figure 11. The HER A-B vertex detector system. The vertex detector system (VDS) has to provide track co-ordinates for reconstructing decay vertices and the impact parameters of decay products. The VDS (Masciocchi 2001) consists of 64 double-sided silicon microstrip detectors assembled in eight super-layers of four quadrants around the proton beam axis (Figure 11). To minimize multiple scattering, the first seven super-layers are placed in a Roman pot configuration. The VDS has successfully operated for several years with interaction rates of up to 40MHz. It has a hit efficiency better than 97%, and a hit resolution of 10 pm. HERA-B and Tevatron 313 (cm) 200 0 -200 cell diameter 10 mm q 5 mm 300 0 -300 (cm) Figure 12. The HERA-B Outer Tracking system: honeycomb drift chambers (top), modular structure (bottom). The tracking system extends over a total length of 11m. The first group of super-layers are situated inside the field of a dipole magnet, while the others are in the field-free region. The particle flux increases towards small polar angles, so the system is divided into inner and outer parts consisting of detector elements with different granularity. The Outer Tracker (OTR) measures charged tracks over most of the geometrical acceptance. It consists of honeycomb drift chambers of two different granularities (Figure 12), covering a total area of about 1000m2. The system has to operate in a high radiation environment, which led to some serious aging problems (Hohlmann 2001). During the year 2000 run, the overall hit efficiency was also somewhat worse than expected, due to HV failures of individual wires. We note that the HERA-B outer tracking system is the first large area detector working in a LHC-like environment. The inner tracker (ITR) measures tracks at small distances from the beam, and provides input for the first level trigger (Gradl 2001). The ITR elements cover the region from 6cm to 25cm around the beam pipe, where the particle flux can reach 106 per square centimeter per second. The detectors are microstrip gas chambers with a gas electron pach 3021pm anode lOpto mitotic 170tan daft electrode 3.0rtan 2nd MalOpheahon step (moor 100-1C00 2.8mra = 1.12 GeV = 2.9 MeV 314 Peter Krian Figure 13. MSGC-GEM detector structure. multiplier stage (GEM), as shown in Figure 13. After an extensive conditioning phase, the operation of the ITR was stable in the year 2000 run, with performance close to the design value. Performances of the complete tracking system for K )s, and A candidates are shown in Figure 14. K ° K.. mass spectrum A p7r A mass spectrum 350 m = 496 MeV 80 70 300 a = 7 MeV 250 50 200 150 30 100 20 50 • • . 0 0 0 4 0.425 0.45 0.475 0.5 0.525 0.55 0.575 0.6 m / GeV 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 m /GeV Figure 14. Performance of the HERA-B tracking system: KS and A signals A ring imaging eerenkov counter (RICH) identifies charged hadrons from the photons emitted in a gaseous C4 F10 radiator (Arifio et al. 2000). The photons are reflected by two mirrors and collected as rings on multianode photomultipliers (Broemmelsiek 1998), using a lens system (Figure 15). To match the variation in occupancy and reduce the cost, 4 and 16 channel versions of the Hamamatsu R5900 multianode PMT are used. The system has typical rates of 1MHz per channel in the hottest parts. It has been in operation since summer 1998, with very few noisy or dead channels (Figure 16). For ultrarelativistic particles, the average number of photons in a ring is 32, and the single photon resolution is arcH = 0.7mrad for the finer, and lmrad for the coarser granulari-ties, in excellent agreement with the design values. Figure 17 shows the probability that Collector lens PMT 150mm HERA-B and Tevatron 315 Field lens Figure 15. HERA-B RICH: radiator vessel (left), light collection system (right) Figure 16. HERA-B RICH: a low occupancy background event with two rings on the photon detector (left), a typical event (right). 316 Peter Krian t 1.2 60.8 - 0.6 • u 0.4- • µ K 0.2 I 0 0 5 /0 15 20 25 30 35 40 45 50 MIX lcut = 0.35 momentum p(GeV/c) Figure 17. HERA-B RICH: particle identification efficiency from JIY: pcp,. a muon from a J/0 decay is recognized as a muon by the RICH, as a function of muon momentum. The performance of RICH is expected to improve with better alignment of the tracking system. The electromagnetic calorimeter (Avoni et al. 2001) measures photon energies, identi-fies electrons and provides an electron pretrigger. The calorimeter is of the "shashlik"-type with three different granularities in order to match occupancies. The measured energy resolution, a(E)/E = 1.7% G 22.5%/a, is close to the design value. The muon detector (Eiges et al. 2001) separate muons from hadrons and gives a muon pretrigger. It consists of four layers of detector elements embedded in iron absorbers. Three types of detector elements (tube, pad and pixel chambers) are used in order to match the required granularity. In the 2000 run, the muon pretrigger provided a good J/t, signal using a fourfold coincidence of signals in the pad chambers. The performance was limited by inefficiencies in certain parts of the detector. The huge background from inelastic proton-nucleon scattering requires a highly selec-tive and efficient trigger. The main idea of the HERA-B trigger (Gerndt 2000, Kreuzer 2001) is to search for a pair of high-pt leptons, with invariant mass consistent with a J/0. At the pretrigger level, data from the calorimeter, muon and tracking systems are used to define regions of interest which correspond to particles with high transverse momenta. Pattern recognition and track reconstruction are performed with dedicated processors at the tracking system stations (Figure 18), starting from the track candidates defined by the pretrigger system. A cut is made on the invariant mass of candidate track pairs The first trigger level aims to reduce the rate by a factor 200. Second and third level triggers refine track candidates and search for vertices. They are designed to give suppression factors of 100 and 10 respectively. A fourth level performs the complete event reconstruction and writes data to storage. The best results in the 2000 run were achieved with a first level trigger requirement of a single high-pt lepton, and a search for the second high-pt lepton in the second level trigger. We note that in this period the first level trigger efficiency was 20% for single muons and 54% for electrons, which is well below the design value of 90%. This is mainly due to inefficiencies in the tracking and muon chambers. HERA-B and Tevatron 317 detector planes ELT processors g—PAD FCAI p —PAD tracker tracker tracker tracker Lx 5— Pre TFU TFU TFU e ,1t — Pre TFU TFU TFU TFU TPM TDU FL "OK" Figure 18. HERA-B first level trigger system: pretrigger candidates are followed through track finding units (TFU), combined into pairs in track parameter units (TPM); before a final decision by the trigger decision units (TDU). 4.2 HERA-B status and plans The experiment was approved in 1995, but the design and construction of detector ele-ments that can cope with the extremely high fluxes proved to be a technological chal-lenge. Problems with the tracking system caused a substantial delay in the schedule, and installation of the full detector was only completed in February 2000. The first phase of commissioning lasted 6 months (HERA-B Collaboration 2000). The performance of the detector for J/ events is shown in Figure 19. After the end of the current HERA shutdown, a continuation of the operation of HERA-B is expected in the second half of 2001. Due to the late start and the trigger inefficiencies, HERA-B is no longer competitive for a measurement of CP violation. In the 2001/2002 running period the experiment will concentrate on the measurement of events which contain a J/7/) meson, for which the detector is optimized. Based on the results of the 2000 run, and detector improvements, it is expected that 1500 reconstructed J/0 events will be collected per hour, resulting in 2.1 million events in the full running period. With these data HERA-B can measure the nuclear dependence of charmonium production, and the bb production cross-section. A suppression of the yield per nucleon of J/W in proton-nucleus, pion-nucleus, and heavy ion collisions is observed in heavy nuclei compared to light nuclei (Leitch et, al. 2000. Alde et al. 1991, Leitch et al. 1995). The effect depends strongly on the xp and pi, of the J/1/). Various mechanisms can contribute to this suppression, but their relative importance is not clear. HERA-B can measure the production of J/1/) mesons down + Unlike sign dimuon pair + Number of JAI; = 2804±58 Width of jhy = (53±1) MeV/c2 2.5 3 I - L 15 4 4.5 5 5 Events I _ 3 Events 318 Peter Kriian Invariant Mass (GeV/e2) -3 0 1 2 Z secondary - Z primary (cm) Figure 19. HERA-B performance: di-muon invariant mass spectrum (top), decay length distributions in J10 mass region (bottom). to xp = —0.4, as well as 01 and xc, using the electromagnetic calorimeter to reconstruct xe J/I(i-y. Running simultaneously with target wires made of different materials reduces the systematic errors. Last year HERA-B took data using Ti and C wires. A measurement of of the production cross-section o-bb is needed, because there are two inconsistent results (Jansen et al. 1995, Alexopoulos et al. 1999). For the related processes, irA bbX, the theoretical predictions underestimate the cross section. The measurement of a-bb at HERA-B will be based on inclusive b -# J/OX compared to direct J/0 production. The J/1/) candidates with decay vertices detached from the target wire will be attributed to decays of b hadrons. It is expected that 340 events of this type will survive all cuts with a small background. The result for o-bb is expected to have a statistical accuracy of 5-10%. HERA-B is the first experiment working in a LHC-like environment, As such it has accumulated valuable experience during construction and commissioning which is vital for the success of the next generation of hadron experiments. After further improvements during the 2001 shutdown, final commissioning of the detector will be completed. 1' 600 500 400 200 100 10 2 -5 -2.5 2.5 5 B hadrons at the Tevatron HERA-B and Tevatron 319 5 BTeV - a second generation B factory The motivation for the BTeV experiment is the expectation that in the next few years the e+e- B-factories will firmly establish the evidence for CP violation in B decays, but that several questions will remain open due to lack of statistics, and the restricted set of b hadron decays that are available at the e+ e- machines. The goal of BTeV is to perform high precision measurements of the CP parameters -y, a, x, in the decays of B°, B1 and B5 mesons, and to search for new physics in rare and flavour changing neutral current decays (Stone 2001). Figure 20. Production of bb pairs at BTeV: dependence of boost on the pseudorapidity (left), correlation of production angles of the bb pair (right). The BTeV experiment (Santoro 1997, Skwarnicki 2001) will exploit the very large production cross section for bb pairs in the forward and backward direction in pp collision at the Tevatron (Stone 1997). The boost factor 0-y becomes very large in this region, compared to the central region covered by CDF and DO. Figure 20 shows that b quark pairs are either both in the forward or both in the backward direction, so a single arm spectrometer similar to HERA-B would in principle be sufficient for flavour-tagged CP measurements. 5.1 BTeV spectrometer To accomplish its goals, the BTeV spectrometer (Sheldon 2001), shown in Figure 21, has to meet the following requirements. It has to efficiently trigger on hadronic B decays, and it needs a high resolution vertex detector to measure B5 oscillations. An excellent particle identification system is required for 7r/K separation, and a high resolution electromagnetic calorimeter for 7r° reconstruction. All the detector components have to survive high interaction rates. Finally, it needs a high capacity data acquisition system that can handle a large number of read-out channels. The BTeV tracking system has three major parts, a vertex pixel detector, a system AV°. Silicon Strips ONWONONN At Pixel Detectors Muon Chamber 320 Peter Kri2an 12 9 6 3 0 3 6 9 12 Electromagnetic meters Ring Imaging Calorimeter Cerenkov Figure 21. BTeV spectrometer with an expanded view of the interaction region. of straw-tube chambers as the main tracking device, and silicon strip detectors in the vicinity of the beam pipe. The pixel detectors are essential to eliminate ambiguities and provide a detached vertex trigger in the high track density region (Artuso 2001). The detectors are expected to have a low noise. The read-out electronics is subject to the same radiation load as the detector, but it has been possible to develop radiation hard electronics using the deep sub-micron process (Christian 2001). The identification of hadrons is essential for the tagging of b-flavour in CP violation and mixing measurements. It is also crucial for the identification of hadronic final states such as B 7+7-. The importance of the latter is illustrated in Figure 22, where the B 7+7- events are obscured by a large background from other, kinematically similar, two-body decays. The kaon and pion momentum range that the identification device has to cover, can be deduced from Figure 23. To achieve the necessary 711( separation in the range 3 < p < 70GeV/c, a RICH counter will be used (Figure 24), with the same radiator gas, C4Fio, as in HERA-B. To supplement the performance in the low momentum region and to improve pl K separation, plans are being discussed to add an aerogel radiator. A novel type of a position sensitive photon counter will be used (Figure 25, the hybrid photon detector (HPD) (Alemi 2000). A high electric field is used to accelerate photo-electrons onto a silicon pixel detector. The HPD is still under development, and is also 500 . I . I . I • 1 1 1 Higher momentum it . , Lower momentum 1000 500 0 o 20 40 60 60 10Q 120 140 Momentum (GeV c) I I • I • • I • HERA-B and Tevatron 321 500 0 5.10 I I I if — v v y • -1--1--/-1-1-- • gr,.••••lvvi .. i - • • • i 1 -1 Ail i twol)ody modes _ _ _ _ B .r.rr • i 'VITYVvi. vein • v 5.20 5.30 M,,,, 5.40 OrWvit 5.50 (GeV/c2) Figure 22. Invariant mass distributions as determined by MC simulation for the B -+ 7r*71-- decay together with the two body background sources, B K7r, B, K K and B, K7r, without particle identification criteria. Koon tag 20 _ 10 - . ' 1 K . K ,...1 exiting absorbed the magnet . in the magnet . - - _ - . ' 0 0 20 Momentum10 (GeV/ci5 Figure 23. Momentum spectra of tagging kaons (left) and pions from B 7r±7r - (right). foreseen for use in the LHCb experiment (Wotton 2000). The calorimeter needs to have high resolution and good radiation tolerance. For this BTeV will use 22000 Lead Tungstate crystals (PbWO4 ), with size 27x 27x 2220mm3. This is the same technology as for the CMS experiment. The crystals are radiation hard, fast scintillators with 99% of the light emitted in less than 100ns. They will be read out with photomultiplier tubes. The energy resolution predicted by a full GEANT simulation is a(E)/E = 0.55% e 1.6%/a(GeV), and the position resolution is given by = 3500/./m/E(GeV) 200µm. The muon detector will be used both to identify muons, and to provide an auxiliary di-muon trigger for checks of the detached vertex trigger, particularly in the commissioning phase of the experiment. It consists of two toroids with three detector sets. ........ ^ ........................................ 10 20 30 Momentum (GeV) 0 Pins to readout chip -1 322 Peter Kriian Figure 24. BTEV RICH: detector layout with aerogel and C4 F10 radiators, spherical mirrors and HPD photon detectors (left), momentum dependence of the Cerenkov angle in the two radiators (right). Figure 25. The HPD design for the BTeV RICH. One of the main components of the BTeV spectrometer is the trigger system (Artuso 2001). The first level trigger is meant to find segments in the entrance and exit of the pixel detector as shown in Figure 26. It will then match segments, get crude momentum estimates, and reject low momentum tracks from further use for vertex determination. In the next step, the primary vertex will be found, and then for each of the tracks which miss the primary vertex, the significance of the detachment parameter, b/o-(b), will be calculated (Figure 27). The goal of the first level trigger is a factor of 100 rejection of bunch crossings which do not produce a bb pair, and an efficiency exceeding 50% for interesting events. As an example, a requirement of two tracks with b I o-(b) > 6 leaves 1% of beam crossings, and has 63% efficiency for B° 7r+7-, 70% for 13- —> D°K- , and 50% for B° J IWO. The dependence of the signal efficiency and the background rejection power on the detachment parameter cut for different numbers of tracks is shown in Figure 28. HERA-B and Tevatron 323 Figure 26. Event in the pixel detector, with the horizontal coordinate compressed by factor 10. Left: raw event, right: segments found at the entrance and exit of the detector. Figure 27. Detachment of a track from the primary vertex. Trigger Efficiency for B.1( Trigger Response for Minimum Bias Events Figure 28. Level 1 trigger efficiency for B, Ds K (left) compared to the efficiency for minimum bias events (right). 324 Peter Kriian The second trigger level will rely on re-fitting the pixel segments, including more hits, and linking these tracks to the downstream tracking systems. This will give an accurate momentum measurement, and a better determination of the detachment parameter of the track. The aim is to achieve a further rejection of a factor of 10 or more at 90% efficiency. Finally a third level will involve more processing, with the goal of reducing the event output rate of 4kHz, with an event size of 40kbyte. 5.2 BTeV physics reach The comparisons shown in Table 2, are meant to illustrate in which respects the BTeV experiment is a second generation B-factory. In the first column we list the expected numbers of events collected by the e+ B-factories with the integrated luminosities of 500f13-1 which are expected by 2005. In the second column we present the amount of data the BTeV experiment is expected to reconstruct in one effective year (107s). The interested reader is referred to a review article (Stone 1997) for further discussions, as well as a comparison between the BTeV and LHCb experiments. Mode e+ e- B-factories with 500f13-1 BTeV yield in 107 s Bd --> 7+7- 850 24000 Bd OK- 700 11000 Bd -# OKs 250 (75 tagged) 2000 (200 tagged) Bd -4 K* Pit 50 2200 B, —> J/01(') - 15000 Table 2. Comparison of expected BTeV yield of reconstructed events in some interesting decay modes with the expectations from the e+ B-factories by the year 2005. 6 Summary We summarize this review by observing that although the hadron production of B mesons has a large cross section, the background levels are high, so elaborate triggering strategies are needed to extract the signal events. In addition an efficient operation of the detector in a high rate environment is required over extended run periods. The results of B physics studies by the CDF experiment prove that B physics can be done at hadron machines. The HERA-B spectrometer components are ready after extensive R& D to develop radiation hardness and high rate tolerance. The trigger of the HERA-B spectrometer is being commissioned, and a year of data taking is expected in 2002. This will provide valuable experience for the next generation of experiments at hadron machines. Finally, the second generation of B factory experiments, BTeV and LHCb, are being developed, and are expected to open a new era of B physics after 2005. HERA-B and Tevatron 325 Acknowledgments I thank K Pitts and M Schmidt for providing me with relevant information on the B physics results, upgrade and expected reach of the CDF experiment, and S Stone for useful discussions on the BTeV project. I congratulate my colleagues from HER A-B for their heroic efforts in mastering a very difficult experiment. Last but not least, I would like to thank the organizers of the School for their hospi-tality, and congratulate them for the excellent scientific and social program. References Abe F et al. [CDF Collaboration], 1988, Nucl. Instrum. Meth. 271 387. Affolder T et al. [CDF Collaboration], 1999, Phys. Rev. Lett. 83 3378. Affolder T et al. [CDF Collaboration], 2000, Phys. Rev. D61 072005. Affolder T et al. [CDF Collaboration], 2000a, Phys. Rev. Lett. 85 4668. Alde D M et al., Phys. Rev. Lett. 66 133. Alemi M et al. [TA2 Collaboration], 2000, Nucl. Instrum. Meth. A 449 48. Alexopoulos T et al., 1999, Phys. Rev. Lett. 82 41. Aririo I et al., 2001, Nucl. Instrum. Meth. A 453 289. Artuso M, [BTeV Collaboration], 2001, Nucl. Instrum. Meth. A 462 249. Avoni G et al., 2001, Nucl. Instrum. Meth. A 461 332. Bedeschi F, 2001, "Heavy flavor production in hadronic collisions," Fermilab-Conf-01-006-E Christian D C et al., 2001, "FPIX2: A radiation-hard pixel readout chip for BTeV," Nucl. Instrum. Meth. A 473 152. Ehret K, 2000, Nucl. Instrum. Meth. A 446 198. Eiges V et al., 2001, Nucl. Instrum. Meth. A 461 104. Gay C, 2000, "B mixing," Ann. Rev. Nucl. Part. Sci. 50 577, hep-ex/0103016. Gradl W, 2001, Nucl. Instrum. Meth. A 461 80 2001. Hall B K et al., 2001, FERMILAB-CONF-01-335. HERA-B Collaboration, 2000, DESY-PRC 00/04. Hohlmann M, 2001, Nucl. Instrum. Meth. A 461 21. Jansen D M et al., 1995, Phys. Rev. Lett. 74 3118. Kreuzer P, 2001, Nucl. Instrum. Meth. A 462 212. Leitch M J et al., Phys. Rev. D 84 3526. Leitch M J et al. , Phys. Rev. D 52 4251. Lohse T et al., 1994, DESY-PRC 94/02. Masciocchi S, 2001, Nucl. Instrum. Meth. A 461 220. Moser H G and Roussarie A, 1997, "Mathematical methods for BO anti-BO oscillation analyses," Nucl. Instrum. Meth. A 384 491. Paulini M, [CDF Collaboration], 1999, "B lifetimes, mixing and CP violation at CDF," Int. J. Mod. Phys. A 14 2791, hep-ex/9903002. Pitts K T [CDF Collaboration], 2000, "Electroweak and B physics results from the Fermilab Tevatron collider," hep-ex/0102010. Santoro A et al., 1997, "BTeV: An Expression of interest for a heavy quark program at CO," FERMILAB-FN-660. Sheldon P D, [BTeV Collaboration], 2001, Nucl. Instrum. Meth. A 462 291. Skwarnicki T, [BTeV Collaboration], 2001, Nucl. Instrum. Meth. A 462 227. Stone S, 2001, these proceedings. 326 Peter Kriian Stone S, 1997, "The goals and techniques of BTeV and LHC-B," hep-ph/9709500. Stone S, 2001, these proceedings. Tanaka M, [BTeV Collaboration], 2001, Nucl. Instrum. Meth. A 462 165. Wotton S A, [LHCb RICH Group Collaboration], 2000, "The LHCb RICH detectors," Nucl. Instrum. Meth. A 453 296. Observation of direct CP Violation in kwon decays Konrad Kleinknecht Johannes Gutenberg-Universitfit, Mainz, Germany 1 Introduction A small matter-antimatter asymmetry of the weak force has been experimentally estab-lished. This CP violation may be related to the small excess of matter from the big bang. The nature of CP violation in the K° system has been clarified after 37 years of exper-imentation as being due to a non-trivial phase in the weak quark mixing matrix which generates "direct CP violation" in the weak Hamiltonian. The experiments demonstrating this direct CP violation are discussed. 2 The big bang and the expanding universe In our present universe, we observe with optical telescopes about 100 billion (10' ) galaxies each of them with 100 billion stars. These stars for most of their life burn hydrogen to Helium, through nuclear fusion. They emit visible light from their surface and neutrinos from their interior. Both of these radiations have been observed from the sun. The analysis of the visible wavelength spectrum reveals black absorption lines due to the absorption of light in the atoms of the outer layers of the sun or other stars. A large step in the understanding of the evolution of the universe was made when Edwin Hubble observed in 1929 that these absorption lines for a given element are shifted towards the red end of the wavelength scale for stars which are very far outside our galaxy, say 100 million light years away. This "redshift" is interpreted as being due to these stars receding froni 327 DOI: 10.1201/9780429187056-8328 Konrad Kleinknecht Figure 1. Galaxies receding from us with velocities indicated by the length of the arrows us. Hubble observed that the size of the redshift, and the velocity of these receding stars is proportional to their distance from us. As an example, there is a galaxy in the Ursa Major which moves away with 1/7 of the light velocity, i.e. 42000km/sec. Figure 1 visualizes the flight velocities of such galaxies, with arrows indicating the size of the recession velocities. Of course, our point of reference, the earth, is by no means a special one. According to relativity theory, each reference system is equivalent in our finite, but unlimited universe. Each galaxy recedes from every other with a velocity proportional to their distance, like two points on a expanding rubber balloon. Turning the time arrow backwards, one con-cludes that these objects must have been very close to each other, about 15 billion years ago. At that time, an incredibly hot fireball of matter and radiation started to expand in the "big bang" . During the big bang, at high temperatures above 1030 degrees Kelvin, matter does not consist only of the stable building blocks which we observe in our old and cold universe, the protons and neutrons of the nucleus, and the electrons and neutrinos. The fundamental constituents of protons and neutrons, the up (u) and down (d) quarks, play their role, but there are also two more massive families of fundamental particles which have been produced and discovered in modern accelerator laboratories over the last few decades. Thus we have altogether three families: 1: up (u) and down (d) quarks, electrons (e) and their neutrinos (v) 2: strange (s) and charm (c) quarks, muons (p) and their neutrinos (v,i ) 3: bottom (b) and top (t) quarks, tau leptons (T) and their neutrinos (VT ) In addition to these matter particles, it was possible to create their antiparticles at CP Violation in kwon decays 329 Figure 2. Production of an electron-positron pair in a hydrogen bubble chamber the time of the big bang . In the universe today there is no indication of antimatter left over from the big bang. If antimatter exists, it must show up through the -y radiation which is produced when antimatter annihilates with matter. In the big bang, matter and antimatter were made at extremely high temperatures and in equal quantities because the forces which are responsible for their production are completely symmetric with respect to matter and antimatter. As an example Figure 2 shows the production of an electron and a positron from a photon hitting liquid hydrogen in a bubble chamber. The curvature of the two trajectories is due to a magnetic field which allows to distinguish between positive and negative electric charges. In a similar way, the other elementary constituents of matter were produced in pairs during the initial phases of the big bang. In this hot fireball, creation and annihilation of particles and antiparticles led to an equilibrium of approximately equal numbers of particles, antiparticles and photons. The temperature (measured in degrees Kelvin) corresponds to the average energy of the particles (measured in electron-volts, eV). The expanding fireball cools down, and below a certain temperature the creation of particle-antiparticle pairs stops while annihilation goes on. If all forces were symmetric with respect to matter and antimatter, there would be no matter left over in the cold phase except for photons. As a result of the expansion of the universe these photons would be shifted to lower energies. This cosmic microwave background was discovered in 1965 by Penzias and Wilson. The wave-length spectrum perfectly matches the Planck radiation of a black body at a temperature of 2.73 degrees Kelvin (Figure 3). r r — 2.73 K blackbody r + FIRAS COBE satellite 31 DMR COBE satellite x UBC sounding rocket o LBL-Italy White Mt. & South Pole O Princeton ground & balloon • Cyanogen optical 10-18 10-22 330 Konrad Kleinknecht Wavelength (cm) 10 1.0 0.1 1 10 100 1000 Frequency (GHz) Figure 3. Frequency distribution of the cosmic microwave background variation, as measured by the COBE satellite This is the echo of the hot photons from the big bang. Today the density of this radiation is 500000 photons per litre. Not only is there no primordial antimatter in our universe. The ratio of nucleons over photons is now only about 10-", whereas it was of order one in the early phases of the universe. A possible explanation was given by Sacharov and Kuzmin. They stated that this small surplus of matter is only possible if . one force violates matter-antimatter symmetry, . baryon number is violated and . the expansion goes through phases when there is no thermodynamic equilibrium. 3 Symmetries Symmetries and conservation laws have long played an important role in physics. The simplest examples of macroscopic relevance are the conservation of energy and momentum, which are due to the invariance of forces under translation in time and space, respectively. This was demonstrated by Emmy Noether. Sacharov postulated that one of the known forces or a new force violated the sym-metry between matter and antimatter, thus producing a small surplus of matter. All the remaining matter annihilated with the corresponding amount of antimatter, and at CP Violation in kaon decays 331 the end we are left with a surplus of 10' of matter compared to the photons. Such a symmetry violation goes against principles which were cherished for centuries. In the domain of quantum phenomena, there are conservation laws corresponding to discrete transformations. One of these is reflection in space by the parity operator P. Invariance of the laws of nature under P means that the mirror image of an experiment yields the same result in its reflected frame of reference as the original experiment in the original frame of reference. This means that "left" and "right" cannot be defined in an absolute sense. Similarly, the particle-antiparticle conjugation operator C, transforms each particle into its antiparticle, whereby all additive quantum numbers change their sign. C in-variance means that experiments in a world consisting of antiparticles will give identical results to experiments in our world. Here again it will be not possible to define in an ab-solute way whether an object consists of antimatter or matter. An anti-atom composed of antinucleons and positrons emits the same spectral lines as the corresponding atom. A third transformation of this kind is the time reversal operator T, which reverses momenta and angular momenta. According to the CPT theorem (Liiders 1954, Pauli 1955), there is a connection between these three transformations such that under rather weak assumptions in a local field theory all processes are invariant under the combined operation CPT. For a long time it was assumed that all elementary processes are also invariant under the application of each of the three operations C, P, and T separately. This assumption was questioned by (Lee & Yang 1956), and the subsequent experiments demonstrated the violation of P and C invariance in weak decays of nuclei and of pions and muons. This violation can be visualized by the longitudinal polarization of neutrinos emerging from a weak vertex: they are left-handed when they are particles and right-handed when antiparticles. Application of P or C to a neutrino leads to an unphysical state (Figure 4). P VL VR .at• VL VL C•P VL VR Figure 4. The mirror image of a left-handed neutrino under P, C, and CP 332 Konrad Kleinknecht The combined operation CP, transforms a left-handed neutrino into a right-handed antineutrino, thus connecting two physical states. CP invariance was therefore considered (Landau 1957) to be a valid replacement for the separate P and C invariance which was no longer true in weak interactions. A unique testing ground for CP invariance are elementary particles called neutral K mesons. They have the property that the particle K° and its antiparticle K° differ in one quantum number, called strangeness, but can mix through a transition mediated by the weak interaction. A consequence of CP invariance for the neutral K mesons was predicted by (Gell-Mann & Pais 1955). It was then known that there was a V° or K° particle that had a short lifetime (10-10 sec). It was predicted that there should be a long-lived partner, and that these two physical particles are mixtures of the two strangeness eigenstates, K° (S = + 1) and K°(S = - 1) that are produced in strong interactions. Weak interactions do not conserve strangeness, but the physical particles should be eigenstates of CP if the weak interactions are CP invariant. With K° = CP K°, the eigenstates are: CP K1 = CP [(K° + K9/0] = (K° + K°)0 = K1 CP K2 = CP [(K° - K9/01 = (K° - K°)/0 = -K2 Because CP (7+7-) = (7+7-) for two 7 mesons in a state with zero angular momen-tum, the decay into 7+71-- is allowed for the K1, but forbidden for the K2. Hence the K2 is predicted to have a longer lifetime, because it must decay to three pions. This was indeed confirmed when the long-lived K2 was discovered. It was discovered by (Christenson, Cronin, Fitch & Turlay 1964), that the long-lived neutral K meson also decays to 7+71--, but with a small branching ratio of ti 2 x 10-3. From then on the long-lived state was called KL because it was no longer identical to the CP eigenstate K2. Similarly, the short-lived state was called Ks. 4 Phenomenology and models of CP violation The phenomenon of CP violation in decays of neutral K° mesons has been with us for 37 years. The first ten years of intense experimentation after the discovery of the decay K L ---> 7+7- were devoted to the observation of other manifestations of the phenomenon, like the decay K L > 7°7° (Gaillard 1967, Cronin 1967), and the charge asymmetry in the decays K L --> 7rIev and KL -> 71-±µT v (Bennett 1967, Dorfan 1967), and to precision experiments on the moduli and phases of the CP violating amplitudes (Kleinknecht 1976). These experimental results excluded a large number of theoretical models proposed to explain CP violation, such that at the time of the London conference (Kleinknecht, 1974) only two classes of models survived. The superweak model postulated a new, very weak, CP violating interaction (Wolfenstein 1964) with AS = 2, whereas milliweak models invoked a small (10-3) part of the normal AS = 1 weak interaction as the source of CP violation. In the milliweak model there is also a direct decay of a Kaon state with CP quantum number -1 into a two-pion state with CP +1 through the weak Hamiltonian. This is called "direct" CP violation, as opposed to "indirect" CP violation by K°/K0 mixing, which is a feature of both models. The key question is can one devise experiments to distinguish between these models? K° K° CP Violation in kaon decays 333 In this context it was very important that a specific milliweak model within the Stan-dard Model was proposed (Kobayashi & Maskawa 1973). At the time of the discovery of CP violation, only 3 quarks were known, and there was no possibility of explaining CP violation as a genuine phenomenon of weak interactions. This situation remained unchanged with the discovery of the fourth quark, because the 2 x 2 weak quark mix-ing matrix has only one free parameter, the Cabibbo angle, and no non-trivial complex phase. Kobayashi and Maskawa remarked that this picture would change if six quarks are present. Then the 3 x3 mixing matrix naturally contains a complex phase, as well as three mixing angles. It is then possible to construct CP violating weak amplitudes from "box diagrams" of the form shown in Figure 5. K° Figure 5. Box diagram for K°—K° mixing connected to CP violating parameter F. A consequence of this model of CP violation is that the relative decay rates for K L 7r+71- --- and KL 7070 are not necessarily equal. This direct CP violation is due to "Penguin diagrams" of the form given in Figure 6. = i(rs — rL)/2 — (ins — inL) d d d d Figure 6. Penguin diagrams for K° 27r decay with direct CP violation. For a quantitative discussion, we use the conventional notation. The eigenstates of CP are: K I = (K° + K°)/ = +CPK I K2 = (Ko 1-0)/v 2 = —CPK2 The physical long-lived (KL) and short-lived (Ks) states are defined as: Ks = (K1 + fsK2)/(1+ ifs12)1/2 KL = (K2 + eLK1)/(1+ 1EL12)1/2 With CPT invariance, cs = c f, = e, where: ImPi2 /2 + 334 Konrad Kleinknecht is given in terms of the elements of the K° — K° mass matrix M and decay matrix F. The phase of E, is arg(c) = arctan(2Am/Fs) = 43.7° ± 0.2°. The experimentally observable quantities are 1 74— I eicb+- = 11+- 1 77001e20" = < 7r+ 7r- VI& > < 71-±7r-ITIKs > < Iro 7rol / I KL < ee1/1/(s > • It can be shown that these amplitude ratios consist of a contribution from CP violation in the box diagrams describing K° — K° mixing, called E above, and another one from CP violation in the penguin diagrams associated with weak K -+ 27r amplitudes, called El: 14_ = E noo = 2€' The quantities 74_, 7100 and 3E form the Wu-Yang triangle in the complex plane. The CP violating decay amplitude E' is due to the interference of a A/ = 1/2 amplitude, A0, and a A/ = 3/2 amplitudes, A2: E = iImA2 2A0 exp[i(62 — 60)] and its phase is given by the 7r7r phase shifts in the I = 0 and I = 2 states, 60 and 62: arg(E') =- (62 — 60 ) + 7r/2 which is measured experimentally to be (61 ± 9)° (Biswas 1981) or (45±9)° (Ochs 1981). The two models discussed above differ significantly in their predictions. In the superweak model there is no direct CP violation in weak decays, E' = 0, and therefore 71+- = 17oo = E. In milliweak models it is likely that el # 0. 5 Theoretical estimates for the parameter c' The parameter c' can be estimated within the weak quark-mixing model of Kobayashi and Maskawa if one infers the magnitude of the quark mixing angles from other experiments, and calculates the hadronic matrix elements for box graphs and penguin graphs. Typical values for are in the range +0.2 — 3.0 x 10-3 for three generations of quarks. A measurement of this quantity to this level of precision therefore becomes crucial for our understanding of CP violation. Since the phase of E' is close to that of E, and since E'/( I << 1, we get: —E' Re(c') = 6 1 (1 1 710o/ 77+-12) Various methods are used to calculate the value of Re(E'/E). Due to the difficulties in calculating hadronic matrix elements, which involve long distance effects, the task turns out to be very difficult. The following results have been obtained recently: CP Violation in kaon decays 335 1. The Dortmund group uses the 1/NC expansion and Chiral Perturbation Theory (XPT). They quote a range of 1.5 x 10-4 < e/c < 31.6 x 10' (Hambye 1999) from scanning the complete range of input parameters. 2. The Munich group uses a phenomenological approach in which as many parameters as possible are taken from experiment. Their result (Buras 1999) is 1.5 x 10' < eV( < 28.8 x10-4 from a scanning of the input parameters and c'/€ = (7.7T) x 10 -4 using a Monte Carlo method to determine the error. 3. The Rome group uses lattice calculation results for the input parameters. Their result is ei/c = (4.71:4) x 10-4 (Martinelli 1999). 4. The Trieste group uses a chiral quark model to calculate ei/f. Their result is 7 x 10-4 < €'/E < 31 x 10 -4 (Bertolini 1998). It is hoped that reliable hadronic matrix elements will be obtained in the near future by lattice gauge theory calculations. A measurement of the double ratio R = 17113o1 2 r(Kt, —> 27r°)/F(KL 7+7-) to a precision of better than 0.3% is required to distinguish between the milliweak and superweak models, and test these theoretical predictions. 6 Experiments on direct CP violation 6.1 The first observation of direct CP violation The first observation of direct CP violation was made in 1988 by a collaboration of physi-cists at CERN. Their experiment, called NA31, was based on the concurrent detection of 707r° and 7r +7- decays from alternating collinear beams of Ks and KL. Kaons with energies around 100GeV were produced by a 450GeV proton beam from the SPS accel-erator at CERN. The energies of the decay products were measured in a combination of a high-resolution Liquid-Argon electromagnetic calorimeter and an iron-scintillator hadronic calorimeter. In the first runs, about 100000 decays of the type KL 7070 and 295000 decays of KL 7+7- were observed, and the result for the CP parameter was Re(c'/E) = (33 + 11) x 10' (Burkhardt 1998). In further data-taking, the number of observed KL 7070 decays was increased to 4 x 105. A similar sensitivity was achieved by an experiment at Fermilab near Chicago, called E731. In 1992/93 the experiments NA31 at CERN and E731 at FNAL presented final results. The CERN result (Barr 1993) of (23.0 ± 6.5) x 10-4 shows evidence for direct CP violation with a significance of more than 3a, whereas the Fermilab result (Gibbons 1993) of (7.4 ± 5.9) x 10-4 is consistent with zero. As a consequence of this disagreement, two new experiments, NA48 at CERN and kTeV at Fermilab, were constructed to resolve the issue. 1ri+-12 F(Ks —* 27r°)/F(Ks 7+7-) 336 Konrad Kleinknecht KL target -14- 219.1 m Drill chamber 1 KS target 972 m Magnet Drift chamber 2 Ortft chamber 3 Aluminium window Beam moelter Drift chamber 4 LKr Hadron calorimeter calorimeter layon amicounter Kevlar window Beam pipe Veto counter 6 Helm tank -m— 9.2 m -41,— 5,4 m I Veto countar 7 7,2m Neutral hodoacope Charged boOoscope -ad 24,1 m 34,0 m Figure 7. Layout of the main detector components of the NA48 experiment. 6.2 The NA48 detector The new CERN experiment NA48 (Figure 7) was designed . to measure all four decay modes concurrently, . to register data at a rate 10 times higher than NA31, . to achieve an improved energy resolution for photons (liquid Krypton calorimeter) and for charged pions (magnetic spectrometer). In the design of the NA48 detector the cancellation of systematic uncertainties in the double ratio is exploited as much as possible. Important properties of the experiment, are: 1. Two almost collinear beams which lead to an almost identical illumination of the detector. 2. The lifetime weighting of the events defined as KL events, to remove systematic errors due to detector acceptance. The KL target is located 126m upstream of the beginning of the decay region. As the decay lengths at the average kaon momentum of 110GeV/c are As = 5.9m and AL = 3 400m respectively, the neutral beam derived from this target gives predominantly KL decays in the decay region. A separate Ks target is located only 6m upstream of the decay region, so this beam is a source of mainly Ks decays. The two beams are almost collinear, with the Ks target situated 7.2cm above the center of the KL beam. The relative angle of the beams is 0.6mrad so that they converge at the position of the electromagnetic calorimeter. The beginning of the Ks decay region is defined by an anti-counter (AKS). which is used to veto Kaon decays occurring upstream of the decay region. The position of the 1o.7m 27m CP Violation in kaon decays 337 AKS also defines the global Kaon energy scale which is directly correlated to the distance scale. The decay region itself is contained in a 90m long evacuated tank. The identification of decays from the Ks beam is done by a tagger consisting of an array of scintillators situated in the proton beam which is directed onto the Ks target. If a proton signal is detected within a time window of ±2ns with respect to a decay, the event is defined as Ks event. The absence of a proton tag defines a KL event. A magnetic spectrometer is used to reconstruct Ks ,L 7+7r- decays. The spectrom- eter consists of a dipole magnet with a transverse momentum kick of 265MeV/c, and four drift chambers which have a spatial resolution of --, 90,am. This leads to a mass resolution of 2.5MeV/c2 . A hodoscope consisting of two planes of plastic scintillator provides the time of a charged event with a resolution of about 200ps. A liquid krypton electromagnetic calorimeter (LKR) is used to identify the four pho-tons from a 707r° event. Liquid krypton has a radiation length of X0 = 4.7cm which allows one to build a compact calorimeter with high energy resolution AE/E = as measured with electrons coming from a KL 7rev. It also has a very good time resolution of less than 300ps and very good linearity. It consists of 13212 cells (2x 2cm2 ) pointing to the average Ks decay position. The transverse spatial resolution is better than 1.3mm. The electromagnetic calorimeter is complemented by an iron-scintillator sandwich calorimeter with a depth of 6.8 nuclear interaction lengths which measures the remaining energy of hadrons for use in the trigger for charged events. A muon veto detector, consisting of three planes of scintillator shielded by 80cm of iron, is used to identify muons and veto KL 7r/iv events. 6.3 Analysis of NA48 data To identify events coming from the Ks target a coincidence window of ±2ns between the proton signal in the tagger and the event time is chosen (see Figure 8a,c). Due to inefficiencies in'the tagger and in the proton reconstruction a fraction asL of true Ks events are misidentified as KL events. On the other hand there is a constant background of protons in the tagger which have not led to a good Ks event. If those protons accidentally coincide with a true KL event, this event is misidentified as a Ks decay. This fraction aLs depends only on the proton rate in the tagger and the width of the coincidence window. Both effects, asL and can be measured in the charged mode as Ks and K1. can be distinguished by the vertical position of the decay vertex (see Figure 8b). The results are arsEE = (1.63 ± 0.03) x10-4 and (4,-; = (10.649 ± 0.008)%. This means that about 11% of true KL events are misidentified as Ks events, however, this quantity is precisely measured to the 10' level. What is important for the measurement of the double ratio R is the difference between the charged and the neutral decay modes .AaLs = ars — n157. Proton rates for both decays modes are measured in the sidebands of the tagging window to determine AaLs. The result is Aacs = (4.3 ± 1.8) x10-4. Several methods have been used to measure AasL , leading to the conclusion that there is no measurable difference between these mistaggings with an uncertainty of ±0.5x Another important correction is the background subtraction. Decays KL 71-e/I and KL 7rptv can be misidentified as K 7T + 71 — decays, as the v is undetectable. However, since the v carries away momentum and energy, these events can be identified by their (b) y-Vertex selected 10 10 10 10 10 1 1 05 0 4 103 10 10 1 Ks+KL 0 0 X X (c) 338 Konrad Kleinknecht 10 10 10 10 10 (a) K + sK L X X 1 II 1111 - 20 -10 0 10 20 Atmin (event-tagger) (ns) - 20 -10 0 10 20 Atmin (event-tagger) (ns) - 20 -10 0 10 20 Atmin (event-tagger) (ns) 10 4 neutral events (d) with tracks 10 10 10 - 0 -5 0 5 10 At (Lkr-Hodoscope) (ns) Figure 8. (a) and (c): minimal difference between tagger time and event time (At,„i„). (b) Atmin for charged K L and Ks events. (d) Comparison between charged and neutral event time. For this measurement decays, selected by the neutral trigger, with tracks are used (7, conversion and Dalitz decays Ks 7r°70D -y7-ye+c-). high transverse momentum and their reconstructed invariant mass. The remaining background can be measured by extrapolating the shape of the background in the in - p,12 plane into the signal region. In this way the charged background fraction leads to an overall correction on R of (16.9 ± 3.0) x10'. A similar extrapolation can be done in the neutral decay mode. Here the background comes from KL -+ 370 decays, where two 7, are not detected. This leads to a misrecon-struction of the invariant 7r° masses. In this case the background leads to a correction of R by (-5.9 + 2.0) x The number of signal events after these corrections are summarised in Table 1. The efficiencies of the triggers used to record neutral and charged events have been determined. Independent triggers are used which accept a down-scaled fraction of events. In the neutral decay mode the efficiency is measured to be 0.99920 ± 0.00009 without measurable difference between Ks and KL decays. The 7+7- trigger efficiency is measured to be CP Violation in kaon decays 339 Event statistics (x 106) NA48 KTeV NA48 KTeV Ks 7r+7r- 22.221 4.52 KL —> 7r+7r- 14.453 2.61 Ks —> 7r°7r-° 5.209 1.42 K L —> 7r°7r° 3.290 0.86 Table 1. Event numbers after tagging correction (NA48) and background subtraction. (98.319 ± 0.038)% for KL and (98.353 ± 0.022%) for Ks decays. Here a small difference between the trigger efficiency in Ks and KL decays is found. This leads to a correction to the double ratio of (-4.5 ± 4.7) x10-4. The error on this measurement is dominated by the total number of events registered with the independent trigger. This error is one of the main contributions to the systematic error on the measurement of R. The distance D from the LKR to the decay vertex is reconstructed using the position of the four -y clusters. From the kinematics of the decay one obtains E E i,j where Ei is the energy of cluster i and rii the distance between cluster i and cluster j. This formula directly relates the distance scale to the energy scale. It is therefore possible to fix the global energy scale with the measurement of the known AKS position in the neutral decay mode. More checks on the energy scale and the linearity of the energy measurement can be performed using the invariant 7r° mass and the known position of an added thin CH2 target in which a 7r- beam produces 7r° -+ 2y. The comparison of all methods gives an uncertainty of ±3 x10-4 in the global energy scale. Another systematic problem is the minimization of acceptance corrections, which is done by weighting the KL events. The difference in the lifetime between Ks and KL events produces a different illumination of the detector. There are more KL events decaying close to the detector, whose decays products are measured at smaller radii closer to the beampipe. NA48 weights the KL events according to the measured lifetime such that the distribution of the z-position of the decay vertices of Ks and KL events and the detector acceptances become equal. Using this method the influence of detector inhomogeneities is minimised and the analysis becomes nearly independent of acceptance calculations by Monte Carlo methods. The price to pay for this reduction in a systematic error is a loss in statistical accuracy. Although the acceptance is almost equal there are nevertheless small differences in the beam geometry and detector illumination between decays coming from the Ks and the K1 , target. These remaining differences are corrected using Monte Carlo methods. to calculate that the deviation of the double ratio R from an input value of 1 is (26.7 ± 5.7) x10'. Summing up the systematic uncertainties of all different sources in quadrature, gives a total of 12.4 x 10-4 in R, arid 2.1 x 10' in e/e. The result of NA48 using the data samples from 1997, 1998 and 1999 is(Fanti 1999, Lai 2001) e'/€ = ( 15.3 ± 2.6) x10 -4 (1) 1 D = MK Vacuum Decay Region Vacuum Window I I :: K1 beams I Regenerator Drift Chambers Trigger Ho&scopes : Hadron Veto with Lead Wall 1 340 Konrad Kleinknecht 6.4 The KTeV detector Muon Muon Analysis Magnet Filter Veto Photon Veto Detectors 20 cmj CSI 120 140 160 180 > Distance from Target (m) Figure 9. Layout of the main detector components of the KTeV experiment The main elements of the KTeV detector (Figure 9) are similar to those in NA48, since both experiments work in a similar environment. The main difference is the way in which the Ks mesons are produced. KTeV uses two parallel well separated kaon beams derived from a single target. In one beam, the K L mesons from the target pass through a collimator and then decay in an evacuated decay region of 30m length. In the other beam the KL mesons traverse a slab of matter, known as a "regenerator". In the regenerator, the K° and K° components of the KL are affected differently by interactions with matter. Therefore the beam emerging behind the regenerator is a slightly different superposition of these two components. In the forward direction, it contains a Ks compone7it which is coherent with the outgoing K L component. This is the regenerated Ks beam used in the experiment. The regenerator alternates between both beams every minute in order to keep the detector illumination identical for the Ks and the KL components. The decay region of the regenerator beam is defined by a lead-scintillator counter at the downstream end of the regenerator. The decay region of the vacuum beam starts at the first photon veto detector. The KTeV magnetic spectrometer consists of four drift chambers, with the magnet providing a momentum kick of pt = 412MeV/c, leading to a mass resolution given by o-(m,.+,- ) = 1.6MeV/c2. For triggering of charged events a scintillator hodoscope is used. The electromagnetic calorimeter consists of 3100 pure Cesium-Iodide (CsI) crystals with a radiation length of X0 = 1.85cm. In the inner region the crystal size is 2.5 x 2.5 c1112 and in the outer region 5.0 x 5.0 cm2. Two beam holes of 15 x 15 cm2 allow the kaon beams to pass to the beam dump. The energy resolution at large photon energies above 10GeV is 0.75% as measured with Kea decays. Ten lead-scintillator "photon veto" counters are used to detect particles escaping the decay volume. The background is further reduced by a muon veto counter consisting of 4 m of steel and a hodoscope. _ .., -1:- • Data IL It H--I-1-11- - .1*------ -41 . _ ... tev ..1---------"-L-i.H.i. - - 0 IT TE0 ... 37co ..._,..•;1--t—H—L., 130 140 150 Vertex Z (m) TT+ TE (b) + + +++ - Slope = - c-1.60 ± 0.63) x10-4 nev- Slope = - C-0.53 ± 0.34) x10-4 7e7t° + 1- + ++ Ape - = ( 0 90 + 1 04) x10-4 : *, - + 37co + ++4.+-.1.+++-. 11-4 +- ++4. + - Slope = = - C 0.14 ± 0.36) x10-4 110 120 130 140 150 Vertex Z (m) 0 110 120 co 150 C aS 0 100 z 0 50 E c\J 0 ZI5 c12000 a) 1000 a) C 0 40 20 0 800 600 400 200 0 "rt's 1.02 N 1.02 crs E 1 O 0.98 1.02 0.98 1.02 0.98 CP Violation in kaon decays 341 Figure 10. (a) Data versus Monte Carlo comparisons of z-vertex distributions. (h) Linear fits to the data/MC ratio of (a). 6.5 Analysis of KTeV data KTeV has similar physical backgrounds to NA48, except that two-pion decays from Ks mesons produced by incoherent regeneration in the regenerator have to be subtracted. Typical numbers for the background fractions are 6.9x10-4 charged background from Ke3 and Ko decays and 27x10-4 neutral background from 370 decays. The background levels in the regenerated beam are 107x10-4 in the neutral mode, which gives rise to a large systematic error, and 7.2x10-4 in the charged mode. The event numbers after background subtraction are given in Table 1. The main difference in the analysis techniques of the two experiments is the treatment of the acceptance correction. KTeV does not use event weighting but uses Monte Carlo studies to correct for detection differences between the Ks and KL decays. Ke3 and 37r0 decays are used to model the detector and the agreement between data and Monte Carlo is good (see Figure 10). The acceptance correction to R. is calculated by Monte Carlo 342 Konrad Kleinknecht simulation to be (231 ± 13) x 10-4. This can be compared to the size of the total effect of cif on R, 168 x 10-4, as measured by KTeV. The main source of systematic uncertainty is a slight disagreement between data and Monte Carlo comparison in the 7+7- decay mode in the vacuum beam. A variation of (-1.60 0.63) x 10-4m-1 has been found which is applied as a systematic error. The energy scale is determined using the known position of the regenerator edge. The comparison of the measured position of the vacuum window with the real position gives an uncertainty of the global energy scale which contributes an error of 4.2 x10-4 on R. The result is obtained by a fit of R.e(e'/E), the regeneration amplitude and phase to the event numbers per energy bin. The result published in 1999 was (Alavi-Harati 1999) EVE = ( 28.0 ± 3.0 (stat) ± 2.8 (sys) ) x10-4 (2) but this was subsequently corrected due to an error in background subtraction: f'/E = ( 23.2 ± 3.0 (stat) ± 3.2 (sys) ± 0.7 (MC) ) x10-4 (3) Woods - E 731A . — 0 Burkhardt NA 31 — _ Gibbons 1 E 731 1 k TeV i Barr NA 31 NA 48 I Patterson E 731 0 1988 1989 1990 1991 1992 1993 1999 2001 Figure 11. Measurements of the parameter F' /e 7 Conclusion The two experiments KTeV and NA48 have definitively confirmed the original observation of NA31 that direct CP violation exists. The results of all published results on Fi/e are shown in Figure 11. Therefore, the CP violation observed in the K meson system is a part of the weak interaction, due to weak quark mixing. Exotic new interactions such as the superweak interaction are not needed. With more 'data, both experiments will reach a precision of 0(2 x 10-4) on e'/e. The theoretical calculations of E'/F within the Standard Model are still not very precise, but this does not change the main conclusion of 50 40 30 20 tA1 W 10 GJ ce 0 -10 CP Violation in kaon decays 343 the experiments, that c' is different from zero and positive, and hence direct CP violation exists. Unfortunately, this milliweak CP violation in the Kobayashi-Maskawa quark mixing matrix is probably not large enough to explain the observed matter-antimatter asymmetry from the big bang. An additional, stronger CP violation is needed for this. There are spec-ulations that this might be due to CP violation in the lepton sector in the early universe. Very heavy Majorana neutrinos could play a role in the formation of an antimatter-matter asymmetry at this time (Buchmiiller 2001). References Alavi-Harati A. et al. (KTeV Collaboration), 1999, Phys. Rev. Lett.,83, 22 Barr G.D. et al. (NA31 Collaboration), 1993, Phys. Lett.,B 317, 233 Bennett S. et al., 1967, Phys. Rev. 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Nucl. Sci.,26, 1 Kobayashi M., Maskawa T., 1973, Prog. Theor. Phys.,49, 652 Cahibbo N., 1963, Phys. Rev. Lett.,10, 531 Lai A. et al., 2001, "A precise measurement of the direct CP violation parameter Re(e' Europ. Phys. Journal Landau L. D., 1957, Nucl. Phys., 3, 127 Lee T. D., Yang C. N., 1956, Phys. Rev.,104, 254 Liiders, G, 1954, Kgl. Danske Videnskab, Selskab, Matfys. Medd., 28(5), 1 Martinelli G., Presentation at Kaon99, Chicago http://hep.uchicago.edu/kaon99/talks/martinelli/ Ochs W., 1981, private communication. from analysis of CERN-Munich data Pauli W, 1955, in "Niels Bohr and the Development of Physics", ed. W. Pauli, p. 30, (Oxford: Pergamon) Wolfenstein L., 1964, Phys. Rev. Lett.,13, 562 Results from the B-meson factories Klaus R Schubert Technische Universitat Dresden, Germany 1 Introduction and short history Two experiments, BABAR at SLAG and Belle at KEK, succeeded this year in presenting new evidence for CP violation, so that 37 years after its discovery in decays of neutral K mesons (Christenson 1964), the effect is now established for the first time in a second system, the decays of neutral B mesons (Aubert 2001a, Abe 2001). Both experiments use double storage rings with asymmetric energies to produce B mesons in e+ e- collisions on the T(4S) resonance. These lectures describe these so-called "B-Meson Factories", the experimental detectors, the special features of experiments on the Y(4S), and the description of CP violation in quantum mechanics and in the Standard Model. The most recent experimental results are discussed. Let me start with a short history of CP violation in B meson decays. The b quark was discovered in 1977 in high-energy proton-tungsten collisions at FNAL (Herb 1977), in the hound bb state T(1,5) = 13,51 , with a mass of 9.46GeV. Soon afterwards, the T(1S) (Berger 1978, Darden 1978a), and "f (2S) (Bienlein 1978, Darden 1978b), were also produced in e+ e- annihilation with the DORIS storage ring at DESY. Both these states, as well as the T(3S), are not heavy enough to decay into the lightest B mesons, B° (bd) and B+ (1)u), and their antiparticles B° and B. . The next bb resonance, the T(4S) with a mass of 10.58GeV, was first produced in Cornell's e+ e- storage ring CESR (Andrews 1980, Finocchiaro 1980). Its decays into B+ B- and B°B° pairs were first seen as an enhancement in lepton production from semileptonic B decays (Bebek 1981, Spencer 1981). Soon after the discovery of the T(1S), it was already clear that B mesons could show large CP-violating effects (Ellis 1977). Plans to look for these effects experimentally led to proposals for B-meson factories with e+ e- luminosities much higher than those achieved by the storage rings CESR and DORIS. The first proposal was worked out at the Swiss National Laboratory PSI (Eichler 1986, Wacker 1988), but was not funded. During this period, the ARGUS experiment at DORIS discovered B°B° oscillations (Albrecht 1987), leading to an enhanced interest in B-meson factories, and in the search for CP violation in B-meson decays. The subsequent proposals at SLAG (Hitlin 1989) and KEK (KEK 1990) were approved in 1993, and construction of the two machines PEP-II and KEN-B started more or less immediately. 345 DOI: 10.1201/9780429187056-9PEP II High Energy Ring (HER) PEP II Low Energy Ring (LER) North Damping Ring e-gun 1111.111MIk 200 MeV injector Linac Positron Return Line Positron Source South Damping Ring 3 km 346 Klaus R Schubert The designs of the detectors BABAR and Belle were finalized and approved in 1995, and the first e+e- collisions in PEP-II took place in July 1998, with the first events recorded by BABAR in May 1999. In July 2000, the first BABAR and Belle results were reported at the International Conference for High Energy Physics at Osaka. In October 2000, PEP-II reaches its design luminosity of 3x1033/cm2 /s, and in July 2001 BABAR and Belle discover CP violation in B-meson decays. 2 PEP-II and KEK-B Figure 1 shows a schematic view of the B-meson factory PEP-II at SLAC. It consists of a double storage ring for electrons of 9GeV and positrons of 3.1GeV located in the 2.2km circumference tunnel of the dismantled PEP storage ring. The new electron ring uses bending magnets from PEP, but all the other components of PEP-II have been built in 1994-1998. The electrons are extracted from the SLAC linac at their nominal energy and then transferred into the HER (high energy ring) of PEP-II. The positrons are produced by high-energy electrons near the end of the linac, returned to the linac start point, damped by synchrotron radiation in two small rings, extracted from the linac at their nominal energy, and then transferred into the LER (low energy ring). The two rings intersect at only one point where the BABAR detector is located. Figure 1. Schematic view of the B-meson factory PEP-II at SLAG. By August 2001, a peak luminosity of = 3.3x 1033/cm2 /s has been reached, (exceed- ing the design value of 3x1033); this corresponds to 3.6 e+e- T(4S) —> BB events per second, is obtained with an electron current of 0.88 A, and a positron current of 1.55A. Averaged over one day the best luminosity has been 220/pb/day, compared to a design value of 135/pb/day. Figure 2 shows the luminosity accumulated up to the end of August 2001. PEP-II has delivered 45 events/fb, and BABAR has recorded 43 events/fb. Out of these, 4/fb were recorded at a center-of-mass energy 40MeV below the T(4S) resonance for background studies, and 39/fb are on the resonance corresponding to 42 x106 BB events. The asymmetric energies of the two beams cause the BB pairs to be produced with a significant velocity in the direction of the high energy beam, unlike in the sym-metric storage rings CESR and DORIS. The Lorentz boost of the Centre of Mass system is: IP(e+ ) + P(e-)1 0-y = = 0.56 . MT(4S) (1) B-meson factory results 347 BABAR II IA PEP-II Delivered 45.3/113 BABAR Recorded 42.9/lb /I= BABAR off-peak 4.05/fb / ..1 fflial 1 i ff / — Delivered Luminosity Rccorded Luminosity — Off Peck I IMPr' MEW 111..111n1111111111Mili. ME ME 9-,% ° c;. 1999 2000 2001. Figure 2. Integrated luminosity of PEP-II up to August 2001. Figure 3 shows the B-meson factory, at the Japanese National Laboratory KEK. It, is a double storage ring for electrons of 8GeV and positrons of 3.5GeV leading to a boost of Ory = 0.45, which is slightly smaller than in PEP-II. The second difference is a beam crossing angle of 22mrad in the interaction region, whereas PEP-II collides u+ and e- beams with zero crossing angle. The non-zero angle could make it possible to reach higher luminosities. Up to July 2001, KEKB has reached a peak luminosity of = 4.5x1033/cm2 /s, but the integrated luminosity of 32.8/fb is slightly lower than PEP-II. There is only one interaction region in KEK-B, which houses the Belle detector. 40 44 42 40 38 36 34 32 D 30 Z.28 26 24 22 1:13) 20 I° 18 E) 16 14 12 10 8 6 4 2 0 TSUKUBA Area HER LER Interaction Region (Belle) 41* 0 cc cc NIKKO Area ccu. CC 11.1 -J OHO Area (3 (TRISTAN Accumulation Ring) Electron ositro RF RF. FUJI Area Figure 3. Schematic view of the B-meson factory KEK-B at KEK. CC 348 Klaus R Schubert 3 BABAR and Belle Figure 4 shows the layout of the BABAR detector (Aubert 2001f). The BABAR Collabo-ration consists of about 520 physicists from 9 countries: Canada, China, France, Germany, Italy, Norway, Russia, UK, and USA. The detector geometry reflects the boosted event topology, with the e+e- interaction region displaced from the centre of the detector. There are six main detector components, a silicon vertex detector (SVX), a drift chamber (DCH), a Cerenkov detector (DIRC), an electromagnetic calorimeter (EMC), a magnet coil, and an instrumented flux return yoke (IFR). IFR Coil Figure 4. Scheme of the BABAR detector. Abbreviations are explained in the text. The SVX reconstructs the decay vertices of the two B mesons. With the boost of /37 = 0.56, the average distance between the production and decay of a B meson is: fi'yer = 260pm. (2) The production point e+ e- -4 T (4S) BB, cannot be reconstructed, and lies somewhere in the bunch-crossing region which has a length a, 1cm. The decay point of a fully reconstructed B meson is found with a resolution a, 70pm, which depends slightly on reconstructed final state. Partially reconstructed B mesons, which are used for flavour tagging, have a vertex resolution a, 180pm. As explained in Section 4, all time- dependent B-meson decay information can be obtained from the distance between the two decay points, without any knowledge of the BB pair production point. This distance has a typical resolution of V702 + 1802 = 190pm, dominated by the tagging vertex. This is below the typical decay length of 260pm in Equation 2, and the B°B° oscillation length: 27/3-ye = 2.0 mm. (3) ,grad Kerilaricar;on-f !Per cupDrt b tort-41)er encpiec? 4. er=0 cetectcrs Carbon-01 ttr support cone Dews Opel - • Mr SCrif Ccslinq ring Upleu fanouts "Hytr Sikssclout Ca CDCN 15o° 8.0 GeV e TOF PID EFC 17° 3.5 GeV e ...,*.•••••••••••/ 0 1 2{m) Cs1 KLM B-meson factory results 349 Figure 5. The Silicon Vertex Detector of BA BAR. The vertex resolution is achieved by tracking charged decay products with five layers of double-sided-readout silicon strip detectors in the geometry shown in Figure 5. The drift chamber (DCH) tracks charged decay products in three dimensions through the magnetic field of 1.5T produced by the solenoid coil. It measures the track momenta, px, py, pz , and their specific ionisation dE/dx. The DIRC determines their velocity by measuring the emission angle of Cerenkov light in 2.5cni thick quartz bars. Combining momentum and velocity information identifies the mass of the charged particle. The EMC measures the directions and energies of photons, identifies electrons, and also determines the direction of Ky, mesons that interact in the calorimeter. The IFR identifies uions, and measures the direction of K2, mesons that do not interact in the EMC. Figure 6. Scheme of the Belle detector. Abbreviations are explained in the text. The Belle detector (Abashian 2001) is shown in Figure 6. The Belle Collaboration has about 300 physicists from 14 countries: Australia, Austria, China, Germany, India, Korea, Japan, Philippines, Poland, Russia, Slowenia, Switzerland, Taiwan, and USA. The 350 Klaus R Schubert 139 12cm Figure 7. Scheme of Belle's Silicon Vertex Detector. detector consists of a silicon vertex detector (SVD), a drift chamber (CDC), a particle identification system (PID), an electromagnetic calorimeter (CsI), and an instrumented flux return for KL and muon identification (KLM). It is quite similar to BABAR, with the main difference being in the particle identification. Instead of a DIRC, where the Cerenkov angle is measured, Belle uses aerogel Cerenkov counters (EFC), which only determine velocity thresholds, and time-of-flight scintillators (TOF), which determine slow velocities. Belle's silicon vertex detector has only three layers, with the geometry shown in Figure 7. An upgrade to five layers is in progress. Figure 8. A fully reconstructed event B°P (0(2S)K(s) )(D*+7-) in BABAR. Figure 8 shows a B°B° event with both B-meson decays fully reconstructed in BABAR. One B meson decays into D*4 71-- with D*+ _y Do7+ Do _y This final state is a flavour eigenstate, i.e. the neutral B was a B° at its time of decay. The other B decays into the final state 0(2S).Kcs' with 0(2S) -4 pt+fr, K(s) -4 7+7-, which is a CP eigenstate that can come from either flavour of neutral B. Such events are very rare, since a single B is fully reconstructed, with all its decay products detected and identified in BABAR, with a probability of only 10-3. B-meson factory results 351 4 Special features of BB pair decays The T(4S) is a bb meson with mass m = (10580.0 ± 3.5)MeV and width F = (14 ± 5)1\1eV (PDG 2000). The cross section for its formation in e+e- annihilation is 1.1nb, and it decays by a strong interaction into BB pairs, 50% .13+13- and 50% B°B° . These final states consists of two and only two particles which decay by weak interactions on a time scale of 10's. The experimental study of these weak decays is influenced by the fact that B mesons from the T(4S) come in pairs. The mass of a B meson is 5279MeV (PDG2000), so in the center-of-mass system of the T(4S), each B has a kinetic energy of only 10580/2 - 5279 = 11MeV, corresponding to a velocity 3* = 0.064, and a momentum p* = 340MeV/c. Let us look first at the determination of the total decay rate of neutral and charged B mesons, through measurements of their lifetimes. us 1.11 - 2.50 wit Figure 9. Decay topology of BB pairs and definition of Az. In the example, a B° decays into D*-71-+ and the other neutral B into three non-identified tracks. Figure 9 sketches the decay topology of a BB pair. The two decay points are measured with the help of the silicon vertex detectors, but the production point is not, measurable. Due to the small value of 0* in the T(4S) frame, compared to the boost factor 07 = 0.56 in BABAR (0.45 in Belle), we can approximate Az = zl - Z2 = /37C(t1 t2) = fhcAt . (4) Therefore, a good measurement of Az provides a measurement of At. The lifetime, r(B±), gives the decay distribution of a B+ 13- pair: d2N e N e_t217. dti dt2 T2 N e-Atir dAt dt2 , T 2 e Jo N -At/, roo e-20 dt217 1, dN 7 for t i > t2, =_ dAt Nr e- Ail-7- f ooAt e-2t21-r (11, ,14 / 2/ T for t2 > t l , dN —2T _N e-iti-t2117 d(t i - t2). N Cl Atlir dAt = 2T (5) 5f1 0.05 u ,Afi (GeV) tl5 0.1 5". 0.05 0 A -0.05 -0.1 400 300 200 100 0 52 , J . r 5.225 5.25 5.275 m,,, (tieVT-) R4BAR ff. 352 Klaus R Schubert Figure 10. Plots of mEs and AE* for B± JI0K ± decays in BABAR. Thus there are decay pairs with positive and negative values of At = t1 — t2 , and the decay law is symmetric, dN(At) = dN(—At), and depends only on the absolute decay time difference, Atl. The first BABAR measurement of T uses pairs where one B is fully reconstructed with its decay vertex at z1 , and the second vertex at z2 is obtained from the intersection of at least two reconstructed tracks from the other B. B+ mesons are fully reconstructed in the decay modes D°71-+, D*°7r±, J/0K+, and ,b(2S)K+. B- mesons are reconstructed in the corresponding charge-conjugated modes. J/0 mesons are reconstructed in their e+e- and u+ µ- decay modes. Figure 10 shows how combinations of J/1/) candidates and K+ tracks are identified as B+ JIY,K+ decays, using the kinematic quantities, "beam-energy-substituted mass": mEs = Orn-r(4s)/2)2 — (13,/1,/, +14)2 , (6) which has to be equal to m(B+) within the experimental resolution of ±2.5MeV, and the centre-of-mass "energy difference": AE* = El/0 + EK — (4S)I 2 (7) which must be zero within the experimental resolution of +151\4eV. Figure 11 shows the mEs distribution of all the P...-7300 [3+ candidates. Within the signal region, there are 93% signal events and 7% background. The At distribution of all events in the signal region is shown in Figure 12. The lifetime 7-(B+) is obtained by fitting the expectation in Equation 5 to the data, folding in the experimental resolution for At. This is described in detail in (Stark 2001). The result (Aubert 2001b) is given in Table 1. The lifetime of the B° is obtained in a similar way, but in this case the presence of B°B° oscillations has to be taken into account. Let us have a quick look at oscillation phenomenology before returning to the r(B°) measurement. A B° meson produced at t = 0 decays at time t either as a B° or as a B°. Owing to second order weak interactions, illustrated in Figure 13, a neutral B meson is a linear superposition 1B) = 011B°) + 021B°) (8) bg. a 354 poo W 250 200 150 100 50 BABAR B-meson factory results 353 > 3000 ,r) 2500 - 2000 tr) a) 1500 C ILI 1000 500 2 5.22 5.24 5.26 5.28 5.3 M ES [GeV] Figure 11. mEs distribution of all Figure 12. At distribution of all fully recon- fully reconstructed B± decay candidates strutted Bt decay candidates in the BABAR in the BABAR lifetime measurement. lifetime measurement. The dotted curve is the background. PDG 2000 BABAR New Average 7-(B±) in ps 1.653 ± 0.028 1.673 + 0.032 + 0.023 1.662 + 0.023 7-(B°) in ps 1.548 + 0.032 1.546 ± 0.032 + 0.022 1.542 + 0.021 T (B+) I T (B°) 1.062 + 0.029 1.082 ± 0.029 + 0.012 1.080 ± 0.021 Table 1. B Meson Lifetime Results (Stark 2001). Vtd*. t Vtb W W Vtb t Vtd. Figure 13. Diagram describing B° B° oscillations. with a time dependence of 01 and 02 determined by initial conditions and the coupled differential equations, z at = zbj where itij = [(mi 1 mi2) (1-1 t F12)1 m2i M22 2 F21 F22 (9) The matrices mij and Fij are Hermitean, with ntji =- mil; and Fji = F. CPT symmetry requires that mil = m22 = m, F11 = F22 = F. There is also an unobservable phase, so the matrix /..tij has only five physical parameters, m, F, I I im121, Re (F12/m12), and Int (1-, 2 /7, 12). There are two and only two solutions of equation (9) with an exponential decay law: BCH' (t) = (pB° oo)e—yHt YH = irrtH + FH/ 2, B°L(t) = (pB° — gB°)e-7 E 1 'YL= imr,+FL12. (10) and vice versa: P(B° B°) = 1 P(B° B°) = 2 _wii +FL W2 [C-F e Lt cos(Amd • t)] 2 [e—rH1 Cr Lt — 2C —(FH+rL)1/2 cos(Arnd • t)] . q p p q 4 1 4 (15) 354 Klaus R Schubert The normalisation requirement is 11'12 + 1q12=1. Again the phase of q/p is not an observable, and the five observable quantities in equation (10) are rrill , Tn L, FH, FL , and lq/pl. Explicit formulae relating these five observables to the five parameters of //,,, can be found in the literature (Branco 1999). By convention (H=heavy, L=light), we have AMd = 171y — Mt > O. The non-exponential decay laws of the two initial states: 63(0) = B°, 74(0) = B°, (12) e —I'll t + e---YLt B° +q e —mi — e-11,1 ____B fie() = °, (13) 2 p 2 07 e-7Ht + e--ytt p C'Ylit - e-nt B° + B°. (14) 0 = 2 q 2 These decay laws include the probabilities P that a B° changes flavour, becoming a B°, are given by: In the Standard Model, CP and T violation in B°B° oscillations is expected to be very small, and these probabilities are almost equal: P(B° B°) = P(B° —> B°) + 0(10-3) <=> ly/pl= 1 + 0(10-3). The lifetime difference should also be less than a percent: FH = FL = r• This leads to the good approximation: P(B° —> B°) = P(B° —> B°) = e-rt[1 — cos(Amd • t)]/2, and a corresponding derivation gives: P(B° —> B°) = P(B° B0) = e-rt[l + cos(Amd • t)]/2. B°B° pairs from T(4S) decays have to be parity eigenstates because of parity con-servation in strong interactions. A two-particle state B°(0)B°(77- — 0), as sketched in Figure 14, is not a parity eigenstate owing to P B°(0)B°(7r — 0) = B° (7r — 0)B°(0). (20) Since the parity of the T(4S) is —1, the allowed state is BO (0) BO Or 0) BO ( R. 0) B0 (0). (21) B-meson factory results 355 B" e+ n-O / e Figure 14. B°(0)B°(7r — 0) is not a parity eiyenstate. It remains in this coherent state until the first of the two B mesons decays. The state of the second B meson is defined at that time t l . If the first decay is a flavour-specific B°, we have: B1(ti) = B°, B2(ti ) = B°, B2(t2) = (t2 — t i )B° +/3(t2 — t )B°, (22) with a (t) and OM from equation (13), where t -= t2 — t1 . Equation 22 describes B° B° pair decays also for the case when only the second B decay at time t2 is flavour-specific. a(t) and [3(0 then describe the time-changing flavour of the first B which decays at t < 0. If both decays are flavour-specific, we have measurements of B°B°, Bo —o, or B°B° pairs as a function of t. In a recent BABAR measurement of Amd (Aubert 2001c), neutral B mesons are fully reconstructed in flavour-specific decay modes, in the same manner as B+ and B- in the lifetime measurement discussed above. The second neutral B is partially reconstructed to obtain its flavour. This flavour measurement, which will be described in more detail in the CP violation Section 5.4, has a certain fraction of wrong flavour determinations: Nwron W = g 0.20. (23) Nright V 7,7. wrong From measurements of the decay vertices, z1 , z2, and equation (4), we obtain At for each reconstructed pair. Including the wrong flavour assignments, and the oscillation formalism with lq/p1= 1, and FL = rH = F, we have the number of unmixed and mixed pairs, and their time-dependent asymmetry a(At): N(B°B°) = [(1 — w)[1 + cos(AmdAt)] + w[1 — cos(AmdAt)]] , = [1 + (1 — 2w) cos(AmdAt)1 , (24) N(B°B°) = N(B°B°) = ne-rl Ati [1 — (1 — 2w) cos(Amd1t)1 , (25) N(B°B°) — N(B°B° ) a(At) = -= (1 2w) cos(AmdAt). (26) N(B°B°) + N(B°B°) The time dependences of Equations 24 and 25 are shown in Figure 15a with perfect flavour-tagging, w = 0. In Figure 15b the effects of w > 0 and the experimental resolution in At are included. The BABAR results are shown in Figure 16. A fit gives: AIN = (0.516 ± 0.016 ± 0.010)/ps. (27) b) -6 -4 -2 0 2 4 6 Decay Time Difference (reco-tag) (ps) 12 10 a 2 -6 -4 -2 0 2 4 6 Decay Time Difference (reco-tag) (ps lor a) ar c. t Unmixed Events ,r. c to 2 s-.t B. i f4 10 I > W I ! t i .. • 1 , . IMixed Events 10 2 10 -20 45 -10 a) bt 10 IS 20 At (ps) a 0.5 0 -0.5 • 10 15 20 1641 (ps) 356 Klaus R Schubert Figure 15. Time dependence of unmixed and mixed pairs of neutral B mesons from the T(4S), (a) without and (b) with experimental resolutions, solid curves = unmixed, dotted curves = mixed. Figure 16. BABAR result on B° B° mixing with fully reconstructed B mesons. The dotted curves indicate the backgrounds. BABAR and Belle have also measured tmd using other methods. A clean sample of "dilepton" events can be obtained, T(4S) teX, where t is a positive or negative high momentum electron or muon. Positive leptons tag the decay of a B+ or a B°, and negative leptons a B- or a B°, with a low mistag fraction w if the momentum cut is high enough. In 23 million BB events, BABAR finds 99000 dilepton events (Aubert 2001d), with a resolution on the vertex separation, a(Az) 250in. The asymmetry N (e+ — N (e± f ± ) a(At) = (28) N(e+e—)+ N(f±t±)' is shown in Figure 17. Its expectation is different from that in Equation 26 and Figure 16 because of the presence of ell— pairs from B+ B- events. Taking this into account, the fit gives the preliminary result Amd = (0.499 + 0.010 ± 0.012)/ps. (29) A compilation of Amd measurements is given in Figure 18. The "world average" of oo 1000 2000 3000 Az (pm) 0.8r 0.51 0.41- 0.3r- 0.2r7 0.9 0.8 0.7r 1 I t• H 11"1 B-meson factory results 357 Figure 17. BA BA R result on B°B° mixing with lepton pairs. A1-111'11 (3+1 pr.I) DEI.PH1 . (4 + 1 prd) 1.3 (3) OPAL (.5) (5 peel) cUI BABAR (2 peel) PEI 1.11 +1+ as erne of also. e after adjustments ARGUS-411.EO (Zd tacasurement•1 IA orld UN erage 0.446±0.020±0.018 ps'' 0.504±i1.019±11.012 ps'l 0.444±0.028±0.028 0.479±0.0111±0.015 ps'i 0.507±0.023±0.019 ps-1 0.495±0.02610.025 ps4 0.5031-0.000±0.011 ps-1 0.463±0.0011±0.016 psrl 0.41111±0.009 pa"' 0.48910.032 ps-i 0.410110.000 ps-' 0.4 0.45 0.5 0.55 Figure 18. Compilation of Arnd results. these measurements is Amd = (0.488 ± 0.009)/ps. (30) When ARGUS discovered B°B° oscillations (Albrecht 1987), they used a time-integrated measurement of dilepton events, which gave: (Amd)2 2(AmD)2 X 2r2 + = — 0.17 ± 0.05, Arrid = (0.47 + 0.10)/ps, (31) where the value of Arrtd has been adjusted for the present value of r = 1/T(B°). The Standard Model expectation is: G2F Val I ,1d1 (32) Alnd = -6---r2 7113M(B°) f(B°)2 B(Y) M(W)2 S ( M(1) ) 121“ 12 112(W) 358 Klaus R Schubert which relates Arad to the CKM matrix element IVtdI with the Fermi constant Gp, a parameter 7113 from perturbative QCD, and a parameter f V.T3 from non-perturbative QCD. The Inami-Lim function S and 7/13 = 0.55 ± 0.01 are known with small errors. The dominant theoretical uncertainty in the relationship between Arrid and IT id low-energy QCD. For this, we will use the lattice QCD result: comes from f (B°) B(B°) = (230 f 25 ± 20) MeV. (33) Let me conclude this section by returning to the B° lifetime measurement. Adding Equations 24 and 25 leads to the result • dN(At) = N(B° B°)+ N(B° B°) = N2 F dAt, F = T(B19' (34) which is as simple as Equation (5) for the determination of T(B+) from B+ 13- pairs. In the same analysis (Stark 2001, Aubert 2001b), BABAR fully reconstructs 7000 neutral B mesons in the modes D(*) '7T, D(*)± e, D(*) aT, and JNK*°, where D(*)± is either a D or a D*. The sample purity of 90% is similar to charged B reconstruction. The other neutral B is partially reconstructed to measure the decay vertex, but without requiring a flavour determination. The fit to the At distribution gives the results in Table 1 (on page 353). The lifetime ratio T(13± )17(B°) is significantly different from unity. This starts to be well understood, see (Buchalla 2001) in this School. 5 CP Violation in B-Meson decays 5.1 CP violation in the Standard Model Understanding CP violation is the main goal of the two experiments BABAR and Belle: . Is there CP violation in B decays and how large is it? . Can this violation be explained by the Standard Model weak interaction? . Does the same explanation hold for the known CP violation in K meson decays? The answers to the second and third questions also require precision measurements of B meson decays without CP violation. Note that the discovery of CP violation in K meson decays does not mean that CP is violated in the Standard Model, since its origin could be in a non-Standard new interaction. In the Standard Model, CP violation is a simple property of the Higgs couplings to quarks if there are more than two families of quarks. Since the three families of leptons and quarks are well established and the Higgs mechanism has been treated in the lecture of (Rosner 2001) at this School, I will not go into details here. The essence is that the Higgs mechanism mixes quarks in a different way from antiquarks. The Glashow doublet, partners of the quark mass eigenstates u, c, and t, are linear combinations d', s', and B-meson factory results 359 b' of the mass eigenstates d, s, and b. These combinations are different for quarks and antiquarks: d' = Vud • d + • s + Val) b, = 17,1 • d + Ves • s + Veb • b = Vtd • d +Vt., • s + Vtb • b , (35) = v:d • d + Vi7, • 3 ± V:b • , 31 = Vc*d • d + Vc*, • 3 ± • b , = V,: • a + vts* • +v: • I) . (36) Here Vii is a unitary quark mixing matrix , named CKM after (Cabibbo 1963) and (Kobayashi Maskawa 1973). CP violation arises only if Vii V. * Since the gauge i3 invariance of the Higgs coupling term in the Standard Model Lagrangian does not ex-plicitly forbid complex coupling matrices, it is expected to lead to a complex matrix Vii. This complexity is a necessary but not sufficient condition for CP violation. Because of unobservable phases, we must have: Im (VikIPlityi ) = J 0. (37) The quantity J is the same for all i j and k 1 (Jarlskog 1985). The CKM matrix has four observable parameters. There are various parametrisations, with experimentalists prefering that of (Wolfenstein 1984) , ( 1 — A2 /2 A AA3(p — in) V = —A 1 — A2/2 AA2 + O(.X4) , A2.\ 67/. (38) AA3(1 — p — —A) 2 1 The four parameters of the matrix, A, A, p, and 77, are 4 of the 18 free parameters of the Standard Model. Six unitarity conditions VVt = 1 can be drawn as triangles (Bjorken 1987) in the complex plane. All six have the same area J/2. The most popular one is shown in Figure 19: v.dvut + vcdv:, + vtdvt: = 0, (39) Vab /(AA3) — 1 + Vtd/(A.\ 3) = 0. (40) Measurements of K and B semileptonic decays, which are not discussed here, give A -= 0.220 + 1%, A = 0.83 + 5%. (41) Information on p and n comes from several observations, as summarized in Figure 20. The figure shows p = pV1 — A2 and 7 = 7W1 — A2, which are different from p and 7/ by about 2.5%. As can be seen from Equation 39, Vtd = A\ 3(1 — p — i7j), together with ['Tub = A\ 3(p — iri), fulfill unitarity more accurately than Equation 40. Consequently the use of p and 71 will become important when the experiments have reached a precision of 3% or better. em(Bo) value with foTTB from lattice 360 Klaus R Schubert 0 1 p .0.4 -0.6 -0.4 At 0 02 O.4 0.0 0. Figure 19. The Unitarity Figure 20. Circular bands for p and ri from measure- Triangle. ments of 1Vubl and 1Vtdl• The inner circles give the re-gions of maximum likelihood, the outer ones give one and two standard deviations. The measurements of F(b ufv) (LEP 1999), and ['(B —> phi) (CLEO 1999), are sensitive to 1Vub l. The measurement of Arnd, discussed in the previous section, and the upper limit on B, mixing, Ants (LEP 2000), are both sensitive to MA I. All these results are consistent, and we conclude that, with a significance of >1.7a, i # 0, and I JI ti 3x10-5 0. These CP conserving measurements already show that CP symmetry is violated in the Standard Model. The significance will increase when the theoretical uncertainties in the circular bands of Figure 20 are reduced. The B-Meson Factories have so far contributed to the measurement of Arnd, but they do not intend to measure Am, which would require running the storage rings on the 7(5S) resonance. They will contribute new results on Vubl starting next year. Two immediate questions arise from Figure 20: • Does = 3x10-5 explain the known value of E K for CP violation in K decays'? . What are the predictions for CP violation in B meson decays? The first answer is yes, as shown in Figure 21. CP violation in K decays has been discussed here by (Kleinknecht 2001). The band in Figure 21 follows from 174- 1 1/001 2.3 x The experimental width of the band is less than 1%, so the width is dominated by the theoretical uncertainty on BK . Adding in the 1K constraint fixes the signs of n and .1 to be positive. Figure 22 shows the result of a maximum-likelihood fit to the experimental results for 11461, 1Ved I and c(K) including theoretical uncertainties in the relationship of these measurements to p and R. As discussed here by (Nir 2001), there are three classes of CP violation which can be observed in the B-meson system; in B°B° oscillations, in decays ("direct CP violation"), and in the interference between oscillations and decay. In the Standard Model with p and 77 as fitted in Figure 22, the largest effects are expected in the third class, where the decay mode is a CP eigenstate. There are some modes where the prediction for the CP asymmetry is very insensitive to theoretical uncertainties, especially dN(B° J/01q) — dN(B° .1/11)10.) [sin 20 ± 0(1%)] sin(AmdAt) , (42) af dN(B° JIOK'D + dN(B° J/y)1q) 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.6 — P I •= 0.8 c(K°) with BK from lattice 0.6 0.4 - 0.2 i B-meson factory results 361 Figure 21. The hyperbolic band shows the relation of p and n to the measured value of ci where )3 has been defined in Figure 19: sin 20 = 112 ( 1 — p)2 Figure 22. Fit result for p and n. (43) 2u(1 — p) and At is the difference between the decay time into J/OK, and the decay time of other B in the B°B° pair. Before coming to the BABAR and Belle results for af ,I would like to discuss briefly the other two classes. 5.2 CP violation in B°B° oscillations The two mass and lifetime eigenstates in the neutral B system have been defined in equation (10). CP conservation in B°B° oscillations is equivalent to 1q/p1 = 1. If CP is violated, we may use the Kaon language for p and q, 1) — q en = p + q p 1+ Eg q 1 — e6 ' p q = 1 + 2Re (03 ) 1 + (44) Because there are several c parameters in Kaon physics, we choose the notation 03 for e in mixing. The imaginary part of e B is unobservable. Since the Standard Model expects Re (E B) to be small, as 1K is in K°K° oscillations, a useful convention is to take Im (03) = 0, giving Ip/q1 = 1 + 2Re (03). CP violation in oscillations shows up as a different number of B°B° and B°13° decay pairs. A straightforward calculation gives am At) = dN(B°B°)(At) — dN(B°B°)(At) 4 • Re (03) ( dN(B°B°)(At) + dN(B°B°)(At) 1+ 1(812 • (45) This asymmetry is independent of At. Table 2 gives results for Re (E B ), including a recent BABAR result (Brandt 2001) which uses 21 million T(4S) events. No CP violation is seen. This is not surprising since the Standard Model expectation is below the sensitivities reached by current experiments. 362 Klaus R Schubert Experiment Reference Method Sample ee/(1 + ieB12) OPAL (Ackerstaff 1998) e+e+le+e- 3M Z° 0.002 ± 0.007 + 0.003 OPAL (Abbiendi 1999) inclusive 4M Z° 0.001 ± 0.014 + 0.003 ALEPH (Barate 2001) combined 4M Z° -0.003 ± 0.007 CLEO (Jaffe 2001) e+ e+ 1 e - e- 10M T(4S) 0.004 ± 0.010 ± 0.002 BABAR (Brandt 2001) f+e+le-e- 21M T(4S) 0.0012 ± 0.0029 + 0.0036 Table 2. Searches for CP violation in B°B° oscillations. 5.3 CP violation in decays BABAR, Belle, and CLEO have searched for direct CP asymmetries in various decay modes. As explained in (Nir 2001), the decay amplitude A(B f) with B = B+ or B° has to be described by at least two diagrams, with different weak phases and with different strong phases, in order to show a direct CP asymmetry: r(T3 - r(B f) . acplf = r(r3 F(B -+ f (46) Table 3 shows the limits on asymmetry values for five decay modes. For details and references, see (Hamel de Monchenault 2001). Again, no indication for CP violation is seen, but this is not in contradiction with Standard Model expectations. Mode BABAR Belle CLEO K+71-- -0.19 ± 0.10 +0.04 + 0.19 - 0.17 -0.04 ± 0.16 K+ 71-° 0.00 ± 0.18 -0.06 + 0.22 - 0.20 -0.29 ± 0.23 KI,7i-+ -0.21 ± 0.18 +0.10 + 0.43 - 0.34 -0.18 ± 0.24 J7 :0K+ -0.009 ± 0.027 + 0.005 +0.018 ± 0.043 + 0.004 K*-y -0.035 ± 0.076 ± 0.012 +0.08 ± 0.13 ± 0.03 Table 3. Search for direct CP violation in hadronic and radiative B decay modes. The K*-y result is an average of K*+ and K*° 5.4 CP violation in interference This third class of CP violation requires decays of neutral B mesons into CP eigenstates, f) = nflf) with 7i f = +1 or - 1 . (47) Examples are f = 7+71--, where ?i f = +1, since the two pions are in a state with e = 0, P = +1, and f = JHuKcs.) , where ?i f = -1. The notation J/1/4(' is a short-hand: in reality B-meson factory results 363 KS and J/t/).K(5), are not CP eigenstates, and what are observed are the decay cascades: B° _4 jiff° with K° 7r+7r- Ro J/070 with K° 7+7r- (48) where the reconstructed state J/0(7+7-) with m7r, = mK is a strict CP eigenstate. The CP eigenvalue is —1 because CP(J/0) = +1, CP(7r7r) = +1, and = 1 between the J/0 and the 7r7r state is required by angular momentum conservation with J(B) = 0, J(J/L) = 1, and J(7r7r) = J(K) = 0. We now introduce the decay amplitudes (f = A1, (fITIT3°) = A1 (49) Using the notations of equations (13) and (14), the time-dependent decay rates dN and dN of initial flavour states 7,b B and Ow are given by dN(t) dN(t) = Using the abbreviated notation: e-nt q e-1H t — e-nt 2 2 (50) (51) (52) e--YHt Al A Al2 2 2 n e-nt 2 + qA1 2 Al r1 = pfi Af , and making the approximations FH = FL = F and lq/pl = 1, a straightforward calculation gives: dN( t) = . 2 • Ile e_ri Ati[i +1 1 + 1 r — ir.f1 rf) .f12 cos(AmdAt) 2Im ( 1 +11712 sin(AmdAt)] , dN(At) — ill.f2 l2 e-IrAti [I. 1 — In11 2 cos(AmdAt) + 2Irn (r1) sin(AmdAt)] . (53) For the case f = J/71)1q, this gives, with a precision of better than 1% (Nir 2001 : rf _ e2is, (54) and we obtain, with the proper normalisation of the decay amplitude A1: re-rlAtl dN (At)2 [1 sin(2/3) sin(AmdAt)] , re-PlAtI dN(At)2 [1 + sin(213) sin(AmdAt)] dN—dN a1(At) dN + dN = sin(20) sin(AmdAt) . (55) The integral of a1 over all At is zero. Therefore, a time-dependent measurement of the two rates is necessary in order to obtain sin 2/3. The experimental determination of sin 2,3 requires eight steps: 1 + Irf 12 1 + Irf 1 2 364 Klaus R Schubert 1. Reconstruct J/7/) mesons in their e+ e- and µ±1.1- decays. 2. Reconstruct neutral K mesons in their 7r*7-- decays. 3. Reconstruct neutral B mesons from the J///) and K candidates. 4. Partially reconstruct the second B meson of the pair in a flavour-specific way. This "tags" the flavour of the first B at the time of decay of the second B. 5. Reconstruct the z coordinates of the two decay vertices to obtain Ai for each event. 6. Determine the fraction w of wrong flavour tags. 7. Determine the vertex resolution function R(Ai - At) between reconstructed and true At values 8. Determine sin 2/3 from a fit to the two time distributions dIVAI) and dN,,,(Af). (Here dN„ = (1 - w) dN + w dN and similarly for dN„,). In the BABAR experiment (Aubert 2001a) with a sample of about 32 Million BB pairs, there are 380 reconstructed and tagged J/WIO events, including about 15% with a K 71-°71-° decay. Steps 1, 2, and 3 have already been discussed in the lifetime section. In addition to J/OK% BABAR has used other charmonium CP eigenstates in their sin 2/3 analysis, 0(2S)Kg. and X,' Kg. with 7if = -1, J/01(2 with 7t f = +1, as well as .//0K°* with K°* -> lics"r° which is a mixture of CP eigenstates with Ti f = -1 and +1, but with a well-determined effective value of 7i f = +0.65 + 0.07 (Aubert 2001e). 2 250 200 B4BAR > I rsi 75 I 0 Da J tiA, viol II( 50 fithgroitild 150 100 728 events purity 94% 273 ev. purity 51% 1116111hiladMili .1E" so 0 , 5200 5210 5220 5230 5240 5250 5260 5270 5280 5290 5300 Beam-Energy Substituted Mass (M.Vtc 2) -20 0 20 40 60 80 AE (Met) Figure 23. Mass distribution for all B Figure 24. Distribution of .,E*, see text, decays into CP eigenstates with a KS in for the B J 10K2 candidates in the the BABAR sin 2/3 measurement. The BABAR sin 2/3 measurement. dotted curve shows the background. The modes with n j = + 1 have r f = -e2i fi, and an asymmetry a f with the opposite sign. The mEs distribution of all events in modes with a Al is shown in Figure 23. The signal of the JNKY, mode is shown in Figure 24. The EMC and IFR. measure the direc-tions of K2 mesons, but not their energies. Using the m(B°) requirement to constrain the If°L energy, leaves only one kinematic constraint, E* =E* (J/ 11))+E* (K2)-111( T4S) /2, P = Nsignal Nsignal Nbackground (56) B-meson factory results 365 which is shown in Figure 24. The background in the KL sample is much higher than in the Ks sample. Background levels are often expressed as purities: The event numbers and purities in Figures 23 and 24 are those before flavour-tagging. Flavour-tagging is achieved by requiring a second B decay vertex with high momentum electrons, high momentum muons, charged Kaons, or other flavour-specific combinations determined using a neural net algorithm. In order to maximize the tagging rate, tags with only one track are accepted, and their decay coordinate z2 is given by the intersection of this track with an assumed B flight line. The neural net tags use poorly identified leptons, slow pions, and fast hadrons. They are grouped into two classes NT1 and NT2. Table 4 gives the tagging efficiency ci, the mistag fraction wi, and the quality factor Qi Tag category ez (%) '11) (%) Qi (%) Leptons 10.9 ± 0.3 8.9 ± 1.3 7.4 + 0.5 Kaons 35.8 + 0.5 17.6±1.0 15.0 + 0.9 NT1 7.8 + 0.3 22.0 ± 2.1 2.5 ± 0.4 NT2 13.8 ± 0.3 35.1 + 1.9 1.2 ± 0.3 Sum 68.4 ± 0.7 26.1 ± 1.2 Table 4. Flavour-tagging in the BABAR experiment, for each tag category i. The quality factors are defined as 2wi) 2 Q = E Qi (57) They determine the statistical error of the fit result: a(sin 20) 1.9 (58) where Nsig is the number of signal events before tagging. Table 5 gives the number of tagged events and their purity for the BABAR experiment (Aubert 2001a). Step 5, the determination of At with equation (4), is the same as in the lifetime and oscillation measurements. Step 6, the determination of the mistag fractions, uses the same sample of fully reconstructed neutral B mesons with identified flavour as in the oscillation measurement. The 9400 events of this "flavour sample" are tagged with the same four tagging categories as the "CP sample" . Since their flavour is known, the tags are easily classified as right or wrong. The results are given in Table 4. Step 7, the determination of the time resolution, is achieved by simultaneously fitting parametrized resolution functions Ri (At - At), which are different for each tagging cat-egory i and each decay mode, to both the flavour sample and the CP sample. This is (a) C.E 0.6 0.2 4 4 At/T 366 Klaus R Schubert Mode N P (%) J/01q(71-±7-) 316 98 JNI-Ccs'(71-°70) 64 94 7,b(2S)I1 67 98 Xei KS 33 97 J/01(*0 50 74 273 51 Sum 803 80 Table 5. Number and purity of tagged events in the BABAR experiment. Figure 25. (a) and (b) show the time dependences for the flavour sample with unmixed (solid curves) and mixed events (dashed curves). (c) and (d) show these curves for the CP sample with B° tags (solid) and B° tags (dashed). (a) and (c) are without At smearing and without mistags, (b) and (d) with smearing and mistags. illustrated in Figure 25. For each i, the observed L\T distributions can be described by dNi (Ai) = f Cr l Ail [1 + Di cos(AmdAt)]R,(Ot— At) dAt, dN2(AI) = f e-rl Atl [1 — Di cos(Arnd At)]Ri(Ai— At) dAt, (59) for the "unmixed" flavour sample events dN I with the opposite-flavour tag, and the "mixed" events dN2 with the same-flavour tag, and by dN3(AT) = f Cr l Ati [1 + D, sin 2[3 sin(Amd At)]R, — At) dAt, dN4 (At-) = f — Di sin 2) sin(AmdAt)I.R.,(A[— At)dAt, (60) B-meson factory results 367 60 40 20 0 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0 5 At (ps) Figure 26. Rates and asymmetry for the g modes in the BABAR measurement for the CP sample events with a B° and a B° tag, dN3 and dN4 , respectively. The "dilution" factors are Di = 1 — 2wi. Since the resolution functions Ri are the same for all four distributions, the fits to the high statistics samples 1 and 2, fix their shapes in the low statistics samples 3 and 4. This combined fit in step 7 is in practice performed together with steps 6 and 8, which determine the mistag fractions wi, and the value of sin 20. The results are sin 2i3 = 0.56 f 0.15 for the Ks modes, 0.70 f 0.34 for J//K L, 0.82 ± 1.00 for JNK°*, where the errors are statistical only. A combination of these results leads to sin 20 = 0.59 ± 0.14 + 0.05 . (61) The systematic error is dominated by the knowledge of the resolution function for Ks modes, and by the knowledge of the CP properties of the background for the K L mode. The rates and the asymmetry af for the Ks modes are shown in Figure 26. If the condition = 1 is not imposed in the fit, the data can be fitted to the expressions in equations (53). This fit, using all modes, gives: = 0.93 ± 0.09 ± 0.03 . (62) Hence there is no indication for direct CP violation in the decay modes used for the sin 2,(3 measurement. In the Belle experiment (Abe 2001), 31M BB events are analysed in a very similar way. The same CP eigenmodes are reconstructed as in BABAR, plus an additional mode, rich's, 200 • Data El BG 0100 0 • • . . 0.5 1 1.5 prs (GeV/c) 50 Lu 0 368 Klaus R Schubert Figure 27. Signal and background in the JIWK2 sample of Belle's sin 2/3 measurement. At (ps) Figure 28. The sin 2/3 result of Belle. (a) Rates for all tagged events; B° tags for = —1 modes are combined with B° tags for rt f = +1 and vice versa. (b) Asymmetry a f for modes with 71.1. = —1, (c) with ?i f = +1, (d) combined asymmetry. which has 7i f = —1. Figure 27 shows Belle's signal for the mode J/01-q, whose purity of 61% is slightly better than BABAR. Another difference is in the tagging strategy, where Belle tries to tag each event, and uses all 1137 reconstructed events of their CP sample in the final fit. Their "tagging efficiency" is e = 1 compared to c = 0.68 in BABAR, but their overall tagging quality is Q = 0.270 ± 0.012, which is very similar to Q = 0.261 + 0.012 in BABAR. Belles's final fit, combined all modes, is shown in Figure 28. Their result is sin 201 = 0.99 + 0.14 + 0.06 , (63) where 01 is the Japanese name for 0. 0 0.2 0.4 B-meson factory results 369 CDF (2000) Belle (Osaka,6.2/fb) Bal3ar (Osaka,Wfb) Belle (11.2M 118) Raliar (23M BB) Belle (31.3M BB) BaBar (32M BB) Average (4 expt.) 0• -0 79+41" .44 0 8 4 1:3)42 ±0.16 4).4541-04344 +11.0 (41 7) 0.12± 0.37±0.09 0.58"J2 +0.°9 -034 - 0.10 0.34± 0.20±0.05 0.99± 0.14±0.06 0.59± 0.14±0.05 0.79± 0.10 -0.5 0 05 1 1.5 2 sin201 Figure 29. Compilation and average of sin 20 measurements. 0.8 0.6 0.4 0.2 0.1 .0.8 -0.4 -0.2 0 0.2 0.4 0.6 0.8 - Figure 30. Fit result from Figure 22 and lines of constant likelihood (max, +la, +20-) for sin 2/3. Figure 31. Final fit to all observations in-cluding sin 2/3. Figure 29 compiles all results for sin 20. The world average is sin 213 = 0.79 ± 0.10 , (64) which is in very good agreement with the ph region favoured by the measurements of I Vubl, and e(K°), as shown in Figure 30. A common fit to 1Vt.d1, e(K°), and sin 2/3 gives the region in Figure 31. With one-standard-deviation errors, this region has p = 0.22+ 0.12 , = 0.34 + 0.06 , J = (2.8 ± 0.6)x10-5 (65) 370 Klaus R Schubert a) VU/cil vub /w/ t) W Vtd d Vtb* u d b) d d d Figure 32. Tree (a) and penguin (b) diagram for 71+ and 7i — decays of B mesons. BABAR has also performed a first measurement (Aubert 2001g) of the angle a in the Unitarity Triangle (see Figure 19 for its definition). sin 2a is obtained from the time dependence of tagged 7+7- decays. If only the tree diagram of Figure 32a contributes to these decays, we would have in complete analogy to Equations 54 and 55 e2za = 1, a70- 0- = sin 2a sin(AmdAt) , (66) where a,r+,- is the asymmetry between dN, the number of 7T+7r — decays with B° tags, and dN, the number with B° tags. However, there is a contribution from the penguin diagram in Figure 32b, which has a different weak and a different strong phase. This leads to direct CP violation, 1r,+,-1 # 1, in addition to CP violation in the interference between oscillations and decays. This means Im ) sin 2a, and instead of equations (55), we have to use equations (53) for the expected time dependence, leading to the asymmetry: a,+,- (At) = cos(AmdAt) + sin(AmdAt), 1 — Ir,c+7,- 1 2 1 + Ir/r+7- 1 2 ' • = 2Im (r7,+„- ) .n(2aeff ). 1 +1r,+--1 = (67) 2 Using the same 32 M BB events as in their sin 20 analysis, BABAR finds the preliminary result • = —0.251:4 + 0.14 , • = 0.031:N + 0.11 . (68) There have been theoretical attempts (Beneke 2001, Luo 2001) to relate sin(2aeff ) to sin(2a), using additional measurements of KT and 7r7r decay rates of neutral B mesons. The central value of the BABAR result, sin(2aeff ) = 0.03, gives L eff = 1° or 89°, which they translate into a close to zero or equal to 75°. The first value does not agree with the preferred plri region from all our previous discussions, but the second one agrees very well. Using the one-standard-deviation errors of the BABAR result in Equation 68 and the aeff to a translation curve in (Luo 2001): a = (75 ± 18)°. (69) This result is drawn with a central and two one-standard-deviation circles in Figure 33. There are no 2a contours because the BABAR result still allows any value of a within two standard deviations. • - t.1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 0.8 0.6 0.4 0.2 B-meson factory results 371 Figure 33. Contours of constant likelihood (max, +loj, from the BABAR sin 2aeff measurement. 6 Conclusion CP violation in the B meson sector has been firmly established in 2001 by two experi-ments, BABAR at SLAC and Belle at KEK. Their observation of a1 0 in charmoniurn final states, has a combined significance of more than seven standard deviations. It is com-patible with the Standard Model explanation of CP violation in the Higgs mechanism. It is also compatible with the CKM matrix parameter values of p and ri as determined from measurements of Mad, 114d1, and e(K). With the combined efforts of theorists and experimentalists, the next years will show how precisely the measurements will determine p and 7/, and if the five independent con-straints on p and n from nbl, IVtdl, e(K), /3, and a will establish a consistent description, or will falsify the assumption that all weak processes including CP violation are well described by the Standard Model. Acknowledgments I thank K Peach and all his co-organizers of SUSSP55 for inviting me to this excellent School and for their friendly hospitality. 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The first was the un-expectedly large lifetime. The second was the mass difference between the two mass eigenstates of the Bd meson system which is about 100 times larger than the mass dif-ference in the neutral K meson system. From the first observation, we learned that the mixing between the second and third families is much smaller than between the first and second families. (The mixing between the first and third families is even more suppressed.) The second observation taught us that the mass of the top quark is much larger than had been anticipated at that time. The main goal of B physics is to study the structure of quark mixing and its role in CP violation. In the Standard Model CP violation can be very naturally accommodated through the complex quark mixing matrix (Kobayashi and Maskawa 1972) defined by four parameters (Nir 2001). All current observations of CP violating phenomena in particle physics are in full agreement with the Standard Model calculations. However, there are still some reasons to speculate about CP violation generated by physics beyond the Stan-dard Model. Firstly, the Standard Model alone cannot account for the large asymmetry between matter and antimatter observed in our universe (Shaposhnikov 1986). Secondly. various extensions to the Standard Model introduce new sources of CP violation. Since CP violation is expected in many B meson decay modes, and the Standard Model can make precise predictions for some of these decay modes, the B meson system appears to be a very attractive place to look for evidence for physics beyond the Standard Model. In the presence of new physics, some assumptions made to extract the Cabibbo-Kobayashi-Maskawa (CKM) parameters are no longer valid. Indeed, the consistency of the CKM picture with current observations could be accidentally due to numerical can-cellation between various effects from new physics. Therefore, it is essential to develop a strategy which allows the new physics and the Standard Model contributions to be clearly disentangled. Then the CKM parameters can be determined in a model-independent way. This is important since we hope that physics at higher energy scale will one day be able to explain the family structure, by deriving the CKM parameters. DOI: 10.1201/9780429187056-10374 Tatsuya Nakada In this article, we discuss how B meson decays can be explored at the LHC in order to obtain a better understanding of the CKM picture, and to look for physics beyond the Standard Model. 2 The CKM picture 2.1 The CKM matrix In the Standard Model, CP violation is naturally introduced by a 3 x 3 complex quark mixing matrix, VCKM, expressed as VCKM = Vud Vcd Vtd Vus Vcs Vis Vub Vcb) • Vtb The charged current of the weak interaction is then proportional to /Pt (1 — ) 14,1Alifit where UL and DL are the left-handed quark operators for the charge 2/3 up-type and the charge —1/3 down-type quarks respectively: UL = (U L CL t L DL = (C L sL) b L • The matrix VCKM is unitary, WV = 1. One of the unitarity relations is: VudVu*b + VcdVel' = 0 • (1) This is illustrated in the complex plane in Figure 1(a). The unitarity condition can be illustrated more easily by translating VedVe't, and Vidl:t to form a closed triangle, as shown in Figure 1(b). The three angles of the triangle, a, /3 and -y (also known as 02, ci and 03 respectively) can be defined as: lid Vtb = tan Vu d Vu*b) = 7r tan_I (Vidl4t) -y = tana' V./ Vi7b) ( 2) cd I cb a) b) Im Vud Vub* Re Vud Vcb* Vtd VII)* Figure 1. (a) One CKM matrix unitarity condition: VudV:b + VedVct + VtdVd: = 0. (b) Elements translated to form a triangle drawn in the complex plane. Vud Vub Vud Vub* Vtd Vtb* * Vcd Vcb a) b) Im Re c) Vtd Vtb* 0 0 1 = VCKM R23 = cos Oi •. = R23 X R13 1 0 0 0 c23 823 0 —523 C23 ) X R12 R13 = C13 ( 0 — si3e46 0 1 0 P i6) , Ci3_ 0 C13 where R12 = —.0 C_ 12 S12 0 e12 0 with sij = sin Oij and B physics at LHC 375 It should be noted that a redefinition of the quark phases results in a rotation of the triangle in which the three angles, a, 0 and -y remain invariant. Violation of the unitarity condition given by Equation (1) is expressed in a graphical form in Figure 2(a) and (b), where the three sides given by VadVu*b, 'Vcd1/,': and VtdVt*b' do not form a closed triangle. Note that a, 0 and 7 defined by Equation (2) still satisfy: a + 0+7=7. If one forms the closed triangle from the length of the three sides, 11/udV,inl, 11/cdI b and Vtb I, the three angles of this triangle, a', /3' and -y' defined in Figure 2(c), are not identical to a, 0 and 7. Therefore, a test of unitarity can be made by comparing the angles defined by the length of the three sides, and the angles measured by CP violating effects (as explained later). Figure 2. (a) and (b), are similar to Figure 1, but unitarity is violated. In (e) a closed triangle is formed using the three sides. A unitary 3 x 3 matrix can be parameterized by four parameters. One possible choice (Particle Data Group 2000) is to use three angles, 812, 823, 013 and one complex phase, S. The standard parameterization for the CKM matrix is then given by: 376 Tatsuya Nakada A parameterization reflecting the observed pattern of the CKM matrix was proposed by Wolfenstein (Wolfenstein 1983). This introduces the parameters: S23 Si3 COS 6 A= P= °12 512523 and expands the matrix elements in powers of A. Neglecting terms proportional to An where n > 5, this gives: 1 — A2/2 ( AA3 (p — i 77) VCKM ti —A — i A2 A5n 1 — A2 /2 AA2 AA3 (1 — p — i 4 —AA2 — i AA'n 1 where /5 and F.) are given by P = p(1 — A2 /2) and = 77(1 — A2 /2). The parameter A is known from light hadron decays to be 0.221 + 0.002 (Particle Data Group, 2000). As seen from Equation 3, the first 2 x 2 sub-matrix is almost unitary: 17„,117,d +14,sVe*, = iA2 A57 N 0 Vudiju,, + VedVe*, = —iA2 A5i1 N 0 and Vud12 Ivusi2 = A4 /4 1 Ycd12 O/c,' = A4/4 With the parameterization given in Equation 3, the imaginary part of Vett becomes negligi-ble in the unitarity relation given by Equation (1), and the phases of the matrix elements are: arg Vtd = — 13, arg Vub = arg Vud = arg Vcb = arg Vtb = 0, arg Vcd = 7r * Vus and 175 are also real, but the imaginary part of 17t, cannot be completely ignored: arg 14, = 8-y + The angles 13, -y and 8-y are functions of p, n and A: = tan-11 p , 7 = tan-1 Ili , =- tan-' A211. (4) — 2.2 Extraction of the CKM parameters From a measurement of IVeb l, the Wolfenstein parameter A can be determined. The matrix element VcbI is extracted from semileptonic (and hadronic) decays of B mesons into charmed meson final states (Stone 2001). In exclusive B meson decays, a description based on the quark tree level process b C is obscured by soft hadronic interactions. This has to be taken into account in order to extract 11:ebi from the data. Significant improvements in understanding these hadronic effects have been made both in theory and experiment. Further progress will come using the high statistics data samples from the BABAR and BELLE experiments (Schubert 2001). The semileptonic decays of B mesons to final states containing only the light u and d quarks are used for the extraction of IVW (Stone 2001). They are generated by the = 512, s13 sin 6 77 = 512523 (3) t c u — _ t C U d b b d B°b 4 d b B° B° B° B physics at LHC 377 tree level process b u+ W-. Unlike the determination of hadronic decays cannot be used since sizeable contributions from the penguin processes, b -+ s and h d, arc present in the decay amplitudes. It is much more difficult to evaluate the effects of strong interactions for 1V„1,1 than for 1V61, and the error on the current value of 1Vubl is totally dominated by the theoretical uncertainties. This will remain the case for some time. While I Kbi and 11461 are determined from decays with tree processes, I icano n bi in accessed indirectly through loop processes such as B°-B° oscillations. The oscillation t the Standard Model is described by the well known box diagrams shown in Figure 3. Due to the large top quark mass and the structure of the CKM matrix, only the top quark Figure 3. Box diagrams describing B°-B° oscillation. contribution needs to be considered in the loop. Neglecting the absorptive part of the box diagrams, the oscillation amplitude for B° -4 B° is calculated to be (Hagelin 1981): G2F f lid B Bd TTI E3,774 1121 = 1 1 1 d d S (X t)(VtdVt'6)2 (5) 1272 where G F, mw and mBd are the Fermi coupling constant, W boson mass and Bd mass. The function S(xt ) is determined from the mass ratio, xt = (mt /mw)2, where mt is the top quark mass, and the QCD correction factor, ri Bd , can be calculated reliably with per-turbation methods. The B meson decay constant, f Bd , and the bag parameter, BBd , have never been measured directly. The decay constant f Bd is given by the transition matrix element between B° and the hadronic vacuum state. Thus it could be experimentally ob- tained from the branching fraction for the leptonic decay, B± T1 VT shown in Figure 4, assuming that IVizb l is known. The decay constants for 13' and B° are expected to be very 13. B+ Figure 4. Diagram for the B+ 7+11, decay. similar. The bag parameter takes into account the difference between the hadronic vac-uum and the actual hadronic states virtually present in the B°-B° oscillation processes. Theoretical estimates exist with large uncertainties due to the difficulties in evaluating the effects of the non-perturbative soft hadronic interactions. The most promising theo-retical approach to obtain f Bd and Bga is the QCD lattice calculation discussed in these proceedings (Davies 2001). The absorptive part in the B°-B° oscillation is due to c and u quarks. It is calculated to be very small compared to the dispersive part and can indeed be ignored. With this 378 Tatsuya Nakada approximation, the mass difference between the two mass eigenstates, Bd and Bd can be derived, Arnd d -mBd = 21 /21I, and the decay width difference between them is negligible: Old = FIB'd — FlBid 0. There is no CP violation in the B°-B° oscillations in this approximation. Note that the phase of the oscillation amplitude is given by the phase 17,2d = -20. The mass difference Auld is experimentally measured as the frequency of B° los- cillations. From the known structure of the CKM matrix, we have Vtbl = 1 with very high accuracy. Therefore, IVtdl can be obtained from Amd. The error on I Vtd1 is totally dominated by the theoretical uncertainties on f i3Vr3d. From the measurements of Vub, Vet, and Vtd, the quantities: \fi) + = 1 1Vubl A2) 1.7,b1 and V(1 75)2 + ,,,c2 1 1-17td1 A 11/eb I can be drawn as circles around (0, 0) and (1, 0) in the fi-i) plane respectively. There are two possible solutions where these circles intersect, but the one with rt > 0 is favoured from the analysis of CP violation in the K°-K° oscillations. The mass difference in the Bs system, Ams, can be evaluated by calculating the Bs°-B° oscillation amplitude using box diagrams in a similar way to the Bd system. A measurement of Am, determines 117es Vt'b'12, which allows: 1 1 -Vtd1 (8) to be used instead of Equation (7). In the ratio: 1 Vtd1 1 Aind X N/mBilBs X \TB3.1 es „ — A ts I A V3Arnd OnadilBd VEad iBd the assumption 1141,1 = 1 is no longer necessary. While the theoretical uncertainty in f B a is quite considerable, the ratio between these quantities for the B, and Bd mesons is theoretically much better understood. Therefore, Equation (8) will have a significantly smaller error than Equation (7). Unfortunately, only an experimental lower limit, is avail-able for Am, at the moment. We now consider CP violation for decay final states which can be produced by both B° and B°. These can be CP eigenstates such as J/iP Ks or non-CP eigenstates such as D*+71-- and its CP-conjugate state D*-7F+. If CP violation is present neither in the oscillation nor the decay amplitudes, it is well known that the only signature of CP violation that can appear is through phases: sin(00 + O f ) sin (Amdt) (6) (7) B physics at LHC 379 in the time-dependent decay rates of initial B° and B°, where 00 is the phase of the B°-B° oscillation amplitude, and O f is the phase of the ratio of the instantaneous decay amplitudes of B° and B° into the particular final state. For the J/11) Ks final state, the b c + W-tree process dominates, but there can also be a b s penguin process (Figure 5). The phase of the tree process is given by Veb, and is 0 in the Wolfenstein approximation. The penguin process is described by a virtual top quark loop with CKM matrix elements, VtbVts*, which has only a small phase -6-y. Therefore, O f = 0, with a theoretical uncertainty less than a few percent. b d b 4 ti Bo d d d Figure 5. Quark diagrams for the B° -# J/11) Ks decay. The final state D* +7r- can be generated by the tree process b c+ W-, followed by W- -+ Ttd, for the B° decay or the doubly Cabibbo-suppressed tree process b + W+ followed by W+ cd for the B° decay, as seen from Figure 6. From the relevant CKM matrix elements, it follows that Of = 'y. Note that only the tree processes contribute. ' S u d B° —B° 1 4 b 7 b u d d 71 d Figure 6. Quark diagrams for the B° and B° D*+7r- decays. The phase of the oscillation amplitude is given by cbo = 213 as discussed before. There-fore the CP violation signatures in B° and B° decays are sin(2/3) sin (Arndt) for J/0 Ks and sin(213 +'y) sin (Arndt) for D*+ 71-- and D*- 7+ . For the sin(213) measurement from J/0 Ks, the average of the recent results from BABAR and BELLE gives (Schubert 2001): sin 2/3 = 0.79 + 0.17 where the error is scaled following the recipe of the Particle Data Group. It is unlikely that any definitive measurement of sin(2fi + 'y), using B D(*±)71-(T) , will be done by the current generation of experiments. 380 Tatsuya Nakada Analogous decays for the Bs° meson system are J/0 0 and J/0 i, which have Of = 0, and Ds+K- which has Of = y. Since the B,()-B°, oscillation phase is —26-y as discussed before, the CP violation signatures would be: sin(-2 5y)sin (Amst) for J/0 0 (or 4077) and sin( —2(5'y + -y) sin (Amst) for DS K and DS K+. No definitive measurements on these quantities will be available from the current gener-ation of experiments. In Figure 7, we summarise the region of /5 and 'F./ allowed by the current data on (AiVeb1), Arnd , and sin(2/3), as well as the one-sided constraint on Amd/Ains from the lower limit on Ams. All the data are interpreted within the framework of the Standard Model, so the constraints may be modified by new physics as discussed earlier. There is a common overlapping region indicating that the data are consistent with the CKM picture, and no sign of new physics is visible within the errors. Note that the sin(20) measurement gives two allowed regions in the 75-11 plane defined in Figure 7, and only the one consistent with the other constraints is drawn. o.8 0.6 0.4 0.2 0 -1 -0.5 0 f3 0.5 Figure 7. Allowed region in the ji-rti plane from1V,,,b Amd, and sin(20) measure- ments, and from the upper limit on Arns. 3 Possible CKM picture in 2006 Let us now speculate about the CKM landscape in 2006. Due to progress in theoretical understanding of the hadronic effect, strongly helped by the large statistics and high quality data from BABAR and BELLE, 117„b /1761 might become known with a relative error of +10%. Once Ants is measured by CDF, 114,11/(A11431) will become known with a relative error of +7%. From CP violation in B° and B° decays to JIW Ks, the combined results from BABAR, BELLE, CDF and DO might yield a result of sin(2f3) with an error as small as 0.03, which is still statistically limited. Estimating the progress in other CP violation channels is more difficult since they have various theoretical uncertainties, and future improvements are difficult to evaluate. Assuming that the currently preferred values of p and ri remain unchanged, the situation expected in 2006 is illustrated in Figure 8. tlmd / J/WKs Ivubl Ivcbl B physics at LHC 381 2006 -1.0 -0.5 Figure 8. Possible situation of the allowed region in the 15-i)" plane in 2006 4 Presence of New Physics Once we allow the presence of physics beyond the Standard Model, the determination of the CKM matrix elements becomes much more complicated. Most of the extensions of the Standard Model introduce new heavy particles. The contributions of these particles to the tree processes should be negligible, but for B°-B° and Bs°-B° oscillations, their contribution to the box processes could be sizeable. New heavy particles could also appear in the penguin processes, but the effects should be smaller than in the box processes, due to the different dependence on the masses of the particles appearing in the loops. In the presence of new physics, the phases for the oscillation amplitudes are modified: og =20+ gp and = -28-y +SNP where 01,/,p and gip are new phases. The new physics has very little effect on the decay amplitude if this is generated by tree processes. Therefore, we assume that the phases of the amplitudes for decay modes such as B° -> J/W Ks, B° D*-F7r-, -> ..1/7,b0 and Bs° D,+,K- remain unchanged. For the CP violation signatures in B° and B° decays, we would then have sin(2/3 + gip) sin Arndt for J/0 Ks (9) and sin(2/3 + SNP y) sin Arndt for D*+71-- and D*-7+. The value of is modified, but it is possible to use both measurements to extract the angle, -y, in a theoretically clean way, even if new physics is present. Similarly for the B° system: sin(-25y + SNP) sin (Amst) for J/0 0 (or J/077) (10) and sin(-2,5-y + SNP + y) sin (Amst) for Ds K- and D; 1.0 0.8 0.6 0.4 tl 0.2 0.0 0.0 15 05 1.0 Amd t Am, J./ 0(s WA!! Ill„b1 F Ad VNeve Physics /2 382 Tatsuya Nakada the value of 87 is modified, but 7 can be cleanly extracted by combining the two CP violation measurements. Once y is determined, p and 7/ are given by: A IVad 1 1Vutd 1 11701 P = COS y, = sin -y A IVebl and the unmodified values of /3 and 67 can be obtained from Equation (4). Hence all the parameters of the CKM matrix can be completely determined even if new physics exists. The contributions from new physics are then determined from Equations (9) and (10). The allowed region in jrrl plane from the CP asymmetry measurement with B° and B° JIW Ks decays is no longer valid. What is measured from the asymmetry is not /3, but /3 + OG/2. Similarly, the region given by the Amd/Anr, measurements is no longer valid since there could be sizeable contributions to both the Am from new physics. The only valid region in Figure 8 is that from the 117„b1 /(AlVebi) measurement, and the apparent consistency seen in Figure 8 could be accidental. Once y is measured in the theoretically clean way explained above, the situation will become clear. Figure 9 illustrates a possible situation where 7 is measured with an accuracy of +5°, and is not consistent with the mixing and CP asymmetry constraints. In this case, the measurements of y reveal the presence of new physics contributions. Bd —4 .T/wKs + DIE B, .1./v4) + D,K —1.0 —0.5 0.0 05 1.0 Figure 9. Possible situation in the 15-1) plane in 2006 in the presence of new physics contributions In the discussion above, we have made an assumption that new physics does not contribute to the decay amplitude. This may not be strictly true for the B° —> ,J K s and Bs° -+ JItY (1) decays, since these decays receive a small contribution from the b -# s penguin process. Whether new physics can appear in the penguin process can be examined by studying CP violation in B° and B° decays into OKs. In the Standard Model the decay amplitude is dominated by a loop with the virtual top quark, so the phase is given by and the CP violation signature is: sin(2/3 + - 26-y + p) sin (Arndt) , (11) where r/SI'cp is the additional phase introduced in the b s penguin process by new physics. By comparing Equations (9) and (11), it is possible to determine whether new physics contributes significantly to penguin processes. 1.0 0.8 0.6 0.4 0.2 0.0 B physics at LHC 383 New physics contributions to the b d penguin processes can be studied from CP violation in B° and B° decaying into K*°K*° or Or° where only the b d penguin process contributes in the Standard Model. Independent studies of b s and b —> d penguins can be made with the B, system. In conclusion, a theoretically clean and model-independent determination of the pa-rameters of the CKM matrix is possible, even if there are new physics contributions to CP violation observables. Once the CKM parameters are determined in a model independent way, the possible existence of new physics can be cleanly established. 5 B experiments at the LHC 5.1 General considerations The goal of B physics in the LHC era (Ahmadov et al, 2000) is to determine the CKM parameters in a model-independent way and to isolate the effect of new physics so that its characteristics can be identified. This calls for an high statistics experiment capable of studying CP violation with both B1,31 and Bs° systems decaying into various hadronic final states. The production cross section for bb quark pairs at the LHC energy is estimated to be approximately 5001ib, which is far larger than at existing machines. The fraction of events with b quarks, abao-inelastic, is about 6 x 10-3, which is similar to the fraction of charm events in present fixed-target charm experiments. Thus, the LHC appears to be a very promising place to perform high precision CP violation measurements in B meson decays. At the LHC, B,, Bs , Be and b-baryons are abundantly produced, in addition to B° and B°. In order to exploit the full potential of the LHC, the following capabilities are needed. 1. A trigger sensitive to both leptonic and hadronic final states. 2. A particle identification system capable of identifying p, K, x, pt and e within the required momentum range. 3. A vertex detector able to reconstruct primary and B vertices very precisely. 4. A tracking system with good momentum resolution. 5. An electromagnetic calorimeter capable of reconstructing r° decays. 5.2 The ATLAS and CMS experiments ATLAS (Armstrong 1994) and CMS (Bayatian 1994) are two general purpose collider detectors designed to perform high pT physics at the LHC, including studies of the top quark, detection of the Higgs boson, and searches for supersymmetric particles. They detect bb quark pairs in the central region of the pp interactions. As already demonstrated by CDF, such general purpose collider experiments can cleanly reconstruct B meson final states containing lepton pairs, such as J/0 Ks, J/00 and f+e-. Both ATLAS and CMS 384 Tatsuya Nakada have excellent muon and electron detection capabilities, and a good vertex resolution, allowing them to collect a high statistics sample of such decays. However, their first level trigger on a high pT lepton is not sensitive to purely hadronic final states. Hadronic decay modes can only be triggered by a semileptonic decay of the other b quark, which has a low efficiency due to the relatively small branching fraction for the semileptonic decay. The two experiments also have no p/K/7r separation capability in the relevant momentum range, although the energy loss, dE/dx, can be used for particle identification at low momenta. In particular they cannot separate kaons from pions in the high momentum region relevant for two-body decay modes of B mesons such as 7F+ 71- and D.K. For these reasons, ATLAS and CMS will not be able to study CP violation with all the final states necessary to perform model-independent analysis. 5.3 The LHCb experiment The LHCb experiment is a single arm forward spectrometer, which covers the region of pp collisions which is not looked at by the two general purpose detectors. The LHCb detector (Amato 1998) is designed to have a trigger that is equally efficient for leptons and hadrons. It will be able to exploit the full b physics potential of the LHC at a much lower luminosity (2 x 1032cm-2s-1) than the nominal luminosity (1034cm-2s-1), and will therefore be able to perform its full physics programme from the beginning of LHC operation. Figure 10. The LHCb spectrometer placed in the LHC Intersection-8 experimental area. B physics at LHC 385 Figure 11. Polar angles of the b- and b-hadrons The detector layout is shown in Figure 10. It resembles a typical fixed target spec-trometer. A vertex detector is placed in "Roman pots" around the interaction region. It is followed by a tracking system, RICH counters with aerogel and gas radiators, a large-gap dipole magnet, a calorimeter system, and a muon system. An existing LEP experimental area will be reused to install the detector. The interaction point is shifted by 11m from the centre of the experimental hall, in order to accommodate the detector elements without extra excavation. The choice of a single arm detector geometry is based on the fact that both the b- and b-hadrons are predominantly produced in the same forward (or backward) cone at high energies. This is demonstrated in Figure 11 where the polar angles of the b- and b-hadrons are shown in events generated with the PYTHIA simulation programme. The polar angle is defined with respect to the beam axis in the pp center-of-mass system. Detecting both b- and b-hadron at the same time is essential for the flavour tag. Further advantages of the forward geometry are: . The b-hadrons produced are faster than those in the central region. Their average momentum is about 80GeV/c, corresponding to a mean decay length of Therefore, a good decay time resolution can be obtained for reconstructed B-mesons. . The spectrometer can be built in an open geometry with an interaction region which is not surrounded by all the detector elements. This allows the vertex detector system to be built with sensors which can be extracted away from the beam during the injection. During data taking, the sensors are positioned closer to the beam in order to achieve a good vertex resolution. . In the forward region, momenta are mainly carried by the longitudinal components. Therefore, the threshold value for the pt trigger can be set low for electrons, muons and hadrons without being constrained by the detector requirements. This makes the pt trigger more efficient than in the central region. . The momentum range required for particle identification is well matched to the Ring Imaging Cherenkov Counters, and the required size for the counters remains affordable. . The open geometry allows easy installation, maintenance and possible upgrades. 386 Ta tsuya Nakada 5.4 The LHCb detector Beam pipe A large vacuum tank with a length of 1.7m and a diameter of lm is placed around the interaction point to accommodate the vertex detector system with its retraction mechan-ics. It has a 2mm Al forward exit window over the full detector acceptance. This part is followed by two conical beam pipe sections; the first part is 1.4m long with a 25mrad opening angle, and the second part is 16m long with a l0mrad opening angle. Except for bellows, flanges and the last 6.3m of the 10mrad cone, the beam pipe is made from Al-Be alloy in order to reduce the radiation length. This is essential to minimize the occupancies of the tracking and RICH systems, and to improve the detection efficiency for photons and electrons. Magnet A dipole magnet with an Al conductor provides a field integral of 4 Tm. The polarity of the field can be changed to reduce systematic errors in the CP-violation measurements that could result from a left-right asymmetry of the detector. The two pole faces form a wedge shape following the spectrometer acceptance, in order to reduce the power consumption. Vertex locator (VELO) A total of 27 stations of silicon microstrip detectors are placed perpendicular to the beam of which 25 stations are used as a vertex detector system. The remaining two stations are used for detecting bunch crossings with more than one pp interaction as a part of the Level-0 trigger (pile-up veto counters). All the stations are split into two halves, covering left and right 180° sections. Each vertex detector station consists of two sensor planes with different strip layouts, one for r and the other for 0 measurements. The pile-up veto counters have only r measurement sensors. The closest distance between the active silicon area and the beam is 8mm. In order to cope with the high radiation dose expected so close to the beam, it is planned to use n-on-n silicon sensors. The silicon detectors are placed in Roman pots surrounded by 250 pm thick aluminium foil, which acts as a shield against RF pickup from the circulating beam bunches. To avoid mechanical collapse, a secondary vacuum is maintained inside the Roman pots. During injection and acceleration, the Roman pot system will be moved away from the beam to avoid interference with the machine operation and accidental irradiation of the detectors. For each bunch crossing, i.e. every 25ns, the signals are read-out and stored in analogue pipeline buffers. Tracking system Because of the high particle density close to the beam pipe, the LHCb tracking detector is split into inner and outer systems. The boundary between the two is chosen so that the occupancy of the outer tracker does not exceed 15%. The outer tracking system uses drift chambers based on a straw cell structure. The straws are made by winding 5mm diameter carbon-loaded Kapton foil around a central anode wire. The drift-time is sampled over 5Ons, i.e.every two bunch crossings. The inner tracking system is made from single sided p-on-n Si strip detectors with a B physics at LHC 387 strip pitch of ,-200//m. Since the sensitive regions of the Si sensor are several centimetres away from the beam the problem of radiation damage is less severe than for the VELO detectors. The RICH detectors The LHCb detector has two Ring Imaging Cerenkov (RICH) systems, with three different radiators, in order to cover the required momentum range, 1 100GeV/c. The first RICH uses aerogel and C4Fio gas as radiators. The second RICH, uses CF4 gas as a radiator. It is placed after the magnet, and is responsible for identifying high momentum particles. In both RICHs the Cherenkov light is reflected by mirrors and detected with planes of Hybrid Photon Detectors (HPD's) placed outside the spectrometer acceptance. Calorimeters The calorimeter system consists of a preshower detector followed by electromagnetic and hadronic calorimeters. It also serves as the initial part of the muon filter system. The cells of the preshower detector are made from two scintillator plates sandwiching 14mm-thick lead plates. The cell size of the preshower detector is matched to the module size of the electromagnetic calorimeter. For the electromagnetic part a Shashlik calorimeter is used since a modest energy resolution is required. The hadron calorimeter is based on a scintillating tile design similar to that used in the ATLAS experiment. Muon detectors For the muon stations Resistive Plate Chambers are used in the region where the charged particle rate is below 1kHz/cm2. For the region with a charged particle rate from 1kHz/cm2 to 100kHz/cm2, Multi Wire Proportional Chambers (MWPC's) are used. For the small region of the first muon station closed to the beam pipe, where the charged particle rate exceeds 100kHz/cm2, triple-GEM chambers or MWPC's with asymmetric gas gaps are being considered. Trigger The LHCb trigger is divided into four decision levels. The Level-0 decision is based on high-pT hadrons, electrons or photons found in the calorimeter system, or muons found in the muon system. Information on these candidates are sent to the Level-0 Decision Unit. The number of primary vertex candidates is determined using the pile-up veto counters in the Vertex Locator and also sent to the Level-0 Decision Unit. Based on all the information, the Level-0 Decision Unit makes an overall Level-0 decision. Events with multiple pp interactions are discarded, since rejection of background is more difficult in those events. The Level-0 trigger provides a modest reduction of minimum bias events by a factor of ,-10. In the Level-1 trigger, data from the vertex detector is used to select events with multiple vertices. In addition, the high-pT candidates used in Level-0 can be combined with the tracks having large impact parameter in the Vertex Locator. This provides a further reduction of —,25 for minimum bias events. After a positive decision of the Level-1 trigger, all the data is read out into an event buffer. At Level-2, a further enhancement of events with b-hadrons can be achieved by com-bining different detector components, e.g. by adding momentum information from the main tracking system to the impact parameter calculation with the Vertex Locator. At 388 Ta tsuya Nakada Level-3, a further decision is made after fully reconstructing the b decays. The LHCb trigger system is designed to cope with the rather small abi,/ainelastIc of x 10-3 at the LHC energies, while still maintaining a high efficiency for events with b hadrons. The strategy is to spread the suppression factors evenly and not to rely on any single trigger selection, in particular at early levels where available information from the detector is limited. This is reflected in the modest suppression factors of 10 and 25 for the ordinary pp interaction events at Level-0 and Level-1, respectively. Simulations of the trigger performance can be relied upon for such a modest suppression. By not heavily relaying on a particular selection criterion, the trigger system is flexible and can be readjusted to the actual running conditions of the experiment. 5.5 Performance of the LHCb detector The benefit of having particle identification and good invariant mass resolution can be demonstrated by reconstructing B3 -* Ds+K- decays. The main background to this decay mode comes from the B3 —> D9+71-- decays which are used to study B,-B, oscillations. Compared to this decay mode, the branching fraction of B3 -+ Do+,1C- is suppressed by a factor of 1/A2 ti 20. The two decay modes have an identical decay topology, and can only be distinguished by invariant mass and particle identification. It should be noted that the momenta of the K- and 71-- from these decay modes are large since they are two-body decays. 1.1111 DsK signal D Tr background 16000 00 Without RICH 0 52 5.3 51 Figure 12. The LHCb simulation results for the reconstructed invariant mass distribu-tions for Ds+.1f - combinations without and with particle identification using RICH. Figure 12 shows the reconstructed invariant mass distributions expected with the LHCb detector for Ds+K- combinations without and with particle identification using RICH. The combination of the good mass resolution, a = 11MeV/c2, and RICH par- B physics at LHC 389 tide identification, mean that the background from B3 —* Ds+71-- decays can be almost completely removed in the reconstructed Ds+K- sample. It should be noted that no CP violation effect is expected in the background decay mode. The decay time resolution for these decays is found to be 40fs. With the particle identification capability, an efficient flavour tag can be obtained using the charged kaons from the b hadron decays. No lepton is required in the analysis for flavour tagging. Therefore, the Level-0 high PT hadron trigger increases significantly the statistics of the flavour-tagged sample of pure hadronic B decay final states, compared to using the high /yr lepton trigger alone. Table 1 summarises the Level-0 trigger efficiencies for various decay modes. Efficiencies are calculated for those events where the initial Decay Mode Level-0 high-pT muon electron hadron Level-0 all combined B° -4 J/0(e±elKs 0.17 0.63 0.17 0.72 B° -4 J/IP(ii+Ft )Ks 0.87 0.06 0.16 0.88 B3 -+ D-.:- K- 0.15 0.09 0.45 0.54 B° -+ 7F+71-- 0.14 0.08 0.70 0.76 Table 1. Level-0 trigger efficiencies for reconstructed and flavour tagged final states. flavour is identified and the final state is fully reconstructed with all the cuts applied to remove background. While J/OKs final states are mainly triggered by the ninon and electron high-PT triggers, the hadron high-py trigger is essential for the other final states. The table also indicates that the Level-0 trigger efficiencies are very high for the events that are useful in the analysis. As a result, the LHCb experiment will run with a luminosity of 2 x 1ecm-2s-1 and still collect 2.4k reconstructed and flavour-tagged B3 and Bs decays into Ds+K- and DS K+ in one year. With these statistics, 'y can be measured with an accuracy of --,10°. By combining the result from the decays of B° and B° into D*+7T- and D*-7i-±, 'y will be measured with a precision of better than 7° with one year of data taking. At a low luminosity, the bunch interactions are dominated by events with only one pp collision. The running luminosity will be locally tuned at the LHCb intersection such that the experiment can run with this optimal luminosity while the other LHC experiments run at the design luminosity. It must be noted that running at lower luminosities has the additional benefit that the radiation damage to the detector is reduced. 6 Conclusions CP violation outside the neutral kaon system has been observed for the first time by BABAR and BELLE experiments, in the decays of B° and B° into the J/1/) Ks final state. Current analysis shows that all the measurements related to b hadron decays are consistent with the CKM picture, including CP violation. Although BABAR, BELLE, CDF and DO will further enhance these studies in the near future, a new generation of experiments at 390 Ta tsuya Nakada the LHC is needed in order to examine the CKM picture in a model-independent way. It should be noted that a dedicated experiment at the Tevatron (Kulyavtsev 2000) would have a similar performance. LHCb is a dedicated detector for B physics at the LHC with particle identification capability and a trigger sensitive to both leptonic and hadronic final states, which should be able to study CP violation in both B° and B° meson systems in many decay modes. Some of those decay modes are essential to determine the CKM parameter in a theoretically clean and model-independent way. This will allow the effect of possible new physics to be identified unambiguously. Acknowledgments I thank the organizers of the school for their hospitality and creating a stimulating envi-ronment. R. Forty and 0. Schneider are deeply acknowledged for many useful comments and suggestions on this manuscript. References Ahmadov, A. et al (2000) Proceedings of the Workshop on Standard Model Physics (and more) at the LHC, CERN 2000-04 . Amato, S. (1998), LHCb Technical Proposal, CERN/LHCC/98-4. Armstrong, W. W. (1994), ATLAS Technical Proposal, CERN/LHCC/94-43. Bayatian, G. L. (1994), CMS Technical Proposal, CERN/LHCC/94-38. Davies, C. (2001) This volume. Groom, D.E. et al. [Particle Data Group] (2000), The European Physical Journal C15, 1 Hagelin, J. S. (1981), Nucl. Phys. B193, 123. For more details, see Buras, A.J, (1999), Proceedings of 14th Lake Louise Winter Institute (Singapore: World Scientific). Kobayashi, M. and Maskawa, K. (1972), Prog. Theor. Phys. 49, 282. Kulyavtsev, A. (2000), BTeV Proposal, BTeV-doc-66-v1 . Nir, Y. (2001) This volume. Shaposhnikov, M.E. (1986), JETP Lett. 44, 364. Schubert, K.R. (2001), This volume. Stone, S. (2001), This volume. Wolfenstein, L. (1983), Phys. Rev. Lett. 51, 1945. Index 391 CDF experiment, 304, 305 charge conjugation (C), 331 charged current interactions, 6, 22 charmonium, 18, 135 chiral supermultiplets, 203 clover quarks, 114 CMS experiment, 383 colour, 3 composite theories, 50 configuration, 110 Constrained MSSM, 210 311, Cosmic microwave background, 329 covariant derivative, 8 CP violation, 23, 358 CP violation and SUSY, 209 CP violation in K decays, 332 CP violation in decay, 362 CP violation in interference, 362 CP violation in mixing, 361 CP violation, direct, 332 CP violation, indirect, 332 CPT theorem, 331 decoupling, 220 discovery of b quark, 345 discretisation errors, 124 dynamical quarks, 117 EiK , 334, 339, 342 6 K1 333 e+e- colliders, 303, 324 effective four-fermion weak Hamiltonian, 21, 35, 61 effective mass, 121 electric dipole moment, 210, 219 electroweak unification, 5, 26 Fermi coupling constant, 5 Fermi theory, 21 fermion doubling problem, 111 field-strength tensor, 10 finite volume errors, 125 flavour, 3 flavour tagging, 365 gauge hierarchy problem, 202 gauge invariance, 8 Abelian gauge theories, 7 algorithms, lattice QCD, 110 03, 137 angle (5-y, 376 angle y, 376, 389 anomalous dimensions, 67 anomalous magnetic moment, 45 asymptotic freedom, 11 ATLAS experiment, 383 auxiliary field, 204 bb production cross-section, 307, 308, 318 B D*1-7-, 379 B K(*)e± , 308, 324 B OKs, 364, 379 B OK(*), 324 B 1r+71--, 324 • , 388 Bd mixing, 307, 355, 356, 377 B, mixing, 307, 308 B, -4 D. , 380 B, 00, 307, 380 B decay constant, 76, 139 B exclusive semi-leptonic decays, 77 B hadronic decays, 89 B inclusive semi-leptonic decays, 84 B lifetimes, 307, 352, 358 B meson factories, 346 B mixing, 140 B semi-leptonic decay, 141 B spectrum, 135 BABAR detector, 348 baryogenesis, 330 B - B mixing, 69 BELLE detector, 349 beta-function, 12 big bang cosmology, 328 bino, 207 bottomonium, 18, 133 BTeV experiment, 319, 324 Cabibbo theory, 26 Cabibbo-Kobayashi-Maskawa matrix, 23, 59, 359, 374 Casimir operator, 15 392 Index gauge invariance, lattice QCD, 108 gauge theories, 7 gauge transformations, 7 gaugino, 206 Gell-Mann matrices, 10 Ginsparg-Wilson quarks, 116 gluino, 207 gluons in lattice QCD, 107 hadron colliders, 303 heavy hadron spectrum, 133 heavy quark effective theory, 72 heavy quark expansion, 80 heavy quark physics, 130 heavy quark symmetry, 73 heavy relativistic quarks, 133 heavy-heavy currents, 77 heavy-light currents, 76 HERA-B experiment, 304, 310 hierarchy problem, 50 Higgs boson, 7 Higgs mechanism, 5, 31 Higgs searches, 41 Higgs, non-standard, 46 Higgsino, 207 hybrid Monte Carlo, 116 importance sampling, 110 infrared fixed point, 13 Isgur-Wise function, 78, 142 K° — K mixing, 332 KTeV experiment, 340 large energy effective theory, 79 Large Hadron Collider (LHC), 383 lattice QCD, 105 LEP experiments, 307 leptons, 2 LHCb experiment, 384 local quark-hadron duality, 82 massive vector bosons, 22 matching lattice matrix elements, 126 matter-antimatter asymmetry, 327 mb, 138 Metropolis algorithm, 110 milliweak models, 332 Minimal Flavour Violating model, 210 minimal subtraction scheme, 17 Minimal Supersymmetric Standard Model, 207 modified minimal subtraction scheme, 17 mSUGRA, 210 it problem, 208 NA48 experiment, 336 neutral current interactions, 6, 35 neutrino mass, 30 neutrino oscillations, 51 New Physics, 381 non-Abelian gauge theories, 9 non-relativistic QCD, 131 operator product expansion, 62 0 suppression, 317 parity, (P), 331 precision electroweak tests, 43 QCD factorisation, 86 quantum chromodynamics, 3, 18 quantum electrodynamics, 5 quark masses, 2 quarkonium, 18 quarks, 2, 328 quarks in lattice QCD, 111 quenched approximation, 117 pin plane, 359, 360, 380 R-parity, 232 R-symmetry, 232 renormalisation, 5, 11 renormalisation group, 66 running coupling constant, 17 sin 2a, 370 sin 2/3, 307, 308, 367, 368, 376, 379 SLD experiment, 307 slepton, 207 smeared operators, 118 soft SUSY breaking, 209 spin-dependent asymmetries, 40 squark, 207 staggered quarks, 115 Standard Model, 1 static quarks, 131 SU(N) generators, 9 supergauge transformation, 230 superGlM mechanism, 214 superpartners, 202 superpotential, 203, 205 superspace, 226 supersymmetry, 201 Index 393 superweak model, 332 symmetry breaking, 30 tan 0, 208 tau lepton, 24 technicolour, 50 three-point function, 119 time Reversal (T), 331 top quark, 41 triviality problem, 50 two-point function, 117 ultraviolet fixed point, 13 unitarity triangle, 60 unitarity triangles, 359, 374 unquenching, 126 vector potential, 8 vector superfields, 227 W decay, 24 W mass, 25 W pair production, 25 weak angle, 6, 36 weak hypercharge, 29 weak isospin, 29 Wess-Zumino gauge, 229 Wilson coefficient, 62 Wilson quarks, 112 wino, 207 Wolfenstein parameterisation, 59 Wolfenstein parameterization, 376 Wolfenstein parametrisation, 359 Z mass, 38 Z width, 38

Summary of the Heavy Flavours Working Group

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Summary of the Heavy Flavours Working Group

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