HADRONS AND QUARK–GLUON PLASMABefore matter as we know it emerged, the universe was filled with the primordialstate of hadronic matter called quark–gluon plasma. This hot soup of quarks andgluons is effectively an inescapable consequence of our current knowledge aboutthe fundamental hadronic interactions: quantum chromodynamics. This bookcovers the ongoing search to verify the prediction experimentally and discussesthe physical properties of this novel form of matter. It provides an accessibleintroduction to the recent developments in this interdisciplinary field, coveringthe basics as well as more advanced material. It begins with an overview ofthe subject, followed by discussion of experimental methods and results. Thesecond half of the book covers hadronic matter in confined and deconfined form,and strangeness as a signature of the quark–gluon phase. A firm background inquantum mechanics, special relativity, and statistical physics is assumed, as wellas some familiarity with particle and nuclear physics. However, the essentialintroductory elements from these fields are presented as needed.This text is suitable as an introduction for graduate students, as well as pro-viding a valuable reference for researchers already working in this and relatedfields.JEAN LETESSIER has been CNRS researcher at the University of Paris since1978. Prior to that he worked at the Institut de Physique Nucleaire, Orsay, wherehe wrote his thesis on hyperon–nucleon interaction in 1970, under the directionof Professor R. Vinh Mau, and was a teaching assistant at the University ofBordeaux. Since 1972 he has made numerous contributions to the research areaof thermal production of particles and applied mathematics.JOHANN RAFELSKI obtained his Ph.D. in 1973 and has been Professor ofPhysics at the University of Arizona since 1987. He teaches theoretical sub-atomic physics both at undergraduate and at graduate level. Prior to this, he helda Chair in Theoretical Physics at the University of Cape Town, and has beenAssociate Professor of Theoretical Physics at the University of Frankfurt. Hehas worked at the CERN laboratory in Geneva in various functions since 1977,and has been instrumental in the development of the relativistic-heavy-ion ex-perimental program. Professor Rafelski has been a Visiting Professor at theUniversity of Paris and other European and US institutions. He is a co-author ofseveral hundred scientific publications, has edited and/or written more than tenbooks, and has contributed numerous popular scientific articles.This title, first published in 2002, has been reissued as an OpenAccess publication on Cambridge Core.CAMBRIDGE MONOGRAPHS ONPARTICLE PHYSICSNUCLEAR PHYSICS AND COSMOLOGY18General Editors: T. Ericson, P. V. Landshoff1. K. Winter (ed.): Neutrino Physics2. J. F. Donoghue, E. Golowich and B. R. Holstein: Dynamics of the Standard Model3. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle Physics,Volume 1: Electroweak Interactions, the ‘New Particles’ and the Parton Model4. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle Physics,Volume 2: CP-Violation, QCD and Hard Processes5. C. Grupen: Particle Detectors6. H. Grosse and A. Martin: Particle Physics and the Schrödinger Equation7. B. Andersson: The Lund Model8. R. K. Ellis, W. J. Stirling and B. R. Webber: QCD and Collider Physics9. I. I. Bigi and A. I. Sanda: CP Violation10. A. V. Manohar and M. B. Wise: Heavy Quark Physics11. R. K. Bock, H. Grote, R. Frühwirth and M. Regler: Data Analysis Techniques forHigh-Energy Physics, Second edition12. D. Green: The Physics of Particle Detectors13. V. N. Gribov and J. Nyiri: Quantum Electrodynamics14. K. Winter (ed.): Neutrino Physics, Second edition15. E. Leader: Spin in Particle Physics16. J. D. Walecka: Electron Scattering for Nuclear and Nucleon Structure17. S. Narison: QCD as a Theory of Hadrons18. J. F. Letessier and J. Rafelski: Hadrons and Quark–Gluon PlasmaIn memory ofHelga E. RafelskiHADRONSANDQUARK–GLUON PLASMAJEAN LETESSIERUniversité Paris 7JOHANN RAFELSKIUniversity of ArizonaShaftesbury Road, Cambridge CB2 8EA, United KingdomOne Liberty Plaza, 20th Floor, New York, NY 10006, USA477 Williamstown Road, Port Melbourne, VIC 3207, Australia314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge.We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence.www.cambridge.org Information on this title: www.cambridge.org/9781009290708DOI: 10.1017/9781009290753© Jean Letessier and Johann Rafelski 2022This work is in copyright. It is subject to statutory exceptions and to the provisions of relevant licensing agreements; with the exception of the Creative Commons version the link for which is provided below, no reproduction of any part of this work may take place without the written permission of Cambridge University Press.An online version of this work is published at doi.org/10.1017/9781009290753 under a Creative Commons Open Access license CC-BY-NC-ND 4.0 which permits re-use, distribution and reproduction in any medium for non-commercial purposes providing appropriate credit to the original work is given. You may not distribute derivative works without permission. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-nd/4.0All versions of this work may contain content reproduced under license from third parties.Permission to reproduce this third-party content must be obtained from these third-parties directly.When citing this work, please include a reference to the DOI 10.1017/9781009290753First published 2002Reissued as OA 2022A catalogue record for this publication is available from the British Library.ISBN 978-1-009-29070-8 Hardback ISBN 978-1-009-29073-9 PaperbackCambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.ContentsPreamble xiI A new phase of matter? 11 Micro-bang and big-bang 11.1 Energy and time scales 11.2 Quarks and gluons 61.3 The hadronic phase transition in the early Universe 81.4 Entropy-conserving (isentropic) expansion 101.5 The dynamic Universe 111.6 Looking for quark–gluon plasma: strangeness 141.7 Other probes of quark–gluon plasma 202 Hadrons 242.1 Baryons and mesons 242.2 Strange hadrons 272.3 Charm and bottom in hadrons 363 The vacuum as a physical medium 373.1 Confining vacuum in strong interactions 373.2 Ferromagnetic vacuum 403.3 Chiral symmetry 433.4 Phases of strongly interacting matter 463.5 The expanding fireball and phase transformation 503.6 QGP and confined hadronic-gas phases 524 Statistical properties of hadronic matter 544.1 Equidistribution of energy 544.2 The grand-canonical ensemble 574.3 Independent quantum (quasi)particles 584.4 The Fermi and Bose quantum gases 614.5 Hadron gas 644.6 A first look at quark–gluon plasma 68viiviii ContentsII Experiments and analysis tools 725 Nuclei in collision 725.1 Heavy-ion research programs 725.2 Reaction energy and collision geometry 785.3 Rapidity 815.4 Pseudorapidity and quasirapidity 855.5 Stages of evolution of dense matter 905.6 Approach to local kinetic equilibrium 955.7 The approach to chemical equilibrium 976 Understanding collision dynamics 1006.1 Cascades of particles 1006.2 Relativistic hydrodynamics 1046.3 The evolution of matter and temperature 1076.4 Longitudinal flow of matter 1087 Entropy and its relevance in heavy-ion collisions 1127.1 Entropy and the approach to chemical equilibrium 1127.2 Entropy in a glue-ball 1167.3 Measurement of entropy in heavy-ion collisions 1207.4 The entropy content in 200A-GeV S–Pb interactions 1227.5 Supersaturated pion gas 1247.6 Entropy in a longitudinally scaling solution 128III Particle production 1308 Particle spectra 1308.1 A thermal particle source: a fireball at rest 1308.2 A dynamic fireball 1378.3 Incomplete stopping 1448.4 Transverse-mass fireball spectra 1488.5 Centrality dependence of m⊥-spectra 1559 Highlights of hadron production 1599.1 The production of strangeness 1599.2 Hadron abundances 1659.3 Measurement of the size of a dense-matter fireball 1719.4 Production of transverse energy 1769.5 RHIC results 178IV Hot hadronic matter 18710 Relativistic gas 18710.1 Relation of statistical and thermodynamic quantities 18710.2 Statistical ensembles and fireballs of hadronic matter 19110.3 The ideal gas revisited 19310.4 The relativistic phase-space integral 19510.5 Quark and gluon quantum gases 199Contents ix10.6 Entropy of classical and quantum gases 20411 Hadronic gas 20711.1 Pressure and energy density in a hadronic resonance gas 20711.2 Counting hadronic particles 21111.3 Distortion by the Coulomb force 21511.4 Strangeness in hadronic gas 21711.5 The grand-canonical conservation of strangeness 21911.6 Exact conservation of flavor quantum numbers 22311.7 Canonical suppression of strangeness and charm 22812 Hagedorn gas 23512.1 The experimental hadronic mass spectrum 23512.2 The hadronic bootstrap 24112.3 Hadrons of finite size 24712.4 Bootstrap with hadrons of finite size and baryon number 25112.5 The phase boundary in the SBM model 254V QCD, hadronic structure and high temperature 25813 Hadronic structure and quantum chromodynamics 25813.1 Confined quarks in a cavity 25813.2 Confined quark quantum states 26213.3 Nonabelian gauge invariance 26713.4 Gluons 27113.5 The Lagrangian of quarks and gluons 27314 Perturbative QCD 27414.1 Feynman rules 27414.2 The running coupling constant 27714.3 The renormalization group 28014.4 Running parameters of QCD 28115 Lattice quantum chromodynamics 28715.1 The numerical approach 28715.2 Gluon fields on the lattice 28915.3 Quarks on the lattice 29015.4 From action to results 29315.5 A survey of selected lattice results 29816 Perturbative quark–gluon plasma 30316.1 An interacting quark–gluon gas 30316.2 The quark–gluon liquid 30616.3 Finite baryon density 30916.4 Properties of a quark–gluon liquid 311x ContentsVI Strangeness 31617 Thermal production of flavor in a deconfined phase 31617.1 The kinetic theory of chemical equilibration 31617.2 Evolution toward chemical equilibrium in QGP 32217.3 Production cross sections for strangeness and charm 32617.4 Thermal production of flavor 33017.5 Equilibration of strangeness at the RHIC and SPS 33718 The strangeness background 34018.1 The suppression of strange hadrons 34018.2 Thermal hadronic strangeness production 34318.3 The evolution of strangeness in hadronic phase 34919 Hadron-freeze-out analysis 35219.1 Chemical nonequilibrium in hadronization 35219.2 Phase space and parameters 35519.3 SPS hadron yields 35719.4 Strangeness as a signature of deconfinement 361References 371Index 389PreambleMost physicists and astrophysicists believe that space, time, and all the matterand radiation in the Universe were formed during the big-bang some 15 billionyears ago. A key challenge is to understand how the Universe we live in todayevolved from the cosmic fireball created in the big-bang. As our understandingof the laws of physics improves, we are able to look further back in time, andunravel the structure of the early Universe and its subsequent evolution.It is widely believed that almost equal amounts of matter and antimatter werecreated in the big-bang, and that most of the antimatter, if not all of it, annihilatedon matter after the Universe had cooled and expanded. This annihilation, whichstarted about 20 µs after the big-bang, occurred after most of the matter we seein the Universe today was already in the form of neutrons, protons, and otherhadrons made of quarks. Before the Universe hadronized, it existed in a phaseof quarks and gluons in which the matter–antimatter asymmetry which makesthe Universe around us today had been a small and insignificant aberration. Weare attempting to recreate this phase today, and to study it in the laboratory.This primordial state of hadronic matter called quark–gluon plasma (QGP)for all purposes an inescapable consequence of our current knowledge about thefundamental hadronic interactions, which is qualitatively rooted in the SU (3)-gauge theory, quantum chromodynamics (QCD). We are seeking to verify thisprediction and to understand this novel form of matter. To accomplish this,we ‘squeeze’ the normal nuclear matter in relativistic nuclear collisions at suf-ficiently high energy. The individual nucleons dissolve, and we hope and ex-pect that their constituents will form the sought-after state, the (color-charged)plasma of freely moving deconfined quarks and gluons.Pertinent experiments are being carried out today at the European Labora-tory for Particle Physics, CERN, located on the French–Swiss border 20 kmnorth of the lake and city of Geneva, and in the USA at the Brookhaven Na-tional Laboratory, BNL, on Long Island, some 100 km east of New York City.The most violent central encounters, in which large chunks of projectile–targetmatter participate, are of particular interest. Therefore, beams of lead and goldxixii Preambleions are made to collide with each other. The available energy in the center-of-momentum (CM) frame exceeds by far the rest energy of each participatingnucleon. In a press release, in February 2000, the CERN laboratory has formallyannounced that it views the collective evidence obtained from seven relativisticnuclear collision experiments as being conclusive proof that some new form ofmatter has been formed:A common assessment of the collected data leads us to conclude that we nowhave compelling evidence that a new state of matter has indeed been created,at energy densities which had never been reached over appreciable volumesin laboratory experiments before and which exceed by more than a factor 20that of normal nuclear matter. The new state of matter found in heavy-ioncollisions at the SPS features many of the characteristics of the theoreticallypredicted quark–gluon plasma.The study of highly excited and dense hadronic matter by means of ultra-relativistic nuclear collisions has been and remains a multidisciplinary area ofresearch, which is subject to a rapid experimental and theoretical evolution. Thisresearch field is closely related both to nuclear and to particle physics, and, ac-cordingly, this book encompasses aspects of these two wide research areas. Itemploys extensively methods of statistical physics and kinetic theory. Lookingback at the early days, it was primarily the theoretical work on multiparticle pro-duction by E. Fermi [121] in the USA, and L. Landau [173, 175] in the USSR,which paved the way to the development in the early sixties [137, 140] of thestatistical bootstrap model description of hadron production by R. Hagedorn.This approach was refined as the understanding of hadronic structure advanced,and ultimately it has been modified to allow for the possibility that individual,confined hadron-gas particles dissolve into a liquid of quarks and gluons, whichwe refer to as the QGP.The multiparticle-production work was primarily the domain of particle physi-cists. However, since the early seventies interest in nuclear ‘heavy-ion’ (notfully stripped heavy atoms) collision experiments at relativistic energies hadbeen growing within the nuclear-physics community. The initial experimentalprogram was launched at the Lawrence Berkeley Laboratory, LBL, at Berkeley,USA, and at the Joint Institute for Nuclear Research, JINR, in Dubna, USSR.At the LBL, a transport line was built to carry heavy ions from the heavy-ion accelerator HILAC to the BEVATRON which was made famous by the dis-covery of antiprotons in the early fifties. This BEVALAC facility permittedthe acceleration of nuclear projectiles to about∗ 1A GeV/c. Lighter projectiles,∗ We follow the convention of presenting the beam energy or momentum per nucleon in the nu-cleus thus: 200A GeV implies a projectile with the total energy 200 × A GeV, or momentum200 × A GeV/c, where A is the number of nucleons in the projectile. We rarely differentiatebetween the units of mass [GeV/c2], of momentum [GeV/c], and of energy [GeV], in the rela-tivistic domain of interest to us in this book. This corresponds to the commonly used conventionwhich sets the units of time such that c = 1.Preamble xiiiwhich could be completely ionized and had more favorable charge over massratios, were accelerated to above 2A GeV/c. At the JINR in Dubna, a simi-lar program of research with an acceleration capability restricted to lighter ionshas been developed. More recently, another heavy-ion accelerator complex, theSIS (SchwerIonenSynchrotron), of comparable energy to BEVELAC, has beenerected at the Gesellschaft für Schwerionenforschung laboratory, GSI, in Darm-stadt, Germany. About the time the more modern SIS started up, the BEVELACclosed down in 1993. The energy scale O(∞) GeV per nucleon yields com-pressed nuclear matter at few times normal nuclear density, and yields final-state particle (spectral) ‘temperatures’ at or below 100 MeV, conditions whichare generally considered inadequate for elementary quarks and gluons to beginto roam freely in the reaction volume.The success of the initial heavy-ion experimental program, specifically thedemonstration of the possibility of studying the properties of compressed andexcited nuclear matter, gave birth to the research programs at the BNL andCERN. Much of this interest has been driven by the hope and expectation that,within the reach of existing elementary-particle-accelerator facilities, one mayfind the point of transition from the hadronic gas (HG) phase of locally confinednucleons and mesons to the new QGP phase in which color-charged quarks andgluons could propagate.The first oxygen beam at 60A GeV was extracted from the Super Proton Syn-chrotron (SPS) accelerator at CERN and met the target in the late autumn of1986, about the same time as the BNL started its experimental program at theAlternate Gradient Synchrotron (AGS) accelerator with a 15A-GeV silicon-ionbeam. Very soon thereafter, the energy of the SPS beam could be increasedto 200A GeV and a sulphur-ion source was added. In order to study the rel-atively large volumes and longer lifetimes expected in dense matter formed incollisions of the heaviest nuclei, an upgrade of the SPS injector system was ap-proved, which, as of 1994, allowed one to accelerate lead (208Pb) ions to 158AGeV. At the BNL, a gold (197Au)-ion beam with energy up to 11A GeV becameavailable at that time. The smaller beam energy per nucleon of the heavier Pbions compared with that for sulphur reflects their smaller ratio of particle chargeto particle mass, given a fixed magnetic field strength used to bend the beam intoa circular orbit in an accelerator.Today, we are redirecting our efforts toward new experimental facilities. Atthe BNL, the Relativistic Heavy Ion Collider (RHIC), completed in 1999 withcolliding nuclear beams at up to 100A GeV, will dominate the experimentallandscape for the foreseeable future. It is allowing the exploration of an entirelynew domain of energy, ten times greater than that of CERN-SPS. The LargeHadron Collider (LHC) project set in the 27-km CERN-LEP tunnel comprisesan important heavy-ion program at energies about a factor of 30 greater thanthose of the RHIC. As this book goes to press, the expectation is that the exper-imental data from the LHC will become available in 2007.xiv PreambleIn this book, our objective is to offer both an introduction and a perspective onthe recent accomplishments and near-term aims of this rapidly developing field.The material derives from our research work, including several reviews, summercourses, and graduate lecture series that we have presented during the past 20years. The selection of material and emphasis represents our personal experi-ence in this rather wide interdisciplinary field of research, that today cannot, inits entirety, fit into a single volume.We assume that the reader is familiar with quantum mechanics, special rel-ativity, and statistical physics, and has been introduced both to nuclear and toparticle physics. However, we recapitulate briefly as needed the essential intro-ductory elements from these fields. We begin with a 70-page overview, followedby more extensive treatment of the core of our personal research experience, andmention other domains of research as appropriate.No book is complete and this book is no exception. We will not address indepth many interesting areas of active current research. We treat the two particleintensity interferometry measurements superficially, and have not discussed theelliptical flow measurements which point to early thermalization. We do notexplore the theoretical models which interpret suppression of charmonium interms of QGP, and only key experimental results from this wide research areaare shown. We do not discuss the production of photons and dileptons, sincethis goes beyond the scope of this book, and also in consideration of the inherentdifficulties in isolating experimentally these QGP signatures. Instead, we haveput a lot of effort into a detailed introductory presentation of hadron physics, asthe title of this book announces.We are hoping that our text can serve both as a reference text for those work-ing in the field and a class text adaptable for a graduate course. One of us (J.R.) has tried out this presentation in the Spring 2001 semester at the Universityof Arizona. This experience further refined our presentation. Doubtless, latereditions will build upon practical experience of how to handle this very diversematerial in a classroom. Rather than conventional homework exercises, we leavein the text topics for further research, ‘We will not discuss further in this book. . . ,’ which students can address in class presentations.We have updated the contents by incorporating advances made up to October2001, including a selection of run 2000 RHIC results. Most of the materialwe present has not yet been covered in any other monograph. Complementarybooks and reports that we found useful are the following.1. The QCD Vacuum, Hadrons and the Superdense Matter. World Scientific,Singapore (1988). E. V. Shuryak presents an early view of the structure ofthe QCD vacuum.2. Finite-Temperature Field Theory. Cambridge University Press (1989). J.Kapusta offers a lucid introduction to the theoretical aspects of QCD andhot quark–gluon matter, an area that has since developed rapidly.Preamble xv3. Thermal Field Theory. Cambridge University Press (1996). M. Lebellacpresents a modern introduction which complements and updates the text ofJ. Kapusta cited above.4. Vacuum Structure and QCD Sum Rules. North Holland, Amsterdam (1992).M. A. Shifman develops a more comprehensive view of the vacuum struc-ture and presents applications of the sum-rule method.5. Particle Production in Highly Excited Matter. Plenum Press, New York(1993), editors H. H. Gutbrod and J. Rafelski. This volume comprises sev-eral comprehensive introductory and survey articles pertinent to interpreta-tion of data.6. Introduction to Relativistic Heavy Ion Collisions. J. Wiley and Sons, NewYork (1994). In this text, L. P. Csernai emphasizes the transport phenomenain the process of collision and presents applications of matter flow mod-els, including an analysis of the LBL, GSI and Dubna energy ranges, thesesubjects are not covered in depth in this book.7. Introduction to High Energy Heavy-Ion Collisions. World Scientific, Singa-pore (1994). C.-Y. Wong emphasizes the role of the parton structure in thecollision, considers model dynamics of color strings and the associated pairproduction mechanisms, and addresses among physical observables morecomprehensively the electromagnetic probes of dense hadronic matter.8. Bose–Einstein Correlations and Subatomic Interferometry. John Wiley, Chich-ester (2000). R. Weiner presents a detailed and technical discussion of theHBT particle-correlation method used to study the space–time geometry inheavy-ion collisions and related topics. Our book offers a short introductionto this monograph.9. Quark–Gluon Plasma. World Scientific, Singapore, Volumes I and II (1990and 1995) editor R. Hwa. Useful collections of articles on a variety of topics,contributed by hand-picked authors.10. Hot Hadronic Matter: Theory and Experiment. Plenum Press, New York(1995), editors J. Letessier, H. H. Gutbrod, and J. Rafelski. This volume,dedicated to Rolf Hagedorn, comprises in particular a comprehensive surveyof the bootstrap model of confined hadronic matter.11. Proceedings of Quark Matter meetings held about every 18 months havein recent years been published in Nuclear Physics A. These proceedingspresent regular comprehensive updates of the experimental results, speckledwith a mostly random assortment of theoretical contributions.12. Proceedings of Strangeness in Hadronic Matter have in recent years beenpublished in Journal of Physics G. These volumes comprise a comprehen-sive survey of the strongly interacting heavy flavor probes of phases ofhadronic matter.13. A very useful reference is the bi-annual reissue of the Review of ParticlePhysics, published as separate issues of Physical Review D, alternating withxvi Preamblethe European Physical Journal and accessible online.A closely related area of research is the study of the properties of quantum chro-modynamics by numerical methods within the lattice-gauge-theory approach.We can barely touch this huge research field in this book. Some standard textsare the following.14. Quarks, Strings, and Lattices. Cambridge University Press (1983), by M.Creutz.15. Quantum Fields on the Computer. World Scientific, Singapore (1992), byM. Creutz.16. Quantum Fields on a Lattice. Cambridge University Press (1994), by I. Mont-vay and G. Münster.17. Proceedings of Lattice meetings, published in Nuclear Physics, are the bestplaces to find the most recent results.The publisher has used its best endeavors to ensure that the URLs for externalwebsites referred to in this book are correct and active at the time of going topress. However, the publisher has no responsibility for the websites and cannot guarantee that a site will remain live or that the content is or will remainappropriate.We would like to thank our friends and colleagues who over the years helpedus reach a better understanding of the material addressed in this book: wethank in particular Drs Mike Danos (Chicago and Washington, deceased), HansGutbrod (GSI), Rolf Hagedorn (CERN), Berndt Müller (Duke University), andEmanuele Quercigh (CERN and Padua).This volume is dedicated to Helga Rafelski. Helga has been a companionfrom day one in the field of relativistic heavy-ion collisions; her presence at thefinale will be sorely missed.Jean Letessier and Johann Rafelski, Paris and Tucson, November 2001.IA new phase of matter?1 Micro-bang and big-bang1.1 Energy and time scalesWhen atomic nuclei, generally called heavy-ions, collide at very high en-ergies, such that the kinetic energy exceeds significantly the rest energy,dense hadronic∗ matter is produced. We refer to these reactions as (ul-tra)relativistic heavy-ion, or nuclear, collisions. The energy density ofhadronic matter with which we are concerned has a benchmark value ofε = 1 GeV fm−3 = 1.8× 1015 g cm−3. (1.1)The corresponding relativistic matter pressure isP 13ε = 0.52× 1030 bar. (1.2)Dense matter with these properties must have existed in the early Uni-verse about 10 µs after the big-bang. It might have been recreated ex-tremely rarely in interactions of very-high-energy cosmic-ray particles.Some astrophysical objects may reach these extreme conditions. It hadbeen speculated that a catastrophic change in the Universe could ensuewhen these conditions are recreated in laboratory experiments, but thesefears have been refuted [85].Experimental study of the physics of the early Universe requires inprinciple a large, practically infinite, volume of matter. For this reason, itis necessary to study high-energy collisions of the heaviest nuclei, ratherthan the more elementary and simpler-to-handle interactions of protonsor leptons. However, we cannot study in the laboratory physical systems∗ In Greek, barys means strong and heavy; leptos is weak, light; mesos is intermediate,and hadros is strong. Hadronic (strong) interactions involve baryons and mesons(heavy and semi-heavy particles) but not leptons, the light and relatively weaklyinteracting electrons, muons, the heavy tau, and nearly massless neutrinos.12 A new phase of matter?larger in volume than lead (Pb) or gold (Au). Hence, it would seem thatwe will not be able to explore experimentally the properties of the phasetransition involving the dissolution of hadronic particles, since it is knownthat genuine phase transitions cannot develop in finite physical systems.However, only for non-relativistic finite systems it is impossible to ob-serve experimentally the discontinuous phase properties. In our case, theability to produce particles from energy and the presence of virtual fluctu-ation effects greatly enhance the number of physical states accessible. Wetherefore hope to identify in collisions of relativistic heavy-ions a (nearly)singular manifestation of a phase transition from the nuclear, hadronicphase to a matter phase consisting of quarks and gluons.We use units in which the Boltzmann constant k = 1. In consequence,the temperature T is discussed in units of energy, which, in this book,are either MeV 2mec2 (me is the electron mass) or GeV= 1000 MeV mNc2 (mN is the mass of a nucleon). The conversion scale of typicaltemperature involves ten additional zeros:100 MeV ≡ 116× 1010 K. (1.3)To appreciate the magnitude of this temperature, let us recall that thecenter of the Sun is believed to be at T = 11 × 106K, and the scale oftemperature of interest to us is in fact 100 000 times greater.In general, the units in this book are chosen such that the numericalvalues = c = 1, e.g., the mass of particles will also be measured in unitsof energy and the energy density can appear as the fourth power of anenergy unit. With the conversion factor c = 0.197 GeV fm, the referenceenergy density in normal nuclei ismNVN= 0.17mN fm−3 0.16 GeV fm−3 = 1.27× 10−4GeV4. (1.4)Experimental results have shown that ultra-relativistic heavy-ion colli-sions lead to the formation of a dense hadronic fireball, well localized inspace, with an energy density exceeding 1 GeV fm−3. Such a spatiallylocalized drop of highly excited, hot, and dense elementary matter will berapidly evolving, indeed exploding, driven by the high internal pressure.The fireball has a short life span characterized by the size of the systemτ 2R/c.In relativistic heavy-ion reactions, the collision energy is shared amongnumerous newly produced hadronic particles. Therefore, in the final statewe observe many soft (low-energy) newly produced hadronic particles,rather than a few particles of high-energy as is the case in hard, elemen-tary interactions. An important objective of our research is the under-standing of the processes that lead to the conversion of kinetic collisionenergy into high particle multiplicity. Because of the large numbers of1 Micro-bang and big-bang 3Fig. 1.1. Top: Lorentz-contracted nuclei collide in the center-of-momentumframe and form a region of dense matter, which evolves into a final state ofhadrons. Bottom: two key differences involving baryon number Nb and totalparticle number N between the micro-bang and the cosmological big-bang.particles produced, many thousands in recent experiments, we believethat this can be accomplished using statistical mechanics. This methodhas the advantage that it does not require a complete description of themicroscopic production and dynamics of particles. It will be introducedin great detail in this book.A qualitative image of the high-energy nuclear-collision ‘micro-bang’ isdepicted in Fig. 1.1: two nuclei are shown, Lorentz-contracted in the di-rection of motion, approaching from two sides and colliding in the center-of-momentum (CM) laboratory frame, forming a region of dense matter(dark-shaded), the fireball. Subsequently, the collective expansion flow offireball matter develops, and evolves in the final state into free streams ofindividual particles, indicated by individual arrows.The temporal evolution of a fireball into a final state comprising a multi-tude of different hadronic particles is similar to, though much faster than,the corresponding stage in the evolution of the early Universe. Relativis-tic heavy-ion collision leads to a rapidly evolving fireball of quark–gluonplasma (QGP), in which the short time scale involved is probed by theequilibration of abundance of quark flavors. We can not hope to be able4 A new phase of matter?to recreate the ‘slow big-bang’ of the Universe in the laboratory in thelast detail. Our objective is to obtain precise information about the physi-cal processes and parameters which govern the rapidly changing hadronicphase. Within a theoretical framework, we can hope to unravel whathappened when the Universe hadronized.The bottom portion of Fig. 1.1 reminds us of the two important dif-ferences between the two ‘bangs’, the big-bang of the Universe and themicro-bangs generated in the nuclear-collision experiments.1. The time scale of the expansion of the Universe is determined by the in-terplay of the gravitational forces and the radiative and Fermi pressureof the hot matter, whereas in the micro-bangs there is no gravitationto slow the expansion, which lasts at most about 10−22 s. The timescale of the heavy-ion collision, indicated in Fig. 1.1, suggests that thesize and the (local) properties of the exploding nuclear fireball mustchange rapidly even on the scale of hadronic interactions, contrary tothe situation in the early Universe. It is convenient to represent theexpansion time constant τU of the Universe in terms of the Newtoniangravitational constant G and the vacuum energy B:τU=√3c232πGB = 36√B0B µs, B0= 0.19GeV fm−3=(195MeV)4.(1.5)The range of values of the ‘bag’ constant B found in the literature,145 MeV < B1/4 < 235 MeV, leads to 66 µs > τU > 25 µs.2. The early radiative Universe was practically baryonless, whereas inthe laboratory we create a fireball of dense matter with a consider-able baryon number Nb per total final particle multiplicity N . Thus,unlike in the early Universe, we expect in a laboratory micro-bang asignificant matter–antimatter asymmetry in particle abundance. Thematter–antimatter symmetry of particle spectra is in turn an importantindicator suggesting that the matter–antimatter symmetry has been re-stored in other aspects.The matter–antimatter-abundance asymmetry is easily overcome the-oretically, since it implies a relatively minor extrapolation of the baryo-chemical potential µb introduced to fix the baryon density. In fact, RHICexperiments at CM-energy 130A GeV per pair of nucleons (√sNN = 130GeV) are already much more baryon–antibaryon symmetric than the SPScondition where√sNN ≤ 17.3 GeV, and the highest RHIC and LHC en-ergies will allow us to extrapolate our understanding from µb/T ≤ 1 toµb/T 1.More difficult to resolve will be the differences in the physics due to thedifferent time scales involved. The evolution of the Universe is slow on1 Micro-bang and big-bang 5Fig. 1.2. Particle energy (temperature) as a function of time in the early Uni-verse. Different evolutionary epochs are shown along with the accessible rangeof accelerator laboratory experiments.the hadronic time scale. Given the value of τU, we expect that practicallyall unstable hadronic particles decay, all hadronic equilibria are fully at-tained, and there is potentially time to develop macroscopic structures inthe ‘mixed phase’ of QGP and hadronic gas (HG), and for weak inter-actions to take place. All this can not occur during the life span of thedense matter created in nuclear collisions.The temporal evolution of the Universe is depicted, in Fig. 1.2, as afunction of time. Beginning with decoupling of neutrinos and nucleosyn-thesis at time O(1) s the evolution of the Universe is well understoodtoday. In comparison, little work has gone into the detailed understand-ing of the earlier period when the nearly symmetric matter–antimatterhadron gas emerged from the quark–gluon phase and evolved into thebaryon Universe in which we find ourselves today. This period spans thetemperature interval 300 MeV < T < 1 MeV, separating the perturbativeQGP epoch from the epoch of decoupling of neutrinos and cosmologicalnucleosynthesis.We see, in Fig. 1.2, that, after about 10µs, the deconfined phase of qu-arks and gluons is transformed into a hot gas of hadrons, namely mesons,6 A new phase of matter?baryons, and antibaryons. Just after that, the evolution of our Universewas marked by a period of baryon–antibaryon annihilation, and, possibly,separation: although we have not been able to observe antimatter in ourgalaxy, or in the neighborhood of our galaxy, it is far from certain thatthere is no antimatter in our Universe.The laboratory study of the formation of the QGP and hadronizationis expected to lead to an understanding of how the hot, baryon- andantibaryon-rich hadron gas evolved after its formation at T 170 MeV.Employing the statistical-physics methods developed in this book, onefinds that the energy fraction of baryons and antibaryons within hadronic-gas matter is about 25% just after the QGP has hadronized in the earlyUniverse, and nearly half of this is comprised of the heavier and un-stable strange baryons and antibaryons. It is believed that this strongantimatter component disappears from the Universe prior to the era ofnucleosynthesis.1.2 Quarks and gluonsBoth quarks and gluons manifest themselves only for a short instant fol-lowing a high-energy interaction, and have never been observed as freeobjects at macroscopic distances from the space–time volume of the reac-tion; they are ‘confined’. Gluons interact only through strong interactionsand pose a great experimental challenge regarding the study of their prop-erties. The measurement of the properties of confined quarks is relativelyeasy, since, in addition to the strong-interaction (color) charge, they alsocarry the electro-weak charges.There are six different flavors of quarks, see table 1.1, two practicallystable flavors referred to as up – for the proton-like quark u, and down –for the neutron-like quark d. We often refer to these two light quarks bytheir generic name q. Light quarks q may be viewed as a single entity withtwo states, up or down. The semi-heavy strange-flavor s-quark decaysdue to electro-weak interaction when it is bound in hadrons, typicallywithin 10−10 s, whereas the heavier charm c and bottom b flavors haveapproximate life spans of 3× 10−13 s and 10−12 s, respectively.These six flavors of quarks form three doublets:(ud),(cs),(tb); Qq =(+23−13).The upper component of a doublet has charge Qq = +23 , in units of theproton charge, whereas the lower component has one unit of charge less, asis also the case for the related lepton doublet comprising the three chargedleptons (electron, muon, and tau) accompanied by their neutrinos. Thereis an antiquark for each quark, carrying the opposite electrical charge.1 Micro-bang and big-bang 7Table 1.1. Properties of quarks: flavor f symbol, flavor name, electrical chargeQf (in units of proton charge Qp), and mass mf at energy scale 2 GeV; see thetext for further discussion of strange-quark massf Quark Qf [Qp] mf(2 GeV)u Up +23 3.5 ± 2 MeVd Down −13 6 ± 3 MeVs Strange −13 115 ± 55 MeVc Charm +23 1.25 ± 0.15 GeVb Bottom −13 4.25 ± 0.15 GeVt Top +23 174.3 ± 5.1 GeVQuarks differ from charged leptons (electrons e, muons µ, and taus τ),and neutrinos (νi, i = e,µ, and τ) by a further internal quantum numberthey must carry, in addition to spin. The presence of this additionalquantum number arises even in the simplest quark models. For example,consider hadronic particles containing three quarks of the same flavor,such as the spin-3/2 baryons:∆++ = (uuu), ∆− = (ddd), Ω− = (sss).The physical properties of these baryons imply that three identical quarksare present in the same S-wave with the same spin wave function. Sincequarks are fermions, they are subject to the Pauli principle. Thus, theremust be an additional way to distinguish the quarks, aside from spin. Thisadditional degeneracy factor has been determined to be gc = 3. It becameknown as the color of quarks – in analogy to the three fundamental colors:red, green, and blue.Color is an internal quantum number, which like the electrical charge, isthought to be the source of a force [123]. It seems that there is no way tobuild an apparatus to distinguish the three fundamental color charges, allcolors must everywhere be exactly equal physically. The theory of colorforces must satisfy the principle of local nonabelian gauge invariance,e.g., invariance under arbitrary local SU(3) transformations in the three-dimensional color space. In other aspects, there is considerable formalsimilarity with quantum electrodynamics (QED). Therefore, the theoryof strong interactions based on such color forces has been called quantumchromodynamics (QCD).The flavor structure and symmetry of quarks and leptons remains amystery today. We also do not have a fundamental understanding of theorigin of quark masses. In table 1.1 we see that quarks of various flavors8 A new phase of matter?differ widely in their ‘current’ mass mf , that is mass which enters theelementary QCD Lagrangian LQCD. The values presented in table 1.1are for the momentum scale 2 GeV.Since quarks are confined inside hadrons, and the zero-point energy ofconfinement is much larger than the masses of light quarks, their massescould not be determined by direct measurement. However, the precisemasses of light u and d quarks do not matter in the study of hadronic in-teractions, being generally much smaller than the pertinent energy scales,The mass of the strange quark ms is barely heavy enough to be de-termined directly in a study of hadronic structure. We adopt, in thisbook, the value ms(1GeV) = 200 ± 20 MeV [150]. In the value ofms reference is made to the scale of energy at which the mass of thestrange quark is measured: akin to the interaction strength, also themass of quarks depends on the (energy) scale. This value of ms corre-sponds to ms(2GeV) ms(1GeV)/1.30 = 154 ± 15 MeV. A somewhatsmaller value ms(2GeV) = 115 ± 55 MeV, see table 1.1, correspondingto ms(1GeV) 150 ± 70 MeV, is the recommended value. The ratherrapid change by 30% of the quark mass between the 1- and 2-GeV scalesis well known, but often not remembered, e.g., the ‘low’ recommendedmass of the charmed quark presented in table 1.1 in fact corresponds tomc(1GeV) = 1.6 GeV, a rather ‘high’ value.1.3 The hadronic phase transition in the early UniverseWe will now show that the ‘freezing’ of quark–gluon ‘color’ deconfineddegrees of freedom is the essential ingredient in determining the conditionsin a transition between phases that has time to develop into equilibrium.The following discussion tacitly assumes the presence of latent heat B inthe transition, and a discontinuity in the number of degrees of freedom,g2 = g1, where ‘1’ refers to the primeval QGP phase and ‘2’ to the finalhadronic-gas state.To find the phase-transition point, we determine the (critical) temper-ature at which the pressures in the two phases are equal. We allow, ina transition of first order, for a difference in energy density ε1 = ε2 asso-ciated with the appearance of latent heat B (the ‘bag constant’), whichalso enters the pressure of the deconfined phase. We consider the Stefan–Boltzmann pressure of a massless photon-like gas with degeneracy gi:Pc ≡ P1(Tc)=π290g1T4c − B, (1.6)Pc ≡ P2(Tc)=π290g2T4c . (1.7)1 Micro-bang and big-bang 9We obtainBT 4c=π290∆g, Tc = B14(90π2∆g)14, ∆g = g1 − g2. (1.8)The transition temperature, in the early Universe, is slightly higher thanthe value seen in laboratory experiments, even though Eq. (1.8) involvesonly the difference in the number of degrees of freedom. For the pressureat the transition we obtainPc = Bg2∆g. (1.9)The pressure, and therefore the dynamics of the transition in the earlyUniverse, depends on the presence of non-hadronic degrees of freedom,which are absent from laboratory experiments with heavy ions.In summary, the phase-transition dynamics in the early Universe isdetermined by(a) the effective number of confined degrees of freedom, g2, at Tc;(b) the change in the number of acting degrees of freedom ∆g, whichoccurs exclusively in the strong-interaction sector; and(c) the vacuum pressure (latent heat) B, a property of strong interactions.In order to understand the early Universe, we need to measure thesequantities in laboratory experiments.Both phases involved in the hadronization transition contain effectivelymassless electro-weak (EW) particles. Even though the critical tempera-ture does not depend on the background of EW particles not participatingin the transition, the value of the critical pressure, Eq. (1.9), depends onthis, and thus we will briefly digress to consider the active electro-weakdegrees of freedom. These involve photons, γ, and all light fermions, viz.,e, µ,νe,νµ, and ντ (we exclude the heavy τ-lepton with mτ T , andwe consider the muon as being effectively a massless particle). Near toT 200 MeV, we obtaingEW=gγ+ 74gEWF = 14.25, (1.10)withgγ=2, 74gEWF =78 × 2× (2e + 2µ+ 3ν) = 12.25,where charged, effectively massless fermions enter with spin multiplicity2, and we have three neutrino flavors – there are only left-handed lightneutrinos and right-handed antineutrinos, and thus only half as manyneutrino degrees of freedom as would naively be expected.In the deconfined QGP phase of the early Universe, we haveg1 = gEW + gg + 74gq. (1.11)10 A new phase of matter?The number of effectively present strongly interacting degrees of freedomof quarks and gluons is influenced by their interactions, characterized bythe strong coupling constant αs, and this book will address this topic indepth,gg = 2s × 8c(1− 154παs),74gq =742s × 2.5f × 3c(1− 5021παs), (1.12)where the flavor degeneracy factor used is 2.5, allowing in a qualitativemanner for the contribution of more massive strangeness; table 1.1. Thedegeneracies of quarks and gluons are indicated by the subscripts s(pin)and, c(olor), respectively. We obtaing1 = 56.5, for αs = 0,∼37, for αs = 0.5,∼33, for αs = 0.6. (1.13)For the QCD perturbative interactions with αs = 0.5–0.6, we see thatg1 35± 2.We now consider the final HG phase of the early Universe: there is nolight, strongly interacting fermion. Aside from three light bosons (pionsπ± and π0), the presence of heavier hadrons contributes at T 170 MeV,and one finds for the hadronic degrees of freedom gh2 5g2 ≡ gEW + gh2 19. (1.14)Thus, we find from Eqs. (1.13) and (1.14),g1 − g2 = ∆g = ∼37, for αs = 0,∼18, for αs = 0.5,∼14, for αs = 0.6. (1.15)For the QCD perturbative interactions with αs = 0.5–0.6, we see thatabout half of the degrees of freedom freeze across the transition in theearly Universe.For the value B1/4 = 190 MeV and αs 0.5, we obtain from Eq. (1.8)a transition temperature Tc 160 MeV. At this temperature, the criticalpressure Eq. (1.9) is found to be Pc 1.4B, and it includes both hadronicand electro-weak partial pressure contributions. The hadronic fractionalpressure present in laboratory experiments and seen in lattice simulationsof gauge theories (compare with section 15.5) is P hc B/4.1.4 Entropy-conserving (isentropic) expansionMuch of the time dependence of an expanding Universe is related to theassumption of adiabatic, i.e., entropy-conserving, expansion dynamics:dE + P dV = T dS = 0, dE = d(εV ),dVV=3 dRR. (1.16)1 Micro-bang and big-bang 11Here, as usual, ε is the energy density in the local restframe, and thethree-dimensional volume element dV scales with the third power of thedistance scale R. We obtain3 dRR= − dεε+ P. (1.17)We will revisit Eq. (1.17) which describes general expansion dynamics ofthe micro-bang.We now relate the expansion dynamics to the velocity of sound, anduse well known relations of thermodynamics which we will discuss in thisbook,dε =dεdPdPdTdT =1v2sσ dT =1v2sε+ PTdT. (1.18)We will revisit the derivation Eq. (1.18) when we study the same physicsoccurring in the expansion of the dense-matter phase formed in heavy-ioncollisions in section 6.3. Using Eq. (1.17), we obtain3 dRR= − 1v2sdTT. (1.19)This equation allows the integralRT 1/(3v2s ) = constant, (1.20)which describes exactly how the temperature decreases in an isentropicexpansion once the equation of state P = P (ε), and hence the velocity ofsound is known.For a relativistic equation of state, v2s =13 and thusR(t)T (t) = constant, V (t)T 3(t) = constant. (1.21)While this result applies to a three-dimensional expansion, it is easilygeneralized to a one-dimensional expansion, such as is expected to applyin ultra-high-energy heavy-ion collisions.1.5 The dynamic UniverseThe (0, 0)-component of the Einstein equation,Rµν − 12gµνR+ Λvgµν = 8πGTµν , (1.22)gives the Friedmann equation which determines the rate of expansion ofthe homogeneous Universe,H2(t) ≡(ṘR)2=8πG3ε+Λv3− kR2. (1.23)12 A new phase of matter?In the last term, k = 0, 1, and −1 for different geometries of the Universe(flat, bubble, and hyperbolic-open); this term is negligible in our consid-erations. Λv is Einstein’s cosmological term, which is playing a similardynamic role to B, but, in comparison, it is of irrelevant magnitude dur-ing the early time period we consider. H is the Hubble ‘constant’ whichvaries with time. Its present-day value, H(t0) = H0, is of considerableinterest and is given in the rangeH0 = 70± 15 km s−1Mpc−1 =0.7± 0.151010 y= (2.2± 0.5)×10−17 s−1.Inserting Eq. (1.17), with V ∝ R3, into Eq. (1.23) and neglecting the lasttwo terms in Eq. (1.23), we find for ε(t)ε̇2 = 24πGε(ε+ P (ε))2. (1.24)We equate the particle energy density and pressure, including the vacuumterm B in the relativistic equation of state for the particle component,ε− B π230gT 4 3(P + B), ε = 3P + 4B. (1.25)Thus,ε̇2 =128πG3ε (ε− B)2, (1.26)which is valid (approximately) both for QGP and for HG phases, but inthe HG phase B = 0.Despite its highly nonlinear nature, Eq. (1.26) has an analytical solu-tion,ε1 = B coth2(t/τU), (1.27)where τU is the expansion time constant we have defined in Eq. (1.5) andthe subscript ‘1’ reminds us that Eq. (1.5) describes the evolution in thequark–gluon phase with B = 0. At t < τU, the energy density rises like1/t2; for t > τU, it would remain constant at the vacuum energy density.Once the transition to HG phase ‘2’ with B → Λv ∼ 0 has occurred,the analytical form of the solution changes; we exploit the singularitycothx → 1/x and obtain the power-law solutionε2 = B(τUt)2=332πG1t2=π230g2T4. (1.28)In the middle of Eq. (1.28), we have substituted τU from Eq. (1.5) to showthat, in principle, B does not enter the solution. On the right-hand sideof Eq. (1.28), we show the energy density of the (radiation-dominated; seeFig. 1.2) Universe, establishing the well-known relation1 Micro-bang and big-bang 13(t2t)2=(TTc)4, T ∝ 1√t. (1.29)Using Eq. (1.8) for Tc, we determine t2, the time when the Universe en-tered the hadronic phase, which we use below in Eq. (1.33). Similarly wealso obtain the behavior of the size of the Universe with time. By insert-ing Eqs. (1.27) and (1.28) into Eq. (1.23), we find how the scale R of theUniverse evolves:R2 ∝ sinh(tτU) ∣∣∣∣1→(tτU) ∣∣∣∣2. (1.30)To determine when the transition between the two phases occurs, andhow long it takes, we consider the pressure in both phases. In deconfinedphase ‘1’ where B = 0, we have, using Eqs. (1.25) and (1.27),3P1 = B[coth2(t/τU)− 4], t ≤ t1, (1.31)while in the post-transition phase3P2 = B(τUt)2, t ≥ t2. (1.32)Equating these two pressures with the critical pressure, Eq. (1.9), we ob-tain3PcB =3g2∆g= coth2(t1τU)− 4 =(τUt2)2. (1.33)Equation (1.33) relates the time t1, when the transition begins, to t2,when it ends, and the fraction of degrees of freedom which are ‘freezing’in the transition, in units of the τU, Eq. (1.5).We show the pressure and temperature in the Universe near hadroniza-tion in Fig. 1.3. The solid line corresponds to αs = 0.6, and B1/4 = 195MeV, for which value τU = 36µs. Dotted lines are for B1/4 = 170 MeVand B1/4=220 MeV; a higher value of B1/4 leads to a shorter time scale,Eq. (1.5).It is straightforward to obtain the values of t1 and t2. Using Eq. (1.14),g2 = 19, and Eq. (1.15), ∆g = 14, we find that the transition is completeat t2 = 0.5τU. The onset of the transition is found at t1 = 0.37τU, andthe transition lasts 0.13τU in this case – the major uncertainty is relatedto the value of g1 − g2 = ∆g. For the central value of B1/4 = 195 MeVwith τU = 36µs, we find that the transition lasts ∆t = (t2 − t1) = 4.7µs.The duration of the hadronization transition is comparable (35%) tothe prior life span of the Universe in the deconfined phase. This time isexceedingly long compared with the time scale of hadronic interactions14 A new phase of matter?µFig. 1.3. Pressure (upper) and temperature (lower part) in the Universe, asfunction of time, in the vicinity of the phase transition from the deconfined phaseto the confined phase. Solid lines, B1/4 = 195 MeV; dotted lines, B1/4 = 170MeV (lower part) and B1/4 = 220 MeV (upper part) all for αs = 0.6.(10−22 s). It allows the decay of all unstable hadronic particles and, poten-tially, the development of domain structures. Moreover, the hadroniza-tion time is also three orders of magnitude longer than the characteristictime of hadronic weak decays, and is even longer than the decay time fora muon. What exactly happens to matter in this last phase transitionin our Universe is not yet known. Studying dense matter in relativisticheavy-ion collisions should help us establish the physical laws governingthis crucial epoch in the development of the Universe.1.6 Looking for quark–gluon plasma: strangenessHow do we look for the phase transition of the primordial state of theUniverse recreated for a short glimpse of time in the laboratory? How canwe distinguish between the reactions involving confined hadronic particlesonly, and those in which we encounter the color-deconfined quarks andgluons?1 Micro-bang and big-bang 15• Is a transient new phase of matter, existing for a brief instant in time,perhaps for no more than 10−22 s, in principle observable? This can bepossible only if time-reversibility is broken more rapidly in the collisionprocess. This will be the tacit assumption we make. How this occurs isone of the great open issues. In some sense on the time scale of 10−23 s‘measurement’ of the colliding system must be occurring, leading to thedecoherence of the many-body quantum state.• How can we observe a new phase of matter that exists for a short time,evolves and ultimately disintegrates (hadronizes) into usual final-stateparticles? At first sight, everything will always appear in the data verymuch akin to a reaction involving only the HG phase.Considerable effort must be put into the understanding of the tempo-ral and spatial evolution of the colliding system. We must identify themeasurable quantities that can depend on the properties of the early anddense stage of collision, allowing us to penetrate the ‘nebula’ of the finalhadronic state.One observable is the quark chemical composition of the fireball ofdense matter, which evolves as new quark flavors, such as strangeness, arecooked up inside the micro-bang fireball. Another observable is the en-tropy content: when quarks and gluons are liberated, the usually ‘frozen’color bonds are broken and an entropy-rich state of matter is formed. Wewill address these two hadronic observables of QGP in greater detail inthis book, and we offer here a first short overview of the related ideas anddiagnostic methods.The quarks ‘q’ (up ‘u’ and down ‘d’) from which the stable matteraround us is made are easily produced as quark–antiquark pairs becausethey have small masses; see section 1.2. Another abundantly added quarkflavor is strangeness, particularly if the deconfined QGP phase of matteris formed. Strangeness was one of the first proposed signatures of thedeconfined phase [220]. The mass of strange quarks and antiquarks is ofthe same magnitude as the temperature T at which protons, neutrons,and other hadrons are expected to dissolve into quarks. This means thatthe abundance of strange quarks is sensitive to the conditions, structure,and dynamics of the deconfined-matter phase. The dominant mechanismfor cooking up strangeness in quark–gluon deconfined matter was foundto be the gluon-fusion reaction gg→ ss̄ [226]. We will address this processin depth in this book. Ultimately the quarks and antiquarks produced inthe fireball of dense matter find their way into a multitude of final-stateparticles, with different quark contents, in the process of hadronization.This situation is illustrated in Fig. 1.4.Detection of strange particles is facilitated by the fact that the mas-sive strange quark decays into lighter quarks. Thus, strangeness-carrying16 A new phase of matter?sssssddddduuu uss s sΩssudss sssu uuudddd dduuugggggggΞggs s ugFig. 1.4. A qualitative image of processes leading to production of (multi)-strange particles: in the QGP phase gluon collisions produce pairs of strangequarks, which are shown to assemble into otherwise rarely made multistrangebaryons (for example Ξ(ssu)) and antibaryons (for example Ω(s̄s̄s̄)).hadrons are naturally radioactive and decay by weak interactions thatoccur, in general, on a time scale that is extremely long compared withthe nuclear-collision times. This makes it relatively easy to detect thestrange particles through the tracks left by their decay products. It isimportant to remember that, unlike the light quarks, strange quarks arenot brought into the reaction by the colliding nuclei. Therefore, we knowfor sure that any strange quarks or antiquarks observed in experimentshave been made from the kinetic energy of colliding nuclei.Should the new deconfined phase of matter be formed, we expect thatfinal abundances of strange particles will be governed by (near) chemi-cal equilibration of strangeness, i.e., that the yield abundance of QGPstrangeness will saturate all available phase-space cells, making it into aq–q̄–s–s̄–g liquid. The total strangeness yielded is thus of considerableinterest and is being measured as a function of the collision energy.The excitation function of strangeness can be qualitatively studied byevaluating the ratio K+(s̄u)/π+(d̄u) shown in Fig. 1.5. Data obtained atseveral experimental facilities is shown: from the KaoS experiment at theSIS/GSI; from the E917 and E866 experiments at the AGS/BNL, fromNA49 and NA44 experiments at the SPS/CERN, and from the STARexperiment at the RHIC/BNL. As long as the production of strange an-1 Micro-bang and big-bang 17π | y = 0+ +K /[GeV]Fig. 1.5. Mid-rapidity K+(s̄u)/π+(d̄u) in reactions of the heaviest nuclei as afunction of the collision energy. See the text for details.tibaryons is small, the K+(s̄u)/π+(d̄u) ratio characterizes accurately therelative abundance of strangeness. This is not the case at RHIC energies,for which the strangeness content found in baryonic degrees of freedomis not negligible; see section 19.4. Using the estimate presented there, wefind thatK+ + 〈s̄〉Yπ+ 0.23,where the second term 〈s̄〉Y is the strangeness content in the baryonicdegrees of freedom.This shows that the ratio of strangeness to hadron multiplicity con-tinues to grow as the collision energy is increased. Since this growth ishere seen to occur relative to the hadron multiplicity, measured in termsof the yield of positively charged pions, this implies that the yield ofstrangeness increases faster than the increase in production of entropy.At low energies the increase in yield of strangeness shown in Fig. 1.5shows the effect of the energy threshold for production of strangeness.18 A new phase of matter?The specific yield of strangeness per participating baryon increases muchfaster. In fact the specific yield of pions increases by 50% on goingfrom AGS to CERN energies, which implies that the yield of stran-geness per baryon continues to grow rapidly in the energy range 5 GeV< s1/2 < 17 GeV.The most interesting qualitative signature of strangeness in QGP is theyield of (multi)strange antibaryons. Given the ready supply of (strange)quarks and antiquarks, otherwise rarely produced (strange) particles willbe emerging from a deconfined phase. In particular, the formation ofantimatter particles comprising strangeness is of interest [215]. Theseparticles can be more readily assembled in the high-density deconfinedenvironment. In Fig. 1.4, we illustrate the sequence of events that leadsto the formation of these particles: microscopic reactions, predominantlyinvolving fusion of gluons, form pairs of strange quarks, of which clustersare formed and emitted.Enhanced production of strange particles has been predicted to occur ina QGP for each strange particle species, and to increase with the strangen-ess content of the particle [164, 215]. Such enhancements in the number ofstrange particles produced per participating nucleon have now been ob-served in, e.g., lead–lead (Pb–Pb) collisions, compared with expectationsarising from studies of proton–proton (p–p) and proton–beryllium (p–Be) collisions, as is shown in Fig. 1.6. The enhancement for a particularparticle is defined as the number of that particle produced per partici-pating nucleon in Pb–Pb collisions, divided by the number produced perparticipating nucleon seen in p–Be interactions [38].In Fig. 1.6 the h− symbol denotes the yield enhancement of negativelycharged hadrons, mainly negative pions. This result implies an enhance-ment by a factor 1.3 for all non-strange hadrons. Such an enhancement isnatural if QGP is formed on account of the breaking of the color bonds,and the associated enhancement in number of accessible degrees of free-dom compared with reaction scenarios involving confined hadrons. Laterin this book we will discuss in depth the issues related to enhanced pro-duction of entropy in the deconfined phase.We further see, in Fig. 1.6, that the production of particles that containone strange quark, such as the neutral kaon K0 and the Λ-particle, isenhanced by a factor of about three; the enhancement factor rises toabout five for the doubly strange Ξ-particle (and its antiparticle, the anti-Ξ), and more than ten for the yield of (Ω + Ω) particles, which containthree strange or antistrange quarks. The particles in the right-hand panelof Fig. 1.6 have no quarks in common with the colliding nucleons.When one is interpreting these results as significant indicators for theformation of the deconfined state, it is important to be able to arguethat both matter and antimatter particles were produced by the same1 Micro-bang and big-bang 19Fig. 1.6. Enhanced production of (strange) hadrons in Pb–Pb 158A-GeV heavy-ion collisions. Enhancement is defined with respect to p–Be collisions. Resultswere obtained by by the CERN WA97 experiment considering particles emittedby a source stationary in the CM frame of reference [38].mechanisms, as would be the case should a deconfined soup of quarks andantiquarks break up into final-state hadrons [42]. To demonstrate this,one studies not only the abundances but also the spectra of the parti-cles produced. In order to reduce the dependence on the flow of matteralong the collision axis, which is related to the collision dynamics, it isconvenient to look only at hadron spectra with momentum componentstransverse to the original collision axis.In Fig. 1.7, we see the spectra of strange baryons (Λ, Ξ, and Ω) andantibaryons (anti-Λ, anti-Ξ−, and anti-Ω−) as functions of the transverseenergy, m⊥ =√m2 + p2⊥. The most significant feature of Fig. 1.7 is thatthe slopes of the spectra for a particle and its antiparticle are very similar.The difference between the particle and antiparticle yields is a result of thequark–antiquark yield asymmetry present. The shape identity of matterand antimatter verifies that the mechanism of production is the same,corroborating the evidence for a common, deconfined source of strangehadrons.Much of the material of this book will be devoted to the developmentof ideas demonstrating that these results are a natural consequence of theformation of the deconfined QGP state. It is important to keep in mind20 A new phase of matter?mT [GeV c–2]Fig. 1.7. Transverse mass spectra of strange and multistrange baryons and an-tibaryons. Results obtained by the CERNWA97 experiment for particles emittedat mid-rapidity [42].that further evidence for the deconfinement of quarks in these reactionsis available. The production of charmonium, i.e., particles containing aheavy charm quark and an antiquark, is another well-studied phenomenon[188].1.7 Other probes of quark–gluon plasmaSince the charm quark is about ten times heavier than the strange quark,at SPS energies pairs of charm quarks can be formed only during the veryearly stages of the collision, as the nuclei begin to penetrate each other.In this early stage the colliding particles have the energy to overcome thehigher energy threshold. If the QGP phase is formed, these charmed qu-arks have less chance of forming a charmonium state, because the gluonspresent within the plasma hinder their binding, or/and break the boundstates. The observed strong suppression of the J/Ψ signal in 158A-GeV1 Micro-bang and big-bang 21J/Observerψparameter bImpactin dense matterJ/ path lengthψµµ−+Fig. 1.8. The CM picture of initial moments of nuclear collision and an illus-tration of the path traveled by J/Ψ in dense matter.Pb–Pb reactions is interpreted as being due to interactions with gluons.We will not develop this interesting topic in this book beyond a briefintroduction, which follows.Seen in the CM frame, the preformed J/Ψ must subsequently travelthrough a fireball of dense matter formed in the collision; see Fig. 1.8.The path in dense matter varies depending on the impact parameter,allowing one to evaluate the interaction (absorption) strength. This effectis particularly strong in most central collisions of nuclei.The experimental J/Ψ yields, seen in Fig. 1.9, show clearly the unusualinteraction strength of J/Ψ, when the path in dense matter is large ascharacterized by large values of the transverse energy generated by parti-cle interactions. The J/Ψ is observed, in the NA50 experiment, by virtueof its decay into µ+µ− [12]. In Fig. 1.9, the yield of J/Ψ is obtained withreference to the yield of the dimuon background, which is believed to scalejust like J/Ψ with the number of participant nuclei. We see, in Fig. 1.9at E⊥ > 50 GeV, a lesser increase in the suppression up to the last binsE⊥ > 120 GeV. It is this rapid two-fold increase in the suppression as afunction of E⊥ which makes an interpretation of these results, in termsof confined matter alone, difficult.Considering quantitatively only the absorptive effect of the QGP, prac-tically no J/Ψ should be observed for most central collisions at the higherenergies available at the RHIC. However, representing the absorption of22 A new phase of matter?3025200 8 020 40 60 100 120 1400510154035 [GeV]ET) / (J/B(DY)σσψ2.94.5208208PbPb 1996 with minimum biasPbPb 1998 with minimum biasPbPb 1996208µµFig. 1.9. Suppression of production of J/Ψ as a function of AA-collision central-ity, characterized in terms of the transverse energy ET produced in the reaction.Results of experiment NA50 [12].the J/Ψ as a result of its interaction with the gluon content of QGP, oneis led to consider the reverse reaction, and both break-up and formationbecome possible in the QGP medium:J/Ψ+ g⇐⇒ c + c̄.Depending on the absolute abundance of open charm, there could beconsiderable in-plasma production of J/Ψ. A quantitative study of thiseffect has recently been carried out [257], resulting in the prediction ofan enhancement in production of J/Ψ in central nuclear collisions in theenergy range accessible at the RHIC. Observation of this new mechanismof J/Ψ formation would constitute proof of the mobility of charmed quarksin the fireball of dense matter.1 Micro-bang and big-bang 23If this changing pattern of J/Ψ suppression turning into enhancementis confirmed experimentally to be present relative to the directly mea-sured yield of ‘open’ charm, with the expected energy dependence andcentrality dependence (peripheral interactions agree with extrapolationsfrom p-induced reactions), this will constitute convincing evidence for theformation of the deconfined form of matter.Charmed hadrons, i.e., particles with ‘open’ charm, have not yet beenobserved directly, but considerable effort is being made today to prepareexperiments at the SPS, RHIC and LHC that will have the capabilityto detect them. The experimental difficulty is the rapid disintegrationof charmed hadrons, requiring an extremely precise tracking of particlesbeginning very near to the interaction vertex. The availability of directmeasurements of the abundance of charm would provide a reliable baselinefor understanding the phenomenon of suppression of production of J/Ψ,and perhaps at higher energies than those at the SPS, enhancement of itsproduction.A few further signatures of deconfinement have also been proposed.Akin to the disappearance of J/Ψ, jets of hadrons produced by hard qu-arks and gluon arising from initial interactions should disappear due tothe enhanced strength of interaction in the ‘colored’ charge plasma of thedeconfined medium [72]. There is very little doubt that this idea works.However, this rapid quenching of hard particles is the mechanism of ther-malization of partons, aside of being a direct signature for a new phaseof matter. This mechanism provides the motivation for the statisticaldescription of the properties of dense matter. Can the thermalization ofpartons be seen as a signature of QGP? We will not address in this bookthis rapidly developing subject.Photons γ and lepton pairs emerging from the decay of virtual photonsγ∗ → e+e− and µ+µ− are, at first sight, the most promising probes ofthe dense hadronic matter [120, 154, 244]. Electromagnetic interactionsare strong enough to lead to an initial detectable signal, with secondaryinteractions being too weak to alter substantially the shape and yieldof the primary spectra. Direct photons and leptons contain informationabout the properties of dense matter in the initial moments of the collision.Of particular interest would be the possible exploration of the initial timeperiod leading to the formation of thermal equilibrium.Therefore, at first, photons and lepton pairs (often also called dileptons)originating from QGP, produced, e.g., by quark–antiquark annihilation,were considered to be the primary diagnostic tool, just as photons area powerful diagnostic for traditional (electromagnetic) plasmas. How-ever, thousands of particles are created in high-energy nuclear collisions.Among those the decay of neutral π0 mesons is known to produce a strong24 A new phase of matter?background. It is very difficult to extract the direct QGP photon signal,which is only a small fraction of all photons produced.Because dilepton formation, compared with formation of photons, re-quires one additional electromagnetic interaction, the yield of dileptonsis considerably smaller, by a factor 300 or more, than the yield of directphotons. However, the dilepton background is also much reduced. Whilethe directly produced photons are present at the level of 5% of the photonsignal observed, the amount of dileptons from formation of dense matteris believed to exceed the background by a factor as large as 3–5. However,dileptons have been shown to have other origins related to properties ofconfined hadronic matter and the observed pattern of dilepton produc-tion, given the large systematic experimental error, is difficult to interpret[22]. We will not dwell on this complex subject in this book.The diagnostic strength of the strangeness signature, in comparisonwith direct photons and dileptons, is that, aside from the overall enhance-ment of abundance, we also have the pattern of enhancement shown inFig. 1.6, and the matter–antimatter symmetry seen in Fig. 1.7. The yieldof strangeness is related to the initial most extreme conditions of the QGPphase, much like the photon and dilepton yields. However, strangenesshas relatively little background. The source of strangeness, gluons, is byfar more characteristic for the new phase of matter than is the sourceof photons and dileptons, which are electrical currents of quarks, presentboth in the confined and in the deconfined state of matter.2 Hadrons2.1 Baryons and mesonsAs the above first and qualitative discussion of signals of QGP formationhas shown, to pursue the subject we next need to understand rather wellthe properties of ‘elementary’ particles, and specifically strongly inter-acting (hadronic) particles. These are complex composites of the moreelementary strongly interacting particles: quarks and gluons. Quark-composite hadronic particles are final products emerging from heavy-ioncollisions, and play an important role in this book – there are two differenttypes of colorless quark bound states: baryons, comprising three valencequarks, and mesons, which are quark–antiquark bound states.Typically, each type of (composite) hadron has many resonances, i.e.,states of greater mass but the same quantum number, which are generallyinternal structural excitations of the lowest-mass state. Normally thelowest-mass states of a given quantum number are stable (that is, if theydecay, it occurs very slowly compared with the time scale of hadronicreactions), but there are notable exceptions, such as the ∆-resonances,2 Hadrons 25which are subject to the rapid hadronic decay ∆→ N+ π, which occurswithin a duration less than the life span of a hadronic fireball.On the other hand, some unstable hadronic resonances can be stableon this time scale. However, by the time relatively distant experimentaldetectors record a particle, all possible hadronic decays and even manyweak decays have occurred. To study particles produced in high-energynuclear collisions, we need to understand their decay patterns well.Because the two light quarks are practically indistinguishable understrong interactions, their bound states reflect a symmetry, referred to asisospin I. In addition one is normally quoting the strange-quark content(strangeness S) of a bound hadron state. Similarly, we refer to moreexotic baryons by quoting the value of their charm content C or bottomcontent B. It is believed that the heaviest quark t can not form hadronicbound states, since its life span is too short for a (quantum) orbit to form.We keep in mind that, for historical reasons, strange baryons, contain-ing strange quarks s, have negative strangeness (hyperon number) S, andthat strange mesons, with a strange quark s, are called antikaons K(sq̄)(we indicate in the parentheses the valence-quark content). Similar con-ventions were adopted for charm and bottom flavors.• BaryonsIn Fig. 2.1, we present baryons made of the four lightest quarks (u, d, s,and c). Were all four quark flavors degenerate in mass, we would haveinstead of the isospin SU(2) an SU(4) flavor symmetry. In Fig. 2.1,the vertical hierarchy is generated by the largest symmetry breaking inthe hadron mass spectrum, introduced by the relatively large mass ofthe charmed quark. The two ‘foundations’ of the SU(4) flavor-multiplet‘pyramids’ are the SU(3) flavor multiplets, arising when three quarks (u,d, and s) are considered: in Fig. 2.1(a) a spin-12 ‘octet’ (eight-fold multi-plicity); in Fig. 2.1(b) a spin-32 ‘decuplet’ (ten-fold multiplicity). SU(2)flavor (isospin) particle multiplets are located along straight horizontallines within the SU(3) flavor multiplet planes.The requirement that quarks form antisymmetric bound states leadsto classification in these two spin–flavor configurations of spin 12+ and32+, respectively, and thus when symbols are repeated, the excited stateis the higher spin state. States with positive intrinsic parity (upper index‘+’) are shown, since negative parity is associated with intrinsic angular-momentum excitation, or the presence of an additional particle compo-nent such as a gluon or, equivalently, a quark–antiquark pair. Therefore,generally such excited negative-parity states can decay rapidly into themore stable positive-parity states.Moving up in the two ‘pyramids’, we replace at each ‘floor’ one of thestrange quarks by a charmed quark, omitting the non-strange baryons.26 A new phase of matter? ++ ccc ++ cc + cc + cc ++ c + c 0 c − 0 + + 0 − − − ++(b) + c ++ c 0n p c0(a)ddcdscudcuscuucuuduusussdssuddddsddddssddsussuusuududd udssscuscdscuucuccsccdcc + cc ++ cc + cc 0 cuuu 0 − − + 0udc + c + c c + 0 c 0 cdcc uccddcudssscsccsss 0 cΞ ΞΞΞΞ ΞΞ ΞΞ ΞΩΩΩΩΩΩ∆ ∆ ∆ ∆Σ Σ ΣΣ Σ ΣΣ Σ ΣΣ Σ ΣΛΛΞ ΞFig. 2.1. Baryons (positive parity, superscript +): (a) generalization of thespin- 12+ octet and (b) generalization of the spin-32+ decuplet classified accordingto their valence-quark content.In the naming convention, the name of the corresponding SU(3) flavorparticles is retained, with the added lower index indicating how manycharmed quarks have been introduced to replace strange quarks. Simi-larly, when bottom-flavor-involving baryons are considered (not shown inFig. 2.1), the lower index ‘b’ is used to indicate how many strange quarkshave been replaced by bottom quarks.•MesonsIn Fig. 2.2, we present mesons made of the four lightest quark–antiquarkpairs (u, d, s, and c). Now the center-planes of the two SU(4) fla-vor multiplets are the well-known SU(3) flavor (u, d, and s) multiplets:Fig. 2.2(a), a spin-0 octet, with the ninth state ηc = cc̄; Fig. 2.2(b), aspin-1 octet, which now also includes the ninth state J/Ψ = cc̄. We recallthat π0 = (uū− dd̄)/√2 and that the heavier η and η′ share the strange-2 Hadrons 27sD0DsD–D0K–+K–K(a)sDDDsD− +K(b)*0K *−*+K*0D 0*D*−*−*+−*+−cdcu−cs−us−ds−su− sd−ud−uc−sc−dc−0J/uc−sc−dc−−cdcu−cs−+D++K 0us−ds−su− sd−du−du−0D′c0ud−K 0*ψ φωρρ ρηηηπ π πFig. 2.2. Mesons (negative parity, superscript −): (a) pseudoscalar 0− and (b)vector 1− mesons, classified according to their valence-quark content.quark component ss̄; there is no pure spin-0 ss̄ state. In the spin-1 octet,case φ0 = ss̄ is a pure strange-quark pair state. Not considered in Fig. 2.2are states containing bottom quarks:B = ±1, bottom mesons: B+ = ub̄, B0 = db̄, B0 = d̄b, and B− = ūb;B = ±1, S = ±1, bottom strange mesons: B0s = sb̄, and B0s = s̄b; andB = ±1, C = ±1, bottom charmed mesons: B+c = b̄c and B−c = bc̄.2.2 Strange hadronsFor the physics of interest in this book, a particularly important familyof particles consists of those which contain strange quarks. Since we willaddress these particles by implying some of their particular properties, wewill summarize in the following their key features, and we comment onthe experimental methods for their detection. A similar comment appliesto heavier flavors (charm and bottom), and we also present a sample ofthese particles below.28 A new phase of matter?πΛpFig. 2.3. A schematic representation of the Λ-decay topological structure show-ing as a dashed line the invisible Λ and the decay ‘V’ of the final-state chargedparticles. Tracks of other directly produced charged particles propagating in amagnetic field normal to the plane of the figure are also shown.• Hyperons Y(qqs) and Y(q̄q̄s̄)Sometimes all strange baryons are referred to as hyperons. We prefer touse this term for singly strange baryons.— The isospin singlet lambda, Λ(uds), is a neutral particle of mass 1.116GeV that decays weakly with proper path length cτ = 7.891± 0.006 cm.The dominant and commonly observed decay isΛ→ p + π−, 63.9± 0.5%.The decay of a neutral particle into a pair of charged particles forms acharacteristic ‘V’ structure as shown in Fig. 2.3. The other importantweak decay,Λ→ n + π0, 35.8± 0.5%,has only (hard-to-identify) neutral particles in the final state.In addition to the 12+ ground state (positive parity, spin 12 :12+), weencounter a 12− resonance Λ∗(1.405) and also a 32− state Λ∗(1.520). Thereis no ‘stable’ positive parity 32+ iso-singlet Λ. Λ∗(1.520) has a remarkablynarrow width Γ = 15.6 MeV, even though the hadronic decay into theNK channel is open. All Λ∗ excited resonance states (13 are presentlyknown, with mass below 2.4 GeV) decay hadronically into two principalchannels:Λ∗→Y+meson(s),Λ∗→N+K.2 Hadrons 29Since the hadronic decays have free-space proper decay paths of 1–10 fm(widths Γ = 16–250 MeV), all these resonances can not be distinguishedfrom the ground state of corresponding flavor content and contribute tothe abundance of the observed ‘stable’ (on hadronic time scale) strangeparticles Λ and K. However, it is possible to measure their yields bytesting whether the momenta of the expected decay products add to theenergy–momentum product of a particle with the ‘invariant’ mass of theparent resonance.The practical approach to the observation of Λ is to detect, in a trackingdevice, the (dominant) decay channel. The two final-state charged parti-cles are pointing to a formation vertex remote from the collision vertex ofprojectile and target. The ancestor neutral particle should point to theinteraction ‘star’ (see Fig. 2.3) and should have the correct mass and lifespan. This approach includes, in a certain kinematic region, the eventswhich originate from the decay of Ks (see below). A well-establishedmethod of analysis of data allows one to distinguish the hyperon decayproducts from kaon decay products [211]. The procedure we describe inthe following is named after Podolanski and Armenteros [208]. It ex-ploits the difference between the phase-space distributions of the pair ofparticles produced.The invisible neutral particle, reconstructed from the charged tracksforming the decay ‘V’, has a definitive line of flight, shown by the dashedline in Fig. 2.3. The magnitude of the transverse momentum q⊥ mea-sured with reference to this line of flight is by definition the same forboth produced, oppositely charged, particles. The longitudinal momen-tum p+‖ = p−‖ for each particle is dependent on a random angular distri-bution of decay products of charges + and −. The relative p‖ asymmetry,a =p+‖ − p−‖p+‖ + p−‖, (2.1)can span the entire range −1 < a < 1 when the decay products havethe same mass, as is the case for decay of kaons. When a proton isproduced in Λ decay, this massive particle carries the dominant fractionof p‖ momentum and the event appears near a → 1. Similarly, the Λdecay populates the domain a → −1. Events appear around the clearlyvisible kinematic q⊥(a) lines shown in Fig. 2.4. For each decay of particlesKs, Λ, and Λ, there is a clear domain where accidental misidentificationof particles is impossible. e+–e−-pair events produce a background foundnear to q⊥ = 0. The entire (q⊥, a) plane is filled with events arisingfrom accidental ‘V’s. Since the physical signal is highly concentratedalong the kinematic lines seen in Fig. 2.4, the signal-to-noise ratio is veryfavorable. As a consequence, the measurement of production of neutral30 A new phase of matter?00.10.20.3− 1 − 0.5 0 0.aKsΛ− ΛqTFig. 2.4. The Podolanski–Armenteros representation of a particle-decay event:the transverse momentum qT in a two-particle decay, as a function of the relativelongitudinal momentum asymmetry a, see Eq. (2.1).strange hadrons using Podolanski–Armenteros analysis is highly reliable.— The isospin triplet uds, uus, dds, i.e., Σ0(1.193 GeV) and Σ±(1.189GeV), varies in its properties. There is only one dominant decay channelfor the Σ− decay,Σ− → n + π−, cτ = 4.43± 0.04 cm.Because there are two isospin-allowed decay channels of similar strengthfor the Σ+,Σ+→p + π0, 51.6%,→n + π+, 48.3%,the decay path is nearly half as long, cτ = 2.4 cm. Σ± have not yetbeen studied in the context of QGP studies, since they are more difficultto observe than Λ – akin to the Ξ decay (see below) there is alwaysan unobserved neutral particle in the final state of Σ± decay. UnlikeΞ decay, the kink that is generated by the conversion of one chargedparticle into another, accompanied by the emission of a neutral particle,is not associated with subsequent decay of the invisible neutral particleaccompanied by a ‘V’ pair of charged particles; see Fig. 2.7 below.2 Hadrons 31Fig. 2.5. The relative yields of the three dominant contributions to the finalyield of Λ: direct production (solid line), Σ0 contribution (dashed line), andΣ∗(1385) contribution (chain line), as a function of T , the temperature of thehot source of these particles.— The neutral Σ0(uds) undergoes a rapid electromagnetic decay:Σ0 → Λ + γ+ 76.96± 0.02 MeV, cτ =(2.2± 0.2)× 10−9 cm,τ =(7.4± 0.7)× 10−20 s.For the observer in the laboratory, this secondary Λ is practically indis-tinguishable from the direct production in the interaction vertex. Con-sequently, all measurements of Λ of interest for this book combine theabundances of Λ and Σ0, including all the higher resonances that decayhadronically into these states, in particular Σ∗(1385).As with Λ, there are several (nine) heavier Σ resonances known atm ≤ 2.4 GeV. When they are produced, all decay hadronically, producingK, Λ, and Σ. Particularly important is the strong resonance Σ∗(1385):Σ∗(1385)→Λ + π, 88± 2%,→Σ+ π, 12± 2%.Thus, 92 ± 3% of Σ∗(1385) decays into Λ. Since the spin of Σ∗(1385) is 32 ,and its isospin is 1, the degeneracy of Σ∗(1385) is six times greater thanthat of Λ. For this reason the final yield of Λ is in fact predominantlyderived from decays of Σ∗, as is shown in Fig. 2.5, in which the chainline describes the partial contribution of Σ∗ to the final yield of Λ. Thesolid line is the directly produced component, and the dashed line is thecontribution of Σ0.32 A new phase of matter?Fig. 2.6. The multiplicative factor allowing one to infer the total yield of qqsstates from the observed yield of Λ as a function of T , the temperature of thehot source of these particles.An important practical issue is the determination of the total yieldof singly strange baryons (qqs), considering that we have available theobserved yield of Λ. It is generally assumed that abundances of the threeisospin-1 states Σ+, Σ− and Σ0 produced in relativistic nuclear collisionsare equal. In the first instance, let us ignore the influence of Σ∗(1385),which we introduce in the next paragraph. Considering the difference inmass between Λ and Σ, ∆m = 77 MeV, and assuming that the relativeabundance yield is appropriately described by the relation,YΣiYΛ=(1 +∆mm)3/2e−∆m/T , (2.2)derived from the relative thermal phase-space size, Eq. (10.50c), one findsfor a reasonable range of values T = 160 ± 15 MeV, implied by thermalhadron production models, that the total yield of hyperons (qqs) is ob-tained by multiplying the experimentally observed yield of Λ by a factorFY = 1.8.This estimate has to be modified in order to allow for the importantrole Σ∗(1385) plays as a contributor to the production of Λ. In Fig. 2.6,we evaluate the yield factor FY allowing for the contributions of Σ0 andΣ∗(1385) to the yield of Λ, using thermal phase space, Eq. (10.51). Thereis very little variation with T in a wide range, 100 MeV < T < 200MeV, and we conclude that, for all practical purposes, one can use theyield factor FY = 1.48 ± 0.03 to estimate the yield of singly strange2 Hadrons 33ΛππpΞFig. 2.7. A schematic representation of the topological structure of decay of Ξ−showing as a dashed line the invisible Λ emerging from the decay kink and thedecay ‘V’ of the final-state charged particles. Tracks of other directly producedcharged particles propagating in a magnetic field normal to the plane of thefigure are also shown.baryons (qqs) from the observed yield of Λ. Naturally, this result holdsfor antibaryons as well. In many publications a slightly larger value,FY = 1.6, is applied.At a smaller level the decays of multistrange Ξ and Ξ also contributeto the yield of Λ and Λ, but this contribution is usually separated exper-imentally.• Cascades, Ξ(qss) and Ξ(q̄s̄s̄)The doubly strange cascades, Ξ0(uss) and Ξ−(dss), are below the massthreshold for hadronic decays into hyperons and kaons, and also just belowthe weak-decay threshold for the π+Σ final state. Consequently, we haveone decay in each case:Ξ−(1321)→Λ + π−, cτ = 4.9 cm,Ξ0(1315)→Λ + π0, cτ = 8.7 cm.The first of these reactions can be found in charged-particle tracks sinceit involves conversion of the charged Ξ− into the charged π−, with theinvisible Λ carrying the ‘kink’ momentum. For Ξ− to be positively iden-tified, it is required that the kink combines properly with an observed‘V’ of two charged particles identifying a Λ decay. This decay topology isillustrated in Fig. 2.7.There are also several Ξ∗ resonances known, which normally feed downin a hadronic decay into the hyperon and kaon abundances:Ξ∗(qss)→ Y(qqs) + K(q̄s).34 A new phase of matter?Fig. 2.8. The relative strength of the two dominant contributions to the finalyield of Ξ: direct production (solid line), and Ξ∗(1530) (chain line), as a functionof T , the temperature of the hot source of these particles.The main exception is the hadronic decay of the spin-32 recurrence of thespin-12 Ξ ground state:Ξ∗(1530) → Ξ + π, Γ = 9.5MeV.This relatively small width corresponds to cτ = 21 fm and this decayoccurs outside of the space–time region of hot matter.Since the spin-32 state is populated twice as often as is the spin-12 groundstate, its relative suppression due to the greater mass (∆m 200 MeV)is by a factor of 0.6, as is shown in Fig. 2.8. Despite this non-negligibledecay contribution, the cascade spectra are, at high transverse momenta,the most representative among measured hadronic spectra of particlesdirectly produced by the fireball of hot and dense matter.• Omegas, Ω−(sss) and Ω−(s̄s̄s̄)There are several primary weak-interaction decay channels leading to therelatively short proper decay path, cτ = 2.46 cm:Ω−(1672)→ Λ +K−, 68%,→ Ξ0 + π−, 24%,→ Ξ− + π0, 9%.The first of these decay channels is akin to the decay of the Ξ−, except thatthe pion is replaced by a kaon in the final state. In the other two options,2 Hadrons 35after cascading has finished, there is a neutral pion in the final state,which makes the detection of these channels impractical. The Λ + K−decay channel is not unique for the Ω−(1672) state; there is a Ξ∗(1690)that also decays in the same way. However, this is a rapid hadronic decaythat occurs in the interaction vertex and can be easily separated from theweak decay of the Ω.The highest known resonance Ω∗(2250) is rather heavy. However, pres-ent experiments do not exclude the possibility of a light resonance, witha dominant electromagnetic decay, Ω∗(M∗ < MΩ +mπ) → Ω + γ. Thepresence of this resonance would greatly affect the theoretical expecta-tions regarding the production of Ω and Ω in heavy-ion interactions. Thestatistical degeneracy due to the high spin of such resonances could eas-ily enhance their abundance, akin to the situation with the Ξ∗ state wedescribed above.• Kaons, K(qs̄) and K(q̄s)— Neutral kaons, KS and KL (m = 497.7 MeV)This is not the place to describe in detail the interesting physics of theshort- and long-lived neutral kaons, except to note that both are orthogo-nal combinations of the two electrically neutral states (ds̄) and (d̄s). Theshort-lived combination KS has cτ = 2.676 cm and can be observed in itscharged decay channel:KS→π+ + π−, 69%,→π0 + π0, 31%.Care must be exercised to separate the KS decay from Λ decay, since inboth cases there are two, a priori unidentified, charged particles in thefinal state, making a ‘V’ originating from an invisible neutral particle; seeFig. 2.4.The long-lived kaon KL, with cτ = 1551 cm, has not been studied inrelativistic heavy-ion-collision experiments. Since both the KL and the KSarise from the time evolution of hadronically produced K(ds̄) and K(d̄s)their abundances are essentially equal.— Charged kaons, K+(us̄) and K−(ūs) ≡ K+ (m = 493.7 MeV)Charged kaons can in principle be observed directly, both in charged-particle-tracking devices and in magnetic spectrometers, since their massdiffers sufficiently from those of the lighter π± and the heavier proton/antiproton. Also, K±(494) decays with cτ = 371 cm, with three principalchannels:K+→ µ+ + νµ, 63.5%,→ π+ + π0, 21.2%,→ π+ + π+ + π−, 5.6%.36 A new phase of matter?The last one, with three charged hadrons in the final state, can be usedto identify charged kaons within tracking devices.Considering the quark content, one sees that the average of the pro-duction of K+(us̄) and K−(ūs) satisfies 〈KS〉 0.5(〈K+〉+ 〈K−〉).• The φ-meson, φ(ss̄)The vector (J = 1) meson φ is believed to be a ‘pure’ bound state of thestrange-quark pair φ = ss̄. With mass 1019.4 MeV, it has a relativelynarrow full width Γφ = 4.43 MeV, since the decay into a pair of kaonsis barely possible in free space. There are two open and relatively slowdecay channels leading to the formation of pairs of leptons,φ→ e+ + e−, 0.03%, Γe+e− = 1.32± 0.05 keV,→µ+ + µ−, 0.025%, Γµ+µ− = 1.3 ± 0.2 keV,allowing the determination of the number of φ-mesons emerging from thehadronic-interaction region. When absolute yields of particles are difficultto determine, one can compare, using the dilepton decay channel, theyield of φ with the yield of non-strange partner mesons ρ(770) (Γe+e− =6.77± 0.32 keV) and ω(782) (Γe+e− = 0.60± 0.02 keV).2.3 Charm and bottom in hadrons• Onium-mesons, cc̄, bb̄ and b̄cA brief summary of the physical properties of the three main onium statesfollows; the first three properties, mass (note the substantial systematicand statistical uncertainty in the mass of the Bc), lifetime and its in-verse the width, are determined by measurement, followed by the bindingenergy with regard to dissociation into mesons, and the average size de-termined within heavy-quark potential models.J/Ψ(c̄c) Bc(b̄c) Υ(b̄b)Mass [MeV] 3097 6400± 390± 130 9460τ [ps] 7.6× 10−9 0.46± 0.17± 0.03 12.5× 10−9Γ [keV] 87± 5 1.4× 10−6 52.5± 1.8Binding energy [MeV] 630 840 1100〈r〉 [fm] 0.42 0.34 0.22The spin-1 J/Ψ(cc̄) has a partial width of 6% (per channel) projectedinto the experimentally observed lepton-pair channels e+ + e−, µ+ + µ−,i.e., Γcl̄l= 5.26 keV; the still narrower bb̄ has Γbl̄l= 0.3 keV, which is2.5% of the total width. We note that J/Ψ is produced very rarely inheavy-ion collisions, e.g., in Pb–Pb interactions at 158A GeV one J/Ψ isproduced for each 3×106 mesons. Production of open charm has not been3 The vacuum as a physical medium 37measured but is at the level of 0.2–0.6 cc̄ pairs per Pb–Pb interaction at158A GeV. Thus, only one in about 2000 cc̄ pairs produced emerges as abound J/Ψ(cc̄) state. The uncertainty in this estimate is at least a factorof two and depends on the centrality of the interaction. It is hoped thatfurther experimental information will become available soon, allowing usto understand this ratio more precisely.The excited state Ψ′ has a yield five times smaller. There has not yetbeen a measurement of production of the other onium states in nuclearcollisions.In the LHC energy range, one can expect that the bound state of b-quarks, the upsilonium Υ(b̄b), will assume a similar role to that which istoday being played, at SPS and RHIC energies, by J/Ψ.The other heavy-quark bound state that is of interest is the Bc(b̄c).This quarkonium state is so rarely produced that it was not discovereduntil very recently [9, 10]. However, it has been studied extensively theo-retically, and the currently reported mass, M = 6.4 ± 0.39 ± 0.13 GeV,is in good agreement with the theoretical quark-potential model expecta-tions. The life span, τ 0.5 ps, cτ 150µm, implies that the currentsilicon pixel detector technology allows one to distinguish the productionvertex from the Bc(b̄c) decay vertex.The conventional mechanism for production of Bc(b̄c) requires the for-mation of two pairs of heavy quarks in one elementary interaction, fol-lowed by the formation of a bound state. The probability of these threeunlikely events occurring in one interaction is not large and hence nei-ther is the relative predicted yield, (Bc + B∗c)/(b, b̄) (3–10) × 10−5 at√sNN = 200 GeV [169]. This small value implies that ‘directly’ producedBc (both in J = 0 and J = 1 channels B∗c and B̄∗c) cannot be observedat the RHIC. On the other hand, an enhancement in production of thisstate is expected in the QGP-mediated recombination [239], which couldlead to a measurable rate of production in nuclear interactions. Since thequark-recombination mechanism of production requires mobility of heavycolor-charged quarks, observation of this new mechanism for the forma-tion of this exotic meson would constitute another good signature of thedeconfined QGP phase.3 The vacuum as a physical medium3.1 Confining vacuum in strong interactionsTheoretical interest in the study of relativistic heavy-ion collisions origi-nates, in part, from the belief that we will be able to explore the vacuumstructure of strong interactions and, in particular, the phenomenon ofquark confinement. The picture of confinement can be summarized as38 A new phase of matter?follows:1. all strongly interacting particles are made of quarks and gluons;2. quarks q and gluons g are color charged [123], but all asymptotic ob-servable physical states they can form are color neutral;3. therefore, the true vacuum state |V 〉 abhors color;4. there is an excited state |P 〉, referred to as perturbative vacuum, inwhich colored particles can exist as individual entities and thereforemove freely;5. |P 〉 differs essentially from |V 〉, the true vacuum, and in particular,it differs by a considerable amount of energy density in the regions ofspace–time in which the |V 〉 structure is dissolved into |P 〉.In the ‘true vacuum’ (in which we live), color-charged quarks and gluonsare ‘confined’. However, under extreme conditions of density and tem-perature, we should reach the crossover to the color-conductive phase ofthe vacuum. In such a space–time domain, nearly free propagation ofcolored quarks and gluons is thought possible. This picture of hadronicinteractions is consistent and indeed justifies the perturbative approachto quantum-chromodynamics (QCD) interactions. It is the foundationthat allows us to describe hadrons as bags, i.e., confined bound statesof quarks, see section 13.2. We also use these simple, but essential, fea-tures in the discussion of the physical properties of the QGP state insection 4.6. The melted color-conductive state |P 〉 is a locally excitedspace–time domain in which quarks and gluons can move around. Thisstate has properties that we would like intuitively to associate with a nor-mal physical state, since it is simple, structureless. We must keep in mindthat the situation is, however, inverted relative to our expectations. Sincequarks and gluons are not observed individually, they cannot propagatein the true vacuum state, thus the true physical ‘ground’ state |V 〉 mustbe complex and structured, and it is the excited state that is simple andstructureless.Vacuum structure keeps the colored particles bound and confined. Qu-ark confinement has not been explained to be a direct result of quark–quark interaction, generated by the color charge and exchange of gluons.Rather, this force determines within a domain of perturbative vacuum |P 〉the structural detail: for the ground state the structure of the hadronicspectrum; at sufficiently high excitation, the properties of the color plasmaof hot quarks and gluons. To be able to move color charges within a re-gion of space, one needs to ‘melt’ the confining structure. For a first-orderphase transition, the two phases have a difference in energy density, thelatent heat B, per unit of volume,B ≡ εQGP(Tcr, Vcr; b)− εHG(Tcr, Vcr; b) ≈ 0.5 GeV fm−3. (3.1)3 The vacuum as a physical medium 39We would like to determine, by studying the QGP phase, the magnitudeof B. So far, only relativistic nuclear collisions can deliver (to a largeregion of space) the required energy and are the best and only tool we havetoday to study the process of melting of the QCD vacuum, see section 5.2.We will discuss the experimental methods further in chapter 5.The vacuum properties of strong interactions can be explored only whenthe locally deconfined state, the QGP, is experimentally established. Inour opinion, the study of the physical properties of the hadronic vacuum,in particular ‘confined vacuum melting’, is the fundamental challenge mo-tivating the high-energy nuclear-collision experimental program. It is rel-evant to note that the key ideas and concepts underpinning the possibilityof finding the vacuum ‘melting’ are robust against change and evolutionof our knowledge: neither the questions about the existence of a true(discontinuous) phase transition between the hadronic vacuum states northe possible quark substructures will greatly influence these considera-tions. All we want is to determine that the color-melted state containsparticle-like quark–gluon excitations with established symmetries and in-teractions.The most interesting property of the true QCD vacuum |V 〉 is that itabhors the color charge of quarks and gluons. However, we are interestedin determining and understanding its other physical properties. The ap-pearance of a glue ‘condensate’ field, i.e., the vacuum expectation valueof the ‘square’ of the gluon field, the so-called field-correlator in the truevacuum state [242, 243], is of particular relevance for the understandingof |V 〉. With the glue fields defined as in section 13.4 we have12F 2 ≡∑a12F aµνFµνa =∑a[ /B 2a − /E 2a ], (3.2)where we use Einstein’s summation convention for repeated Greek indices.The value of F 2 is obtained by studying QCD sum rules [197, 198, 242,243], and is in agreement with the results obtained numerically usinglattice-gauge-theory methods [100, 101]:∆F 2 ≡ 〈V |αsπF 2|V 〉−〈P |αsπF 2|P 〉 (2.3± 0.3)× 10−2GeV4, (3.3)= [390± 12 MeV]4.Here, αs = g2s /(4π) is the coupling constant for the strong interaction.Since in empty space the vacuum state is field-free, i.e., the vacuum expec-tation value of the gauge field vanishes, the appearance of a non-vanishingvacuum expectation value of the square of the gauge field in Eq. (3.3) isa quantum effect without a classical analog.40 A new phase of matter?3.2 Ferromagnetic vacuumWe now describe a model and discuss other properties of the vacuumstate that are related to the remarkable result Eq. (3.3). Because of thenon-abelian nature of color charges, the quanta that mediate the colorforce, gluons, can themselves interact by means of exchanging gluons.Since gluons are massless, there is no energy gap that would stabilizetheir number. An attractive force between them will induce a majorrealignment in the perturbative wave function, i.e., |P 〉, of the many-bodygluon system.Upon inserting Eq. (3.2) into Eq. (3.3), we see that the color B-field(magnetic) fluctuations dominate the color E-field (electrical) fluctua-tions:∆∑a/B 2a = ∆∑a/E 2a + 2[390± 12MeV]4. (3.4)Here, ∆ is defined as on the left-hand side of Eq. (3.3). The naturalinterpretation of this equation is that the true vacuum structure is pre-dominantly magnetic. Indeed, an instability of the perturbative vacuumof QCD toward the formation of a ferromagnetic structure, was discov-ered early on in the development of QCD [56, 187, 236]. This effect hasbeen shown to arise due to the attractive magnetic spin–spin interactionof gluons [35, 199, 200]. This spontaneous ferromagnetic instability par-allels, in many important aspects, the instability in QED vacuum in thepresence of constant electro-magnetic (EM) fields.In QED, in the presence of a constant electrical field E, there is anonvanishing probability of spontaneous particle-pair formation, with theprobability per unit time and volume given by [240]w =α/E2π2∞∑n=11n2exp(−nπm2|e /E|). (3.5)The electromagnetic fine-structure constant, α 1/137, is relativelysmall, and the massm of the lepton (electron) produced is large comparedwith the laboratory fields available. Thus, in fact, this process has neverbeen observed. The physical origin of the QED vacuum instability residesin the fact that, in a constant infinitely extending field, we can always finda potential difference between two distant points that exceeds the pairmass, and thus spontaneous pair production can ensue [219]. Schwinger’srate Eq. (3.5) is arising in such a description from the process of quantumtunneling through a barrier that the potential V =∫d/x/E implies, andtherefore it can be adapted with ease to the study of QCD [88, 132].This mechanism is serving as the basis for particle production within thecolor string models, in which breaking of the color-electrical-flux tube3 The vacuum as a physical medium 41connecting rapidly separating quarks provides the mechanism for particleproduction [39].In the case of QED, the particles produced are screening the field source,and the vacuum-state energy still has a local minimum around the per-turbative vacuum-state configuration with vanishing EM fields. In thisregard the situation is different in QCD, in which there is a ferromagneticinstability. To understand this QCD magnetic instability, recall that, ina constant magnetic field of magnitude B, a particle with spin projectionσ and orbital momentum l = 1, 2, 3, . . ., with reference to the directionof B has the Landau energyE2lσ = m2 + k2‖ + 2gsB(l + σ +12), (3.6)and the effective degeneracy isgB =V 2/32πgsB. (3.7)For σ = −12 (leptons, quarks), the lowest energy level for k‖ = l = 0 isat E20 = m2, as is seen in Eq. (3.6). However, for spin-1 gluons, stateswith σ = −1 display an instability whereby E20 becomes negative forgB > m2 = gsBcr. For gluons with mg = 0, this occurs for an arbitrarilysmall value of B. Therefore, the spectrum of Landau states begins at aminimum momentum, k‖ >√gsB for the relevant case of gluons withσ = +1 and l = 0. This has a profound impact on the zero-point energyof the vacuum.The sum over all (stable) modes of particles (+) and antiparticles (−)yields the vacuum energy, that is the expectation value of the Hamiltonianin the perturbative state |B〉 in the presence of the magnetic field B:〈B|H|B〉 = (−)2σ 12(∫∑+E+(B)−∫∑−E−(B))≡ Eσ0 (B)V. (3.8)The coefficient of the zero-point energy density Eσ0 reflects the spin-statis-tics relation. The appearance of the lowest-angular-momentum statesof the minimum allowable momentum leads for the gluon fields (aftersubtraction of the perturbative state and renormalization) to [35, 199, 200]EQCD0 (B) =b02(gsB)24πlog((gsB)2Λ2), (3.9)where b0 = (1/2π)(11nc/3− 2nf/3) > 0 is as given by Eq. (14.14). Equa-tion (3.9) proves that the vacuum state acquires an instability in the limitat B = 0, since the vacuum energy does not exhibit a minimum in thislimit. We find a new minimum of the vacuum energy at a finite value of42 A new phase of matter?(gsB)2. The scale of the ‘condensation’ field is determined by the renor-malization scale Λ.While these results prove the instability of the perturbative state |P 〉,given the variational approach the ferromagnetic-vacuum model may be avery poor approximation to the actual vacuum structure of |V 〉. Thoughthe energy of the |P 〉 vacuum is lowered, and we find a minimum at afinite value of the magnetic field B, it cannot be expected that we have,within this crude model, reached the lowest energy corresponding to thetrue state |V 〉. Even so, Eq. (3.9) allows a first estimate of the latentenergy involved in melting the (magnetic) QCD vacuum structure to bemade:BB ≡ −E(Bmin) =b08π(gsB)2min <∼ B. (3.10)BB is seen as the variational approximation to the true value B. Thevalue at the minimum underestimates the true gain in energy within amore accurate vacuum structure model. To determine the scale of themagnetic field near the minimum of the energy density, we take as theaverage value of the square of the vacuum magnetic field the vacuumexpectation value of the field operator squared, Eq. (3.3):12π2(gsB)2min ≡ δαsπ〈V |F aµνFµνa |V 〉. (3.11)δ is a positive number by definition. It can not be bigger than unity. Theexample of the quantum oscillator expression for 〈x2〉 suggests that it isprobably small relative to unity. The nonperturbative energy density ofthe vacuum state Eq. (3.10) is then of the magnitudeB >∼11− 23nf8〈V |εαsπF 2|V 〉 δ 2.5 GeV fm−3. (3.12)We also note the Curie-point (the temperature at which the magneticferric structure melts) of the magnetic QCD state at temperature Tcr B1/4B , at which one finds a strong first-order phase transition [192].We infer from this exploration of a magnetic-vacuum model that theperturbative QCD vacuum |P 〉 is unstable for T < Tcr, and that the tran-sition to the true vacuum state involves a considerable release of latentheat. However, the quantitative results discussed here are merely provid-ing an order-of-magnitude estimate. In fact, many other more complexsemi-analytical models of the QCD vacuum structure were developed, ofwhich the other most often addressed case is the instanton vacuum. Inthis approach, one draws on the (infinite) degeneracy of the unstructuredstate. A more thorough discussion of this model is offered in the mono-graph of Shuryak [245].3 The vacuum as a physical medium 433.3 Chiral symmetryThe light u and d quark masses, which we have considered in table 1.1,are just slightly different when they are measured on the energy scaleassociated with the QCD vacuum structure, which is of the order of afew hundred MeV. This opens up an interesting interplay between theeffective flavor symmetry of QCD and the vacuum properties. Recall thatup and down quarks satisfy the relativistic Dirac quantum field dynamics,Eq. (13.79),(iγµ∂µ −m)Ψ = 0, (3.13)from which there arise two identities,∂µjµ+≡∂µ(ūγµd) = i(mu −md)ūd, (3.14)∂µj5µ+ ≡∂µ(ūγµγ5d) = i(mu +md)ūγ5d, (3.15)where u and d are the Dirac spinor-field operators representing the twolight-quark flavor fields of current-quark masses mu and md, respectively.The subscript ‘+’ reminds us that these currents ‘lift’ the ‘down’ quarkto the ‘up’ quark; in the quantum-field-theory formulation this current isan iso-raising operator that increases the electrical charge by +|e|.When the quark masses are equal, the isospin-quark current is con-served in Eq. (3.14), which implies that the Hamiltonian is symmetricunder transformations that mix equal mass ‘u’ with ‘d’ quarks; this isan expression of the isospin-SU(2) symmetry of strong interactions; thissymmetry is broken by the electromagnetic and weak interactions, andby the difference in current-quark masses mu = md, as seen in Eq. (3.14).In case that the light quark masses were to vanish, by virtue of Eq. (3.15),the pseudo-vector isospin-quark current would also be conserved. Thus,when we are dealing with physical situations in which the current quarkmasses can be neglected, each isospin quark doublet operator q ≡ (u, d)must be invariant under transformations that comprise two ‘isospin rota-tions’ associated with the two current-conservation laws.When we are motivated by the physical properties of weak interactions,it is common to study the left- and right-handed quark fieldsqL,R ≡ 12(1± γ5)q.The reader is reminded that, for the right-handed case, the spin rotatesright-handedly around the propagation axis, that is the spin and momen-tum vectors are pointing in the same direction; the ‘helicity’ is positive. Itcan be shown, on general grounds, that, for massless fermions, the helicityis conserved.On forming the sum and difference of Eqs. (3.14) and (3.15), one findsthat both the right- and the left-handed doublets form conserved iso-currents; thus the overall symmetry is SU(2)L × SU(2)R. This is the44 A new phase of matter?so-called chiral symmetry, i.e., ‘handedness’ symmetry. It is important toremember that this symmetry can be exact only if the masses of u andd quarks vanish exactly, and electro-weak interactions that distinguishthe light quark flavor can be neglected. Since mu +md = 5–15MeV, weexpect this nearly exact chiral symmetry to be manifesting itself stronglyat the hadronic energy scale O(1) GeV, literally wherever we ‘look’. Yet,there is no sign of the corresponding symmetry in the hadronic spectrum;there are no double doublets of hadronic parity states, e.g., we know thatthere is only one isospin doublet of nucleons (proton–neutron), not two:the second, chiral-symmetry-motivated opposite-parity, isospin doublet ofnucleons is not observed. It would seem that chiral symmetry is badlybroken by strong interactions, presumably the mass difference of quarkssomehow matters.However, the Adler–Weisberger sum rules, which relate weak and strongproperties, confirm the presence of the intrinsic SU(2) × SU(2) symme-try in the elementary Hamiltonian. We refer to the recent discussion ofWeinberg for a more comprehensive introduction to this rather importantmatter [268]. Nambu resolved this conflict between weak and strong in-teractions by proposing that the required symmetry-breaking mechanismis part of the structure of the strongly interacting vacuum state, and thephysical hadron spectrum can indeed break the intrinsic (almost) chiralsymmetry of the Hamiltonian [195]. Weinberg is of the opinion that theimmediate acceptance of QCD as the dynamic theory of strong inter-actions was very much the result of a rather natural implementation interms of practically massless ‘current’ u and d quarks (see Eqs. (3.14) and(3.15)) of these contradictory properties of weak and strong interactions.The Nambu breaking of chiral symmetry in the hadronic spectrum re-quires that, in the limit that the quark masses vanish exactly, there wouldbe an exactly massless Goldstone boson, a particle with quantum num-bers of the broken symmetry, thus spin zero, negative parity, and isospinI = 1. Since the chiral symmetry of the strong-interaction Hamiltonianis not exact, the lowest-mass particle with these quantum numbers, thenearly massless pion state, expresses the properties of the massless Gold-stone meson of strong interactions.One could argue that the finite pion mass noticeable on the scale ofhadronic interactions is removing from the hadronic spectrum most ofthe signature of chiral symmetry. The missing parity doublets of allstrongly interacting particles are a ‘direct product’ of the Goldstone bo-son (pion) with all elementary hadron states. This, in turn, implies thatmany features of the hadronic spectrum, and possibly of the vacuumstructure, should depend on the small, and seemingly irrelevant, currentquark masses we see in Eqs. (3.14) and (3.15). How this could happen isnot understood.3 The vacuum as a physical medium 45We show now that, in the limit of vanishing quark masses, we expectthe pion mass also to vanish. This behavior plays an important role in theconceptual understanding of the vacuum structure of strong interactions.We consider matrix elements of the pseudo-scalar and the pseudo-vectorquark currents between the vacuum state and one pion state,〈π+(p)|ū(x)γµγ5d(x)|V 〉≡−i√2pµfπeipµxµ, (3.16)〈π+(p)|ū(x)γ5d(x)|V 〉≡ i√2gπeipµxµ, (3.17)where pµpµ = m2π = (139.6 MeV)2. The form of the right-hand sides ofEqs. (3.16) and (3.17) arises by virtue of the Lorentz symmetry propertiesof the (true) vacuum state |V 〉 and the π+ state |π+(p)〉. We considerthe divergence ∂µ of Eq. (3.16). Using relation Eq. (3.15), the followingwell-known result is found:m2πfπ = (mu +md)gπ. (3.18)The matrix element fπ is well known, since it governs the weak-inter-action decay of pions, see, e.g., Weinberg, and the value gπ is determinedby sum rule methods [242, 243]:fπ = 93.3MeV, gπ (350MeV)2. (3.19)Equation (3.18) implies thatmu +md = 0.1mπ, (3.20)a somewhat unexpected result in the present context, since the (current)quark masses are found to be much lighter even than that of the ‘massless’pion.Weinberg also presents an in-depth discussion of the exploration of theNambu–Goldstone structure of the vacuum, in terms of symmetry rela-tions between current-matrix elements (current algebra). In our context,the most important vacuum property involving quarks is the Gell-Mann–Oakes–Renner (GOR) relation, which, adapted to the quark language (seesection 31 of [280], or [125]), implies a relation with the quark fluctuations(condensate) in the true vacuum:m2πf2π = 0.17× 10−3GeV4 −12(mu +md)〈ūu+ d̄d〉+ · · · . (3.21)On dividing Eq. (3.21) by Eq. (3.18), we obtain−fπgπ = 12〈ūu+ d̄d〉|1GeV ≡12〈q̄q〉 = −(225± 9 MeV)3, (3.22)where we have used the values of fπ and gπ given in Eq. (3.19). WhenEq. (3.22) is combined with Eq. (3.21), and some of its generalizations, wecan determine the values of current quark masses [105]. This shows how46 A new phase of matter?the use of matrix-element properties and sum rules allows one to establishthe physical values of the light quark masses presented in table 1.1.We have introduced, in section 3.2, the condensates of glue, and above,of quark fields as if these were two quite independent physical effects ofstrong interactions. There remains an important question: is there arelation between glue-condensate melting (confinement-to-deconfinementtransformation of the vacuum) and quark-condensate melting (the restora-tion of chiral symmetry)? One could be tempted to infer that the chiralsymmetry-breaking features in QCD and gluon condensation have littlein common. However, studies of restoration of symmetry of the vacuumat high temperature [103] have yielded contrary evidence: the two differ-ent vacuum structures of QCD always disappear together in the numer-ical studies as, e.g., the temperature is varied [103]. Model calculations[106, 107, 109, 251] employing mean-field configurations of gauge fields inthe QCD vacuum suggest that it is the presence of the glue-field con-densate which is the driving force causing the appearance of the quarkcondensate.The mechanism connecting the two structures in the QCD vacuum(glue condensate, Eq. (3.3), and chiral condensate, Eq. (3.22)) is a majorunsolved theoretical problem of strong-interaction physics. We will notpursue further in this book this interesting subject, which is undergoingrapid development.3.4 Phases of strongly interacting matterIt is expected that, in nuclear collisions at relativistic energies, we at-tain conditions under which the structured confining vacuum is dissolved,forming a domain of thermally equilibrated hadronic matter comprisingfreely movable quarks and gluons. A qualitative sketch of the phase dia-gram of dense hadronic matter is shown in Fig. 3.1. The different phasespopulate different domains of temperature T and baryon density ρb pre-sented in units of the normal nuclear density in heavy nuclei, ρ0 16 nucle-ons fm−3. For high temperatures and/or high baryon density, we have thedeconfined phase. If deconfinement is reached in the nuclear-collision re-actions, it ‘freezes’ back into the state containing confined hadrons duringthe temporal evolution of the small ‘fireball’, as indicated by the arrowsin Fig. 3.1. Almost the entire ρ–T region can be explored by varying thecollision energy of the colliding nuclei.The most difficult domain to reach experimentally is the one of lowbaryon number density, at high T , corresponding to the conditions per-taining in the early Universe. This demands extreme collision energies,which would permit the baryon number to escape from the central rapid-ity region, where only the collision energy is deposited; see chapter 5.3 The vacuum as a physical medium 47Fig. 3.1. The regions of the principal forms of hadronic matter are shown in thebaryon-density–temperature plane. Their exploration with various acceleratorsis indicated, as are the domains relevant for cosmology and astrophysics. Alsoindicated is the behavior of the quark condensate.The vertical arrow at the lowest temperatures, in Fig. 3.1, correspondsto the case of the stellar explosion of a supernova. Rather low-energycollisions at the AGS lead to such baryon-dense environments, which aremore similar to nova explosions than they are to the early-Universe big-bang (horizontal arrow), which is better simulated by RHIC and futureLHC experiments. In between these two extremes, we find the SPS con-ditions. The specific beam capabilities of the various accelerator facilitiesare complementary; section 5.1.There are three regions indicated in Fig. 3.1 by the quark condensate,the expectation value of the quark fields 〈q̄q〉, see Eq. (3.22), and 〈qq〉.The attractive quark–quark interaction present in some of the two-particlechannels allows this di-quark color-condensate to form at low tempera-ture and high quark density. We will not discuss the extensive workwhich recently addressed the properties of ‘cold’ quark matter, in which a‘color–flavor’ locking (pairing) of quarks introduces yet other interestingstructures in the deconfined state [31]. This color-superconductive phasehad already been proposed early on in the study of properties of quarkmatter [53].48 A new phase of matter?Recent studies have confirmed that the temperature at which such acolor–flavor-locked phase of quark matter could exist is too low for anexploration in present-day laboratory experiments involving relativisticheavy-ion collisions [207]. At the temperature of interest in our studies,T > 100 MeV, the quark pairing will be largely dissolved. Work on thissubject is rapidly evolving, for its current status we refer the reader to arecent review [227].Where exactly an equilibrium transition between two phases occurs isdetermined from Gibbs’ conditions for phase equilibrium. These establishthe boundary between the physical phases considered, for bulk matterembedded in heat and particle-number ‘baths’. These baths supply energyand particles to maintain given thermodynamic conditions. Althoughthe circumstance of a nuclear-collision fireball is very different, the logicinherent in Gibbs’ conditions will guide our understanding. The firstcondition isP1 = P2, (3.23)which assures that there is no physical force acting on the phase boundary.We will momentarily return to discuss what happens when the phaseboundary is in (relativistic) motion, see Eq. (3.28) below. The secondGibbs condition isT1 = T2, (3.24)which assures that there is no radiative transport of energy between thephases.How these conditions define the phase boundaries is illustrated in theP–V diagram in Fig. 3.2. The pressure in two phases (QGP and HG) isconsidered at fixed temperature T (and at given conserved baryon num-ber b) as a function of volume V , at variable baryon density ρb = b/V .We distinguish three domains in Fig. 3.2:1. the HG region for V > V2 (corresponding to ρb < ρ2), where thepressure rises modestly with the reduction of the volume;2. the QGP region where the hadrons have disappeared at V < V1 (cor-responding to ρb > ρ1); and3. the Van der Waals regime in the intermediate region from V1 to V2; away to understand this domain is, e.g., that, at V1, the progenitors ofindividual hadrons begin to emerge in the QGP phase in the form of alocalized cluster of quarks. Similarly, beginning at V2 and with decreas-ing volume, one can, e.g., consider clustering of individual hadrons intoquark-drops [220].Because clustering of hadrons leads naturally to the formation of drops ofthe QGP-like phase, we refer to the coexistence region, between V1 and3 The vacuum as a physical medium 49V 1 V 21P = P22VQGPHGb = constantT = constantP1Fig. 3.2. The p–V diagram for the QGP–HG system, at fixed temperatureand baryon number; dashed lines indicate unstable domains of overheated andundercooled phases.V2, also as the mixed phase, comprising a mixture of hadrons and dropsof QGP or perhaps hadron-like clusters of quarks and free quarks. To findout at which pressure the transformation between the phases occurs, ata given temperature T (and fixed baryon number b), we find the value ofthe pressure, P1 = P2 ≡ P12, connecting the volumes V1 and V2, requiringthat the work done along the metastable branches vanishes:∫ V2V1(P − P12) dV = 0. (3.25)The integrand is shown shaded in Fig. 3.2.This Maxwell construction can be repeated for different values of band T , and the set of resulting points 1 and 2 forms then two phase-boundary lines shown on the left-hand side in Fig. 3.3, in the (ρb–T )plane. The Maxwell-construction line, seen in Fig. 3.2, is the verticalline connecting at fixed temperature T the two different values of baryondensity found – in general, a jump in baryon density (and energy densityand entropy density) is encountered if a first-order phase transition occurs.The region of high T , at fixed ρb, is associated with the deconfined QGPand the region of small T with HG. The shape of the phase boundaryis expressing the fact that a baryonless hadronic-gas phase cannot existat a high enough temperature, and that dense compressed cold baryonmatter will transform into the deconfined quark matter phase of quarkmatter.50 A new phase of matter?Mixed phasebHG12T QGPρNuclear matterbRHICSPSHGQGPT µNuclear matterAGSFig. 3.3. Left: the regions of QGP and HG in the (ρb–T ) plane are separated bya band in which the phases coexist. The Maxwell-construction line correspondingto Fig. 3.2 is also shown, as is the path for the evolution of the Universe. Right:the same in the (µb–T ) plane. The qualitative evolution of fireballs of densematter created at the AGS, SPS, and RHIC is shown.When we conserve the baryon number b on ‘average’ and introduce thebaryochemical potential µb as a variable, the representation of the phaseboundary changes. According to the third Gibbs condition,µ1 = µ2, (3.26)the two chemical potentials must have the same value at each given T inorder to assure that no transport of particles across the phase boundaryoccurs. Given Eq. (3.26), there is just a simple line separating any twohadronic phases in the (µb–T ) plane, as is shown on the right-hand sidein Fig. 3.3. There is a discontinuity of the energy density, baryon density,and entropy density across the phase boundary.3.5 The expanding fireball and phase transformationThe lines shown on the right-hand side in Fig. 3.3 suggest the possibleevolution of the fireball of dense matter formed in a heavy-ion collision. Ifa QGP fireball were indeed formed in the micro-bang, it will not expandalong a fixed-temperature trajectory such as is encountered under theisothermal conditions of a heat bath. In our case, instead, entropy is the(nearly) conserved quantity for an isolated system subject to ideal flow.The evolution at constant entropy per baryon corresponds nearly to astraight line in the µb–T diagram in the domain of QGP (dark shaded),since the dimensionless ratio entropy per baryon is a function of otherdimensionless variables – which, in the absence of significant scales otherthan µb, and T is µb/T . A considerable change in temperature must occurduring the evolution of a fireball, as is indicated in qualitative terms bythe trajectories shown in the right-hand panel of Fig. 3.3.3 The vacuum as a physical medium 51A phase transition that is ‘strong’, i.e., involves significant changesin physical properties, will be easier to find. As a strong transition, weunderstand a case with, e.g., a jump in magnitude of the energy, or baryondensity. Should such a strong phase transition separate the two phases,the super-cooling effect of a rapidly expanding (exploding) fireball of finitesize could be much more pronounced.We recall that both the QGP and the HG phases have metastablephase branches indicated by dashed lines in Fig. 3.2. However, now welook at these at constant entropy and variable temperature. Therefore,these domains are referred to as the undercooled plasma (continuation of1 in Fig. 3.2), or the overheated hadronic-gas states (continuation of 2in Fig. 3.2). Thus, the pressure of the QGP phase can evolve to be wellbelow the transition pressure.In fact, when a fireball of dense quark–gluon matter (phase 1) rapidlyexplodes, driven by the high internal temperature and pressure, it is pos-sible that it continues even beyond P = 0. Namely, the fluid-flow motionof quarks and gluons expands the domain of deconfinement by exercisingagainst the vacuum component in the total pressure a force originatingfrom the collective velocity /vc.Let P and ε be the pressure and energy density of the deconfinedphase in the local restframe, subject to flow velocity /vc = (v1, v2, v3).The pressure-tensor component in the energy–momentum tensor (com-pare with Eq. (6.6)) isT ij = Pδij + (P + ε)vivj1− /v 2 . (3.27)The rate-of-momentum-flow vector /P at the surface of the fireball is ob-tained from the energy-stress tensor Tkl:T̂ · /n = P/n+ (P + ε)/vc(/vc ·/n)1− /v 2c. (3.28)The pressure and energy comprise particles (subscript p) and the vac-uum properties:P = Pp − B, ε = εp + B. (3.29)Equation (3.28) for the condition T̂ · /n = 0 readsB/n = Pp/n+ (Pp + εp)/vc(/vc ·/n)1− v2c, (3.30)and it describes the (equilibrium) condition under which the pressure ofthe expanding quark–gluon fluid is just balanced by the external vacuumpressure. On multiplying by /n, we find52 A new phase of matter?B = Pp + (Pp + εp)κv2c1− v2c, (3.31)where we introduced the geometric factor κ:κ =(/vc · /n)2v2c. (3.32)κ characterizes the angular relation between the surface-normal vectorand the direction of flow. Under the condition Eq. (3.31), the total QGP-phase pressure P = Pp−B, Eq. (3.29), is negative, as we set out to show.Expansion beyond P → 0 is in general not possible. A surface region ofa fireball that reaches condition Eq. (3.31) and continues to flow outwardmust be torn apart. This is a collective instability and the ensuing disinte-gration of the fireball matter should be very rapid. We find that a rapidlyevolving fireball that supercools into the domain of negative pressure isin general highly unstable, and we expect that a sudden transformation(hadronization) into confined matter can ensue under such a condition.It is important to note that the situation we have described could ariseonly since the vacuum-pressure term is not subject to flow and alwayskeeps the same value.3.6 QGP and confined hadronic-gas phasesWe next seek to qualitatively understand the magnitude of the tempera-ture at which the deconfined quark–gluon phase will freeze into hadrons.The order of magnitude of this transition temperature (if a phase changeoccurs) or transformation temperature (if no phase transition occurs, seeFig. 3.1) is obtained by evaluating where a benchmark value for the energydensity occurs:εH 3PH = 1 GeV fm−3.The generalized Stefan–Boltzmann law (Eqs. (1.6) and (4.66)) describesthe energy density ε and pressure P as functions of the temperature T ofa massless relativistic gas:P SB =13εSB =π290gT 4. (3.33)The quantity g is the number of different (relativistic) particle statesavailable and is often called the ‘number of degrees of freedom’ or ‘degen-eracy’. In the deconfined phase,g ≡ gg + 74gq, (3.34)3 The vacuum as a physical medium 53which comprises the contribution of massless gluons (bosons) and quarks(fermions). The relative factor 2 × 78 =74 expresses the presence of par-ticles and antiparticles (factor 2) and the smaller fermion phase space,compared with the boson case, given the exclusion principle (factor 78 ,section 10.5).We use the degrees of freedom of quarks and gluons many times inthis book. Here, we ignore the role of interactions. Gluons carry colorand spin, and so do quarks, which, in addition, come in two (nf = 2)flavors u and d; see table 1.1. Since at high temperatures the flavor countmay include the strange quark, we leave nf as a variable. We obtain thefollowing degeneracy in a QGP:gluons: gg=2(spin)× (N2c − 1)(color) = 2× 8 = 16, (3.35a)quarks: gq=2(spin)×Nc(color)× nf(flavor) = 2× 3× nf. (3.35b)When the semi-massive strange quarks are present, the effective numberof ‘light’ flavors is 2.5. Thus, g 40 in Eq. (3.33), to be compared withjust two directions of polarization for photons.For a massless ideal quark–gluon gas, we findTH = 160 MeV, for εH = 1.1 GeV fm−3.Hagedorn introduced this critical temperature in his study of the boil-ing point of hadronic matter [140]. Numerical simulations obtained byimplementing QCD on a space–time lattice are available for zero baryondensity, and these results confirm that, at approximately TH, there is aphase transformation between confinement and deconfinement [160]. Onealso finds a rapid change in the behavior as the number of quarks andtheir masses ms and mq are varied.The resulting complexity of the phase structure is shown, in Fig. 3.4,as a function of ms and mq. In this qualitative representation, we lookat the plane spanned by the light-quark mass mq = mu = md and thestrange-quark mass ms. On two boundaries of Fig. 3.4 these masses areinfinite. Only near the origin is the effective number of massless flavorsthree, along the diagonal we have three massive flavors. Depending onthe actual values of quark masses, different phase properties emerge [162].The theoretical finding that a smooth crossover between the confinedand deconfined phases is a possibility raises the question of how to un-derstand qualitatively the gradual onset of color (quark, gluon) mobil-ity. A gradual change implies that free quarks can coexist with confinedhadrons. This then also suggests that liberation of quarks is possiblesince permanent confinement could be assured only at zero temperature,a mathematical limit. For any finite excitation of the system, quark mo-bility remains, akin to the transition of an atomic gas to an electron–ion54 A new phase of matter?Fig. 3.4. The nature of finite-temperature QCD phase structure as a functionof quark masses mq and ms.plasma. However, experimental searches for quarks have not succeeded[186]. The experimental limits which were set suggest that confinementis a fundamental physical property. This being the case, we are of theopinion that, in the physical world, the transformation from the confinedto the deconfined phase is a discontinuous phase transition, most likelyof first order. For this reason, we placed the physical quark-mass pointwithin the region of first-order phase transition in Fig. 3.4. This topicalarea is undergoing a rapid evolution.4 Statistical properties of hadronic matter4.1 Equidistribution of energyThe physical tools required to describe in further detail the properties ofhot hadronic matter are much like the usual ones of statistical physics,which we briefly introduce and review. A more detailed analysis willfollow.Consider a large number N of identical coupled systems, distinguish-able, e.g., by their energies Ei. To simplify the matter, we assume that4 Statistical properties of hadronic matter 55the energies Ei can assume only discrete values, and that there are Kdifferent ‘macro’ states such that K N . Some of the energies of themacro states will be equal, i.e., most are occupied more often than once,and in general ni times. The total energy,E(N) =K∑i=1niEi, (4.1)is conserved. Further below, we will introduce also conservation of adiscrete quantum number, e.g., the baryon number. We note anothersubsidiary condition arising from the definitions:K∑ini = N. (4.2)Without an additional quantum number, systems with the same energyEi are equivalent, i.e., in the language of quantum statistics, indistin-guishable.The distribution n = {ni} having the same energy Ei can be achievedin many different ways. To find how many, consider the relationKN =(x1 + x2 + · · ·+ · · ·xK)N |xi=1=∑nN !n1!n2! · · ·nK !xn11 xn22 · · ·xnKK |xi=1. (4.3)The normalized coefficients,W (n) =K−NN !∏Ki=1 ni!, (4.4)are the relative probabilities of realizing each state in the ensemble n,with ni equivalent elements. The practical way to find the most probabledistribution n is to seek the maximum of lnW , Eq. (4.4), subject to theconstraints Eqs. (4.1) and (4.2),A(n1, n2, . . ., nK) = lnW (n)− a∑ini − β∑iniEi, (4.5)characterized by two Lagrange multipliers a and β. We differentiateEq. (4.5) with respect to the ni:∂∂ni[− ln(ni!)− nia− βniEi]∣∣∣∣n̄m= 0. (4.6)Insofar as all n̄i 1, we can use the relationddk[ln(k!)] ≈ ln(k!)− ln[(k − 1)!](k)− (k − 1) = ln k. (4.7)56 A new phase of matter?One obtains for the maximum of the distribution Eq. (4.4), i.e., for thestatistically most probable distribution n̄ = {n̄i}, the well-known expo-nentialn̄i = γe−βEi , (4.8)where the inverse of the slope parameter β is identified below as thetemperature:T ≡ 1/β. (4.9)The supplementary condition Eq. (4.2), given the set n̄, now reads∑in̄i = γK∑i=1e−βEi = N. (4.10)The quantity γ,γ ≡ e−a, (4.11)as seen in Eq. (4.10), controls the total number of members of the ensembleN . It is the chemical fugacity introduced in section 1.1. We will meetboth statistical parameters T and γ many times again in this book.Employing Eq. (4.8), we find for the energy E(N), Eq. (4.1),E(N) =∑in̄iEi = γ∑iEie−βEi . (4.12)On dividing E(N) by N , we obtain the average energy of each member ofthe ensemble:E(N)N≡ E(N) = γ∑i Ei e−βEiγ∑i e−βEi ≡ −ddβlnZ. (4.13)We introduced here the canonical partition function Z:Z =∑iγe−βEi . (4.14)Unlike the microscopic (micro-canonical) approach in which the energyfor each member of the ensemble is fixed, in the statistical ‘canonical’approach, one studies the most likely distribution of energy and otherphysical properties among members of the ensemble. These propertiesare controlled solely by the statistical parameters β and γ which are theLagrange multipliers related to the conservation of energy, and the num-ber of members of the ensemble.4 Statistical properties of hadronic matter 574.2 The grand-canonical ensembleWe will now relax the assumption that only energy is equipartioned as itis exchanged between macro systems. In the grand-canonical approach,we seek the most probable statistical distribution allowing for the flow ofa discrete quantum number between the individual members of the statis-tical ensemble. In passing, we also show how the mathematical frameworkof the grand-canonical approach offers a convenient path to the evalua-tion of the canonical ensemble quantities, when discrete conservation lawsapply.We proceed in every detail as before, but need to characterize eachmacro state by an additional discrete number, and we need to introducea further subsidiary condition to assure that this (baryon) number isconserved:N∑i=1nbibi = b(N) ≡ Nb̄i. (4.15)Here, b̄i is the average number of baryons in each ensemble member con-sidered. The condition Eq. (4.15) introduces a further constraining La-grange parameter into Eq. (4.6), which we write in the form − lnλ. Inthis way the generalization of Eq. (4.6) is∂∂nbi[− ln(nbi !)− nbia− βnbiEi + lnλnbibi]∣∣∣∣n̄m= 0. (4.16)Proceeding as in section 4.1, we obtain the most probable distribution ofn̄i asn̄bi = γλbie−βEi , (4.17)where the number of particles is controlled by the fugacity factor λ andthe factor γ = e−a, see Eq. (4.11). It is common practice to introduce thechemical potential µ:µ ≡ T lnλ, λ = eβµ = eµ/T . (4.18)The chemical potentials shown have physical meaning, and determine theenergy required to add/remove a particle at fixed pressure, energy, andentropy. Following the method that led us to Eq. (4.13), we obtainE(N) = γ∑i;b Ei λbie−βEiγ∑i;b λbie−βEi≡ − ddβlnZ. (4.19)We have introduced the grand-canonical partition function Z:58 A new phase of matter?Z(V, β, λ) = γ∑i;bλbie−βEi . (4.20)Z is in fact also a ‘generating’ function for the canonical partition functionZb,Zb(V, β) =12πi∮db1λb+1Z(β, λ), (4.21)where Zb(V, β) describes a system with a fixed baryon number b. Thepath of integration in Eq. (4.21) leads around the singularity at λ = 0; itis often chosen to be the unit circle.We can also evaluate the average value of b for the grand-canonicalpartition function:b̄=∑i;b bi λbie−βEi∑i;b λbie−βEi,=λddλ(ln∑i;bγλbie−βEi)≡ λ ddλlnZ(β, λ). (4.22)4.3 Independent quantum (quasi)particlesElementary quantum physics allows a simple evaluation of the grand-canonical partition function. The discrete energies Ei of the physicalsystems in the statistical ensemble we introduced above are now to beunderstood as eigenenergies with eigenstate |i〉 of a quantum HamiltonianĤ:Ĥ|i〉 = Ei|i〉. (4.23)Since the (conserved-baryon-number) operator b̂ commutes with the Ham-iltonian, [b̂, Ĥ] = 0, the eigenstates can furthermore be characterized bytheir baryon number (and strangeness, and other discrete quantum num-bers that are constants of motion, but we restrict the present discussionto the baryon number only). We haveb̂|i, b〉 = b|i, b〉. (4.24)The grand-canonical partition function, Eq. (4.20), can be written asZ ≡∑i,b〈i, b|γe−β(Ĥ−µb̂)|i, b〉=Tr γ e−β(Ĥ−µb̂), (4.25)≡∑n〈n|e−β(Ĥ−µb̂−β−1 ln γ)|n〉.4 Statistical properties of hadronic matter 59The great usefulness of this relation is that the trace of a quantum op-erator is representation-independent; that is, any complete set of micro-scopic basis states |n〉 may be used to find the (quantum) canonical orgrand-canonical partition function. This allows us to obtain the physi-cal properties of quantum gases in the, often useful, approximation thatthey consist of practically independent (quasi)particles, and, eventually,to incorporate any remaining interactions by means of a perturbativeexpansion.The reference to quasi-particles is made since, e.g., in a medium, massesof particle-like objects can be different from masses of ‘elementary’ par-ticles. Generally there will be collective excitation modes characterizedby a mass spectrum. In this respect, dense hadronic matter behaveslike any dense-matter system. As long as there is a set of well-definedexcitations, it really does not matter whether we are dealing with realparticles or quasi-particles, when we compute the trace of the quantumpartition function Eq. (4.25). Putting it differently, even though we com-pute the properties of a ‘free’-particle quantum gas, by choosing a suitablequasi-particle basis, we accommodate much of the effect of the strong in-teractions between particles.The ‘single (quasi)particle’ occupation-number basis is the suitable onefor the evaluation of the trace in Eq. (4.25). In this approach, each macrostate |n〉 is characterized by the set of occupation numbers n = {ni} ofthe single (quasi)particle states with baryon charge bi of energy εi, andthe state energy is given by En =∑i niεi. The sum over all possiblestates corresponds to a sum over all allowed sets n: for fermions, eachni ∈ 0, 1 and for bosons, ni ∈ 0, 1, 2, . . .,∞:Z=∑ne−∑∞i=1 niβ(εi−µbi−β−1 ln γ),=∑n∏ie−niβ(εi−µbi−β−1 ln γ). (4.26)In this case the sum and product can be interchanged:∑n∏ie−niβ(εi−µbi−β−1 ln γ) =∏i∑ni=0,1...e−niβ(εi−µbi−β−1 ln γ). (4.27)To show this equality, one considers whether all the terms on the left-hand side are included on the right-hand side, where the sum is not overall the sets of occupation numbers n, but over all the allowed values ofoccupation numbers ni. For fermions (F, Fermi–Dirac statistics) we canhave only ni = 0, 1, whereas for bosons (B, the Bose–Einstein statistics)ni = 0, 1, . . .,∞. The resulting sums are easily carried out analytically:lnZF/B = ln∏i(1± γe−β(εi−µbi))±1= ±∑iln(1± γλbie−βεi). (4.28)60 A new phase of matter?The plus sign applies to F, and the minus sign to B; fermions have Paulioccupancy 0, γ, of each distinct single-particle state, and bosons haveoccupancy 0, γ1, 2γ2, . . ., ∞. The factor γn arises naturally since wehave not tacitly set the occupancy of each single-particle level to unity asis commonly done when absolute chemical equilibrium is assumed.For antiparticles, the eigenvalue of b̂, in Eq. (4.28), is the negative ofthe particle value. Consequently, the fugacity λf̄ for antiparticles f̄ isλf̄ = λ−1f . (4.29)It is convenient to also introduce this change in sign into the definitionof the chemical potential, see Eq. (4.18), and to introduce particle andantiparticle chemical potentials such thatµf = −µf̄ . (4.30)These relations, Eqs. (4.29) and (4.30), will often be implied in what fol-lows in this book. The microscopic (quasi)particle energy is denoted byε in Eq. (4.28). For a homogeneous space–time, it is determined in termsof the momentum /p in the usual manner:εi =√m2i + /p 2. (4.31)In order to make any practical evaluations of Eq. (4.28), we need tointerpret the level sum∑i with some precision. If energy is the onlycontrolling factor then we carry out this summation in terms of the single-particle level density σ1(ε, V ):∑i[. . .] =∫dε σ1(ε, V )[. . .]. (4.32)To obtain σ1, i.e., the number of levels in a box of (infinite) volume V = L3per unit of energy ε, we note that quantum mechanics does not allow acontinuous range of /p in Eq. (4.31).We consider a box L3 with periodic boundary conditions and obtainthe complete set of plane-wave states ψ having the required periodicity,ψ ∝ ei(pα· X) = eipα( X+nL), (4.33)where /n = (n1, n2, n3) with n = 0, ±1, ±2, . . .. This fixes the allowed /pαtoL/pα · /n = 2πk, k = 0, ±1, ±2, . . .. (4.34)This can be satisfied only if/pα =2πL(k1, k2, k3), with ki = 0, ±1, ±2, . . .. (4.35)4 Statistical properties of hadronic matter 61To sum over all single-particle states, we sum over all ki. The numberof permitted states equals the number of lattice points in the ‘inverse’ or‘phase-space’ k-lattice. In the limit of large L,[number of states in d3k] =(L2π)3d3p =V(2π)3d3p. (4.36)Thus, we obtain the single-level density, Eq. (4.32):∑i[. . .] =∫dεV d3p(2π)3 dε[. . .]. (4.37)We keep in mind that, in general, the replacement of the discrete-levelsum implies, in the limit of infinite volume of the system the phase-spaceintegral∑i→ g∫d3x d3p(2π)3. (4.38)Discrete quantum numbers, such as spin, isospin, and flavor, contribute anadditive component of the same form in the sum over the single-particlestates, which gives rise to the degeneracy coefficient g in Eq. (4.38). Asidefrom the volume term shown in Eq. (4.38), there is, in general, also acorrection that has the form of a surface term. The magnitude and signof the surface term depend on the physical problem considered. We willnot pursue this topic further in this book; for a general discussion of thissubject see, e.g., [51, 52].4.4 The Fermi and Bose quantum gasesAllowing for the presence both of particles and of antiparticles, the quant-um-statistical grand partition function Eq. (4.28) for a particle of mass mand degeneracy g can be written explicitly aslnZF/B(V, β, λ, γ) = ±gV∫d3p(2π)3[ln(1± γλe−β√p2+m2)+ ln(1± γλ−1e−β√p2+m2)]. (4.39)The second term in Eq. (4.39) is due to antiparticles. The well-known‘classical’ Boltzmann limit arises from expansion of the logarithms, i.e.,when it is possible to consider the exponential term as small relative tounity:lnZcl(V, β, λ, γ) = gV∫d3p(2π)3γ(λ+ λ−1)e−β√p2+m2 . (4.40)62 A new phase of matter?We will often use the (normalized) particle spectrum, the average rela-tive probability of finding a particle at the energy Ei, which is the coeffi-cient of Ei in Eq. (4.13). Using Eqs. (4.8) and (4.10) we obtainwi ≡n̄iN=e−βEi∑j e−βEj= − 1β∂∂Eiln∑jγe−βEj = − 1β∂∂EilnZ. (4.41)The single-particle spectrum that follows from Eq. (4.28) is easily evalu-ated,fF/B(ε;β, λ, γ) =1γ−1λ−1eβε ± 1 , (4.42)where the plus sign applies for fermions, and the minus sign for bosons.For antiparticles, we replace λ by λ−1. The classical Boltzmann approxi-mation arises again in the limit in which it is possible to neglect the term±1 in the denominator, i.e., when the phase-space abundance is small,fF/B → fcl = γλe−βε, (4.43)where λ → 1/λ for antiparticles. More generally, for γλe−βε < 1, thisStefan–Boltzmann spectral shape can be written as an infinite series:fF/B = ±∞∑n=1(±γλe−βε)n. (4.44)We consider, as an example, the spectra and yield of gluons, a specialcase of interest to us among bosons. Their behavior is similar to the caseof photons (gγ = 2) but gluons have an eight-fold greater color degeneracy(gg = 16). Both photons and gluons do not have an antiparticle partner,and their number is unrestricted by particle/antiparticle conservation;hence λ → 1. We obtainlnZγ,g = −gγ,gV∫d3p(2π)3ln(1− γe−βε), (4.45)where ε = ε(/p ) =√m2 + /p 2 → |/p |, except when we consider a non-vanishing thermal gluon mass in the medium. The particle occupationprobability isfγ,g(ε) =1γ−1eβε − 1 =∞∑n=1γne−nβε, γ < eβm. (4.46)4 Statistical properties of hadronic matter 63The gluon (and photon) particle densities areργ,g ≡Nγ,gV=1Vlimλ→1λddλlnZγ,g = gγ,g∫d3p(2π)3fγ,g. (4.47)Using the series expansion from Eq. (4.46), we can explicitly evaluate thisintegral, substituting np/T for x term by term:ργ,g =gγ,g2π2T 3∞∑n=1γnn3∫ ∞0dxx2e−√(nm/T )2+x2 . (4.48)For m → 0 and γ → 1, we obtain the well-known Stefan–Boltzmannequilibrium limit:ργ,g =gγ,gπ2T 3ζ(3), (4.49)with the Riemann zeta-function ζ(3) 1.202; see Eq. (10.66b). UsingEq. (10.50a), for the general case, to evaluate the integral, we obtain aninfinite sum over terms containing the Bessel function K2 (also called theMcDonald function), which is discussed in section 10.4:ργ,g =gγ,gπ2Tm2∞∑n=1γnnK2(nm/T ). (4.50)Many other properties of the quark–gluon gas are discussed in section 10.5.The statistical method is a powerful tool to deal with the physics we ad-dress in this book. Looking back, we recognize that we have assumed thepresence of sufficiently many (weakly) interacting (quasi)particles in thisdiscussion of basic results of statistical physics. Two important questionscome to mind.• In our context, the practical question is that of how statistical physicsworks when we have a few hundred (at the SPS), or a few thousand (atthe RHIC) particles experiencing a limited number of collisions each. Inthis book, we will answer this question by consulting the experimentalresults, and our finding is that the thermal particle spectra describeexperimental data very well.• It seems that, perhaps, we could derive statistical-physics laws for anytype of many-object system – could it be that the statistical partitionfunction even describes the behavior of investors on Wall Street? Letus clearly identify what specific tacit physical feature makes a systemof particles so much simpler to understand than a crowd of investors.An appropriate economical toy model, in our context, would consist oftaking a ‘conserved’ number of Wall Street investors who, in view oftheir frequent interactions, should see their investments equipartitioned64 A new phase of matter?into an exponential wealth distribution, provided that all members ofthe same wealth class are, basically, indistinguishable, a hypothesismany of our colleagues agree with. All the above equations apply, withEi being now the wealth range of ni investors. To compute anythingwith precision we need, however, to specify the meaning of the discretesum,∑i; we need to know the number of ‘investors’ per unit of ‘wealth’.In case of physical particles, this level density Eq. (4.37) is implicit inour understanding of the many particle phase space, which allows usto convert the symbolic expressions into quantitative equations. Weare not able to generalize this naively to non-physics applications ofstatistical physics.4.5 Hadron gasParticularly important in our study is the hadronic ‘gas’ (HG) matterconsisting of individually confined hadronic particles. Although relativis-tic dynamics is required, we can consider the classical (Boltzmann)-gaslimit Eq. (4.40) since, in a very ‘rich’ multicomponent phase, each parti-cle species has a rather low ‘non-degenerate’ phase-space abundance. Inother words, at sufficiently high temperature, a high density of hadronicparticles can arise as a consequence of many hadron species contribut-ing, and does not in general imply a quantum degeneracy of the phasespace. However, even in the HG phase, it is possible to encounter (pion)quantum degeneracy, which requires full quantum statistics, Eq. (4.39).To see why the classical Boltzmann distribution almost always suf-fices in the hadronic gas phase of matter, consider the denominator ofthe quantum distribution, Eq. (4.42): even for the least-massive hadronicparticle, the pion, the expansion of the denominator of quantum distribu-tions makes good sense. For a range of temperatures up to T < 150 MeVwherein confined hadrons exist, we find exp(−Eπ/T ) < exp(−mπ/T ) < 1.The limits of the Boltzmann approximation are tested when, e.g., thephase space is oversaturated, i.e., γπ > 1, or when the baryo-chemicalpotential compensates for the mass term which could occur in extremelydense baryonic systems.We present next a brief survey of the properties of a hadronic Boltz-mann gas, and refer to chapter 10 for further developments. We considera series expansion of the logarithmic function in Eq. (4.39):lnZ =∞∑n=11nZn. (4.51)Each term comprises contributions from all contributing bosons Bf and4 Statistical properties of hadronic matter 65fermions Ff :Zn =∑Bfgfγnf (λnf + λ−nf )V∫d3p(2π)3e−nβεf+(−)n+1∑Ffgfγnf (λnf + λ−nf )V∫d3p(2π)3e−nβεf . (4.52)The single-particle energy εf entering Eq. (4.52) depends on the mass mfof particle f, Eq. (4.31). Since the mass spectra of hadronic bosons andfermions are quite different, n > 1 quantum corrections do not cancel out.In the Boltzmann limit, the first term n = 1 is retained in Eq. (4.51)and there is no distinction between the Bose and Fermi ideal gases in this‘classical’ limit, as seen in Eq. (4.40):lnZcl =∑fgfγ(λf + λ−1f )V∫d3p(2π)3e−βε(p) ≡ Z(1). (4.53)The last definition reminds us that the right-hand side of Eq. (4.53) is thepartition function arising for a single particle enclosed in a given volume.This is not an entirely ‘classical’ expression. We note thatZcl =∞∑k=01k!(Z(1))k, (4.54)which expresses the fact that the partition function comprises the additivecontributions of terms for k microscopic particles. However, the quantumindistinguishability is retained in the factor 1/k! – only with this quan-tum factor can one obtain the correct ‘classical’ Maxwell distribution ofatoms in a gas. This issue marked strongly the pre-quantum-mechanicsdevelopment of statistical physics in the Boltzmann era since there wasno easy explanation why this factor was needed.We already know the momentum integral appearing in Eq. (4.53), fromEq. (4.48),lnZcl =β−3V2π2∑fgfγ(λf + λ−1f )W (βmf), (4.55)where we used the function W (x) = x2K2(x), shown in Fig. 10.1 onpage 197.Using Eq. (4.55), we obtain the properties of a hadronic gas in the clas-sical (Boltzmann) limit. The ‘net’ (particle minus antiparticle) particledensity, Eq. (4.22),ρf =T 32π2∑fgfγ(λf − λ−1f )(βmf)2K2(βmf), (4.56)66 A new phase of matter?pressure,Pcl =TVlnZcl =T 42π2∑fgfγf(λf + λ−1f )(βmf)2K2(βmf), (4.57)and energy density,εcl = −1V∂∂βlnZcl =T 42π2∑fgfγf(λf + λ−1f )×[3(βmf)2K2(βmf) + (βmf)3K1(βmf)]. (4.58)comprise the sum over all particle fractions. In Eq. (4.56) we obtained thedifference between numbers of particles and antiparticles. The relationbetween partition function and pressure which we introduced in Eq. (4.57)is discussed in section 10.1, see Eq. (10.11). To obtain Eq. (4.58), we useddx2K2(x)/dx = −x2K1(x).The relativistic limits of Eqs. (4.57) and (4.58) arise in view of theproperties of the Bessel function, Eqs. (10.47) and (10.50b),K2(x)→ 2/x2and K1(x)→ 1/x, and only the K2 term contributes:Pcl→T 4π2∑fgfγf(λf + λ−1f ),εcl→3T 4π2∑fgfγf(λf + λ−1f ). (4.59)In the case of fermions, the Pauli exclusion principle decreases the particledegeneracy below the classical value. The energy and pressure shown inEq. (4.59) are reduced in the relativistic limit by the Riemann η-functionfactor η(4) = 78π4/90 = 0.9470. On the other hand, since bosons are‘attracted’ to each other, one finds a greater than classical degeneracy,expressed in the relativistic limit by the factor ζ(4) = π4/90 = 1.0823.For a relativistic hadron gas, comprising a similar number of fermionsand bosons, this quantum effect averages out. Thus, when we speak of aneffective number of degrees of freedom (also effective degeneracy) in HG,we will use as a basis the classical expression Eq. (4.59):gPeff ≡ π2PT 4, gεeff ≡π23εT 4. (4.60)When T m for all particles, or, equivalently, when T is the only relevantenergy scale, we have gPeff gεeff .We consider, in Fig. 4.1, how gPeff and gεeff look in a simple hadronic gas,as functions of T . Solid lines correspond to gεeff, and dashed to gPeff. Thethin lines are for the classical Boltzmann pion gas (gπ = 3, mπ 1404 Statistical properties of hadronic matter 67Fig. 4.1. Effective numbers of degrees of freedom from energy density (solidlines) and pressure (dashed lines), Eq. (4.60), for a Boltzmann pion gas (thinlines), and gas comprising Boltzmann pions, nucleons, kaons, and ∆(1232) forγi = 1 and λi = 1, as functions of temperature T .MeV); thick lines also include the four kaons K, the nucleons N, and thedeltas ∆(1232), and for N and ∆ their antiparticles (gK = 4, mK 495MeV; gN = 4, mN 939 MeV; and g∆ = 16, m∆ 1232 MeV) evaluatedwith all γ = 1, as appropriate for chemical equilibrium, and λ = 1, fora nearly baryon-free system, as appropriate for the early Universe. Wesee for the pion-only case (thin lines) the expected high-T limit, whichis nearly reached already at T mπ. However, because of the relativelyhigh hadron masses, the effective number of degrees of freedom keepsrising even at T 300 MeV toward its maximum for this example, whichis near 50. We also note that the energy density approaches its relativisticlimit faster than does the pressure, a point to which we shall return inEq. (10.58).We draw two important conclusion from results seen in Fig. 4.1.• Since pions are several times lighter than the next heavier hadronicparticle, they determine rather exactly the properties of a hadron gasat ‘low’ temperature below T (mπ/2) MeV, as is seen in Fig. 4.168 A new phase of matter?(the case that the net baryon and strangeness density is zero). Even atT mπ, the pion fractional pressure is still the dominant component.• The influence of the numerous massive hadronic particles rapidly gainsin importance with rising temperature. At low temperature, the quan-tum corrections (not shown in Fig. 4.1) are in fact more importantthan the contributions of heavier particles since 2mπ < mh, h = π.For 2βmπ < 1 in the HG phase, with gπ = 3, (for derivation compareEq. (10.62)), we havePπh 3T 42π2(γπλπW (βmπ) +116γ2πλ2πW (2βmπ) + · · ·). (4.61)As the temperature increases, the small quantum correction remains aminor effect compared with a rise due to excitation of numerous heavyhadron states.4.6 A first look at quark–gluon plasmaWe consider next the properties of the QGP, modeled initially as an idealchemically equilibrated gas of quarks and gluons, including the effect ofconfining vacuum structure. In the study of the quark-and-gluon gas,our task is considerably simplified by the observation that the gluons andlight u and d quarks are to all intent massless particles, at least on thescale of energies available in the hot plasma, i.e., T ≈ 200 MeV.Since the energy density is, in general terms, given by (see Eqs. (10.7)and (10.11))ε = − ∂∂β1VlnZ(β, λ), (4.62)in the absence of any dimensioned scales,1VlnZ(β, λ) = β−3f(λ), (4.63)and we findε = 3β−4f(λ) = 3TVlnZ(β, λ) = 3P. (4.64)The presence of masses of quarks, and in general scaled variables, breaksthis perhaps most used relationship of relativistic gases. It applies tofermions, bosons, and classical gases. Equations (10.58)–(10.60) show howthe presence of masses reduces the pressure below ε/3. Put differently,massive particles are less mobile at a given temperature, and thus thepressure they can exercise is smaller than ε/3; the energy density ε is‘helped’ by the presence of masses, and is closer to the relativistic limit.4 Statistical properties of hadronic matter 69In the limit βm = m/T 1, the phase-space integrals of ideal quantumgases are easily carried out. We can effectively neglect the particle massm compared with the high momenta that occur. We also omit, at first,chemical potentials. We obtain for the energy densityEF,BV=g2π2∫ ∞0p2 dppeβp ± 1 =gβ−42π23!∞∑1(±1)n−1n4. (4.65)The infinite sums are the zeta and eta Riemann sums, see Eqs. (10.66a)–(10.67b), which for bosons give the well-known Stefan–Boltzmann result:PB|m=0 =TVlnZB|m=0 =gπ290T 4 =13εB ≡EB3V. (4.66)We have made explicit the result ε = 3P , see Eq. (4.64), which is validwhen the mass of particles is small relative to their energy (masslessparticles or ultra-relativistic gas). For fermions, the alternating sum inEq. (4.65) introduces a relative reduction factor, which is 78 , see Eq. (10.67b).However, allowing for the presence of antifermions, the energy density andpressure have to be multiplied by an extra factor of two, and become infact greater by a factor 74 :εF ≡EFV=gπ23074T 4 = 3PF. (4.67)For fermions, the inclusion of a finite chemical potential is of impor-tance. In the limit m → 0, the Fermi integrals of the relativistic quan-tum (degenerate) quark gas can be evaluated exactly at finite µ, seeEq. (10.73):PF|m=0 =TVlnZF|m=0 =g(πT )490π2(74+15µ22(πT )2+15µ44(πT )4). (4.68)Since in the domain of freely mobile quarks and gluons the vacuum isdeconfined, a finite vacuum energy density (the latent heat of the vacuum)arises within the deconfined region, as we have discussed at length insection 3.1. This also implies that there must be a (negative) associatedpressure acting on the surface of this volume and attempting to reducethe size of the deconfined region. These two properties of the vacuumfollow consistently from the vacuum partition function:lnZvac ≡ −BV β. (4.69)On differentiating Eq. (4.69) as in Eqs. (4.57) and (4.58), we in fact findthat the perturbative vacuum region is subject to the (external) pressure−B while the internal energy density is +B relative to the outside volume.70 A new phase of matter?Fig. 4.2. P = 0 in the quark–gluon liquid in the (µb–Tc) plane. Dotted (fromright to left): breakup conditions of the liquid for expansion velocities v2 =110 ,16 ,15 ,14 and13 .The partition function (i.e., pressure) of the quark–gluon phase is ob-tained after we combine contributions from quarks, gluons, and vacuum:TVlnZQGP≡PQGP = −B +845π2c1(πT )4+nf15π2[74c2(πT )4 +152c3(µ2q(πT )2 +12µ4q)]. (4.70)We have inserted the quark and gluon degeneracies as shown in Eqs. (3.35a)and (3.35b). The interactions between quarks and gluons manifest theirpresence aside from the vacuum-structure effect, in the three coefficientsci = 1, see Eqs. (16.1) and (16.2), [91]:c1=1−15αs4π+ · · · , (4.71a)c2=1−50αs21π+ · · · , (4.71b)c3=1−2αsπ+ · · · . (4.71c)4 Statistical properties of hadronic matter 71One can evaluate the pressure Eq. (4.70) by choosing values for B andαs. It turns out that the value of the running strong-interaction couplingconstant αs changes rather rapidly in the domain of interest to us, andhence one needs to employ a function αs(T ), see Fig. 14.3 on page 286.Then, also allowing for the latent heat B, a surprisingly good agreementwith lattice results in section 15.5 is found, this is shown in Fig. 16.2 onpage 307. This comparison hinges strongly on an understanding of αs(T ),and inclusion of B.Drawing on these considerations, we show the QGP-phase pressure con-dition PQGP → 0 in Fig. 4.2. The solid line denotes, in the (µb–Tc) plane,where PQGP = 0 in a stationary quark–gluon phase. The dotted linescorrespond (from right to left) to the condition Eq. (3.31) for flow veloc-ities v2 = 110 ,16 ,15 ,14 and13 for which an exact spherical expansion withκ = 1, see Eq. (3.32), was used. The last dotted line to the left cor-responds to an expansion with the velocity of sound of relativistic (i.e.,effectively massless) matter. For small baryo-chemical potentials, theequilibrium phase-transition temperature of a non-dynamically evolvingsystem is somewhat greater than that shown here at the intercept of thesolid line at µb = 0. The actual value is Tc 170 MeV, as it occurs atfinite pressure balanced by hadrons, compare with Fig. 3.2. Looking atthe high-flow-velocity curves in Fig. 4.2, we see that an exploding QGPfireball can supercool to T 0.9Tc.IIExperiments and analysis tools5 Nuclei in collision5.1 Heavy-ion research programsThe energy content available in the nuclear collision is the main factor inwhich experimental facilities differ from each other. The ultra-relativisticnuclear-collision systems we are considering are identified in table 5.1. Forthe maximum possible mass number up to Amax 200, we show the fixed-target maximum beam energy per nucleon EmaxP [A GeV]; for colliders, wepresent in this line the equivalent projectile energy. Similarly, we show theCM energy in the nucleon–nucleon system√sNN [GeV], which is twice thenominal beam energy of the RHIC and LHC collider systems. We alsoshow the total√sAA [GeV] energy in the interaction region, allowingfor the maximum mass number A of the beam. The final line refers tothe rapidity ‘gap’ ∆y. We will discuss these variables in the followingsections.∆y is defined as the difference between the rapidities of projectile andtarget. In laboratory fixed-target experiments, yt = 0, and ∆y is therapidity of the projectile yp. Using the definition of rapidity Eq. (5.4), wehavecosh∆y = Ep/mp. (5.1)For head-on interactions occurring at rest in the laboratory, at the colliderfacilities, ∆y/2 is the projectile (target) rapidity of each beam, which isevaluated using, e.g., Eq. (5.1) again.A convenient way to represent the data of table 5.1 is shown in Fig. 5.1:the solid line depicts the CM energy per pair of nucleons,√sNN, asa function of the rapidity y. The horizontal distance between the twobranches of the solid line is the projectile–target rapidity gap ∆y. Theshaded areas correspond to the accessible CM energies,√sNN, at ex-725 Nuclei in collision 73AGSLHCRHICSPSFig. 5.1.√sNN (vertical axis) of various accelerators as a function of the projec-tile and target rapidities seen from the CM frame. Shaded areas: energy rangesaccessible at the various accelerators.perimental facilities that are in operation and under construction to-day.As the energy increases, the rapidity gap ∆y between projectile andtarget opens up as we see in Fig. 5.1. In the central rapidity region, wecan study conditions of matter without having to account for particlesspilled from the projectile and target fragments, which are known exper-imentally to spread over about two units of rapidity. Two extreme casesare illustrated qualitatively in Fig. 5.2, in which we sketch the distribu-Table 5.1. Parameters of existing ultra-relativistic heavy-ion beam facilities andthose under construction.AGS AGS SPS SPS SPS RHIC RHIC LHCStart year 1986 1992 1986 1994 1999 2000 2001 2006Amax28Si 197Au 32S 208Pb 208Pb 197Au 197Au 208PbEmaxP [A GeV] 14.6 11 200 158 40 0.91×104 2.1×104 1.9×107√sNN [GeV] 5.4 4.7 19.2 17.2 8.75 130 200 6000√sAA [GeV] 151 934 614 3.6×103 1.8×103 2.6×104 4×104 1.2×106∆y/2 1.72 1.58 2.96 2.91 2.22 4.94 5.37 8.7774 Experiments and analysis tools(a)(b)dE/dydB/dyyFig. 5.2. Rapidity distributions of energy (solid lines) and baryon number(dashed lines) (in a qualitative representation): (a) for a ‘Transparent’ reactionmechanism; and (b) for full stopping in the collision.tions of the energy (solid lines) and baryon number (dashed lines), asfunctions of the rapidity.In the baryon-punch-through case, shown in Fig. 5.2(a), which was in-vestigated by Bjørken [73], see section 6.4, the colliding nuclei are leaving atrail of energy between the projectile–target rapidity, but the baryon num-ber continues to move out of the collision zone, apart from the down-shiftin projectile and target rapidities necessary for conservation of energy.The stopping limit, implicit in the work of Fermi [121] and described byLandau [173, 175], is shown in Fig. 5.2(b): both the particle multiplicity(energy) and the baryon number are centered around the central rapidityyCM. The projectile and target baryons will, under the most extremecircumstance of complete stopping, lose all memory about the initial state,and in this limit there should in particular be little, if any, differencebetween the distributions of energy and baryon number in the longitudinaland transverse directions with respect to the collision axis.We now survey the nuclear-collision experiments that are currently op-erating or under development. These include in particular the CERN–SPS heavy-ion program which continues a 15-year-long tradition in theso-called North Area (NA) in fixed-target mode with energy range up to200A GeV for up to A 100 and dropping to 158A GeV for neutron-richprojectiles such as Pb. At higher energies, we have the beginning of theexperimental program at the RHIC collider, and in the near future there5 Nuclei in collision 75will be one at the LHC. The RHIC allows head-on collisions of two Auions, each carrying energy in the range (10–100)A GeV. Results from theinitial 65A-GeV run will be described and a glimpse of first 100A-GeVresults is presented as this book goes to press. Compared with the SPSenergies, the available CM energy per nucleon has been increased by anorder of magnitude and accordingly the densities of matter reached aremore extreme. Since the laboratory frame is also the CM frame in acollider experiment, the greatly increased particle density is distributedmore evenly in all spatial directions.We begin with the CERN–SPS research program∗.• The experiment NA45.2 investigates primarily the production of elec-tron–positron pairs and of direct photons, continuing the research pro-gram of NA45 carried out with S-beams. Both experiments observedielectron pairs and compare results with expectations based on p–preactions. The current experimental set-up consists of a double spec-trometer covering a region near mid-rapidity with full azimuthal cov-erage. Electrons are identified in two ring-imaging Čerenkov detectors(RICH).• The experiment NA49, which had as its predecessor experiment NA35,uses several time-projection chambers (TPCs) for large-acceptance track-ing of charged particles. Its current objective is to explore in greaterdetail the excitation function of strangeness near the possible thresh-old for the formation of the QGP phase. NA49 is at present the onlyexperiment at the SPS capable of measuring many global observablesrequired to characterize the nature of heavy-ion collisions as the energyis varied.• The experiment NA57 continues the research program of experimentsWA97, WA94, and WA85, all of which studied the production of (multi)strange hadrons in the central rapidity region, with particular emphasison the production of strange antibaryons. NA57 is completely differ-ently instrumented compared with the WA series and provides an im-portant cross-check for all the results. It comprises silicon pixel trackingof hadrons in a magnetic field, and its results are based primarily onreconstruction of decays of strange hadrons.• The experiment NA60 attempts detection of charmed hadrons, to com-plement the earlier study of suppression of production of J/Ψ by itspredecessors NA50 and NA38. The NA50 muon spectrometer is comple-mented by a completely redesigned target area using radiation-tolerantsilicon pixel detectors. The NA50 experiment studied dimuons pro-∗ For further details consult the following CERN web pages:http://greybook.cern.ch/programmes/SPS.html; and see also http://greybook.cern.ch/programmes/EXP NAM.html, for all CERN experiments, including those completed.76 Experiments and analysis toolsduced in Pb–Pb and p–A collisions. The muons are measured in theformer NA10 spectrometer, which is shielded from the target region bya beam stopper and absorber wall. The observed muons traverse 5 mof BeO and C.• The completed experiment WA98 will be repeatedly mentioned in thatwhich follows: WA98, which had as predecessors WA80 and WA87,which addressed the measurement of photons, but also measured theglobal production of charged hadrons. It comprised, in particular, a10 000-module lead-glass spectrometer, which now is incorporated intothe PHENIX detector (see below), yielding high-precision data on π0and η at mid-rapidity within a large range of transverse momenta 0.3GeV/c > P⊥ > 4.5 GeV/c for π0. Detailed comparison of photons withthe production of charged particles allowed also an evaluation of thephoton enrichment potentially due to direct radiance from QGP.We now turn to review the experimental research program at the BNL.Four experiments are at present taking data at the RHIC†. They aredesigned to allow both a survey of the reactions occurring in this hithertounexplored condition of matter and an in-depth study of the propertiesof the deconfined QGP phase. We review the first results from the year-2000 run in section 9.5. The experiments currently under way are thefollowing.• BRAHMS (Broad Range Hadron Magnetic Spectrometer)is designed to measure hadronic particles inclusively (that is, to mea-sure one particle at a time irrespective of what else is happening, whenthe system is triggered), over a wide range of rapidity (0 < η < 4)and transverse mass (up to 30 GeV). It consists of two (forward andmid-rapidity), magnetic focusing charged-particle (π±, K±, p, p̄) spec-trometer arms, which can be set to the desired angular acceptancewindow.• PHENIX (Pioneering High Energy Nuclear Interaction Experiment)is a detector optimized to observe photons and dilepton pairs (γ, e±and µ±). It comprises a central detector made of an axial field magnetand two almost identical arms placed on the left and right of the mag-net, each covering a window of ±0.35 units of pseudorapidity. Eacharm comprises several detector subsystems: the important goal of thecentral detector is observation of dielectrons at high mass resolution, al-lowing one to detect changes in the properties of decaying vector mesons(e.g., J/Ψ → e+e−, φ → e+e−). The electro-magnetic calorimeter al-lows one to measure low-p⊥ photons near y = 0. Hadron detection inthe silicon vertex detector, for −2.65 < η < 2.65, will allow studies of† For RHIC experiments, see http://www.rhic.bnl.gov.5 Nuclei in collision 77the distribution of charged hadrons (without identification of particles)on an event-by-event basis. First results from this subsystem obtainedin the RHIC 2000 run have recently been published [16].• PHOBOS, a scaled down ‘satellite’ of MARS (ModularArray forRHICSpectra),is a very small (in comparison) arrangement of silicon-based detec-tors that will allow one to study low-momentum particles within thecomplete (pseudo)rapidity interval −5.4 < η < 5.4, aiming to exploreglobal event structure. PHOBOS has published the first results fromthe RHIC 2000 run on particle multiplicity [49], as well as the RHIC2001 run [50].• STAR (Soleonoidal Tracker at RHIC)is a (large) 4π primarily hadronic-particle detector, with a 4-m-diameterand 4-m-long solenoidal 0.5-T magnetic-field volume, comprising as themain charged-particle-tracking device a TPC with inner radius 50 cmand outer radius 200 cm, with 45 planes of tracking. This allows apseudorapidity coverage of −2 ≤ η ≤ +2, and the design allows for alower particle-momentum cutoff at 60 MeV/c. In addition, the innersilicon vertex tracker (SVT) is surrounding the interaction region be-tween 5 and 15 cm, facilitating observation of the production of stran-geness. The time-of-flight array, 2.5 m from the primary interac-tion vertex, will help identify charged particles. The outside electro-magnetic calorimeter (EMC) aims to measure jets of particles, fluctua-tions, and high-p⊥ phenomena. The high tracking resolution facilitatesreconstruction of unstable hadronic resonances. First results on centralproduction of antiprotons [19] and anisotropy of particle multiplicity(elliptical flow) [15] have been published.Still much more extreme matter conditions will be reached when theLHC collider is completed (http://lhc.web.cern.ch/lhc/) and the equiva-lent laboratory energy of EmaxP 2×1016A eV reaches into the domain ofhighest cosmic-particle energies, where the cosmic flux begins to decreaseunusually rapidly. This ‘knee’ in cosmic flux as a function of the energyis below the high end of the LHC energy. At the LHC there will be ini-tially three major detectors, ATLAS, CMS, and ALICE. ALICE is thededicated heavy-ion experiment. CMS is intended to measure dileptonspectra under heavy-ion operation conditions. The ATLAS collaborationis exploring the potential of its detector in the heavy-ion environment.• ALICE (A Large Ion Collider Experiment)‡. It comprises a TPC as amain tracking device of charged particles with an inner radius of 1 mand an outer radius of 2.5 m, and a length along the beam direction‡ See for further details the web page http://www1.cern.ch/Alice.78 Experiments and analysis tools1020304050607080901002 4 6 8 10 1232S 200 GeV/ A c16O 200 GeV/ A c16O 60 GeV/ A c4He 12C 4.5 GeV/ A c16O 60 200 GeV/ − A c AT1/3 + AP1/3 σprod1/2 [mb1/2]Fig. 5.3. The square root of the inelastic reaction cross section,√σ, as a func-tion of the geometric size of interacting nuclei, A1/3T +A1/3P , for various collisionpartners, after [37].of 5m, covering the pseudorapidity interval −0.9 < η < 0.9. The high-resolution inner tracking system consists of five concentric cylindricallayers with radii from 7.5 to 50 cm around the beam pipe and allows thestudy of decays of charmed particles. An electro-magnetic calorimeterand a dilepton arm complement this very large and universal detector.5.2 Reaction energy and collision geometryOn intuitive grounds, we expect that, for the short-range hadronic inter-actions, the collision geometry determines the amount of matter partic-ipating in nuclear collisions. The collision geometry is a very importantand carefully explored subject. For an in-depth discussion of the impor-tance of collision geometry, we refer the reader to the extensive body ofwork for hadron–hadron and hadron–nucleus interactions [184, 190].The earliest experimental heavy-ion results confirmed the role of thissimple geometric picture of nuclear-collision reaction dynamics [201]. Thereaction radius, defined as the square root of the reaction cross section,rises linearly with the geometric size of the colliding nuclei, describedby the sum of their radii, which is proportional to A1/3, as is shown in5 Nuclei in collision 79aFig. 5.4. A geometric illustration of nuclear collision. Left: small sulphur oraluminium nuclei colliding with much heavier lead or gold targets (correct rela-tive scale). Right: symmetric collision of large nuclei at impact parameter a.Fig. 5.3. This result confirms that the colliding nuclei need to ‘touch’each other for local deposition of energy and baryon number to occur.We show the central collision of sulphur or aluminium nuclei collidingwith much heavier lead or gold targets in Fig. 5.4 (left-hand side). Onthe right-hand side, the symmetric slightly off-center collision with leador gold is illustrated using the correct relative scale – we can see howimportant it is to assure that, in this system, the collision is geometricallyas central as possible, in order to minimize the number of spectator (non-interacting, or partially interacting) nucleons. In symmetric collisions,only in a quite rare situation in which the impact parameter a is very smalldo we truly have the benefit of the largest possible region of interactionof the projectile and target, and do not encounter complications arisingfrom spectator matter ‘polluting’ the experimental data.A quantity of considerable importance is the energy content of the col-liding system, which must be, by virtue of conservation of energy, theenergy content of the final-state many-body system. The Lorentz invari-ant quantity we can form from the energy and momentum of the collidingprojectile (p) and target (t) is√spt ≡√(Ep + Et)2 − (/pp + /pt)2. (5.2)In the CM frame where by definition /pp + /pt = 0,√spt is recognizedas the available energy content of the projectile–target reaction, the CMenergy. The quantity√s is thus the available reaction energy. Since it isan invariant,√s can be evaluated in any reference frame. It is natural togeneralize this definition to any number of particles:√s(n) ≡√√√√( n∑i=1Ei)2−(n∑i=1/pi)2. (5.3)For n = 1, we see that√s is just the mass of a particle, i.e., its energycontent at rest. The conservation of energy assures that, when a particle80 Experiments and analysis tools4781115135353363Fig. 5.5. A streamer-chamber picture of a S–Ag collision taken at 200A GeV(NA35 experiment [115]) showing the multiplicity of charged particles bent upand down in magnetic fields (with decays of neutral strange particles identifiedby superposed dashed lines).decays, the final state comprising any number of particles n has the same√s. Conversely,√s(n) is also the (Lorentz invariant) mass of the ancestorsystem of the final-state n-body system, as determined by the momentumfour vectors pµi = (Ei, /pi) of the particles produced.This final-state energy described by Eq. (5.3) must be delivered by thecolliding nuclei, see Eq. (5.2).√s(n) is also the invariant intrinsic restenergy (mass) of the fireball of dense matter, measured in terms of theparticipating energy and momentum of the colliding nuclei. Both thesemeasures are jointly used in experiments to characterize a collision in-teraction: for example, the absence of the forward energy/momentum ofthe beam in the so-called zero-degree calorimeter (ZDC) can be corre-lated to the energy found in particles emitted in a direction transverseto the collision axis, see section 9.4, in order to define the geometric cen-trality of the collision. We will not follow these procedures further inthis book, also since each experimental group applies a slightly differentmethod.In the fixed-target experiments, the longitudinal momentum is largelydue to the Lorentz transformation from the CM frame to the laboratoryframe. This longitudinal momentum is in general considerably greaterthan the transverse momentum component, and particles are focused for-ward along the collision axis, as seen in Fig. 5.5 [115]. In this streamer-5 Nuclei in collision 81θpp Tp LϕFig. 5.6. The decomposition of particle momentum /p (shown in the CM frame)into the parallel pL and perpendicular p⊥ components. Note the inclination angleθ of /p and the azimuthal angle ϕ of p⊥ (this is a qualitative presentation).chamber picture of a S–Ag collision at 200A-GeV of the NA35 experi-ment, we see all particles in a cone to the right of the interaction vertexto which the charged-particle tracks are pointing. We also see that cen-tral collisions of S–Ag nuclei at 200A-GeV lead to the production of manysecondary particles. Both positive and negative particles are bent in theapplied magnetic field pointing normal to the plane of the picture. Severalsimultaneous photographs taken from various directions allowed precisetracking of charged particles.Not all particle tracks go through the interaction vertex at the left-handedge of Fig. 5.5: a few particle tracks, highlighted by dotted lines, belongto the V decays of neutral (strange) particles, see Fig. 2.3 on page 28.Low-momentum particles winding up as spirals in the high (1-T magni-tude) magnetic field do not originate from the primary high-energy-vertexinteractions.5.3 RapidityWe will now introduce the key kinematic variables that relate particlemomentum to the dynamics that is occurring in the heavy-ion reaction.Each particle momentum decomposes, as shown in Fig. 5.6, into a longi-tudinal component (pL) and a transverse component (/p⊥) with referenceto the collision axis. We note, in Fig. 5.6, the inclination angle θ of theparticle against the collision axis. Also shown is the azimuthal angle ϕ ofthe two-dimensional vector /p⊥.82 Experiments and analysis toolsThe longitudinal momentum of a particle is an inconvenient variable,since it depends on the velocity of the CM frame with reference to thelaboratory frame, as the appearance of Fig. 5.5 reminds us. For theanalysis and understanding of the experimental results, it is necessary tobe able to view the physical results from the CM frame, e.g., to transformthe coordinate system to the CM frame of reference. The introductionof the rapidity y, replacing pL, allows one to considerably simplify theselection and changing of the reference frame. This is due to the factthat the variable y is defined to be additive under successive Lorentztransformations along the same direction, as we shall see in Eq. (5.14):it can be understood as the ‘angle’ of the (hyperbolic) rotation in (3 +1)-dimensional space. The ‘angle’ y is defined in terms of energy andmomentum by the equationsE = m⊥ cosh y, pL = m⊥ sinh y, (5.4)where m⊥ is the transverse ‘mass’:m⊥ =√m2 + /p 2⊥. (5.5)We note that Eqs. (5.4) and (5.5) are consistent with the relativisticdispersion relation (energy–momentum relation):E =√m2⊥ + p2L =√m2 + /p 2⊥ + p2L. (5.6)The variable y (and m⊥ ≥ m) replaces pL (and |/p⊥|), which are usuallydefining the momentum of a particle. The azimuthal angle ϕ of the vector/p⊥, see Fig. 5.6, is the third variable required in the complete definitionof /p.The relation between velocity and rapidity is obtained from Eq. (5.4):vL ≡cpLE= c tanh y. (5.7)Thus, in the non-relativistic limit, vL → cy.§ Equation (5.7) also impliesthatcosh y =1√1− v2L≡ γL, sinh y = γL vL, (5.8)where γL is the (longitudinal) Lorentz contraction factor. Sincetanh−1 z =12ln(1 + z1− z), (5.9)§ Even though we like to work with units that do not explicitly introduce the velocity oflight c, whenever the non-relativistic limit is discussed, it is convenient to reintroducec explicitly into the equations, as shown above.5 Nuclei in collision 83we obtain from Eq. (5.7)y =12ln(1 + vL1− vL)=12ln(E + pLE − pL)= ln(E + pLm⊥). (5.10)Lorentz ‘boosts’ are the Lorentz transformations with one of the three(x, y, z) Cartesian coordinate directions employed as the reference axis forthe transformation. To verify the additivity of rapidity under a sequenceof Lorentz boosts mentioned earlier, we consider the transformation ofthe momentum vector under a change of the reference frame along thecollision axis. Under such a transformation, the transverse momentumand the transverse massm⊥ are not changed. The energy and longitudinalcomponent of momentum transform according toE′ = γc(E + vc pL), p′L = γc(pL + vcE). (5.11)Here and below, the ‘primed’ quantities are seen by an observer in thelaboratory system which moves with the velocity vc with respect to theCM frame of reference, in which the energy E and momentum pL aremeasured. Noting that the rapidity yc of the transformation satisfiesEq. (5.8), we obtaincosh yc = γc, sinh yc = γc vc, (5.12)and, upon introducing Eq. (5.4), we find for Eq. (5.11)E′ = m⊥ cosh(y + yc), p′L = m⊥ sinh(y + yc). (5.13)It is now evident that the rapidity y′ seen in the laboratory system isgiven in terms of the CM rapidity y byy′ = y + yc. (5.14)It is this simple result which gives the rapidity variable its importance asa tool in the analysis of particle-production data. For example, in fixed-target experiments, we can study particle spectra using y as a variablewithout an explicit transformation to the CM frame of reference, anddeduce from the rapidity spectra the point of symmetry corresponding tothe CM rapidity. In symmetric collisions with fixed targets, the CM framehas to be in the middle between the rapidities of projectile and target;the CM rapidity is half of the rapidity of the projectile yCM = yp/2. Inthis case, the particle-rapidity spectrum must be symmetric around yCM.This allows one to complement measured particle spectra: if these areavailable for, e.g., y ≥ yCM, a reflection at the symmetry point yCM givesus the part of the spectrum with y ≤ yCM, or vice-versa.Understanding the actual value of yCM is of particular interest in ‘asym-metric’ collisions of heavy ions, i.e., those involving two different nuclei,84 Experiments and analysis toolswhich we continue calling ‘projectile and target’, though such a distinctionis irrelevant in our following argument, since the result will be symmet-ric between the two colliding nuclei. We recollect that, considering thedefinition Eq. (5.4), we also haveE ± pL = m⊥e±y. (5.15)The total energy and momentum of the colliding system is obtainedfrom the total energy and momentum of colliding nuclei:E = Ep + Et, pL = pp + pt. (5.16)Using Eq. (5.10) for these values of E and pL, we obtain the rapidityof the frame of reference in which the combined longitudinal momentumvanishes. For collinear collisions, the transverse momentum also vanishes,and this is the rapidity of the CM frame. Using Eq. (5.10),yCM =12ln(Ep + Et + pp + ptEp + Et − (pp + pt)). (5.17)We use now Eq. (5.15) for the rapidities of projectile and target and obtaina manifestly projectile–target-symmetric expression:yCM =12ln(mpe+yp +mte+ytmpe−yp +mte−yt). (5.18)We now consider the asymmetric collisions both for collider and forfixed-target experiments: another way to write Eq. (5.18) offers immediateunderstanding of the physics involved. We take the factor eyp in thenumerator and the factor e−yt in the denominator out of the logarithmand obtainyCM =yp + yt2+12ln(mp +mte−(yp−yt)mt +mpe−(yp−yt)). (5.19)In most cases of interest, we have yp − yt 0 and thusyCM yp + yt2− 12ln(mtmp)+m2t −m2p2mpmte−(yp−yt) + · · · . (5.20)In general, the first two terms largely suffice. In the way we wroteEq. (5.20), we chose the usual convention to call the more massive nucleusthe ‘target’. Two cases of explicit interest in Eq. (5.20) are the collidermode yp = −yt, and a stationary target yt = 0 (up to Fermi motion inthe stationary target nucleus).For asymmetric collisions, the precise magnitude of mt is determinedin part by the value of the impact parameter, see Fig. 5.4. Hence the5 Nuclei in collision 85CM rapidity, Eq. (5.20), becomes dependent on the impact parameter.The magnitude of the shift in asymmetry of rapidity arising can be easilyestimated: in collisions in which the projectile with AP emerges fully inthe target AT, all of the projectile nucleons participate: Ap = AP < AT,while the number of target participants At isAt ∝ A2/3p A1/3T . (5.21)Thus,yCM yp + yt2− 16ln(ATAP). (5.22)For light-on-heavy-ion collisions such as of S on Pb, the expected andobserved shift in mass asymmetry of rapidity (the last term in Eq. (5.22))is noticeable (0.3 units).5.4 Pseudorapidity and quasirapidityIn the study of production of charged hadrons, e.g., in section 9.2, we willsee that observed particles are often not identified, and hence we do notknow their masses, which are required in order to determine the rapidityof particles Eq. (5.10), given the momentum measured by deflection ofparticles within a magnetic field. On the other hand, mass can be negligi-ble compared with the momenta carried by the particles, especially so infixed-target experiments. Consequently, we now consider what happenswith the rapidity spectra when the mass of a particle is small relativeto the momentum, and the momentum alone determines the energy of aparticle, e.g.,E =√p2 +m2 → p. (5.23)In analogy to Eq. (5.4), a simpler variable, the ‘pseudorapidity’ η of aparticle is introduced,p = p⊥ cosh η, pL = p⊥ sinh η, (5.24)which, with Eq. (5.10), leads toη =12ln(p+ pLp− pL)=12ln(1 + cos θ1− cos θ)= ln(cotθ2). (5.25)Here, θ is the particle-emission angle relative to the beam axis, see Fig. 5.6.In Fig. 5.7, we see for the range of pseudorapidity of interest to us (upto η = 9, the maximum value seen in Fig. 5.1) how the angle θ varies withthe pseudorapidity. A massless particle emitted transversely at η = y = 086 Experiments and analysis toolsFig. 5.7. The emission angle θ in radians as a function of the pseudorapidity η.has θ = π/2; η = 3 corresponds to θ = 0.1 rad, (≡0.1 × 180o/π = 5.7o).The projectile–target fragmentation region at the LHC, where η 8.5,corresponds to θ = ±4.5× 10−4 rad (≡ ±0.025o).From Eqs. (5.4) and (5.24), we obtain the implicit relations betweenpseudorapidity and rapidity:m⊥ sinh y = p⊥ sinh η, E tanh y = p tanh η. (5.26)We see from these relations that the pseudorapidity is always greater thanthe rapidity:sinh ηsinh y=m⊥p⊥=√1+m2p2⊥> 1,tanh ηtanh y=Ep=√1+m2p2> 1. (5.27)More massive particles that have not been identified appear in a pseu-dorapidity particle spectrum at greater values of η than do the lighterpions.In order to establish a precise relation between pseudorapidity andrapidity, we replace in Eq. (5.25) the (longitudinal) momentum usingEqs. (5.4) and (5.6) to obtainη =12ln√m2⊥ cosh2 y −m2 +m⊥ sinh y√m2⊥ cosh2 y −m2 −m⊥ sinh y. (5.28)5 Nuclei in collision 87Similarly, to determine rapidity in terms of pseudorapidity, we simply re-place the momenta in the definition of rapidity, Eq. (5.10), using definitionEq. (5.24):y =12ln√m2 + p2⊥ cosh2 η + p⊥ sinh η√m2 + p2⊥ cosh2 η − p⊥ sinh η. (5.29)Taking the logarithm of the first expression in Eq. (5.27), we obtain theshift in pseudorapidity relative to rapidity:δη ≡ η − y = 12ln(1 +m2p2⊥)+ ln(1− e−2y1− e−2η). (5.30)The leading term is the only term remaining for large η and it establishesan upper limit for the shift δη. The difference δη = η − y between thepseudorapidity and the rapidity, as a function of pseudorapidity, is shownin Fig. 5.8, which was obtained by inserting Eq. (5.29) into Eq. (5.30).Thick lines are for p⊥ = 0.3 GeV, thin lines for p⊥ = 0.5 GeV; solidlines are for nucleons, chain lines for kaons, and dashed lines for pions.We see that, when η ≥ 3, the first term in Eq. (5.30) in fact sufficesto approximate the ‘shift’ in pseudorapidity which approaches a fixedmaximum.For sufficiently large p⊥ > m, when a particle’s rest mass can be ne-glected, the shift δη becomes negligible. For pions the error associatedwith considering the pseudorapidity instead of rapidity in hadronic re-actions can often be ignored since the mass is usually smaller than thetypical momentum cut – and thus δη < 0.1 is seen at pseudorapidity η = 3for p⊥ > 0.3 GeV. Moreover, the use of pion-quasirapidity yπ, which wediscuss next, eliminates this shift completely. On the other hand, use ofpseudorapidity seems not to be advisable for situations in which contribu-tions from more massive particles are of importance, unless, as Eq. (5.30)suggests, the p⊥ cut is well above the mass of the particle. We see, inFig. 5.8, that, for nucleons, taking the transverse momentum cut at 0.3GeV, one encounters a shift of more than one rapidity unit at η = 3, theSPS value.Since, in the upper SPS energy range (see table 5.1), the pion abun-dance dominates the hadron abundance, it has become common practiceto show the distribution of hadrons as a function of pion-quasirapidity yπ,presuming that all hadrons observed are pions, as is done in Fig. 9.6 onpage 166. One assumes, in lieu of the correct definition for each particle,the expression as if this particle had the mass of a pion:pL = p⊥ sinh η → pL =√p2⊥ +m2π sinh yπ. (5.31)88 Experiments and analysis toolsFig. 5.8. The difference between pseudorapidity and pion-quasirapidity π(dashed lines), kaons K (chain lines), and nucleons N (solid lines) as a func-tion of pseudorapidity η, for p⊥= 0.3 GeV (thick lines) and p⊥= 0.5 GeV (thinlines).Following the derivation of Eq. (5.30), we obtainδy ≡ yπ − y =12ln(1 +m2 −m2πp2⊥ +m2π)+ ln(1− e−2y1− e−2yπ). (5.32)In Fig. 5.9, we see the solution of the above equation as a function ofthe pion-quasirapidity. Lines are for p⊥ = 0.5 GeV(bottom line, smallestshift), p⊥ = 0.3 GeV (middle line), and for p⊥ = 0.1 GeV(top, largestshift); solid lines are for nucleons and chain lines for kaons. Again, whenδy ≥ 3, the first term in Eq. (5.32) nearly suffices to approximate the‘shift’ in rapidity for pions as shown in Fig. 5.9 for increasing yπ, it ap-proaches a fixed value, which for p⊥ < mπ is significant. The quasirapid-ity distribution for nucleons experiences a widening by ±1.9, and that forkaons widens by ±1.3 units of rapidity.As we see in Figs. 5.8 and 5.9 and Eqs. (5.30) and (5.32), the error inmeasurement of rapidity grows with decreasing p⊥ of the particle. Forkaons and nucleons, in the range of p⊥ within which the pseudorapidity is5 Nuclei in collision 89Fig. 5.9. The difference between pion-quasirapidity and rapidity as a functionof quasirapidity yπ for kaons K (chain lines), and nucleons N (solid lines), frombottom to top for p⊥ = 0.5 GeV, p⊥ = 0.3 GeV, and p⊥ = 0.1 GeV.failing to be a good variable, the number of particles produced increaseswith decreasing p⊥. Thus, in fact, use of pseudorapidity or quasirapiditycan be significantly misleading when one wants to understand both thespectral shape and the hadron yield.In this context, we recall that a study of the distribution of heavyparticles (nucleons and kaons) can be based on the difference betweenthe distributions of positively and negatively charged particles, which isrelatively easy to measure:d(N+ −N−)dyπ=d(π+ − π−)dyπ+d(p− p̄)dyπ+d(K+ −K−)dyπ. (5.33)The physics, in Eq. (5.33), is that the yield of pions is nearly charge sym-metric (this has been observed at the SPS for p⊥ > 0.3 GeV [77]) andthe first term cancels out. In the remainder, we have an initial measureof the quasirapidity distribution of protons and kaons. At the SPS, bothprotons and kaons contribute in Eq. (5.33). At the RHIC, the abundanceof charged kaons is the dominating contribution, but only at the level of1%–3% of all charged particles. As noted above, at the SPS, the canceling90 Experiments and analysis toolsout of charged-pion yield is not exact at low p⊥ < 0.25 GeV, as a directmeasurement has shown [77]: in the Pb–Pb collision system, we have 20%(n− p)/(n + p) asymmetry in the number of protons and neutrons. Thischarge asymmetry translates into a relatively strong π+/π− asymmetryat small p⊥, but disappears for p⊥ > 0.2 GeV.5.5 Stages of evolution of dense matterSince hadronic interactions are strong, we can hope and expect that localequilibrium conditions can be approached in experiments involving heavyions. This is in particular the case if we characterize the essential physicalproperties of elementary matter in term of local, position-dependent pa-rameters. The local average energy of each particle characterizes the localtemperature T . (Local) chemical potentials µi need to be introduced inorder to regulate the average particle and/or quark-flavor density.These parameters express different equilibration processes in the fire-ball, and in general there is a considerable difference between thermal andchemical equilibrium.• In order to establish thermal equilibrium, equipartition of energy amongthe different particles present has to occur in the collisional processeswhich lead to the statistical energy distribution. It is important tonote that (local) thermal equilibrium can be achieved solely by elasticscattering. We will call the time scale on which these processes occurτth. The use of temperature T as a parameter presupposes that thermalequilibrium has (nearly) been established.• Chemical equilibration requires reactions that change numbers of par-ticles, and it is more difficult and thus slower to become established.There are also two quite different types of chemical equilibria.— Relative chemical equilibration, just like the commonly known casein chemistry, involves reactions that distribute a certain already exis-tent element/property among different accessible compounds. Use ofchemical potentials µi presupposes, in general, that the particular rela-tive chemical equilibrium is being considered. We call the relevant timescale τ relchem.— In relativistic reactions, particles can be made as energy is convertedinto matter. Therefore, we can expect to approach (more slowly) theabsolute chemical equilibrium. We call the relevant time scale τabschem.We characterize the approach to absolute chemical equilibrium by afugacity factor γi for particle ‘i’. We often study the evolution of γi inthe collision as a function of time, since absolute chemical equilibriumcannot generally be assumed to occur.5 Nuclei in collision 91Some authors introduce separate chemical potentials for particles andantiparticles, µ±i = ±µi + T ln γi, [188, 189]. This is equivalent to ourapproach. However, it is still common to see in the literature that a equi-librium is assumed, γi = 1, with particles ‘instantaneously’ reaching theirabsolute chemical-equilibrium abundances. Such an approach cannot ingeneral be justified. We see this on considering the relation between therelaxation times,10−22 s > τ exp τabschem > τ relchem > τth, (5.34)where τ exp is the life span of the expanding fireball of dense matter, whichis of the same magnitude as the time light needs to traverse the largestnuclei. In such a rapidly evolving system, we cannot assume that absolutechemical equilibrium, γi = 1, has been attained.In order to illustrate the difference between absolute and relative chem-ical equilibrium better, let us consider some examples.• We consider the baryon number, the globally conserved property ofdense hadronic matter. Locally, the global conservation implies a balanceof inflow against outflow, viz., there are no sources or sinks of baryonnumber. Generally, one always associates a conserved quantity with thepresence of a chemical potential, here the chemical potential µb which con-trols the difference in number of all baryons and antibaryons. A changein the energy of the system, according to the first law of thermodynamics,is then given bydE = −P dV + T dS + µb db. (5.35)However, the addition of a baryon–antibaryon pair to the system will notbe noted in Eq. (5.35), since the baryon number b remains unchanged!At this point, we are not at liberty to add or remove a pair: in writingdown Eq. (5.35), we implicitly assumed what we have above called abso-lute chemical equilibrium – there is a bath of baryon number in whichour system is immersed, and hence a full phase-space occupancy of allavailable phase-space cells, and there is no place for an extra pair. Bychanging the chemical potential µb, we can regulate the difference innumber of baryons and antibaryons present in the system, but densitiesof baryons and antibaryons move together, absolute equilibrium is as-sumed while relative chemical equilibrium controls the relative number ofparticles by virtue of the value of the chemical potential. If we changethe baryon number by one at fixed volume and entropy, then accordingto Eq. (5.35), there is a change in energy by µb.• Next, we look at the abundance of strangeness in the baryon-rich HGphase. There is no strangeness ‘bath’ and, initially, we have no stran-geness, therefore we will be making pairs of s and s̄ quarks; there isplenty of phase space available to fill, and we are far from absolute92 Experiments and analysis toolschemical-strangeness equilibrium. To make ss̄ pairs in a HG, there aremany possible reactions, section 18.2, classified usually as the direct- andassociate-production processes. In the associate-production process, apair of strange quarks is shared between two existent hadrons, of whichone is a baryon, typically a nucleon N, which becomes a hyperon Y:π+N↔ K+Y.In a direct-production process, a pair of strangeness-carrying particles isformed directly via annihilation of two mesons, akin to our Gedankenex-periment in which we are adding a pair to the system:π+ π ↔ K+K.Here, a pair of strange particles is made in the form of a pair of kaons,K+K−. With these two reaction types alone it could be that popula-tions of strange mesons and baryons evolve differently. However, themeson carrier of the s quark, K−, can exchange this quark rather fast, viaexothermic reaction with a nucleon, forming a hyperon:K− +N↔ π+Y.This reaction establishes relative chemical equilibrium by being able tomove the strange quark between the two different carriers, sq̄ mesons andsqq baryons.Reactions establishing the redistribution of existent flavor, or the abun-dance of some other conserved quantity, play a different role from thereactions that actually contribute to the formation of this flavor, or otherquantum number, and facilitate the approach to absolute chemical equi-librium. Accordingly, the time constants for relaxation are different, sincedifferent types of reaction are involved.Apart from the different relaxation times associated with the differenttypes of thermal and chemical equilibria, there are different time scalesassociated with the different fundamental interactions involved. For ex-ample, the electro-magnetic interactions are considerably slower at reach-ing equilibrium than are the strong interactions governing the evolution ofdense hadronic fireballs created in ultra-relativistic heavy-ion collisions.All the important time constants for relaxation in heavy-ion collisionsarise from differences in mechanisms operating within the realm of stronginteractions. Therefore, the separation of time scales is not as sharp asthat between the different interactions, though a clear hierarchy arises,as we noted in Eq. (5.34).Under weak interactions, there is, in comparison, an extremely slowtransmutation of quark (and lepton) flavors, involving a much longerelectro-weak equilibration time. Such long times are not available in the5 Nuclei in collision 93micro-bang process, in contrast to the big-bang case. We considered theseelectro-weak degrees of freedom in section 1.3, since the life span of theUniverse at the time of the hadronic phase transition exceeds all typicalrelaxation times for the weak interaction.The chemical equilibration, and hence the chemical composition of thefireball, evolve along with the temperature of the fireball. The followingstages occur in heavy-ion collision.1. The initial quantum stage.The formation of a thermalized state within τth is most difficult tounderstand, and is subject to intense current theoretical investigation.During the pre-thermal time, 0 ≤ t < τth, the properties of the collisionsystem require the study both of quantum transport and of decoher-ence phenomena, a subject reaching today far beyond the scope of thisvolume. We assume, in this book, that the thermal shape of a (qu-ark, gluon) particle-momentum distribution is reached instantaneouslycompared with the time scales for chemical equilibration in Eq. (5.34).This allows us to sidestep questions regarding the dynamics occurringin the first moments of the heavy-ion interactions, and we explore pri-marily what happens after a time¶ τ0 ≡ τth 0.25–1 fm/c. The valueof τ0 decreases as the density of the pre-thermal initial state increases,e.g., as the collision energy increases. At τ0 gluons g are, due to theirgreater reactivity, at or near to the local chemical equilibrium.2. The subsequent chemical equilibration time.During the inter-penetration of the projectile and the target lasting noless than ∼1.5 fm/c, diverse particle-production reactions occur, allow-ing the approach to chemical equilibrium by light non-strange quarksq = u, d. As the energy is redistributed among an increasing numberof accessed degrees of freedom, the temperature drops rapidly.3. The strangeness chemical equilibration.A third time period, lasting up to5 fm/c, during which the productionand chemical equilibration of strange quarks takes place. There is areduction of temperature now mainly due to the expansion flow, thoughthe excitation of the strange quark degree of freedom also introduces anon-negligible cooling effect.4. The hadronization/freeze-out.The fireball of dense matter expands and decomposes into the final-state hadrons, possibly in an (explosive) process that does not allowre-equilibration of the final-state particles. The dynamics is stronglydependent on the size of the initial state and on the nature of theequations of state.¶ The time τth is often called τ0 in the literature, and we will use this notation as well,though the subscript ‘th’ is more specific about the evolution step considered.94 Experiments and analysis toolsThroughout these stages, a local thermal equilibrium is rapidly estab-lished and, as noted, the local temperature evolves in time to accom-modate change in the internal structure as is appropriate for an isolatedphysical system. We have a temperature evolution that passes throughthese series of stages:Tth the temperature associated with the initial thermal equilibrium,↓ evolution dominated mainly by production of q and q̄;Tch chemical equilibrium of non-strange quarks and gluons,↓ evolution dominated by expansion and production of s and s̄;Ts condition of chemical equilibrium of u,d and s quark flavors,↓ expansion, dissociation by particle radiation;Tf temperature at hadron-abundance freeze-out,↓ hadron rescattering, reequilibration; andTtf temperature at thermal freeze-out, T = T (τ exp).We encounter a considerable decrease in temperature. The entropy con-tent of an evolving isolated system must increase, and this is initiallyrelated to the increase in the number of particles within the fireball andlater also due to the increase in volume. However, in the later stagesdominated by flow, the practical absence of viscosities in the quark–gluonfluid implies that there is little additional production of entropy. The finalentropy content is close to the entropy content established in the earliestthermal stage of the collision at t < τ0, despite a drop in temperature byas much as a factor of two (under current experimental RHIC conditions)during the evolution of the fireball.Except for the unlikely scenario of a fireball not expanding, but sud-denly disintegrating, none of the temperatures discussed above corre-sponds to the temperature one would read off the (inverse) slopes ofparticle spectra. In principle, the freeze-out temperature determines theshape of emission and multiplicity of emitted particles. However, thefreeze-out occurs within a local flow field of expanding matter and thethermal spectrum is to be folded with the flow which imposes a Doppler-like shift of Ttf : we observe a higher temperature than is actually locallypresent when particles decouple from flowing matter (kinetic or thermalfreeze-out). The observable temperature T⊥ is related to the intrinsictemperature of the source:T⊥ 1 + /n · /vtf√1− /v 2fTtf →√1 + vtf1− vtfTtf . (5.36)This relation must be used with caution, since it does not apply in thesame fashion to all particles and has a precision rarely better than ±10%.We study the shape of m⊥-spectra in section 8.5.5 Nuclei in collision 955.6 Approach to local kinetic equilibriumIn the above discussion, the formation of a space–time-localized fireballof dense matter is the first key physics input. The question we wish toaddress now is that of how this fireball can possibly arise from a rathershort sequence of individual reactions that occur when two, rather small,gas clouds of partons, clustered in nucleons, bound in the nucleus, collide.Indeed, at first sight, one would be led to believe that the small cloudscomprising point-like objects would mutually disperse in the collision,and no localized, dense state of hadronic matter should be formed. Atbest, it was suggested in some early work, the two colliding ‘eggs’ shouldemerge from the high-energy interaction slightly ‘warmed’, but still largely‘unbroken’.Two remarkable properties of hadronic interactions are responsible forjust the opposite, deeply inelastic, behavior:• the multiparticle production in hadron–hadron collisions; and• the effective size of all hadrons expressed in term of their reaction crosssections.What appears to be a thin system of point-like constituents is effectivelyalready a volume-filling nucleon liquid, which will undergo, in a colli-sion, a rapid self-multiplication with particle density rising and individ-ual scattering times becoming progressively much shorter than the overallcollision time.Ultimately, as the energy available in collision is increased, the hadronparticle/energy density will reach values at which the dissolution of thehadronic constituents into a common deconfined domain will become pos-sible, and indeed must occur according to our knowledge about stronginteractions. While we do not really know whether deconfinement of had-rons is not a general mechanism operating already at AGS energies, seetable 5.1, there is today no experimental evidence that this low-energyrange suffices. In contradistinction, a significant number of results ob-tained at the SPS energy range can be most naturally interpreted interms of the formation of a deconfined space–time domain, section 1.6.We note that, per participant, there are as many as 7–10 further hadronsproduced at SPS energies. This implies that there are thousands of quarksand gluons in the space–time domain of interest, and hence considerationof a ‘local’ (in space–time) equilibrium makes good sense.There are many ways to estimate the particle number. We can use thenumber of final-state hadrons and evaluate the numbers of constituentquarks and antiquarks, or we can take the available energy content anddivide it by the estimated energy per particle (quark, gluon). Both pro-cedures give O(10 000) particles for the case of Pb–Pb collisions at 158A-GeV (√sNN = 17.2 GeV). Of these, not all particles can be causally96 Experiments and analysis toolsconnected, i.e., not all these particles can influence each other in classicaldynamics, and local equilibrium is a feature obviously involving a causallyconnected region only.A suitable measure of the causally connected size is offered by the initialdecoherence time τ0, which also determines the size of the decoherencevolume, R0 τ0. This has to be scaled up by the ensuing expansionfactor, see the discussion below Eq. (6.35). For τ0 = 0.5 fm, we can expectabout 5%–10% of all particles (500–1000) to be causally connected, whichimplies that the causal ‘range’ of rapidity is an interval ∆y 1. ∆y ariseson considering the final-state rapidity distribution, see Fig. 9.6 on page 166and Fig. 9.19 on page 184. In any case, the concept of a local equilibriummakes good sense.When we are talking about thermal equilibria, we must first establishmore precisely what these words mean. We will implicitly always refer to‘local’ equilibrium. The thermalization of the momentum distributions isdriven by all scattering processes, elastic as well as inelastic, because allof them are associated with transfer of momentum and energy betweenparticles. The scattering time scale, for particles of species i, is given interms of the collision length l byτi,scatt =〈lv〉i=1∑j〈σijvij〉ρj, (5.37)where the sum in the denominator is over all particle species (with den-sities ρj) available, σij and vij are the (energy-dependent) total crosssections and relative velocities, for a process scattering particles i and j,and the average is to be taken over the momentum distributions of theparticle considered.It is not hard to ‘guestimate’ the time scale governing the kinetic equi-libration in the QGP. The typical particle-collision time (the inverse ofthe collision frequency) is obtained from Eq. (5.37) above. Given the par-ticle densities and soft reaction cross sections, with the relative velocityof these essentially massless components being the velocity of light c, wefind for the QGP scattering time,τQGPi = 0.2–2 fm, with ρi = 2–10 fm−3, σi = 2–5mb, (5.38)as a range for different particles of type i, with the shorter time applyingto the early high-density stage. This is about an order of magnitudeshorter than the time scale for evolution of the fireball, which is derivedfrom the spatial size of the colliding system: for the largest nuclei, inparticular the Pb–Pb or Au–Au collisions, over a wide range of energy,we expectτ exp RAc 5–8 fm/c. (5.39)5 Nuclei in collision 97The achievement of kinetic equilibrium must be visible in the energyspectra of the particles produced, as we shall discuss below in section 8.1.This behavior, as we argue, can be understood in qualitative terms forthe case of nuclear collisions. However, it remains to date a mysterywhy in some important aspects thermal models succeed for the case ofp–p reactions. In particular, the exponential fall off of particle spectra,suggesting thermal equilibrium, has been noted with trepidation for aconsiderable time.Hagedorn evaluated this behavior in the experimental data some 35years ago [140, 145] and he developed the statistical bootstrap model(chapter 12), which assumes a statistical phase-space distribution (sec-tion 12.2). Hagedorn called it preestablished or preformed equilibrium:particles are produced in an elementary interaction with a probabilitycharacterized by a universal temperature. We can today only speculateabout the physical mechanisms.For example, it has been proposed that vacuum-structure fluctuationslead to color-string tension fluctuation, and thus the resulting string-breaking produces thermal hadrons [65]. Another informally discussedpossibility is the presence of intrinsic chaotic dynamics capable of rapidlyestablishing kinetic equilibrium. We cannot pursue further in this bookthese ideas about the process of initial thermal equilibration.Sometimes, the fact that we do not fully understand thermalization inthe p–p case is raised as an argument against the possibility of conven-tional equilibration in nuclear collisions. We do not think so. In fact, ifthe p–p case leads to thermal hadrons, we should have a yet better ther-malization in the A–A case. Thus, a microscopic model that is adoptedto extrapolate from p–p to A–A collisions should respect the concept ofthe hadronic preestablished equilibrium, else it is not going to be fullysuccessful, see section 6.1.5.7 The approach to chemical equilibriumThe approach to chemical equilibrium is, in comparison with the thermalcase, better understood. Firstly, we must consider which particles can beexpected to have reached equilibrium and which not, and this requiresa kinetic description. Though, in general, one is tempted to think of abuild-up of chemical abundance of different quark flavors, the approach toabsolute chemical equilibrium need not always occur from ‘below’, and/orthe measured quark yields can be in excess of chemical equilibrium; sec-tion 19.4.At the collision energies available at the RHIC and LHC, the moremassive charm c and bottom b quarks (see table 1.1) are produced inthe initial interaction, reaching and even exceeding the yield expected in98 Experiments and analysis toolsabsolute chemical equilibrium in thermalized deconfined matter. In QGPthe chemical equilibration of these flavors occurs exceedingly slowly, anda significant excess of abundance is expected. A similar situation canarise with strangeness in presence of rapid cooling, from T > 250 MeVto T 150 MeV, which preserves the high initial thermal yield. Inthe early Universe, the well-known example of chemical nonequilibriumoccurring, despite thermal equilibrium being established, is the freeze-outof abundances of light nuclear isotopes.Of particular interest, in the physics of QGP, is that the saturation(‘absolute’ chemical equilibration) of the phase space of strange particlesrequires just the life span of the QGP. This is, in part, due to the rela-tively large threshold for the production of strange quarks and, in part,because for practical purposes most strangeness needs to be produced inthermal energy collisions – direct initial-state production of strangenessis of course quite prevalent but at the level of 10%–30% of the final-stateequilibrium yield of strangeness, as long as only the normal processes ofhadron collisions contribute to direct production of strangeness.In the QGP phase, there is no need to redistribute strange quarksamong different carriers and relative chemical equilibrium is automati-cally established. More generally, in the HG phase the relative chemicalequilibrium is more easily attained than is the ‘absolute’ chemical equi-librium, due to the strangeness-exchange cross sections being greater thancross sections for its production.The population master equation,2τ ichemρeqidρidt= 1−(ρiρeqi)2, (5.40)describes the population evolution of strangeness (and charm, etc.) withinthe scattering theory; chapter 17. τ ichem is the time constant for chemicalrelaxation. The quadratic term on the right-hand side, in Eq. (5.40),arises from, e.g., annihilation of strangeness, ss̄ → XX, which rate isestablished by detailed balance consideration of two-body reactions. Inthe first instance, one has not ρ2i but ρiρ̄i, where ρ̄i is the s̄ density.However, since in heavy-ion collisions only hadronic reactions producestrangeness, we maintain the condition ρi = ρ̄i and Eq. (5.40) follows. Thesolution of Eq. (5.40) approaches equilibrium exponentially for t → ∞:ρi = ρeqi tanh[t/(2τichem)]→ (1− e−t/τichem)ρeqi . (5.41)The chemical equilibration (relaxation) time constant τ ichem, for particlespecies i, is computed as an inverse of the invariant reaction rate per unitvolume Ri:τ ichem =ρeqi2Ri. (5.42)5 Nuclei in collision 99In Eq. (5.42), ρeqi is the chemical-equilibrium density. Ri is the rate atwhich the system ‘chases’ ρeqi ; the ratio is a characteristic time whenthe chase is over. The factor 2 in Eq. (5.42) is introduced to assure thatthe approach to equilibrium due to two-body reactions is governed by anexponential function with the time-decay parameter τ ichem, as is seen onthe right-hand side of Eq. (5.41).In terms of the reaction cross section, the invariant reaction rate perunit of time and volume is obtained from (see section 17.1)Ri(x)=∑a,b,X∫ ∞(mi+mX)22λ2(s) ds∫d3ka(2π)32Ea∫d3kb(2π)32Eb× fa(ka, x)fb(kb, x) σ̄ab→iX(√s) δ[s− (ka + kb)2]. (5.43)where, see Eq. (17.10),λ2(s) =[s− (ma +mb)2][s− (ma −mb)2].In Eq. (5.43), we are neglecting Pauli or Bose quantum effects (suppressionor stimulated-emission factors) in the initial and final states. Consideredhere, is the inelastic production process a + b → i + X. fa(ka, x) andfb(kb, x) are the phase-space distributions of the colliding particles, andσ̄ab→iX(√s) is the energy-dependent cross section for this inelastic chan-nel. The ‘bar’ indicates that the dependence on transfer of momentum(scattering angle) is averaged over.We will further study this integral for thermal distributions in sec-tion 17.1. However, given the importance of the final result Eq. (17.16),we record it here for the simplest case of a relativistic Boltzmann momen-tum distribution,Ri(x) =∑a,b,X∫∞w0dw λ2σ̄ab→iX(w)K1(w/T )4Tm2am2bK2(ma/T )K2(mb/T ), (5.44)where w =√s is the CM energy and w0 = mi + mX is the reactionthreshold. This formula is presented in this form in [164], Eq. (5.7); it isstated there for the special case in which the reacting particles a and bare identical bosons, which, to avoid double counting of indistinguishablepairs of particles, requires an extra factor 12 , which is not included inEq. (5.44). The interesting ma,b → 0 limit follows considering Fig. 10.1and Eq. (10.47). It is implemented with a replacement of each factorm2K2(m/T ) by 2T 2 in Eq. (5.44), and λ2 → s, which reduces Eq. (5.44)to the result presented in [226], Eq. (2); [67] lacks the factor 1/T .We see explicitly, in Eq. (5.44), the mass threshold in the s-integrationoccurring for inelastic (particle-producing) rates. A high threshold com-bines with the exponentially small K1 Bessel function, see Eq. (8.7), to100 Experiments and analysis toolsreduce the strength of inelastic hadronic particle-production rates, whichare usually much smaller than the total rates of reaction (scattering).For this reason, the time scale of chemical equilibration is, in general,considerably longer than the thermal one.6 Understanding collision dynamics6.1 Cascades of particlesThe principal shortcomings of the near-statistical-equilibrium method,combined with ideal flow of hadronic fluid in the study of heavy-ion col-lisions, are the following:• we do not have a long-lived, large region of hot hadronic matter to lookat, and some features of the collision are certainly not well equilibrated;• we need to establish the physical conditions at the initial time τ0; and• the system considered is subject to rapid evolution and all thermalproperties are actually fields, i.e., we have a position-dependent localtemperature T (/x), etc.Hence, a lot of effort continues to be committed to the development ofa better understanding of the initial interaction dynamics, and its sub-sequent description within microscopic kinetic-scattering models. Theresearch field of the study of computer-code ‘event generators’ is vast andundergoing development. Consequently, in this book, we will enter intodiscussion of kinetic models only as matters of example and/or principle.We survey the rapidly developing field in order to offer an entry point forfurther study.For a novice in this very rapidly changing panorama, the best next stepis to look at the progress of the working group which has been monitor-ing the development of the computer codes with the objective of ensuringthat reasonable quality control is attained.OSCAR (Open Standard Codes and Routines)‖. OSCAR begun in June1997 to resolve the lack of common standards, documentation, versioncontrol, and accessibility in many transport codes. These transport codesfor relativistic heavy-ion collisions differ from computer codes in otherareas of physics, where numerical methods are only technical tools usedto solve specific equations that define the physics. The source code ofa nuclear-collision transport model often implements extra physical as-sumptions and dynamic mechanisms that go beyond the equations usedto motivate its design. These algorithms often undergo evolution withtime, and the very large number of phenomenological parameters also‖ See: http://www-cunuke.phys.columbia.edu/people/molnard/mirror-OSCAR/oscar.6 Understanding collision dynamics 101makes it difficult to pinpoint the relevant physical input controlling theobserved computational result. Since the code itself defines the physicalcontent of the model, it is essential to be able to closely scrutinize theactual algorithms used.The list of codes currently either maintained or/and accessible, withthe meaning of the acronyms, and principal authors is as follows∗∗.Correlation buildersCRAB – Correlation After Burner, by S. Prat.HydrodynamicsBJ-HYDRO – Relativistic Hydrodynamics with Bjørken Geometry, by A.Dumitru and D. H. Rischke.Partonic/string transportHIJING – Heavy Ion Jet INteraction Generator, by M. Gyulassy and X.-N. Wang [266].HIJING/B-anti-B – HIJING/Baryon Junction, by S. Vance and M. Gyu-lassy.MPC – Molnar’s Parton Cascade, by D. Molnár.neXus – by H.-J. Drescher and K. Werner.PCPC – Poincaré Covariant PartonCascade, by V.Boerchers, S.Gieseke,G. Martens, J. Meyer, R. Kammering and C. C. Noack.VENUS – by K. Werner.VNI – by K. Geiger, R. Longacre and D. Srivastava [130, 131].VNIb – by S. A. Bass.ZPC – Zipping Parton Cascade, by B. Zhang.String/hadronic transportART – Another Relativistic Transport, by B.-A. Li and C.-M. Ko.BEM – Boltzmann Equation Model, by P. Danielewicz.BNC – Burn and Crash, by S. Pratt.HSD – Hadron String Dynamics, by W. Cassing.JAM –Jet AA Microscopic Transport Model, by Y. Nara.JPCIAE – Jetset Pythia CIAE (China Institute of Atomic Energy), byB.-H. Sa and A. Tai.LEXUS – Linear Extrapolation of ultra-relativistic Nucleon–NucleonScattering, by S. Jeon.LUCIAE – Lund CIAE,, by A. Tai and B.-H. Sa.RQMD – Relativistic Quantum Molecular Dynamics, by H. Sorge [250].UrQMD – Ultra-relativistic Quantum Molecular Dynamics, by S. A. Bass[274].∗∗ For more details see http://www-cunuke.phys.columbia.edu/people/molnard/mirror-OSCAR/oscar/models/list.html.102 Experiments and analysis toolsTransport toolsGCP – General Cascade Program, by Y. Pang.PYTHIA, JETSET, and LUND, mentioned above, are programs forthe generation of high-energy-physics events, i.e., for the description ofcollisions at high energies between elementary particles such as e+, e−, p,and p̄ in various combinations. Together, they contain theory and modelsfor a number of aspects of physics, including hard and soft interactions,parton distributions, initial- and final-state parton showers, multiple in-teractions, fragmentation, and decay.Development of JETSET, the first member of the ‘Lund Monte-Carlo’family, was begun in 1978. The most extensive of these programs isPYTHIA. Over the years, these two programs have more and more cometo be maintained in common. In the most recent version, they have there-fore been merged into one, under the PYTHIA label. The current versionis PYTHIA 6.1, by T. Sjöstrand.††The common feature within transport-cascade models is that they pic-ture a multiscattering process as a succession of binary collisions anddecays, each well separated in space–time. For such an approach to havea chance of success, we must be in a physical situation dominated by well-separated collisions, the so-called collision regime. It is rather easy to seewhere this collision regime will occur in nuclear collisions: the de Brogliewavelength of one of the incident particles, and its (classical) mean freepath in the medium, have to be compared with each other in order toidentify the collision partners.For example, at low energy, the de Broglie wavelength can be as large asthe radius of the nucleus, so the dynamics will be dominated by the scat-tering of all nucleons, not by two-body collisions. As the energy increases,the resolving power increases and one also crosses particle-productionthresholds and enters the multiple-scattering process involving elastic andinelastic nucleon–nucleon collisions, as well as collisions between the had-rons produced. We call this energy region the ‘hadronic-cascade’ region;as extensive studies of the data show, at AGS energies, (10–15)A GeV,this is the dominant reaction mechanism. At higher energies, the deBroglie wavelength of the projectile becomes smaller than even a fractionof the size of a nucleon. The interaction will therefore involve the partonsubstructures – we reach the ‘partonic-cascade’ region.In microscopic transport models describing the collision event (eventgenerators), two primary mechanisms are used in order to describe evo-lution dynamics including production of particles: the nonperturbativeproduction involving strong fields with field string-breaking, see Eq. (3.5),†† For more information, see http://www.thep.lu.se/torbjorn/Pythia.html.6 Understanding collision dynamics 103and the production due to reactions caused by collisions of individual par-ticles:• Programs using primarily measured and extrapolated hadronic crosssections, e.g., RQMD [250], ARC [202, 203], QGSM [83], and UrQMD[274].• Programs using perturbative QCD reactions, e.g., VNI [130, 131] andHIJING [266]; the main differences between these two models are thefollowing: VNI is a Monte-Carlo implementation of a parton-cascademodel (PCM) in which the time evolution of heavy-ion collisions is sim-ulated by the parton cascading, whereas HIJING assumes the Glaubertheory in the description of A–A collisions and handles the soft processon the basis of the string model.• There are also hybrid models such as the PHC [196], a parton–hadron-cascade model, which is an extension of the hadronic-cascade model in-corporating hard partonic scattering based on HIJING. However, prac-tically all generators mentioned have, in some ways, taken the hybridapproach.The hadronic-event generators are more suitable for lower AGS ener-gies, and can be extended to SPS energies by introducing novel reactionmechanisms. The partonic generators are more geared to RHIC and LHCenergies, but again, with some fine tuning, can be applied to the SPSenergies. The SPS energy range is so difficult to cover, since p-QCDseems not to be well defined at such ‘soft’ energies, but the hadron cas-cade alone clearly cannot describe this energy range properly. The hybridmodel (PHC [196]) is therefore more able to handle that energy domain.There are major uncertainties in the hadronic-cascade models relatedto the impossibility of measuring reaction all relevant cross sections, sec-tion 18.2, and the necessity to introduce particle-production mechanismswell beyond the scope of the model (color ropes, for example, in RQMD).The perturbative QCD reactions in the deconfined phase are, on otherhand, well determined in terms of elementary processes. The major uncer-tainty arises from the soft-QCD properties: for small transfers of energy,the QCD processes become very strong, and the issue of what physicalmechanism is indeed responsible for the soft cutoff arises. This is rem-iniscent of the fact that we do not understand, in terms of QCD, the(inelastic) low-energy processes: e.g., the inelastic N–N cross section. Itis for this reason that considerable attention was given to the color-stringmechanism of particle production, which can be tuned to describe verywell the nucleon–nucleon inelastic interactions within the LUND familyadaptation (see below) to nuclear collisions, the FRITIOF model [40, 205].Unfortunately, both the scope and the extent of this introductory bookdo not allow us to pursue in detail how these approaches differ. The104 Experiments and analysis toolsabove remarks can, however, serve as an entry point to further reading,for an up-to-date report, see [271]. The reader should be aware that abook could be written just on the subject touched on the surface in thissection.In closing this discussion, we wish to note that it is of course of interestto check how far the microscopic dynamic models are leading to near-equilibrium thermal and chemical conditions. Several studies exploringparticle production and momentum distributions have revealed a verygood approach to chemical and thermal equilibrium [80, 249]. This resultreally confirms that the large nuclear-reaction system, at the energiesconsidered, disposes of sufficiently many degrees of freedom, and thatstatistical near-equilibrium methods are able to characterize the final statereached in the reaction. These results do not imply that the conditionscreated in the event generator are those observed experimentally.6.2 Relativistic hydrodynamicsIn the hydrodynamic description of the evolution of matter, rather thanindividual particles, one considers the flow of particles in a volume ele-ment. Therefore, we consider as one of the dynamic equations the conser-vation of (e.g., baryon) number-density current, along with the conserva-tion of energy–momentum flow. These flows are described in terms of thelocal flow field /v(/x, t), or equivalently in terms of the 4-velocity vector ofthe flow uµ:dxµdτ≡ uµ(x) = γ(1, /v), dtdτ≡ γ = 1√1− /v 2. (6.1)We see that, in general,uµuµ ≡ u2 = 1, uµ = gµνuν . (6.2)We use Einstein’s summation convention for repeated Greek indices (im-plied summation over time ‘0’ and space ‘1, 2, 3’), and work in flat space–time gµν = gµν , with the metric convention gµν ≡ diag(1,−1,−1,−1).There is a simple relation between the 4-divergence of the 4-velocity andthe 4-divergence of the density arising from the conservation of current.We write a conserved current jµ in terms of the local density ρ:∂µ(ρuµ) = ρ∂µuµ + uµ∂µρ = 0, ∂µ =∂∂xµ. (6.3)The proper time τ coordinate of the local volume element and laboratoryframe coordinates are related by the Euler relation:ddτ= uµ∂µ = γ(∂∂t+ /v · /∇), ∂µ ={∂∂t, /∇}. (6.4)6 Understanding collision dynamics 105We divide Eq. (6.3) by ρ and obtain, using Eq. (6.4),∂µuµ = − 1ρidρidτ≡ 1τ exp. (6.5)We suggest, at the end of condition Eq. (6.5), that it is a suitable def-inition of the expansion life span of the system, Eq. (5.34). In fact, asEq. (6.32) below is showing, this is exactly true (taking freeze-out propertime) for the case of longitudinal flow in one spatial dimension.Aside from the baryon number also, the flow of energy is considered inthe hydrodynamic description of the time evolution. The hydrodynamicenergy–momentum-flow equation, isfν ≡ ∂Tµν∂xµ, Tµν = (ε+ P )uµuν − gµνP. (6.6)The form of Tµν we present is suitable for adiabatic (entropy-conserving)flow of matter when the external force density vanishes, fν → 0, a pointto which we will return momentarily.The condition for energy–momentum conservation, Eq. (6.6), involvesfour equations. One of the equations can be made to look like the conser-vation equation Eq. (6.5): multiplication by uν of Eq. (6.6) yields, usingEq. (6.2),uµ∂µε+ (ε+ P )∂µuµ = 0. (6.7)If the pressure P = 0, this is the continuity equation Eq. (6.5) for theenergy density. To make this obvious, we writeε+ Pε∂µuµ = −1εdεdτ. (6.8)For P = 0, the energy flow (uµε) is not conserved. For P > 0, there is atransfer of the energy content of matter to the kinetic energy of the flowof matter. The expanding matter cools. In the rare situation that P < 0(see section 3.5), the transfer of energy goes from kinetic energy of flowback to the intrinsic energy density ε.Equation (6.7) is equivalent to Eq. (1.17), which we recognize usingEq. (6.5),1ε+ Pdεdτ=1ρidρidτ,dεε+ P= d(ln ρi) = −3ṘR, (6.9)noticing that the local density scales with ρ ∝ 1/V ∝ 1/R3.106 Experiments and analysis toolsThe other three equations which follow from Eq. (6.6) determine thevelocity field /v(/x, t), Eq. (6.1),∂/v∂t+ (/v · /∇)/v = −1− v2ε+ P(/∇P + /v ∂P∂t), (6.10)which form is obtained for the three spatial components i = 1, 2 and 3in Eq. (6.6). Naturally, a solution of the hydrodynamic equations can beobtained only when the equation of state P (ε) is known, or equivalently,ε(T ) and P (T ) are given.As we have noted, the hydrodynamic-flow equation Eq. (6.6) conservesentropy. To show this, we consider again a contraction with uν of thehydrodynamic equations, but this time, we proceed in a different fashion.In the following the sum over the repeated index i is implied, which de-notes more than one conserved particle number; in the simplest case, itis the baryon number,uν(∂µTµν)=∂µuµ(P + ε)− uµ∂µP=∂µuµ(Tσ + µiρi)− uµ∂µP, (6.11)where we have used the Gibbs–Duham relation, see Eq. (10.26). Aftersome reordering of Eq. (6.11), we obtain0 = T∂µ(σuµ) + µi∂µ(ρiuµ) + σuµ∂µT + ρiuµ∂µµi − uµ∂µP. (6.12)The first term is the conservation of entropy flow which we are lookingfor,∂σµ∂xµ= 0, σµ = σuµ, (6.13)and thus other remaining terms in Eq. (6.11) should cancel out. Thesecond term is the conservation of current flow, Eq. (6.3); it vanishesnaturally.The last three terms in Eq. (6.11) all contain the total proper time localderivative, Eq. (6.4). After multiplication with dτ , we see that, for themto cancel out, we must have0 = σ dT − dP + ρi dµi → 0 = S dT − V dP + bi dµi. (6.14)On multiplying by the volume V , we find the relation on the right-handside. A more convenient way to consider Eq. (6.14) is0 = d(TS − PV + biµi)− (T dS − P dV + µi dbi), (6.15)where, according to the Gibbs–Duham relation Eq. (10.26), the left paren-thesis is just dE; hence, we recognize Eq. (6.15) as the first law of ther-modynamics, Eq. (10.12), which proves Eq. (6.13).6 Understanding collision dynamics 107For a more complete discussion of the relativistic hydrodynamics, werefer the reader to the monograph by Csernai [98], as well as introductorysections in Gravitation and Cosmology by Weinberg [267], who consid-ers generalization of Tµν with dissipative terms. A generalization of theadiabatic hydrodynamic expansion to include production of entropy hasrecently been proposed [114].6.3 The evolution of matter and temperatureIn our following discussion, we consider a reduction of Eq. (6.8). Intro-ducing the velocity of sound,1v2s≡ dεdP∣∣∣∣S=constant, (6.16)we obtaindεdτ=dεdPdPdTdTdτ=1v2sσdTdτ, (6.17)where we have used, in the limit of an extensive system at fixed chemicalpotential (here zero), for the entropy density, σ,σ ≡ SV=dPdT, (6.18)which follows from Eq. (10.16). On substituting Eq. (6.17) into Eq. (6.8),we find∂µuµ = − Tσε+ P1v2s1TdTdτ. (6.19)The Gibbs–Duham relation, Eq. (10.26), allows us to write Eq. (6.19)in the form∂µuµ = − 11 + (µb/T )b/S1v2s1TdTdτ. (6.20)We have introduced b/S = νb/σ, the inverse of the entropy per baryon,which is a constant of motion in ideal fluid dynamics: ideal flow conservesthe entropy content and, of course, the baryon number is also conserved.Moreover, in an ideal gas of quarks and gluons, without a significant in-trinsic dimensional scale, the ratio µb/T of the two-dimensional statisticalvariables is also not evolving with proper time. In this case, the velocityof sound, v2s =13 , is also exactly constant.Using the conservation of baryon flow (ρi → νb in Eq. (6.5)) on theleft-hand side of Eq. (6.20), we obtain:1νbdνbdτ=11 + (µb/T )b/S1v2s1TdTdτ. (6.21)108 Experiments and analysis toolsEquation (6.21) allows an exact integer for 1/v2s = constant. Consideringthe relativistic quark matter 1/v2s = 3,νbν0b=(TT0)3/(1+µbT bS ). (6.22)Some readers may wonder how it is possible that T , rather than µb,controls the evolution of the baryon density. This, of course, is just anoptical illusion. Namely,TT0=Tµbµbµ0bµ0bT0=µbµ0b,where the last equality arises since T/µb does not change during theisentropic evolution of the ideal quark–gluon gas. In this case a morepalatable way of writing Eq. (6.22) isνbν0b=(µbµ0b)3/(1+µbT bS ). (6.23)It is interesting to observe that, for a baryon-dense fireball of quarkmatter, possibly formed in 10–40A-GeV fixed-target heavy-ion collisions,the deviations from the νb ∝ µ3b law are quite significant. However, atthe SPS and RHIC, the initial conditions established assure that thisrelationship is valid: S/b > 35 (the SPS value; it is certainly larger at theRHIC) is seen to be relatively large compared with µb/T (1.4 at theSPS and <0.1 at the RHIC).Although we were able to extract the behavior of the baryon densityfrom ideal-flow equations in the case of a relativistic gas of particles, theactual objective, namely the determination of the proper time variationof any of the quantities involved, has not been accomplished. We willobtain T (τ) in a very special, but interesting case, in the next section.6.4 Longitudinal flow of matterA special case of interest is the reaction picture invoking a rapid flow ofmatter along the collision axis, the so-called Bjørken scenario [73]. Forthis simple picture of the reaction to apply, we need to assume that1. the colliding particles had so much energy that the flow of energy andmatter after the collision remains unidirectional along the original col-lision axis; and2. the transverse extent of the system is so large that the existence of theedge of matter in a direction transverse to the collision axis is of littlerelevance.6 Understanding collision dynamics 109An interesting aspect of this ‘punch’-through limit, seen in Fig. 5.2(a), isthat the baryon number which is attached to the colliding valence quarkswill also be leaving the interaction region, continuing to travel along thecollision axis. Even though a trail of energy is deposited in the centralrapidity region, the hope is that, at the highest energies, we should beable to recreate the baryon-free conditions of the early Universe.After the time τ0 = 0.25–1 fm/c has passed since the initial contactbetween the Lorentz-contracted nuclear pancakes (the laboratory-frameview), the thermalized matter begins its evolution as indicated in figureFig. 6.1. Each particle involved in the reaction has at its later freeze-outa ‘proper’ age τf since ‘birth’:τf =∫ f0dτ, dτ2 = dt2 − d/x 2. (6.24)If all particles move with a constant velocity (e.g., c) along a commonlongitudinal direction z, and assuming that all particles have the sameproper time at freeze-out, in laboratory coordinates the freeze-out time,tf , and the freeze-out space coordinates, zf , form a hyperbola,tf =τf√1− v2f, zf = vfτf√1− v2f, τ2f = t2f − z2f , (6.25)as shown in the body of Fig. 6.1. The trajectory of each particle is astraight line z = vt, leading from the interaction point to the freeze-outlocation on the hyperbolic, τf = constant, surface.The Minkowski space–time evolution of the ultra-relativistic collision isthen rather simple: soon after the collision has occurred (see the CM-timesnapshots to the left, beginning at the bottom in Fig. 6.1), the baryonnumber of the nuclei begins to separate (black lines along the light conein Fig. 6.1), leaving in the intermediate region a trail of energy, presumedto be in the baryon-number-free QGP phase. The nuclei are trailed toright and left by the expansion of the energy they deposited at the instantof collision.As the distance between the projectile and the target increases, thecontinued longitudinal expansion of every volume element reduces thelocal energy density/temperature until it is so low that individual had-rons can emerge (the chemical-freeze-out condition). As we shall see, thetemperature will depend only on the proper time, Eq. (6.33), not on therapidity. Therefore, a phase transition or transformation, as the case maybe, and particle freeze-out occurs along given τ = constant space–timehyperbolas. In the graphic representation in Fig. 6.1, it is assumed thatmost of the time the QGP phase prevails, with a short period of freeze-out and hadronization, before final-state hadrons free-stream out of theinteraction region.110 Experiments and analysis toolsSpaceTimeQGPFreeze-outFig. 6.1. A space–time-image illustration of a heavy-ion collision in the ultra-relativistic (Bjørken) collision limit. Left: Lorentz-contracted nuclei collide andseparate as a function of the laboratory time (vertical axis). Right: the light coneestablishes the causality limit for the flow of energy, which fills the space–timedomain between the separating nuclei.The spatially central region is obviously at rest in the symmetric CMframe and will emit particles around yc = 0. As we go away from thespatial center, the velocity of the local energy flow under the freeze-outcondition increases, reaching the speed of the baryonic matter at thematter-trailing edge (upper right/left-hand edges of the light cone). Wesee that the rapidity of particles emitted and the relative position in spaceare correlated. Also, in laboratory time, the earliest particles to be emit-ted emerge at central rapidity, the latest at projectile–target rapidities,as can be seen in Fig. 6.1.This discussion suggests that the natural variables in the study of thedynamics of longitudinally expanding matter are the proper time τ(t, z)which characterizes the parabolas in Fig. 6.1, and the space–time rapid-ity‡‡ y(t, z),τ = (t2 − z2)1/2, y = 12ln(t+ zt− z), (6.26)with the inverse relation‡‡ The reader is reminded that the variable y is not the spatial coordinate, but thespace–time rapidity variable, and that only the spatial coordinate z enters into thatwhich follows.6 Understanding collision dynamics 111t = τ cosh y, z = τ sinh y. (6.27)We also record that∂y∂t= − zt2 − z2 ,∂y∂z=tt2 − z2 ,∂τ∂t=t(t2 − z2)1/2,∂τ∂z=−z(t2 − z2)1/2, (6.28)and∂t∂y= τ sinh y,∂t∂τ= cosh y,∂z∂y= τ cosh y,∂z∂τ= sinh y. (6.29)These transformations imply for the volume element thatdt dz = τ dτ dy. (6.30)The 4-velocity field of some volume element at proper time τ isuµ ≡ dxµdτ= (cosh y, 0, 0, sinh y), u2 = 1. (6.31)Equation (6.31) implies that∂µuµ =∂u0∂t+∂u3∂z=∂y∂t∂cosh y∂y+∂y∂z∂sinh y∂y=1τ. (6.32)All these relations become considerably more complex when one allows forflow in the transverse direction. For a velocity field including transversecylindrical flow see Eq. (8.20).We will now describe the Bjørken ‘scaling’ solution for the (1 + 1)-dimensional hydrodynamics [73]. The discussion above Eq. (6.26) sug-gests that one ought to use space–time rapidity and proper time as vari-ables when one is considering a one-dimensional relativistic hydrodynamicmodel. We substitute Eq. (6.32) into Eq. (6.20) and obtain (µb 0)v2sτ= − 1TdTdτ. (6.33)This important result allows us to understand how fast the temperatureis changing during the ‘scaling’ one-dimensional hydrodynamic evolutiondescribed. We encountered a related result in the study of the adiabatic(isentropic) expansion, see section 1.4.Perhaps the most cited equation of (1+1)-dimensional hydrodynamicsarises when, in Eq. (6.8), we use Eq. (6.32) (see Eq. (21) in [73]):112 Experiments and analysis toolsε+ Pτ+dεdτ= 0. (6.34)For ε(T ) and P (T ), this implies that T is a function of τ , but not ofy. This important result originates from the assumption that the propertime τ of a fluid volume element is as given in Eq. (6.26), and it is inparticular independent of the transverse coordinates.For a (nearly) relativistic gas v2s 13 , and the decrease of the temper-ature is slow. Explicitly, integrating Eq. (6.20), we obtain, assuming thatthe velocity of sound changes slowly,T = T0(τ0τ)v2s, (6.35)where the initial temperature T0 is established at an initial (proper) timeτ0, at which local thermal equilibrium has been established and the isen-tropic hydrodynamic expansion begins. In order to decrease the temper-ature by a factor two, we need the time τ 8τ0.In a more realistic evolution of a fireball, which allows for transverseexpansion, the expansion cooling is faster [58, 163]; see section 6.2. Onthe other hand, one also must allow for a less than fully relativistic soundvelocity. The properties of the equation of state obtained on the latticesuggest that, in the vicinity of the phase transition, i.e., for T < 2Tc,there are significant deviations from ideal-gas behavior. A seemingly smallchange in vs matters: we note that, when vs 0.5 (recall that 1/√3 0.58), for the scaling solution Eq. (6.35), the time needed to decrease thetemperature by a factor of two increases two-fold to τ 16τ0.7 Entropy and its relevance in heavy-ion collisions7.1 Entropy and the approach to chemical equilibriumEntropy is a quantity characterizing the arrow of time in the evolutionof a physical system – in every irreversible process the entropy increases.In elementary interactions, and in particular those involving relativisticcollisions of two large atomic nuclei, there is considerable production ofparticles and hence of entropy. A number of questions arise naturally inthis context:1.When and how is entropy produced in a quantum process, such as anuclear collision?2. How is production of hadronic particles related to production of en-tropy?3. How does one measure the entropy produced in the reaction?7 Entropy and its relevance in heavy-ion collisions 113In the deconfined phase the color degree of freedom is ‘melted’. There-fore, the specific entropy content per baryon (S/b), evaluated at somegiven (measured) values of statistical parameters, is generally greater inthe deconfined state than it is in to the confined state. Entropy can onlyincrease, and thus, once an entropy-rich state has formed, we have anopportunity, by measurement of the entropy created in the heavy-ion col-lision, to determine whether the color bonds of valence quarks present inthe collision have been broken.The final entropy content of the hadronic particles emerging has toexceed the initial entropy of the thermal state. In fact, quantitative stud-ies show that very little additional entropy is produced during the entireevolution of a fireball, after the initial thermalization stage. For this rea-son, the final hadronic state conveys key information about the initialthermal state of dense and hot hadronic matter. For example, in theexpanding quark–glue fireball the quasi-entropy-conserving evolution hasbeen confirmed within a model study involving parton cascade [129]. Thefinal-state entropy is largely produced in the first instant of heavy-ioncollision.The entropy can be obtained using the momentum-distribution functionfB,F:SB,F =∫d3x∫d3p(2π)3[±(1± fB,F) ln(1± fB,F)− fB,F ln fB,F] . (7.1)The upper sign + is for bosons (B) and the lower sign − is for fermions(F), which is somewhat counterintuitive, but in fact in agreement withFermi and Bose statistics. We are reminded of this change by the changein the usual sequence of letters ‘F, B’ in the subscript.There are two well-known ways to obtain Eq. (7.1). It follows (up tonormalization) from Boltzmann’s H-function in the study of momentumequilibration. It also arises naturally on rewriting Eq. (10.25) in terms ofthe single-particle distribution function Eq. (4.42). Since in this approachthe statistical definition of entropy, which corresponds to the thermaldefinition, is used, the normalization is fixed by the laws of thermody-namics.Entropies of different particles add, and the entropy of particles andantiparticles adds as well. The entropy of fermions, in Eq. (7.1), vanishesin the pure quantum-state limit for T → 0, since the value of the particle-occupancy probability is either unity or zero. The ‘classical’ Boltzmannlimit arises when fB,F 1. In this case, with fB,F → f ,Scl ≡∫dωf ln(ef)=∫dω(f − f ln f),∫dω ≡∫d3x∫d3p(2π)3. (7.2)114 Experiments and analysis toolsGenerally, expression Eq. (7.1) is presented as the generalization of thewell-known classical micro-canonical definition Eq. (7.2) to quantum gases;however, this language leads to the (frequent) omission of the first term onthe right-hand side, which is the number of particles present, and whichcomprises 25% of the total entropy of the relativistic gas.We will study the entropy of a statistical gas in more detail in sec-tion 10.6. We note that, for massless particles (quarks, gluons), the en-tropy per particle following from Eq. (7.1) and Eq. (7.2) isSN∣∣∣∣Bm=0= 3.61,SN∣∣∣∣clm=0= 4,SN∣∣∣∣Fm=0= 4.20. (7.3)The effect of the finite pion mass is to increase the entropy per particle,see Eq. (10.79) and Fig. 10.4 on page 206. Each pion (a boson) emergingcarries away just about 4 units of entropy from the source as long as thetemperature is T mπ. These results are for a chemically equilibratedsystem. In general, below equilibrium at a given fixed temperature, theentropy density is lower, but the entropy per particle is higher than that inEq. (7.3), and the opposite is true for a system above chemical equilibrium,γ > 1; section 7.5.For an isolated system, like our hadronic fireball, a very importantphysical property is that relatively little entropy is generated in the ap-proach to chemical equilibrium, both from above and from below. Thishappens since the change in number of particles consumes or releases ther-mal energy and this changes the temperature. Thus, even after chemicalequilibration, the final-state entropy content is closely related to the initialentropy of the thermal state generated in the collision.To demonstrate this, we obtain the shape of the particle-occupancyprobability f of particles in the isolated fireball from the Boltzmann H-theorem result, i.e., the principle that a physical system evolves towardmaximum entropy for a given energy and number of particles in the sys-tem. We seek to maximize the entropy Eq. (7.1) subject to these con-straints, i.e.,T (f) =∫dω [±(1± f) ln(1± f)− f ln f − (αf + βεf )f ] , (7.4)as a functional of the distribution shape {f}. Here, dω is the phase-spaceintegral seen in Eqs. (7.1) and (7.2).The results are the standard Fermi and Bose distributions Eq. (4.42),including the chemical nonequilibrium factor γ = e−α which allows thatthe particle number is fixed independently from the temperature,fF,B =1eβε+α ± 1 . (7.5)7 Entropy and its relevance in heavy-ion collisions 115This demonstrates that an isolated system in the presence of suitableinternal dynamics, e.g., two-body elastic collisions, will evolve towardthe kinetic-equilibrium statistical Bose/Fermi distributions, which maybe in chemical nonequilibrium expressed by γ = 1, depending on theinitial energy and number of particles. When inelastic particle-productionprocesses are occurring, we further expect that also chemical equilibriumshould be reached, but on a slower time scale; see section 5.5.Now, we are ready to show that the entropy content of a chemicallynot fully equilibrated system is nearly the same as the entropy contentof a system in equilibrium. For the Boltzmann approximation, we canshow this analytically. The factor γ becomes a normalization factor thatdescribes the average occupancy of the phase space relative to the equi-librium value; the additive term describes how the entropy per particlechanges as the occupancy changes. We find in the Boltzmann limitN = γN|eq, (7.6)E = γE|eq, (7.7)S = γS|eq + ln(γ−1)γN|eq. (7.8)For massless particles, the phase-space integrals are easily performed andone obtains, see chapter 4,N 0 = aV γT 3, (7.9)E0 = 3aV γT 4, (7.10)S0 = 4aV γT 3 + ln(γ−1)aV γT 3, (7.11)where a = g/π2 and g is the degeneracy. One easily finds how, for E0 =constant, the entropy varies as a function of γ,S ∝ γ1/4(4− ln γ). (7.12)This functional has a very weak maximum at γ = 1. For example, atγ = 2 the entropy is 98.3% of the value at γ = 1.One could imagine that an important change in number of particles isrequired when γ increases by say a factor of ten from 0.1 to 1. However,since the total energy and volume of the system (and hence the energydensity) do not vary, we obtain a result that contradicts our intuition.Namely, at a high initial temperature the phase space is much greaterand, in the Boltzmann approximation, the number of particles scales withγT 3, Eq. (7.6). Since E/V ∝ γT 4, we obtainN|E/V ∝ γ1/4. (7.13)Thus, a ten-fold increase in γ is accompanied by a 1.8-fold increase innumber of particles.116 Experiments and analysis toolsAt this point, it is perhaps wise to briefly review the more familiarcase of a fixed-temperature environment (a heat bath) in order to un-derstand better the difference from the isolated-fireball system. Considerthe ‘black-body’ radiator: a thermally insulated box with a small emis-sion hole, for which the loss of energy due to radiation through the hole isexternally compensated by keeping the temperature constant: the spec-trum of the emitted radiation displays the Planck shape which minimizesthe free-energy content F of the photon gas, at a fixed temperature ofthe walls – this spectrum is arising from interaction of the photons withthe walls, with the spectrum and number of photons changing due to ab-sorption and re-emission by the walls. Recalling now that F = E − TS,we can combine Eqs. (7.6) and (7.8), which gives, in the Boltzmann limit,F = −aV T 4γ[1 + ln(γ−1)], (7.14)with a minimum at γ = 1. However, now a change by a factor of two inγ, at fixed T , leads to a change by 35% in the value of the free energy andan even greater change in entropy. Clearly, at fixed T , the equilibriumpoint γ → 1 is much better defined than it is at fixed E .The lack of sensitivity of entropy to chemical equilibration for an iso-lated fireball assures that there is ample room to generate nonequilibriumparticle yields during the dynamic evolution of the system. Given that thesystem we are considering is actually subject to a dynamic evolution, withexpanding volume V , it is natural to expect that chemical equilibrium isan exception rather than a rule.7.2 Entropy in a glue-ballWe are now ready to examine in detail the simplest system of dynamicinterest to us. We consider an initially thermal glue–parton ball far fromparticle-abundance equilibrium. There are glue interactions that are pro-ducing particles, driving the system to chemical equilibrium while thetemperature decreases, due to sharing of a fixed available amount of ther-mal energy by an ever larger number of constituents. We assume, in theexample below, that, when chemical equilibrium is reached, the glue-gasstate is at T = 250 MeV.The intuitive expectation is that a lot of entropy is produced whilethis system evolves toward the particle-abundance equilibrium. However,this is not so [179]. The reason is that, as the equilibrium in particle-number abundance is approached, we must adjust the temperature of thesystem. There is a subtle balance between the different effects, and theresult is that we find considerable constancy of the entropy of the isolatedparticle-producing system.7 Entropy and its relevance in heavy-ion collisions 117Fig. 7.1. The entropy density S/V (units fm−3), at a fixed energy densityE/V = 2.66 GeV fm−3, for mg = 0 (solid line), and at E/V = 1.89 GeV fm−3for mg = 0.450 GeV (dashed line), for a (gluon) Bose gas, as a function of thechemical occupancy γ, with T (γ = 1) = 250 MeV.We consider both the massless-gluon case and the case of thermallymassive gluons, choosing for the thermal gluon mass mthg = 0.450 GeV,see Fig. 16.3 on page 308. A massless-gluon (Bose) gas, with g = 16,has an energy density E/V = 2.66 GeV fm−3, at T = 250 MeV, at thechemical-equilibrium point γ = 1. The massive gas, at the chemical-equilibrium point γ = 1 at T = 250 MeV, has E/V = 1.89 GeV fm−3.In Fig. 7.1, we see the entropy density S/V (units fm−3) as a func-tion of γ; the solid line is for massless gluons and the dashed line is formassive gluons. The maximum in entropy at γ = 1 is shallower thanwould be the case for a Boltzmann gas. The curves end at the singu-larity of the Bose distribution function, γ = 1 for massless gluons, andγc = emg/T ∼ 2.7 (beyond the range shown in Fig. 7.1), which valuescannot be exceeded.The vertical line, to the left in Fig. 7.1, shows that the entropy contentof the hot-glue system at γ = 0.1 is already nearly 90% of the chemical-equilibrium entropy. The ‘hot’ glue is at this point at T 453 MeV formg = 0, and at T = 426 MeV for mg = 0.450 GeV.118 Experiments and analysis toolsFig. 7.2. The temperature T as a function of the chemical occupancy γ formassless gluons with E/V = 2.66 GeV fm−3 (solid line) and mthg = 0.450 GeV,E/V = 1.89 GeV fm−3 (dashed line). The equilibrium point γ = 1 has beenchosen to occur at T = 250 MeV.In order to maintain a fixed value of E/V , the temperature T andphase-space occupancy γ are not independent, and, as a function of γ,the temperature T drops rapidly, which is shown in Fig. 7.2. The dashedline, corresponding to the case of massive gluons, coincides with the solidline (massless gluons) at γ = 1, T = 250 MeV, by token of the judiciouschoice of E/V . As Fig. 7.2 shows, the temperature can drop rapidly inthe process of chemical equilibration of the gluon gas.It is interesting to note that, when the chemical cooling, seen for smallγ in Fig. 7.2, is fastest at small γ, the collective flow of the QGP fireballshould not yet be established. Therefore, it is probable that the initial-state cooling is due to chemical processes. The mechanism for a chemicalequilibration of the hot initial glue phase which is faster than the volumeexpansion has been proposed to be inherent in the multi-glue-productionreactions [247], gg→ ggg, gggg, . . ..The number of gluons changes relatively slowly, in particular consider-ing massive (thermal) gluons, as can be seen in Fig. 7.3. In the case of7 Entropy and its relevance in heavy-ion collisions 119Fig. 7.3. The particle density N/V (units fm−3) as a function of the chemicaloccupancy γ. Lines are as in Fig. 7.1.massless gluons (solid line), when γ increases by a factor of ten from 0.1to 1, the number of gluons increases five times slower. This increase isconsiderably more modest for mthg = 0.450 GeV (dashed line in Fig. 7.3).To understand the greater change seen in Fig. 7.3 compared to Fig. 7.1 itis important to know that the entropy per particle is noticeably greaterthan four for a system far from chemical equilibrium, Fig. 7.8.The process of chemical equilibration of glue involves, apart from anincrease in the number of gluons, a change in the momentum distribution.In Fig. 7.4, we compare the spectra of gluons initially at γ = 0.1 withequilibrium γ = 1. At equilibrium, the temperature is T = 250 MeVand we see that the massless- (solid line) and massive-gluon (dashed line)spectra coincide (lines ending at E 3.5 GeV). The ‘missing’ gluons, atlow energies, contribute to the difference in energy density (2.66 versus1.89 GeV fm−3 for massless and massive, mg = 450 MeV, gluons, respec-tively). The relatively slowly falling spectra are for the early hot-gluenonequilibrium stage at γ = 0.1, at which for massless gluons T = 453MeV and for massive gluons T = 426 MeV (values determined for fixedvolume and energy of the fireball).120 Experiments and analysis toolsFig. 7.4. Spectra of massless (solid lines) and mg = 450 MeV (dashed lines),gluons prior to (γ = 0.1, lines reaching to right lower corner) and at (γ = 1, T =250 MeV) chemical equilibrium.We have shown that the dynamics of chemical equilibration is oftencounterintuitive. In particular, when one is considering the dynamics ofan isolated fireball, we learned that the entropy varies little if chemi-cal equilibrium is not maintained. We infer that in a rapid evolution,an isolated system can evolve away from chemical equilibrium, even ifthis means that the chemical entropy is not at its maximum. On theother hand, since the entropy content is not a sensitive probe of chemical-equilibrium properties we can, in the following study of the experimentalentropy production, proceed as if chemical equilibrium were maintained,without loss of generality.7.3 Measurement of entropy in heavy-ion collisionsThe final-state entropy content is visible in the multiplicity of particlesproduced. In the HG and QGP phases of dense hadronic matter, theentropy content is in general different. The entropy content per partici-pant (specific entropy) offers a method to distinguish these two different7 Entropy and its relevance in heavy-ion collisions 121hadronic phases. At a temperature T > Tc 160MeV, the QGP isthe phase with the higher specific entropy; this difference occurs becauseof the liberation of the color degrees of freedom in the color-deconfinedQGP phase. The question of whether it is possible to measure the en-tropy per baryon in the fireball arises. A measure of entropy must countthe total production of particles, while the number of participants canbe measured using the positive-hadron multiplicity, which comprises, inparticular, protons participating in the reaction.It has been argued that, in the SPS energy range, the ratio of net chargemultiplicity to the total charged multiplicity comprises information aboutthe specific entropy content of the matter phase in the fireball [182, 183]:DQ ≡N+ −N−N+ +N−. (7.15)DQ is understood to be a function of rapidity when considering particledistribution in rapidity, rather than the total abundance number. In gen-eral, DQ can be measured, it is an easy task if particles are not identified.The sum of positive and negative hadron multiplicity can be identified bythe sign of the curvature in a magnetic field, and the emission angle; seeFig. 5.5 and Eq. (5.25).A first estimate of this ratio isDnsQ ≈AfNπ0.751 + 0.75 (Af/Nπ), (7.16)where Af is the number of baryons in the fireball. We have consideredonly non-strange particles ‘ns’, i.e., pions and nucleons, and assumed thatthe system is symmetric in isospin, that is half of the participants in thefireball Af are protons, and all yields of pions are equal: Nπ+ = Nπ− =Nπ0 = Nπ/3.As we see in Eq. (7.16), DQ is indeed a measure of the baryon-to-pion ra-tio and thus of entropy per participant. However, this estimate Eq. (7.16)is wrong by two partially compensating factors of order 2: both higher-mass non-strange resonances and charged strange particles must also beconsidered. We recall the significant kaon contribution seen in Eq. (5.33).Theoretically, such calculations require knowledge of the relative abun-dances of particles for higher-mass resonances as well as for strange parti-cles. This can be determined in a statistical model of chemical freeze-outas a function of a few parameters, in particular T . Similarly, the entropyper baryon in the fireball is given as a function of the same statisticalvariables. Both the charge-multiplicity ratio DQ and the specific entropyS/Af are known functions of the statistical parameters.As Eq. (7.16) suggests, the quantityCQ ≡ DQSAf∝ DQNπAf(7.17)122 Experiments and analysis toolsCQ disintegrationHG–resonanceFig. 7.5. The product CQ = DQ(S/b) before (upper curves) and after (lowercurves) resonance disintegration, as a function of λq, for fixed λs = 1 ± 0.05 andconserved zero strangeness in an equilibrated HG.should be a structure constant that depends somewhat on the mixtureof hadronic flavors, mass spectrum and similar general hadron-spectrumproperties, but should be largely independent from the statistical prop-erties of the system. Once the value of CQ has been established withina theoretical model, it should then apply in general – a value, CQ 4.5,was found [182] for a chemically equilibrated system, see the upper line inFig. 7.5. There is in addition the effect of hadron-resonance decays, andthis increases the final-state multiplicity of charged hadrons. Accordingto Eq. (7.15), the value of DQ diminishes. In consequence, the observablevalue of CQ 3 is seen to apply to the lower lines in Fig. 7.5.7.4 The entropy content in 200A-GeV S–Pb interactionsSince DQ is generally a small number, it has not been studied extensively.Experiment EMU05 at CERN–SPS has exposed a lead target located in7 Entropy and its relevance in heavy-ion collisions 123_0.085 0.01+S Pb at 200 GeV − A DηFig. 7.6. Emulsion data for the charged-particle-multiplicity ratio DQ obtainedin central S–Pb collisions at 200A GeV as a function of pseudorapidity η. Thebold black line is drawn to guide the eye.front of a photographic emulsion to a 200A-GeV beam of sulphur atoms.Since a magnetic field was present, the charge of particles produced wasdetermined and thus the experimental value of DQ, as a function of pseu-dorapidity η, could be determined by evaluation of the angle of emission ofcharged hadrons from the interaction vertex, which fixes η, see Eq. (5.25),and the polarity of charged particles.Selecting the most central events with charge multiplicity N+ +N− >300, corresponding to a total central hadron multiplicity in the range450–1000, in the central (pseudo)rapidity region, the valueDQ = 0.088± 0.007, η 2.5± 0.5,is found [104, 117]. The distribution DQ(η) is shown in Fig. 7.6. Thepseudorapidity distribution is flat in the central region, ∆η = (η− 2.6) ±1.5. This suggests partial transparency and the presence of longitudinalflow in the rapidity distributions of protons and K+; see section 8.3.124 Experiments and analysis toolsInspecting Fig. 7.5 and Fig. 7.6 we arrive at a first estimate of theentropy content of a fireball formed in these interactions:Sb=3± 0.10.088± 0.007 = 34± 4.The sources of systematic error involved in the use ofDQ to fix the entropyinclude the difference in the distribution between rapidity and pseudora-pidity, and the uncertainty about yields of strange charged hadrons, whichvary with the degree of chemical equilibration of strangeness.This high value of specific-entropy content in highly central S–Pb 200A-GeV interactions, 40% higher than expected, suggests that an entropy-rich (deconfined) state has been created [182]. Since a high specific-entropy content can be found in the HG phase at smaller values of λq,i.e., smaller baryon density, it is important in comparison to the HG tohave a good understanding of the baryochemical potential. The value ofµb = 3µq, see Eq. (4.18), needs to be reduced by nearly a factor of two,to about µb = 100–120MeV, before the entropy content of HG becomesconsistent with these experimental data. Such a small baryochemical po-tential is not in agreement with many measured yields of hadrons [176]. Asthis simple case shows it is the simultaneous consideration of several ob-servables (charged-particle asymmetry combined with specific (strange)particle ratios) which allows understanding of the physics of heavy-ioncollisions.7.5 Supersaturated pion gasThe excess of hadron multiplicity (entropy) is a consistently observed phe-nomenon: the data obtained for Pb–Pb collisions supports this strongly.We have seen in Fig. 1.6 a significant excess of hadron multiplicity inhigh-energy A–A reactions compared to p–p and low-energy A–A reac-tions. An important question is how this excess of abundance can betheoretically described in terms of the final-state hadron phase space.During hadronization, hadrons need to acquire the excess entropy aris-ing from broken color bonds of QGP. We now look for the most entropy-rich hadron gas and consider the super-saturated massive (pion) Bosegas, in which the chemical potential, i.e., the abundance fugacity γ, isnearly compensating for the suppressing effect of the mass, which occursat γe−m/T → 1.For pions, composed of a light-quark–antiquark pair, it is convenient touse as the abundance fugacityγπ = γ2q. (7.18)7 Entropy and its relevance in heavy-ion collisions 125Fig. 7.7. Pion-gas properties N/V for particles, E/V for energy, and S/V forentropy density, as functions of γq at T = 142MeV.This allows one to express, in terms of the abundance of pions, the relationto the abundance of quarks at hadronization, see chapter 19. We studythe momentum-space distribution for pions of the formfπ(E) =1γ−2q eEπ/T − 1, Eπ =√m2π+ p2. (7.19)The range of values for γq is bounded from above by the Bose singularityγcq:γq < γcq = emπ/(2T ). (7.20)For γq → γcq, we encounter condensation of pions, the lowest-energy statewill acquire macroscopic occupation. Formation of such a condensate‘consumes’ energy without consuming entropy of the primordial high-entropy QGP phase, and thus is not likely to occur.In Fig. 7.7, we show the physical properties of a pion gas as functionsof γq, for a gas temperature T = 142MeV [181]. We see (solid line) thata large range of entropy density can be accommodated by varying theparameter γq. A super-saturated pion gas has an entropy density rivalingthat of the QGP at the point of transformation into hadrons, as we seeon comparing it with Fig. 16.7 on page 315, for T = 140–160MeV.126 Experiments and analysis toolsFig. 7.8. Specific pion-gas properties E/N for energy, S/N for entropy perparticle, and E/S for energy per unit of entropy, as functions of γq at T =142MeV.The presence of chemical nonequilibrium reduces and potentially elim-inates discontinuities at the phase transition, promulgating rapid evolu-tion. This implies that, in particular, the sudden hadronization of anentropy-rich QGP should lead to the limiting value γq → γcq, since otherways of increasing the entropy content involve secondary processes withrelatively slow dynamics amongst hadron degrees of freedom. In fact, inan adiabatic equilibrium transformation, one allows an increase in V T 3,characterizing the entropy content, either by expanding the volume V , orinvoking a rise of T (reheating), or both. Another remarkable feature ofthe chemical-nonequilibrium mechanism is that a first order phase tran-sition may appear in other observables more like a phase transformationwithout large fluctuations.It is important to remember that, in the hadronization of a quark–gluonphase, it is relatively easy to ‘consume’ excess energy density, simply byproducing a few extra heavy hadrons. However, such particles, being infact non-relativistic at the temperature considered, are not effective con-7 Entropy and its relevance in heavy-ion collisions 127Fig. 7.9. Pion-gas properties (N , number of particles; E, energy; and S, en-tropy) relative to chemical equilibrium as functions of γq for T = 142MeV.tributors to pressure and entropy. As we see now, the super-saturated piongas is just the missing element needed in order to allow rapid hadroniza-tion.The specific properties (E/N , S/N , and E/S) of the pion gas are shownin Fig. 7.8, as functions of γq. We see a monotonic decrease of energyand entropy per particle while the energy per unit of entropy increasesreaching the condition E/S > T , which plays an important role in sec-tion 19.1. We see that the entropy per pion drops as γq increases, and, atthe condensation point γq → γcq, we can add pions without an increase inentropy. Figure 7.8 can be better understood by considering the relative(to chemical equilibrium) physical properties shown in Fig. 7.9. Thereis, in particular, a large growth in the number of pions, which yield isenhanced at fixed temperature by a factor 3.6 at the condensation point– it is this feature that allows one to hadronize the QGP into a super-saturated pion gas at low, supercooled temperature, accompanied by theexperimentally observed excess of pions. We also see, using the energy asan example, that the Boltzmann approximation, i.e., simply a yield factor128 Experiments and analysis toolsγ2q is not producing qualitatively wrong results, even though the increasein number of particles is underestimated by 50%.7.6 Entropy in a longitudinally scaling solutionWe now study the relationship between hadron multiplicity and initialconditions reached for the case of very-high-energy collisions, for whichthe scaling hydrodynamic solutions can be used to understand the flowof matter. The hydrodynamic expansion of an ideal fluid is entropy-conserving, Eq. (6.13). What this means, in case of the longitudinal ex-pansion, is best seen by considering, in Eq. (6.3), the entropy currentσµ = σuµ, and using Eq. (6.32) and the Euler relation Eq. (6.4):0 =∂σµ∂xµ=στ+dσdτ=1τd(τσ)dτ. (7.21)We haveτσ(τ) = σ(τ0)τ0 = σ(τf )τf = constant, (7.22)where τσ is a constant of evolution independent of rapidity y.The physical meaning of conservation of entropy, and thus conservationof τσ(τ), becomes clear on remembering the volume element Eq. (6.30).In the locally at rest frame of the fluid (the comoving frame) we have∆S = σ dz dt = στ dy dτ, (7.23)whence,ddτ(dSdy)=ddτ(τσ) = 0,dSdy= constant, (7.24)where, in the last equality, we have used the result Eq. (7.22). dS/dy isindependent of y and not a function of τ , i.e., it is independent of thefreeze-out condition.Since entropy is characteristic of the particle yield, Eq. (7.24) impliesthat the particle multiplicity is flat in rapidity, and is not evolving, i.e., itwill not depend on the (uncertain) freeze-out condition. It is establishedduring the initial period of time when the entropy density is built upas the system approaches local thermal equilibrium. Qualitatively, thisresult is shown in Fig. 5.2; baryons punch through and in between thereis a flat distribution in y of particle abundance, since the entropy densityper unit rapidity is constant.It is important to appreciate that, as a matter of principle, the initial en-tropy density reached in A–A collisions remains naturally undetermined,it arises from microscopic-entropy-producing reactions occurring prior to7 Entropy and its relevance in heavy-ion collisions 129the onset of the entropy-conserving hydrodynamic expansion. One cantry (and the diverse codes we described in section 6.1 do this) to modelthe A–A collisions on the basis of the behavior of p–p reactions, but it isfar from certain that such an approach will be successful. In other words,we cannot use scaling arguments, Eq. (7.24), to understand how big avalue of particle rapidity density we can expect to find. The microscopicphysics we introduce explicitly (or sometimes implicitly) into the dynamicmodel determines the final-state particle multiplicity.We now relate the observed final-state particle multiplicity to the initialentropy density. Employing the volume element shown in Eq. (7.23), andusing the conservation law Eq. (7.24), we obtainσ0 ≡dS0dV0=1F⊥1τ0dS0dy0=1F⊥τ0dSdy∣∣∣∣f. (7.25)The transverse surface isF⊥ = π(1.2 fm)2A2/3, (7.26)as given by the geometry of the collision, at zero impact parameter.From Eq. (7.25), we obtain, for the initial-state entropy density,σ0 =1π(1.2 fm)2A2/3τ0432dNchdy, (7.27)where we have assumed that a final-state particle consumes on average 4units of entropy (see Fig. 10.5), and that the charged-particle multiplicityis two thirds of the total.IIIParticle production8 Particle spectra8.1 A thermal particle source: a fireball at restThe longitudinally scaling limit in production of hadrons, section 6.4,applies at the RHIC and at higher collision energies. At the SPS and AGSenergy ranges, table 5.1, it is natural to explore the other reaction picture,the full-stopping limit. In this case all matter and energy available in thecollision of two nuclei is dumped into a localized fireball of hot matter.Even at the highest SPS energies many experimental results suggest thatsuch a reaction picture is more appropriate than the (1 + 1)-dimensional-flow picture.The m⊥ spectra we have seen in Fig. 1.7 on page 20 provide a strongencouragement to analyze the collision region in terms of the formationof a thermalized fireball of dense hadronic matter. The high slopes seenstrongly suggest that the dynamic development in the transverse direc-tion is very important. The pattern of similarity seen for very differentparticles is what would be expected to occur in hadronization of a nearlystatic fireball, and thus this case will be the first one we explore. However,we note that this is solely an academic exercise since SPS results provideample evidence for rather rapid v 0.5c transverse expansion. One canrecognize this important physical phenomenon only once the propertiesof the stationary fireball matter are fully understood.We consider a space–time-localized region of thermal hadronic matteracting as a source of particles, yielding naturally a Boltzmann spectraldistribution. The thermal equilibrium is strictly a local property, withdifferent temperatures possible in different space–time domains. The ne-cessity that there is also a local thermal pressure implies that a fireball isin general a dynamically evolving object with local flows of matter, whichwe shall study further below. The virtue of this model is that the spectra1308 Particle spectra 131and abundances of particles can be described in terms of a few parametersthat can be measured.The thermal analysis of the experimental results differs in many key as-pects from the microscopic-transport methods introduced in section 6.1.These contain as inputs detailed reaction data and their extrapolations,including often enough hypothetical reaction cross sections and novelmechanisms without which the experimental results cannot be describedcompletely. The attainment of thermal equilibrium is, in these calcula-tions, a result of many complex reactions. For the N–N collisions theappearance of the thermal particle distributions in the final state is stillinexplicable in terms of such dynamic microscopic models. For this reasonalone, a microscopic dynamic approach cannot lead to an understandingof the thermalization of fireball matter. Moreover, since the physicalthermalization processes are faster than those operating in present-daynumerical transport codes, microscopic transport theory delays the ther-malization of collision energy available in heavy ion reactions and thuswill in general fail at predicting observables of interest which dependon (early and rapid) thermalization. As long as these issues are beingstudied, an empirical thermal model allowing for flow of matter and non-equilibrium abundances of particles offers considerable advantages for theunderstanding of experimental data.We first aim to relate the experimental rapidity and transverse-massspectra to the particle distribution of the fireball. We consider the dif-ferential particle-momentum distribution, e.g., near the surface of thefireball,Ed3Nd3p≡ f(E, pL), (8.1)where the presence of pL in the argument reminds us that an emittedparticle could remember the collision axis; the distribution need not beintrinsically spherically symmetric as is implied when only the energy ofthe particle is considered. The coefficient E is introduced in Eq. (8.1)for convenience, it assures that the quantity f is invariant under Lorentztransformations. This is understood on reexpressing the left-hand side ofEq. (8.1) in terms of the invariant variables m⊥ and y. Given Eq. (5.4),at constant p⊥, we finddy =dpLE. (8.2)Since p⊥ dp⊥ = m⊥ dm⊥, considering Eq. (5.5) the Lorentz-invariant mo-mentum-space volume element isd3pE= dym⊥ dm⊥ dϕ. (8.3)132 Particle productionIt is important to remember, looking at the spectra, that, while p⊥ > 0,we have m⊥ > m.We consider at first as the intrinsic distribution the simplest exponentialBoltzmann-type thermal shape:f(E, pL)→ CEe−βE = Cm⊥ cosh y e−βm⊥ cosh y, (8.4)with C = gV/(2π3) and Eq. (5.4) is used on the right-hand side to replacethe particle energy by transverse mass and rapidity. Since we have aLorentz-invariant distribution, a change of the frame of reference alongthe pL axis, e.g., from the laboratory frame to the CM frame, is amountingto a shift along the rapidity y axis of the particle spectrum consideredto be centered around the CM rapidity yCM = 0, see section 5.3. Thedifferential particle spectrum which we obtain isd2N(y,m⊥)m2⊥ dm⊥ dy= C∫dϕ cosh y e−βm⊥ cosh y. (8.5)To obtain the transverse-mass spectra, we need to integrate Eq. (8.5)over the applicable rapidity acceptance (often referred to as ‘rapidity win-dow’):1m⊥dN(y,m⊥)dm⊥= C∫dϕ∫ y+y−dym⊥ cosh y e−βm⊥ cosh y. (8.6)For a wide (see below) rapidity window, we can extend the limit of theintegration to infinity, since the argument is a rapidly decreasing expo-nential function. We use Eqs. (10.44) and (10.45) and obtainK1(z) =∫ ∞0dt e−z cosh t cosh t,→( π2z)1/2e−z(1 +38z− 15128z2. . .). (8.7)We obtain for the full rapidity coverage1m3/2⊥dN(y,m⊥)dm⊥∝ e−βm⊥(1 +38βm⊥. . .). (8.8)For a narrow rapidity window, δy = y+−y−, surrounding yCM, one simplysubstitutes, in the integral Eq. (8.6), cosh(δy/2) by 1 and the result is1m⊥dN(y,m⊥)dm⊥∝ e−βm⊥ . (8.9)In both cases, Eqs. (8.8) and (8.9), we have a (nearly) exponential transverse-mass spectrum, provided that the preexponential factors in the spectra8 Particle spectra 133Fig. 8.1. The saturation of particle yield for a fireball at rest within a rapiditywindow: dashed line, y± = yCM± 0.5 and solid line y± = yCM± 1, as a functionof βm⊥; see the text for details.are properly chosen. The result is not at all what one is naively temptedto use when one is fitting invariant cross sections, i.e., to simply take anexponential fit of the cross-section data: the choice is either to include thefactor 1/m1/2⊥ (compare with Fig. 8.9 and also Fig. 8.8), or to multiply by1/m⊥ for a truly narrow rapidity window, as we see in Fig. 1.7 on page 20.The question thus is that of how narrow the ‘narrow’ rapidity windowmust be for the factor to be as given in Eq. (8.9). We note that, in addi-tion to the width of the typical experimental acceptance of 0.5–1 rapidityunit, one has to keep in mind that there is, in principle, a superposi-tion of contributions to the spectra occurring due to longitudinal flow inthe source, which effectively widens the rapidity acceptance domain. Weshow, in Fig. 8.1, the ratioRI ≡∫ y+y−dy cosh(y − yCM) e−βm⊥ cosh(y−yCM)∫ +∞−∞dy cosh(y − yCM) e−βm⊥ cosh(y−yCM), (8.10)of the rapidity integral Eq. (8.6) with the full rapidity coverage, as a func-tion of βm⊥. Results shown are for a rapidity window of one unit (dashedline, y± = yCM±0.5) and two units (solid line, y± = yCM±1) of rapidity,centered around yCM.We see that, for an experimental rapidity window of one unit (e.g.,−0.5 < y − yCM < 0.5) (dashed line) and for a typical ‘high’ transverse134 Particle productionmass m⊥ 1.5 GeV at T → T⊥ = 1/β = 230–300 MeV, we wouldhave reached nearly 80%–90% of the full rapidity integral, justifying useof a result with a full rapidity window coverage – note that adding insmearing of flow (the solid line) means that more than 99% of the spectralstrength will be effectively included. Consequently, we find that the testfor applicability of Eq. (8.8) is βm⊥ cosh(δy/2) > 8, with δy/2 comprisingan estimate of the flow.We next consider the thermal rapidity spectra. We now integrateEq. (8.4) over the full range of transverse mass,dN(y,m⊥)dy= C∫dϕ∫ ∞mdm⊥m2⊥ cosh y e−βm⊥ cosh y, (8.11)to obtain the rapidity distribution shown in the top portion of Fig. 8.2,for the case mπ <∼ β−1 < mK (here β−1 ≡ T = 160 MeV).The thin lines in Fig. 8.2 apply to spectra of massless particles, dashedlines are for pions (m = 138 MeV), chain lines are for kaons (m = 497MeV), and the thick solid line depicts data for nucleons (m = 938 MeV).Since the experimental acceptance in p⊥ cannot, for practical reasons,begin with p⊥ = 0, we have also shown in Fig. 8.2 what happens to thesespectra when only particles with p⊥ > pmin⊥ are included, with minimummomentum cutoffs shown at pmin⊥ = 0.3, 0.5, and 1 GeV. We notice that,when pmin⊥ < 0.5 GeV, the maximum peak for massless particles is nearlyhalf as high as that for nucleons, and, correspondingly, the widths of thedistributions vary considerably with particle mass.This change in relative abundance of the different particles increaseswith pmin⊥ (we changed the scale of the drawing by a factor of three tomake the small remaining particle abundance more visible for pmin⊥ =1 GeV). The lighter particles disappear more rapidly and the relativeabundance of the heavier ones is increased in the sample. Moreover, allshapes become increasingly more similar, resembling more and more thenucleon spectrum. We note that for pmin⊥ = 0.3, the half-width parameter,for most particles, is within the range 0.6 < σ < 0.7.We see that, in the ideal situation of a thermal Boltzmann-like emitter,the rapidity spectra of identified particles are very narrowly distributedaround the ‘central’ rapidity, with the distribution of the massive parti-cles being narrower than that of lighter particles when all p⊥ are included,which difference disappears when the minimum m⊥ for the different par-ticles are (nearly) the same. Since the width of the rapidity spectra isjust half as large as was seen for the width for negative hadrons, h−, seeFig. 9.6, there must be some other contribution to the width, that is ingeneral believed to be the flow: the small source is not stationary, and itssize and all other properties evolve rapidly in time, an effect we addressin section 8.4.8 Particle spectra 135Fig. 8.2. Normalized thermal rapidity particle spectra (quantitative) for aBoltzmann (exponential) energy distribution with β−1 = 160 MeV and set-ting yCM = 2.92 as appropriate for the highest SPS energy: massless particles(thin lines), pions (dashed lines), kaons (chain lines), and nucleons (solid lines).The effect of the minimum transverse-momentum cutoff on particle yield andshape of distribution is illustrated: we show pmin⊥ > 0, > 0.3, > 0.5, and > 1GeV. Note the change of scale for the last (bottom) case.Even though the above example of a thermal source is no more thana case study, we have learned much about the possible shape of the ra-pidity spectra of a well-defined, localized thermal source. This leads tothe practical question of what spectral shape arises when identification of136 Particle productionparticles is not possible. In such a case, one generally studies pseudorapid-ity distributions. We keep in mind that the relatively easily measurablepseudorapidity, Eqs. (5.24) and (5.25), arises from the rapidity, Eqs. (5.4)and (5.10), in the limit m → 0, and thus, in cases when an appreciableyield of nucleons and even kaons is present, significant distortions in thespectra occur.We now discuss this quantitatively and evaluate the shape of the ther-mal pseudorapidity distribution. Since η is not a good Lorentz variable,we study the specific example of the spectra in the laboratory frame inwhich the target is at rest and the projectile had 158A GeV. We takeas the input spectrum the rapidity shape of the thermal source definedabove in Eq. (8.11). To proceed with the change of variables, we need toexpress the CM energy and momentum of the distributions in terms ofthe laboratory pseudorapidity. Using Eq. (5.4),E = m⊥ cosh(y′ − yCM) = E′ cosh yCM − p′L sinh yCM, (8.12a)pL = m⊥ sinh(y′ − yCM) = p′L cosh yCM − E′ sinh yCM. (8.12b)With the help of Eq. (5.24), we eliminate E′ and p′L, using the pseudora-pidity η′ with reference to the laboratory frame, and p⊥,E =√m2 + p2⊥ cosh2 η′ cosh yCM − p⊥ sinh η′ sinh yCM, (8.13a)pL = p⊥ sinh η′ cosh yCM −√m2 + p2⊥ cosh2 η′ sinh yCM. (8.13b)We also obtain1p⊥dpLdη′= cosh η′ cosh yCM −p⊥ sinh η′ sinh yCM cosh η′√m2 + p2⊥ cosh2 η′, (8.14)for the (frame-of-reference-dependent) integration Jacobian relating theCM longitudinal momentum and laboratory pseudorapidity (see Eq. (8.2)for comparison).We now are ready to evaluate the pseudorapidity particle distribution.Proceeding in the same way as when we obtained Eq. (8.11), i.e., integrat-ing over the azimuthal angle and the transverse momentum, and effectingthe change of the integration variable from longitudinal momentum topseudorapidity, we obtain, in the laboratory frame, the pseudorapiditydistributiondNdη′≡ 2π∫dp⊥ p2⊥(dp′Lp⊥ dη′)f(E, pL)E. (8.15)The arguments of the distribution are as given in Eqs. (8.13a) and (8.13b)and the volume element is given by Eq. (8.14).8 Particle spectra 137Equipped with this result, we can explore quantitatively the case ofthe exponential, thermal-like distribution, Eq. (8.4). The explicit form ofthe laboratory pseudorapidity distribution, including a necessary Lorentzcontraction factor (cosh yCM)−1 arising from the Lorentz transformationof the volume V of the source, takes the formdNdη′=2πC∫ ∞pmin⊥dp⊥ p2⊥e−β[cosh yCM√m2+p2⊥cosh2 η′−p⊥ sinh yCM sinh η′]× cosh yCMcosh η′− tanh yCM sinh η′ p⊥ cosh η′√m2+ p2⊥ cosh2 η′. (8.16)A simple test of this not-so-simple expression is its normalization, whichis easily (numerically) verified by integrating over η′ at given m and β,for various values of yCM.This distribution is shown in Fig. 8.3, which parallels Fig. 8.2 with thesame conventions and parameters. On comparing Figs. 8.2 and 8.3, we seethat the rapidity and pseudorapidity spectra agree exactly for masslessparticles, since the pseudorapidity is the rapidity, in this case. The nearlymassless pions are visibly little changed in spectral shape. With progres-sively increasing mass, the pseudorapidity spectra differ more from therapidity spectra and, in particular, their center shifts to higher pseudora-pidity.There is a notable deformation of the symmetric shape accompaniedby considerable widening – the peak is only 60% of the height of the orig-inal rapidity spectrum for pmin⊥ = 0. In practical situations, the small-p⊥particles are eliminated, which we allow for by means of a cutoff in pmin⊥ .We see that now the spectral shapes appear progressively less shiftedfrom their rapidity form; the pseudorapidity and rapidity shapes becomemore similar, although a residual shift remains for the heaviest parti-cles (nucleons). Thus the pseudorapidity–rapidity difference is primarilya low-momentum phenomenon. As the p⊥ cutoff increases, the relativestrength of the particle spectra changes and, in particular, there is con-siderable relative enrichment of the contributions of the heaviest particlescompared with those of light particles.8.2 A dynamic fireballNaturally, a fireball at rest is not what we are likely to encounter in thehighly dynamic situation of colliding nuclei. We now look at the modifi-cations introduced by the presence of a local flow of matter. As before avolume element in a fireball is the particle source, but now this volume isin motion, typically due to a local flow originating from a (hydrodynamic)138 Particle productionFig. 8.3. Particle spectra as a function of laboratory pseudorapidity η′, forβ−1 = 160 MeV, yCM = 2.92; massless particles (thin lines), π (dashed lines),K (chain lines), and N (solid lines). Results for various minimum transverse-momentum cutoffs are shown: pmin⊥ > 0, > 0.3, > 0.5, and > 1 GeV. Note thechange of scale for the last (bottom) case.expansion. We will refer to this collective flow velocity below simply as/v. We would like to know how the statistical distribution appears to alaboratory observer. However, when we refer to a statistical phase-spacedistribution, we always imply an observer at rest in the local ‘intrinsic’frame of reference. The ‘intrinsic i’ particle energy Ei and momentum8 Particle spectra 139/p i are measured in the local non-flowing frame of reference of a movingvolume element of the fireball.We determine now how the intrinsic thermal spectrum appears to anarbitrary Lorentz observer, such as a laboratory observer is. The physicalidea is to express the intrinsic thermal phase space in terms of Lorentz co-variant quantities, and than to use variables associated with any observer,e.g., a laboratory-frame observer. There are several approaches possible,and we proceed in the first instance to consider as in section 12.3 theTouschek invariant phase-space measure [143, 261]:V0 d3pi(2π)3e−Ei/T → Vµpµ(2π)3d4p 2δ0(p2 −m2)e−pµuµ/T . (8.17)The flowing volume element Vµ = V0uµ is as observed in the laboratoryframe. V0 is the comoving volume element in the local rest frame. δ0 isthe Dirac delta function for the positive (energy) roots only, . It is aninvariant function for all proper Lorentz transformations. The left-handside of Eq. (8.17), is written in terms of the intrinsic variables, but theright-hand side is not frame-dependent and we can read it in the frameof reference of the laboratory observer.We see the invariant measure introduced in Eq. (8.3):2δ0(p2 −m2) d4p =d3pE= m⊥ dm⊥ dy dφp = p⊥ dp⊥ dy dφp. (8.18)The particle momentum defined with reference to the collision axis hasthe explicit formpµ = (m⊥ cosh y, p⊥ cosφp, p⊥ sinφp, m⊥ sinh y), (8.19)(in cylindrical coordinates) where we omit superscript p for the variablesy and m⊥ =√m2 + p2⊥, which, as usual, are understood to refer to theobserved particle. We recall the usual relations,m⊥m= γ⊥ = cosh y⊥,p⊥m= v⊥γ⊥ = sinh y⊥,which allow us to write Eq. (8.19) in the formpµm= (cosh y⊥ cosh y, sinh y⊥ cosφp, sinh y⊥ sinφp, cosh y⊥ sinh y).This suggests that we introduce such a cylindrical representation of thevelocity field as well,uµ=(cosh yv⊥ cosh yv‖ , sinh yv⊥ cosφv, sinh yv⊥ sinφv, cosh yv⊥ sinh yv‖),u2=1. (8.20)140 Particle productionIt is straightforward, in these coordinates, to obtain uµpµ required toconstruct the spectra in Eq. (8.17),uµpµ = γv⊥[m⊥ cosh(y − yv‖)− p⊥v⊥ cosφ], (8.21)where γv⊥ = cosh yv⊥ = 1/√1− v2⊥, φ = φp−φv, and the variables y, m⊥,and p⊥ refer to the rapidity, the transverse momentum, and the transversemass of the observed particle.The explicit form of the invariant spectrum which generalizes Eq. (8.5),isd2Nm2⊥ dm⊥ dy=∫dφ γv⊥(2π)3(cosh(y − y‖)−p⊥m⊥v⊥ cosφ)× exp{−γv⊥[m⊥ cosh(y − y‖)− p⊥v⊥ cosφ]/T}. (8.22)For a suitable choice of the coordinate system in which the x axis ispointing in the direction of the transverse flow vector, φv = 0, the particle-emission angle is the azimuthal angle of integration φ = φp. We use therange 0 < φ ≤ π, which has to be counted twice to include the partπ < φ ≤ 2π. The φ integrals we encounter are analytical:1π∫ π0e±a cosφ dφ = I0(a),1π∫ π0e±a cosφ cosφdφ = ±I1(a). (8.23)It is helpful to remember that I0 ‘looks like’ a cosh function, and I1 likea sinh function, and the analogy goes further withI1(a) =∂I0(a)∂a. (8.24)However,cosh a= I0(a) + 2I2(a) + 2I4(a) + 2I6(a) + · · · , (8.25)sinh a=2I1(a) + 2I3(a) + 2I5(a) + · · · , (8.26)whereIn(a) =1π∫ π0ea cosφ cosnφdφ =∞∑k=0(a/2)2k+nk!(n+ k)!. (8.27)8 Particle spectra 141Using Eq. (8.23) in Eq. (8.22), we obtaind2Nm2⊥ dm⊥ dy=γv⊥(2π)2(cosh(y−y‖) I0(p⊥v⊥/T )−p⊥m⊥v⊥I1(p⊥v⊥/T ))× exp[−γv⊥m⊥ cosh(y − y‖)/T]. (8.28)This is the statistical particle spectrum seen in the laboratory frame andoriginating in a volume element of a fireball having two velocity compo-nents y‖ and v⊥, and emitting particles at the local temperature T . Ifthe laboratory frame is not the CM frame, we need to shift the rapidityy → y − yCM. This is the particle spectrum of final-state hadrons arisingif the matter in the entire volume of the fireball froze out suddenly.This volume-style statistical phase-space hadronization based on workcarried out by Touschek [261], differs from the approach of Cooper andFrye [96], which allows for the dynamics of the particle-emitting surface.One imagines an opaque fireball, and each surface element is the parti-cle source. The physical idea is thus to couple the intrinsic (statistical)particle spectrum to the Lorentz-covariant surface dynamics. The de-velopments till 1993 are well documented in [238]. It was subsequentlydiscovered that the radiation formula was non-positive definite and a gen-eralization was proposed [119].The particle phase space is written using a covariant surface in 4−1 = 3space–time dimensions. Consequently, apart from the flow, there is yetanother velocity that describes how the hadronization surface moves, e.g.,the surface may be flowing outward, but a rapid ‘peeling’ of matter maymove the boundary of the particle-producing volume inward. Moreover,over the history of the particle production (freeze-out from the surface),the surface may have both positive and negative velocities, and thus itcan be difficult to make sure that a particle is actually emitted ratherthan absorbed in the fireball.We now illustrate the difficulty inherent in dealing with the problemof emission of particles, which continues to be actively studied. First, werecall how, in the Touschek approach, the particle density in phase spacehas been written in a Lorentz-invariant way asd6Nd3x d3p=g(2π)3e−uµpµi /T , (8.29)where uµ is the 4-flow velocity. In the volume-hadronization approach,we have made the following qualitative steps:Ed3Nd3p=dNdφdym⊥ dm⊥=g(2π)3e−uµpµi /T[Ei d3x → pµi Vµ ∝ pµi uµ]. (8.30)142 Particle productionWhen surface emission dominates the particle spectra, the other wayto proceed isEd3Nd3p=g(2π)3e−uµpµi /T[Ei d3x →∫dτ d3Σµf pµ i δ(dτ−√dt2f −d/x 2f)], (8.31)where xµf = (tf , /xf) are the freeze-out surface coordinates, and Σµf is thethree-dimensional hypersurface of the Minkowski volume element, char-acterized by a unit 4-vector normal to the surface uµf , i.e., the 4-velocityof the freeze-out surface:nµs =dxµfdτ=dtfdτ(1,d/xfdtf). (8.32)Equation (8.31) arises since we wish to sum the emission spectrum overthe contributions made by each surface element d3Σ over its (proper timeτ) history. For a fireball at rest, we have nµf = (1, 0, 0, 0), d3Σµf = dtf d2xf ,and /vf = d/xf/dtf = 0. Noting that the δ-function simply sets the propertime to the freeze-out time, we obtainEd3Nd3p=g(2π)3e−Ei/TEiSf ∆tf , (8.33)where ∆tf is the length of (proper) time during which the emission ofparticles occurs, and Sf is the size of the surface.For simple geometries, we can usedτ δ(dτ − dtf√1− d/x 2f /dt2f)= δ(1− dtfdτ√1− d/x 2f /dt2f)→ 1,and we find the conventional Cooper–Frye formula:Ed3Nd3p=dNdφdym⊥ dm⊥=g(2π)3∫Σfe−Ei/T piµ d3Σµ. (8.34)d3Σµ is the normal surface vector for the four-dimensional space–timevolume boundary, from which the emission of particles occurs,d3Σµ ≡ εµνλρ∂Σν∂u∂Σλ∂v∂Σρ∂wdu dv dw, (8.35)where u, v, w is a suitable set of three locally orthogonal coordinates. Incylindrical coordinates (u, v, w) = (rf , φf , zf) andΣµf = (tf , rf cosφf , rf sinφf , zf). (8.36)8 Particle spectra 143The freeze-out time tf is independent of the angle φf due to the assump-tion of cylindrical symmetry and thus we have tf(rf , zf). The covariantvolume element isd3Σcylµ =(1,−∂tf∂rfcosφf ,−∂tf∂rfsinφf ,−∂tf∂zf)rf drf dφf dzf . (8.37)We use the momentum vector of a particle pµi in cylindrical coordinates,Eq. (8.19), and obtaindτdtfpµ d3Σcylµ = rf drf dφf dzf[m⊥(1u0fcosh y − 1u‖fsinh y)− 1u⊥fp⊥ cosφ], (8.38)where as before φ = φf − φp.The 4-velocity of the freeze-out surface isdxµ/dτ = uµf = (u0f , u⊥f cosφf , u⊥f sinφf , u‖f ).This additional surface dynamics influences the observed spectra, eventhough we are dealing with a preexponential factor only. The transverse-mass spectra contain an additional factor, compared to Eq. (8.22),d2Nm⊥ dm⊥ dy→∫dφ γv⊥(2π)3(1− /v−1fr · /pE)×(cosh(y − y‖)−p⊥m⊥v⊥ cosφ)(8.39)× exp{−γv⊥[m⊥ cosh(y − y‖)− p⊥v⊥ cosφ]/T},where/v−1fr · /p ≡∂tf∂rfp⊥ cosφ+∂tf∂zfpz. (8.40)Particles are emitted from the surface of fireball volume, and thusthe phase space is (2 + 2)-dimensional when the number of particlesis counted. For this dimensional reason there is one power of m⊥ lessin Eq. (8.39), than there is in Eq. (8.22). When vfr → c the prefactorin Eq. (8.39) is able to compensate for this effect and both methods candescribe the experimental hadron spectra with similar precision.We proceed to show how the longitudinal flow, and, in section 8.4, thetransverse flow, influence particle spectra. Numerical study shows thatthe two flows are practically independent from each other, and it has144 Particle productionydn/dyydn/dyFig. 8.4. On the left, the abundance of Λ in S–S collisions at 200A GeV, asa function of rapidity. The squares are the results for N–N collisions scaledup by the pion-multiplicity ratio. On the right, corresponding results for theabundance of Λ. Data produced by the NA35 collaboration [24].become commonplace to study particle spectra as if either only parallelor only transverse flows were present: in a study of rapidity spectra,only y‖ is considered, while v⊥ is ignored; in a study of m⊥ spectra, thelongitudinal flow y‖ is not considered.8.3 Incomplete stoppingConsidering that, at very high collision energies, the longitudinal scalingbehavior is expected, see section 6.3, whereas in collisions of large nuclei atmoderate energies a central fireball is more appropriate, it is natural thatthe real world is observed to be much more complex than these simple‘asymptotic’ models.A nice example of the case in which the baryon number just does notpunch through is seen in the central 200A-GeV S–S collisions at theSPS. We show, in Fig. 8.4, the production yields of Λ (left-hand side)and Λ (right-hand side) hyperons as functions of rapidity. The opencircles in Fig. 8.4 are the directly measured data. The particle spec-tra must be symmetric around the CM rapidity, since this is a symmet-ric collision system. For this reason the open black circles are obtainedby reflecting the measured data points (solid black circles) at the valuey = 2.96.The spectra arising from N–N interactions at the same energy areshown in Fig. 8.4 (open squares). These N–N-interaction comparisondata have been multiplied by the rapidity-dependent pion-multiplicity-enhancement factor, which accounts for the increase in production of8 Particle spectra 145Fig. 8.5. Rapidity spectra of massless QGP quanta with flow: dotted, no flowy‖ = 0; short-dashed, y‖ = 0.5; long-dashed y‖ = 1; and solid, y‖ = 1.5 [178].pions per nucleon observed on comparing S–S with N–N reactions. Inthe target (y = 0.75 ± 0.25) and projectile (y = 5.25 ± 0.25) fragmen-tation regions, this procedure gives a good agreement between yields ofΛ particles in S–S and N–N scaled by the pion multiplicity. This sug-gests that, in the target/projectile fragmentation regions, the productionof Λ has the same origin in both cases, presumably from individual N–Ninteractions.However, in the central rapidity region in Fig. 8.4, new mechanismsof production of Λ and Λ are clearly visible. Inspecting the yield of Λ,we see considerable localization at central rapidity of a particle madeentirely from constituents not brought into reaction, Λ(ūd̄s̄). Naturally,there must have been associated localization of the energy. The Λ rapidityspectrum is, in contrast, relatively flat. Λ(uds) contains, aside from thestrange quark made in the reaction, constituent quarks brought into thecollision region by the projectile and target. Were the punch through ofthe light (u, d) quark content complete, we should see for Λ a distributionsimilar to Λ, both in shape and in yield.We consider now the rapidity spectra in the presence of a longitudinalflow y‖, evaluating the m⊥ integral in Eq. (8.22). The challenge is todescribe a diversity of rapidity spectra of observed hadrons, which arevery strongly varying between certain particles. The different behaviorsare shown schematically in Fig. 8.5. We see how the thermal rapidityspectra of massless quanta in the deconfined phase, with m = 0, (andwith T = 145 MeV, and v⊥ = 0.52, which values matter little), vary.146 Particle production020406080100 3− 2− 1−ycmdn/dyFig. 8.6. Rapidity spectra of baryons 〈b− b̄〉 observed by experiment NA49 [43]in central (5%) Pb–Pb interactions at√sNN = 17.2 GeV (solid circles, directmeasurement; open circles, reflection at yCM). Stars are rapidity spectra ofbaryons for S–S interactions obtained by NA35 at√sNN = 18.4 GeV, for the 3%most central events, scaled with participant number 352/52.These gradually ‘flow’ apart as y‖ is increased from y‖ = 0 (dotted line)to y‖ = 1.5 (solid line) in steps of 0.5.Comparing with Fig. 8.4, we see that both limits (the central productionof Λ and the flat distribution of Λ) are seen in Fig. 8.5. How can this behappening in the same reaction? In order to obtain different types of flowfor different particles, we presume in the following illustrative examplethat hadrons arise from a mix of three quark fluids. The incoming valencequarks of colliding nuclei are retaining some (vvalence‖ = 0) memory of theoriginal motion along the collision axis, and constitute the projectile andtarget fluids. However, all newly made pairs of quarks have practically nomemory (vpair‖ 0) of the initial condition of colliding matter, they areformed near yCM and are constituents of the third fluid. In particular,pairs of strange quarks made in the plasma do not flow in the longitudinaldirection.For the protons produced, this model implies that all three quarksremember the incoming flow and their distribution should follow the solidor long-dashed line (depending on y‖). For Λ, with one strange quark, weconsider a mix of two thirds weight in the spectrum with flow and onethirds without. For particles like K+(us̄), we take a 50%–50% mix, and,for all newly made particles like Λ and Φ, we assume that only no-flowcomponents contribute. To describe baryon rapidity spectra in Pb–Pb8 Particle spectra 147Fig. 8.7. Schematic representations of rapidity particle spectra within a thermalmodel with flow, parameters chosen for√sNN = 17.2 GeV [178].collisions at√sNN = 17.2 GeV, reported by NA49 [43], see Fig. 8.6,we assume that the longitudinal flow is y‖ ±1, and choose this valuewithout attempting to fit the spectra.The proton rapidity spectrum is shown in the bottom panel in Fig. 8.7.We usedmq = 0, T = 145 MeV, and v⊥ = 0.52c, which parameters hardlymatter and could be chosen very differently; these values were taken inview of the m⊥ spectra we discuss below in section 8.4. We average overpositive and negative flows y‖ = ±1, since the collision in the CM frameinvolves both. The strange-quark content of the Λ which contributes withrelative strength 33% suffices to yield a flat distribution – see the secondpanel from the bottom in Fig. 8.7. The shape of the central rapidityplateau is in agreement with the results seen in NA49 data, as well aswith those shown above in Fig. 8.4 for the NA35 S–S collisions.148 Particle productionIn the third panel from the bottom, corresponding to the 50%–50% flowmix such as would be appropriate for K+(us̄), the resulting rapidity shapeis already peaked at the central rapidity. Finally, in the top panel, weshow a prototype of the rapidity distribution arising for all hadrons madefrom completely newly made particles such as Λ and Φ. The measured Φspectrum is again in qualitative agreement with this result [21].A comparison of the baryon distributions between Pb–Pb and S–S colli-sion, seen in Fig. 8.6, suggests that y‖ is about 0.4 units of rapidity largerin the lighter collision system. Even though it is seemingly a small change,this opens by 50% the gap between the fluids, as we saw in Fig. 8.5, anda more pronounced central-rapidity reduction in abundances of certainparticles is present for S–S compared with Pb–Pb collision systems.8.4 Transverse-mass fireball spectraThe experimental study of the rapidity spectra is complemented by stud-ies of particle-abundance distributions in the direction transverse to thecollision axis. Under a Lorentz transformation along the collision axis, p⊥remains unchanged and thusm⊥ =√m2 + /p 2⊥is invariant. Transverse-mass m⊥-particle spectra are therefore not di-rectly distorted by flow motion of the fireball matter along the collisionaxis, and also no further consideration of the CM frame of reference is nec-essary, which in fixed target experiments is rapidly moving with respectto a laboratory observer.There is also a great difference in the physics when we evaluate rapid-ity and transverse-mass spectra. As discussed in section 8.3, the rapidityspectra help us understand the degree of stopping and transparency ofmatter in collision, whereas the m⊥ spectra offer insights into thermal-ization of matter after collision, and evolution of flow. In that sense m⊥spectra are often more interesting and also a greater challenge to describein an ab initio study. Within the statistical model the focus in studyingm⊥ spectra is on determining the local temperature and transverse flowof the evolving fireball matter.One could consider the particle spectra as functions of transverse-momentum p⊥, but the regularities occurring for transverse-mass spectrafor different particles suggest that the spectra have a thermal character.Therefore m⊥ is a better variable to use in heavy-ion collisions, at leastwhen m⊥ is not too big: particles with high values of m⊥ > 4 GeV, atthe temperatures we consider, are potentially produced in initial hardscattering of partons. These decay in yield as a power law, and hence8 Particle spectra 149dominate the exponentially decaying thermal particle yields at high m⊥.Since hard parton scattering knows nothing about the mass of the finalhadron observed, a better variable to look at to evaluate these processes isp⊥. However, for m⊥ m, there is little difference between p⊥ and m⊥,so we conclude that m⊥ is overall the more suitable variable to considerin heavy-ion collisions. One of the surprising early results obtained atRHIC is the absence of high-p⊥ particles in central interactions [17]. Thissuggests an effective parton thermalization mechanism.In order to study the thermal properties in the fireball as ‘reported’by the emitted particles, we analyze m⊥ spectra of many different had-rons. The range of m⊥, on the one hand, should not reach very largevalues, at which hadrons originating in hard parton scattering are rele-vant. On the other hand, we do need relatively small m⊥, in order for thenon-exponential structure associated with transverse flow and resonancedecays to emerge.The transverse-mass spectra of hyperons, which we have seen in Fig. 1.7on page 20, are potentially very important in understanding and in mod-eling of the exploding QGP fireball. We have already in the S–Au 200A-GeV collisions the appearance of the exponential thermal spectra. Thecentral-rapidity high-transverse-mass spectra of strange particles, K0s , Λ,and Λ, given by the CERN–SPS WA85 collaboration, m−3/2⊥ dNi/dm⊥,are shown in Fig. 8.8. The factor m−3/2⊥ is introduced in view of the formof Eq. (8.8). The resulting shape, shown in Fig. 8.8 on a semi-logarithmicdisplay, can be fitted with a straight line. This exactly exponential behav-ior is initially surprising, considering that Eq. (8.8) required summationover the entire range of rapidity, given the rapidity acceptance range ofWA85 limited to central ∆y < 1 interval. However, effective summationover a wider range of y occurs, given the presence of collective longitu-dinal flow of matter. Similar results were also reported from the relatedwork of the WA94 collaboration for S–S interactions [8].We see, in Fig. 8.8, in the region of transverse masses presented, 1.5GeV < m⊥ < 2.6 GeV, not only that the particle spectra are exponential,∝ exp(−m⊥/T⊥), but also that the behaviors of all three different parti-cles feature the same inverse slope, T⊥ = 232 ± 5 MeV. This is not theactual temperature of the fireball, as noted earlier. The lower emissiontemperature of these particles, Ttf , is blue-shifted by the flow as is seen inEq. (8.39), and can be approximately understood in terms of the Dopplerfactor in Eq. (5.36).The same shape of m⊥ spectrum appears in results from the WA80 col-laboration results, for the neutral hadrons π0 and η. In Fig. 8.9, we showthe S–Au and S–S WA80 results at 200A GeV [28, 29, 234, 235], multiply-ing the invariant cross sections by the power m−1/2⊥ in order to establish150 Particle productionRΛ–1S WΛ11010106541.5 2.5 32–3/2ΛT [GeV]T [arb. units]T m dN/dmmKs0Fig. 8.8. Strange-particle spectra for Λ, Λ, and KS [225]. The line connectingthe Λ and Λ spectra, denoted R−1Λ , shows how at fixed m⊥ the ratio RΛ ofabundances of these particles can be extracted. Experimental WA85 results at200A GeV [104, 116, 117].a direct correspondence between the representations of the data of ex-periments WA85 and WA80. To determine the required multiplicativefactor, we note that the particle-production cross section dσ is controlledby the geometry of the collision, see section 5.2, and thus is the geometricinteraction surface, σinel, multiplied by the yield of particles dN . UsingEq. (8.3) we obtainm−1/2⊥ Ed3σd3p= σineldN2πm3/2⊥ dm⊥ dy. (8.41)Like WA85, the WA80 experiment also presented data for the centralregion in rapidity, 2.1 < y < 2.9, and no further adjustment is needed inorder to make the results exactly comparable.8 Particle spectra 151Fig. 8.9. Neutral-particle π0 and η spectra (invariant cross sections divided bym1/2⊥ ) in the central-rapidity interval 2.1 < y < 2.9 [225]. Upper solid line, S–Authermal spectrum with temperature T = 232 MeV; lower solid line, S–S, T = 210MeV. Experimental data at 200A GeV courtesy of the WA80 collaboration [28,29, 234, 235].The upper straight line (S–Au collisions) in Fig. 8.9 is the same expo-nential as we saw in Fig. 8.8, for the three different WA85 strange-particlespectra. While the WA85 data covered the interval 1.5 GeV < m⊥ < 2.5GeV, the thermal exponential shape continues, in the WA80 data, throughthe highest data point at m⊥ = 4GeV. The lower solid line in Fig. 8.9is for S–S 200A-GeV interactions and is drawn with T = 210 MeV. Thechoice of S–S temperature is based on the WA94 results obtained fromtheir spectra of strange antibaryons [8]. It is noteworthy that the WA80particle spectra shown in Fig. 8.9 span seven decades, and that over 5–6 decades the thermal spectral shape for neutral hadrons is in excellentagreement with the strange-particle spectral shape. We note that the risein the yield of neutral mesons at low m⊥ 0.5 GeV is expected. It is dueto secondary contributions to the yield of particles by decay of hadronic152 Particle productionTable 8.1. Inverse (net) proton slopes T⊥ for various reaction systems at 200AGeV (158A GeV for Pb–Pb), increasing in size from left to rightReaction p–S p–Au d–Au O–AuT⊥ 154± 114 163± 5 172± 5 219± 5y interval 0.5 ≤ y ≤ 3.0 0.5 ≤ y ≤ 3.0 0.5 ≤ y ≤ 3.0 0.5 ≤ y ≤ 3.0Reaction S–S S–Ag S–Au Pb–PbT⊥ 235± 9 238± 2 276± 48 308± 15y interval 0.5 ≤ y ≤ 3.0 0.5 ≤ y ≤ 3.0 3.0 ≤ y ≤ 5.0 y 3resonances. While at high m⊥ pion, kaon, and hyperon slopes agree, atsmall m⊥ < 0.8 GeV the pion spectrum is much steeper. Since mostpions are produced at these m⊥, a global fit to the pion data yields aninverse slope parameter which is much smaller than the value we can seein Fig. 8.9.There is a clear difference between T⊥ inverse slopes pertinent to differ-ent collision systems, and a systematic trend is visible: T⊥ increases withthe volume of the reaction zone. We show, in table 8.1, the m⊥ inverseslopes of participating (net) protons for a number of collision systemsstudied by the NA35/NA49 collaboration [26, 43]. The shape of the m⊥spectra has been fitted to the simple formdNdm⊥ dy∼ mα⊥ exp(−m⊥/T ). (8.42)The results presented in table 8.1 were obtained with α = 1. Severaleffects contribute to an increase of T⊥ with increasing size of the collid-ing system. With increasing number of participants in the collision, thefireball of dense matter becomes less transparent and thus colliding mat-ter can be compressed more at a given collision energy. Moreover, largersystems have more time to develop the outward flow under the (higher)internal pressure, acquiring a greater collective velocity. Thus, what wesee is that the initial fireball of the collision system is getting hotterand denser with increasing collision volume, which leads to a longer, andmore violent explosion. This in turn enhances the transverse velocity atthe time of production of particles. An in-depth analysis, which requiresconsideration of other particles apart from the (net) production of pro-tons, confirms that the systematic increase of the inverse m⊥ slope withincreasing size of the system is associated with an increasing velocity ofexpansion of the source. The intrinsic temperature of emission from thefireball remains at the level of T 160 MeV [59, 60, 176, 259].8 Particle spectra 153Table 8.2. Inverse slopes T⊥ for various strange hadronsΛ Λ Ξ− Ξ+ Ω− +Ω+ φPb–Pb 289± 2 287± 4 286± 9 284± 17 251± 19 305± 15S–W 233± 3 232± 7 244± 12 238± 16To reach the most extreme conditions, collisions of the heaviest nucleiare required, and thus much of the experimental effort has gone into study-ing the Pb–Pb collision system. The highest inverse slopes are reportedfor several strange baryons and antibaryons by the WA97 collaboration[42]. Results presented in table 8.2 were obtained using, in Eq. (8.42),α = 1 for Pb–Pb, and α = 32 for the S–W collision system. The corre-sponding spectra are shown in Fig. 1.7 on page 20. The most interestingresult seen in table 8.2 is that there is practically the same inverse slopefor baryons and antibaryons of the same type. This confirms the resultreported by the WA85 collaboration for S–W interactions [6, 118], as isalso shown in table 8.2.The data point in table 8.2 for the φ(ss̄), in Pb–Pb collisions, is from theevaluation of the kaon-decay channel by the NA49 collaboration [21]. Thisdata point disagrees with a preliminary result Tµµ = 227 MeV, which wasreported by the NA50 collaboration and obtained in the dimuon-decaychannel [212].If strange baryons and antibaryons were to be produced in an envi-ronment of baryon-rich confined matter, the difference in interactions ofantibaryons, which have a large annihilation cross section at small mo-menta, should be visible as a baryon–antibaryon difference in the spectralshape, in particular at small m⊥. The absence, to a very high precision,of any transverse-mass spectral asymmetry between strange baryons andantibaryons is a very important item of experimental evidence for a com-mon mechanism of production of strange baryons and antibaryons by asource such as a QGP fireball which treats matter and antimatter in thesame way. In order to suppress interactions within a hadronic-matterphase possibly formed after the QGP state hadronizes, either a suddenbreakup of the fireball, arising after considerable super-cooling, or se-quential evaporation of hadrons in time, without formation of a hadronphase, is required. This symmetry between matter and antimatter has notbeen reproduced in transport models, in which confined hadron degreesof freedom appear.In Fig. 8.10, we see, for Pb–Pb collisions at 158A GeV, results shownin table 8.2 along with other inverse slopes T⊥ ordered as functions ofparticle mass. Several different results are shown for pions, which arise154 Particle productionFig. 8.10. Mass dependences of inverse slopes observed in Pb–Pb interactionsat 158A GeV; symbols indicate the experiment from which data is drawn, ascoded in the figure.for different (low) m⊥, y windows considered in different experiments.Even so, there is some unresolved variance between different pion results.Ten different hadronic particles with 0.9 GeV < m < 1.5 GeV exhibita common inverse slope indicated by the horizontal dashed line. Thereis general agreement that the increase in the slope seen on comparingpions, kaons, and baryons (obtained within an overlapping range of p⊥,not of m⊥) is due to the presence of a strong transverse flow of matterfrom which these particles originate [59]. The observation of a thermalcharmonium spectrum (the point at m = 3.1 GeV) both in S- and Pb-induced reactions [11, 14], with m⊥ slopes similar to those for the otherheavy hadrons, suggests that thermalization of hadrons is a universalphenomenon in heavy-ion collisions. The thermal shape of the observedcharmonium spectra is somewhat surprising considering the ‘standard-model’ reaction picture of suppression of charmonium, see section 1.6.The highest value of T in Fig. 8.10, at m = 1.9 GeV, for the deuteron,confirms that these particles are not produced thermally. Production ofdeuterons is believed to arise predominantly from the final-state interac-tion between nearly free-streaming nucleons. The inverse slope for Ω andΩ (at m = 1.6 GeV) seems to be about two standard deviations belowexpectation. It is understood to be due to excess production of Ω and Ωat low p⊥; see Fig. 8.11.8 Particle spectra 155Fig. 8.11. Thermal analysis of Λ (left) and Ω+Ω (right)m⊥ spectra for variouscentralities of collision [259].8.5 Centrality dependence of m⊥-spectraA study of transverse-mass spectra [259] has been performed for the pre-cisely known strange-hadron spectra of the experiment WA97 [42], re-ported for several centrality bins; this data, with all centrality bins com-bined, is shown in Fig. 1.7 on page 20. The shapes of the various particlespectra depend in a complex, nonlinear, but unique, way, on the temper-ature used, and on the velocity of transverse flow, and these parametersare determined universally for all particles considered in each collisioncentrality.In an early study of hadron spectra it was suggested that spectra alonecould not separate Ttf and vtf , as these quantities are highly correlated;see the Doppler formula Eq. (5.36). As we will see, these two parame-ters can be determined without any need for other experimental input,when precise experimental data are available for m⊥ spectra reachingdown to relatively low values of p⊥. This is possible for the followingreason: it is assumed that, after hadronic resonances have decayed, theirdecay products do not rescatter from surrounding matter, thus the non-thermal spectrum is combined with the primary thermal spectrum toform the final observed spectrum. By choosing the yield of resonancesto be determined by the temperature seen in the spectrum, the shapeof the computed spectrum becomes a highly nonlinear function of tem-perature and velocity. Since there is a minimum transverse momentumrequired in order to observe a particle, the yields above pmin⊥ depend alsoon the transverse-flow velocity. This method assumes that the chemical(particle-production) freeze-out temperature Tf is assumed to be nearlyequal to the thermal (spectrum-shaping) freeze-out temperature Ttf . Theresults obtained are consistent with this assumption.156 Particle productionThe finalm⊥ distribution for particles is composed of directly producedparticles and decay products originating in the ‘root’ particle R decayingto the observed particle X and any Z, with variables R(M,MT, Y ) →X(m,m⊥, y) + Z [238]:dNXdm⊥=dNXdm⊥∣∣∣∣direct+∑∀R→X+ZdNXdm⊥∣∣∣∣R→X+2+···. (8.43)Only first-generation, and only two-body, decays were considered, as isappropriate for the hyperons and kaons. The decay contribution to theyield of X isdNXdm2⊥ dy∣∣∣∣R=gRbRX4πp∗∫ Y+Y−dY∫ MT+MT−dM2T Jd2NRdM2TdY. (8.44)Here, gR and bRX are the R-degeneracy and branching into X, and p∗ =√E∗2 −m2 with E∗ = (M2 −m2 −m22)/(2M), are the energy and mo-mentum of the decay product X in the restframe of its parent R. Thelimits on the integration are the maximum values accessible to the decayproduct X:Y± = y ± sinh−1(p∗m⊥),MT± =ME∗m⊥ cosh∆Y ± p⊥√p∗2 −m2⊥ sinh2∆Ym2⊥ sinh2∆Y +m2,andJ = M√P 2Tp2⊥ − (ME∗ −MTm⊥ cosh∆Y )2,where ∆Y = Y − y.The primary particle spectra (both those directly produced and parentsof decay products) are derived from the thermal Boltzmann distribution.As discussed earlier in this chapter, in general the longitudinal flow doesnot significantly influence m⊥ spectra. Thus it is possible, in order tosimplify the evaluation of integrals, to disregard longitudinal flow and toallow spherical symmetry of the transverse flow. A second hadronization-surface ‘velocity’ seen in Eq. (8.39), v−1f ≡ dtf/dxf , was considered. Thusthe thermal distribution of directly produced and parent particles R hadthe formd2Ndm⊥ dy∝(1−/v−1f · /pE)γm⊥ cosh y exp[−γET(1−/v · /pE)], (8.45)where γ = 1/√1− v2.8 Particle spectra 157Multiplicity bin100110120130140150160170180190200T [Gev]5 x statistical K0 errorStatistical K0 errorChemical fitT limitsMultiplicity bin0.300.500.700.901.10Chemical fitflow limitsvvfFig. 8.12. The thermal freeze-out temperature T (left), flow velocity v (bottomright), and breakup (hadronization hyper-surface-propagation) velocity vf (topright) for various collision-centrality bins. The upper limit vf = 1 (dashed line)and chemical-freeze-out-analysis limits for v (solid lines) are also shown. Forthe temperature, results obtained with increased error for kaon spectra are alsoshown.Simultaneous analysis of the spectra of Λ, Λ, Ξ, Ξ, Ω + Ω, and KS =(K0 + K0)/2 in four centrality bins was performed. In each centralitybin pronounced minima in T, vf and v plane are observed for the totalstatistical error:χ2 =∑i(F theoryi − Fi∆Fi)2,evaluated relative to the experimental precision of measurement ∆Fi ofthe result Fi. The chemical parameters, which are not well determinedby a momentum-distribution fit, are not varied. Since the statistics ofkaons was very high, and thus the statistical precision of data potentiallywas significantly greater than systematic error, also a global fit with afive-fold-increased kaon error was performed [259].Some of the resulting m⊥ spectra for particles are shown, in Fig. 8.11,in each part for the four bins separately. On the left-hand side, we seeas an example the Λ spectrum. The description of the shape, in all fourcentrality cases, is very satisfactory, also for all other particles consid-ered, except for Ω + Ω in the right-hand panel in Fig. 8.11. In all fourcentrality bins for the sum Ω+Ω, the two lowest m⊥ data points are un-derpredicted. This low-m⊥ excess explains why the inverse m⊥ slopesfor Ω and Ω are reported to be smaller than the values seen for allother strange (anti)hyperons in Fig. 8.10. This behavior suggests thatsoft Ω and Ω are produced in a significant manner by mechanisms be-yond the statistical model, which we discuss further at the end of sec-tion 19.3.158 Particle productionThe parameters shaping the spectral form, which arise in this descrip-tion of hyperon m⊥ spectra, are shown in Fig. 8.12, on the left-hand sidethe thermal freeze-out temperature Ttf , and on the right-hand side thetransverse velocity v (bottom) and the breakup (hadronization) speed pa-rameter vf (top). The value of vf is near to the velocity of light, which isconsistent with the picture of a sudden breakup of the fireball. The hor-izontal lines delineate the ranges of the result of chemical particle-yieldanalysis, and are similar to those presented in table 19.3 on page 360. Therange of values of T seen in this table is slightly different as the resultspresented were updated. The m⊥-spectral-shape analysis is found to beconsistent with the purely chemical analysis of strange and non-strangehadron production.An important objective of this complex analysis is to see whether differ-ent centrality bins yield results consistent with the same physics. Thereis no indication, in the left-hand panel of Fig. 8.12, of a significant orsystematic change of T with centrality, or dominance of the result bythe kaon spectrum alone. The resulting temperature, here dominated bythe thermal shape, agrees with the temperature obtained from analysisof particle yields alone, table 19.3, which is dominated by the chemicalfreeze-out temperature.The flow (expansion) velocity v (lower part of the right-hand panel ofFig. 8.12), even though it is flat to within the experimental error, reveals aslight but systematic increase with centrality, and thus size of the system.This is expected, since the more central events involve a greater volumeof matter, which allows more time for the development of the flow.For all four centralities these results show that there is no need to in-troduce a two-staged freeze-out; in fact, we can conclude that Ttf Tf .The myth of unequal thermal and chemical freeze-out temperatures isrooted in the high temperature obtained in chemical-equilibrium anal-ysis of hadron yields. However, results of such an analysis of the ex-perimental data lack the required statistical confidence, even though thesystematic behavior of the particle production data is well reproduced,as we shall discuss at the end of section 9.2. Specifically, once a fullydescriptive set of parameters is introduced, allowing for precise data de-scription, thermal and chemical freeze-out conditions are found to be thesame.The results of the analysis described in this section are consistent withstrange hadrons being produced by the new state of matter at CERNin all centrality bins explored by the experiment WA97, i.e., for num-bers of participants greater than 100. The low-centrality fifth bin, nowbeing studied by experiment WA57, see Fig. 9.5 [87], exhibits differentcharacteristics, with less enhancement of the production of multistrangehadrons.9 Highlights of hadron production 159ydn/dyydn/dyFig. 9.1. The abundance of 1.6Λ + 4KS + 1.6Λ as a function of rapidity. Onthe left, S–S; on the right, S–Ag (open circles are the directly measured data).The triangles are reflected data points for S–S and reflected interpolated dataemploying S–S and S–Ag. The squares in the S–S case are the results for N–Ncollisions scaled up by the pion-multiplicity ratio; for S–Ag these are the scaled-up p–S results. Data courtesy of the NA35 collaboration [128].9 Highlights of hadron production9.1 The production of strangenessStrangeness is a valuable tool for understanding the reaction mechanism,since it has to be made during the collision. The question is that ofhow it is produced. In terms of experimental information, the first thingwe would like to establish is whether the mechanism producing stran-geness involves a hot fireball at central rapidity, or whether perhaps alot of strangeness originates from the projectile/target-fragmentation re-gion.Results of the experiment NA35 [128] are shown in Fig. 9.1 as functionsof rapidity for the case of S–S 200A-GeV collisions. We consider the over-all abundance of 〈s + s̄〉. The open circles are the measured data points,the open triangles are the symmetrically reflected data points, and squareson the left-hand side are the results of N–N (isospin-symmetric nucleon–nucleon) collisions scaled up by the ratio in pion multiplicity, whereas onthe right-hand side the p–S results are scaled up. We show the rapidityyield obtained by integrating the transverse-mass m⊥ distribution for thetotal yield of strangeness:d〈s + s̄〉dy= 1.6dΛdy+ 4dKSdy+ 1.6dΛdy. (9.1)We note that, on doubling the KS yield, we include KL, and, on doublingagain, we add both K+ and K−, which explains the factor 4.160 Particle production−WA94Ξ + Ξ Λ + Λ+UA5 AFSARGUSCLEOHRSMARK IITASSOTCPp p−+0.20.10+Ξ /Λ−e e− p p−Ξ /Λ+ Ξ /Λ−Ξ /Λ+ Ξ /Λ−S W− S S−WA85Fig. 9.2. The ratio (at fixed p⊥) of (multi)strange baryon–antibaryon particleabundances, measured in the central rapidity region for 200A-GeV S–S/W col-lisions, compared with ratios obtained in lepton- and nucleon-induced reactions.Data assembled by the WA85/94 collaboration [1–5, 7].In Fig. 2.6 on page 32, we saw that a factor 1.5 allows one to extrapolatethe yields of Λ and Λ to include all singly strange hyperons. However,there is also strong production of multistrange Ξ and Ξ, with Ξ−/Λ 0.1 and Ξ+/Λ 0.2, as is seen in Fig. 9.2. Assuming similar rapiditydistributions for Ξ− and Λ and Ξ+ and Λ, and remembering that, foreach charged Ξ, there is its neutral isospin partner, this implies that thecoefficient of Λ should have been 1.9, and that of Λ 2.3, in order to accountfor the production of multistrange hyperons. Thus the enhancement inproduction of strange particles seen in Fig. 9.1 is slightly (by 6%)understated.The difference between scaled N–N and S–S data is most pronouncedat central rapidity (yCM = 2.97), and it disappears within one unit of theprojectile- and target-fragmentation regions. At central rapidity y 3,a new source of strangeness not present in the N–N-collision system con-tributes. The S–S system is relatively small, thus stopping is small, and itis quite impressive that the enhancement in production of strangeness isobserved only at central rapidity. Since other SPS experiments involvedheavier nuclei and/or lower energy, and thus certainly involved more stop-9 Highlights of hadron production 161ping, we can be sure that the excess of strangeness at the SPS originatesat central rapidity.We show, in the right-hand panel of Fig. 9.1, similar results for S–Agcollisions: the open circles are the measured points, open triangles areestimates based on S–S and the ‘reflected’ S–Ag results, and the opensquares are pion-multiplicity-scaled p–S results. An enhancement in pro-duction of strangeness is also seen here, though the asymmetry of thecollision system makes it more difficult to understand the effect quantita-tively.The abundant strangeness, in the central rapidity region, is at the originof the effective production of multistrange antibaryons. The WA85 andWA94 collaborations [1–5, 7] explored the relative abundances at centralrapidity of the various strange baryons and antibaryons produced in S–Wand S–S reactions. The central-rapidity particle ratios have been obtainedat p⊥ ≥ 1GeV. The results for relative abundances are reported whenyields of particles of unequal masses are compared, both with p⊥ ≥ 1GeV and using as cutoff a fixed value of m⊥ ≥ 1.7GeV. The resultsat fixed p⊥ are shown in Fig. 9.2. In the left-hand panel, we see theannihilation and production results from e+–e− and p–p̄ reactions, in themiddle panel, we see the ISR–AFS p–p measurement of Ξ/Λ, which is afactor of five below the S–W and S–S result, even though the ISR energy√sNN was nearly four times higher than is available at the SPS. In S–Sand S–W interactions a clear enhancement of the Ξ+/Λ ratio is observed,which has been predicted as a signature of the QGP [215, 226].A more extreme picture of the enhancement is found when we comparethe yields of various hyperons at fixed m⊥, as would be done in a thermalmodel considering coalescence of quarks to give hadrons. This meansthat, on comparing, e.g., Ξ with Λ, we are looking at particles at differentp⊥. The experimental results reported by the WA85 collaboration, forS–W interactions at 200A GeV, are at ‘fixed m⊥’:Ξ−Λ + Σ0∣∣∣∣∣m⊥= 0.4± 0.04 , Ξ−Λ + Σ0∣∣∣∣m⊥= 0.19± 0.01. (9.2)We introduce the average singly strange hyperon yield Y(qqs), which atfixed m⊥ is the same for all components:Y = Λ = Σ0 = Σ+ = Σ−. (9.3)Thus, the actual antihadron ratio is indeed twice as large as that mea-sured:Ξ−Y∣∣∣∣∣m⊥ 0.8± 0.1 . (9.4)162 Particle productionS–S2101 1 0p–Ap–pS–AgS–Au2Λphdn/dy dn/dy dn/dy Fig. 9.3. The ratio of the rapidity density dn/dy for Λ/p̄, measured at centraly, as a function of the negative-hadron central-rapidity density dn/dy|h− . NA35collaboration [46, 47, 138].We now consider the quark content:ΞY∣∣∣∣m⊥=s̄s̄q̄s̄q̄q̄∣∣∣∣m⊥=s̄q̄∣∣∣∣m⊥ 0.8. (9.5)We see that, at the time of production of antihyperons, there has beencomparable availability of antiquarks at high momentum, ū = d̄ = 1.2s̄.This result is hard to explain other than in terms of QGP, for which,at the values of statistical parameters applicable here, near to chemicalequilibrium all three antiquark flavors are at nearly equal abundances.Since the abundance of light quarks comprises valence quarks, it is twiceas large, as can be seen in Fig. 10.3 on page 203, u = d 2.5s, consistentwith the half as large baryon ratio seen on the right-hand side of Eq. (9.2).These results involving the abundances of multistrange antibaryons havenot been explained in terms of hadron-cascade models.Consistent with this result is the observation of the NA35 collaboration[24, 25, 54] regarding the Λ/p̄ ratio. In Fig. 9.3, we show this ratio as afunction of the negative-hadron rapidity density dn/dy|h− at central y.The p–p and p–A reactions are at small values of dn/dy|h− , whereas9 Highlights of hadron production 16301234Λ ΛΞ− Ξ+ [S W]/ / [p W]/ − 〈 〉 − 〈 〉Y Y h − −Fig. 9.4. Ratios of abundances of particles (Λ, Λ, Ξ and Ξ) normalized withrespect to the abundance of h−: S–W results divided by p–W results at 200AGeV in the rapidity window 2.5 < y < 3 for 1.4 GeV/c < p⊥ < 3 GeV/c. WA85collaboration [41].the S–S, S–Ag, and S–Au reactions are accompanied by a relatively highdn/dy|h− . We observe that there is an increase in this ratio by nearly afactor five, and, even more significantly, the abundance of the heavier andstrange Λ is similar to if not greater than the abundance of p̄.The enhancement in production of strange hyperons and antihyperonscan be studied by comparing it directly with the yield seen in p–A inter-actions. For this purpose, one obtains specific yields of strange particles‘sp’, normalized with respect to the yield of negative hadrons h−. Thesecan be compared with such yields in proton-induced interactions, i.e., welook at the enhancement Eis in production of a strange particle i definedasEis ≡Y isp(S–A)Y isp(p–A), (9.6)where i = (Λ,Λ,Ξ,Ξ). The results are presented in Fig. 9.4 [41]. We seean enhancement of this specific yield in nuclear interactions, comparedwith p–A collisions, and this enhancement increases with increasing stran-geness content. The stronger enhancement in production of multistrangehadrons is expected for hadronization involving enhancements in yieldand density of strangeness compared with p–A interactions.164 Particle productionFig. 9.5. Yields Y per wounded nucleon 〈N〉 in Pb–Pb relative to p–Be collisionsfrom WA97 [41] (solid data points), and from NA57 for Ξ− and Ξ−[108] (opendata points).This presentation of enhancement in production of strange hadronsunderstates the magnitude of the effect since it is diluted by the overallenhancement of the yield of h−, which is also expected to arise due tothe enhancement of entropy in QGP, see section 7.1. Therefore a methodto extrapolate the yields from p–A interactions to A–A interactions wasdevised, by scaling with the number of participating (wounded) nucleons〈N〉 [41]. The Pb–Pb 158A GeV results of the WA97 collaboration areshown in Fig. 9.5, using the p–Be interaction results as reference. Solidpoints are the four centrality bins considered. We have presented someof these results in Fig. 1.6 on page 19. The absolute enhancement inproduction of strange particles compared to the p–Be extrapolated yieldsincreases strongly with the strangeness content.In order to understand whether there is a threshold for the enhance-ment to occur, the experiment NA57 has repeated the measurement ofthe experiment WA97, and has extended the reach by studying more pe-ripheral collisions. In the most peripheral fifth data bin the number ofparticipants is 60. At this time, only the Ξ− and Ξ− NA57 data for Pb–Pb are available, as is seen in Fig. 9.5. The rapid drop in enhancement ofthe production of Ξ− is most remarkable, and, if it is confirmed in, e.g.,Ω results, this can be seen as definitive evidence for a rather sudden onsetof the formation of QGP as a function of the size of the system.9 Highlights of hadron production 165We do not yet have a similar measurement for the onset of the enhance-ment as a function of energy. The specific production of strangeness perhadron is shown in Fig. 1.5 on page 17. However, there is a rapid changein hadron yield with energy, thus this result, as we have discussed in sec-tion 1.6, is not fully representative of the energy dependence of strangenessproduction. Another complication with the study of the energy depen-dence is the presence of the energy threshold for antibaryon production.Thus the energy dependence of the enhancement in production of strangeantibaryons has to be evaluated, not with reference to the p–A reactionsystem, but entirely within the A–A system. The low-energy measure-ment, presumably under pre-QGP conditions, can be used to establish abasis against which the production at higher energy can be studied. Insuch a measurement, we can hope and expect to see a sudden onset withenergy of the yield of multistrange (anti)baryons, if indeed a new state ofmatter is being created.Another important topic in the production of strange hadrons, is thesymmetry of particle and antiparticle spectra seen in Fig. 1.7 on page 20.As discussed, see table 8.2, the fitted inverse-slope parameters agree atthe level of 1%, when statistics is good enough: TΛ = 289 ± 3 MeV, tobe compared with TΛ = 287± 4 MeV [42]. There is no evidence for anydifference at small momenta, for which the annihilation reaction wouldbe most significant. Thus we can literally ‘see’, in Fig. 1.7, that theseparticles are escaping from the central fireball without further interactionwith hadronic gas.9.2 Hadron abundancesDespite the relative smallness of the S–S collision system, and the highestavailable fixed-target heavy-ion energy, the remarkable difference betweenΛ and Λ rapidity distributions, shown in Fig. 8.4, proves that neither isthe baryon-punch-through Bjørken reaction picture applicable, nor do wesee stopping of the valence quarks in the central region, as we discussedin section 8.3. How does the situation look for nonstrange hadrons? InFig. 9.6, we see the rapidity distribution of negative particles h−, whichcomprise π−, K−, and p̄, shown per collision event for the 5% most centralPb–Pb 158A-GeV collisions (full triangles), 3% most central S–S 200A-GeV collisions (stars, multiplied by factor 6.5), and the N–N interactions(full dots, multiplied by a factor 176) [43]. Detailed study of the chemicalcomposition of hadrons produced in these experiments, section 19.3, leadsto an estimate that the π− make up 92% of these particles, K− makeup 6.6%, and p̄ contribute 1.1% in Pb–Pb fixed target interactions at159A GeV.166 Particle production050100150200250 3− 2− 1− 0 1 2 3y π CMdN /dy h – πPb Pb central 5%−S S x 6.5 central 3%−N N x 176−Fig. 9.6. The pseudorapidity distribution of negative hadrons (sum of π−, K−and p̄) in Pb–Pb, scaled S–S, and N–N collisions [43].The abundance of h− produced in the Pb–Pb reactions can be obtainedfrom Fig. 9.6. On fitting the h− distribution to a Gaussian shape,dndη= n0e−(y−yCM)2/Γ,σ22= Γ, (9.7)one finds σ 1.4 (half width at half maximum). The integral of thedistribution gives the total multiplicity of all negative hadrons, after anadditional minor correction [43]: 〈h−〉 = 695± 30. Thus, in geometricallycentral Pb–Pb 158A-GeV interactions, the total hadron multiplicity isnearly 2400 per event, when we allow for positively charged and neutralhadrons. Since the maximum negative-hadron abundance at y = 0 isdh−/dy|max 200, we also have dnh/dy|max 680 in these events.To compare with Pb–Pb data, the S–S yields have been scaled up by afactor 6.5, this factor arising from the ratio of participants, which are inPb–Pb measured to be 352± 12, and in S–S 52± 3. A further factor 0.96is introduced by the NA49 collaboration to account for the difference incollision energy, which is somewhat higher in the S–S system. However,no correction for the fact that S–S is proton–neutron symmetric and Pb–Pb asymmetric was made. This correction may be significant since π− isthe carrier of the excess in valence d quarks, and there are 80 more d–d̄present than u–ū. Given that 〈h−〉 700, an enrichment of π− by valencequarks of projectile and target could reach a non-negligible level of 10%.9 Highlights of hadron production 167Were this ‘isospin’ correction introduced, it would perhaps be difficult todistinguish the Pb–Pb from S–S results in Fig. 9.6, and both are notablygreater than the N–N-based expectations: the observed yields have beenscaled by the factor 176 which is the ratio of Pb–Pb to N–N participants.We conclude that, both in Pb–Pb and in S–S collisions, we observe asimilar per participant central rapidity excess in hadron multiplicity, andthus also an excess in production of entropy; see chapter 7.The relevance of the 40% excess in production of h− in A–A collisionsat SPS energies is amplified by the opposite behavior seen at the lowerAGS energies, for which there is a 20% suppression compared with thescaled nucleon–nucleon results. One can qualitatively argue about theAGS result obtained at much lower energies as follows: whereas in N–N reactions all pions produced reach the detectors, pions produced in aseries of N–N collisions at the AGS are deposited in dense baryonic matter,where some can be absorbed. Moreover, a notable amount of the availablecollision energy is used to do work to compress colliding nuclear matter,and this energy is not available to produce pions, a point already notedin early work with relativistic heavy ions [233]. We will not pursue theimplications of the suppression of pion production at fixed-target collisionenergies below 15A GeV, for the study of the nuclear-matter equations ofstate.Since there is a change in pattern of behavior of pion production asa function of the collision energy, we consider in a more systematic waywhether this can be understood in terms of a general change in patternof behavior as a function of collision energy. Gaździcki [126] proposed toexplore this effect as a function of the Fermi-energy variable [121, 172, 174]F ≡ (√sNN − 2mN)3/4(√sNN)1/4, (9.8)where√sNN is the CM energy for a nucleon–nucleon pair and mN is thenucleon mass. There are several advantages in using F as an energyvariable. The measured mean multiplicity of pions in N–N interactionsis approximately proportional to F [127]. In the Landau model [172,174], both the entropy and the initial temperature of the matter (for√sNN 2mN) are also proportional to F .In Fig. 9.7, we see the average yield of all pions 〈π〉 per average par-ticipating nucleon 〈Np〉. The data is from [127], with the most recentresults presented in [254]. The lower straight line follows open diamonds,which are results from N-N interactions, whereas the upper line followsthe high-energy SPS (squares) and RHIC (open crosses) results. Thecross over from the one behavior to the other is seen at the lower range ofSPS energies, whereas the AGS results (triangles) indeed fall below the168 Particle productionFig. 9.7. The per-participant average yield of pions 〈π〉/〈Np〉 as a function ofthe Fermi-energy variable F , results from A–A and N–N interactions.N–N results. Both high-energy SPS and RHIC results are seen to followthe ‘high-entropy’ branch (see chapter 7), which differs clearly from thelow-entropy reactions at the AGS and in N–N collisions.Not only the yield of pions harbors a mystery. The shape of the SPSrapidity distribution for hadrons, see Fig. 9.6, is not fully understoodtoday, and we can not convincingly explain why there is so little differ-ence in shape among the three reactions shown. Generally, one wouldexpect the h− yield in S–S reactions to be ‘wider’ in rapidity than thatfor Pb–Pb collisions. Instead, what we see in Fig. 9.6 is that the rapid-ity shape of h− produced in N–N reactions is the same as that observedin Pb–Pb reactions, apart from an additional central-rapidity contribu-tion. However, we recall the qualitative study seen in Fig. 8.5, alongwith the observation that about half of all pions observed are actuallydecay products of hadronic resonances. The dilemma in understandingthis distribution is in fact one of the reasons that encourages us to focuson the study of particle spectra that are fully made of newly created mat-ter, such as Λ, see Fig. 8.4. These are clearly a more sensitive, and lessmodel-dependent, probe of novel physics occurring in the central rapidityregion.9 Highlights of hadron production 169π+p/pη/πo(K + K )/2Ko+−K /K+− K/ΛoK/Λo Λ/(p p)−p/T = 170 MeVpΛ/Λ/Λ Ξ /Λ− Ξ /Λ+ Ξ /Ξ−+10110−110−2Particle ratios(+) − (−)/(+) + (−)φ/(ρ + ω)d/pΩ/ΞT = 160 MeVFig. 9.8. Particle ratios (experimental dots) seen in A–A 200A-GeV reactionsfor various particle species (shown in horizontal order), compared with the pre-diction of the thermal model for two different freeze-out temperatures and quarkfugacity λq = 1.42 [82].Further below, we will return to consider, in Fig. 9.19, the rapidity dis-tribution of all charged hadrons observed at the nearly eight-fold higherRHIC energy. There is some spreading of the distribution, which neednot be entirely due to the rapidity-versus-pseudorapidity effect we dis-cussed in Fig. 8.3, originating most probably from the expected onset oftransparency and outflow of baryon number from the central region.The spectra of many identified hadronic particles have been measuredover sufficiently large ranges of rapidity and transverse mass to allow ex-trapolation to cover all of the relevant kinematic domain, and the totalparticle-production yield can be established. The total yields of parti-cles are not dependent on the deformation of the spectra arising from thecollective flow motion within the source. Consideration of relative abun-dance ratios eliminates biases from the various experimental set-ups, inparticular the event trigger bias cancels out.We show the compilation of CERN (200A-GeV) and AGS (14A-GeV)data in Figs. 9.8 and 9.9, in a procedure in which chemical-yield equi-librium of hadron abundances is assumed in a statistical model. In some170 Particle production-110-210-310-410-5p/pπ++K /−πK /− Ko10/π+K /K+−Λ/(p + n)Λ/Λπ/(p + n)Ξ /Λ−+−(K + K )φ/Particle ratios101d/(p + n)d/pT = 120 MeV T = 140 MeVFig. 9.9. Particle ratios seen in A–A 14A-GeV reactions (solid lines) for vari-ous particle species compared with the prediction of the thermal model for twodifferent freeze-out temperatures and quark fugacity λq = 4 [82].cases seen in these figures, the experimental errors are smaller than thesize of the experimental ‘dot’.We see that the particle ratios span typically several decades, yet thesystematic pattern of theoretical and experimental results coincides. Thus,the production of hadronic particles occurs, without doubt, near to thechemical equilibrium. We note that the two chemical freeze-out (particle-production) source temperatures used were T = 160 and 170 MeV for200A-GeV data and T = 120 and 140 MeV for the AGS 14A-GeV results.Other parameters used in Figs. 9.8 and 9.9 include, for 200-GeV data, thequark fugacity λq = 1.42 and, for 14-GeV data, λq = 4. Conservation ofstrangeness is imposed as a constraint, i.e., the number of strange quarksand antistrange quarks, in different hadrons, balances exactly. We willdiscuss how to perform this calculation in chapter 11.The first impression we have is that we see a rather good system-atic agreement in the behavior of the particle yields with this statisti-cal equilibrium-abundance model: almost all gross features of the data,for both sets, are well reproduced. Before we proceed, let us thereforepause to wonder if we should abandon the kinetic, i.e., collisional theory9 Highlights of hadron production 171of particle production, and focus solely on the experimental fact that theobserved hadronic multiplicities are the result of a preestablished statis-tical distribution, which works so well. In a sense, this finding confirmsa 15 year-old prediction that such a result can be naturally explained interms of a dynamic theory of a transient deconfined state hadronizing ina coalescence model [166]. Only a detailed study of the subtle deviationsin hadron yields from precise statistical equilibrium yields allows one tounderstand the hadronization mechanism [69], and therefore ultimatelyalso to explore the properties of the hadronizing QGP state.Indeed, looking closer at Figures 9.8 and 9.9, we see systematic devia-tions involving, in particular, (multi)strange particles: in the 200A-GeVdata the yield anomalies mostly involve strange antibaryons. The netdeviations in the total hadron yields are in fact greater – for example,were the chemical freeze-out condition set to reproduce, in the 200A-GeVcase, the ratio Λ/(p − p̄) exactly, we would have enhanced the disagree-ment in the ratio Λ/p̄ further. There is clear evidence, in these twofigures, that yields of strange particles require greater attention, beyondchemical-equilibrium mode, and we devote much of our effort in this bookto understanding the physics behind this phenomenon.Figures 9.8 and 9.9 demonstrate that the yields of strange antibaryonscompared to non-strange hadrons in general vary between 50%–150% ofthe chemical-equilibrium yield. This strangeness ‘fine structure’ yieldvariation is one of the reasons that the measurement of abundances ofrarely produced (strange)antibaryons is an excellent diagnostic tool inthe study of the properties of the dense hadronic matter, as we havediscussed above and in section 2.2. For this reason, we will discuss theproduction of strangeness and strange antibaryons as signature of QGPin considerably more detail in part VI.9.3 Measurement of the size of a dense-matter fireballAn important aspect of hadron-production studies is the measurementof two-particle Bose–Einstein correlations, which permits one to evaluatethe size of the space–time region. Also Fermi–Dirac correlations can beconsidered, but practical considerations have favored the measurement ofthe positive-boson interference. The two-particle intensity interferometryoriginates from the ambiguity in the path between the source and the de-tector for indistinguishable quantum particles. The two-particle intensitymethod was developed by Hanburry-Brown and Twiss as means of deter-mining the dimensions of distant astronomical objects, and is referred toin short hand as ‘HBT’; see [75] and references therein.HBT analysis is today a wide subject of specialization, which could fillthis book. We will briefly introduce the method of analysis and present172 Particle productionDetectorAB21SourceFig. 9.10. HBT interference: detectors A and B see quantum waves emittedfrom different source locations 1 and 2 differently.some recent results. In Fig. 9.10, we illustrate an uncorrelated sourceof particles π1 and π2, with momentum p1 and p2, emitted from somepoints 1 and 2, respectively, within an emission region (the shadowedregion in Fig. 9.10). Particles are counted at points A and B. When twoobserved quantum particles are identical, two indistinguishable historiesare possible, drawn with full and dashed lines, respectively, in Fig. 9.10.The intensity-interference pattern is observed as an enhancement inthe number of like-particle (boson) pairs originating from a single source,normalized with respect to a random sample of particles from two differentinteractions. This enhancement is studied by means of the two-particlecorrelation function,C2(pA, pB) =ρ(A,B)ρ(A) ∗ ρ(B) , (9.9)where the numerator represents events with particles registering in bothdetectors, and the denominator the number of pairs of uncorrelated par-ticles. If no correlation in particle intensity exists, the counts in bothdetectors are independent, which means that ρ(A,B)→ ρ(A) ∗ ρ(B).The correlation function is in principle dependent on the momenta ofboth particles observed. A first simple measure of the size of the sourceis obtained by considering C2 for similar transverse momenta of bothparticles. Summing over all other variables (‘projecting’), one finds that,as a function of the difference in transverse momenta /q⊥ = /pA,⊥ − /pB,⊥,C2(pA, pB) exhibits a clear correlation peak near q⊥ = 0.More generally, the shape of the enhancement as a function of the avail-able momentum variables contains information on the geometric source9 Highlights of hadron production 173parameters and thus both the size and the shape of the source, and, whenmodels are considered, also its dynamic evolution. The difference in mo-mentum for pions /q = /pA − /pB, can be decomposed using as a referencevector the sum of the pion momenta (the pair momentum) /p = /pA + /pB,as well as the collision axis of nuclei. The ‘longitudinal’ direction ‘l’ withcorresponding difference in momentum ql is also referred to as the ‘beam’direction, as is appropriate for fixed-target experiments.The ‘out’ component of the transverse-momentum-difference vector /q⊥(as before, transverse with respect to the beam axis) is the projectiononto the pion-momentum axis of /q⊥:qo =/q⊥ · /p⊥|/p⊥|. (9.10)The ‘side’ component qs is the remaining second component of /q⊥ and itsmagnitude isqs =√/q 2⊥ − q2o. (9.11)In the fits of the correlation functions, one likes to sharpen the definitionof the longitudinal (beam) component, considering that /p is in generalnot normal to the axis,q2l = q2z − q2o +poqo − pzqzp2o − p2z, (9.12)where qz is the magnitude of the difference in momenta for the pair alongthe longitudinal (beam) axis, and pz is the same component of the sumof momenta of the pair.The correlation C2 is fitted to the form comprising three source-shapeparameters Ri:C2 = D(1 + λe−(q2oR2o+q2sR2s+q2l R2l )). (9.13)Here∗, 0 ≤ λ ≤ 1, and, for the ideal HBT situation, λ = 1. Othergeometric parametrizations have been considered, and also further inter-ference terms between the geometric parameters in Eq. (9.13) have beenintroduced [272].The interpretational situation in heavy-ion collisions is complicatedby the finite lifetime, and the strong dynamic evolution of the particle-emitting source. Thus detailed interpretation of the observed correlationsbetween the particles produced requires development of model-dependentunderstanding, and a considerable amount of effort continues to be de-voted to the interpretation of the data. Generally, the following hypothe-ses are made regarding the source of particles:∗ In this section λ is not a fugacity.174 Particle production1. emission of particles is chaotic;2. correlated particles do not arise primarily from decay of resonances,though a strong resonance input is expected for pion correlations;3. particles do not interact subsequent to strong-interaction freeze-out –corrections for Coulomb effects are often applied; and4. kinematic correlations, e.g., conservation of energy–momentum, are ofno relevance.A considerable wealth of available experimental results leads to a fewconclusions of relevance to the understanding of the reaction mechanismsoperating in relativistic nuclear collisions.• No evidence is found for a major expansion of the hadronic fireball,which would be required, e.g., for a (long-lived) mixed (HG/QGP) in-termediate phase. The nuclear-collision geometry determines the sizeof the source for pions and kaons.• The size of the particle source is similar though a bit smaller for strange(kaons) than it is for non-strange particles (pions). Thus the conditionsfor production of these rather different particles are surprisingly similar.• There is a proportionality of the central hadron multiplicity yield tothe geometric volume of the source.• Evidence for the occurrence of transverse flow of the particle source isseen.These results are consistent with a reaction picture in which the (de-confined) fireball expands and then rather suddenly disintegrates andhadronizes. In such a process even the momentum freeze-out of final-state particles occurs at a relatively early stage of the evolution of stronglyinteracting matter.Figure 9.11, compiled by the STAR collaboration [18], shows the exper-imental results, i.e., parameters introduced in Eq. (9.13), as functions ofthe collision energy√sNN for pion-intensity interferometry. These resultsof diverse experimental groups (see the top of the figure) are only com-patible with a compact pion source being present at all reaction energies.While this is expected for the lower AGS energies, at which the fireballof nuclear matter is expected to have nuclear size, the actual slight de-crease in size seen at the RHIC with√sNN = 130 GeV and CERN–SPSwith√sNN = 17 GeV implies that, despite a rapid observed expansion ofthe fireball, there is even more sudden production of hadrons without anextended period of hadronization.As a function of collision energy, we see in Fig. 9.11 that the parameterλ ≤ 1 falls smoothly and rapidly from unity (the ideal expected value foran incoherent source) to about 0.5 at the RHIC; this decrease is attributedpartially to an increase in the fraction of pions arising from hadron reso-nances at higher energies. λ is also affected by several experiment-specific9 Highlights of hadron production 175Fig. 9.11. Sizes of fireballs of excited nuclear matter derived from pion-correlation analysis, compilation and RHIC result by the STAR collaboration[18].background effects and thus the physics of this behavior is not exploredin depth.The two parameters Rs and Ro correlate most directly to the geometryof the emitting source. This is illustrated in Fig. 9.12. The source of pionsis here presumed to be a shallow surface structure; the ‘out’ direction istoward the eye of the observer. If the source is longitudinally deformed,and the observer is at a more transverse location, the effect is amplified176 Particle productionFig. 9.12. Surface hadronization offers a possible explanation for why the HBT‘side’ radius Rs can be larger than the radius measured out toward the eye ofthe observer, Ro.by the geometry of the source. In contrast to this situation, simulationsinvolving a long-lived phase of pion matter with pions originating from thevolume of the fireball of dilute matter lead to Ro/Rs > 1. Experimentalresults, seen at the bottom of Fig. 9.11, deviate from this expectationmost clearly at RHIC energies.Ro/Rs 1 signals a rather short duration of pion production: one canshow that, when spatio-temporal correlations vanish, δ ≡√R2o −R2s isa measure of the life span of the emitting source, which, as can also beseen in Fig. 9.11, is not as large as the equilibrium hadronization modelsrequire. The source volume R2sRl is found to increase along with thetotal produced multiplicity of particles, as the centrality of the collisionis varied at RHIC energies.In addition to the overall behavior shown in Fig. 9.11, the STAR col-laboration notes that the size parameters decrease significantly with in-creasing m⊥ of particles. In that regard, the m⊥ dependence at the RHICis similar to, but stronger than that observed in central Pb–Pb collisionsat the CERN–SPS facility. This suggests that the emission of hadrons atthe RHIC is occurring from a more rapidly expanding surface source thanis the emission at the SPS.9.4 Production of transverse energySo far, we have been describing production of hadronic particles and par-ticle spectra. However, it is almost always simpler to measure the total‘flow’ of energy contained in the various particles, rather than abundances9 Highlights of hadron production 177of the many kinds of particles. The energy distribution can be used todetermine the extent to which energy from the longitudinal motion ofthe colliding nuclei participates in the nuclear interaction. This is mosteasily done by considering the energy emitted in the transverse direc-tion.Since fragments of projectile and target contaminate the longitudinalflow, it is the transverse to the beam axis component of energy which isconsidered as a suitable measure of the amount of CM energy, Eq. (5.2),made available in the reaction for production of particles. Therefore, onestudies the distribution of the energies of the particles, weighted by thesine of their angle θi with the beam axis (see Fig. 5.6), which is called thetransverse energy,E⊥ =∑iEi sin θi. (9.14)The resulting distribution of transverse energy produced, dE⊥/dθ, as afunction of the angle θ, can be converted into a distribution in pseudora-pidity η by employing Eq. (5.25).Experimentally, dE⊥/dη is determined with the help of a segmentedcalorimeter: particles entering a segment, covering a range of θ, deposittheir energy, which is determined by exploiting various mechanisms ofinteraction of particles in matter – hence the name ‘calorimeter’ whichderives from the name of a common heat measuring device. In fixed-targetexperiments, the laboratory-frame angle is not very large, see Fig. 5.7, andthus the calorimeter is typically located relatively far away, in front of thebeam axis.We show the transverse-energy-distribution data reported by the exper-iments WA98 [191] and NA49 [27], for 158A-GeV Pb–Pb (fixed-target)CERN experiments in Fig. 9.13. The key feature of this result is thatthere is a pronounced peak in the transverse energy distribution, slightlyforward of the rapidity value y = 2.9. The shift in pseudorapidity distribu-tion is a result of the definition of pseudorapidity; see Fig. 5.8 on page 88.This well-peaked distribution is consistent with the expectations basedon the observed negative-particle distribution shown in Fig. 9.6.The study of the transverse energy spectra has systematically beencarried out for many systems and collision impact parameters, by nu-merous groups. It is worth noting that, for relatively small projectiles,doubling the mass of the projectile increases the geometric number ofparticipants, in small-impact-parameter collisions, by approximately afactor 22/3 1.59, corresponding to the increase in area of the impactsurface on a large target. For example, an increase in E⊥ by about afactor 1.6 was observed for the S–Au reactions relative to the O–Au re-actions [201].178 Particle productionFig. 9.13. The transverse energy distribution as a function of pseudorapidityfor Pb–Pb fixed-target collisions at 158A GeV, with a central collision trigger.Combination of NA49 results [27] and WA98 results [191].The three main trends observed on the SPS energy scale are• the increase in the transverse energy with increasing mass of the col-liding system,• the increase in transverse energy with increasing energy, and• the increase in the transverse energy with the number of participatingnucleons, derived from the geometric centrality of the colliding nuclei;see section 5.2.The measurement of transverse energy at the RHIC has produced a ratherunexpected result, which we will discuss next.9.5 RHIC resultsWith the first physics run at the RHIC, in 2000, a new domain of collisionenergy has been reached. These results were obtained at√sNN = 130 GeVand have produced some surprises when one compares them with SPSresults. One is the discovery that the size of the fireball is barely differentfrom that at the SPS, section 9.3, the other addresses the suppression9 Highlights of hadron production 179pN100 200 300 400 [GeV]〉ch / dN〉 〈⊥ dE 〈00.20.40.60.811.21/2PHENIX = 130 GeV s1/2WA98 = 17.2 GeV sFig. 9.14. The transverse energy per charged particle (solid dots) as a functionof the number of participants at RHIC (PHENIX, 130 GeV) and at SPS (WA98,Pb–Pb fixed-target collisions at 158A GeV).of hard parton production noted in section 8.4. In Fig. 9.14 we see theconstancy of the transverse-energy yield per charged hadron produced.Once the number of participants exceeds 100, there is no difference fromthe results we presented in Fig. 9.13 for the most central collisions, whenresults are expressed per participant, both PHENIX and WA98 resultsare shown in Fig. 9.14. This agreement between two different energyregimes is natural should the hadron-production mechanism at the RHICand SPS be the same, as would be expected if a new state of matterwere formed, hadronizing in both cases under similar conditions. Thedifference between the SPS and the RHIC is in the hadron-multiplicityyield, which is related to the total entropy available to hadronize.There is more total transverse energy produced at the RHIC at cen-tral rapidity, than there is at the SPS, and this is seen on consideringthe pseudorapidity density of transverse energy per pair of participants,shown in Fig. 9.15. We note that the number of colliding pairs is half ofall participants, i.e., in case of p–p reaction, there are two participantsand one pair, and thus in this case the experimental data can be shownas measured.We thus conclude that the extra deposition of energy per unit of rapidityat the RHIC is converted into extra hadronic particles, which explainsthe remarkable result we saw in Fig. 9.14. It will be most interesting to180 Particle productionpN0 100 200 300 400) [GeV]p/(0.5 Nη /d ⊥dE0123 1/2PHENIX = 130 GeV s 1/2WA98 = 17.2 GeV sFig. 9.15. The transverse-energy pseudorapidity density per pair of participantsas a function of the number of participants, obtained at RHIC (PHENIX) andat SPS (WA98) [16].see whether this trend continues in the near future; that is, whether, atthe highest RHIC energy,√sNN = 200 GeV, the transverse energy perhadron produced will remain constant and only an increase in productionof hadronic particles will be observed.We now turn to the excitation function of hadron production (de-pendence on√sNN), and we include the first results at 200 GeV fromRHIC. The central-rapidity charged-hadron yield per pair of participantsis shown as a function of the collision energy√sNN (on a logarithmic scale)in Fig. 9.16. We see three experimental heavy-ion-multiplicity yields atthe RHIC: Au–Au results at√sNN = 56, 130, and 200 GeV (filled blackdata points) [50], CERN–SPS NA49 Pb–Pb results at√sNN = 17.2 =2× 8.6 GeV and 4.3 GeV (open circles), and the low-energy AGS results.This is compared with p–p̄ inelastic-collision results of UA5 (CERN) andCDF (Fermilab). The interpolation line for the p–p̄ results defines refer-ence yields used in Fig. 9.17. The importance of the RHIC results is clear,since without these one could argue that the top-energy SPS point is inagreement with the p–p̄ line, which, given RHIC results, we recognize tobe near a crossing point of two very different types of behavior. We recallthat some of these data are also shown in Fig. 9.7.The maximum-energy result from the RHIC (√sNN = 200 GeV) fallson a nearly straight line, which begins near the intercept√sNN = 1 GeV,9 Highlights of hadron production 181 [GeV]NN1/2s10 102103〉p N0.5/〈|<1η||η/dchdN012345s22.5 0.25 ln + 0.023 ln− sPHOBOS 200 GeVRHIC 130 GeV (combined)PHOBOS 56 GeVNA49 (SPS Pb Pb)−E866/E917 (AGS)) p−UA5 (p) p−CDF (pFig. 9.16. The charged-hadron multiplicity per pair of participants at centralrapidity as a function of√sNN, on a logarithmic display. Shown are results forthe 6% most central Au–Au collisions at the RHIC (PHOBOS) at√sNN = 56,130, and 200 GeV (filled black data), SPS Pb–Pb results (open circles), AGSresults (open diamonds), along with the high-energy p–p̄ data (open squares andtriangles), which are fitted to an empirical formula [50].and follows within error all other experimental heavy-ion points. Thus,to a remarkable accuracy, the central-rapidity multiplicity in heavy-ioncollisions is described by the empirical relationdNchdη= (1.6± 0.1)12Npart log(√sNNGeV). (9.15)The overall yield of particles produced is increasing faster than linearlywith log√sNN. However, the decrease in stopping just compensates forthe increase in rapidity density, distributing the increase in particle yieldover a wider range of (pseudo)rapidity. If this simple scaling, Eq. (9.15),were to continue to the LHC energy range, the rapidity density per par-ticipant would be ‘only’ 6–7 per pair of participants. For the 6% mostcentral events, corresponding to Npart = 365 in Pb–Pb interactions at theLHC, a relatively low dNch/dη 2500 charged particles yield per unit ofpseudorapidity is thus expected, based on this simple extrapolation.We now consider the charged particle yield at central rapidity, per par-ticipant pair. The dependence on the number of participants Npart, shownin Fig. 9.17, (from the PHENIX collaboration) agrees with the results ob-tained by the PHOBOS collaborations [48, 281]. In Fig. 9.17, we also seeto the left the UA5 p–p̄ (√s = 130 GeV) interpolated value. The periph-182 Particle productionpN0 50 100 150 200 250 300 350 400)p(0.5 N⁄/dηchdN00.511.522.533.544.5PHENIXUA5Fig. 9.17. Production of charged hadrons at central rapidity in Au–Au collisionsat√sNN = 130A GeV, obtained at various collision centralities, and presentedper pair of participants as a function of the number of participants (solid circles,the uncertainty is shaded) [281]. The solid square is the interpolated result fromUA5 p–p̄.eral yield in Au–Au interactions for 50–100 participants extrapolates wellto this point. A slight increase in the specific yield of charged hadrons isnoted for the most central collision. Overall, an increase of 50% in spe-cific yield of hadrons per participant is observed on comparing N–N withp–p̄ reactions. This very characteristic behavior allows discrimination be-tween models of hadron production. This is a topic in rapid evolutionwhich we will not further pursue at this time.The primary reason to move to the highest accessible nuclear collisionenergies is the desire to create a matter–antimatter-symmetric state ofdense matter akin to the conditions present in the early Universe. Abaryon-free QGP state should be accessible in LHC experiments. How-ever, at√s = 130A GeV at the RHIC, considerable matter–antimatterasymmetry is still observed. A measure of the baryon content is ob-tained by inspecting the central-rapidity antiproton-to-proton ratio p̄/p.In Fig. 9.18, devised by the STAR collaboration [19], to the right, we seethat, in the mid-rapidity region, this ratio is appreciably different fromunity. In view of the systematic behavior seen in the p–p interactions(open symbols), this is not unexpected, though there was some hope thata rapid onset of longitudinal expansion of matter could precipitate thecreation of the baryon-free region at the RHIC.The low-energy ‘AGS’ point is showing the production threshold, theobserved small ratio is not visible on the scale of the figure. We note9 Highlights of hadron production 18300.20.40.60.810 102p− /p ratioSTARSPSAGS√s−−NN (GeV)Fig. 9.18. The mid-rapidity antiproton-to-proton ratio (p̄/p) measured in cen-tral heavy-ion collisions (filled symbols) and p–p collisions (open symbols).a marked increase of the antimatter-to-matter ratio on going from SPSto RHIC energies. As expected, the conversion of kinetic energy avail-able in the interaction into hadron multiplicity, and here specifically an-tibaryons, is more effective in A–A reactions at RHIC energies than it isin p–p and even matter–antimatter p–p̄ interactions, as can be seen inFig. 9.16.The (pseudo)rapidity shape of the charged-particle distribution, seeFig. 9.19, as measured by the PHOBOS collaboration, displays a flattop, as could be expected in the punch-through case, see Fig. 5.2 onpage 74. The presence of a slight central dip could be in part dueto pseudorapidity being used as a variable, see section 8.1. For themost central 3% collisions one finds a charged-hadron multiplicity of〈h+ + h−〉 = 4100 ± 100 (statistical)± 400 (systematic), within the in-terval |η| ≤ 5.4 [230]. This is nearly a 3-fold increase compared with theSPS yield (for h− see Fig. 9.6), while the collision CM energy is 7.5-foldhigher. This implies that a high fraction of the collision energy is availablefor production of particles at RHIC energies. This fraction is less thanfor the SPS due to greater transparency at higher energy.The charged-hadron rapidity distributions, shown in Fig. 9.19, are seento fall within the rapidity gap between projectile and target rapidities. Wesee again the physics motivation to desire a rapidity separation, which isavailable at the RHIC collider: particles produced at central rapidity can-not be confounded with contributions from fragmentation of the projectileand target.184 Particle productionFig. 9.19. The rapidity distribution of charged hadrons in Au–Au collisions at√sNN = 130A GeV obtained at various collision centralities implying numbersof participants 〈N〉 =102, 216, and 354 for open circles, squares, and diamonds,respectively. Phobos collaboration [230].Turning briefly to strangeness we note that the STAR experiment re-ported the resultdNK+dy∣∣∣∣y=0= 35± 3.5, dNK−dy∣∣∣∣y=0= 30± 3.Allowing for strangeness in neutral kaons and hyperonsds̄dy∣∣∣∣y=0=dsdy∣∣∣∣y=0> 100.This very large abundance of strangeness has to be compared with theyield of non-strange hadrons:dπ+dy dπ−dy 235.The primary number of mesons is 175, considering resonance cascading,see section 7.3. Strangeness is thus reaching near symmetry with lightflavors. We will return to a full analysis of this interesting subject insection 19.4.It is quite clear, given the RHIC results, that the nucleus–nucleus colli-sions differ substantially from elementary hadronic collisions such as p–pand p–p̄ in their hadron-production efficiency. The conditions reached at9 Highlights of hadron production 185the RHIC are clearly more extreme than at SPS. The√sNN = 130 GeV,RHIC central-rapidity hadron yield is about 40% higher at 7.5-fold higherCM collision energy than that seen at SPS, as can be seen in Fig. 9.16.Inspecting Eq. (7.27) and noting that τ0, the initial thermal equilibrationtime, is most likely shorter at the RHIC than it is at the SPS, we con-clude that the initial entropy density is at least 40%, and probably morethan that, greater than that at the SPS. Along with the entropy density,given that the energy per particle seen at the RHIC is similar to resultsobserved at CERN, see Fig. 9.14, we expect a similar enhancement ofinitial energy density reached during the 2000 RHIC run at 130A GeV,compared with CERN top-energy results.Can we estimate more precisely the initial energy density produced atthe RHIC? In order to convert Eq. (7.27) into a relation for the initialenergy density, we can use E/S = ε/σ, the energy per unit entropy avail-able. For a relativistic (massless) gas with P = ε/3 at negligible baryondensity, the Gibbs–Duham relation, Eq. (10.30), implies that ε0 = 34T0σ0.Thus, Eq. (7.27) also means thatε0 =A1/3π(1.2 fm)2T0τ094dNch/(0.5A)dy. (9.16)This slightly unusual form Eq. (9.16), as stated here, shows that theinitial energy density, apart from an increase due to an enhancementin multiplicity density, also increases at RHIC compared to SPS due totwo likely changes in the initial condition: a shortening of the initialthermalization time τ0 for the more dense initial state formed at the higherRHIC energy, and also an associated increase of the initial temperatureT0 at which thermalization has occurred. Of course, it is very difficult topin down quantitatively these two contributions to the initial increase inenergy density. On the other hand, combined they could be as importantas the increase in the energy density due to an increase in the final stateparticle multiplicity.In order to arrive at an estimate for ε0, we take T0 = 300 MeV and τ0 =1 fm, and, for A = 350, we use the result seen in Fig. 9.17, dNch/(0.5A) =3.5. We obtain ε0 = 16 GeV fm−3. Lattice calculations seen in Fig. 15.3on page 300 suggest [159]: ε0 11T 40 , which yields for ε0 = 16 GeV fm−3a temperature T0 = 325 MeV. Given our ignorance of the value of τ0 andremaining uncertainties in lattice studies of QGP equations of state, weestimate that the energy density and temperature reached at the RHICare ε0 15–20 GeV fm−3 and T0 320–330 MeV ≥ 2Tc, where Tc, thecritical temperature for deconfinement, is estimated to be about 160 MeV[159]; section 15.5.The ‘Bjørken energy formula’ used often in such estimates arises fromEq. (9.16) by the substitution 3T0Nch/(0.5A) → 〈E〉. 〈E〉 is the mean186 Particle productionenergy per pair of participants,ε0 A1/3π(1.2 fm)2τ0d〈E⊥〉dη. (9.17)This expression leads to a lower energy-density estimate, since it does notaccount for the factor T0/Tf implicitly present in Eq. (9.16) and omittedin Eq. (9.17).IVHot hadronic matter10 Relativistic gas10.1 Relation of statistical and thermodynamic quantitiesThe first law of thermodynamics describes the change in energy dE of asystem in terms of a change in volume dV and entropy dS:dE(V, S)=−P dV + T dS, (10.1)T =(∂E∂S)V, P = −(∂E∂V)S. (10.2)The coefficients of the first law are the temperature T and the pressure P .Both can be introduced as the partial derivatives of the energy E(V, S). Eis a function of the extensive variables V and S, i.e., variables that increasewith the size of the system. Below, we include into this consideration thebaryon number, see Eq. (10.12), which is also an extensive variable.The free energy,F (V, T ) ≡ E − TS, (10.3)is the quantity in which, as indicated, the dependence on the entropy isreplaced by the dependence on temperature, an intensive variable thatdoes not change with the size of the system. Namely,dF (V, T ) = dE − T dS − S dT = −P dV − S dT, (10.4)and, as a consequence of the transformation Eq. (10.3), we obtain in anal-ogy to Eq. (10.2)S = −(∂F∂T)V, P = −(∂F∂V)T. (10.5)For an extensive system with F ∝ V , a very useful relation for the entropydensity σ follows from Eq. (10.5):187188 Hot hadronic matterσ ≡ SV=∂P∂T. (10.6)Since the free energy F depends on T = 1/β, we relate it to the sta-tistical functions introduced in chapter 4. To establish the connection ina more quantitative manner, we obtain the energy E in terms of the freeenergy F . We substitute S into Eq. (10.4), in order to arrive at a relationfor E(V, T ):∗E = F (V, T ) + TS(V, T ),= F (V, T )− T ∂∂TF (V, T ) ≡ − ∂∂(1/T )(−FT). (10.7)This relation has the same form as Eq. (4.13), and we deduce thatF (V, T ) = −β−1[lnZ(V, β) + f ]. (10.8)The integration constant f could be a function of V . However, by defini-tion of P , Eq. (10.5), and using Eq. (10.8),P =∂∂V(β−1 lnZ)+ T∂f∂V. (10.9)In the definition of the partition function Eq. (4.14), the individual ener-gies Ei depend on the volume V . We obtainP =−∑i(∂Ei/∂V )e−βEi∑i e−βEi + T∂f∂V= −〈∂Ei∂V〉+ T∂f∂V. (10.10)Only if ∂f/∂V vanishes does the proper relation between work and pres-sure arise: the work done on the system, when the volume V is decreasedby dV (dV is negative), is equivalent to the mean value of the change ofall the energy levels brought about by a change of the volume. A con-stant f is a physically irrelevant ambiguity in the relationship Eq. (10.8)between the free energy F and the canonical partition function Z, andcan be discarded. Thus, Eqs. (10.8) and (10.9) readF (V, T ) = −β−1 lnZ(V, β), βP = ∂ lnZ∂V, β−1 = T. (10.11)This well-known equation establishes the bridge between the thermal(T, F, and P ) and statistical (β and lnZ) quantities. The volume Vis present in both formulations, but, in fact, since lnZ and F are exten-sive in V for infinite volumes, V in general disappears from many furtherconsiderations.∗ Clearly E(V, S) is not the same function as E(V, T ), which is indicated by statingthe variables on which E depends, rather than introducing a new symbol.10 Relativistic gas 189We now allow for the presence of a conserved (baryon) number b,i.e., we consider E(V, S, b). This necessitates the introduction of the(baryo)chemical potential µ. µ is the incremental energy cost requiredto change the baryon number at fixed pressure and entropy:dE(V, S, b) = −P dV + T dS + µdb, (10.12)P = −(∂E∂V)S,b, T =(∂E∂S)V,b, µ =(∂E∂b)V,S. (10.13)The coefficients P, T, and µ are, as before, fixed in terms of the partialdifferentials of E(V, S, b) with respect to its three variables.The familiar generalization of the free energy Eq. (10.4), often called thethermodynamic potential, F(V, T, µ), is defined by the transformationF(V, T, µ) ≡ E(V, S, b)− ST − µb. (10.14)On evaluating the differentials as in Eq. (10.4), we indeed see that, as sug-gested on the left-hand side of Eq. (10.14), the thermodynamic potentialis a function of V, T , and µ,dF = −P dV − S dT − b dµ, (10.15)whereP = −(∂F∂V)T,µ, S = −(∂F∂T)V,µ, b = −(∂F∂µ)V,T. (10.16)A series of arguments that has allowed us to establish Eq. (10.11) fixes arelation between F(V, T, µ) and the grand partition function Z(V, T, λ):F(V, T, µ) = −β−1 lnZ(V, β, λ), β = 1/T, λ = eµ/T . (10.17)The thermodynamic pair of variables (T, µ) is often used for describingthe properties of F , instead of the grand-canonical statistical quantities(β and λ). To do this, it is quite important that appropriate attention bepaid to the simple relationµ = µ(λ, β) = β−1 lnλ. (10.18)Consider, for example, the expression for the energy. WithZ(V, β, λ) = Z̃(V, T, µ), (10.19)we obtainE = − ddβlnZ(V, β, λ), (10.20)= T 2ddTln Z̃(V, T, µ) + ddµln Z̃(V, T, µ) dµ(λ, β)dβ∣∣∣∣β=T−1. (10.21)190 Hot hadronic matterThe second form is clearly much different from the simple statisticalrelation Eq. (10.20). However, given Eq. (10.18), we havedµ(λ, β)dβ∣∣∣∣β=T−1= −Tµ, (10.22)and thus the last term in Eq. (10.21) is µdF/dµ. We hence obtain a formof the the important Gibbs–Duham relation, see Eq. (10.26) below,E(V, T, µ) = F(V, T, µ) + TS(V, T, µ) + µb(V, T, µ), (10.23)where the baryon number and entropy areF(V, T, µ) = −P (T, µ)V, (10.24a)b = − ddµF(V, T, µ) = λ ddλlnZ(V, β, λ), (10.24b)S = − ddTF(V, T, µ) = ddTT ln Z̃(V, T, µ). (10.24c)The expression for the entropy, Eq. (10.24c), takes a much more com-plex form in terms of statistical variables. Namely, Eq. (10.23) impliesthatS =1T(E −F − µb) = lnZ − β ∂ lnZ∂β− (lnλ)λ ∂ lnZ∂λ. (10.25)In an extensive system, we can greatly simplify Eq. (10.23). We replaceF by −PV , Eq. (10.16), and obtain the usual form of the Gibbs–Duhamrelation:P = Tσ + µν − ε, σ = SV, ν =bV, ε =EV. (10.26)For completeness of the discussion, we mention now two more quan-tities, the enthalpy H(P, S, b) and the Gibbs free energy G(P, T, b). Toobtain these two quantities with a new mix of variables, we continue theprocess of replacement of variables. We recall that, at first, we movedfrom E(V, S) → F (V, T ) and subsequently from E(V, S, b) → F(V, T, µ),i.e., we replaced the extensive variables by the intensive variables. Theone extensive variable left is the volume itself. Since we address in thisbook an isolated system (a fireball) that can expand its volume with en-tropy and baryon number remaining nearly constant, elimination of V infavor of P would seem a logical step and indeed the statistical (partition-function) analog to G(P, T, b) is the recently proposed generalization ofthe (grand) canonical partition function to the pressure partition functionΠ(P, β, b) [142]. We will not pursue this interesting subject further in thisbook.10 Relativistic gas 191However, we note that, for a given pressure, entropy, and baryon num-ber, it should be convenient to introduce the enthalpy,H(P, S, b) ≡ E(V, S, b) + PV . (10.27)The volume occupied by the system is obtained as the change of H withrespect to pressure at fixed S and b:dH = V dP + T dS + µdb. (10.28)In thermal physics, a quantity often considered in the study of a freelyexpanding isolated system is the specific enthalpy per particle, the so-called heat function h,Hb≡ h = εν+Pν. (10.29)The Gibbs free energy G is introduced to facilitate consideration of evo-lution, not at a constant entropy, but in a ‘heat bath’, i.e., at a giventemperature, though at a fixed baryon number (not in a ‘baryon bath’):G(P, T, b) ≡ E + PV − TS = µ(P, T, b) b. (10.30)Both the enthalpy H(P, S, b) and the Gibbs free energy G(P, T, b) havenot yet been used much in the study of heavy-ion-fireball dynamics.10.2 Statistical ensembles and fireballs of hadronic matterWe extend the discussion of physical ensembles introduced in chapter 4.The concept of an ensemble consisting of weakly coupled physical systems,M = {Mi, i = 1, . . ., N}, was introduced by Gibbs and Boltzmann.It helped to establish a conceptual foundation of statistical physics. Alarge number, N → ∞, of such systems is normally considered. Theotherwise negligible interactions between individual systems Mi are suchthat both energy and (conserved) quantum numbers (such as, e.g., thebaryon number) can be exchanged between the systems. This establishesa ‘bath’ of energy and baryon number, in which each individual systemis immersed, and with which it can equilibrate its properties.When we examine the microscopic properties in the ensemble, such asenergies of individual members, we speak of a micro-canonical ensem-ble. Furthermore, we distinguish between the canonical ensemble andthe grand-canonical ensemble: in both cases, we have adopted a statisti-cal distribution in energy. However, in the canonical ensemble, we stilltreat discrete quantum numbers (particle number, baryon number, etc.)microscopically, whereas in the grand-canonical ensemble, we have alsoadapted the statistical-ensemble distribution for the discrete properties192 Hot hadronic mattersuch as baryon number. Of course, when we have more than one con-served discrete quantum number, any of these properties can be treatedin ‘canonical’ or ‘grand-canonical’ way.A colloquial way to explain the difference between the ensembles isto say that, in the micro-canonical ensemble, we consider each individualsystemMi as being decoupled from the others. In the canonical ensemble,we allow only for coupling of energy, and in the grand-canonical ensemble,we allow for exchange of energy and quantum number (particle number).Practically, we maintain the picture of an ensemble of many weakly cou-pled systems M in place, but, for describing physical properties withinmicro-canonical, canonical, or grand-canonical ensembles, we use differentphysical variables.Said differently, in a theoretical description, it is our choice how wecharacterize the properties of the system as long as there is a precisemathematical transformation we can use to make a transition betweenthe different descriptions of the same physical situation, and the descrip-tion of choice is what is most convenient. There is an exception to this‘convenience principle’: the color-confining nature of strong interactionsimposes color neutrality on all ‘drops’ of QGP we consider, thus, in prin-ciple, we may not use a color-grand-canonical ensemble, and, if it is used,the question to consider is that of whether results obtained in this waymake good physical sense.What is, in our context, the individual ensemble element Mi? Canwe view it as a single hadron, or do we have to take the entire drop ofhighly excited hadronic matter formed in the nuclear collision as being theelement in the Gibbs ensemble? In chapter 4, we have wondered if a singleparticle can be seen as the element of the ensemble. This is motivatedby the fact that, in our physical environment, the number of particles isnot fixed, and their variable number is an expression of the sharing of thetotal energy and baryon number (or other conserved number). In supportof this point of view, we will next show in section 10.3 that, allowingfor a change in numbers of particles, the state of maximum entropy atfixed energy and baryon number is the conventional statistical-equilibriumdistribution.We know that there are physical processes of particle production thatallow conversion of energy into particles, such that their yields reach(chemical) equilibrium. The well-known Boltzmann collision dynamicsassures that the momentum distributions are equilibrated in (binary) col-lisions. Microscopic processes of particle production and interaction canestablish, in a particle ensemble Mi, a distribution that is normally asso-ciated with ensemble elements consisting of larger drops of matter.Each high-energy heavy-ion reaction forms a many-body system, a fire-ball, which evolves into a final state with thousands of particles. The10 Relativistic gas 193study of the average rather than individual microscopic properties of sucha large system makes sense, if the distribution of individual properties ofthe sub-components has a ‘peaked’ shape. What this means is that mostindividual objects considered should be found near to their common av-erage, just a few may be far from it. On intuitive grounds, it seemsthat otherwise many-body systems equilibrate exceedingly slowly, if atall. Since there are many examples of dynamic systems that do not sat-isfy this criterion, we see that in general there is a priori no guaranteethat strongly interacting confined matter will ever equilibrate. However,experimental results suggest that strongly interacting particle systems ofpractically any size approach statistical equilibrium very rapidly. Whythis is the case remains today an open issue, see section 5.6.In the following, we develop further the physical properties of idealrelativistic gases introduced in section 4.4. We assemble useful formulas,including in the discussion chemically nonequilibrated gases, which havebeen treated only sparingly before.10.3 The ideal gas revisitedThe additivity of different gas fractions (i.e., flavors) ‘f’ originates from theadditive property of the logarithm of the partition functions Eq. (4.19):Z =∏fZf , lnZ =∑flnZf . (10.31)For an ideal Fermi gas, such as a quark gas in the deconfined phase, wehave for each flavor, as seen in Eqs. (4.38) and (4.39),lnZF = gFV∫d3p(2π)3[ln(1 + γλe−βε) + ln(1 + γλ−1e−βε)], (10.32)where the degeneracy factor is, e.g., gF = gsgc and comprises gs = 2 forspin-12 degeneracy and gc = 3 for color. For bosons with degeneracy gB, inprinciple, we must allow for the possibility of condensation (macroscopicoccupancy) in the lowest energy state ε0:lnZB = −gBV∫d3p(2π)3[ln(1− γλe−βε) + ln(1− γλ−1e−βε)]− gB[ln(1− γλe−βε0) + ln(1− γλ−1e−βε0)]. (10.33)We will not address further in this book the condensation phenomena andwill not pursue further the last term in Eq. (10.33).Differentiating with respect to the energy of the particle, see Eq. (4.41),we obtain the single-particle distribution functions. For the fermions and194 Hot hadronic matterantifermions seen in Eq. (4.42), respectively, we havefF(ε, µ) =1γ−1eβ(ε−µ) + 1, (10.34a)f̄F(ε, µ) =1γ−1eβ(ε+µ) + 1, (10.34b)and similarly for bosons and antibosons:fB(ε, µ) =1γ−1eβ(ε−µ) − 1, (10.35a)f̄B(ε, µ) =1γ−1eβ(ε+µ) − 1. (10.35b)We will also use the short-hand notationf±F,B = fF,B ± f̄F,B, (10.36)since these combinations occur in evaluations of statistical properties ofgases.The particle densities areρF ≡NFV=1VλddλlnZF = gF∫d3p(2π)3f−F , (10.37a)ρB ≡NBV=1VλddλlnZB = gB∫d3p(2π)3f−B . (10.37b)These distributions determine the local equilibrium particle densities, forexample, the local density of quarks and antiquarks given by the integralof the Fermi distribution, Eqs. (10.34a) and (10.34b):nq =∫d3p(2π)311 + γ−1i λ−1i eε(p)/T→ γiλi∫d3p(2π)3eε(p)/T , (10.38a)nq̄ =∫d3p(2π)311 + γ−1i λieε(p)/T→ γiλ−1i∫d3p(2π)3eε(p)/T . (10.38b)The Boltzmann limit, which is applicable when the phase-space cells havesmall overall occupancy, is also indicated in Eqs. (10.38a) and (10.38b).In this limit, the chemical-abundance factors enter as coefficients of thedistributions. We note that, while the chemical potential enhances theabundance of particles, it suppresses the abundance of antiparticles.In order to obtain other statistical properties, such as, e.g., the en-ergy content of the system, one can also apply the rule that the occu-pation functions Eqs. (10.34a)–(10.35b) and Eq. (4.46) can be folded with10 Relativistic gas 195the quantity of interest. To obtain the energy density, we fold with thesingle-particle energy ε the sum of particle and antiparticle spectra. Thecorrectness of this prescription is seen on evaluating the derivative of lnZwith respect to β, Eq. (10.20):εB = gB∫d3p(2π)3εf+B , (10.39a)εF = gF∫d3p(2π)3εf+F , (10.39b)εg = gg∫d3p(2π)3εfg. (10.39c)The gluon distribution fg is seen in Eq. (4.46). The total energy densityis the sum of all contributing terms:ε =∑iεi. (10.40)10.4 The relativistic phase-space integralTo evaluate the properties of ideal relativistic gases, we need to evalu-ate the relativistic momentum integral, which appears in all phase-spaceintegrals in a similar form. To do this we consider the definition of theBessel function Kν ,Kν(z) =√π(z/2)νΓ(ν + 12)∫ ∞1e−zt(t2 − 1)ν− 12 dt, - ν > −12 , (10.41)valid for |arg z| < π/2. We used before the case ν = 1, Eq. (8.7), whicharises on substituting in Eq. (10.41) t → cosh t. The connection to theclass of integrals which we now require is obtained by recognizing thatz = βm and substituting into Eq. (10.41):t →√p2 +m2/m. (10.42)With ε =√p2 +m2, we obtainKν(βm) =√πΓ(ν + 12)(β2m)ν ∫ ∞0p2νεe−βε dp. (10.43)On integrating by parts with the relation∂∂pe−βε = −β pεe−βε,196 Hot hadronic matterwe obtain (ν > 12)Kν(βm) =√πΓ(ν − 1/2)1m(β2m)ν−1 ∫ ∞0p2ν−2e−βε dp. (10.44)We recall thatΓ(12) =√π; Γ(32) =√π/2; Γ(52) =32Γ(32); . . ..Two interesting limits arise from the well known series expansion of theBessel function Eq. (10.41).• The non-relativistic limit in which we use p/m as the small parameter:Kν(z)→√π2ze−z(1 +4ν2 − 18z+(4ν2− 1)(4ν2− 9)2!(8z)2+(4ν2− 1)(4ν2− 9)(4ν2− 25)3!(8z)3+ · · ·). (10.45)We note that this expansion is rather slowly convergent. The specialcase of interest to us isK1(z)K2(z)= 1− 321z+1581z2− 1581z3+1351281z4+O(z−5). (10.46)• The relativistic limit, in which the mass is negligible relative to thetypical energies, and thus effectively m 0. For the relevant two cases,we have, for z → 0,K1(z)=1z+[ln(z2)+ γE]z2+[ln(z2)+ γE −54]z316+ · · ·,K2(z)=2z2− 12−[ln(z2)+ γE −34]z28−[ln(z2)+ γE −1712]z496+ · · ·,leading toK1(z)K2(z)=z2+[ln(z2)+ γE]z34+ · · ·. (10.47)We recall thatγE = limn→∞n∑k=11k− lnn = 0.577 215 664 9 . . .is the Euler constant.10 Relativistic gas 197Fig. 10.1. The relativistic distribution function W (x) = x2K2.With this preparation, we are now in a position to study properties ofrelativistic gases. Let us first look at the classical limit, the first, ‘classical’term in the expansion of the partition function Eq. (4.51). For n = 1, weobtain, from Eq. (4.53),lnZcl = Z(1) =∑fγf(λf + λ−1f )Z(1)f , (10.48)withZ(1)f = gfV∫d3p(2π)3e−βε(p) ≡ gfβ−3V2π2W (βmf). (10.49)The sum, in Eq. (10.48), includes all Fermi and Bose particles.We encounter in Eq. (10.49) the function Eq. (10.44), with ν = 2,W (βm) ≡ β3∫e−βεp2 dp = (βm)2K2(βm), (10.50a)→ 2, for m → 0, (10.50b)→√πm32T 3e−m/T , for m T, (10.50c)where, in the last limit, we exploited the large-argument limit shown inEq. (10.45). As can be seen from Fig. 10.1, the change between the twoasymptotic limits occurs near βm = 1.In the classical (Boltzmann) limit indicated by superscript ‘cl’ the num-ber of particles of each species, Eq. (4.56), is given by198 Hot hadronic matterN |cl = λ ∂∂λlnZcl = λZ(1) = gλβ−3V2π2W (βm), (10.51)where we have combined the factors γ±1λ → λ, since, when only one par-ticle species is considered, one fugacity suffices. The most useful and oftenquoted property of a relativistic gas is the average energy per particle:EN∣∣∣∣cl = −(∂/∂β) lnZclλ(∂/∂λ)∂lnZcl = 3T +mK1(βm)K2(βm) . (10.52)The fugacity coefficients λ for the particles cancel out. To obtain Eq. (10.52),we exploited the property of the function W = x2K2,ddxW (x) = −x2K1(x), (10.53)arising from the recursion relation of the K-functions,ddxKν(x) = −Kν−1(x)−νxKν(x), (10.54a)written in the formddx(xνKν(x)) = −xνKν−1(x). (10.54b)• In the relativistic limit βm → 0, we can use Eq. (10.47) and obtainEN∣∣∣∣clm=0= 3T. (10.55)Equation (10.55) can be improved; see Eq. (10.68). However, further re-finement in the limit m/T → 0 requires that quantum statistics be con-sidered.• In the non-relativistic limit βm 1, the ratio appearing in Eq. (10.52)is as given in Eq. (10.46), and we obtainEN∣∣∣∣clnr= m+32T(1 +54Tm− 54T 2m2+4564T 3m3· · ·),mT> 1. (10.56)Note that, to obtain the correct first 3T/2 term in the non-relativisticlimit, the next-to-leading term in Eq. (10.46) needs to be considered. Theslow convergence of the series, Eq. (10.56), is also to be remembered,i.e., the non-relativistic limit requires a truly a non-relativistic m Tcondition. For m T , the relativistic limit offers a better approximation.We continue with a more thorough discussion of the energy per baryonin the next subsection, addressing there, in particular, the differencesarising for bosons and fermions.10 Relativistic gas 19910.5 Quark and gluon quantum gasesIn the deconfined QGP phase, we have to consider the quantum natureof the effectively massless, relativistic quark and gluon gases. Many ofthe results for quantum gases that we require arise in terms of a seriesexpansion of which the Boltzmann approximation is the first term. Weare, in particular, interested in the properties of the equation of state, i.e.,the relation between the energy density ε, Eq. (10.40), and the pressureP , Eq. (10.11).Integrating by parts Eqs. (10.32) and (10.33),±∫d3p(2π)3ln(1± γλe−βε) = β3∫d3p(2π)3|/p | ∂ε∂|/p | fF,B (10.57)=β3∫d3p(2π)3/p2εfF,B,where the factor 13 arises from the p2 dp momentum integral. We haveused the (relativistic-dispersion) relation Eq. (4.31) in the last equality.We obtain for the pressure Eq. (10.11), noting Eqs. (10.32), (10.33), and(10.36), and using Eq. (4.31) to eliminate the momentum,3P = gF∫d3p(2π)3(ε−m2ε)f+F + gB∫d3p(2π)3(ε−m2ε)f+B ≤ ε. (10.58)Since the particle-occupation probabilities f+B,F are always positive, theterms proportional to m2 in Eq. (10.58) always reduce the pressure. Forthis reason the maximum absolute value of the ideal-gas pressure, forgiven thermal parameters, is subject to the relativistic boundε− 3P = gFm∫d3p(2π)3(mε)f+F + gBm∫d3p(2π)3(mε)f+B ≥ 0. (10.59)The right-hand side of Eq. (10.59) is cast into the form which is naturalconsidering the trace of the energy–momentum tensor of quantum fields.It can be evaluated using Eq. (10.43), when the series expansion of quan-tum distributions exists. The leading (Boltzmann) term isε− 3P = gT42π2x3K1(x), x = m/T. (10.60)For high temperatures relative to (vanishingly small) mass, we find therelativistic equation of state,3P → ε, for β m → 0, (10.61)corresponding to the maximum mobility of particles. Massive particlesmove slowly relative to the velocity of light and are far away from this200 Hot hadronic matterlimit. In fact, the pressure in the normal world around us is vanishinglysmall, seen on the scale of energy density comprising the rest mass. Thisis expressed in the power of 30 arising in Eq. (1.2), which separates thepressure on Earth from that in the QGP.We turn now to consider the energy per particle in relativistic quan-tum gases in more detail. We employ the series expansion appearing inEq. (4.51), and obtain, by expanding Eqs. (10.32) and (10.33) for eachparticle species,lnZ = β−3V2π2∞∑n=1gnλnn4(nβm)2K2(nβm). (10.62)We have combined the factor (−)n+1 for fermions with the degeneracyg to form the factor gn. This expansion Eq. (10.62) can not be used ifthe condition m − µ < 0 arises. This happens, in particular, for mass-less quarks at finite baryon density. We will be able to deal with thisinteresting case exactly for m → 0; see Eq. (10.74). In the HG phase,for a very narrow parameter range, allowing in particular condensationof kaons, the expression Eq. (10.62) is also not valid. Apart from theseexceptions, Eq. (10.62) can be used as the basis for the evaluation of theproperties of hot, strongly interacting matter.Using the series expansion Eq. (10.62), the quantum generalization ofthe classical particle number, Eq. (10.51), isN = λ∂∂λlnZ = β−3V2π2∞∑n=1gnλnn3(nβm)2K2(nβm). (10.63)As noted, the masses and fugacities are such that m− µ > 0, so that theseries expansion exists. The relativistic limit m → 0 is nowNV∣∣∣∣Bm=0=gT 3π2ζ(3), (10.64)NV∣∣∣∣Fm=0=gT 3π2η(3). (10.65)We have introduced the Riemann zeta functionζ(k) =∞∑n=11nk. (10.66a)We note thatζ(2) =π26, ζ(3) 1.202, ζ(4) = π490. (10.66b)10 Relativistic gas 201For a Fermi occupation function, the signs of the terms in the sums inEq. (10.63) are alternating, which leads to the eta functionη(k) =∞∑n=1(−1)n−1 1nk= (1− 21−k)ζ(k), (10.67a)and thusη(3) =34ζ(3) = 0.901 5, η(4) =78ζ(4) =7720π4. (10.67b)The generalization of the energy per particle, Eq. (10.52), to quantumstatistics yieldsEN= 3T∞∑n=1gnλnn4((nx)2K2(nx) +13(nx)3K1(nx))∞∑n=1gnλnn3(nx)2K2(nx), (10.68)where x = mβ. For the non-relativistic limit, the Boltzmann approx-imation Eq. (10.56) is quite appropriate, resulting in Eq. (10.56). Theultra-relativistic limit with m → 0 yieldsEN∣∣∣∣Bm=0= 3Tζ(4)ζ(3)= 2.70T, (10.69)EN∣∣∣∣Fm=0= 3Tη(4)η(3)= 3.15T. (10.70)The factor 78 seen in Eq. (10.67b) enters Eq. (10.70) and is the source ofthe reduction of the number of fermionic degrees of freedom in a Fermigas compared with that in a Bose gas. However, we have to allow for thepresence both of quarks and of antiquarks; thus the radiation term in thequark gas is actually 74 times as large as that in the gluon gas, apart fromthe other statistical flavor and color multiplication factors.In Fig. 10.2, we show the variation of the energy per particle in units ofm, as a function of T/m. The solid line depicts the Boltzmann limit; thelong-dashed line, fermions with λ = 1; and the short-dashed line, bosons.Asymptotic conditions are indicated by dotted lines corresponding to thelimits Eqs. (10.55), (10.69), and (10.70). For finite mass, we see at T → 0the non-relativistic linear rise common to all three cases, Eq. (10.56). Thetransition from non-relativistic to relativistic behavior occurs within theentire temperature domain shown in Fig. 10.2.Our discussion of the properties of gases of relativistic particles cannotbe complete without a review of the particularly interesting case of a free202 Hot hadronic matterFig. 10.2. The energy per particle in units of m, as a function of T/m. Solidline: Boltzmann limit; long-dashed line: fermions; short-dashed lines: bosons.Asymptotic conditions are indicated by dotted lines. Quantum gases are evalu-ated with fugacities γ, λ = 1.gas of massless quarks. In particular, at finite baryon density, for which wecannot expand the Fermi distribution function in the presence of strongquantum degeneracy, this analytically soluble case offers the only practicalmethod for studying the behavior of an ideal gas of quarks. To see this,let us assume that we are at a finite positive chemical potential, whichmeans that there is a net number of quarks present. Fermi distributionsas functions of ε/T both for particles, Eq. (10.34a), and for antiparticles(dashed), Eq. (10.34b), are shown in Fig. 10.3, for a typical situation ofµ/T = 0.5 (that is, λ = 1.65).Let us restate the mathematical problem more precisely. The grandpartition function of the Fermi system, Eq. (10.32), can be written, usingEq. (10.57), in the form3TVlnZF = gF∫d3p(2π)3/p 2ε(1eβ(ε−µ) + 1+1eβ(ε+µ) + 1). (10.71)Our usual series expansion would work for the momentum range such thatε > µ. For the massless case, we would have to split the integral into two,and expand in a slightly different fashion. On approaching this problemin this way, one actually discovers that, for m = 0, the partition functioncan be exactly integrated.10 Relativistic gas 203Fig. 10.3. A comparison of particle (solid) and antiparticle (dashed) Fermi dis-tribution functions, as functions of ε/T for µ/T = 0.5.We substitute the arguments of fF and f̄F with x = β(ε± µ):3TVlnZF=gF2π2T 4(∫ ∞β(m−µ)dx[(x+ µ/T )2 − (m/T )2]3/2ex + 1+∫ ∞β(m+µ)dx[(x− (µ/T ))2 − (m/T )2]3/2ex + 1). (10.72)A systematic expansion in m/T was carried out in [113]. We consider theleading term for m = 0. For what follows it is important to note that, inEq. (10.72), one of the factors under the integral is, for m → 0,[(x± µ/T )2 − (m/T )2]3/2 → (|x± βµ|)3.The range of the integrals is now split to be from ±βµ → 0 and from0→ ∞. The final-range integrals can be recombined to give an elementarypolynomial integral,∫ 0−βµdx|x+ βµ|31 + ex−∫ βµ0dx(x− βµ)31 + ex=∫ βµ0dx(βµ− x)31 + e−x+∫ βµ0dx(βµ− x)31 + ex,=∫ βµ0dx (βµ− x)3 = (βµ)44, (10.73)204 Hot hadronic matterwhere we have changed variable from x to −x in the first integrand.This term is the usual Fermi-integral contribution remaining in the limitT → 0. The remaining infinite-range integral is evaluated by expansionin power series along the lines of the method shown in section 10.5, and itleads, in a straightforward way, to the first two terms in large parenthesesin the following final result:lnZF|m=0 =gFV β−36π2(7π460+π22ln2 λ+14ln4 λ). (10.74)The net quark density follows immediately from Eq. (10.74):ρq ≡ nq − nq = 3ρq1Vλ∂∂λlnZF|m=0,ρq =gFβ−36π2(π2 lnλ+ ln3 λ)=gF6(µT 2 +µ3π2). (10.75)At zero temperature, the second term is the well-known expression for thedegenerate Fermi gas. However, already at a modestly high temperatureT > µ/π, the first term dominates. In the range of parameters of interestto us, when λq 1.2–2.5 and T > 140 MeV this is always the case. Theresulting proportionality of the quark (i.e., baryon) density to the chemi-cal potential, and the accompanying quadratic temperature dependence,offer a very counterintuitive environment for a reader used to workingwith cold Fermi gases.We refer to section 4.6 for the energy and pressure of the quark andgluon gases, and a more thorough discussion the properties of a QGP isgiven in chapter 16.10.6 Entropy of classical and quantum gasesWe consider next the single-particle entropy associated with hadrons.We recall the expressions for entropy presented in section 7.1, Eq. (7.1)for a Fermi–Bose gases and Eq. (7.2) for a Boltzmann gas. We use theGibbs–Duham relation Eq. (10.26) as well as the statistical-physics ana-log, Eq. (10.25) and obtainSN=PV + ETN− µT=lnZ − β ∂∂βlnZλ∂∂λlnZ− lnλ. (10.76)We will consider several cases of physical interest and note that, forpions, even at a temperature m/T 1, the relativistic Boltzmann limit10 Relativistic gas 205is of interest, while the non-relativistic limit is of interest primarily forunderstanding the entropy of the baryon contribution.• The classical-gas case: the partition function is, in the Boltzmann ap-proximation, proportional to the fugacity; see, e.g., Eq. (4.40). ThusEq. (10.76) simplifies toSN∣∣∣∣cl = 1 + ETN − µT = 1−β∂∂βlnZλ∂∂λlnZ− lnλ. (10.77)Here, lnλ = µ/T , with the understanding that λ, in the present context,is synonymous with the abundance fugacity γ, and in what follows for thepion gas µ ≡ T ln γ.By inserting Eq. (10.52) into Eq. (10.77), we obtainSN∣∣∣∣cl= 4 + βmK1K2 − µT . (10.78)We consider first the limit m/T → 0, Eq. (10.47):SN∣∣∣∣cl 4 + m22T 2 +[ln( m2T )+ γE) m44T 4 − µT + · · ·, mT → 0. (10.79)An expansion suitable for the non-relativistic case, m/T 1, can also beobtained using Eq. (10.46):SN∣∣∣∣cl 52 + mT + 158 Tm − 158 T 2m2 + 135128 T 3m3 − µT + · · ·, mT 1. (10.80)Numerical calculation shows that the Boltzmann specific entropy ismonotonically falling, as shown in Fig. 10.4 by the solid line, towardthe asymptotic value S/N = 4 (dashed line), Eq. (10.78). The two ap-proximants, Eqs. (10.79) and (10.80), are depicted as dotted lines. Bothfourth-order approximants describe the exact result well in general, exceptnear to the physically interesting case m/T 1, for which the entropyper particle is S/N = 4.4, in the absence of a chemical potential, i.e., fora chemically equilibrated classical (Boltzmann) gas.• The low-density nucleon gas case: we consider the non-relativistic ex-pansion, m/T > 1, but we need to retain in our consideration the baryonnumber fugacity. Moreover, it is the entropy per baryon rather than theentropy per particle which is of interest. Recalling that in the Boltzmannapproximation lnZcl ∝ (λ+λ−1) = 2 cosh(µ/T ), and b = λ(d/dλ) lnZcl ∝(λ− λ−1) = 2 sinh(µ/T ), Eq. (10.77) now implies thatSb∣∣∣∣clN=(4 + βmK1(βm)K2(βm))coth(µT)− µT. (10.81)206 Hot hadronic matterFig. 10.4. The entropy per particle of a classical (Boltzmann) gas. Dashed line,asymptotic value S/N = 4; dotted line, fourth-order approximants, Eqs. (10.79)and (10.80).Using the asymptotic expansion Eq. (10.46), we obtainSb∣∣∣∣clN=52(1 +34Tm− 34T 2m2+2764T 3m3+ · · ·)coth(µT)−µ−m coth(µ/T )T. (10.82)•We find the entropy for quantum quark and gluon gases using the Gibbs–Duham relation in the form Eq. (10.76) and the relativistic equation ofstate Eq. (10.61). The entropy isS|m=0 =4PVT−∑fµfTNf =4E3T−∑fµfTNf , (10.83)where the sum over different kinds of component f is implied, E =∑f Ef ,etc. For each component, we obtain in the relativistic limit m/T 1,and, for µ = 0, dividing by N , and using Eqs. (10.69) and (10.70),SN∣∣∣∣Bm=0= 4ζ(4)ζ(3)= 3.61, (10.84)SN∣∣∣∣Fm=0= 4η(4)η(3)= 4.20. (10.85)11 Hadronic gas 207To obtain the complete dependence on m/T , we use Eq. (10.68). Foreach particle species, subject to the existence of the series representationof the integral, as addressed earlier, the result isSN= 4∞∑n=1(uλ)nn4((nx)2K2(nx) +14(nx)3K1(nx))∞∑n=1(uλ)nn3(nx)2K2(nx),− lnλ, (10.86)where x = mβ. u = −1 for fermions and u = 1 otherwise. For the non-relativistic limit x > 1, one can use Eq. (10.46) in Eq. (10.86) to obtainSN=∞∑n=1(uλ)nn4(nx)2K2(nx)I(nx)∞∑n=1(uλ)nn3(nx)2K2(nx)− lnλ, (10.87)I(nx) =52+ nx+1581nx− 1581(nx)2+1351281(nx)3+ · · · , (10.88)which, for n = 1, yields the result Eq. (10.82), once we rearrange terms oftwo components to include particles and antiparticles and divide by thebaryon number (particle–antiparticle difference).For the case of a vanishing chemical potential, the non-relativistic Boltz-mann approximation, Eq. (10.80), is quite appropriate. In Fig. 10.5, wecompare the entropy per particle, evaluated at zero chemical potential(λ = 1), for the Fermi (long-dashed line), Bose (short-dashed line) andBoltzmann (solid line, see Fig. 10.4) particles.11 Hadronic gas11.1 Pressure and energy density in a hadronic resonance gasWe now consider the physical properties of a hadronic, confined phase,such as energy density, pressure, and abundances of various particles,assuming that we have a locally thermally and chemically equilibratedphase. Although full chemical equilibrium is most certainly not attain-able in the short time of the nuclear-collision interaction, see chapter 5,this study provides very useful guidance and a reference point for under-standing the properties of hadronic matter out of chemical equilibrium.There are two ways to look at a hadronic gas: the first is that we canstudy its properties using the known hadronic states. This approach will208 Hot hadronic matterFig. 10.5. Comparison of entropy per particle for Fermi and Bose gases, theirclassical Boltzmann limit, as a function of T/m: long-dashed Fermi gas, short-dashed Bose gas, solid line Boltzmann classical limit.lead to difficulties when and if the temperature is high, since the contribu-tion of high-mass resonances is apparently not convergent. Even thoughthe population of each such state is suppressed exponentially by the Boltz-mann factor e−m/T , the number of states rises exponentially with mass,and compensates for this effect. This phenomenon was noticed almost 40years ago. This led to the development of the statistical-bootstrap model(SBM) and the Hagedorn-gas model, which we will address in chapter 12.In the physically most relevant hadron-gas domain, 70 MeV > T > 170MeV, each distinguishable hadron distribution is far from quantum de-generacy. Therefore, we can use the Boltzmann approximation. Theonly exceptional case is the pion, which, when necessary and appropri-ate, will be considered as a Bose particle. Each of the hadronic states isconsidered as a separate contributing fraction in the thermal and chem-ically equilibrated gas phase, with all fugacities set at λ = 1 (no netquantum numbers, e.g., b = 0 etc.). The result is shown in Fig. 11.1.We included 4627 (counting spin and isospin degeneracy) hadronic stateslisted by the particle data group (PDG) [136]. No doubt many morehadronic resonances exist. However, as the mass of the new resonancesincreases, they become more difficult to characterize, given the densebackground of the neighboring resonances, and normally increasing decaywidth, both of which effects are reducing the signal-to-noise ratio in theexperiment.11 Hadronic gas 209Fig. 11.1. The energy density (solid line) and pressure (dashed line) in units ofT 4 for all known hadrons (on a logarithmic scale) as functions of temperatureT . All fugacities are set to unity.The worrying fact is that the energy and pressure seem to grow wellbeyond the values spanned by the lattice calculations; see section 15.5.This happens since we have allowed very many hadrons to be present inthe same volume. Even though each kind is relatively rare, the large num-ber of resonances implies a considerable total particle density. However,hadrons are not point-like, and in some sense the presence of particles fillsthe space available. In the context of the statistical bootstrap, we willargue in section 12.3 that each hadron occupies a fraction of the spacialvolume. This qualitative argument leads to a correction that relates thephysically observable P and ε to the point-particle result (subscript ‘pt’)we have so far studied [144]:P =Ppt1 + εpt/(4B), ε =εpt1 + εpt/(4B). (11.1)The energy density of a hadron is assumed to be 4B, where, as before,B is the bag constant, and we recall the benchmark value, B1/4 = 171MeV, corresponding to 4B = 0.45 GeV fm−3. This excluded volumemodifies and limits the growth both of ε and of P with temperature.The magnitude of the effect depends on details of the implementationand on the parameters used. However, ε/P is little influenced by thisphenomenological uncertainty.The dynamics of HG matter described in, e.g., the hydrodynamic ap-proach in section 6.2 depends in a critical way on the ratio of the inertia210 Hot hadronic matterFig. 11.2. The ratio of energy density and pressure for a hadronic gas as afunction of the temperature T . Dotted line: pure pion gas; solid lines: gascomprising pions, nucleons, kaons, and ∆(1232), from bottom to top for λq =1, 1.2, 1.4, 1.6, 1.8, and 2.(energy density) to force (pressure) . In Fig. 11.2 for several simple cases,we show the HG ratio ε/P , as a function of temperature. The dottedcurve is for the pure pion gas, and we see how the relativistic equation ofstate is approached for T > 100 MeV. Remarkably, a very different resultis seen once heavy hadrons are introduced. The solid lines include, apartfrom pions, a few more massive states: nucleons, kaons, and ∆(1232). Thesolid lines from bottom to top are for λq = 1, 1.2, 1.4, 1.6, 1.8, and 2. Werecognize that increasing λq (i.e. increasing the massive-baryon compo-nent) leads to a greater ratio of inertia to force. This result is clearlyindependent of the (schematic) finite-volume correction we introduced inEq. (11.1). A fully realistic calculation of this situation is presented inFig. 11.3, for the case λs = 1.1 and γs/γq = 0.8 for λq = 1 to 2 in steps of0.2 from bottom to top, and γq = 1 (dashed lines), or γq = emπ/(2T ) (fulllines). Imagine that a hadron phase is formed from a deconfined QGP atsome temperature T > 140 MeV. In view of these results, we then expectan accelerating flow of matter as the ratio of inertia to force decreases,until a minimum is reached at T = 90 MeV. At this point, the HG phasemost likely ceased to exist, in the sense that the distance particles travel11 Hadronic gas 211Fig. 11.3. The energy density over pressure for a hadronic gas with statisticalparameters λs = 1.1 and γs/γq = 0.8, with λq = 1 to 2 in steps of 0.2 frombottom to top, and γq = 1 (dashed lines), or γq = emπ/(2T ) (full lines).between scattering had become too large. At the time of writing thisbook, it is not clear whether the hadronic phase is present at all, sincein the sudden breakup of a QGP a direct transition to free-streaminghadrons produced sequentially in time can be imagined. However, theseresults apply certainly to the case in which no QGP is formed, namely atsufficiently low collision energy.11.2 Counting hadronic particlesThere are several discrete quantum numbers of a hadron gas that are con-served and require introduction of independent chemical potentials. Thechemical potentials for conservation of baryon number and strangeness,µB and µS, are the best known. Alternatively, and more convenientlyfor our purposes, one can use the quark chemical potentials µq and µs forlight and strange quarks, respectively. We will often differentiate betweenthe u and d quarks, and use µu and µd.This choice of quark chemical potentials is a matter of convenience andis made in order to facilitate the translation of QGP-phase variables intoHG-phase variables; in no way does it assume deconfinement of quarks.212 Hot hadronic matterThe relationship between the two sets of chemical potentials, quark-basedand traditional hadron-conserved-quantum-number based, is given by thenatural relationsµb = 3µq, µb = 3T lnλq, (11.2a)µS = µq − µs, µs = µb/3− µS, (11.2b)where the minus signs are due to the conventional assignment of stran-geness −1 to the strange quark. Expressed in term of the fugacities wehave:λb = λ3q, λS = λq/λs. (11.3)Bh and Sh are the baryon number and strangeness of hadron ‘h’, andits chemical potential can be written either in terms of µb and µS, or interms of µq and µs:µh = Bhµb + ShµS, (11.4a)µh = νqhµq + νshµs, (11.4b)where νqh and νsh count the numbers of light and strange valence quarksinside the hadron, respectively, with antiquarks counted with a minussign. By adapting the quark-based chemical potentials for hadrons, werecognize the fact that, in the quark model, the quantum numbers ofhadrons are obtained by adding the quantum numbers of their constituentquarks.The particle numbers are more directly addressed in the partition func-tion in terms of fugacities. Since the fugacities are obtained by exponen-tiating the chemical potentials, Eq. (4.18), the fugacity of each hadronicspecies is simply the product of the fugacities of the valence quarks. Weview a hadron as simply a carrier of the valence quarks, which determinethe fugacity and chemical potential of each particle. For example, we havep : µp = 2µu + µd, λp = λ2uλd ;n : µn = µu + 2µd, λn = λuλ2d ;Λ : µΛ = µu + µd + µs, λΛ = λuλdλs ; etc.We distinguish between the up and down quarks by introducing sepa-rate chemical potentials µu and µd, which is tantamount to introduction ofthe chemical potential µQ related to the conservation of electrical charge.In view of the quark baryon number 13 and the quark charges −13 and +23 ,the relations between the chemical potentials areµu ≡ 13µb +23µQ, µd ≡13µb −13µQ. (11.5)11 Hadronic gas 213The average of µu and µd is the quark chemical potential µq:µq ≡µu + µd2. (11.6)The definitions Eqs. (11.5) and (11.6) imply a modification of Eq. (11.2a),µq → µq = 13µb +16µQ, (11.7)which is rarely considered. It arises from the fact that a quark systemcontaining (nearly) equal numbers of u and d quarks would still have anet (positive) charge of a sixth the total number of u and d quarks, arisingfrom the electrical charge of the proton in the initial state formed by thecolliding nuclei.The asymmetry in the number of u and d quarks is best described bythe quantityδµ = µd − µu = −µQ, (11.8)where the negative sign in the last equality reminds us that the d quarkhas negative charge. Inverting Eq. (11.7), we obtainµb = 3µq(1 +16δµµq). (11.9)In a free-quark gas with µq < πT , we have, in view of Eq. (10.75),µd ∝ 〈d− d̄〉, µu ∝ 〈u− ū〉, (11.10)where the net number (number of quarks minus that of antiquarks) oflight quarks enters. In a QGP, we find the remarkably simple relation[216]16δµµq=13µd − µuµd + µu=13〈d− d̄〉 − 〈u− ū〉〈d− d̄〉+ 〈u− ū〉=n− pA, (11.11)with A = n + p, and n and p are the neutron and proton contents of thematter which formed the QGP phase.For the case of greatest asymmetry available, in Pb–Pb collisions, wehave δµ/(6µq) = 0.21. In the HG phase a similarly sized effect forδµ/(6µq) to that in a QGP is found, considering this issue numerically;see figure 1 in [183]. Especially in studying yields of individual particles,the specific quark u and d content can play a noticeable role. To see this,let us compare the u and d fugacities:λdλu= eδµ/T = λδµ/µqq . (11.12)For Pb–Pb interactions under baryon-rich conditions, a λd/λu ratio sig-nificantly different from unity results. Some dilution of this phenomenon214 Hot hadronic matterwill occur if a QGP is formed due to the contribution of hadronizinggluons, which do not differentiate between the two light u and d flavorstates. We conclude that the u–d asymmetry can not be completely ig-nored when one is considering the abundances of hadronic particles inbaryon-rich fireballs.Flavor-changing weak interactions are too slow to matter on the timescale of heavy-ion collisions. The strong and electro-magnetic interactionsdo not mix the quark flavors u, d, and s. These are separately conserved onthe time scale of hadronic collisions. Only the number of quark–antiquarkpairs of the same flavor changes; that is, pairs can be produced or annihi-lated. The fugacities we call γq and γs serve to count the number of pairsof light and strange quark, respectively, at any given time. In general,these pair-abundance fugacities are rapidly evolving in time, in contrastto the fugacities λq and λs. In fact, for entropy-conserving evolution of afireball of QGP, the fugacity λq is nearly constant, and, as we now shalladdress, as long as local conservation of strangeness is maintained, λs 1.Comparing the QGP with the HG phase, the value of the strangenessfugacity λs is in a subtle and important way different. Given the mobilityof individual quarks in the QGP phase, and ignoring the influence ofelectrical charge in this qualitative discussion, the phase space of both sand s̄ quarks must be the same, irrespective of the baryon content. Tobalance the s and s̄ distributions, we have λs = 1, irrespective of thevalue of λq, see, e.g., Eq. (4.42). It is instructive to check the phase-space integral describing the density of strangeness in order to appreciatethese remarks, and to recall the precise physical difference between thefugacities λs and γs:〈ns〉 − 〈ns̄〉 =∫d3p(2π)3 1γ−1s λ−1s exp(√p2+m2sT)+ 1− 1γ−1s λs exp(√p2+m2sT)+ 1. (11.13)We note the change in the power of λs between these two terms, andrecognize that this integral can vanish only for λs → 1. We discussin the following section the small but significant asymmetry in λs dueto the Coulomb charge present in baryon-rich quark matter: long-rangeelectro-magnetic interactions influence strange and antistrange particlesdifferently, and a slight deviation λs > 1 is needed in order to compensatefor this effect in the QGP phase.11 Hadronic gas 215Now, let us look at the HG phase. Strange quarks are bound in statescomprising also light quarks. The presence of a net baryon number as-sures that there is an asymmetry in abundance of light quarks and an-tiquarks, and thus also, e.g., of strange baryons and antibaryons, withmore hyperons than antihyperons being present. Owing to this asym-metry, strangeness cannot be balanced in the HG with the value λs = 1,unless the baryon density vanishes locally. We will address this importantissue in a more quantitative manner in section 11.4. We have learned thata determination of λs = 1 in the hadron-abundance analysis is indicatingproduction of these hadrons directly in a breakup of a QGP phase, sincea value different from unity is expected when a HG phase breaks up.11.3 Distortion by the Coulomb forceIt has been recognized for a long time that the Coulomb force can be ofconsiderable importance in the study of relativistic heavy-ion collisions.It plays an important role in the HBT interferometry method of analysisof the structure of the particle source [57, 209]; section 9.3. The analysisof chemical properties is also subject to this perturbing force, and in con-sideration of the precision reached experimentally in the study of particleratios, one has to keep this effect in mind.We consider a Fermi gas of strange and antistrange quarks allowingthat the Coulomb potential VC is established by the excess charge ofthe colliding nuclei. Within a relativistic Thomas–Fermi phase-space oc-cupancy model [193], and for finite temperature in a QGP, we have asgeneralization of Eq. (11.13) [177]〈Ns〉 − 〈Ns̄〉 =∫Rfgsd3r d3p(2π)3(11 + γ−1s λ−1s e(E(p)−13VC(r))/T− 11 + γ−1s λse(E(p)+13VC(r))/T), (11.14)which clearly cannot vanish for VC = 0, in the limit λs → 1.In Eq. (11.14), the subscript Rf on the spatial integral reminds us thatonly the classically allowed region within the fireball is covered in theintegration over the level density; E =√m2 + /p 2, and, for a uniformcharge distribution within a radius Rf of charge Zf ,VC =−32Zfe2Rf[1− 13(rRf)2], for r < Rf ;−Zfe2r, for r > Rf .(11.15)216 Hot hadronic matterOne obtains a rather precise result, for the range of parameters ofinterest to us, using the Boltzmann approximation:〈NBs 〉 − 〈NBs̄ 〉 = γs(∫gsd3p(2π)3e−E/T)×∫Rfd3r(λseVC/3T − λ−1s e−VC/3T). (11.16)The Boltzmann limit allows us also to verify and confirm the signs: theCoulomb potential is negative for the negatively charged s quarks withcharge 13 , which is made explicit in the potential terms in all expressionsabove. We haveλ̃s ≡ λsO1/3C = 1, OC ≡∫Rfd3r eV/T∫Rfd3r. (11.17)OC < 1 expresses the Coulomb deformation of strange quark phase space.OC is not a fugacity that can be adjusted to satisfy a chemical condi-tion, since consideration of λi, i = u, d, s, exhausts all available chemicalbalance conditions for the abundances of hadronic particles, and allowsintroduction of the fugacity associated with the Coulomb charge of quarksand hadrons; see section 11.2. Instead, OC characterizes the distortion ofthe phase space by the long-range Coulomb interaction. This Coulombdistortion of the quark phase space is naturally also present for u and dquarks, but appears less significant given that λu and λq are empiricallydetermined. On the other hand this effect compensates in part the u–dabundance asymmetry effect we have discussed in Eqs. (11.5)–(11.12).Choosing T = 140 MeV and ms = 200 MeV, and noting that thevalue of γs is practically irrelevant since this factor cancels out in theBoltzmann approximation, see Eq. (11.16), we find for Zf = 150 that thevalue λs = 1.10 is needed for Rf = 7.9 fm in order to balance the Coulombdistortion. One should remember that the dimensionless quantities ms/Tand RfT determine the magnitude of the effect we study. Chemical freeze-out at higher temperature, e.g., T = 170 MeV, leads for λs = 1.10 tosomewhat smaller radii, which is consistent with the higher temperatureused.The influence of the Coulomb force on chemical freeze-out is relevant incentral Pb–Pb interactions, wheras for S–Au/W/Pb reactions, a similaranalysis leads to a value λs = 1.01, which is little different from thevalue λs = 1 expected in the absence of the Coulomb deformation ofphase space. Another way to understand the varying importance of theCoulomb effect is to note that, while the Coulomb potential acquires inthe Pb–Pb case a magnitude comparable to the quark chemical potential,it remains small on this scale for S–Au/W/Pb reactions.11 Hadronic gas 21711.4 Strangeness in hadronic gasWe now describe the abundance of strange particles in the hadronic-gasphase. This is, compared with the QGP, a very complicated case, sincethere are many particles which are carriers of ‘open’ strangeness. More-over, strong interactions result in the presence of numerous hadronic reso-nances with open strangeness. The postulate of the dominance by hadronresonance formation of hadron–hadron interactions [140] allows a vastsimplification of the theoretical treatment. Regarding the hadronic-gasphase as a mixture of various non-interacting hadronic-resonance gases,all information about the interaction is contained in the mass spectrumτ(m2, b) which describes the number of hadrons of baryon number b ina mass interval dm2. We will address this postulate in more detail inchapter 12. Within this approach to strong interactions, the logarithmof the total partition function is additive in its strange and not strangesectors, so long as the various gas fractions interact mainly via formationof hadronic resonances. We then havelnZ = lnZnon-strange + lnZstrange. (11.18)In the grand-canonical description, one finds that the non-strange had-rons influence the strange ones by providing a background value of sta-tistical parameters, such as the baryochemical potential µb, which areaccessible to direct measurement. We conclude that, in order to under-stand abundances of strange particles, it is sufficient to consider lnZstrange.In the Boltzmann approximation, it is easy to write down the partitionfunction for the strange-particle fraction of the hadronic gas, Zs. In-cluding the possibility of an only partially saturated strange phase spacethrough the factor γs, and similarly γq for light quarks, but suppressingfor simplicity the isospin asymmetry δµ, Eq. (11.8), we havelnZHGs =V T 32π2[(λsλ−1q + λ−1s λq)γsγqFK + (λsλ2q + λ−1s λ−2q )γsγ2qFY+(λ2sλq + λ−2s λ−1q )γ2s γqFΞ + (λ3s + λ−3s )γ3sFΩ]. (11.19)In the phase-space function Fi all kaon (K), hyperon (Y), cascade (Ξ),and omega (Ω) resonances plus their antiparticles are taken into account:FK =∑jgKjW (mKj/T ); Kj = K,K∗,K∗2, . . ., m ≤ 1780 MeV ,FY =∑jgYjW (mYj/T ); Yj = Λ,Σ,Σ(1385), . . ., m ≤ 1940 MeV ,FΞ =∑jgΞjW (mΞj/T ); Ξj = Ξ,Ξ(1530), . . ., m ≤ 1950 MeV ,218 Hot hadronic matterFΩ =∑jgΩjW (mΩj/T ); Ωj = Ω,Ω(2250) . (11.20)The gi are the spin–isospin degeneracy factors, W (x) = x2K2(x), seeEq. (10.50a) and Fig. 10.1, where K2 is the modified Bessel function,Eq. (10.44).We need to understand, in terms of experimental observables, the chem-ical properties of the fireball at the time of hadron production. Themethod of choice is the study of particle ratios [167, 216]; section 9.1. Inorder to obtain the mean abundances of various strange particles, we in-troduce for each species its own dummy fugacity (which we subsequentlywill set equal to unity). The explicit expressions for these ratios turn outto be very simple, and one quickly deduces from the following examplesthe principles which allow one to construct any ratio:〈nΛ̄〉〈nΛ〉= λ−4q λ−2s ; (11.21a)〈nΞ̄〉〈nΞ〉= λ−2q λ−4s ; (11.21b)〈nΩ̄〉〈nΩ〉= λ−6s ; (11.21c)〈nK+〉〈nK−〉= λ−2s λ2q ; (11.21d)〈nK〉〈nΛ〉= λ−2s λ−1q γ−1qFKFΛ. (11.21e)In a more colloquial notation found also in this book, one omits 〈n〉, usingas the symbol for the particle considered the subscript only.The baryochemical potential, or more simply, the quark fugacity λq,can be deduced from the above stated ratios. Best for this purpose isto consider the ratios not involving quark-pair fugacities γq and γs. Anytwo ratios containing only λq and λs can be combined to evaluate thesequantities. Since many more than two ratios are available, a check of theprocedure is possible. This, in fact, constitutes a strong confirmation ofthe validity of phase-space characterization of particle yields. Postponingdetailed discussion to chapter 19, we note that all groups that appliedthis method to study the chemical properties have found extremely goodconsistency. This implies that the production of particles as different askaons K and anticascades Ξ occurs by a similar mechanism, and nearly atthe same instance in the evolution of the fireball; these particles know ofeach other, either due to processes of rescattering in the HG phase, or sim-ply because they have been produced directly with yields correspondingto the relative size of the phase space.11 Hadronic gas 219An example of the consistency relation can be obtained by combiningthe ratios of cascades, lambdas, and kaons,Ξ/ΞΛ/Λ=K+K−, (11.22)which is very well satisfied in all measurements of which we are aware.It is important to note that Eq. (11.22) applies a full ‘4π’ yield. For thecentral-rapidity particle yield ratio, a correction containing the influenceof the velocity of expansion of the fireball has to be applied.Although the proper determination of the chemical properties is bestachieved in a global fit of hadron yields, it is important that we see howthe physics of this determination works. It can be seen that the multitudeof strange hadrons allows us to determine the value of λs in many differentways, for example,Λ/Λ(Ξ/Ξ)2= λ6s (11.23)and similarlyΞ/Ξ(Λ/Λ)2= λ6q. (11.24)This estimate produces an answer for the value of these parameters thatvery accurately agrees with results of global fits.It is equally easy to fix the ratio γs/γq since comparison of hyperonsof unequal strangeness content always yields this pair fugacity ratio. Thedifficulty is that we have to understand the ratio of phase spaces of thevarious baryons, which is controlled by the temperature, when we considerthe full yield. A first estimate is obtained by comparing in the same m⊥range, e.g., Λ and Ξ. How this is done is shown in Fig. 8.8 on page 150.Even then, the feed from higher resonances is important and temperatureremains an input into the determination of γi.11.5 The grand-canonical conservation of strangenessUsing the partition function Eq. (11.19), we can calculate the net stran-geness by evaluating〈Ns〉 − 〈Ns̄〉 = λs∂∂λslnZHGs . (11.25)We find〈ns〉 − 〈ns̄〉 =T 32π2[(λsλ−1q − λ−1s λq)γsγqFK220 Hot hadronic matter+(λsλ2q − λ−1s λ−2q )γsγ2qFY+2(λ2sλq − λ−2s λ−1q )γ2s γqFΞ+3(λ3s − λ−3s )γ3sFΩ]. (11.26)In general, Eq. (11.25) must be equal to zero since strangeness is aconserved quantum number with respect to the strong interactions, andno strangeness is brought into the reaction. The possible exception isdynamic evolution with asymmetric emission of strange and antistrangehadrons. The grand-canonical condition,〈ns〉 − 〈ns̄〉 = 0, (11.27)introduces an important constraint, i.e., it fixes λs in terms of λb (andcharge λQ when the later is considered).Equation (11.25) can be solved analytically when the contribution ofmultistrange particles is small:λs|0 = λq√FK + γqλ−3q FYFK + γqλ3qFY. (11.28)This relation between the strange chemical potential µs|0 = T lnλs|0 andthe baryochemical potential µb = 3T lnλq is shown for γq = γs = 1 inFig. 11.4. To understand Fig. 11.4, we note that the term with λ−3q =e−µb/T in Eq. (11.28) will tend to zero as µb gets larger and the term withλb will dominate in denominator. Thus, λs ∝ λ−2/3b , i.e., µs ∝ −23µb forlarge µb. At small µb, in particular, for relatively small temperatures, thehyperon contribution is small and we see µs ∝ 13µb. Putting it differently,Eq. (11.28) knows that, in a baryon-rich HG phase, qs̄ (K+,K0) kaons arethe dominant carriers of s̄ quarks, whereas qqs (Λ,Σ) hyperon states arethe main carriers of s quarks at finite baryon density. We see that thecompetition between strangeness content in the four classes of strangen-ess carriers determines, at each temperature T , the location where oneobtains a nontrivial µs = 0 at finite µb, and the QGP property λs = 1 isaccidentally present in the HG phase. There is no such nontrivial solutionat sufficiently high temperature. For T > 200 MeV and γq = γs = 1, onlynegative strangeness chemical potential is seen in Fig. 11.4.The line in the (µb–T ) plane corresponding to µs = 0 is the divide be-tween positive and negative values of the strangeness chemical potentialin a strangeness-balanced hadronic gas. The relation between µb and Tcorresponding to µs = 0, i.e., λs = 1, arising from Eq. (11.26) when netstrangeness vanishes, can be solved analytically allowing for the effect ofmultistrange baryons and antibaryons. First, we note that for λs = 1,11 Hadronic gas 221Fig. 11.4. The strange-quark chemical potential µs versus the baryon chemicalpotential µb in a strangeness-neutral grand-canonical chemically equilibratedHG. The long-dashed line corresponds to T = 150 MeV, the solid line to T = 200MeV, and the short-dashed line to T = 300 MeV. The dotted line is the limitingcurve for large T , computed here at T = 1000 MeV.there is always an exact balance between Ω and Ω and this term disap-pears. The coefficient of the hyperon FY contribution, when it is writtenin the formλ−2q − λ2q =(λ−1q − λq) (λ−1q + λq),allows us to cancel out a common factor λ−1q − λq present in all terms,along with γqγs. We obtainµb = 3T ln(x+√x2 − 1), 1 ≤ x = FK − 2γsFΞ2γqFY. (11.29)This result is shown in Fig. 11.5. We have chosen to consider the nonequi-librium condition γq = emπ/(2T ) corresponding to the maximum entropycontent in a hadronic gas, as could be emerging from hadronization of anentropy-rich QGP phase. The solid line is for γs = γq, while the dashedlines span the range γs = 0.8–2.8 in steps of 0.2, from right to left.Below and to the left of this separation line in Fig. 11.5, we have posi-tive strangeness chemical potential in a strangeness-balanced HG phase,222 Hot hadronic matterFig. 11.5. The condition of vanishing strangeness at λs = 1 in a hadronic gas,evaluated for its maximum entropy content, i.e., with γq = emπ/(2T ). Solid line,γs = γq; dashed lines from left to right are for γs = 0.8–2.8 in steps of 0.2.whereas above and to the right we have negative strangeness potential.The importance of this observation is that the relative yield RΩ ≡ Ω/Ω,Eq. (11.21c), is strongly sensitive to the sign of µs. At present, we knowthat, at the SPS top energy, both in S–W and in Pb–Pb interactions,RΩ ≥ 1 and thus λs ≥ 1 and µs ≥ 0; the allowed range of T–µb is belowand to the left in Fig. 11.5. Indeed, all analyses of the abundances ofparticles of which we are aware have yielded results in this domain of Tand µb.We further denote, in Fig. 11.5, the area below and to the left as s < s̄,whereas the domain above and to the right is denoted as s > s̄. What weindicate is that, for λs = 1, the resulting phase space of strange particleswould add up to satisfy these conditions within these domains of T andµb. To recognize the importance of this condition consider that a QGP isevaporating hadrons. Below and to the left, with s < s̄, the evaporationfavors emission of antistrangeness and this allows the accumulation of anexcess of strangeness in the evaporation remnant; this is the process called‘strangeness distillation’ [134, 135].11 Hadronic gas 223The result of the distillation is the production of strangelets, dropsof quark matter with an unusually high abundance of s quarks. Sincestrange quarks are negatively charged, such states would have unusuallysmall charge relative to their mass; indeed in the limit of equal abundanceof u, d, and s, a strangelet would be neutral. Many searches for long-lived(on the scale of strong interactions) strangelets have been performed,without success [44, 45, 232], suggesting that such states are not stablewith respect to strong interactions. If they are produced, strangelets aredissociating into strange hadrons rather rapidly. One will note that, in thedecay of hadronic strangelets, production of multistrange baryons, and inparticular Ω, would be well above the normal expectations. Since thestatistical yield of Ω is very small, even a small yield of strangelets couldlead to visible distortions of the otherwise rarely produced Ω. An excessof Ω over the statistical-model expectations is seen at the top energy ofthe SPS; see section 19.3 and Fig. 8.11.11.6 Exact conservation of flavor quantum numbersWe now consider, in more detail, what effect the exact conservation ofquantum numbers, such as strangeness or baryon number, has on the sizeof the available particle phase space. As is intuitively clear, only whenthe yield numbers are small, can this lead to a noticeable effect. In thegrand-canonical approach, flavor conservation, expressed by Eq. (11.27),is not exact. In other words, strangeness, even baryon number, is con-served on average but not exactly. We will focus our interest on the caseof newly-produced flavors (strangeness and charm) since the number ofpairs of quarks produced can be sufficiently small to warrant this. Theexact conservation of the baryon number (or the light-quark flavors) is ofparticular interest in the study of the small collision systems.When the number of strange-quark pairs is relatively small, Eq. (11.27)has to be replaced by the sharper ‘canonical’ conservation condition,〈ns − ns̄〉 = 0. (11.30)According to Eq. (11.30), the net strangeness vanishes exactly in eachphysical system we study. This introduces a correlation between the phasespace of particles and of antiparticles and thus, in general, the chemicalequilibrium yield of, e.g., the pairs of strange quarks evaluated underconstraint Eq. (11.30), is smaller when compared with that expected whenEq. (11.27) is considered.We are, in particular, interested in understanding under which condi-tions the canonical and grand-canonical yields are equal, and how thegrand-canonical yields are altered by the physical constraint Eq. (11.30)[218]. For strangeness, this amounts to finding the yield for which we can224 Hot hadronic matterstudy the colliding system as if it had infinite size. In the context of theproduction of charm, the yields (almost) always remain relatively small,and use of the canonical formulation is necessary in order to evaluatethe expected chemical-equilibrium yields, even when the grand-canonicalapproach applies for all the other observables.The grand partition function, Eq. (10.48), in the Boltzmann limit, canbe written as a power series:Zcl = eZ(1)f =∞∑n=01n!(Z(1)f)n. (11.31)To emphasize that any flavor (in particular s, c, and b) is under con-sideration here, we generalize slightly the notation s → f. The flavorand antiflavor terms within Z(1)f are additive, and we consider at firstonly singly flavored particles in Eq. (11.19), adopting a simplified andself-explanatory notation:Z(1)f = γ(λf F̃f + λ−1f F̃f̄), F̃i =V T 32π2Fi. (11.32)Combining Eq. (11.32) with Eq. (11.31), we obtainZcl =∞∑n,k=0γn+kn!k!λn−kf F̃nf F̃kf̄ . (11.33)When n = k, the sum in Eq. (11.33) contains contributions with un-equal numbers of f and f̄ terms. Only when n = k do we have contribu-tions with exactly equal number of f and f̄ terms. We recognize that onlyn = k terms contribute to the canonical partition function:Zf=0cl =∞∑n=0γ2nn!n!(F̃f F̃f̄)n = I0(2γ√F̃f F̃f̄). (11.34)The modified Bessel function I0 is well known, see Eqs. (8.23) and (8.27).The argument of I0 has a physical meaning, it is the yield of flavor pairsNGCpair in the grand-canonical ensemble, evaluated with grand-canonicalconservation of flavor, Eq. (11.27). To see this, we evaluate〈Nf〉 − 〈Nf̄〉 = λf∂∂λflnZ fcl = γ(λf F̃f − λ−1f F̃f̄) = 0. (11.35)We obtain, see Eq. (11.28),λf |0 =√F̃f̄/F̃f , (11.36)11 Hadronic gas 225and thuslnZ fcl∣∣∣λf=λf |0= 〈Nf〉+ 〈Nf̄〉 = 2γ√F̃f F̃f̄ ≡ 2NGCpair, (11.37)which is just the argument of the I0 function in Eq. (11.34). In the grand-canonical-ensemble approach the (average) number of pairs NGCpair is ex-tensive in volume, since F̃i ∝ V .In order to evaluate, using Eq. (11.34), the number of flavor pairs inthe canonical-ensemble, we need to average the number n over all thecontributions to the sum in Eq. (11.34). To obtain the extra factor n, weperform the differentiation with respect to γ2 and obtain the canonicalensemble f-pair yield,〈NCEf 〉 ≡ γ2ddγ2lnZ f=0cl = γ√F̃f F̃f̄I1(2γ√F̃f F̃f̄)I0(2γ√F̃f F̃f̄)= NGCpairI1(2NGCpair)I0(2NGCpair). (11.38)where we have used Eq. (8.24). The first term is identical to the resultwe obtained in the grand-canonical formulation, Eq. (11.37). The secondterm is the effect of exact conservation of flavor.The intuitive derivation of the canonical constraint we have presentedfollows the approach of [218]. This can be generalized to more com-plex systems using the projection method [229, 262]. This method can beapplied to solve more complex situations, for example inclusion of mul-tistrange hadrons, conservation of several ‘Abelian’ quantum numbers[61, 102] (such as strangeness, baryon number, and electrical charge), andthe problem of particular relevance in this field, the exact conservationof color: all hadronic states, including QGP, must be exactly color ‘neu-tral’ [111, 112]. The solution of this ‘nonabelian-charge’ problem is mostinteresting but reaches well beyond the scope of this book.For the case of ‘Abelian’ quantum numbers, e.g., flavor or baryon num-ber, the projection method arises from the general relation between thegrand-canonical and canonical partition functions implicit in Eq. (4.20):Z(β, λ, V )cl =∞∑nf=−∞λnfZf(β, V ;nf). (11.39)In the canonical partition function Zf , some discrete (flavor, baryon)quantum number has the value nf ≡ f. The inverse of this expansion226 Hot hadronic matteris given in Eq. (4.21). On making the substitution λ = eiϕ we obtainZf(β, V ;nf) =∫ 2π0dϕ2πe−infϕZ(β, λ = eiϕ, V ). (11.40)In the case of the Boltzmann limit, and including singly charged parti-cles only, we obtain for the net flavor nf from Eq. (11.33)Zf(β, V ;nf) =∞∑n,k=0γn+kn!k!∫ 2π0dϕ2πei(n−k−nf)ϕF̃nf F̃kf̄ . (11.41)The integration over ϕ yields the δ(n − k − nf) function. Replacing n =k + nf , we obtainZf(β, V ;nf) =∞∑k=0γ2k+nfk!(k + nf)!F̃ k+nff F̃kf̄ . (11.42)The power-series definition of the modified Bessel function If isInf (z) =∞∑k=0(z/2)2k+nfk!(k + nf)!. (11.43)Thus we obtainZf(β, V ;nf) =(F̃fF̃f̄)nf/2Inf(2γ√F̃f F̃f̄). (11.44)The case of nf = 0 which we considered earlier, Eq. (11.34), is reproduced.We note that, for integer nf , we have Inf = I−nf , as is also evident in theintegral representation Eq. (8.27). We used nf as we would count thebaryon number, thus, in flavor counting, nf counts the flavored quarkcontent, with quarks counted positively and antiquarks negatively. Thisremark is relevant when the factors F̃f and F̃f̄ contain baryochemicalpotential.When the baryon number is treated in the grand-canonical approach,and strangeness in the canonical approach, there is potential for math-ematical difficulties. These can usually be avoided by considering themeromorphic expansion of the partition function Eq. (11.39). Insertingthe explicit form Eq. (11.32) we obtainZcl eγ(λf F̃f+λ−1f F̃f̄) =∞∑nf=−∞λnff(F̃fF̃f̄)nf/2Inf(2γ√F̃f F̃f̄). (11.45)Multistrange particles can be introduced as additive terms in the expo-nent in Eq. (11.45). This allows us to evaluate their yields [148]. However,11 Hadronic gas 227the canonical partition function is dominated by singly strange particlesand we will assume, in the following, that considering only these sufficesto obtain the effect of canonical conservation of flavor. In order to findyields of rarely produced particles such as, e.g., Ω(sss), we show the omegaterm explicitly:Zf(β, V ;nf = 0) =∫ 2π0dϕ2πeF̃feiϕ+F̃f̄e−iϕ+λΩe3iϕF̃Ω+···. (11.46)The unstated terms in the exponent are the other small abundances ofmultiflavored particles. The fugacities not associated with strangeness,as well as the yield fugacity γs, are incorporated in Eq. (11.46) into thephase-space factors F̃i for simplicity of notation.The number of Ω is obtained by differentiating lnZf(β, V ) with respectto λΩ, and subsequently neglecting the subdominant terms in the expo-nent,〈nΩ〉 F̃ΩI0∫ 2π0dϕ2πe3iϕeF̃feiϕ+F̃f̄e−iϕ. (11.47)The result of the integration is easily read off the meromorphic expansion,Eq. (11.45), to be Zf(β, V ;nf = −3), Eq. (11.44). This result is easilyunderstood, the three strange quarks in the particle observed are balancedby the background of singly strange particles (kaons and antihyperons),〈nΩ〉 F̃Ω(F̃fF̃f̄)−3/2I3(2NGCpair)I0(2NGCpair). (11.48)We recall that, according to Eq. (11.36), the middle term is just the fu-gacity factor λ3s . The first two factors in Eq. (11.48) constitute the grand-canonical yield, while the last term is the canonical Ω-suppression factor.A full treatment of the canonical suppression of multistrange particleabundances in small volumes has been used to obtain particle yields inelementary interactions [60].Similarly, one finds that the suppression of Ξ abundance has the factorI2/I0, whereas, as discussed for the general example of the flavor-pairyield, the yield of single strange particles is suppressed by the factor I1/I0.The yield of all flavored hadrons in the canonical approach (superscript‘C’) can be written as a function of the yield expected in the grand-canonical approach in the general form〈sκ〉C = F̃κ(F̃fF̃f̄)κ/2I|κ|(2NGCpair)I0(2NGCpair)= 〈sκ〉GCI|κ|(2NGCpair)I0(2NGCpair), (11.49)with κ = ±3, ±2, and ±1 for Ω, Ξ, and Y and K, respectively. On theleft-hand side in Eq. (11.49) the power indicates the flavor content in the228 Hot hadronic matterparticle considered, with negative numbers counting antiquarks. We note,on inspecting the final form of Eq. (11.49), that the canonical suppressionof particle and antiparticle abundances is the same, certainly so when westudy systems with several pairs present. In very small systems, one mayneed to evaluate the quantum distributions including multistrange parti-cles in order to obtain precise results. A particle/antiparticle asymmetrycan occur if baryon/antibaryon asymmetry applies.The simplicity of Eq. (11.49) originates from the assumption that thecontributions of singly strange particles to conservation of strangenessare dominant. This assumption is consistent with the neglect of quantumstatistics. In fact, on expanding the Bose distribution for kaons, one findsthat the next-to-leading-order contribution, which behaves as strangenessns = ±2 hadrons, is dominating the influence of all multistrange hadrons.Our study is consistent with the Boltzmann statistics assumed here; morecomplex evaluation taking multistrange hadrons into account, but consid-ering kaons as Boltzmann particles, is theoretically inconsistent.11.7 Canonical suppression of strangeness and charmThe canonical flavor yield suppression factor,η ≡I1(2γ√F̃f F̃f̄)I0(2γ√F̃f F̃f̄) = I1(2NGCpair)I0(2NGCpair)< 1, (11.50)depends in a complex way on the volume of the system, or, express-ing it alternatively, on the grand-canonical number of pairs, NGCpair. Thesuppression function η(N) ≡ I1(2N)/I0(2N) is shown in Fig. 11.6 as afunction of N . For N > 1, we see (dotted lines) that the approach to thegrand-canonical limit is relatively slow; it follows the asymptotic formη 1− 14N− 1128N2+ · · · , (11.51)whereas for N 1, we see a nearly linear rise:η = N − N32+ · · ·. (11.52)Overall, when the the yield of particles is small, we have, using Eq. (11.52),NCEpair = (NGCf )2. (11.53)Hagedorn was puzzled by this quadratic behavior of the particle yield,being concerned about rarely occurring astrophysical processes of pairproduction. In his 1970/71 CERN lectures [141], he asked how the yield11 Hadronic gas 229Fig. 11.6. Solid line: the canonical yield-suppression factor as a function of thegrand-canonical yield of particles N . Dotted lines: asymptotic expansion formspresented in the text.of particles can be Y ∝ e−2m/T , when the threshold for production of apair is relevant, and another time Y ∝ e−m/T , when the statistical yieldis evaluated. This is the grand-canonical yield for m > T , as seen insection 10.4,NGC =gf2π2T 3V√πm32T 3e−m/T , (11.54)whereas when the yield of particles is small, e.g., when m T , thecanonical result applies:NCE =g2f8π3T 3m3V 2e−2m/T . (11.55)We see that the Hagedorn puzzle has been resolved. The reaction volumeis an important factor controlling which of the two results Eqs. (11.54)and (11.55) should be considered in a given physical situation.We next consider whether there is any effect of QGP compared withHG in the study of canonical conservation of strangeness. The possibledifference would arise from the different sizes of the phase space for stran-geness in these two phases of matter. In the Boltzmann limit, the flavorand antiflavor phase space in the symmetric QGP is:F̃f = F̃f̄ = gfV∫d3p(2π)3e√p2+m2fT =3V Tm2fπ2K2(mf/T ). (11.56)230 Hot hadronic matterFig. 11.7. Volume needed for one strange quark pair using grand-canonicalcounting as function of λq for T = 160 MeV, γq = 1, γs = 1, Vh = (4π/3) 1 fm3.Solid line: hadron gas phase space, dashed line: quark phase space withms = 160MeV.For the hadronic phase, it is derived from Eqs. (11.19) and (11.20). Wehave, counting strange-quark content as positively ‘flavor charged’ as be-fore,F̃f = λ−1q F̃K + λ2qF̃Y, (11.57)F̃f̄ = λqF̃K + λ−2q F̃Y. (11.58)All these quantities F̃i are proportional to the reaction volume.The interesting result, seen in Fig. 11.6, is that the suppression of yieldis at the level of 30% when one pair of particles would be expected to bepresent in grand-canonical chemical equilibrium; the suppression meansthat instead we find that the true phase-space yield is 0.7 pairs. Actually,in p–p interactions at 158 GeV/c projectile momentum, the analysis ofexperimental results yields 0.66± 0.07 strange pairs [277].Pursuing this line of thought, but also to obtain a reference regardingthe magnitudes involved for strangeness, we consider how big a volumewe need in order to find (using grand-canonical-ensemble counting) onepair of strange particles. In the hadronic phase space, with λs chosen toconserve strangeness, we haveVVh=2π2VhT 3γqγs√(FK + λ3qFY)(FK + λ−3q FY). (11.59)11 Hadronic gas 231Fig. 11.8. Canonical yield enhancement at large volumes compared with thecase of unit hadron volume Vh = 43π fm−3. Solid line, QGP phase; dashed line,HG.For p–p interactions, we consider T = 160 MeV, and the elementaryhadronic volume is chosen to be Vh = 43π fm3. The applicable value of λq,if statistical methods are used, is close to unity. The result is shown asthe solid line in Fig. 11.7, as a function of λq, for γq = 1 and γs = 1. Thedashed line is the corresponding result for the QGP strange-quark phasespace, which naturally does not depend on λq, and has been obtainedby choosing ms = 160 MeV. Just a little less than one hadronic volumesuffices; one finds one pair in Vh for ms = 200 MeV.We show in Fig. 11.8 the canonical strangeness-suppression factorEq. (11.50), both for a QGP (solid line) and for a HG (dashed line). Wehave converted the suppression η into an enhancement by normalizing atη(V = Vh). For the QGP, we take ms = 160 MeV, whereas, for a HG,we take µb = 210 MeV. Both phases are considered at T = 145 MeV.Since the strangeness content in QGP is greater than that in HG, thereis less ‘catching up’ to do and the overall yield is increased by factor 1.8,whereas for HG, we find an increase by a factor of three. Practically allof this enhancement occurs when the reaction volume increases to five,i.e., for rather small reaction systems.We now look at the suppression of multistrange particle abundances bythe factors η3(N) = I3(2N)/I0(2N), for Ω, and η2(N) = I2(2N)/I0(2N),for Ξ. For small values of N , we obtainηκ ≡Iκ(2N)I0(2N)→ Nκ 1κ!(1− κκ+ 1N2). (11.60)232 Hot hadronic matterFig. 11.9. Canonical yield-suppression factors Iκ/I0 as function of the grand-canonical particle yield N . Short-dashed line: the suppression of triply-flavoredhadrons; long-dashed line: the suppression of doubly-flavored hadrons; and solidline, the suppression of singly-flavored hadrons.This result is easily understood on physical grounds: for example, whenthe expected grand-canonical yield is three strangeness-containing pairs,it is quite rare that all three strange quarks go into an Ω. This is seenin Fig. 11.9 (short-dashed curve), and in fact this will occur about atenth as often as we would expect from computing the yield of Ω, ig-noring the canonical conservation of strangeness. The other lines inFig. 11.9 correspond to the other suppression factors; the long-dashed lineis η2(N) = I2(2N)/I0(2N) and the solid line is η(N) = I1(2N)/I0(2N).They are shown to be dependent on the number of strange pairs expectedin the grand-canonical equilibrium, denoted in Fig. 11.9 as N .It has been proposed to exploit the canonical suppression which growswith strangeness content to explain the increase in production of strangehadrons seen in Fig. 1.6 on page 19, when the per-participant yield inA–A interactions is compared with that from p–Be interactions [228]. Adirect comparison of the reduction factors ηκ is possible. Choosing asthe reference point the yield N 1, the claim is that one can comeclose to explaining the enhancement in production of three out of fivestrange hadrons seen in Fig. 1.6. The reader should notice that the en-hancement effect is derived from the suppression of the base yield in thesmall reference system. We obtain this effect by rebasing the resultsshown in Fig. 11.9 to the strangeness yield observed in p–p reactionsevaluated within canonical formulation; see Fig. 11.10. The three casesstudied in Fig. 11.9 are seen, where the dotted lines are derived from11 Hadronic gas 233Fig. 11.10. Canonical yield-suppression factor of Fig. 11.9 expressed as enhance-ment factors Ei, i = 1, 2, 3 as functions of the canonical-pair-particle yield NCE.Solid line: E1, the enhancement of singly-flavored hadrons, relative to the yield0.66± 0.07, expected in p–p reactions. Similarly, long-dashed line: E2, enhance-ment of doubly-flavored hadrons; and short-dashed line: E3, enhancement oftriply-flavored hadrons. Dotted lines correspond to the errors arising from theerror in the strangeness yield, to which the results are normalized.the error in the reference yield of production of strangeness in p–p reac-tions.We see, in quantitative terms, the strength of the canonical effect, espe-cially for multistrange hadrons, and its rapid rise with the yield of stran-geness [223]. The canonical-enhancement effect rises rapidly but smoothlyand saturates at the grand-canonical yield in rather small systems. Thegrand-canonical chemical-equilibrium yield is reached for systems com-prising ten strangeness pairs and for reaction systems about six timesgreater than the p–p system, considering that the yield of singly-strangeparticles is enhanced by a factor three, as is seen in Fig. 11.10. This resultis inconsistent with the experimental results from the NA52 experiment[153], which reveal an abrupt threshold for enhancement of productionof strangeness at 50 participants, just where the WA57 team recentlyreported a sudden onset of enhancement in yield of Ξ [108]. Given thesensitivity of the results shown in Fig. 11.10 to the strangeness referenceyield, it is natural to conclude that the explanation of strange-hadronenhancement offered in [228] is based on a fine tuned p–Be strangenessyield, not cross-checked with the (at-present-unavailable) experimentalyield.We addressed, with such a great precision, the canonical chemical-equilibrium yields of strange particles expected to originate from small234 Hot hadronic matterFig. 11.11. The canonical yield of pairs of open charm quarks 〈nc〉pair per unitvolume as a function of volume, in units of Vh = 43π fm3. Solid line, QGP withmc = 1.3 GeV; dashed line, HG at µb = 210 MeV, both phases at T = 145 MeV.systems, since hadron yields observed in p–p and p–p̄ interactions, andeven in hadron jets produced in LEP e+–e− reactions are remarkably closeto the expectations for chemical equilibrium [62], allowing in the analysisfor the canonical suppression and including the effects of quantum degen-eracy. This thorough analysis results in a not completely satisfactory χ2per degree of freedom (= 61/21). Yet a reader of this thorough reportwill have the impression that a modern-day Maxwell’s demon must be atwork, generating canonical chemical equilibria for hadrons in all these el-ementary interactions, and abundances of strange quarks within a factorof two of absolute chemical equilibrium.On the other hand, a demon that works for strangeness should also workfor charm. The yield of charm in Pb–Pb interactions is estimated fromthe lepton background at 0.5 pairs per central collision [13]. We can usethe small-N expansion, Eq. (11.60). The corresponding A–A canonicalenhancement factor, compared with p–A, is NAA/NpA 100A. (Here Nis now the yield of ‘open’ charm rather than strangeness.) The measuredopen-charm cross sections, however, scale with the number of participants,and there is no space for a large canonical enhancement/suppression ofproduction of charm. To be more specific, we show, in Fig. 11.11, the spe-cific yield of charm 〈nc〉pair per unit volume as a function of the volume.The canonical effect is the deviation from a constant value and it is sig-nificant, O(100). Even at V = 500Vh the infinite-volume grand-canonicallimit is not yet attained, for the case of the larger phase space of QGP(solid line), the total yield of charm is just one charm pair. The absolute12 Hagedorn gas 235yield in both phases is strongly dependent on the temperature used, hereT = 145 MeV. In QGP, we took mc = 1.3 GeV. The phase space of aHG includes all known charmed mesons and baryons, with abundances oflight quarks controlled by µb = 210 MeV and µs = 0.Although, by choosing a slightly higher value of T , we can easily in-crease the equilibrium yield of charm in a HG to the QGP level [133],this does not eliminate the effect of canonical suppression of productionof charm if chemical equilibrium is assumed for charm in the elementaryinteractions. We are simply so deep in the ‘quadratic’ domain of the yield,see Eq. (11.60), that playing with parameters changes nothing, since weare constrained in Pb–Pb interactions by experiment to have a yield ofcharm of less than one pair.It is natural to argue that the very heavy charm quarks are not in chem-ical equilibrium, and that their production has to be studied in kinetictheory of collision processes of partons. However, this means that thereis no twenty-first-century Maxwell’s demon with control of charm, and,of course, also not of strangeness. The production and enhancement ofcharm and strangeness in heavy-ion collisions is in our opinion a kineticphenomenon. To study it, we should explore a wide range of collisionvolume and energy. The objective is to determine boundaries of the high,possibly QGP-generated, yields.12 Hagedorn gas12.1 The experimental hadronic mass spectrumOne of the most striking features of hadronic interactions, which wasdiscovered by Hagedorn [140], is the growth of the hadronic mass spectrumwith the hadron mass. With the 4627 different hadronic states we haveused in the study of properties of HG in section 11.1 [136], it is reasonableto evaluate the mass spectrum of hadronic states ρ(m), defined as thenumber of states in the mass interval (m, m + dm). We represent eachparticle by a Gaussian, and obtain ρ(m) by summing the contributions ofindividual hadronic particles:ρ(m) =∑m∗=mπ,mρ,...gm∗√2πσm∗exp(−(m−m∗)22σ2m∗). (12.1)Here, gm∗ is the degeneracy of the hadron of mass m∗ including, in partic-ular, spin and isospin degeneracy, and σ = Γ/2, Γ = O(200) MeV beingthe width of the resonance. The pion, with mπ σ is a special case, andis set aside in such smoothing of the mass spectrum. Downward modifi-cation of its mass has a great impact on properties of HG and is thus notallowed.236 Hot hadronic matterFig. 12.1. Dashed lines are the smoothed hadronic mass spectrum. The solidline represents the fit Eq. (12.2) with k = −3, m0 = 0.66 GeV, and T0 = 0.158GeV. Long-dashed line: 1411 states of 1967. Short-dashed line: 4627 states of1996.We compare the logarithm of the resulting smoothed mass spectrumfor the hadronic particles known in 1967 (long-dashed line) with that forthose known in 1996 (short-dashed line) in Fig. 12.1. We see that, in the20 years following Hagedorn’s last study of the phenomenon, the newlyclassified hadron resonances have improved the exponential behavior. Werefer to a hadronic gas with an exponential mass spectrum as a Hagedorngas. The solid line in Fig. 12.1 represents a fit using the empirical shapeρ(m) ≈ c(m20 +m2)k/2 exp(m/T0) (12.2)with k = −3. This value is preferred in the statistical-bootstrap model,section 12.2. However, many other values of k fit the mass spectrum well.The inverse slope T0 and the preexponential power k are correlated in afit of the mass-spectrum data and we present, in table 12.1, the resultsfor several choices of k. We shall show that the value of k determinesthe behavior of the thermodynamic quantities of a gas of hadrons whenT → T0 and its value is of some relevance.12 Hagedorn gas 237Table 12.1. Fitted parameters of Eq. (12.2) for given kk c m0 T0−2.5 0.83479 0.6346 0.16536−3.0 0.69885 0.66068 0.15760−3.5 0.58627 0.68006 0.15055−4.0 0.49266 0.69512 0.14411−5.0 0.34968 0.71738 0.13279−6.0 0.24601 0.73668 0.12341−7.0 0.17978 0.74585 0.11489The mass spectra for fermions ρF(m) and bosons ρB(m) can differ, and,using these two functions, the generalization of Eq. (10.62) readslnZHG =β−3V2π2∞∑n=1ρn(m)1n4(nβm)2K2(nβm) , (12.3)whereρn(m) ≡ ρB(m)− (−1)nρF(m) , (12.4)The Boltzmann approximation amounts to keeping in Eq. (12.3) the termwith n = 1, in which caseρ(m) ≡ ρ1(m) = ρB(m) + ρF(m). (12.5)To understand how the parameter k influences the behavior of the Hage-dorn gas, we now introduce the asymptotic form Eq. (10.45) with the firstterm only, and consider the (classical, ‘cl’) Boltzmann limit,lnZclHG = cV(T02π)3/2∫ ∞M0mk+3/2e(m/T0−m/T ) dm+D(T,M0), (12.6)where M0 > m0 is a mass above which the asymptotic form of K2 holds,and where D(T,M0) is finite. Because of the exponential factor, theintegral is divergent for T > T0, and the partition function is singular atT0 for a range of k.The pressure and the energy density for T → T0 areP (T )→(1T− 1T0)−(k+5/2), for k > −52 ,ln(1T− 1T0), for k = −52 ,constant, for k < −52 ;(12.7)238 Hot hadronic matterandε →(1T− 1T0)−(k+7/2), for k > −72 ,ln(1T− 1T0), for k = −72 ,constant, for k < −72 .(12.8)The energy density goes to infinity for k ≥ −72 , when T → T0, and inthis range falls the result of the statistical-bootstrap model with pointhadrons, Eq. (12.35). Therefore T0 appears as a limiting temperature forsuch a hadronic system [140].Interestingly, the partition function and its derivatives may be singularat T = T0 even when the volume of the system is finite, unlike the moreconventional situation, with a true singularity expected only if the volumeis infinite. However what is actually needed is an infinite number ofparticipating particles, which in the conventional situation can occur onlyfor V → ∞. In relativistic statistical physics, particles are produced, and,for an exponential mass spectrum, an infinite number of particles arisesalready in a finite volume, for a sufficiently singular value of k and point-like hadrons, and T → T0. When hadrons of finite volume are considered,we find in section 12.3 that the energy density remains finite at T = T0,independently of the value of the mass power k in the hadronic massspectrum Eq. (12.2).The reader will wonder whether the seemingly small difference betweenthe exponential mass spectrum, and the so-far-known hadron mass spec-trum, seen in Fig. 12.1 for m > 1.5 GeV, matters. We now comparethe energy and pressure of HG evaluated using individual hadrons, thinlines in Fig. 12.2 (see also Fig. 11.1), with the results obtained using theanalytical mass spectrum defined by Eq. (12.2), with parameters given intable 12.1. The vertical dotted line shows the limiting temperature fork = −3. Comparing the thick lines (exponential mass spectrum) withthe results including known hadrons only (thin lines), we see in Fig. 12.2significant differences both for ε/T 4 (a factor of four for k = −3) andP/T 4 (a factor of two for k = −3) in the physically relevant domainT 150 MeV. The various thick lines correspond to values of k listed intable 12.1 and can be assigned by noting at which value of temperatureT0 the singular behavior arises.One would be tempted to conclude that, without full knowledge ofthe hadronic spectrum, we cannot use individual hadrons in the study ofthe properties of the HG, and hence evaluation of the total multiplicityof hadronic particles, as, e.g., is required in order to obtain Fig. 9.8 onpage 169. There is, however, another effect, which counterbalances the12 Hagedorn gas 239Fig. 12.2. The energy density ε/T 4 (solid lines) and pressure P/T 4 (dashedlines) (on a logarithmic scale) for a hadronic gas with a smoothed exponentialmass spectrum, with values of k = −2.5 (the most divergent thick line) to k = −7in steps of 0.5. The thin lines were obtained by using the currently knownexperimental mass spectrum. All fugacities γ and λ = 1.effect of missing hadron resonances. When the finite size of a hadron is in-troduced, e.g., according to Eq. (11.1), significant decreases in magnitudeof energy density, pressure, and number of particles at a given tempera-ture ensue. For the value B1/4 = 190MeV, corresponding to 4B = 0.68GeV fm−3, a value we introduce to reproduce lattice QCD results in sec-tion 16.2, we show in Fig. 12.3 that there is practical agreement betweenthe exponential mass-spectrum properties with finite-volume correction(thick lines) and the point hadron gas evaluated using known hadrons(long-dashed thin line for ε/T 4 and dotted thin line for P/T 4). Consid-ering that the population of very massive resonances is not going to riseto full chemical equilibrium in nuclear collisions, along with the uncer-tainties in the finite-volume correction, (e.g., choice of B) the remaining15%–20% difference between the resonance gas with finite-volume correc-tion and the point gas of known hadrons is not physically relevant. Onthe other hand, this clearly is the level of precision (theoretical system-atic error) of the current computation of abundances of hadrons using240 Hot hadronic matterFig. 12.3. The same as Fig. 12.2 but thick lines now show the gas of hadronswith an exponential mass spectrum including the finite-volume correction withB = (190MeV)4.the known hadron spectrum. Most of this remaining systematic errordisappears when hadron-abundance ratios are evaluated.A cross check of the validity of the energy density and pressure obtainedeither by summing the physically known spectrum of point hadrons, or byemploying the exponentially extrapolated spectrum of finitely sized had-rons is obtained by comparing them with lattice-gauge results. Resultspresented in section 15.5, in Fig. 15.3, show that, at the critical temper-ature, ε/T 4c 6.5. On comparing this with results shown in Fig. 12.3,we see that this result is consistent with the exponentially extrapolatedresults for −3.5 ≤ k ≤ −2.5 corrected for finite hadron volume, with150 MeV Tc 165 MeV, the center of the range of lattice simulations.Using only the known point hadrons, a slightly larger value of Tc 171MeV is found. Comparison of pressure, shown in Fig. 12.3, with the lat-tice result in Fig. 16.2 is more difficult but clearly the results are also inqualitative agreement.We have learned that the use in the field of heavy-ion collisions of a gasof point hadrons is justified because the contributions of probably stillunknown hadronic resonances and the excluded-volume effect approxi-mately cancel out. These remarks apply to all values of k we considered,12 Hagedorn gas 241though clearly the values k = −3.5,−3, and −2.5 are privileged by thecomparison with lattice-gauge-theory results, and the value k = −3 isalso central to the statistical-bootstrap model. We will now address thistheoretical framework which leads to the exponential mass spectrum. Westress that our measure of singularity k refers in this book always to thepoint-particle theory; consideration of the finite hadronic volume removesthe singular behavior of the hadronic energy density.12.2 The hadronic bootstrapTo study interacting hadrons in a volume V , we first consider the N -particle level density σN (E, V ), a generalization of Eq. (4.37). σN gener-ates the N -particle partition function,ZN (β, V ) =∫σN (E, V )e−βEdE. (12.9)For the non-interacting case, the number of states σN of N particles isobtained by carrying out the momentum integration Eq. (4.36) for eachparticle, keeping the total momentum /P = 0 and the energy E fixed. Wedivide by N ! for indistinguishable particles of degeneracy g and obtainσN (E, V ) =gNV N(2π)3NN !N∏i=1∫δ( N∑i=1εi − E)δ3( N∑i=1/pi)d3pi, (12.10)where the single-particle energy Eq. (4.31) is εi =√p2i +m2.If an interaction between these particles is such that they form a boundstate with mass m∗ and nothing else happens, then the level density ofthis new system, including the effect of interaction, would be describedas a mixture of ideal gases, one of mass m and the other of mass m∗. Thelogarithm of the partition function of such a system is additive, and theinteraction in the gas of the mass m is accounted for by allowing for thepresence of the second gas of mass m∗.Beth and Uhlenbeck [64] formulated this argument more precisely forthe case in which the interaction leads to the formation of a resonance ina scattering process, e.g.,π+N→ ∆→ π+N.In such a case, the Oth partial wave will be at large distances,ψ<(r, p) ∼1prsin(pr − Oπ2+ η<(p)), (12.11)where η<(p) is the phase shift due to scattering.242 Hot hadronic matterTo simplify, we argue in a manner similar to the study of the leveldensity above Eq. (4.37). We consider a large sphere of radius R. Thewave function Eq. (12.11) should vanish at r = R:pR− Oπ2+ η<(p) = n<π ; n< = 0, 1, 2, . . .. (12.12)n< labels the allowed two-body spherical momentum states {p0, p1, . . .}.The density of states of angular momentum O at p is ∆n −72 .12.3 Hadrons of finite sizeIn the first presentation of the SBM results and methods, we have consid-ered point-like hadrons in an arbitrary volume V . For a dilute gas, thisis a good approximation. However, when we formulated the bootstraphypothesis, we dealt with a system that has the density of the ‘inside’of a hadron. We now generalize this approach and introduce the volumeof the constituent cluster in the spirit of the quark model of hadrons,and confinement; see section 13.1. In the following we will also allow forclusters of finite baryon number.The natural volume V (m) of a hadron cluster is to be proportional tothe cluster mass,V (m) =m4B , (12.36)where B, which has the dimension of energy density, is the bag constant;see Eq. (13.9). This equation is valid in the restframe of the cluster. Fora cluster with 4-momentum pµ, Eq. (12.36) takes the formV µ(m) =pµ4B , (12.37)which defines the proper 4-volume V µ of the particle. In the clusterrestframe, Eq. (12.37) reduces to Eq. (12.36) and therefore is its uniquegeneralization. Each object of 4-momentum pi can be given a volume248 Hot hadronic matterV µi . All clusters have the same proper energy density ε0 = 4B. In arelativistically covariant formulation, we also consider the energy, andthe inverse temperature in terms of 4-vectors [261]:E → pµ = (p0, /p) ; pµpµ = p · p = m2, (12.38)1T→ βµ = (β0, /β) ; βµβµ = β · β = β2 =1T 2. (12.39)Note that the four-dimensional vector product p · p = (p0)2 − (/p)2 isrecognized by the absence of the vector arrow. In this notation,Z(V, T ) =∫ ∞0σe−E/T dE →∫ ∞0σe−β·pd4p. (12.40)We now need to obtain the covariant form of the N -finite-sized-particlelevel density σN (p, V, b) Eq. (12.10). These particles occupy ‘available 4-volume’∆µ = V µ −N∑i=1V µi . (12.41)∆ is the volume in which the particles move as if they were point-like,while in reality they have finite proper volumes and move in V . The leveldensity of extended particles in the volume V must be identical to that ofthe point-like particles in the available volume ∆. This means that, for asystem with baryon number b and 4-momentum p,σN (p, V, b) ≡ σNpt(p,∆, b), (12.42)where ‘pt’ refers to point-like particles. Equation (12.42) is, in spirit, aVan der Waals correction, which introduces a new repulsive interactioninto the system of hadronic resonances.The generalization to an invariant phase-space volume isV d3p(2π)3⇒ 2Vµpµ(2π)3δ0(p2 −m2) d4p. (12.43)To go back from the invariant form to the restframe we needδ0(p2 −m2) d4p =d3p2p0. (12.44)Then, in the restframe of the volume V , Eq. (8.17),2Vµpµ(2π)3δ0(p2 −m2) d4p = 2V p0d3p2p0=V(2π)3d3p. (12.45)12 Hagedorn gas 249Therefore, in the Boltzmann approximation, and assigning to each clus-ter the degeneracy g → τ(m2i , bi)dm2i with intrinsic baryon content bi,Eq. (12.10) generalizes to [143]σN (p, V, b)=1N !∫δ4(p−N∑i=1pi)δK(b−N∑i=1bi)×N∏i=12∆ · pi(2π)3τ(m2i , bi)δ0(p2i −m2i ) d4pi dm2i . (12.46)τ(m2i , bi) is the mass spectrum of a cluster with baryon number bi in themass interval [m2i , dm2i ]. It is the analog of ρ(m2i ) in Eq. (12.20). Thediscreet conservation of baryon number is assured by the Kronecker-δKfunction.The micro-canonical Lorentz-invariant density of states of a systemmade of any number of clusters, each cluster having any baryon numberbi, with −∞ < bi < ∞, readsσ(p, V, b)=∞∑N=11N !∫δ4(p−N∑i=1pi)∑{bi}δK(b−N∑i=1bi)×N∏i=12∆ · pi(2π)3τ(m2i , bi)δ0(p2i −m2i ) d4pi dm2i . (12.47)In Eq. (12.47), the contributing states are subdivided into any number ofsubsets corresponding to any partition of the total 4-momentum pµ andthe total baryon number b.The canonical partition function, for a fixed baryon number b, is theLaplace transform of the level density given by Eq. (12.47),Z(T, V, b)=∫e−β·p d4p∞∑N=11N !∫δ4(p−N∑i=1pi)∑{bi}δK(b−N∑i=1bi)×N∏i=12∆ · pi(2π)3τ(m2i , bi)δ0(p2i −m2i ) d4pi dm2i , (12.48)from which we obtain the grand-canonical partition function, Eq. (4.20),defined byZ(T, V, λ) =∞∑bλbZ(T, V, b) =∞∑b=−∞λb∫e−β·pσ(p, V, b) d4p, (12.49)where λ is the baryon-number fugacity corresponding to the baryonicchemical potential µ: λ = expµ/T .250 Hot hadronic matterTo implement Eq. (12.42), we postulateZ(T, V, λ)→ Z(T, 〈V 〉, λ) = Zpt(T,∆, λ). (12.50)Equation (12.50) permits us to calculate everything for fictitious pointparticles in ∆ and afterwards obtain the correct quantities by eliminating∆ in favor of a computed, average value 〈V 〉. Use of 〈V 〉 instead of Vconstitutes an approximation, and a lot of effort over the years, since thisapproach was first proposed [143], has gone into remedying this step ina consistent statistical-physics approach, and into generalizing the ideacontained in Eq. (12.42). A state-of-the-art calculation is given, e.g., in[170]. However, the original and physically simple model presented hereoffers all the required understanding without the ballast of mathematicalcomplexity, and yields sufficiently precise results.Zpt(T,∆, λ) can be written in the formZpt(T,∆, λ)=∞∑N=11N !∫δ4(p−N∑i=1pi)e−β·p d4p (12.51)×∞∑b=−∞λb∑{bi}δK(b−N∑i=1bi)N∏i=12∆ · pi(2π)3τ(p2i , bi) d4pi.The momentum δ4 function permits us to do the d4p integration and theδK permits the summation over b. The integrand thereafter splits into Nindependent identical integrals, and the sum over N yields an exponentialfunction. Taking its logarithm, we obtainlnZpt(T,∆, λ) ≡ lnZ(T, 〈V 〉, λ) = Z1(T,∆, λ), (12.52)whereZ1(T,∆, λ) ≡∫2∆ · p(2π)3τ(p2, λ)e−β·p d4p, (12.53)withτ(p2, λ) =∞∑b=−∞λbτ(p2, b). (12.54)All information about the interaction is contained in the ‘grand-canonical’hadronic mass spectrum τ(m2, λ).We obtain now the relation between 〈V 〉 and ∆ in the restframe. Weuse Eq. (12.37) to find the expectation value of the volume:〈V µ〉 = ∆µ + pµ4B → ∆+〈E〉4B∣∣∣∣restframe. (12.55)12 Hagedorn gas 251The energy density ε(β, λ) can be obtained from Eq. (12.52), for the en-ergy:〈E〉 = − ∂∂βlnZ(β, 〈V 〉, λ) = − ∂∂βlnZpt(β,∆, λ). (12.56)Since lnZpt is linear in ∆, the last term is equal to ∆εpt(β, λ); hence,ε(β, λ) =∆εpt(β, λ)〈V 〉 . (12.57)Inserting Eq. (12.55) into Eq. (12.57) and solving for 〈E〉, we findε(β, λ) =εpt(β, λ)1 + εpt(β, λ)/(4B), (12.58)which we have used in Eq. (11.1).We can use Eq. (12.58) in Eq. (12.55) to obtain a more explicit relation-ship between the volume V and the available volume ∆:〈V 〉=∆(1 +εpt(β, λ)4B), (12.59a)∆= 〈V 〉(1− ε(β, λ)4B). (12.59b)This procedure can be followed for the baryon density, pressure and, inprinciple, other statistical quantities:ν(β, λ) ≡ 〈b〉〈V 〉 =νpt(β, λ)1 + εpt(β, λ)/(4B), (12.60)P (β, λ) =Ppt(β, λ)1 + εpt(β, λ)/(4B). (12.61)12.4 Bootstrap with hadrons of finite size and baryon numberAs explained in section 12.2 a system of total mass m, when it is com-pressed to its natural volume Vc(m), becomes one of the particles countedin the hadronic mass spectrum (see Fig. 12.4). By the same token, anuclear cluster with baryon number b compressed to its natural volumeVc(m, b) becomes a cluster appearing in the mass spectrum τ(m2, b). Thebootstrap hypothesis can now be expressed by writingσ(p,∆, b)|〈v〉→vc(m,b) ⇐⇒ τ(p2, b), (12.62)252 Hot hadronic matterwhere ⇐⇒ means ‘corresponds to’ in a way to be specified.With the condition Eq. (12.62), the statistical-bootstrap-model equa-tion for τ arises from Eq. (12.47):Hτ(p2, b)=Hgbδ0(p2 −m2b) +∞∑N=21N !∫δ4(p−N∑i=1pi)×∑{bi}δK(b−N∑i=1bi)N∏i=1Hτ(p2i , bi) d4pi, (12.63)whereH ≡ 2m20(2π)34B . (12.64)This equation is obtained, by first separating the ‘input particle’ (cor-responding to N = 1) in Eq. (12.47), and then making the following re-placement, where Eq. (12.37) is used:σ(p, Vc, b)⇒2Vc(m, b) · p(2π)3τ(p2, b)⇒ 2m20(2π)34B τ(p2, b), (12.65a)2∆ · pi(2π)3τ(p2i , bi)⇒2m20(2π)34B τ(p2i , bi). (12.65b)The factors m2 and m2i have been absorbed into the definition of τ(p2i , bi).Either H or m0 may be taken as the new free parameter of the model.The first term in Eq. (12.63), the ‘input-particle’ term, comes from thecluster structure: if clusters consist of clusters, which consist of clusters,and so on, this should end at some ‘elementary’ particles, here a hadronof baryon number b and of mass mb. Typically, the input consists of thepion for the b = 0 term and the nucleon for b = ±1. The similarity ofEq. (12.63) to Eq. (12.20) allows us to repeat all the steps we made insolving Eq. (12.20), to obtain from Eq. (12.63) the asymptotic form of thehadronic mass spectrum. We introduce two functions ϕ(β, λ) and Φ:ϕ(β, λ)≡∫e−β·p∞∑b=−∞λbHgbδ0(p2 −m2b) d4p,=2πH∞∑b=−∞λbgbm2bK1(mbβ)mbβ, (12.66)andΦ(β, λ) ≡∫e−β·p∞∑b=−∞λbHτ(p2, b) d4p. (12.67)12 Hagedorn gas 253Once the set of ‘input particles’ is introduced, ϕ(β, λ) is a known function,while Φ(β, λ) is unknown. By applying the double Laplace transform(integration over p and summation over b) used in the definition of ϕ(β, λ)and Φ(β, λ) to the Eq. (12.63), we obtainΦ(β, λ) = ϕ(β, λ) + eΦ(β,λ) − Φ(β, λ)− 1. (12.68)This implicit equation for Φ can be solved again without regard to theactual dependence on β and λ. WritingG(ϕ) ≡ Φ(β, λ), (12.69)we obtainϕ = 2G(ϕ)− exp[G(ϕ)] + 1, (12.70)which is Eq. (12.25). The graphical solution found in section 12.2 showsthat G(ϕ, λ) has a square-root singularity atϕ(β, λ)→ ϕ0 = ln(4/e), (12.71)which defines a critical curve βcr(λ) in the (β, λ) plane. In the vicinity ofthis curve,G(ϕ) ≈ G0 + constant×√ϕ0 − ϕ, (12.72)and, therefore,Φ(λ, β βcr) = Φ0 − C(λ)√β − βcr. (12.73)As we have shown in section 12.2, this square-root singularity fixes thepower k of m in the hadronic mass spectrum at k = −3 and we obtainτ(m2, λ) ∝ m−3eβcr(λ)m. (12.74)For λ = 1, βcr = β0 and we recover the usual form of the hadronic massspectrum. However, the generalization obtained gives a solution for anyvalue of λ.Given the mass spectrum in Eq. (12.52), we can calculate all the usualthermodynamic quantities. For this, we need to write down lnZpt. Theformal similarity of Eq. (12.53) and Eq. (12.67) yields a relation betweenlnZpt and Φ that is best expressed in the restframe of ∆ and β:lnZpt(T,∆, λ) = −2∆(2π)3H∂∂βΦ(β, λ). (12.75)254 Hot hadronic matterThe point-like quantities are derived from lnZpt as given by Eq. (12.75):εpt(β, λ) =2(2π)3H∂2∂β2Φ(β, λ), (12.76)νpt(β, λ) ≡λ∆∂∂λlnZpt(β,∆, λ) = −2λ(2π)3H∂∂λ∂∂βΦ(β, λ), (12.77)Ppt(β, λ) ≡T∆lnZpt(β,∆, λ) = −2T(2π)3H∂∂βΦ(β, λ). (12.78)All the above point-particle quantities involve derivatives of Φ(β, λ);they become singular at ϕ = ϕ0. Explicitly, using Eq. (12.72), whichcontains a square-root singularity, we have:∂∂βΦ(β, λ) =dGdϕ∂ϕ∂β→ constant× ∂ϕ∂β1√ϕ0 − ϕ. (12.79)Therefore, ϕ → ϕ0 implies point-particle infinities for all of the abovequantities, with the second derivative required in εpt being the most sin-gular. On comparing the degrees of divergence of the numerator anddenominator in Eqs. (12.58), (12.60), and (12.61), we see that the energydensity and the baryon density are finite, while the pressure vanishes, onthe critical curve. The overcompensation of the pressure is seen alreadyin Fig. 12.3, which was evaluated with a model hadron mass spectrum.This behavior of the pressure reflects the fact that we have countedonly the pressure generated by the clusters and, as we shall see in thefollowing subsection, all clusters coalesce on the critical curve, and hencethe pressure of a single large cluster vanishes. This of course is an artifact,since at that point we should have included the internal cluster pressure,since the single cluster we find is in the QGP-type state. We will notintroduce in this book the required generalization which can be found in[213].12.5 The phase boundary in the SBM modelWe have seen that the singular point of the solution to the bootstrapequation is located at the value ϕ0 = ln(4/e) and that the critical curvein the (β, λ) plane is defined byϕ(β, λ) = ϕ0 = ln(4/e). (12.80)Its position depends, of course, on the actual content of ϕ(β, λ), i.e., on thefundamental set of ‘elementary particles’ {mb, gb} and the value of theconstant H, Eq. (12.64). In the case of three elementary pions (π+π0π−)12 Hagedorn gas 255b= 4 , = 0QGP00.510.05 0.10 0.15 [GeV]0TT B Pµ [GeV]Hadron gasεFig. 12.6. The SBM critical curve in the (µb, T ) plane. In the shaded region,the theory is not valid because we have neglected Bose–Einstein and Fermi–Diracstatistics.and four elementary nucleons (spin ⊗ isospin) and four antinucleons, weobtain, from Eq. (12.66), the relationϕ0(βcr, λcr) = 2πHTcr[3mπK1(mπTcr)+ 4(λcr +1λcr)mNK1(mNTcr)]. (12.81)The condition of Eq. (12.81), written in Tcr = 1/βcr and µcr = Tcr lnλcr,yields the critical curve shown in Fig. 12.6, drawn for H = 0.724 GeV−2.For µ = 0, the curve ends at T = T0, which becomes the maximum phasetransition temperature instead of a limiting temperature.Our system consists, for small T and µ, of nucleons and nuclei. Forincreasing T , creation of pions sets in and finally also creation of baryon–antibaryon pairs, as well as (not included here) creation of strange had-rons. If the latter is taken into account, the input set of ‘elementary par-ticles’ has to be enlarged. This changes slightly the position of the criticalcurve and the equations of state of hadron matter, since T0 is of the orderof the pion mass, while the other particles have larger masses and make lit-tle contribution to ϕ(β, λ). More precisely, each new conserved quantumnumber (strangeness, charm, . . . ) gives rise to another corresponding fu-gacity λ; hence the singularity is defined by ϕ(β, λ1, λ2, λ3, . . ., λn) = ϕ0as a hypersurface in an (n + 1)-dimensional space. Since, however, inphysical situations, generally only the baryon number is different from256 Hot hadronic matterzero, we have to consider only the intersection of this hypersurface withthe (T , µb) plane. That procedure yields the curve which was said to belittle different from the one shown in Fig. 12.6.What does the SBM hadron gas do when it approaches the criticalcurve? As the point-particle quantities εpt, νpt, and ppt diverge, one sees(by comparing degrees of divergence when ϕ → ϕ0) thatε(βcr, λcr) → 4B, (12.82a)ν(βcr, λcr) → νcr(βcr, λcr) = 0, (12.82b)P (βcr, λcr) → 0, (12.82c)∆(βcr, λcr) → 0, if 〈V 〉 = 0, (12.82d)〈V (βcr, λcr)〉 → ∞, if ∆ = 0, (12.82e)where βcr and λcr are the values along the critical curve.As noted already, see Eq. (12.36), the energy density of our clusters wasconstant and always equal to 4B. Equation (12.82a) suggests that, on thecritical curve, the whole hadron system has condensed into one giant clus-ter, witnessed by the vanishing of the pressure; one can explicitly see that,for any given external volume 〈V 〉, the number 〈N〉 of particles (clusters)contained in it goes to zero on the critical curve: indeed, introducing thefugacity ξ relative to the number of clusters, Eq. (12.52) can be written:Zpt(β,∆, λ) = Z(ξ)pt (β,∆, λ, ξ)|ξ=1 ≡∞∑N=01N !(ξ lnZpt)N . (12.83)Hence, with Eq. (12.75),〈N〉 = ξ ∂∂ξlnZ(ξ)pt∣∣∣∣ξ=1= lnZpt = −2∆(2π)3H∂∂βΦ(β, λ), (12.84)and, with Eq. (12.59a),〈N〉〈V 〉 =Bπ3H∂Φ(β, λ)/∂β1 + εpt(β, λ)=⇒critical curve0, (12.85)because εpt contains a second derivative of Φ(β, λ). It follows that, fromEqs. (12.61), (12.78), and (12.85),P 〈V 〉 = 〈N〉T, (12.86)that is, our hadron gas obeys, formally, the ideal-gas equation of state forthe average number of clusters 〈N〉; 〈N〉 is not a constant, but a functionof β and λ.In the bootstrap model of hadronic gas, our finding is that the criticalcurve limits the HG phase; approaching it, all hadrons dissolve into a giant12 Hagedorn gas 257Fig. 12.7. The physical interpretation of the different regions of the (T, µ) planeaccording to the statistical-bootstrap model of hadronic matter.cluster. The gradual change in the structure of hot hadronic matter isillustrated in Fig. 12.7; at low T, µ we have a dilute pion gas of essentiallypoint-like pions. With an increase in T and/or µ, progressively denserhadron matter is formed, and hadron proper volume becomes relevant.Near the phase boundary hadrons coalesce into large clusters comprisingdrops of QGP.In SBM the singular curve is reached with finite energy density 4B. Inthe hadron phase, ε(β, λ) < 4B and β ≤ βcr. For ε > 4B, we enter intoa region that cannot be described by the thermodynamics of the SBM.Indeed in this region, β ≥ βcr and the partition function Zpt(β,∆, λ) andall densities become complex. This region cannot be described withoutmaking assumptions about the inner structure and dynamics of the ‘el-ementary particles’ {mb, gb} – here pions and nucleons – entering intothe input function ϕ(β, λ). In other words, to continue, we need to con-sider the hot hadron interior made of quarks and gluons. Assuming thatwe have a phase transition between a HG and a QGP, the evolution ofthe system in the P–V diagram is qualitatively illustrated in Fig. 3.2 onpage 49. In order to make this picture quantitative we need to explore,within the realm of quantum chromodynamics, the hadron structure andthe behavior of a gas of quarks and gluons with color interactions.VQCD, hadronic structure and hightemperature13 Hadronic structure and quantum chromodynamics13.1 Confined quarks in a cavityA hadronic particle, according to section 3.1, is a quark-filled bubble, a‘swiss-cheese’ hole, in the structured vacuum. The highly excited dropof QGP is indeed much akin to the picture of an individual, colorlesshadron, except that it is the thermal pressure that acts against the vac-uum pressure, not the quantum pressure. As a first step in a more detaileddiscussion of the QGP phase, we briefly discuss how this approach allowsus to understand properties of individual hadrons.In the quark-bag model of hadronic structure, colorless qqq baryons orq̄q mesons are embedded in the structured vacuum sea. In a calculationalframework proposed by Bogoliubov [78], independent quarks confined bya static Lorentz-scalar potential with infinite walls were considered. Thisis ensuring permanent ‘confinement’ of the constituents within a givenvolume. The interest in this approach grew only after it was understoodthat the confining potential is not to be derived from quark–quark in-teractions, but that it arises from the repulsion of colored quarks by thestructured QCD vacuum state.The structure of hadrons emerges on considering a static sphericalstate in which residual quark–quark interactions are introduced. ThisMIT-bag model is able to capture most features of the hadron spectrum[92, 93, 99, 151]. Our limited objective is to extract from a study of thehadronic spectrum information about the latent heat of the vacuum B,and the mass of the strange quark ms. To accomplish this we will notneed to introduce in this book improvements addressing the restorationof translational invariance, and the absence of chiral symmetry; see sec-tion 3.3. For further details on chiral symmetry and the bag model, werefer the interested reader to [94, 256, 258].25813 Hadronic structure and quantum chromodynamics 259Assuming that quarks are moving independently inside a region ofspace, the mass Mh of the bound-state system of quarks (bag) comprisesthe kinetic energy of confined quarks, as well as the volume energy of thedisturbance in the vacuum,Mh(Rh) =∑i εiRh+4π3R3hB + δEV +∆Mmag, (13.1)where εi is the (dimensionless) eigenvalue energy coefficient of the ithquark in a static cavity of radius Rh. Since we are considering the lowestpossible hadronic states, this term becomes proportional to the numbern = 2 or 3 of valence quarks and antiquarks in mesons or baryons, re-spectively:∑i εiRh→ n ε0Rh. (13.2)We have further introduced, in Eq. (13.1), the finite-volume correctionto the vacuum energyδEV =z0Rh(13.3)in Mh(Rh). Since this is a term independent of the number of quarks inthe bag, not requiring a dimensioned constant, a judicious choice of thenumber z0 allows for many other effects.A residual interaction must be introduced in order to describe the en-ergy splitting between the baryon octet with j = 12+ and the decupletwith j = 32+, see Fig. 2.1, and similarly between the nonets of pseudo-scalar j = 0− and the vector mesons j = 1−; see Fig. 2.2 on page 27. ACoulomb-like interaction could not accomplish this, since it can not distin-guish among the different angular-state multiplets. Akin to the magneticfield which splits the spin states, an interaction of magnetic (hyperfine)type is needed,∆Mmag =∑i>j〈αsrijtai taj/σi · /σj〉=1Rh∑i>jcijhij . (13.4)This is the usual form with the spin-Pauli matrices /σi, Eq. (13.27), andspecific to the color interaction, SU(3) generators ta, Eq. (13.58). Theimportant feature of this color-magnetic hyperfine interaction is that itdoes reflect correctly the signs, and even the magnitude, of the splittingsbetween various hadronic multiplets.The coefficients cij , in Eq. (13.4), are found by evaluatingcij = 〈h |(ta/σ)i · (ta/σ)j |h〉 . (13.5)260 QCD, hadronic structure and high temperatureThis is done by methods developed in the study of hyperfine QED inter-actions, and we defer this discussion. hij , in Eq. (13.4), are the transition-matrix elements of magnetic moments, which incorporate the coupling αs,but where we have taken out the dominant dimensional factor 1/Rh.The mass of the hadronic state is dependent on the size parameter Rh.In the absence of any other dimensioned constants (such as quark masses),but with a constant ah containing quark and interaction contributionsspecific to each hadron state, we haveMh(Rh) =4π3R3hB +ahRh. (13.6)There is a clearly defined minimum in Eq. (13.1) as a function of Rh, atwhich the forces associated with the vacuum and quarks balance. Thephysical state has a mass associated with this minimum:∂Mh∂Rh= 0 . (13.7)This condition,ah4πR4h− B = 0, (13.8)is equivalent to the pressure equilibrium point between the internal Fermipressure and the exterior vacuum pressure (negative pressure seen fromthe interior).Reinserting the result of Eq. (13.7) into Eq. (13.6), we find that thevolume and mass of a quark-bound state are related:Mh = 4BVh, RhMh = 43ah, Mh =43a3/4h (4πB)1/4. (13.9)Aside from the bag constant, B, the hadron-state-specific value of ahdetermines the value of each hadron mass. To determine ah in the studyof hadronic spectra based on quark-cavity states, section 13.2, the fiveparameters εq and εs (the energies of light and strange quarks), and hqq,hqs, and hss (the (transition) magnetic moments seen in Eq. (13.1)) are setto the values expected from the structure of the unperturbed bag model,and the values of the elementary parameters αs and ms are fitted, alongwith z0; thus one looks at four parameters. However, much more preciseinformation on the magnitude of the bag constant, B, is obtained in anapproach in which one takes Eq. (13.1) as the starting point. In Eq. (13.1),aside from B, six more parameters enter: z0, εq, and εs; and hqq, hqs, andhss. One of the results of such an approach is the verification of how wellhadronic structure is described by quark-cavity states.Owing to the coincidence that the strange-quark mass and hadron ra-dius have the same scale msRh 1, there is considerable sensitivity to13 Hadronic structure and quantum chromodynamics 261the exact value of the strange quark ms, which determines the energy ofthe strange quark:εfits =√m2s +x2sR2h. (13.10)Considering the running of the mass of the strange quark predicted byQCD, see Fig. 17.4 on page 328, the effective mass of the strange quarkshould vary with the hadronic size, e.g., according toms(Rh) ≡ m0s ln(πRhΛh). (13.11)The variation of the mass of the strange quark with the quark momentumintroduces the eighth parameter Λh.These considerations are quite successful at describing the hadronicspectrum [20]. Aside from the effect of the quark mass in the strangehadrons, the differences in mass between the various hadronic multipletsarise from the differences in the quantum numbers which influence thevalue of cij , Eq. (13.5). Thus, the value of ah, Eq. (13.6), depends both onthe quantum numbers of the multiplet and on the quark-flavor content.A unique fit with a significant confidence level arises, yieldingεfitq = 1.97± 0.02. (13.12)The value of εfitq is close to the massless-quark value x0 = 2.04 derivedfrom solution of the Dirac equation in a cavity, see section 13.2. Thisresult is thus providing an empirical foundation for the bag model ofhadrons.The values m0s = 234± 14MeV and Λh = 240± 20MeV are found [20].The effective mass of the strange quark ms(Rh) varies between 170 MeVin kaons, which are relatively small (RK = 0.5 fm), and about 320 MeV instrange baryons. Using a fixed mass for the strange quark, ms = 280MeV[99], was obtained. The QCD-motivated variability with hadron size ofms in each hadron leads to a much better fit to the hadronic spectrum,and yields for the value of the vacuum energy BB = (171MeV)4, 4B = 0.45GeV fm−3. (13.13)The bag constant is larger than the original MIT result, BMIT = (145MeV)4. The main reason for the difference from the MIT fit arises fromthe allowed variation in mass of the strange quark with the size of thehadron. There is a remaining systematic dependence of the results forB on the here assumed behavior, Eq. (13.11). However, Eq. (13.13) isin much better agreement with the value of B required to describe thelattice-pressure results, for which we need a still larger value of B, seeEq. (16.12).262 QCD, hadronic structure and high temperatureSince the matrix elements hqq, hqs, and hss comprise αs, a value for αsis not determined within this procedure. We note that in the cavity bagmodel, αs = 0.55 is found.∗13.2 Confined quark quantum statesSince the short-range interactions among quarks can be ignored in thefirst instance, the ‘independent’ quark wavefunction ψq obeys the Diracequation,iγµ∂µψq −Mψq + (M −m)ΘVψq = 0, (13.14)where ΘV = 1 inside the quark bag and ΘV = 0 outside. Inside thevolume, the dynamics of quarks is governed by the (small) mass m, whileoutside it is subject to the mass M . Since, despite a great effort, we havenot discovered free quarks, M must be very large, and the idea of colorconfinement indeed requires M → ∞, in order to have quarks confinedforever.However, that limit is not trivial as it turned out. In a series of pub-lications in 1974–75, the MIT-bag model was developed [92, 93, 151] ina way that creates the framework for quark dynamics with confinement,M → ∞. Novel physical concepts were introduced since the confinementcondition broke conservation of energy–momentum at the confinementboundary. Namely, in the limit M → ∞, there is no quark quantumwave outside of the hadron volume, and, in order to have a stable phys-ical state, the internal quantum pressure of the confined quarks must bebalanced at the confinement boundary by some new external pressurepointing inward.In a Lorentz-covariant formulation, along with pressure, there is alsoenergy density, which had to be lower outside of the bag than inside,in order to have the inward-pointing pressure. The improvement of theenergy–momentum tensor inside the volume region occupied by quarksincludes the covariant bag term:Tµν = Tµνfields + gµνB. (13.15)On comparing this with, e.g., Eq. (6.6), we indeed see that the bag termincreases the energy density, and decreases the pressure within the volumeoccupied by quantum particles.On physical grounds, the size of the system is determined by balancingthe internal pressure against the external pressure, and thus B has toenter into the dynamics of quark fields. We follow, in the next few lines,∗ In some literature, a four-fold greater value of αs = 2.2 is quoted; the difference arisesfrom the factor 12which relates λa to the generator ta in Eq. (13.56).13 Hadronic structure and quantum chromodynamics 263the summary of the situation presented in [256]. We will need the surface∆S-function, which arises from the volume function in a familiar way,∆S = −nµ∂µΘV, (13.16)where nµ is the outward space-like normal to the surface of the bag. Thestatic spherical cavity, in spherical coordinates, reads as usual:δ(Rh − r) =dΘ(Rh − r)dr. (13.17)The action which fully accounts for all physical aspects isS =∫d4x[(∑q12ψqiγµ↔∂ µψq − B)ΘV − 12(∑q ψqψq)∆S], (13.18)where ψq = ψ†qγ0, and γµ are Dirac matrices, Eq. (13.26).To obtain the dynamic equations, we perform variation of the actionseeking its stationary point:ψq → ψq + δψq, ψq → ψq + δψq. (13.19)Furthermore, the geometry-defining volume ΘV and surface ∆S functionschange under variation,ΘV → ΘV + ε∆S, ∆S → ∆S − εnµ∂µ∆S. (13.20)For a spherical cavity with nµ = (0, r̂), the following three equations ofmotion give the stationary point of the action:the Dirac equation,iγµ∂µψq(x) = 0, x ∈ V; (13.21)the linear boundary condition,iγµnµψq(x) = ψq(x), x ∈ S; (13.22)the quadratic boundary condition,12nµ∂µ(∑qψq(x)ψq(x))− B = 0, x ∈ S. (13.23)A solution satisfying the boundary condition, Eq. (13.22), satisfies therequirement that the normal flow of quark current through the surfaceof the bag vanishes. To see this, we write this condition, along with theadjoint form:iγµnµψq |S= ψ |S, −iψqγµnµ |S= ψ |S . (13.24)264 QCD, hadronic structure and high temperatureTo obtain the adjoint form, we used γµ † = γ0γµγ0. We construct theoutward current at the surface:nµjµ |S= nµ(ψγµψ) |S= ±ψψ |S= 0. (13.25)Since the right-hand side of Eq. (13.25) is both positive and negative ofthe same value, as can be seen by using one of the two forms of Eq. (13.24),it must be zero.Since there is no flow of probability through the surface, the linearboundary condition guarantees confinement of quarks. Moreover, thisboundary condition allows us to determine the eigenenergies of quarks ina cavity of a given size Rh, as we shall discuss. On the other hand, thequadratic boundary condition Eq. (13.23) allows us to find the size of thesystem, establishing a balance of forces at the surface; see Eq. (13.8).We now proceed to obtain examples of solutions of the bag-model dy-namic Eq. (13.21), in the static-cavity approximation. γµ are the usualcovariant Dirac matrices and we use the Bjørken–Drell conventions [74]:γ0 ≡ β ≡[I2 00 −I2], γi ≡ βαi ≡[0 σi−σi 0]. (13.26)Here, I2 is a unit 2 × 2 matrix, and /σ = (σ1, σ2, σ3) are Pauli’s spinmatrices:σ1 =[0 11 0], σ2 =[0 −ii 0], σ3 =[1 00 −1]. (13.27)We are interested in the lowest-energy solution of the Dirac equation,Eq. (13.14), for a stationary spherical cavity. We consider the wave func-tionψq(/r, t) = qn(/r)e−iωntτq. (13.28)τq is the flavor (e.g., isospin or SUf(3)) part of the independent parti-cle wave function, and ωn is the nth-state eigenenergy. The stationaryquark 4-spinor wave function q(/r) satisfies the equation (suppressing thequantum number(s) subscript ‘n’)(/α · /p+ βm− ω)q=0,((m− ω)I2 /σ · /p/σ · /p (m+ ω)I2)(quqd)=0, (13.29)where qu are the upper and qd the lower quark 2-spinor components.13 Hadronic structure and quantum chromodynamics 265When there is no potential inside the bag, each of the four components‘k’ of the spinor q has to satisfy the energy–momentum condition obtainedby ‘squaring’ the Dirac equation, i.e., the Klein–Gordon equation:[ω2 −m2q − (i /∇)2]qk = 0. (13.30)We recall the spherical decompositions/∇ = r̂ ∂∂r− i r̂r× /L, /L = /r × i/∇, (13.31)/∇ 2 = d2dr2+2rddr−/L2r2, (13.32)and recognize that the components of the Dirac cavity solution have theformqk = Njl(x r/Rh)l∑µ=−lclµjjz(k)Ylµ(Ω), (13.33)where x is obtained from an eigenvalue condition, and Ylµ(Ω) are the usualspherical functions of fixed angular momentum l, µ. The Clebsch–Gordancoefficients clµjjz are fixed by construction of a spinor spherical wave ofgood total angular momentum j and its z-axis projection, j ≤ jz ≤ −j.Equation (13.30) now implies thatω ≡ εRh=√x2R2h+m2q. (13.34)The no-node, lowest-energy quark-cavity solution isqu0 = N0j0(x r/Rh)χs, j0(z) =sin zz. (13.35)We use the 2-spinor χs, s = ± 12 for spin-up and -down quarks:χ1/2 =(10), χ−1/2 =(01). (13.36)To obtain the corresponding lower components of the Dirac spinor, weuse Eq. (13.29):qd =1ω +m/σ · i /∇qu. (13.37)The spherical decomposition Eq. (13.31), along with /Lqu0 = 0, and thespherical-Bessel-function property,j1 = −ddzj0 =sin zz2− cos zz, (13.38)266 QCD, hadronic structure and high temperatureyields the lowest angular (j = 12) quark wavefunction Eq. (13.28):ψq(/r, t) = N j0(xr/Rh)χsxRh(ω +mq)j1(xr/Rh)iσrχse−iωtτq. (13.39)The radial spin matrix has been introduced:σr = r̂ · /σ, σ2r = I. (13.40)Equation (13.39) can easily be cast into the form Eq. (13.33), with thelower components qd having l = 1.The linear boundary condition, Eq. (13.22), at the surface of the bag,that is at a given radius Rh of the bag, takes the form−i(/γ · /n)ψ |s= ψ |s, (13.41)which eigenvalue condition will now be used to fix the value of x. Thesurface-normal vector for a spherical bag is /n = r̂. The boundary condi-tion reads−iσrqdχs |r=Rh= quχs |r=Rh , iσrquχs |r=Rh= qdχs |r=Rh . (13.42)Using Eq. (13.39), we obtainj0(x) =x√x2 + (mqRh)2 +mqRhj1(x), (13.43)which takes the explicit form [93]1− x cotx =√x2 + (mqRh)2 +mqRh. (13.44)WhenmqRh → 0, the lowest eigenvalue is x0 = 2.04. The (kinetic) energyof a massless quark in the confining radius Rh isωbag(mq = 0) =2.04Rh. (13.45)The first radial excitation found, on solving Eq. (13.44) for the second-lowest eigenvalue, is relatively high, with x1 = 5.40, more than doublingthe kinetic energy.For massive quarks we haveωbag =√x20 + (mqRh)2Rh, 2.04 < x0 < π. (13.46)As indicated in Eq. (13.46), with increasing mqRh, x0 increases, neverreaching the singularity of cotx at x0(mqRh → ∞) = π.13 Hadronic structure and quantum chromodynamics 267The quark-wave-function normalization N , in Eq. (13.39), is easily ob-tained:∫d3r ψqγ0ψq = 4πN2R3h∫ 10dz z2(j20(xz) +x1− x cotxj21(xz)).We have used the eigenvalue condition Eq. (13.44) and substituted z =r/Rh. The quadratic boundary condition Eq. (13.23) isB = 12ddr(∑qψqψq))∣∣∣∣∣Rh. (13.47)We leave it to the reader as an exercise to show that Eq. (13.47) is indeedequivalent to the intuitive requirement that the total energy contained inthe bag volume be at a minimum with respect to the radius of the bag,Eq. (13.7).13.3 Nonabelian gauge invarianceThe color hyperfine magnetic interaction, Eq. (13.4), is the residual forcebetween quarks in the perturbative vacuum. It defines the hadron spec-trum. To understand this force properly, we need to understand quantumchromodynamics and, in particular, the quark–quark interaction better.Akin to spin, the color charge of quarks is an internal quantum number,but it resembles in its properties more the electrical charge, so much sothat one also speaks of ‘color charge’: like electrical charge, color chargeis thought to be the source of a force field. Since we have so far not beenable to build an apparatus to distinguish among the three fundamentalcolors, akin to the way electro-magnetic fields differentiate the spin andcharge states, all colors must appear on an exactly equal footing.Therefore, the theory of color forces, i.e., quantum chromodynamics(QCD), is based on the principle of gauge invariance extended to includeinvariance under arbitrary rotations (redefinition of color-principal ‘axis’)in the three-component color space. The resulting theory of strong inter-actions based on color forces has been called quantum chromodynamicsin order to underline its formal similarity to quantum electrodynamics(QED). The form of the QCD Lagrangian is the generalization of QEDrequired when one is considering invariance under local gauge color trans-formations.Before introducing QCD, let us recall how, within QED, the princi-ple of gauge invariance operates. It allows for changes in the electro-magnetic potential, leaving the electro-magnetic fields Fµν , i.e., /E and/B, unchanged. The electro-magnetic potential Aµ = (A0, /A) is thus not268 QCD, hadronic structure and high temperaturedefined uniquely. We further recall that a measurement, in general, doesnot allow one to observe the phase factor of a quantum wave.We now show how a change in the choice of the quantum phase isrelated to the change in gauge of the potential Aµ. The effect of a localchange in the phase of the wave function,ψ → ψ′ = e−iαψ, (13.48)∂µψ → ∂µ(e−iαψ) = e−iα[∂µψ − (i∂µα)ψ], (13.49)can be compensated for by the simultaneous gauge transformation of theelectro-magnetic potential Aµ:Aµ(x)→ A′µ(x) = Aµ(x) +1e(∂µα). (13.50)This occurs if the quantum fields and potential are ‘minimally coupled’:[(∂µ + ieAµ)ψ]′ = (∂µ + ieA′µ)ψ′ = e−iα(∂µ + ieAµ)ψ. (13.51)We see that the generalized covariant derivative,∂ν → Dν = ∂ν + ieAν , (13.52)remains gauge invariant, up to an overall phase factor e−iα.We will now generalize the principle of gauge invariance to the case ofQCD. The additional difficulty is the non-commutative, i.e., nonabelian,aspect of the transformation, which is associated with the fact that thereis not just one but several charges, i.e., colors. which a particle can carry:the usual 4-spinor wave function of a spin-12 particle, ψ, becomes in ourcase a component of a 12-spinor in color space:Ψ = ψrψgψb. (13.53)As long as the RGB (red, green, blue) ‘color’ is not observable, the color-gauge transformation generalizing Eq. (13.48) can, apart from introducinga phase, also rotate (mix) the color components of the wave function. Thearbitrariness of the quantum wave is now expressed by the transformationΨ→ Ψ′ = VΨ, V † = V −1, det(V ) = 1. (13.54)Since the complex rotations of a three-dimensional spinor-vector are de-scribed by unitary 3 × 3 matrices V = (vik) of unit determinant, thesymmetry group of the gauge transformations is SUc(3), where the sub-script reminds us of color, and will be omitted where confusion with flavorsymmetry SUf(3) is unlikely.13 Hadronic structure and quantum chromodynamics 269The flavor SUf(3) symmetry arising from the near degeneracy in energyof the three ‘light’ u, d, and s quarks is quite different in its nature. Thecolor symmetry permits us to rename locally the color of quarks. Theflavor symmetry is a global symmetry: once a definition of flavor has beenchosen at CERN it applies at the BNL, as long as both laboratories belongto a region of the Universe occupying the same vacuum state. However,even this situation has an exception, namely when, after chiral symmetryhas locally been restored in a high-energy heavy-ion collision, the chiral-symmetry-breaking vacuum is reformed, and the dynamic processes thatare occurring could lead to a physical vacuum state in which the definitionof flavor is different from that already established in the remainder of theworld. We will not further discuss in this book these disoriented chiralstates, which are a current topic of research.Any unitary matrix V with det(V ) = 1, Eq. (13.54), can be written asthe imaginary exponential of a hermitian traceless matrix L:V = exp(iL), L† = L, tr(L) = 0. (13.55)All traceless hermitian 3 × 3 matrices can be expressed as linear combi-nations of the eight generators ta of the Lie group SU(3), using eight realvariables θa:L =128∑a=1θaλa, ta ≡12λa. (13.56)The ‘fundamental’ (Gell-Mann) 3× 3-matrix representation of the SU(3)algebra is well known. There are n2c − 1 = 8, with nc = 3 for SUc(3),λa, a = 1, . . ., 8, matrices:λ1 =(0 1 01 0 00 0 0), λ2 =(0 −i 0i 0 00 0 0), λ3 =(1 0 00 −1 00 0 0),λ4 =(0 0 10 0 01 0 0), λ5 =(0 0 −i0 0 0i 0 0), λ6 =(0 0 00 0 10 1 0),λ7 =(0 0 00 0 −i0 i 0), λ8 =1√3(1 0 00 1 00 0 −2). (13.57)The λa have been constructed in close analogy to the Pauli matrices σi,Eq. (13.27), and the ta are analogous to the spin matrices si = σi/2. Thefirst three λa correspond (up to the added third trivial row and column)to the Pauli matrices. While λ8 is the second traceless diagonal 3 × 3270 QCD, hadronic structure and high temperaturematrix we can construct, the pairs λ4, λ5 and λ6, λ7 are generalizationsof the SU(2) σ1 and σ2 matrices, which are similar to λ1 and λ2.The following commutation and anticommutation relations can be usedto define the algebra of the SU(3) group:[ta, tb] = i∑cfabctc, (13.58){ta, tb} = (1/3)δabI3 +∑cdabctc, (13.59)where I3 is the 3 × 3 unit matrix. fabc and dabc are the antisymmetricand symmetric ‘structure constants’, respectively, of the Lie group SU(3),which can be determined in a straightforward fashion from Eqs. (13.58)and (13.59). One of their frequently used properties isn2c−1∑k,l=1fiklfjkl = ncδij . (13.60)Another important relation, which is related to the definition of the colorcharge, is found by considering the trace of Eq. (13.59):tr(tatb) = 12δab. (13.61)A second frequently needed representation of the SU(3) algebra interms of the 8 × 8 matrices is called ‘adjoint’ representation. This rep-resentation plays with regard to the eight-component glue field a similarrole to that which the fundamental representation plays with regard tothe quark field. Pushing the analogy to the spin, we are looking for theequivalent of spin-1 representation. The matrix representation of genera-tors in the adjoint representation is(Ta)bc = −ifabc, a, b, c = 1, . . ., 8. (13.62)Ta satisfy the same algebra as the generators ta of the fundamental rep-resentation, Eqs. (13.58) and (13.59). The trace of the product of twoTa, an analogous result to Eq. (13.61), follows from Eq. (13.60) in view ofEq. (13.62):tr(TaTb) = ncδab. (13.63)We now make the color-rotation matrix V space-dependent, V → V (x),allowing that the eight real parameters ϑa, in Eq. (13.56), depend on x,V (x) = e−iϑa(x)ta , (13.64)13 Hadronic structure and quantum chromodynamics 271with summation over a repeated group index such as a being implicitlyunderstood henceforth. In analogy to Eqs. (13.48) and (13.49), the localnonabelian gauge transformation of matter fields isΨ→ Ψ′ = VΨ, (13.65)∂µΨ→ ∂µΨ′ = V [∂µΨ+ V †(∂µV )Ψ], (13.66)where we have used Eq. (13.54). Since V †V =1, we have(∂µV †)V =−V †(∂µV ). (13.67)Instead of the term (i∂µα), in Eq. (13.49), we have now a matrix termV †(∂µV ); the entire expression is multiplied by a matrix V , Eq. (13.64),rather than a phase factor eiα.13.4 GluonsWe introduce now the dynamic color-potential field Aµ. To proceed inanalogy to QED, we have to couple the color potential to a product ofquark–antiquark spinors. Thus, Aµ must be a 3×3 matrix, given the colorstructure of the spinor Ψ. The three-component quark wave functionforms a triplet (fundamental) representation of the color group SU(3),while the wave function of an antiquark forms an antitriplet. From theproduct of a triplet and an antitriplet, intuitively, we can understandthat one can form an SU(3) singlet; what remains is an octet of states.In analogy to spin, for which the product of two spin-12 particles can bea singlet (S = 0) or a triplet (S = 1) state, we write3c × 3̄c = 8c ⊕ 1c. (13.68)Since the color-gauge-field quantum must have the same quantum num-bers as a quark–antiquark pair, if it is to be able to be the product oftheir annihilation, it must contain at least eight non-matrix fields; theninth singlet field corresponds to the colorless case. In analogy to the realfield Aµ in QED, we now choose a hermitian matrix Aµ to represent themassless gluons, in the form of a linear combination of eight Gell-Mannmatrices with eight real Yang–Mills fields Aaµ(x):Aµ(x) = 12Aaµ(x)λa = Aaµ(x)ta. (13.69)We proceed in analogy to Eq. (13.50): if the potential Aµ changes undera local color-gauge transformation according toAµ → A′ = VAµV † +1gV (∂µV †), (13.70)272 QCD, hadronic structure and high temperaturethe minimally coupled derivative, ∂µ + igAµ, remains invariant in formunder the gauge transformation, up to the nonabelian phase factor(∂µ + igA′µ)Ψ′ = V (∂µ + igAµ)Ψ. (13.71)To show this, we have to use Eq. (13.66) and remember Eq. (13.67).It is often helpful to check the form of equations, and in particular rel-ative signs, by remembering that the transition from QED to QCD equa-tions can be effected with the introduction of a gauge-covariant derivative,both for quark and glue fields, defined byDν = ∂ν + igtaAaν = ∂ν + igAν . (13.72)In view of Eqs. (13.66) and (13.71), we have shown that this covariantderivative transforms under nonabelian gauge transformations asDν → D′ν = VDνV †. (13.73)Similarly, the eight-component field-strength tensor,Fµν = taF aµν , (13.74)defined as,Fµν = ∂µAν − ∂νAµ + ig [Aµ,Aν ] , (13.75)or equivalently,F aµν = ∂µAaν − ∂νAaµ − gfabcAbµAcν , (13.76)remains up to a phase form-invariant under a local color-gauge transfor-mation, i.e.,Fµν → F′µν = V FµνV †. (13.77)This transformation property is most easily proved once we realize that wecan define the field-strength tensor using the commutator of the covariantderivative, Eq. (13.72),[Dµ,Dν ] ≡ igFµν , (13.78)which verifies Eq. (13.77) in view of Eq. (13.73), remembering propertiesof V , Eq. (13.54).13 Hadronic structure and quantum chromodynamics 27313.5 The Lagrangian of quarks and gluonsThe complete gauge-invariant Lagrangian of quantum chromodynamicsis thenLQCD =∑fΨfγµ(i∂µ − gAµ)Ψf −mfΨfΨf − 12 tr(FµνFµν). (13.79)Here, the summation over the different quark flavors f has been madeexplicit. The form similarity to the Lagrangian of QED,LQED = ψγµ(i∂µ − eAµ)ψ −mψψ − 14FµνFµν , (13.80)is evident.In view of Eq. (13.61), we have an exact correspondence between thegluon and photon terms:14FµνFµν =12(B2 − E2)∣∣∣∣QED,12tr(FµνFµν) =14∑aFµνa Faµν =12∑a(BaBa − EaEa)∣∣∣∣∣QCD. (13.81)The key difference is in the nonlinear term entering into the definitionEq. (13.76) of F aµν , which is quadratic in the color potentials Aaµ.The analog of the Maxwell equations may be derived for the color fieldfrom the QCD Lagrangian, Eq. (13.79):[Dν ,Fµν ] = gjµ. (13.82)The color indices of the matter fields Ψ define the matrix current jµ ob-tained from LQCD,jaµ = ΨγµtaΨ. (13.83)We can write Eq. (13.82) in the more conventional form∂νFµνa = gjµa + gJµa , (13.84)where we encounter the gluon current Jµa :Jµa = fabcAbνFµνc , Jµ = −i[Aν ,Fµν ]. (13.85)The nonlinear term, Eq. (13.85), placed on the right-hand side ofEq. (13.84), shows that the color field acts also as its own source. In otherwords, the quanta of the color field, gluons, carry color charge themselves.This is the source of the substantial physical difference between QCD andQED. From Eq. (13.84), one can further see that jaµ does not obey a con-tinuity equation, which means that the color charge of quarks alone is not274 QCD, hadronic structure and high temperatureconserved. This is not surprising since quarks can emit or absorb gluons,which carry color. Only if we add the color charge of the gluon field,represented by the second term on the right-hand side of Eq. (13.84), is aconserved color current obtained.In view of the similarity in form of LQCD to LQED, many of the well-known other formal properties carry over. We refrain from systematicallydeveloping this here, even though we will call upon these similarities asneeded in further developments.14 Perturbative QCD14.1 Feynman rulesThe nonabelian gauge theory of quarks and gluons, proposed in sec-tion 13.5 and called QCD, has widely been accepted as the fundamentaltheory of strong interactions, with both quarks and gluons being the car-riers of the strong-interaction charge [123]. The evidence for the validityof QCD as a dynamic theory governing hadronic reactions is overwhelm-ing, and this is not the place where this matter should be argued. Rather,we will show how QGP-related practical results can be derived from thecomplex theoretical framework. There are many books dealing with moreapplications of QCD and the interested reader should consult these forfurther developments [110, 194, 280].Akin to QED, QCD is a ‘good’ renormalizable theory. QCD is knownto be also an asymptotically free theory, viz., the running coupling con-stant αs, see Eq. (14.12), is a diminishing function as the energy scaleincreases. Therefore, the high-energy, or, equivalently, the short-distancebehavior is amenable to a perturbative expansion. On the other hand,perturbative QCD has ‘fatal’ defects at large distances, which are signaledby the growth and the ultimate divergence of αs as the scale of energydiminishes (infrared ‘slavery’). Consequently, at any reasonable physicaldistance of relevance to the ‘macroscopic’ QGP, we have to deal with anintrinsically strongly coupled, nonperturbative physical system. A per-turbative treatment ignores this, and, in principle, must be unreliable inproblems in which the confinement scale becomes relevant. The questionof when exactly this occurs will be one of the important issues we willaim to resolve, using as criterion αs ≤ 1.The perturbative approach, which applies to short-distance phenomena,has been tested extensively in high-energy processes. When the ‘short dis-tance’ grows and approaches 0.5 fm, the perturbative expansion of QCDmay still apply insofar as its results are restricted to the physics occur-ring in the deconfined, viz., QGP, phase. The rules of perturbative QCDfollow the well-known Feynman rules of QED, allowing for the glue–glue14 Perturbative QCD 275interactions. However, unlike QED, in which the expansion parameteris α/π = O(10−3), we must deal in case of QCD with ‘strong’ coupling,which is nearly 30–100 times greater than that in QED. Therefore, evenwhen a perturbative description is suitable, some effort to reach the re-quired precision is often necessary, involving the inclusion of higher-orderFeynman diagrams.The quadratic terms ‘ψ2’ and ‘A2’ in the Lagrangian LQCD define freequark and gluon fields which are described by propagators of the sameform as those for electrons and photons in QED. The terms of thirdand fourth order in LQCD give rise to interaction vertices among thefree propagators of quarks and gluons. There is a quark–gluon vertex,and three-gluon and four-gluon vertices. Propagators and vertices can becombined to generate Feynman diagrams in all possible ways.One technical difference arises between QED and QCD, which is asso-ciated with the difficulties of gauge theories with regard to gauge fixing.To ensure gauge invariance of QCD, it is convenient to introduce into theperturbative expansion fictitious (virtual) particles called Fadeev–Popov(FP) ghosts, which never appear in physical states, but are to be in-cluded in all virtual processes. FP fields carry color, and satisfy Fermistatistics even though they propagate like spin-zero particles – hence thename ‘ghost’. The complete Lagrangian in a Lorentz-covariant gaugereadsLQCD=−14(Fµνa Faµν)2 + ξ(∂µAaµ)2 + φ̄(∂µδab + gfasbAsµ)∂µφb+∑fΨf[γµ(∂µ + gAsµ)−mf]Ψf , (14.1)where the summation over color indices is implied and the second termin Eq. (14.1) is the gauge-fixing term and the third term formed withthe scalar fields φ introduces the Fadeev–Popov ghosts. We note thatit is possible to work in a non-Lorentz-covariant gauge and to obtainresults without introducing ghosts, instead using longitudinal and trans-verse gluons as identifiable degrees of freedom, with longitudinal gluonsnot present in any asymptotic physical states. For further discussion, werefer to relevant textbooks [110, 194, 280].We skip the technical details regarding the development of the pertur-bative QCD, and only collect for the convenience of the reader and furtherreference the building blocks of the perturbative expansion required forevaluation of Feynman diagrams, i.e., propagators and vertices of QCD,presented in a self-explanatory notation. Latin indices refer to color de-grees of freedom, Greek indices to space, q to quarks, g to gluons, andFP to the ghost field.276 QCD, hadronic structure and high temperatureWe havethe quark propagator,(Sαβ(p))ab =(δabγp−m+ iε)αβ, (14.2)the gluon propagator,i(Dµν(k))ab =δabgµνk2 + iε, (14.3)the quark–antiquark–gluon vertex,(Γµ,cqqg)abαβ = g(tc)ab(γµ)αβ , (14.4)the three-gluon vertex,(Γµνσg3)abc = gfabc[gµν(k − p)σ + cyclic permutations], (14.5)the four-gluon vertex,(Γµσντg4)abcd = g2fiabficd[(gµσgντ − gµτgνσ) + cyclic permutations],(14.6)the ghost propagator,(GFP(k))ab =δabk2 + iε, (14.7)and the gluon–ghost–ghost vertex,(ΓµgFP)abc = −gfabcpµ. (14.8)14 Perturbative QCD 277For any given process, for which a Feynman diagram is drawn usingthe lines and vertices as illustrated above, the above list allows one tocompose the mathematical expression for the amplitude of the process.Very few additional rules need to be remembered, such as integration overthe ‘spare’ momentum variables in the diagram, an overall coefficient (−1)for each fermion and ghost loop, and the absence of ghost propagators thatdo not begin and end in a vertex.While the forms of the above-stated propagators and vertices changein a finite-temperature environment, which is mainly being addressed inthis book, the structure of the perturbative expansion generated by thesequantities remains the same. The construction of a matrix element andcross section requires wave-function-normalization factors, and flux fac-tors that are all quite standard in this context, and available in numerousintroductory textbooks. In these aspects, there is no difference betweenQCD and any other theoretical framework, such as QED. However, we re-call that, in order to obtain a cross section, we average over initial states,and sum over final states, which now will include, in particular, the colordegree of freedom.14.2 The running coupling constantThe free-gluon propagator, which, like in QED, is proportional to 1/q2,implies that the ‘free’ color force falls off like 1/r. The gluon propagatoris, even in the perturbative vacuum, modified by scattering from virtualquark–gluon fluctuations. This ‘dressing’ of the propagator leads to therunning, q2-dependent coupling constant, and the dressed physical gluonpropagator is expected to significantly differ from the free one. In order tosee how this comes about, we consider the loop diagrams correspondingto the virtual and momentary creation of a pair of colored particles in thevacuum. For the contribution of fermions to the polarization loop, thisprocess is very similar to the case of QED.There are two more elementary processes that contribute to the covari-ant form of perturbative QCD, namely the gluon loop and the ghost loop,The vacuum-polarization loop Π(q2) comprises these three terms:It is customary to sum the chain of higher-order diagrams, containingseries of all different loops,D(q2) = D0 + D0ΠD0 + D0ΠD0ΠD0 + · · ·278 QCD, hadronic structure and high temperatureD(q2) =D0(q2)1−Π(q2)D0(q2), (14.9)such that the gluon is dressed by the consecutive interactions with the vac-uum polarization. The effect of the quark, glue, and ghost loops combinesin the coefficient b0, Eq. (14.14), of the vacuum-polarization function. In-troducing a high momentum (ultraviolet) cutoff to secure the convergence,for massless quarks and gluons, Π(q2) takes the formΠ(q2) =g2b08π(−q2) ln(−q2µ2), (14.10)where µ is a reference momentum absorbing the cutoff, and defined bythe renormalization condition: Π(q2 = −µ2) = 0.The value of µ2 introduces, in general, a dependence on renormaliza-tion. In this order, this dependence can be absorbed into the choice ofthe value of the coupling constant at a given transfer of momentum; seeEq. (14.13) below. The overall sign of Π, Eq. (14.10), is related to thesign of b0, and is opposite to that of the polarization function of fermionsalone (QED) as long as the flavor number nf < (11/2)nc. The sign of thepolarization function is quite important, as we shall see.With the help of Eq. (14.10), we obtaing2iD(q2) =1q2g21 +g2b08πln(−q2µ2) . (14.11)The last factor acts as a momentum-dependent modification of the strongcoupling constant g2. It is therefore convenient to introduce the ‘running’coupling constant αs(q2),αs(q2) =2b0 ln(−q2/Λ2), (14.12)withΛ2 = µ2 exp(− 8πb0g2), αs =g24π. (14.13)The above expression applies for positive b0, and, in this case, we see that,for an increasing q2, the physical coupling constant αs(q2) decreases; QCDis asymptotically free.In the case of QED, the sign of b0 is opposite, and the effective couplingconstant is finite at large distances, i.e., at small q2, and increases withq2. The reference scale µ2 in Eq. (14.11) can be chosen to be at zeromomentum (infinite distance) and this corresponds to the usual definition14 Perturbative QCD 279of the electron charge. This choice is not possible in case of QCD, sincethe interaction strength becomes infinite for µ2 → 0.Λ is the dimensional parameter which emerges in perturbative QCD.The original dimensionless coupling constant g2 has been absorbed inthe scale Λ governing the change of the running coupling constant αs, inthe process of transition from Eq. (14.11) to Eq. (14.12). This so-called‘transmutation’ of the dimensionless scale of strength into a dimensionedstrength parameter of the interaction also absorbs the scale dependenceintroduced by the choice of µ2.In the limit that all quark masses vanish, mi → 0, Λ is the only dimen-sional parameter of the theory of strong interactions. It is believed thatthe world of hadrons (except the pion) is not decisively dependent on thescale of the quark mass. Thus, it seems that Λ alone controls the mass andthe size of the massive hadrons (nucleons and heavy non-strange mesons).To understand this, we would need to express the vacuum structure interms of Λ, a problem which has not been resolved.The measurable dimensioned parameter Λ2 determines the strength ofthe interaction at a given momentum scale. This approach applies quiteaccurately for energy–momentum scales above the mass of the b quark, aswe shall see later in Fig. 14.1, where the value of Λ 90±15 MeV applies.At small q2, i.e., at ‘large’ distances, the coupling constant Eq. (14.12)diverges within the perturbative approach. The magnitude of the strongcharge must be defined at some finite momentum scale, which has in recentyears, been chosen to be the mass of the Z0 boson, µ ≡ MZ0 91.19 GeV.Since we have more than one quark, the important coefficient b0 iscomposed of a term proportional to the number of ‘active’ flavors nf , i.e.,those with |q2| > 4m2f . The number of colors nc = 3 enters the glueloop: in the gluon loop diagram, each external leg requires the triple-glue vertex, Eq. (14.5), which invokes relation Eq. (13.63) or equivalentlyEq. (13.60) for two external gluon legs. b0 for SUc(3) assumes the formb0 =12π(113nc −23nf). (14.14)The spin s of particles contributing to the vacuum polarization is foundto be the key ingredient controlling the sign of b0 [149],bs0 =(−)2s2π((2s)2 − 13), (14.15)which leads to Eq. (14.14), introducing s = 1 for gluons and s = 12 forquarks. Equation (14.15) shows why for s = 0, 12 the same (negative) signappears, whereas for gluons with s = 1 there is a change of sign. Photonsdo not interact with photons and hence this issue did not arise in QED.280 QCD, hadronic structure and high temperature14.3 The renormalization groupThe question of what happens as we carry out the same procedures inhigher orders in perturbation theory now arises. A considerable amountof effort went into designing a scheme for computing the observable ef-fects in QED, and this experience has been generalized to the more com-plex case of QCD. We will restrict ourselves to a few elements of therenormalization-group approach relevant to our presentation, sidestep-ping many interesting and intricate questions, which are addressed in,e.g., [110, 194, 280].The renormalization-group approach allows us to understand the vari-ation of the physical observables in terms of the momentum dependenceof the coupling constant αs. A functional dependence is found demandingthat the result of a physical measurement (say a cross section σ) be in-variant with respect to the process of renormalization, and, in particular,the observable (cross section) can not depend explicitly on the choice ofthe ‘(re)normalization’ point µ2,µddµσ(pi;αs,m;µ) = 0. (14.16)Accounting for both a direct and an indirect dependence on µ inEq. (14.16),(µ∂∂µ+ µ∂αs∂µ∂∂αs+ µ∂m∂µ∂∂m+ · · ·)σ = 0. (14.17)It is convenient to defineµ∂α∂µ≡ β(αs) (14.18)and− µm∂m∂µ≡ γ(αs), (14.19)and thus:(µ∂∂µ+ β(αs)∂∂αs− γ(α)m ∂∂m+ · · ·)σ = 0. (14.20)Equation (14.20) allows us to understand the behavior of the observableσ, and it establishes the behavior of σ under simultaneous changes ofreference scale µ, coupling constant, and mass.Equations (14.18) and (14.19) establish how the parameters of QCDvary once they are known at some given scale. Therefore, precise knowl-edge of the renormalization functions β and γ is, to a large degree, equiv-14 Perturbative QCD 281alent to finding a practical ‘solution’ of QCD. For this reason, these quan-tities have attracted a lot of attention. We will look at the perturbativeresults only, in terms of a power expansion in αs [255]:βpert =−α2s[b0 + b1αs + b2α2s + · · ·], (14.21)γpertm = αs[w0 + w1αs + w2α2s + · · ·]. (14.22)For the SU(3) gauge theory with nf fermions only, the first two terms(two ‘loop’ orders) are renormalization-scheme-independent. When a de-pendence on the renormalization scheme arises, this means that compen-sating terms, which remove scheme dependence, are obtained on evaluat-ing in the same scheme the physical process considered. For what followsin this book, this so-called two-loop level of perturbative expansion forβpert and γpertm is sufficient.We haveb0=12π(11− 23nf), b1 =14π2(51− 193nf), (14.23)w0=2π, w1 =112π2(101− 103nf). (14.24)The number nf of ‘active’ fermions depends on the scale µ. Assumingthat the two lightest quarks are effectively massless,nf(µ) = 2 +∑i=s,c,b,t√1− 4m2iµ2(1 +2m2iµ)Θ(µ− 2mi), (14.25)with values of mi evaluated, in principle, for the energy scale being con-sidered. There is a very minimal impact of the values of quark-massthresholds in Eq. (14.25), on the running behaviors of the coupling con-stant and quark masses.14.4 Running parameters of QCDFor the purpose of QGP studies, we are interested in understanding howthe strength of the QCD interaction and the quark mass change withthe energy scale. The simplest way to obtain this result is to integratethe first-order differential equation, Eqs. (14.18) and (14.19), given ini-tial values of αs(M) and mi(M), using the perturbative definition of thefunctions β and γ, Eqs. (14.21) and (14.22), in terms of the perturbativeexpansion Eqs. (14.23) and (14.24).For the determination of the coupling constant, it has become commonto refer to the value of αs(MZ = 91.19GeV). We use, in Fig. 14.1, thevalue [136] αs(MZ) = 0.1182 ± 0.002 (thick solid lines). The thin solid282 QCD, hadronic structure and high temperatureFig. 14.1. αs(µ) (top section), the equivalent parameter Λ0 (middle section),and mr(µ) = m(µ)/m(MZ) (bottom section) as functions of the energy scale µ.The initial values are αs(MZ) = 0.118 (thick solid lines) and αs(MZ) = 0.115(thin solid lines). In the bottom section, the dots indicate the strangeness-pair-production thresholds for ms(MZ) = 90MeV, while crosses indicate charm-pair-production thresholds for mc(MZ) = 700MeV.lines are for αs(MZ) = 0.115 arising from analysis of decays of heavyquarkonium (bb̄), and addressing an energy scale closer to our directinterest in this book.As can be seen in the top portion of Fig. 14.1, the variation of α(2)s < 1(the upper index indicates the level of perturbative expansion used; seeEq. (14.21)) with the energy scale is substantial. We note the rapid changein αs(µ) at, and below, µ = 1 GeV. This is not an unexpected result.However, the fact that a solution with αs ≤ 1 governs the energy scale1 GeV is important, since the formation of the strange quark flavor occursin a hot QGP phase formed in experiments at 160–200A GeV (SPS–14 Perturbative QCD 283CERN) at around 1 GeV. We can thus use the methods of perturbativeQCD to study this process.Among the parameters in Eq. (14.25), only the mass of the bottomquark plays a (hardly) noticeable role; the results shown were obtainedfor ms = 0.16GeV, mc = 1.5GeV, and mb = 4.8GeV [225]. When mb ischanged by 10%, the error on a low energy scale is barely visible. Otherquark masses have less significance since the error has less opportunityto ‘accumulate’ in the solution of the differential equation as the energyscale decreases in the integration of Eq. (14.18).As expected and seen in the top portion of Fig. 14.1, in the soft QGPdomain 0.8 GeV < µ < 3 GeV, it is impossible to use a constant value ofαs. More surprisingly, the frequently used approximate inverse-logarithmform Eq. (14.12) for αs is nearly equally inappropriate. To see this, wedefine a quantity Λ0(µ),αs(µ) ≡2b−10 (nf)lnµ2/Λ20(µ), Λ0(µ) = µ exp(− 1b0αs(µ)), (14.26)where αs(µ) is obtained in two-loop or higher-order perturbation expan-sion of the β-function using Eq. (14.18). The form of Eq. (14.26) is chosento be identical to the one-loop form, compare with Eq. (14.13). Using theresult for αs shown in the top portion of Fig. 14.1, we obtain Λ0(µ) seenin the middle section of the figure.If the one-loop form Eq. (14.12) with a constant Λ0 were a good approx-imation to αs, we should see a sequence of step functions, dropping at eachheavy-quark threshold. In fact, above the bottom threshold, for µ > 2mb,this is nearly the case. However, below the charm threshold, for µ < 2.5GeV, where practically all QGP action is occurring, we see a rather rapidchange in Λ0(µ), which drops from a value near Λ0(1GeV) 300 MeVtoward Λ0(3GeV) 200 MeV.It is common to refer to the number of active quarks by using an upperindex on Λ0(µ), thus Λ(3)0 refers to the range 1 GeV < µ < 2mc, and Λ(5)0refers to µ > 2mb, and below the top threshold. We also see, in Fig. 14.1,that Λ(5)0 90 ± 15 MeV. This value of Λ(5)0 derived from a comparisonwith the one-loop solution should not be mixed up with Λ(5) = 205± 25MeV, which is the value required to describe αs in the analytical two-loopsolution, Eq. (14.28) [136].To understand how quark masses depend on the energy scale, given αs,we integrate Eq. (14.19). Substituting m(µ) = mr(µ)m(MZ), we recog-nize that mr(µ) is a multiplicative factor applicable to all quark masses.mr(µ) is shown in the bottom portion of Fig. 14.1. All quark masses‘run’ according to this result. A quark mass given at the scale µ = MZincreases by factor 2.2 at scale µ = 1 GeV, as the dotted lines drawn to284 QCD, hadronic structure and high temperatureguide the eye show. Near to µ 1GeV, the quark-mass factor mr(µ)is driven by the rapid change of αs. For each of the different functionaldependences αs(µ), a different function mr is found, and two results arepresented, corresponding to the two cases considered in the top sectionof Fig. 14.1.Since αs refers to the scale of µ0 =MZ, it is a convenient reference pointalso for quark masses. The value ms(MZ) = 90 ± 18MeV correspondsto strange-quark mass ms(1GeV) 195 ± 40MeV, i.e., ms(2GeV) 150±30MeV, at the upper limit of the established range seen in table 1.1on page 7. Similarly, we consider mc(MZ) = 700 ± 50MeV, for whichvalue we find the low-energy mass mc(1GeV) 1550 ± 110MeV, i.e.,mc(2GeV) 1200 ± 85MeV, which is also at the upper limit of theaccepted range, table 1.1.For quark-pair production, the intuitive energy scale to consider is arange near to twice the (running) quark mass. Since, below√s = 1 GeV,the mass of the strange quark increases rapidly, the pair-production thresh-old is considerably greater than 2ms(1GeV) 400 MeV. The dots in thebottom portion of Fig. 14.1 show where the strangeness threshold is found,and this is at 2mr(2ms)ms = 611MeV for αs(MZ) = 0.118. The stran-geness threshold is where αs 1 and we can expect, considering that thephase space for pair production opens up at about 3m, that strangeness isproduced predominantly in an energy domain accessible to perturbativetreatment.For charm, the threshold shift due to running mass occurs in the oppo-site direction: since the mass of charmed quarks for µ = 1GeV is above1GeV, the production-threshold mass is smaller than 2mc(1GeV) 3.1GeV; the production threshold is found at ∼2mthc 2.3 GeV, and the cor-responding values of mr are indicated by crosses in the bottom portion ofFig. 14.1. In other words, we expect that, near the threshold, there is aslight enhancement in production of charm related to a reduction of thethreshold, while the coupling strength is at αs(2mc) 0.3.The inclusion of higher-order terms in the perturbative expansionEq. (14.21) does not influence the behavior of αs. This is shown inFig. 14.2, in which a study of αs is shown. To obtain the solid line,the full current ‘scheme-dependent’ knowledge about the perturbative β-function is employed. The four-loop β-function obtained in the modi-fied minimum-subtraction scheme (MS) was used [136]. On the otherhand, Eq. (14.21) demonstrates that there is a considerable sensitivity tothe initial value αs(MZ). If αs(MZ) were to increase, the evaluation ofthe coupling strength in the ‘low’-energy domain µ 1 GeV of interesthere would become impossible, or at best unreliable, see the dotted linesin Fig. 14.2 above the solid line. In fact, we do not present many re-14 Perturbative QCD 285Fig. 14.2. α(4)s (µ) as a function of the energy scale µ for a variety of initialconditions. Solid line, αs(MZ) = 0.1182 (see the experimental point, whichincludes the error bar at µ = MZ); dotted lines, sensitivity to variation of theinitial condition.sults in this domain since the renormalization-group-evolution equation,Eq. (14.18), becomes numerically unstable when the four-loop perturba-tive β-function is used.Interestingly, a 20% reduction in αs(MZ) leads to a ‘good’ αs(0.1GeV).The distance scale 1/µ at which QCD becomes unstable is not just 1 fm,but, as this study shows, an intricate functional of the strength of thefundamental interaction, which has reliably been established only in re-cent years. An essential prerequirement for the perturbative theory of theproduction of strangeness in QGP, which we will develop in section 17.3,is the relatively small value αs(MZ) 0.118.For studying thermal processes in QGP at temperature T , the proposedinteraction scale is, see Eq. (16.11),µ ≡ 2πT 1GeV T/Tc,for Tc 160MeV. We can expect considerable sensitivity in this lowrange of µ to the exactness of the functional form of αs(µ), and it isnecessary to use the precise function αs(µ). In Fig. 14.3, the solid linebounded by error lines corresponds to the exactly computed two-loop αswith physical quark thresholds, Eq. (14.25), and with αs(MZ) = 0.1181±286 QCD, hadronic structure and high temperatureFig. 14.3. αs(2πT ) for Tc = 0.160GeV. Dashed line, αs(MZ) = 0.119; solid line= 0.1181; chain line = 0.1156. Dotted line, approximate two-loop solution, givenin Eq. (14.28), with the choice Λ = 150 MeV.0.002, evaluated for the thermal scale, and expressed in terms of T/Tc.The range of experimental uncertainty in αs(T ), due to uncertainty inαs(MZ), is delimited by dashed and chain lines bordering the solid linein Fig. 14.3. A good approximation is obtained fitting αs(T ) with alogarithmic form,αs(T ) αs(Tc)1 + C ln(T/Tc), C = 0.760± 0.002, for T < 5Tc. (14.27)The value αs(Tc) = 0.50−0.05+0.03 applies in the two-loop description withµ = 2πT and Tc = 0.16 GeV (see Fig. 14.3).A popular approximation of αs, which incorporates the next term be-yond the one-loop logarithmic term Eq. (14.26), isα(2)s (µ) 2b0L̄(1− 2b1b20ln L̄L̄), L̄ ≡ ln(µ2/Λ2). (14.28)α(2)s agrees, using the standard value Λ(5) = 205±25 MeV, with the exactsolution shown at the top of Fig. 14.1, but only for µ > 2mb. On theother hand, when one is studying thermal properties of a QGP at a lowenergy scale the use of Eq. (14.28) below µ = 2mb introduces consider-able error, as can be seen in Fig. 14.3. Equation (14.28) is represented15 Lattice quantum chromodynamics 287by the dotted line, and misses the exact result by a factor of two, forTc < T < 1.75Tc, the effective range of observables emerging from SPSand RHIC experiments. The experimental error in determination of αs istoday considerably smaller. This large difference between exact and ap-proximate result arises, in part, because the value of Λ(5) used to obtainthe thermal behavior was adjusted to be Λ(4) = 0.95Tc 0.15 GeV. Thisvalue would be correct if Tc were indeed around 210 MeV, as has beenthought for some time.The high sensitivity of physical observables to αs, makes it imperativethat we do not rely on this approximation. Yet a fixed value αs = 0.25(instead of αs = 0.5) derived from this approximation is still often used instudies of the phase properties of QGP, loss of energy by jets of partons,thermalization of charmed quarks, thermal production of strange quarks,etc. Such a treatment of thermal QCD interaction underestimates byas much as a factor of four the interaction with the QGP phase, andthus the speed of these processes. In most cases, this mundane factormatters, and we see that an accurate evaluation of αs at the appropriatephysical scale is required in order to establish the correct magnitude ofthese results.15 Lattice quantum chromodynamics15.1 The numerical approachThe perturbative approach to QCD lacks the capability to describe thelong-distance behavior, which is essential for understanding the QGP–HGtransformation. We need a more rigorous approach in order to charac-terize the physical mechanisms at the origin of color confinement, andthe transition to the deconfined state of hadronic matter. A suitablenonperturbative approach is the numerical study of QCD on a lattice(L-QCD).L-QCD is a vast field that is evolving very actively. We will limitour presentation to a pedestrian guide to the language used in this field,along with a report on a few key results of greatest importance to us. Wewill not be embarking on a thorough introduction to the theoretical andnumerical methods. For a survey of the historical developments until theearly eighties we refer to the monograph by Creutz [97], and for a summaryof recent theoretical advances, and many numerical results addressing hotQCD, we refer the reader to the recent survey by Karsch [159].The particular usefulness of the lattice-gauge-theory formulation is thatit allows one to numerically carry out Feynman path integrals which rep-resent expectation values of quantum-field-theory operators. Specifically,the expectation value of an operator O, including both glue and quark288 QCD, hadronic structure and high temperaturefields, is〈O〉 =∫ [dAµ dψ̄ dψ]Oe−i∫L(A,ψ̄,ψ) d4x∫ [dAµ dψ̄ dψ]e−i∫L(A,ψ̄,ψ) d4x . (15.1)These integrals are over all values of the gluon and quark fields at all pointsin space and time. However, most parts of the domain of the integralare unimportant – in ‘weak coupling’ (i.e., when perturbative expansionmakes good sense), only paths close to the classical paths (classical-fieldsolutions) are important. In order to do a path integral efficiently, it is es-sential to sample more densely the domains that give large contributions,see Eq. (15.2) below, and this obviously then poses a practical challenge.The integration measure of the path integral is indicated by[dAµ dψ̄ dψ].The goal of computations, in lattice QCD, is to evaluate Eq. (15.1) numer-ically by evaluating the integrand at selected lattice points representingits domain.The functional integral in Eq. (15.1), expressed on the lattice, meansthat we are integrating the fields at each lattice site and lattice link, andthe domain of the integral has accordingly a very high dimensionality.The method of choice for doing such integrals numerically is the Monte-Carlo (random-choice) method. However, a considerable complication inapplying this method arises since e−i∫L d4x is not in general a positivereal number: aside from the i factor in the exponent, it is a functionalof quark fields, which have to be represented by anticommuting numbers:ψxψy = −ψyψx. This problem can be solved since the dependence onψ and ψ̄ of Eq. (15.1) has the form of a polynomial times a Gaussian.Therefore, the quark portion of the path integral can be done analytically.This integral yields a ‘Fermi determinant’ FD, which changes for eachconfiguration of the gauge (gluon) fields considered. We will address theform of FD in section 15.4.In order to allow a Monte-Carlo integration procedure for the gaugefields, the explicit i in the exponent in Eq. (15.1) is combined with dt,and the integral is considered in ‘imaginary’ time, or, as it is usually said,Euclidian space. It is generally believed that some, if not all, physicalresults can be analytically continued from the real- to the imaginary-timeaxis. Even so, the Fermi determinant remains real only for zero chemicalpotential, and we can use as a probability for sampling the Monte-Carlointegralρ(A) = FDIe−∫La d4xI . (15.2)La is the gluon part of the Lagrangian in imaginary time and subscriptI indicates that the quantities have been suitably modified by the trans-formation t → it.15 Lattice quantum chromodynamics 289The physical system is restricted to a finitely sized box, which intro-duces an infrared (long-distance) cutoff at the size of the box, L. Thecontinuous space and time is represented by a lattice, which introducesan ultraviolet cutoff (i.e., distance) at the lattice spacing† O. In goingfrom the continuum to the lattice, derivatives are replaced by finite dif-ferences. This replacement must be done in a gauge-invariant way, andhence one often refers to lattice gauge theory (LGT) or, in our context,lattice quantum chromodynamics (L-QCD).In what follows, we will describe how to deal with dynamic gluons andquarks. Since the presence of quarks requires that the Fermi determinantFD be evaluated, this imposes a need for much more computational ef-fort than in the case of the ‘pure-gauge’ lattice which comprises gluonsonly. In an intermediate step we can study quark operators in a non-fluctuating gauge-field background; this is the ‘quenched’-quark approx-imation, which excludes the contributions of particle–antiparticle pairs.The full calculation then has ‘dynamic’ quarks.15.2 Gluon fields on the latticeReplacement of continuous space–time xµ by a lattice xµ = Onµ mustbe accomplished in a gauge-invariant manner, and, as with any otherregulator, in order to be able to interpret the results, the regulator, i.e.,lattice spacing, must be removed (O → 0) after a finite result has beenobtained. In other words, contact with the real physical world exists only,in the continuum limit, when the lattice spacing is taken to zero; this limitmust be reached in natural fashion in any formulation. Moreover, we mustalways be aware that, on the lattice, we sacrifice Lorentz invariance, andhave to be vigilant about the fates of all internal symmetries, which wedesire to preserve. A suitable approach was devised by Wilson [273].The action, an integral over the Lagrangian, is replaced by a sum oversites:βS =∫dxL → O4∑nLn. (15.3)β reminds us that all calculations are carried out in a four-dimensionalEuclidian world, and β corresponds to the time dimension, or, as we shallsee for equilibrium thermodynamics, the usual relation β = 1/T applies.The generating functional used to obtain many of the results implied by† It is common to call the lattice spacing a. To avoid conflicts of notation with thecolor indices of QCD, we chose the symbol , which is not used as often, though itshould not be confounded with the angular-momentum eigenvalue employed earlier.290 QCD, hadronic structure and high temperatureEq. (15.1) is now an ordinary integral over all lattice fields and sites φin:Z =∫ (∏i∏ndφin)e−βS . (15.4)In the specific case of interest to us, quantum chromodynamics, thisintegral does not comprise the gauge fields Aaµ; these are represented byfundamental variables Uµ(n), which live on the links connecting point xnand xn + Oµ of a d = 4 dimensional space [273],Uµ(n) ≡ eigνRe tr {1− Uµ(n)Uν(n+ µ̂)U †µ(n+ ν̂)U †ν (n)}. (15.7)βSW is called the ‘Wilson action’. The volume element in Eq. (15.4) issimply an integral over the group elements:[∏i∏ndφin]→[∏ndUn]. (15.8)Summation over all group elements amounts to a projection of the argu-ment in the integral onto its color-singlet component.15.3 Quarks on the latticeThe Euclidian fermion action in the continuum (in four dimensions) isS =∫d4x [ψ̄(x)γµ∂µψ(x) +mψ̄(x)ψ(x)] . (15.9)15 Lattice quantum chromodynamics 291A ‘naive’ lattice formulation is obtained by replacing the derivatives bysymmetric differences:SnaiveL =12O∑n,µψ̄nγµ(ψn+µ − ψn−µ) +m∑nψ̄nψn . (15.10)The elementary solution of the associated dynamic equations, i.e., thepropagator, isG(p) =Oiγµ sin(pµO) +mO=−iγµO sin(pµO) +mO2∑µ sin2(pµO) +m2O2→ 1iγµpµ +m. (15.11)We identify the physical spectrum through the poles in the propagator atp0 = iE:sinh2(EO) =∑jsin2(pjO) +m2O2. (15.12)The lowest-energy solutions, as expected yielded for p = (0, 0, 0) the usualE ±m, but there are many other degenerate solutions yielding thisvalue of E, at Op = (π, 0, 0), (0, π, 0, ), . . ., (π, π, π). This is a model for 16light fermions, not one. More generally, when fermions are discretized inthis way on a d-dimensional lattice, they double and produce 2d species.Initially, two ways to deal with this problem were developed. The‘Wilson fermions’ [273], and the ‘Kogut–Suskind (staggered) fermions’[168]; more recently, also a five-dimensional formulation with ‘domain-wallfermions’ [155] has been considered. Wilson fermions are implemented byadding a second-derivative-like term,SW = − r2O∑n,µψ̄n(ψn+µ − 2ψn + ψn−µ), (15.13)to Snaive, Eq. (15.10). The parameter r must lie between 0 and 1; r = 1is almost always used and ‘r = 1’ is implied when one speaks of using‘Wilson fermions’. The propagator isG(p) =−iγµ sin(pµO) +mO− r∑µ[cos(pµO)− 1]∑µ sin2(pµO) +{mO− r∑µ[cos(pµO)− 1]}2 . (15.14)It has one pair of ‘low-energy’ poles at pµ (±im, 0, 0, 0). The other polesare at p r/O. In the continuum limit, these states become infinitelymassive.292 QCD, hadronic structure and high temperatureThis makes all but one of the fermion species heavy (with heavy massesclose to the cutoff 1/O), and we have, in principle, the required discretiza-tion method. However, for the nf flavor QCD addition of SW, Eq. (15.13)breaks the SU(nf)L × SU(nf)R chiral symmetry; see section 3.3. Thesize of the symmetry breaking is proportional to the lattice spacing andonly close to the continuum limit O → 0 does the explicit chiral-symmetrybreaking become small. At any finite lattice spacing a proper represen-tation of the chiral symmetry of the massless theory becomes a subtlefine-tuning process.Despite the computational problems related to implementation of chi-ral symmetry, there is also some advantage with this formulation. Wil-son fermions are closest to the continuum formulation – there is a four-component spinor on every lattice site for every color and/or flavor ofquark. Therefore, the usual rules apply to the formulation of currents,and states. Explicitly, the Wilson-fermion action for an interacting theoryisOSW=∑nΨ̄nΨn−κ∑nµ(Ψ̄n(r−γµ)Uµ(n)Ψn+µ+Ψ̄n(r+γµ)U †µΨn−µ).(15.15)We have rescaled the fields ψ =√2κΨ, and have introduced the ‘hoppingparameter’ κ−1 = 2(mO+ 4r).In studies of properties of QGP, another description of quarks on alattice has been used more extensively. In the staggered-fermion method,a one-component staggered-fermion field rather than the four-componentDirac spinors is used. The name staggered is used since Dirac spinors andquark flavors are constructed by combining appropriate single-componentfields from different lattice sites. Staggered fermions also break the chiralsymmetry, but there remains a U(1) × U(1) symmetry, which comprisesmuch of the physics of chiral symmetry. Moreover, explicit chiral sym-metry is present for mq → 0, even for finite lattice spacing, as long as allflavor masses are degenerate. On the other hand, flavor symmetry andtranslational symmetry are mixed together, which poses problems, sincein the real world, the flavor symmetry is broken.Since exact chiral symmetry and broken flavor symmetry are importantphysical phenomena influencing the physics of high-temperature QCD, seeFig. 3.4 on page 54, a third approach to place quarks on a lattice is cur-rently being developed. The domain-wall formulation of lattice fermionsis expected to support accurate chiral symmetry, even at finite latticespacing. In this new fermion formulation, it seems that it will be possi-ble to more easily simulate two-flavor, finite-temperature QCD near thechiral phase transition. For further theoretical details, we refer to [265],and the first exploratory hot-QCD calculations are reported in [90].15 Lattice quantum chromodynamics 293Another area of rapid development is the search for a most appropriate‘improved’ action for each of the important applications of L-QCD. Theoriginal Wilson action for the gauge fields is not unique, since the prin-ciple of gauge invariance leads to the building block, the plaquette, butnot to the actual form of the action made from plaquettes. Consequently,the form of Wilson-fermion action can also be ‘improved’. Improvementscan perform better in one area than in another, since they address theproblems encountered in extracting physics from extensive numerical cal-culations.15.4 From action to resultsOnce we have quark fields on the lattice, as noted at the end of sec-tion 15.1, we must deal with their anticommuting nature. We carry outthe integral over the Fermi fields in the path integral Eq. (15.4). For nfdegenerate flavors of staggered fermionsZ=∫[dU ][dψ][dψ̄] exp(− βS(U)−nf∑i=1ψ̄M(U)ψ)(15.16)=∫[dU ](detM(U))nf/2exp(−βS(U)). (15.17)In order to make explicit the positive-definite nature of the Fermi deter-minant FD appearing as the preexponential factor in Eq. (15.17), we willbe writing it asFD = det(M †M)nf/4.Recalling that a determinant is a product of eigenvalues, we can expressits logarithm as a sum of logarithms of eigenvalues, i.e., a trace, and wewriteZ =∫[dU ] exp(−βS(U)− nf4tr ln(M †M)). (15.18)The major computational problem dynamic fermion simulations faceis inverting the fermion matrix M for any change in any of the gluon-link fields U . M has eigenvalues with a very large range – from 2π tomq O – and, in the physically interesting limit of small mq, the matrixbecomes ill-conditioned. Just a few years ago, it had been possible onlyto study quenched fermions, i.e., to proceed ignoring the second term inthe exponent in Eq. (15.18). Today, it is possible to compute at relativelyheavy values of the quark mass and to extrapolate to mq = 0.We will not enter further into practical discussion of how to do thehigh-dimensional Monte-Carlo integrals; neither shall we discuss the many294 QCD, hadronic structure and high temperatureingenious algorithms that are in use. This is clearly a field of its ownmerit, and would fill more than this volume. However, even if somebodydid provide all numerical answers, we still would need to take a few furthersteps. Perhaps the most important physical consideration is the approachto the continuum limit, O → 0. Any calculation by necessity will be donewith a finite lattice spacing O. The lattice spacing O is an ultraviolet cutoff,and it is the variable which now provides the scale to the QCD couplingconstant g and quark masses mi.The continuum limit, in which we are interested, requires the limitO → 0, holding physical quantities fixed, not the input (‘bare’) action pa-rameters. In section 14.3, we have shown that the input parameter g, thebare coupling of quantum chromodynamics, is replaced by the runningcoupling constant αs measured at some given scale (typicallyMZ). Whenonly one dimensioned parameter is present, in the absence of quarks orwhen all quark masses are set to zero, the situation is simple. For exam-ple, in order to evaluate hadron masses on the lattice, one computes thedimensionless combination Om(O). One can determine the physical mean-ing of the lattice spacing by fixing one hadron mass from experiment.Then other dimensional quantities can be predicted.Consider, as an example, the ratio of two hadron masses:Om1(O)Om2(O)=m1(0)m2(0)+O(O) + · · · . (15.19)The leading term does not depend on the value of the ultraviolet-cutoff.One of the goals of a lattice calculation aiming to obtain the physicalproperties is to separate an ultraviolet-cutoff-distance scale-dependentremainder from the physical observable. One says that the calculation‘scales’ if the O-dependent terms in Eq. (15.19) are zero or small enoughthat one can extrapolate to O = 0. All the O-dependent terms are ‘scaleviolations’. To be able to make extrapolations, the results for severalvalues of lattice spacing O are required.The precision with which we can extract the physics will obviously de-pend on how small the lattice cutoff is. However, the lattice must covera sufficiently large region of space–time for the physical question we areaddressing. We cannot study the proton of size R = 1 fm without havinga few lattice distances within its radius, and a lattice of a few fermis.Repeating this basic cell domain infinitely using periodic boundary con-ditions helps, but cannot much reduce these requirements.A summary of the lattice-gauge-theory conditions and procedures whichwe have introduced is presented in table 15.1.For the study of hadron masses, as long as fundamental symmetries arerespected, the physical size of the lattice should be O 0.1–0.2 fm, and therepeating cell ought to have the size 5 fm. The computer power of today15 Lattice quantum chromodynamics 295Table 15.1. A summary of the procedure for L-QCD simulationsDo the path integral:quark integrals analyticallygluon + Fermi-determinant integrals numericallyRestrictions:Imaginary timeChemical potential = 0quark–antiquark symmetry makes the determinant realApproximations:Restrict volume L ∞ (infrared cutoff)Introduce lattice O ≥ 0 (ultraviolet cutoff)Need: O ltypical Llimit: scale O → 0Need low quark masses,Difficulty: tcomputer ∼ (mq)−2limit: scale mq → 0allows 323 × 16, or 243 × 48, lattices. In view of the physical differencebetween time and space, the time, i.e., inverse-temperature, dimensionof the lattice can be chosen to be smaller than the spatial extent of thelattice. On such lattices, the hadron spectra that emerge nowadays arequite realistic; see section 15.5 below.However, such lattices may not be large enough to describe preciselymany of the interesting properties of the QGP. We need to describe twodifferent quark mass scales (u and d, and s) while maintaining chiralsymmetry in the light-quark sector, and treating an odd number of qu-ark flavors (both staggered and Wilson quarks favor the presence of aneven number of flavors). This task has not been resolved completely, andthe properties of QGP we will discuss retain a systematic uncertainty.Moreover, a many-body system such as QGP should have many collec-tive modes of excitation. Given the size of the physical lattices studied,collective oscillations with wave lengths greater than a few fermis arenot incorporated. Although this does not influence in a critical way theproperties of the equations of state, such long-range oscillations are oftencarriers of flows, e.g., of heat. Therefore, the study of transport propertiesof the QGP phase on the lattice is not yet possible.The euclidian lattice is indeed ideal for simulating high-temperatureQCD since, in this case, there is a direct correspondence between theimaginary time and temperature – the path-integral weight is, in fact,the partition function with ONt = 1/T = β. The statistical-physics prop-296 QCD, hadronic structure and high temperatureerties are the operators which are fairly simple, e.g., the energy densityinvolves only one point in space–time. One has to remember that otherlattice studies of the hadron mass spectrum involve two point functions,whereas weak-interaction matrix-element computations typically involvethree point functions since we have to create a hadron from the vacuumat t = 0, act on the state with an operator at t = t1, and then annihilatethe hadron at t = t2.Among the quantities which are studied in high-temperature QCD arethe Polyakov loop [273], the chiral condensate ψ̄ψ, Eq. (3.22), the energyand pressure, screening lengths of color-singlet sources, the potential be-tween static test quarks, and the response of the quark density to aninfinitesimal chemical potential. Of these, the Polyakov loop and ψ̄ψ arethe most intensively studied. ψ̄ψ is the order parameter for chiral sym-metry breaking. It is nonzero under ordinary conditions T → 0, and weexpect it to vanish when chiral symmetry is restored for T > Tc. Looselyspeaking, the Polyakov loop has the value e−F/T , where F is the free en-ergy of a static test quark. In pure SUc(3) gauge QCD, the Polyakov loopis zero at low temperatures, indicating confinement of the test quark, andnonzero at high temperature.It is understood today that dynamic quarks make a big differencein high-temperature QCD, and the ‘quenched’ approximation has beenfound to be in general unsuitable. Looking at the energy of free quarksand gluons (the Stefan–Boltzmann law, see section 4.6) even with nf = 2flavors of light quarks, we find that the 16 = 8c × 2s gluon degrees offreedom are dominated by 21 = 2f × 3c × 2s × 74 equivalent quark degreesof freedom. The thermal properties of quarks dominate those of glu-ons. Quenched quarks are known to exclude the particle–antiparticle-pairfluctuations in the vacuum. Thus, if quenched quarks are used, some im-portant physical processes are forbidden. For example, consider a quark–antiquark pair connected by a string of color flux. With quenched quarks,when the distance grows, we encounter an ever-growing linear potential –if pair fluctuations are excluded, this string never breaks. In the presenceof dynamic quarks, when the string is long enough, there is enough energyto create a quark–antiquark pair, which breaks the string, forming twomesons.It is easy to find, in the numerical Monte-Carlo integration, the tran-sition (crossover) to some novel high-temperature behavior in a latticesimulation, though it is very difficult to ascertain the nature of the tran-sition. To vary the temperature with a fixed number of lattice spac-ings in the time direction, the lattice spacing O is changed by varyingthe coupling g. This works because g is the coupling constant definedon the scale of the lattice spacing. In an asymptotically free theory,the coupling decreases for shorter length scales. Therefore, decreasing15 Lattice quantum chromodynamics 2975.2 5.3 5.40.100.20.3m /T = 0.08q LLχ5.2 5.3 5.40.10.20.30.40.50.6m / T = 0.08qχ m0Fig. 15.1. Deconfinement and restoration of chiral symmetry in two-flavor L-QCD. Open circles: on the left-hand side the Polyakov loop L, which is theorder parameter for deconfinement in the pure gauge limit (mq → ∞); and onthe right-hand side quark chiral condensate ψ̄ψ, which is the order parameterfor chiral-symmetry breaking in the chiral limit (mq → 0), shown as a functionof the coupling β = 6/g2. Also shown are the corresponding susceptibilitiesχL ∝(〈L2〉 − 〈L〉2)(left) and χm = ∂〈ψ̄ψ〉/∂m (right) which peak at the samevalue of the coupling [159].g, or increasing 6/g2 (6 = 2nc), makes O smaller and the temperature,T = (NtO(g))−1, higher. As the temperature is increased through thecrossover, ψ̄ψ drops and the Polyakov loop increases, see Fig. 15.1. ThePolyakov loop and ψ̄ψ change at the same temperature, indicating that‘deconfinement’, and restoration of chiral symmetry are happening at thesame temperature.Little is known with certainty about the nature of the crossover betweenthe confined (frozen) phase and the new phase suggested by Fig. 15.1. Inparticular, we cannot yet be sure what kind of phase transition or trans-formation is encountered, see Fig. 3.4 on page 54 and the related discus-sion. It is fairly well established, from lattice simulations, that there is afirst-order phase transition in the pure gauge limit, and for three-masslessquarks. As a quark mass is lowered from infinity this transition disap-pears, and there may be a continuous crossover from the low-temperatureregime to the high-temperature regime. But even if there is no phasetransition, the crossover is fairly sharp. This can be seen by consideringthe inverse screening lengths for qq̄ sources with the quantum numbersof the pion π and its opposite-parity partner σ. At high temperatures,they become very close, with the remaining small difference being due tothe explicit breaking of symmetry originating from the quark mass. This298 QCD, hadronic structure and high temperatureand other quantities show that the high-temperature regime does indeedhave the expected characteristics of the QGP.Only recently have calculations progressed far enough to lead to firmresults about properties of the QGP. To find the temperature of thecrossover in physical units, the lattice spacing must be determined bycomputing some physical quantity, such as the ρ-meson or the nucleonmasses. The π mass is not a good choice of scale, since it can be madearbitrarily small by making the quark mass small, as we have seen in sec-tion 3.3. Though the value of the crossover temperature one finds is stillin doubt for real-life quarks, at present the opinion of experts we shalldiscuss in the next section is biased toward a value of 160MeV [160], nearthe Hagedorn temperature; see chapter 12.15.5 A survey of selected lattice resultsThere are many lattice results related to QCD properties we study, ad-dressing diverse questions such as hadronic masses, matrix elements, andphysical properties of hot QCD. Given the rapid development of the fieldwhich promises to render any presentation quite rapidly obsolete, we fo-cus our attention on ‘stable’ results that are most relevant in the contextof this book and, in particular, the study of equations of state of QGP.We will not further discuss in this section the intricate extrapolations(continuum limit, massless-quark limit) which form part of the process ofevaluation of the bare numerical results, and which we described abovein section 15.4.The running of the gauge coupling constant has now been tested forthe case of two massless Wilson fermions [76]. The lattice results com-pare very well with the renormalization-group result, as can be seen inFig. 15.2. These results are, at present, still mainly of academic interestand are different in detail from Fig. 14.1, since, in the range αs < 0.4,we actually have to include s, c, and b quarks, in order to compare withexperiment. On the other hand, the fact that the running is seen as ex-pected in the theoretical evaluation of two-light-flavor QCD reassures usregarding the validity of the findings we presented in Fig. 14.1.The study of hadronic masses allows one to draw conclusions aboutthe input quark masses. The CP-PACS collaboration has recently com-pleted an extensive evaluation using its dedicated (peak) 614-GFLOPS(giga-floating-point-operations) computer [33]. The lattice action andsimulation parameters were chosen with a view to carrying out a pre-cise extrapolation to the continuum limit, as well as scaling in the chiralmq → 0 limit for dynamic up- and down-quark masses. However, thestrange quarks were considered in a quenched approximation, which en-tails an ‘uncontrolled’ error.15 Lattice quantum chromodynamics 299Fig. 15.2. Dots, lattice-gauge theory with two light fermions for αs(µ), com-pared with the perturbative three-loop result (line); parameters are chosen suchthat, for Λ = 1 GeV, αs(MZ) = 0.118; results of the ALPHA collaboration [76].Evaluation of the masses of kaons and/or φ allows one to determinethe mass of the strange quark. The results found using dynamic u andd quarks are mMSs (2GeV) = 88+4−6 MeV (K input) and mMSs (2GeV) =90+5−11 MeV (φ input), which are about 25% lower than the values foundwith quenched u and d quarks. The low value for the mass of the strangequark is well within the accepted range; see table 1.1 on page 7. Theconsistency of these two results is quite remarkable. Moreover, using themass of the K meson to fix the strange-quark mass, the difference fromexperiment for the mass of the K∗ meson is 0.7+1.1−1.7%, and that for the φmeson 1.3+1.8−2.5%. When the φ meson is used as input, the difference inthe mass of the K∗ meson is less than 1%, and that for the mass of theK meson is 1.3± 5.3%. The masses of (multi)strange baryons are, withinmuch larger computational error, also in agreement with the experimentalvalues.Should this relatively low mass for the strange quark be confirmedwhen dynamic strange quarks are introduced, the speed of production ofstrangeness at low temperatures T Tc in QGP as perhaps formed at in-termediate SPS energies would dramatically increase. Strangeness coulddevelop into a highly sensitive ‘low energy’ probe of formation of QGP,even when the initial conditions reached are near to the critical tempera-ture. In this context, it is interesting to note the steep rise and thresholdof the strangeness-excitation function shown in Fig. 1.5 on page 17.300 QCD, hadronic structure and high temperatureFig. 15.3. The energy density of hadrons obtained with staggered fermions di-vided by T 4, as a function of T/Tc. Stefan–Boltzmann limits of a non-interactinggas of quark and gluons are indicated with arrows for each case considered.In our context, an even more important recent lattice advance has beenthe extensive study of 2, 2 + 1 and 3 flavors in hot QCD [159–161]. Thebehavior of the energy density ε/T 4 is presented in Fig. 15.3 as a functionof temperature T/Tc, obtained with staggered fermions. The Stefan–Boltzmann values expected for asymptotically (high-T ) free quarks andgluons are shown by arrows to the right, coded to the shades of the threeresults presented: two and three light flavors (up and down, respectively),for which the quark mass is mq = 0.4T and a third case (dark line), inwhich, in addition to the two light flavors, a heavier flavor ms = T isintroduced. The temperature scale is expressed in units of the criticaltemperature Tc, as appropriate for each case (Tc changes with the num-ber of flavors). The value of Tc is where one observes a rapid changein the behavior of the quark condensate/susceptibility and, at the samelocation, one sees also the onset of deconfinement in the Polyakov loop;see Fig. 15.1.We see that, around T/Tc = 1, the number of active degrees of freedomrapidly rises, and the energy density attains as early as T = 1.2Tc thebehavior characteristic of an ideal gas of quarks and gluons, but witha somewhat reduced number of active degrees of freedom. The energydensity in the deconfined phase can be well approximated byεQGP (11–12)T 4. (15.20)15 Lattice quantum chromodynamics 301 0 1 2 3 4 100 200 300 400 500 600T [MeV] P/T 4PSB/T 43 flavors2 + 1 flavors2 flavors0 flavor Fig. 15.4. The pressure of hadrons obtained with staggered fermions in unitsof T 4 as a function of T .The critical temperature Tc has the valueTcmρ= 0.20± 0.01, Tc = 154± 8 MeV, for nf = 3, (15.21)where, in addition to the statistical error quoted above, a systematicerror at a similar level, associated with uncertainties in the scaling be-havior, is expected. In case of two flavors, the value Tc = 173 ± 8 MeVis found. These results are consistent with calculations performed withclover-shape-improved Wilson action (see section 15.3) by the CP-PACScollaboration [32] for two flavors. In Fig. 15.4, we present the behavior ofP/T 4 as a function of temperature, with the temperature scale derivedfrom Eq. (15.21). The expected Stefan–Boltzmann limits are shown byarrows. Apart from the cases of 2, 2 + 1 and 3 flavors, we see also the‘pure gauge’ case with zero flavors.The conclusion we draw from these results is that the lattice-QCDevaluation has matured to the level of being able to offer informationdirectly relevant to the physical properties of hot QCD. These results,in particular, show that there is a rapid phase transformation or even afirst-order phase transition at Tc = 163± 15 MeV.We see, in Fig. 15.4, that the phase-transition temperature decreasessignificantly with increasing number of flavors. However, the shapes of allcurves scale similarly. This is shown in Fig. 15.5, in which the temperaturescale is expressed in units of Tc and the pressure in terms of the ideal302 QCD, hadronic structure and high temperature0.00.20.40.60.81.0 1.5 2.0 2.5 3.0 3.5 4.0T/Tc P/PSB3 flavors2 flavors2 + 1 flavors0 flavor Fig. 15.5. The pressure of hadrons obtained with staggered fermions dividedby the appropriate Stefan–Boltzmann limit, as a function of T/Tc.Stefan–Boltzmann pressure. The zero-flavor pure gauge case is fastestgrowing toward the limiting value, but still at a considerably slower ratethan that for the energy density we have seen in Fig. 15.3.An ideal massless relativistic gas should satisfy the relation ε−3P = 0,Eq. (4.64). The difference in behavior, comparing Fig. 15.3 with Fig. 15.5,must originate from the presence of variables with dimensioned scales. Weencountered two such (related) variables, the vacuum property B and theparameter Λ controlling the magnitude of the running variables (αs andmi). The deviations of pressure from the Stefan–Boltzmann ideal-gasbehavior seen in lattice results, in particular ε− 3P = 0, are in direct orindirect fashion related to these quantities. We will quantify this in thefollowing section.We note that, using the Gibbs–Duham relation, Eq. (10.26), we canrelate the change in the pressure, seen in Fig. 15.4, to the differenceε− 3P . We generalize slightly the argument presented in Eqs. (4.62) and(4.64). We consider the free-energy densityf = −TVlnZ(T, V ). (15.22)P = −f for an infinite system. Moreover, the entropy density σ = ∂/∂Tf ,see Eq. (10.6). We find, employing Eq. (10.26) at zero baryon density,ε− 3PT 4= TddT(PT 4). (15.23)16 Perturbative quark–gluon plasma 303As soon as we understand the gentle slope of pressure P in Fig. 15.4,i.e., the right-hand side of Eq. (15.23), we will also understand the differ-ence between the behaviors of energy and pressure, the left-hand side ofEq. (15.23), noted on comparing Fig. 15.3 with Fig. 15.5.We will show, in section 16.2, that this non-ideal-gas behavior can beinterpreted as resulting from perturbative quark–gluon interactions andthe presence of the latent heat of the vacuum B. An equivalent expla-nation invoking the presence of quasi-particles with mass, and quantumnumbers of quarks and gluons, will also be considered.16 Perturbative quark–gluon plasma16.1 An interacting quark–gluon gasAs explained in section 14.1, the interactions between quarks and gluonsare contained in the QCD Lagrangian Eq. (13.79), improved by gauge-fixing and FP-ghost terms Eq. (14.1). Strictly considered, the rules forFeynman diagrams we presented in Eqs. (14.2)–(14.8) are applicable toprocesses in perturbative vacuum, whereas to compute thermal proper-ties of interacting quark and gluons, we are working in matter at finitetemperature T and chemical potential µ. The generalization required isdiscussed in detail in the textbook by Kapusta [157].A lot of effort in the past few decades has gone into the development ofthe perturbative expansion of the partition function. The series expan-sion, in terms of the QCD coupling constant g, has been carried out to or-der [(g/(4π)]5 = (αs/π)5/2/32 [282]. This series expansion, which was de-veloped using as reference the perturbative vacuum in empty space, doesnot appear to lead to a convergent result for the range of temperatures ofinterest to us [36]: the thermodynamic properties vary widely from orderto order, oscillating quite strongly around the Stefan–Boltzmann limit. Ithas therefore been claimed that the perturbative QCD thermal expansionhas a zero-range convergence radius in αs.Our following considerations will be limited to the lowest-order pertur-bative term combined with the vacuum energy B and allow an excellentreproduction of the key features of lattice results. It remains to be un-derstood why this is the case. It is not uncommon to encounter in aperturbative expansion a semi-convergent series. The issue then is howto establish a workable scheme. It is, for example, possible that a differ-ent scheme of perturbative approach, in which the QCD parameters (αsand masses) are made nonperturbative functions of the medium using anin-medium renormalization group, would yield a better converging seriesin αs.Considering the inconclusive and rapidly evolving landscape of thermal304 QCD, hadronic structure and high temperature12− 12−12+181Fig. 16.1. Feynman diagrams contributing to the equation of state of the QGPin order αs. Wavy lines represent gluons, solid lines represent quarks, and dashedlines denote the ghost subtractions of non-physical degrees of freedom.QCD, we will in this book not explore the subject beyond the study of theconsequences of the lowest-order thermal corrections. The lowest-ordercontributions are obtained by evaluating the graphs shown in Fig. 16.1.Evaluation of these diagrams at high temperature for massless quarks andgluons is possible analytically [91]. For massless quarks and antiquarksone finds the following two terms in the partition function:lnZq(β, λ)=gV6π2β−3[(1− 2αsπ)(14ln4 λq +π22ln2 λq)+(1− 50αs21π)7π460], (16.1)where g = nsncnf = 12, for ns = 2s+1 = 2, nc = 3, and nf = 3. The firstterm in parentheses is c3, Eq. (4.71c), and the second is c2, Eq. (4.71b).The quark fugacity λq is related to the baryon-number fugacity, as dis-cussed in Eq. (11.3). The glue contribution islnZg(β, λ) =8π245β−3(1− 15αs4π), (16.2)where the last term in parentheses is c1, Eq. (4.71a). Finally, the vacuumcontribution can be added in the formlnZvac(β) = −βBV. (16.3)This term insures that the energy density, inside the bag, is positive andsimultaneously that the pressure exerted on the surface of the bag isnegative. The total grand partition function islnZQGP = lnZq + lnZg + lnZvac. (16.4)This equation was presented explicitly in section 4.6, Eq. (4.70). The re-sulting perturbative interactions between quarks and gluons are obtainedby differentiating Eq. (16.4) with respect to V , β, and λq; see section 10.1.Another important consequence of interactions in a conductive colorplasma is the change in location of poles of particle propagators. Rather16 Perturbative quark–gluon plasma 305than of particles, one than speaks of quasi-particles, in our instance ther-mal quarks and gluons. The idea of quasi-particle thermal gluon andthermal quark masses is rooted in the desire to characterize, in a simpleway, the behavior of quark and gluon correlators (propagators) evaluatedinside the color-conductive medium. The monograph by Kapusta [157]offers an excellent early introduction to this still rapidly evolving subject,and we will restate here a few results of immediate interest.Allowing for interaction to lowest order in αs, a relation between theenergy E and momentum p = |/p| (a dispersion relation) of quasi-quarks isfixed by the location of the pole of the quark propagator Eq. (14.2) [270]:E2 = /p 2 + (mTq )2[E + p2p− E2 − p24p2ln(E + pE − p)]. (16.5)For p → ∞, with E → p+m2/(2p), we recognize the quantity(mTq )2 =4π3αsT2 (16.6)as the mass parameter, establishing the relation with the momentum andcontrolling the quasi-quark phase space.Near to p = 0, the long-range oscillations described by Eq. (16.6) requiremore attention. They have the energyEq(p → 0) =m(/p → 0, ω;T )√2+p3√2p23m(/p → 0, ω;T ) + · · ·. (16.7)It is evident from the above that the ‘thermal’ mass of the quark in amedium may be defined also by considering this zero-momentum limit[270], rather than the high-momentum limit Eq. (16.6) adopted here.However, the domain p → 0 is not important in the counting of states inthe phase space, considering the momentum-volume factor d3p = 4πp2 dp.We conclude that, in the study of the phase space of light thermal quarksin plasma, we should use for the light-quark energy the high momentumlimitEq √p2 + (mTq )2, q = u,d.A similar discussion arises for the behavior of gluons. The collectiveoscillations in the plasma with the quantum numbers of gluons behave,at high momentum, according toEg √p2 + (mTg )2,where(mTg )2 = 2παsT 2(1 +nf6), (16.8)306 QCD, hadronic structure and high temperaturewhereas near p → 0Eg(p → 0) =√23mTg . (16.9)The thermal mass of gluons introduced in Eq. (16.8) is not the Debyescreening mass limiting the effective range of interactions in plasma. Thisquantity corresponds to the static limit in the behavior of the longitudinalgluon-like plasma oscillations [269]:(mDg )2 = 4παsT 2(1 +nf6). (16.10)16.2 The quark–gluon liquidWe consider the properties of the QGP allowing for the lowest-order inter-actions with a temperature-dependent interaction strength, and vacuumpressure B. We refer to this model as the quark–gluon liquid [147]. Nat-urally, we hope and expect to reproduce lattice-QCD results [160]. Thereis considerable sensitivity to the value of αs(µ) and it is necessary to useits precise form; see section 14.2.Along with many other authors, we adopt the relationµ 2πT (16.11)between the scale of the QCD coupling constant and the temperature ofthe thermal bath. On the one hand, the right-hand side is close to theaverage collision energy of two massless quanta at T , and on the other,the relation Eq. (16.11) makes the thermal-QCD expansion least sensitiveto the renormalization scale [283].In Fig. 16.2, the ‘experimental’ values are the numerical lattice simu-lations [160], see section 15.5, for 2 (diamonds), 3 (triangles), and 2 +1 (squares) flavors. The non-interacting Stefan–Boltzmann quark–gluongas, Eq. (4.70), with ci = 1 and with the bag constantB = 0.19GeVfm−3 (16.12)is shown as thin lines, dotted for the case of three flavors and dashedfor the case of two flavors. We see that the effect of vacuum pressuredisappears as T → 2Tc, and that the lattice results differ significantlyfrom those for the free gas, even at T = 4Tc.To obtain agreement with the lattice results, it is necessary to intro-duce perturbative coefficients ci, Eqs. (4.71a)–(4.71c), with numericallycomputed αs(µ = 2πT ) [147]. The thick lines seen in Fig. 16.2 were ob-tained allowing for αs(2πT ) shown in Fig. 14.3 with Tc = 160MeV. Tofind the behavior near to T = Tc, the only ‘free’ choice we can make is16 Perturbative quark–gluon plasma 307Fig. 16.2. Lattice-QCD results [160] for P/T 4 at λq = 1 (for 2 (diamonds),3 (triangles), and 2 + 1 (squares) flavors) compared with free quark–gluon gaswith bag pressure B = 0.19GeV fm−3, thin dotted (three flavors) and dashed(two flavors) lines. Thick lines (agreeing with lattice data) are derived from thequark–gluon liquid model: dotted line, 3 flavors; solid line, 2 + 1 flavors; anddashed line, 2 flavors.the value of B, and this is the reason why that particular number wasselected in Eq. (16.12). To achieve the agreement with lattice results seenin Fig. 16.2, the relevant relation isBT 4c= 2.2, B1/4 = 1.22Tc. (16.13)The precise relationship between the scale µ and T has negligible impacton the result shown, as long as the natural order of magnitude seen inEq. (16.11) is maintained. Within the simple model we introduced insection 1.3 to describe the phase transition, Eq. (16.13) implies nearly thecorrect number of degrees of freedom freezing in the transition, ∆g 20.It has been shown that it is also possible to reproduce the lattice resultsusing fine-tuned thermal masses (see table I in [204]). In Fig. 16.3, weshow the light-quark (solid thick line) and gluon (dashed thick line) ther-mal masses which were used to fit the lattice data. The actual thermalquark and gluon masses, defined in Eqs. (16.6) and (16.8), are also shownin Fig. 16.3, as thin lines (dashed for gluons) obtained using αs(µ = 2πT ),from Fig. 14.3. We conclude that the thermal masses required to de-scribe the reduction of the number of degrees of freedom for T > 2Tc are308 QCD, hadronic structure and high temperatureFig. 16.3. Thermal masses fitted to reproduce lattice-QCD results for nf = 2[204], thick solid line for quarks, and thick dashed line for gluons. Thin lines,perturbative QCD result for αs(µ = 2πT ).just the perturbative QCD result. Importantly, this means that thermalmasses express, in a different way, the effect of perturbative quantum-chromodynamics, and thus, for T > 2Tc, we have the option of usingEq. (16.1), or the more complex thermal-mass approach.However, in the temperature domain T < 2Tc, in which the vacuumpressure B is relevant, see Fig. 16.2, the thermal mass required to fit latticeresults, as can be seen in Fig. 16.3, is quite different from the perturbativeQCD result, and we believe that the interpretation of lattice results inthis phenomenologically important domain is much less natural than thequark–gluon-liquid approach. The introduction of thermal masses, inorder to describe the behavior seen in Fig. 15.5, is expressing just thesame fact that the pressure must be some function of the parameter Λwhich is controlling the magnitude of the running αs, and that additionalphysics, such as the vacuum pressure B, is required in order to understandthe behavior of the QGP, for T < 2Tc.We have seen that the suppression of the number of degrees of freedom,seen in the QGP pressure can be well described by a first-order thermalQCD result, either using thermal masses, or more directly using the first-order corrections seen in Eqs. (16.1) and (16.2). However, to describe thebehavior for T → Tc, we should invoke nonperturbative properties of thevacuum. As discussed at the end of section 15.5, once the variation of16 Perturbative quark–gluon plasma 309P/T 4 with T has been described, the deviation of lattice results from theideal-gas law, ε− 3P → 0, is also understood.16.3 Finite baryon densityIt is in the consideration of the finite baryon density that the issues re-garding how to model the lattice results we raised in the last section aremost relevant. We believe that it is appropriate to obtain the propertiesof QGP in a manner allowing the magnitude of the color interaction tobe controlled by the energy scale which depends on the baryon-chemicalpotential. The dependence of the scale µ of αs on the fugacity we adoptis [264]µ = 2√(πT )2 + µ2q = 2πβ−1√1 +1π2ln2 λq. (16.14)Like with Eq. (16.11), there is no exact mathematical rationale for this ex-pression; it is entirely based on intuition and the particle-energy behaviorseen in studies of thermal QCD.We note that Eq. (16.14) implies that−β ∂αs(β, λq)∂β= µ∂αs∂µ,λq∂αs(β, λq)∂λq=lnλqπ2 + ln2 λqµ∂αs∂µ, (16.15)T∂αs(T, µq)∂T=π2T 2(πT )2 + µ2qµ∂αs∂µ.The derivative of the QCD coupling constant can be expressed as, Eqs. (14.18)and (14.21),µ∂αs∂µ= −b0α2s − b1α3s + · · · ≡ βpert2 . (16.16)βpert2 is the beta-function of the renormalization group in the two-loop ap-proximation, with bi defined in Eq. (14.23). βpert2 does not depend on therenormalization scheme, and solutions of Eq. (16.16) differ from higher-order results by less than the error introduced by the experimental un-certainty in the measured value of αs(MZ); see section 14.3.310 QCD, hadronic structure and high temperatureWe are now prepared to study physical properties of the quark–gluonliquid. The energy density is obtained from Eq. (4.70):εQGP = −∂lnZQGP(β, λ)V ∂β,= 4B + 3PQGP − β∂αs∂β3∑i=1∂ci∂αPQGP∂ci. (16.17)We find thatεQGP − 3PQGP = A+ 4B, (16.18)whereA = (b0α2s + b1α3s )[2π3T 4+nf5π18T 4+nfπ(µ2qT2+12π2µ4q)], (16.19)and PQGP is stated explicitly in Eq. (4.70). We see in Eq. (16.18) theinteresting propertyεQGP − 3PQGPT 4→ π18(12 + 5nf)(b0α2s + b1α3s ) + 4BT 4, µq → 0, (16.20)where the thermal interaction (the first term) is determining the behaviorat T 2Tc, Fig. 16.2.A convenient way to obtain the entropy and baryon density uses thethermodynamic potential F ; see Eq. (4.70) and chapter 10. For the quark–gluon liquid, we haveF(T, µq, V )V= −TVlnZ(β, λq, V )QGP = −PQGP(T, µq), (16.21)with entropy density sQGP and baryon density ρb, which is a third of thequark density:sQGP = −dFV dT, ρb = −13dFV dµq. (16.22)The entropy density issQGP =32π245c1T3 + nf(7π215c2T3+c3µ2qT)+A π2Tπ2T 2 + µ2q. (16.23)The coefficients ci are defined in Eq. (4.71a) and are the same as inEqs. (16.1), (16.2), and (4.70). The baryon density isρb =nf3c3(µqT2 +1π2µ3q)+13A µqπ2T 2 + µ2q. (16.24)16 Perturbative quark–gluon plasma 31116.4 Properties of a quark–gluon liquidThe behavior of the quark–gluon liquid is of particular interest in thestudy of• initial conditions corresponding to an early instant in time during theheavy-ion collision when the light quarks and gluons are in approximatechemical equilibrium, but strange quarks have not yet been produced,thus nf = 2; and• properties of the fireball of quarks and gluons at the time of its breakupnear to T = Tc, at which strangeness is practically equilibrated, andnf = 2.5.Given the good agreement with the lattice results for vanishing chemicalpotential, the following study offers a quantitative description of the rela-tionship between the temperature and physical properties reached in thedeconfined phase, but with an unknown systematic error when propertiesinvolving baryon density are physically relevant (e.g., in the AGS energyrange).In the left-hand panels of Fig. 16.4, we show the physical properties atfixed energy per baryon, in the range 2 GeV≤ E/b ≤ 15 GeV, as functionsof temperature, while in the right-hand panels we study the behavior atfixed value of the (dimensionless) entropy per baryon 10 ≤ S/b ≤ 60. Inpanels (a) and (b) of Fig. 16.4, as we step from line to line from left toright, the energy per baryon is incremented by 1 GeV; in panels (d) and(e) the entropy per baryon is incremented by 5 units; in panel (c) we stepfrom bottom to top incrementing by 1 GeV; and in panel (f) from bottomto top by 5 entropy units. The light-dashed boundaries are obtained fromthe conditions• on energy density εq,g ≥ 0.5GeV fm−3 (excluding here the latent heatof the vacuum B 0.19GeV fm−3), and/or• baryochemical potential, µb ≤ 1GeV.We highlight the result for E/b = 8.5GeV by using a thick dashed lineon the left in Fig. 16.4, and that for S/b = 40 by using a thick solid lineon the right.This systematic exploration should allow one to assess the behaviorof the quark–gluon liquid possibly created in collisions performed in theenergy range between those of the AGS and SPS accelerators, and com-prising chemically equilibrated u and d quarks and gluons. The particlemultiplicity in the final state tells us that the entropy per baryon is at thelevel of 40 units for the high-energy range of the SPS; see section 7.4. Thecorresponding temperature which we read from panel (c) for E/b < 8.5GeV is T < 280 MeV. Evaluation of properties of the final abundances ofparticles, section 19.3, shows that it is easier to deposit baryon number312 QCD, hadronic structure and high temperatureFig. 16.4. Left: lines corresponding to fixed energy per baryon E/b = 2 to15 GeV in steps of 1 GeV: (a) baryo-chemical potential µb (highest E/b at thebottom), (b) baryon density n/n0 in units of equilibrium nuclear density (highestE/b at the bottom), and (c) S/b, the entropy per baryon (highest E/b at thetop). Right: lines corresponding to fixed entropy per baryon S/b = 10 to 60 insteps of 5, from top to bottom: (d) µb, (e) n/n0, and (f) E/b; see the text forfurther details.than energy in the fireball, and, in general, the initial energy per baryonis smaller than the collisional kinematic limit. Taking E/b 7 GeV, weobtain Tch 220. Before the light-quark flavor has been equilibrated, thetemperature of gluons could be as high as Tth 250 MeV, for the SPStop energy.We see in Fig. 16.4 panels (a) and (d) the appropriate ranges of thebaryo-chemical potential; and in panels (b) and (e) the baryon density inunits of equilibrium nuclear density, n0 = 0.16 fm−3. The dotted linesin panels (c) and (d) show where Pq,g − B = 0: there, the entropy perbaryon at fixed energy per baryon reaches its maximum as a functionof temperature, and the energy per baryon at fixed entropy per baryonreaches its minimum. In an equilibrium transition, the QGP transforms16 Perturbative quark–gluon plasma 313Fig. 16.5. The analog of Fig. 16.4 for the RHIC energy domain. Left, lines atfixed energy per baryon E/b = 10 to 190GeV in steps of 20GeV. Right, lines atfixed entropy per baryon S/b = 50 to 500 in steps of 50; see the text for moredetails.for T above this condition, since the ‘external’ hadron pressure needs tobe balanced. In an exploding system, the breakup occurs at T below thiscondition, since the flow of the quark–gluon liquid adds to the pressureworking against the vacuum; see section 3.5.In Fig. 16.5, a similar discussion of the RHIC physical conditions isshown, following the same pattern as Fig. 16.4, except for the use oflogarithmic scales. On the left-hand side, the energy range is now 10GeV ≤ E/b ≤ 190 GeV, and lines are in steps of 20GeV. On the right-hand side, the specific entropy range is 50 ≤ S/b ≤ 500, in step of 50units. The lines become denser toward higher energy or entropy. Thedotted lines in panels (c) and (f) indicate where the particle pressure isbalanced by the vacuum pressure.In the RHIC run at√sNN = 130 GeV, the final hadron phase space atcentral rapidity, the intrinsic local-rest-frame energy per baryon is E/b 25 GeV, while the entropy content is S/b 150, the uncertainties in bothvalues are of 10%–15%, compare with table 19.4 on page 367. The exact314 QCD, hadronic structure and high temperatureFig. 16.6. The energy density as a function of temperature for a quark–gluonliquid. Top, at fixed E/b; bottom, at fixed S/b; left for the SPS and right forthe RHIC energy domain. Limits are energy density ε = 0.5 GeV fm−3 andbaryo-chemical potential µb = 1GeV. Lines: left top for fixed E/b = 2 to 15GeV in steps of 1 GeV; left bottom, at fixed S/b = 10 to 60 in steps of 5; righttop, at fixed E/b = 10 to 190 GeV in steps of 20 GeV; and right bottom, at fixedS/b = 50 to 500 in steps of 50.values of E/b and S/b depend on the way one accounts for the influenceon the yield analysis of unresolved (at the time of writing) weak decaysof hyperons and assumptions made about chemical equilibria. Despitethis substantial increase compared to the energy and entropy contentseen at the SPS, the value of Tch consistent with this result is only 30–40 MeV higher than that at the SPS. For this reason, the increases inthe particle multiplicity we have discussed in section 9.5 are relativelymodest.The rise of energy density with temperature at fixed E/b and S/b isshown in Fig. 16.6, for the expected domain of parameters at the SPSon the left and for the domain of RHIC parameters on the right; thelines follow the same key as that used in Figs. 16.4 and 16.5. We seethe expected rise with T 4, but the narrow band of values associated withbaryon content is quite striking. In fact, in the right-hand panel thedifferent lines coalesce, energy density is effectively solely a function of Tfor E/b > 10 GeV and S/b > 50.These results indicate clearly how the presence of baryon-rich quarkmatter possibly formed at lower collision energies augments the entropyand energy content. At large baryon density these results depend strongly16 Perturbative quark–gluon plasma 315Fig. 16.7. The entropy density in chemically equilibrated QGP at λq = 1 asa function of temperature: solid line, nf = 2; long-dashed line, nf = 2.5; andshort-dashed line, ‘pure glue’ nf = 0.on the validity of the extrapolation made from µb = 0 of the quark–gluonliquid and there is no way to estimate the associated systematic erroruntil lattice results of comparable quality for µb = 0 become available.However, it would appear that at above E/b > 3 GeV, conditions suitablefor formation of deconfined state are present.Entropy plays a very important role in the study of scaling one dimen-sional hydrodynamic expansion, and the hadron yield in the final stateoffers a reliable measure of the product σ0τ0, Eq. (7.27). In Fig. 16.7, theentropy density σ is shown as a function of temperature. The solid line isfor the case of an equilibrated light-quark–glue system in the limit of van-ishing chemical potential. We note that initially the entropy rises fasterthan the asymptotic T 3 behavior, since the quantum chromodynamicsinteractions weaken, and there is an increase in the effective number ofacting quark and gluon degrees of freedom. Thus, the drop in entropydensity on going toward the hadronization condition is considerable. Inorder to preserve the entropy content in the fireball when the QGP fire-ball expands, from T 300 MeV toward 150 MeV, a volume growth bya factor of nine must occur.The ‘pure-glue’ case (short-dashed line) contains as expected about halfof the entropy when one makes a comparison at equal temperature. Theaddition of strangeness expressed by increasing nf = 2 to 2.5 adds about10% to the entropy content.VIStrangeness17 Thermal production of flavor in a deconfined phase17.1 The kinetic theory of chemical equilibrationStrangeness, and more generally heavy-flavor quarks, can be producedeither in the first interactions of colliding matter, or in the many ensuingless-energetic collisions. The mass of the strange quark ms is comparablein magnitude to the typical temperatures reached in heavy-ion interac-tions, and the numerous ‘soft’ collisions of secondary partons dominatethe production of strangeness, and naturally, of the light flavors u and d.The masses of charm and bottom quarks are well above typical tem-peratures; these quarks are predominantly produced in the hard initialscattering. This process remains today a topic of current intense studyboth for the elementary and for the nuclear collisions [124]. We will notdiscuss it further in this book.At the time at which the strange flavor approaches chemical equi-librium in soft collisions, the back reaction is also relevant. The quantum-mechanical matrix element driving a two-body reaction must be, channelby channel, the same for forward- and backward-going reactions. Theactual rates of reaction differ since there are usually considerable differ-ences in statistical and phase-space factors. However, the forward andbackward reactions will balance when equilibrium yields of particles areestablished. This principle of detailed balance can sometimes be used toevaluate reaction rates.The net change in yield of flavors f and f̄ is given by the differencebetween the rates of production and annihilation. The evolution in thedensity of heavy quarks in QGP can be described by the master equationdNf(t)d3x dt=dN(gg, qq̄→ f f̄)d3x dt− dN(f f̄ → gg, qq̄)d3x dt. (17.1)31617 Thermal production of flavor in a deconfined phase 317When locally at a point in space there is exact balance between the twoterms on the right-hand side of Eq. (17.1), chemical equilibrium has beenestablished, which, as we recall, is the state of maximum entropy; seesection 7.1.Each of the terms in Eq. (17.1) expresses a change in number of particlesper unit of 4-volume and it is a Lorentz-invariant quantity. We takeadvantage of this to write∂µjµf ≡∂ρf∂t+∂/v ρf∂/x= ρ2g(t)〈σv〉gg→f f̄p + ρq(t)ρq̄(t)〈σv〉qq̄→f f̄p− ρf(t)ρf̄(t)〈σv〉f f̄→gg,qq̄p . (17.2)The left-hand side describes the change in the local particle density in-cluding the effect of flow; the right-hand side is another way to expressthe change in number of particles in terms of individual reactions, as weshall show, see Eq. (17.7).The momentum-averaged cross section of reacting particles is〈σvrel〉p ≡11 + I12∫d3p1∫d3p2 σ12v12f(/p1)f(/p2)∫d3p1∫d3p2 f(/p1)f(/p2). (17.3)The factor 1/(1 + I12) is introduced to avoid double counting of indis-tinguishable pairs of particles, I12 = 1 for an identical pair of bosons(gluons, pions), otherwise I12 = 0. Some authors introduce this factorinto the kinetic equation Eq. (17.2). Considering that the cross section isobtained as an average over all reaction channels, the implicit sums overspin, color, and any other discrete quantum numbers can be combined inthe particle density,ρ =∫d3p(2π)3f(/p) =∫d3p(2π)3∑i=s,c,...fi(/p). (17.4)We have suppressed, in the above discussion, the dependence of the phase-space distributions f(/p, /x, t) on the spatial coordinates, as well as theirevolution with time.In general terms, we need to obtain f(/p, /x, t) for gluons and light qu-arks from a solution of a transport master equation such as the Boltzmannequation. However, this introduces a large uncertainty due to our greatignorance of the early collision (quantum) dynamics. Moreover, a sevendimensional evolution equation for f(/p, /x, t) cannot yet be handled withthe available computing power without simplifying assumptions invokingspherical symmetry. Moreover, the uncertainty about the initial temper-ature, initial yield of strangeness from pre-equilibrium reactions, and thepoorly known mass of the strange quark ms introduce significant uncer-318 Strangenesstainties into calculations of the yield of strangeness, and limit the needfor very precise methods.We proceed to simplify by the use of two assumptions, which followfrom the discussion we presented in section 5.5.• The kinetic (momentum-distribution) equilibrium is approached fasterthan the chemical (abundance) equilibrium [23, 231, 246]. This allowsus to study only the chemical abundances, rather than the full momen-tum distribution of the (strange) quark flavor.• Gluons equilibrate chemically significantly faster than does strangeness[276]. We consider the evolution of the population of strangeness onlyafter gluons have (nearly) reached chemical equilibrium.In view of these assumptions, the phase-space distribution fs can becharacterized by a local temperature T (/x, t) of a (Boltzmann) equilibriumdistribution reached for t → ∞, f∞s , with normalization set by a phase-space-occupancy factor:fs(/p, /x; t) γs(T )fTs (p), fTs (p) = e−√m2s+p2/T , (17.5)where fTs is the equilibrium Boltzmann momentum distribution. Equa-tion (17.5) invokes in the momentum independence of γs the first assump-tion. The factor γs allows the local density of strange quarks to evolveindependently of the local temperature.Using the Boltzmann momentum distribution Eq. (17.5) in Eq. (17.3),we are performing a thermal average of the cross section and relativevelocity, and the result is a thermally averaged cross section, a functionthat depends on T instead of√s. Some books refer to this as thermalreactivity; we will often call it the thermal cross section:〈σvrel〉T ≡11 + I12∫d3p1∫d3p2 σ12v12fT1 (p)fT2 (p)∫d3p1∫d3p2 fT1 (p)fT2 (p). (17.6)This thermal cross section is dependent on T , and on the masses of re-acting particles, and its physical dimension is volume per time. We willoften drop the subscripts T and rel, since the only average to which werefer in a cross section is ‘thermal’, and, in this context, the velocity isalways relative.The thermal reaction rate per unit time and volume, R12(T ), is ob-tained as follows: consider that a single particle ‘1’ enters at velocity v12a medium of particles ‘2’; the rate of reactions is 〈σvrel〉Tρ2. If per unitvolume there are N1 particles, i.e., we have a density ρ1, thenR12(T ) ≡ 〈σvrel〉Tρ1ρ2. (17.7)17 Thermal production of flavor in a deconfined phase 319The densities ρ1 and ρ2 arise from the momentum integration of the Boltz-mann distributions f1 and f2, Eq. (17.4), and contain the degeneracy fac-tors g1 and g2. This rate Eq. (17.7) is Lorentz invariant, i.e., all observersagree by how much the number of particles changes per (invariant) unitvolume in space–time.The following evaluation of R12 applies to reactions occurring in con-fined and deconfined matter. However, for almost all particles (exceptpions) in the hadronic gas, it is sufficient to use the Boltzmann momen-tum distribution function, since the phase-space cells are nearly empty,while the density of particles arises from the numerous resonances en-countered (see section 12.1). In QGP, we must in general use the Boseand Fermi distributions, as appropriate, which will complicate the resultsslightly.We recall here the (Mandelstam) variables s, t, and u characterizing,in an invariant way, the two-particle reaction 1 + 2→ 3 + 4,s = (p1 + p2)2 = (p3 + p4)2, (17.8a)t = (p1 − p3)2, (17.8b)u = (p2 − p3)2, (17.8c)s + t+ u =4∑i=1m2i , (17.8d)where the 4-momenta pµ = (Ep, /p) are used with Ep =√/p2 +m2.√s isas usual the total CM energy and t is the invariant generalization of thescattering angle.The cross section for reaction of two particles to give n final-state par-ticles is computed according toσ12v12E1E2 =∫ n+2∏i=3d4pi δ(p2i −m2i )Θ(p0i )× δ4(p1 + p2 −n+2∑i=3pi)|M|2. (17.9)|M(s, t)|2 is the reaction-matrix element obtainable, for perturbative pro-cesses, using the Feynman rules described in section 14.1. The relativevelocity of two collinear particles, which is used in the definition of thecross section, is∗v122E12E2 ≡ 2λ1/22 (s),= 2√s− (m1 +m2)2√s− (m1 −m2)2. (17.10)∗ λ1/22 (s) has nothing to do with a fugacity.320 StrangenessThe invariant reaction rate, Eq. (17.7), thus isR12 =∫d3p1(2π)32E1d3p2(2π)32E2f1 f21 + I12σ12v12 × 2E1 × 2E2 (17.11)=g1g2(2π)6∫ ∞sthds2λ1/22 σ121 + I12(∫d3p12E1d3p22E2e−E1/T e−E2/T δ(s−(p1+p2)2)),where we have inserted a (dummy) integration over s. The lower limitsth of the integration over s is the threshold for the reaction, usuallysth = (∑imi)2, the sum of masses of the final state created.We are also interested in understanding at which values of√s the pro-duction processes occur. We will evaluate the p1 and p2 momentum inte-grals first, which will leave us with a final integral over√s in Eq. (17.11).We can present the rate of production as an integral over the differentialrate dRi/ds, where i refers to the reaction channel considered:R ≡∑i∫ ∞sthdsdRids≡∑i∫ ∞sthds σi(s)Pi(s). (17.12)σi(s) is the cross section of the channel. The factor Pi(s), which has thesame dimension as the invariant rate R, is interpreted as the number ofcollisions per unit volume and time, and corresponds to the expressionin the second line in Eq. (17.11), with the channel i corresponding to thecollision of particles {1, 2}.In order to evaluate the p1 and p2 momentum integrals in Eq. (17.11), itis convenient to introduce, for the Boltzmann distributions, the 4-vectorof temperature, Eq. (12.39), in the local restframe β = (1/T, 0) and towrite the (invariant) factor in large brackets in Eq. (17.11),[· · ·]=∫d4p1 d4p2 δ4(p− p1− p2)δ0(p21−m21)δ0(p22−m22)×∫d4p e−β·pδ(p2− s), (17.13)with δ0 being the δ-function restricted to positive roots of the argumentonly (compare with Eq. (12.45)). A dummy integration over p = pA + pBallows one to rearrange the terms in a way that separates the expres-sion into the two factor integrals. The first is known as the two particleinvariant phase-space integral ‘IMS2’ [86]:∫d4p1 d4p2 δ4(p− p1 − p2)δ0(p21 −m21)δ0(p22 −m22) =π2√λ2s. (17.14)17 Thermal production of flavor in a deconfined phase 321λ1/22 (s) is as defined in Eq. (17.10). The second integral can be obtainedby evaluating Eq. (10.43), it appeared previously in Eq. (12.22):∫d4pe−βpδ0(p2 − s) =2πβ√sK1(β√s). (17.15)The invariant reaction rate, Eq. (17.7), isR12 =g1g232π4T1 + I12∫ ∞sthds σ(s)λ2(s)√sK1(√s/T ). (17.16)So far, we have not addressed the quantum nature of colliding particles.The difficult case is that of a pair of light quarks reacting at finite baryondensity, for which one of the distributions cannot be expanded. Since, inthis case, the mass of the light quark is negligible, one of the integralscan be done analytically. The integral of interest, which is obtained afterperforming the angular integrals in Eq. (17.11), isK1(√s/T )→ 1T√s∫ ∞0dp1∫ ∞0dp2Θ(4p1p2 − s)fq(p1)fq̄(p2), (17.17)with (compare with Eqs. (10.34a) and (10.34b))fq(p) =1γ−1q λ−1q ep/T + 1, fq̄(p) =1γ−1q λqep/T + 1.Assuming that γq/λq < 1 (baryon-rich matter), we can expand the distri-butions for antiquarks and obtain the generalization of K1 in Eq. (17.16):K1(√s/T )→∞∑l=1(−)l+1γlqlλlq∫ ∞0dp1√sexp(−l s4Tp1)γ−1q λ−1q ep1/T + 1, (17.18)which has to be evaluated numerically.In the special case that all chemical factors are unity (or otherwise allowthe expansion), we expand Eq. (17.18) again to obtainK1(√s/T )→∞∑l,n=1(±)l+n√s l∫ ∞0dp1 exp(−l s4Tp1− np1T). (17.19)We have allowed for Fermi and Bose distributions, recalling the expansion1eE/T ∓ 1= ±∞∑n=1(±)ne−nE/T .322 StrangenessThe upper sign in Eq. (17.19) is for bosons and the lower for fermions.We now use∫ ∞0dx e−a/(4x)−bx =√abK1(√ab), (17.20)and obtain in the case γq = λq = 1 the generalization of K1 in Eq. (17.16):K1(√s/T )→∞∑l,n=1(±)l+nK1(√lns/T )√ln. (17.21)We see how the powers of√s cancel out, leaving only the slowly converg-ing pre-factor. It turns out that many terms in the sum of l and n arerequired in order to arrive at a precise result.17.2 Evolution toward chemical equilibrium in QGPThe conservation of current used in Eq. (17.2) applies to the laboratory‘Eulerian’ formulation. This can also be written with reference to the in-dividual particle dynamics in the so-called ‘Lagrangian’ description: con-sider ρs as the inverse of the small volume available to each particle. Sucha volume is defined in the local frame of reference (subscript ‘l’) for whichthe local flow vector vanishes, /v(/x, t)|local = 0. For the considered volumeδVl being occupied by a small number of particles δN (e.g., δN = 1), wehaveδNs ≡ ρs δVl. (17.22)The left-hand side of Eq. (17.2) can be now written as∂ρs∂t+∂/v ρs∂/x≡ 1δVldδNsdt=dρsdt+ ρs1δVldδVldt. (17.23)Since δN and δVl dt are Lorentz-invariant quantities, the actual choiceof the frame of reference in which the right-hand side of Eq. (17.23) isstudied is irrelevant and, in particular, it can be considered in the localrest frame. The last term in Eq. (17.23) describes the effect of volumedilution due to the dynamic expansion of matter. The other term on theright-hand side is then interpreted as the evolution of the local density inproper time of the volume element.We continue to use the first form of Eq. (17.23) and evaluate the lo-cal change in number of particles. We introduce ρ∞s (T ) as the (local)chemical-equilibrium abundance of strange quarks which arises at t → ∞,thus ρs = γs(t)ρ∞s . We use the Boltzmann equilibrium abundance, sec-tion 10.4,δNs = δV γsρ∞s = [T3 δV ]γs3π2z2K2(z). z =msT, (17.24)17 Thermal production of flavor in a deconfined phase 323In an entropy-conserving evolution, e.g., subject to (ideal) hydrodynamicflow, section 6.2, the first factor on the right-hand side in Eq. (17.24) (insquare brackets) is a constant in time, δV T 3 = δV0 T 30 = constant. Wenow substitute in Eq. (17.23) and obtain, using Eq. (10.54b),∂ρs∂t+∂/v ρs∂/x= Ṫ ρ∞s(dγsdT+γsTzK1(z)K2(z)), (17.25)where Ṫ = dT/dt. Only a part of the usual flow-dilution term is left, sincewe implemented the adiabatic volume expansion, and study the evolutionof the phase-space occupancy in lieu of the particle density.We include the collision term seen in Eq. (17.2) and two channels, thefusion of gluons and light-quark–antiquark fusion into a pair of strange(or equivalently charm) quarks:Ṫ ρ∞s(dγsdT+γsTzK1(z)K2(z))= γ2g(τ)Rgg→ss̄ + γq(τ)γq̄(τ)Rqq̄→ss̄−γs(τ)γs̄(τ)(Rss̄→gg +Rss̄→qq̄). (17.26)Similar equations can be formulated for the evolution of γg and γq. Knowl-edge of the dynamics of the local temperature, along with the requiredinvariant rate of production R(T ), allows one to evaluate the dynamicbehavior of occupancy fugacities γi(t).Since only weak interactions convert quark flavors, on the hadronic timescale we have γs,q(τ) = γs̄,q̄(τ). Moreover, detailed balance, arising fromthe time-reversal symmetry of the microscopic reactions, assures that theinvariant rates for forward/backward reactions are the same, specificallyR12→34 = R34→12, (17.27)and thusṪ ρ∞s(dγsdT+γsTzK1(z)K2(z))= γ2g(τ)Rgg→ss̄(1− γ2s (τ)γ2g(τ))+ γ2q(τ)Rqq̄→ss̄(1− γ2s (τ)γ2q(τ)). (17.28)When all γi → 1, the right-hand side vanishes; chemical equilibrium isestablished.In order to be able to evolve the population of (strange) quarks we needto understand the population of gluons, i.e., γg. Several workers haveconsidered the glue approach to equilibrium in perturbative processessuch as gluon splitting, e.g., gg→ ggg [71, 252, 253, 275]. They find that,in the thermal environment, there is not enough production of gluonsto reach chemical equilibrium. Accordingly, there are too few gluons to324 Strangenessderive the approach to chemical equilibrium of light quarks, and, evenmore so, of strange quarks. The initial conditions for such a full-kinetic-evolution study are relatively unreliable, but can not alter this conclusion.We pursue an entirely different point of view in this book, based on thebelief that strong multigluon processes gg → ng, n > 3 can equilibratethe abundance of gluons faster [278]. Therefore, we assume that gluonchemical equilibrium is reached rapidly, O(1 fm), on a scale not muchlonger than the time required to reach thermal equilibrium. Our choiceof an initial condition for consideration of strange flavor equilibration isγg = 1. In the study of evolution of strange (and charm) quarks we alsotake γq = 1. A large error, in this last assumption, is without significancefor what follows, since gluons dominate the thermal production of strangequarks.Given the chemical equilibrium of gluons and light quarks, we obtainthe dynamic equation describing the evolution of the local phase-spaceoccupancy of strangeness (and, in analogy, charm),2τsṪ(dγsdT+γsTzK1(z)K2(z))= 1− γ2s . (17.29)As discussed at the end of section 5.7, we introduce the relaxation timeτs of chemical (strangeness) equilibration as the ratio of the equilibriumdensity that is being approached and the rate at which this occurs,τs ≡12ρ∞s(Rgg→ss̄ +Rqq̄→ss̄ + · · ·) . (17.30)The factor 12 is introduced by convention, in order for the quantity τs(T )to describe an exponential approach to equilibrium, Eq. (5.41).One generally expects that γs → 1 monotonically as a function of time.However, Eq. (17.29) allows the range γs > 1, for it incorporates thephysics of a rapidly expanding high yield of strangeness created in theearly stage at high T . At a high background temperature, the evolutionγs(t) → 1 produces a high yield of particles, which corresponds, at thelower temperature established after expansion of the system, to γs > 1.One finds that thermal annihilation of flavor cannot keep up with therapid evolution of a fireball of QGP, and an overabundance will generallyresult. Annihilation is slow, since the density of strange and antistrangequarks is about four times smaller than the density of gluons (an effect ofcolor and mass) and the rate of annihilation for strange quarks is 16 timesslower. With a relaxation time for the production of strangeness of 1.5–10fm (depending on temperature), see Fig. 17.11 below, the relaxation timefor ss̄ annihilation is 20–150 fm, so practically all strangeness is preservedon the time scale of 5 fm of a QGP fireball. For charm this argumentis much stronger, given the greater effect of mass. Once it has been17 Thermal production of flavor in a deconfined phase 325Fig. 17.1. The statistical equilibrium densities of strange or antistrange quarkswith ms = 160MeV (solid line) and of charmed or anticharmed quarks withmc = 1500MeV (dashed line), as functions of temperature T .produced, heavy flavor has no time to annihilate and reequilibrate. Thisis a very important feature that makes the yield of strangeness a ‘deep’probe of the deconfined phase.Said differently, the high abundance of strangeness (or charm) formedin the high-temperature QGP stage over-populates the available phasespace at lower temperature, when the equilibration rate cannot keep upwith the cooling due to expansion. We will quantify this effect in moredetail in section 17.5. In the kinetic equation Eq. (17.29), this is seenmost clearly by considering the case T < ms. In this limit, 1/τs becomessmall, the dilution term (second term on the left-hand side in Eq. (17.29))dominates the evolution of γs. For the massive charm quarks T mc, soexpansion dilution can generate a very large phase-space overabundance,compared with the equilibrium yields expected in hadronization.To grasp the sensitivity of these remarks to the early QGP stage,we look at the equilibrium densities of strange (ms = 160MeV) andcharmed (mc = 1500MeV) quarks, shown in Fig. 17.1. We note that, forT 250MeV, the equilibrium abundance of strangeness exceeds one s̄ qu-ark for each fm3 of matter, and that charm reaches this for T ≥ 450MeV.326 Strangenessqsqggggggs(a) (b) (c)(d)s sssFig. 17.2. Leading order Feynman diagrams for production of ss̄ (and similarlycc̄) by fusion of gluons and pairs of quarks (q = u,d).Looking at charm, we note that, when the system expands and coolsrapidly from T = 450 to 150 MeV, the volume grows by less than a factorof 70, see the entropy density in Fig. 16.7, while the equilibrium densityof charm declines by factor 10 000 and the charm saturation factor γc in-creases by the factor 150, to preserve the yield of charm. This abundanceof charm is a product of the first interactions, not of thermal processes,except possibly at the LHC in a very-high-T scenario.17.3 Production cross sections for strangeness and charmThe production processes involving quark and gluon degrees of freedomin QGP areu + ū→ s + s̄, d + d̄→ s + s̄, (17.31a)g + g→ s + s̄. (17.31b)These three processes describing perturbative production of pairs ofstrange quarks are represented to lowest order in Fig. 17.2, and have tobe summed incoherently. These lowest-order diagrams were studied inthe early eighties, for the quark process [68] and for the gluon process[226], employing fixed values of αs = 0.6 and ms = 160–180MeV.The evaluation of the lowest-order Feynman diagrams shown in Fig. 17.2yields the cross sections [95]:σqq̄→f f̄(s)=8πα2s27s(1+2m2fs)w(s), w(s) =√1− 4m2fs, (17.32a)17 Thermal production of flavor in a deconfined phase 327Fig. 17.3. Production cross sections for strangeness in leading order: (a) forαs = 0.6 and ms = 200MeV; (b) for running αs(√s) and ms(√s), with αs =0.118. Solid lines, qq̄→ ss̄; dashed lines, gg→ ss̄.σgg→f f̄(s)=πα2s3s[(1 +4m2fs+m4fs2)ln(1 + w(s)1− w(s))−(74+31m2f4s)w(s)]. (17.32b)Inspecting Fig. 17.3(a), we see that the magnitudes (up to 0.4mb) of bothtypes of reactions considered, quark fusion and gluon fusion, are similar.At this stage, it is not immediately apparent that gluons dominate theproduction of flavor.The magnitude of the cross section of interest is normalized by αs. Toobtain Fig. 17.3(a), we took αs = 0.6. While the value seems reasonable, avalue of αs = 0.3 would lengthen the relaxation time of strangeness, τs ∝α−2s , by a factor of four, nearly beyond the expected life span of the QGPfireball. Thus, we must improve the determination of αs. There are twonatural ways to do this; the easier one is to adopt the functional αs(T ) seenin Fig. 14.3. However, in such an approach, two-body collisions occurringat very different√s but in a thermal bath at the same temperature T areevaluated with the same value of αs. Only for the thermal production of328 StrangenessFig. 17.4. Running masses for αs = 0.118: (a) the mass of the strange quarkms, for which the dot indicates the production thresholds for pairs of strangequarks for ms(MZ) = 90 MeV; (b) the mass of the charmed quark mc, forwhich the cross indicates production thresholds for pairs of charmed quarks formc(MZ) = 700 MeV.charm does this approach turn out to yield the same result as does themore complex, but more precise, consideration of an appropriate valueof αs for each collision, governed by the applicable αs(µ), Fig. 14.1, withµ √s.This second method, in which for each collision in the thermal bathan appropriate coupling strength is selected, is necessary for studyingthe production of strangeness in order to account for the growth of thecross section for soft scattering. The increase of cross section in softcollisions is, however, largely balanced by the concurrent suppression ofthe cross section due to the increase in mass of the strange quark mson the soft momentum scale. We adopt the running-mass and coupling-constant results presented in chapter 14, for αs(MZ) = 0.118. In Fig. 17.4,the running masses of the strange and charm quarks mi(µ), i = s, c, forms(MZ) = 90MeV and mc(MZ) = 700MeV, derived from Fig. 14.1, areshown. These values imply that ms(1GeV) 200MeV and mc(1GeV) 1.55 GeV.17 Thermal production of flavor in a deconfined phase 329µFig. 17.5. Cross sections for leading order production of charm for the runningαs and running mass of the charmed quark with mc(MZ) = 0.7GeV: the solidline is for fusion of pairs of light quarks, whereas the dashed line is for gg→ cc̄.The energy scale of greatest interest for studying the production ofstrangeness is certainly µ = 2πT 1–2 GeV, which is precisely the regionof rapid change of the value of ms. Below√s = 1GeV, the mass of thestrange quark increases rapidly and the threshold for producing pairsof strange quarks increases to above 2ms(1GeV). Half of the thresholdenergy is indicated by the black dot in Fig. 17.4(a). The pair-productionthreshold is, section 14.4,2mths (µ = 2mths ) = 611MeV, ms(MZ) = 90MeV, αs(MZ) = 0.118.For charm, the running-mass effect plays differently, since the naivethreshold for production of charmed quarks 2mc(2GeV) > 2GeV. Therunning of the mass has the effect of reducing the effective threshold. Formc(MZ) = 700MeV, the production threshold is found, rather than at3.1GeV, at2mthc (µ = 2mthc ) = 2.3GeV, mc(MZ) = 700MeV, αs(MZ) = 0.118.The cross, in Fig. 17.4(b), indicates the position of half of the thresholdenergy. Even this small reduction in threshold enhances the productionof charm at low energy and especially so in the thermal environment weare considering.In Fig. 17.3(b), we have presented the cross sections for productionof strangeness Eqs. (17.32a) and (17.32b), evaluated using the runningQCD parameters obtained in sections 14.3 and 14.4, identifying µ →√s.330 StrangenessOn comparing this with the ‘conventional’ result seen in Fig. 17.3(a), agreater threshold and more rapid decline of the cross sections are noted.Near√s = 1 GeV, in both cases, the fusion of gluons (dashed line)dominates, even though at lower energy the quark-pair fusion reaction(solid line) has a stronger peak. Similarly, for production of charm, we seein Fig. 17.5 the cross sections for fusion of gluons (dashed line) and pairsof quarks (solid line), for production cross sections which were computedfor the running αs(MZ) = 0.118 and running charmed-quark mass withmc(MZ) = 700 MeV. These cross sections are a factor 100 smaller thanthose for strangeness, at the level 1–2 µb. This is due to the fact thatan eight-fold greater√s is required, given that σ ∝ 1/s, and a reductionin the effective coupling strength. The smallness of the cross sectionfor production of charm is the reason why thermal production of charmbecomes relevant only at T → 1 GeV. Inspecting Fig. 17.5, we can alsoclearly understand the great sensitivity of the direct production of charmin non-thermal parton collisions to the value, and running, of the mass ofthe charmed quark: using a production threshold at 3 GeV, we cut 40%of the available strength of the cross section.The use of scale dependent QCD parameters, αs and mf , f = s, c, withµ ∝√s amounts to a re-summation of many QCD diagrams comprisingvertex, and self energy corrections. A remaining shortcoming of thermalproduction evaluation is that up to day, there has not been a study of thenext to leading order final state accompanying gluon emission in thermalprocesses, e.g., gg→ ss̄ + g. In direct parton induced reactions, this nextto leading order effect enhances the production rate by a factor K = 1.5–3. This causes a corresponding increase in the rate of production, anda reduction in the thermally computed chemical equilibration time ofstrangeness and charm.17.4 Thermal production of flavorThe thermal production processes occur over a wide range of√s. Thereare two factors determining this. Aside from the cross section, the collisionfrequency is the determining factor. We have introduced the thermal col-lision frequency per unit time and volume Pi(s) in channel i, Eq. (17.12).Employing the result Eq. (17.16) and the discussion of quantum correc-tions which followed, we obtain, setting g1 = g2 = 16, I12 = 1 and λ2 = s2(for massless gluons)Pg =4Ts3/2π4∞∑l,n=11√nlK1(√nl sT). (17.33)17 Thermal production of flavor in a deconfined phase 331Fig. 17.6. The collision distribution functions as functions of√s: (a) for quarksand (b) for gluons, computed for temperature T = 260MeV, λq = 1.5 (dottedlines, amplified by a factor of eight); T = 320MeV, λq = 1.6 (dashed lines,amplified by a factor of four); and T = 500MeV, λq = 1.05 (solid lines). In allcases γq, γg = 1.For quark processes, using Eq. (17.18) for γq/λq < 1, and setting g1 =g2 = 6, I12 = 0 and λ2 = s2 (for massless quarks), and taking the resulttwice to allow for the two quark flavors which can undergo incoherentreactions,Pq|µq=0 =9Ts3/24π4∞∑l=1(−)l+1γlqlλlq∫ ∞0dp1√sexp(−l s4Tp1)γ−1q λ−1q ep1/T + 1. (17.34)In Fig. 17.6, the distribution functions for the collision frequencyEqs. (17.33) and (17.34) are presented as functions of√s for gluons (b),and a qq̄ pair of light quarks (a). The temperatures correspond to a rangeof possible initial fireball temperatures at the SPS and RHIC: T = 260MeV (dotted, λq = 1.5), 320 MeV (dashed, λq = 1.6), and 500 MeV332 StrangenessFig. 17.7. The differential thermal production rate for strangeness dRs/ds, withT = 250, λq = 1.5 for gluons (dashed line), and qq̄→ ss̄ (solid line, includes twointeracting flavors), for the running αs(MZ) = 0.118 and running mass of thestrange quark ms(MZ) = 90 MeV ± 20% (thin lines).(solid λq = 1.05). Gluons and light quarks are assumed to be in chemicalequilibrium, γq, γg = 1.There is a shift in the maximum of the distribution of Pq,g to higher√swith increasing temperature. The collision frequency for gluons is aboutfive times greater, Pg 5Pq, than that for a pair of quarks, and this isthe origin of the glue dominance of thermal production of strangeness.Moreover, the peak in the gluon collision frequency Pg is more coincidentwith the peak in the cross section, as a function of√s, Fig. 17.3(b). Thisfurther amplifies the gluon dominance. This combined enhancement effectcan be seen in the thermal differential production rates,dRfds=∑i=q,gσif(s)Pi(s), f = s, c, (17.35)shown for thermal production of strangeness, in Fig. 17.7 for T = 250MeV, and for thermal production of charm, in Fig. 17.8 for T = 500MeV. Gluons (dashed lines) dominate quark-pair processes (solid lines),which are contributing at the level of 15%. The uncertainty in mass of thestrange quark has significant impact; thin lines bordering thick lines showthe effect of 20% variation in the value of the quark mass considered.Since formation of charm occurs in the domain T mc, near to thethreshold, where the cross section for fusion of a pair of quarks domi-nates gluon fusion, the gluon dominance of the production rate is not17 Thermal production of flavor in a deconfined phase 333Fig. 17.8. The differential thermal production rate for charm dRc/ds, with T= 500MeV, λq = 1 for gluons (dashed lines) and qq̄ → cc̄ (solid line, includesthree interacting flavors), for the running αs(MZ) = 0.118 and running mass ofthe charmed quark mc(MZ) = 700 MeV ± 7% (thin lines).as pronounced as it is for strangeness. Only for T ≥ 500MeV does theglue fusion pick up strength and clearly dominate the thermal productionof charm. Yet, even at T = 500 MeV, the rates for charm are a factor100 smaller than those for strangeness at T = 250 MeV, and thermalproduction of charm is expected to be irrelevant at the RHIC.The differential production rate can easily be integrated, and we showthe results in Fig. 17.9 for strangeness, and in Fig. 17.10 for charm. InFig. 17.9, we see that the early results (dotted line) [226] are found withinthe uncertainty in mass of the strange quark (a smaller mass leads to abigger value of R). A yet greater value of R should result after the K-factor has been introduced, describing the next-to-leading-order effects.The rate of production of strangeness per unit volume and time is at thelevel of unity at temperatures reached at the RHIC, and production ofstrangeness is very abundant.The thermal production of charm could be significant at the LHC rel-ative to the direct first-parton-collision production, if temperatures wellabove T = 500 MeV are reached. We see, in Fig. 17.10, that the produc-tion rate for charm changes by six orders of magnitude as the temperaturevaries between 200 and 600MeV. This sensitivity to the initial tempera-ture is due to the exponential suppression with m/T > 1. In turn, thisimplies that the thermal production of charm can become important atsufficiently high temperature.334 StrangenessFig. 17.9. Thermal production rates for strangeness Rs in QGP (thick solidline), calculated for λq = 1.5, αs(MZ) = 0.118, and ms(MZ) = 90 MeV, as afunction of temperature. Chain lines show the effect of variation of the mass ofthe strange quark by 20%. The dotted line shows comparison results for fixedαs = 0.6 and ms = 200MeV.Fig. 17.10. Thermal production rates for charm Rc in QGP (solid line), calcu-lated for λq = 1.05, αs(MZ) = 0.118, and mc(MZ) = 0.7GeV, as a function oftemperature. Chain lines show the effect of variation of mass of the charmedquark by 7%. Dotted lines are comparison results for fixed αs = 0.35 andmc = 0.9GeV (upper) and αs = 0.4 and mc = 1.5GeV (lower).17 Thermal production of flavor in a deconfined phase 335Insertion of the rates Ri into Eq. (17.30) allows us to obtain the timeconstants for chemical relaxation τs and τc. It should be stressed that,in an actual kinetic evaluation of the production of flavor, the relaxationtime τf enters only when we relate the actual yield of flavor to the ex-pected yield ρ∞f . Namely, the back reaction, f f̄ → gg, qq̄, is driven by theactual density of strangeness, whereas the forward rate, ignoring Pauliblocking, is not affected by the equilibrium yield at all. Which mf is usedin Eq. (17.30) to define ρ∞f is physically irrelevant, as long as the samevalues of mf and ρ∞f are used both in the definition of τf , Eq. (17.30), andin γf(t) = ρf/ρ∞f . In Fig. 17.11, we see τs evaluated with ms(1GeV) = 200MeV. The range of the assumed 20% uncertainty in ms(MZ) is indicatedby the hatched areas. The initial predictions obtained 20 years ago [226]at fixed values αs = 0.6 and ms = 200 MeV (the dotted line in Fig. 17.11)are well within the band of values related to the uncertainty in mass ofthe strange quark. The approximate formula obtained in [226],τf =1.6α2sγ2gTmf/T emf/T[1 + (99/56)T/mf + · · ·], (17.36)allows a quick estimate of the expected relaxation time in all the en-vironments discussed in this subsection. We have added the pre-factorγ−2g relevant in case the dominant source of heavy flavor, gluons, is notin chemical equilibrium. We see that the equilibration time lengthensaccordingly.Thermal nonperturbative effects on the relaxation of strangeness werestudied by introducing thermal, temperature-dependent, particle masses[70]. After the new production rates, including the now possible gluon de-cay, were added up, the total rate of production of strangeness was foundto be little changed compared with the free-space rate. This finding waschallenged [34], but a further reevaluation [66] confirmed that the ratesobtained with perturbative glue-fusion processes are describing preciselythe rates of production of strangeness in QGP, for the relevant tempera-ture range T > 200 MeV. We can thus assume today that the ‘prototype’strangeness-production processes seen in Fig. 17.2, re-summed using therenormalization-group method, are dominating the rates of production ofstrangeness in QGP.The poor knowledge about the mass of the strange quark makes itpossible that the actual relaxation time for strangeness is even smaller.In quenched-lattice calculations, see section 15.5, a much smaller valuems(MZ) 50 MeV is found. The thin-dotted line in Fig. 17.11 gives thecorresponding result for ms(MZ) = 50 MeV. We see that the relaxationtime is already small enough to allow chemical equilibration at T < 200MeV. Moreover, next-to-leading-order effects (the K-factor) should fur-ther reduce the chemical relaxation constant.336 StrangenessFig. 17.11. The QGP chemical relaxation time for strangeness τs, for αs(MZ) =0.118 with ms(MZ) = 90 MeV and ρ∞s (ms 200MeV) (thick line). Hatchedareas show the effect of variation of the mass of the strange quark by 20%. Thefat-dotted line shows comparison results for fixed αs = 0.6 and ms = 200MeV.The thin-dotted line shows the result for ms(MZ) = 50 MeV.While precise evaluation of the production of strangeness at tempera-tures as low as Tc 160 MeV is not reliable within the scheme we havepresented, it is highly probable that the combined effect of lowms and theK-factor would ensure that near-chemical equilibrium for strangeness candevelop as soon as the QGP phase can be formed. As a result, the stran-geness energy excitation function, seen in Fig. 1.5 on page 17, can thenbe interpreted as due to the onset of deconfinement already in collisionsbelow SPS energies. We see that, despite 20 years’ work on strangeness,we still have many new, interesting insights to gain.In Fig. 17.12, the chemical relaxation time for charm is shown in the ex-tended interval through T = 1000 MeV. Since charm is considerably moremassive than strangeness, there is less uncertainty in the extrapolation ofthe running QCD coupling constant. There is also less relative uncertaintyin the value of the mass of the charmed quark, shown by the hatched area.We also see (dotted lines) the results for fixed mc and αs with parametersselected to border high- and low-T limits of the results presented. In thehigh-T limit, the choice (upper dotted line) mc = 1.5GeV, αs = 0.4 isappropriate, whereas to follow the result at small T (lower dotted line),we take a much smaller mass mc = 0.9GeV, with αs = 0.35.The important result, see Fig. 17.12, is that, above T = 700 MeV,the relaxation time drops below to 10 fm; the curves flatten. At gluon17 Thermal production of flavor in a deconfined phase 337Fig. 17.12. Solid lines show the thermal relaxation constant for charm in QGP,calculated for running αs(MZ) = 0.118, mc(MZ) = 0.7GeV and ρ∞c (mc 1.5GeV). Lower dotted line, for fixed mc = 0.9GeV and αs = 0.35; upperdotted line, for fixed mc = 1.5GeV and αs = 0.4. The hatched area shows theeffect of variation mc(MZ) = 0.7 GeV ± 7%.collision energies of several GeV, it is quite natural to expect that thenext-to-leading-order effects enhance the cross section for production ofcharm by a factor of two, and this reevaluated relaxation time wouldcorrespond to a true value of a few fermis only. At this juncture in time,it is quite impossible to be sure how important the thermal componentis in the production of charm at the LHC. On the other hand, even thefirst parton collisions are expected to produce 200 ± 50% cc̄ pairs, andthus, in one way or another, charm certainly will be the novel-physicsfrontier at LHC energies, replacing strangeness as the flavor signature ofnew physics.17.5 Equilibration of strangeness at the RHIC and SPSGiven the relaxation constant τs, we evaluate the thermal yield of stran-geness in the QGP which arises on integrating the kinetic equationEq. (17.29). Since this requires as input initial conditions the temporalevolution of the fireball, results are somewhat model-dependent. Indeed,there is considerable difference of opinion among groups regarding thewell-studied RHIC system [71, 221, 252, 253, 275], since the experimentaldata which would narrow down the models is only now being obtained.The most important issue on which the various groups differ is the directly338 Strangeness(or indirectly) assumed gluon content. In this book, we assume rapidchemical equilibration of gluons, which is not reached in studies relyingon kinetic evolution by implementing the lowest-order gg → ggg gluonfragmentation.For the RHIC conditions, we present a qualitative model with a cylin-drical longitudinal flow, and transverse expansion [221]. We assume thatthe transverse flow of matter occurs at the velocity of sound for relativis-tic matter v⊥ c/√3 = 0.58c. For a purely longitudinal expansion, thelocal entropy density scales according to S ∝ T 3 ∝ 1/τ ; see Eqs. (6.35)and (7.22). The transverse flow of matter accelerates the drop in entropydensity. To model this behavior without too great a numerical effort, con-sidering the other uncertainties, the following temporal-evolution functionof the temperature was proposed:T (τ) = T0(1(1 + 2τc/d)(1 + τv⊥/R⊥)2)1/3. (17.37)Considering various values of T0, the temperature at which the gluonequilibrium is reached, the longitudinal dimension is scaled according tod(T0) = (0.5GeV/T0)3 × 1.5 fm. (17.38)This adjustment of the initial volume V0 assures that the different evolu-tion cases refer to a fireball with a similar entropy content. The followingresults are thus a study of one and the same collision system, and thecurves reflect the uncertainty associated with unknown initial conditionsof a fireball of QGP with identical, (but large by current standards) en-tropy content.The numerical integration of Eq. (17.29) is started at τ0 = 1 fm/c, thetime at which thermal initial conditions are reached. A range of initialtemperatures 300 MeV ≤ T0 ≤ 600 MeV, varying in steps of 50 MeV,is considered. Since the initial p–p collisions also produce strangeness,to estimate the initial abundance a common initial value γs(T0) = 0.15is used. For T0 = 0.5GeV, the thickness of the initial collision regionis d(T0 = 0.5)/2 = 0.75 fm. The initial transverse dimension in nearlycentral Au–Au collisions is taken to be R⊥ = 4.5 fm. The initial volumeof QGP is 190 fm3, which, at the temperature of T0 = 0.5GeV, implies,according to results seen in Fig. 16.7, a total entropy content of S =38 000. We divide this by the specific entropy content per hadron in thefinal state, S/N = 4; see section 10.6. We see that the primary final-state hadron multiplicity has implicitly been assumed to be 9500. Thisis somewhat above results seen even during the RHIC run at√sN = 200GeV, for which we estimate, for the 3% most central events, a total hadronmultiplicity, after resonance cascading, of 8000.17 Thermal production of flavor in a deconfined phase 339Fig. 17.13. The evolution of QGP phase-space occupancy for strangeness γs:(a) as a function of time and (b) as a function of temperature for ms(1GeV) =200MeV; see the text for details.The evolution with time in the plasma phase is followed up to thebreakup of the QGP at a temperature TRHICf 150 ± 5 MeV. The nu-merical solution of Eq. (17.29) for γs is shown as a function both of timet, in Fig. 17.13(a), and of temperature T , in Fig. 17.13(b). This evolu-tion is physically meaningful until it reaches the QGP breakup condition.Since the results for higher temperatures are also displayed, the readerwho prefers hadronization at T = 170 ± 5 MeV can easily draw his ownconclusions.We see in Fig. 17.13 the following phenomena.• A steep rise at early times, showing actual production of strangeness,which is followed by a dilution-driven increase of γs near the breakuptemperature.• Widely different initial conditions (but with similar initial entropy con-tents) lead to rather similar chemical conditions at chemical freeze-outof strangeness.• Despite the use of a high mass of ms = 200 MeV, we find that strange-ness nearly equilibrates chemically, and that the dilution effect allows incertain cases a small over-population of the strange-quark phase spaceeven in the strangeness-dense QGP.• For a wide range of initial conditions and final freeze-out temperaturea narrow band of final result is seen, 1.10 > γs(Tf) > 0.9.Since strangeness is more easily made in the ‘hot’ era in glue–glue inter-actions, we can estimate that, if the abundance of glue had been at thetime 70% chemically equilibrated, then γs 0.5. This high sensitivity to340 Strangenessthe glue density is at the origin of the claim that measuring the yield ofstrangeness probes the presence and abundance of glue, which is a specificproperty of QGP.In the model calculations presented, the fireball begins to expand inthe transverse direction instantly at the full velocity. For this reason,the initial drop in temperature is very rapid. This defect also makes thetransverse size at the end of the expansion too large, R⊥ R0+tf/√3 9fm, compared with the results of HBT analysis, Fig. 9.11. This can easilybe fixed by introducing a more refined model of the transverse velocity,which needs time to build up. The yield of strangeness may slightlyincrease in such a refinement, since the fireball will spend more time nearto the high initial temperature.The RHIC results presented are typical for all collision systems. In thetop SPS energy range, the initial temperature reached is certainly less (by10%–20%) than that in the RHIC 130-GeV run, and the baryon number inthe fireball is considerably greater; however, the latter difference matterslittle for production of strangeness, which is driven by gluons. A modelsimilar to the above yields γQGPs 0.6–0.7, the upper index remindsus that in this section the strangeness occupancy factor γs refers to theproperty of the deconfined phase. The experimental observable directlyrelated to γQGPs is the total yield of strangeness per participating baryon.We will return to discuss the significance of these results in section 19.4.18 The strangeness background18.1 The suppression of strange hadronsSince the matter around us does not contain valence strange quarks, allstrange hadrons produced must contain newly made strange and antis-trange quarks. If strangeness is to be used as a diagnostic tool for inves-tigating QGP, we need to understand this background rate of productionof strange hadrons. In that context, we are interested in measuring howoften, compared with pairs of light quarks, strange quarks are made. Onedefines for this purpose the strangeness-suppression factor†Ws =2〈ss̄〉〈uū〉+ 〈dd̄〉. (18.1)In Ws, all newly made ss̄, uū, and dd̄ quark pairs are counted.If strangeness were to be as easily produced as light u and d quarks,we would find Ws → 1. To obtain the experimental value for Ws, a care-ful study of produced hadron yields is required [277]. Results shown in† We chose Ws in lieu of the usual symbol λs, which clashes with the strangenessfugacity.18 The strangeness background 34100.10.20.30.40.50.6102103√ [GeV]sWsp p −−p p– e+ e−−A A′−Fig. 18.1. Ws as a function of the center-of-mass energy√s [GeV] (for heavyion collisions the nucleon–nucleon center of mass energy√sNN) [61].Fig. 18.1 were obtained using a semi-theoretical method [61], in whichnumerous particle yields are described within the framework of a statisti-cal model, and the computed hadron yields are analyzed in terms of theWróblewski procedure [277].We see in Fig. 18.1 that, in elementary p–p, p–p̄, and e+–e− collisions,the value Ws 0.22 is obtained. In order to estimate the possible influ-ence of the annihilation process on p–p̄ reactions, two values of Ws areshown per energy, calculated with initial valence quarks and antiquarks(lower points) and without (upper points). For e+–e− interactions, theleading s quarks in e+e− → ss̄ have been subtracted [61]. Strangeness isthus relatively strongly suppressed. On the other hand, we also see that,in nuclear A–A′ collisions, Ws more than doubles compared with that inp–p interactions considered at the same energy.To explain the two-fold increase in yield of strangeness a kinetic modelof particle production requires a shift toward production of strangenessin all particle-formation processes. In other words, for modeling the en-hanced yields of strangeness within a variety of approaches, see section 6.1,in each model a new reaction mechanism that favors production of stran-342 Strangenessgeness must be introduced. Even at this relatively elementary level ofcounting abundances of hadrons, new physics is evident. In a model withthe deconfined phase this new reaction mechanism is due to the presenceof mobile gluons, which, as we have seen, are most effective at makingpairs of strange quarks. Moreover, because the conditions created in theQGP become more extreme with increasing collision energy, e.g., the ini-tial temperature exceeding substantially the mass of the strange quark,we expect an increase in Ws.Several among the cascade models we listed in section 6.1 have beentuned to produce not only enough strangeness, but also the observedantinucleons and Λ. However, once this has been done, these hadronicmodels predict wrong abundances of the rarely produced particles such asΞ and Ω. We are not aware of any kinetic hadron model with or without‘new physics’ that is capable of reproducing the pattern of productionof rare hadrons, along with the enhancement in production of strange-ness and hadron multiplicity. Moreover, if rapidity spectra are modeled,usually the transverse momentum spectra are incorrect, or vice-versa. Itseems impossible, in a collision model based on confined hadron inter-actions, to find sufficiently many hadron–hadron collisions to occupy byhadrons the large phase space (high p⊥, high y) filled by products ofnuclear collision. If indeed a non-QGP reaction picture to explain heavy-ion-collision data exists, the current situation suggests that some essen-tial reaction mechanism has been overlooked for 20 years. In short, a lotmore effort needs to be expended on hadron models in order to reach sat-isfactory agreement with the experimental results, even regarding rathersimple observables such as hadron multiplicities, transverse-energy pro-duction, and yield of strangeness.Another way within the kinetic approach, and within the realm of quiteconventional physics, to acquire an excess of strangeness in heavy-ioncollisions compared with elementary reactions is to cook dense hadronicmatter for a long time. In hadronic gas, strangeness can be producedin unusual (but not novel) reactions such as π + π → K + K. Evenin the presence of an abundant yield of pions, the high-mass threshold,compared with the temperatures we are studying here, suggests that theirreaction is ‘slow’. Thus, an important piece in the puzzle is to know howlong it takes for enhancement of strangeness to be created in a kinetic HGfireball. Otherwise, the fact that the final state in all reaction scenarioslooks more or less the same will make the argument that the enhancementin production of strangeness is a signature of new physics difficult.We therefore will consider, in the following section, the dynamics ofproduction of strangeness in thermal processes in HG [164, 165], and weestablish the time scales involved. Since the strange quarks are producedas constituents (‘valence quarks’) of the usual hadronic states, the direct18 The strangeness background 343mechanisms of production will not populate all hadronic states as thephase-space distribution would demand, and therefore, aside from theproduction, we will also encounter the (relatively rapid) process of stran-geness exchange (redistribution) in hadronic quark-exchange reactions.In general, the observed abundances of strange hadrons, and especially ofthe rarely produced ones, will be the result of multi-step processes.18.2 Thermal hadronic strangeness productionIn a thermally equilibrated central HG fireball the standardN + N→ N+Y+K (18.2)nucleon-based reactions for production of strangeness are, surprisingly,not very important. The hyperon Y = qqs may be either the iso-scalarΛ = uds or the iso-triplet Σ = (uus,uds,dds), and the kaon K may beeither K+ = s̄u or K0 = s̄d, which are found experimentally in one of the(almost) CP eigenstates KS and KL.There are three reasons for the relative unimportance of the reactionEq. (18.2):1. The energy threshold, viz.,√sth = mN +mY +mK − 2mN ∼ 670MeV, (18.3)is considerably higher than, e.g., the energy threshold for productionof strangeness in reactions between a pion and a baryon:π+N→ K+Y, √sth ∼ 540MeV . (18.4)2. Pions are the most abundant fireball particles, with the pion-to-baryonratio of the central-rapidity region considerably exceeding unity in allcentral collisions at above 10GeV per nucleon. It is for this reason thatthe reactionπ+ π̄ → K+ K̄,√sππth = 710MeV, (18.5)between two pions, which has also a rather high threshold, is found tobe important at temperature T > 100 MeV, as we shall discuss.3. The final-state phase space of the two-particle reaction system aboveis more favorable than that of the three-body final state required inEq. (18.2).The common reaction feature of the hadronic production of strangenessis the qq̄→ ss̄ reaction, illustrated for the case of Eq. (18.4) in Fig. 18.2.Note that three of the five (light) quarks are spectators, a qq̄ pair isannihilated, and an ss̄ pair is formed. The experimental value of this344 StrangenessπYKP Fig. 18.2. The production of strangeness in reactions of the type π + N →K + Y in the HG phase. Solid lines indicate the flow of light quarks and thedisappearance of one q̄q pair, the dashed line is for the added s̄s pair.so-called Okubo–Zweig–Izuka-rule forbidden cross section is about 0.1–0.5 mb in the energy region of interest here (just above the threshold),thus of a magnitude comparable to the cross section for the QGP processes– but the threshold is considerably lower in the QGP processes, and mostof the thermal collisions occur in this lower-energy region, as we haveshown in Fig. 17.6.Apart from production of strangeness, we also have ‘strangeness-ex-change’ reactions, as depicted in Fig. 18.3: we see that, in the process, thealready existent strange quark can be moved from one particle ‘carrier’to another, but the number of strange quarks remains unchanged. Weshow, in Fig. 18.3, the most relevant class of exchange reactions:K̄ + N→ π+Y. (18.6)In an exchange reaction, new hadrons that are difficult to make in directreactions can be produced. For example, the reactionΛ + K̄→ π+ Ξ (18.7)produces the doubly strange particles Ξ(ssq). More generally, exchangereactions distribute strangeness into all accessible particle ‘carriers’, andhelp establish the relative chemical equilibrium.For every strangeness-reaction, there is the same type back-reaction,and one of the two has to be exothermic. Thus 50% of strangeness-exchange reactions are exothermic. Exothermic cross sections do notvanish at small collision energies, and the thermal average is actuallynearly a constant O(1 mb), which is related to the geometric size andstructure of hadrons. Consequently, thermally averaged reaction rates forexchange of strangeness are, especially at low temperatures, much greaterthan rates of production of strangeness. This assures that strange quarks18 The strangeness background 345s P K πΛ0-s Fig. 18.3. An example of a strangeness-exchange reaction in the HG phase:K− + p→ Λ + π0. Solid lines, flow of u and d quarks; dashed line, exchange ofan s quark between two hadrons.produced in HG are rapidly distributed among many hadronic states, evenif the absolute chemical equilibrium is not reached.On the other hand, the cross section for production of strangeness inHG is relatively (to QGP) slow, being suppressed by smaller particle den-sities and the higher threshold of the production process. Some authors,pursuing explanation of the enhancement in production of strangenesswithin a HG scenario, try to overcome this in terms of partial restorationof chiral symmetry, which allows hadron masses to melt in part, loweringthresholds for production of strangeness; see [241] for the current statusof the search for this effect. We will not pursue further in this book theseinteresting developments.In the study of the evolution with time of the production of strangehadrons in the HG fireball, similarly to the case of QGP, only the totalnumbers of particles will be considered. We are assuming that the ther-malization (kinetic equilibration) is a rapid process, in comparison withthe relaxation time constant for strange hadrons, and also in comparisonwith the lifetime of the fireball. To quantitatively develop the kinetic evo-lution abundances of strange hadrons in the HG, we need to use a largenumber of hadronic cross sections.Only limited experimental information on cross sections is available,since it is often impossible to study in the laboratory some of the pro-cesses which involve relatively short-lived particles (which of course arequite infinitely long-lived on the scale of hadronic collisions). There is noreliable theoretical framework allowing one to evaluate cross sections notaccessible to direct measurement, since our understanding of the hadronicstructure is incomplete. Consequently, we consider a method facilitatinguse of the accessible experimental information, in order to be able toestimate required reaction cross sections.346 StrangenessThe cross section times the flux factor (velocity) for two-body reactions1+2→ n, in which n particles are produced in the exit channel, takes theform Eq. (17.9). The generalization of the thermally averaged productioncross section Eq. (17.16) to n final-state particles, can be cast into theformRn = ρ1ρ2〈σ12v12〉=g1g2(2π)5T|M(1 + 2→ n)|2(2π)3n−3×∫ ∞sthds√s IMS(s; 2)IMS(s;n)K1(√s/T ), (18.8)where the n-particle invariant phase space is a generalization of the twoparticle invariant phase space Eq. (17.14):IMS(s;n) =∫ n∏i=1d4pi δ0(p2i −m2i )δ4(p1 + p2 −n∑ipi). (18.9)IMS is implicitly a function of the masses of the particles.Many studies have shown that the hadronic-reaction matrix element|M(1 + 2 → n)|2 for hadronic processes is insensitive to the relativelyslow changes of√s in the region of interest to us, and it can be assumedto be constant. For this reason, it is outside of the integral over s inEq. (18.8). This is the crucial detail which permits us to relate manythermal cross sections to each other: Eq. (18.8) allows us to infer, froman example known experimentally, the thermal average of a family ofcross sections. All we need do is adjust the applicable threshold andparticle masses. Given the high range of temperatures we are considering,the isospin-breaking effects are relatively unimportant, and one does notdistinguish between the u and d quarks. Consequently, one can studyisospin-averaged cross sections. Appendix B of [164] gives a useful listingof relevant reactions.We show now how the method works: the isospin-averaged, thermallyaveraged associate rate of production of strangeness, 〈σ(πN→ K)vπN〉,can be determined from experimentally known cross sections. It allowsus to infer the strength of other associate production processes,〈σ(πY → KΞ)vπY〉= 〈σ(πN→ KY)vπN〉P1, (18.10a)P1=〈σ(πY → KΞ)vπY〉〈σ(πN→ KY)vπN〉, (18.10b)where P1 in Eq. (18.10b), with the assumption of constant |M|2, dependson T and µb only through its dependence on the particle phase space.The thermal averages of cross sections falling under the categories‘strangeness exchange’ and ‘baryon annihilation’ can also be dealt with18 The strangeness background 347 3−10 2−10 1−10100 150 200T (MeV)Fig. 18.4. The thermally averaged ππ → KK̄ reaction cross section: the solidline is for a constant matrix element (see the text); the dashed line is for aconstant value of the reaction cross section, σ = 3 mb [164].in this way. The various strangeness-exchange cross sections are relatedto 〈σ(K̄N → Yπ)vK̄N〉 and diverse baryon-annihilation cross sections to〈σ(pp̄→ 5π)vpp̄〉, which are known experimentally. Similarly, the matrixelement |M|2 for reactions in which particles and antiparticles are inter-changed is the same, and the average thermal cross sections of the reversereactions are given by those of the forward reactions times appropriatephase-space factors.This method leaves us with one important channel, Eq. (18.5), for pro-duction of strangeness in the thermally equilibrated hadronic gas, whichcan neither be measured nor be related to known reactions. Its strengthhas been derived from the reactions pp → ppππ, ppKK̄ with the helpof dispersion relations. Protopopescu et al. [210] have found that, abovethe threshold, pion-based production of strangeness has an approximatelyconstant cross section of 3± 1 mb. The thermally averaged cross section348 StrangenessT v〈ρ 〉 [mb]10–410–310–110–250 100 150 2006 K Y →7 K →πππππΞπΞ1 N K Y→3 K→π Ωπ Ξ2 Y K→ ΞΩ4 K K→5 K N Y→6724135 [MeV]Fig. 18.5. Thermally averaged cross sections for formation and exchange ofstrangeness in HG 〈σv〉/c, based on the assumption of universal invariant matrixelements [164].using as input a constant cross section of 3 mb is shown in Fig. 18.4 bythe dashed line, as a function of temperature. In comparison, the solidline describes the result of an evaluation in which a constant matrix ele-ment |M|2 was used with the strength equal to the associate reaction forproduction of strangeness π+N→ K+Y.In Fig. 18.5, we show the thermally averaged cross sections for forma-tion and exchange of strangeness obtained by Koch et al. [164] within theframework outlined above. Note that the thermally averaged strangeness-exchange reactions which are capable of building up multiply strange had-rons are O(1 mb) at all temperatures. This is consistent with the intuitionthat this value should be at the level of a third of the geometric cross sec-tion, multiplied by the probability of exchanging one of the quarks, whichis at the level of 10%, certainly not 100%, as is sometimes assumed bysome authors. Even though (by coincidence) reactions 5 and 6 agree instrength, reaction 7, which is important for understanding the formation18 The strangeness background 349of Ω, is considerably smaller. This strongly breaks the SU(3) flavor sym-metry, and slows down the formation of the all-strange Ω and Ω in HG,compared with the formation of other strange baryons and antibaryons.We further see in Fig. 18.5 that strangeness-formation reactions in HGare dominated at high temperature by the ππ → KK̄ reaction. Up totemperatures near T = 150MeV, strangeness-exchange reactions 5 and6 are more than an order of magnitude faster than is the productionof strangeness. Thus, relative chemical equilibrium can be established‘instantaneously’ in singly and doubly strange hadrons, but not for Ω andΩ, during the growth in abundance of strangeness in the fireball. For thesame reason, it makes good sense to expect relative chemical equilibriumamong strange hadrons to occur in an equilibrium model of hadronizationof QGP, except for Ω and Ω.18.3 The evolution of strangeness in hadronic phaseTo address the issue of equilibration of strangeness in HG in a quantitativemanner, we study the temporal evolution in the framework of a masterequation. We consider all two-body reactions, which are predominant,dρi(t)dt=∑AB〈σvAB→i〉TρAρB −∑C〈σviC→X〉Tρi(t)ρC, (18.11)where the collision of particles A + B leads to production of the strangeparticle i, and i can be annihilated in collisions with (strange) parti-cles C. Assuming the thermal momentum distribution of all particlesEq. (17.16) permits one to compute the evolution of particle populationsusing Eq. (18.11), once the thermal cross sections of the reaction areknown; see section 18.2. Appendix A of [164] gives the explicit formof the master equation Eq. (18.11).In HG antibaryons are difficult to produce, considering that the directpair-production thermal cross section for π + π → N + N̄ is suppressedby a high threshold. Intermediate steps are necessary in order to buildup heavy mesons, e.g., ρ and ω resonances, which are more capable ofproducing a baryon–antibaryon pair, and which also benefit from a largerelementary cross section. A recent kinetic-theory study of reactions in-volving fusion of three mesons into antibaryons suggests that this channelmay be the dominant source of antibaryons [89].In order to establish an upper limit for the abundance of multistrangebaryons in HG, the results shown below were obtained assuming thatthe non-strange antibaryons are at the chemical HG equilibrium yield atthe initial time, at which the evolution of the master equation is con-sidered. In this approach, the build-up of strange antibaryons is a func-tion of the developing abundance of strangeness, and the effectiveness350 Strangenessof strangeness-exchange reactions, which as we have seen is particularlyweak for the formation of the most enhanced Ω. Even allowing full oc-cupancy of non-strange-antibaryon phase space, the relaxation time foryields of strange antibaryons in HG turns out to be much too long tobe relevant in a dynamic heavy-ion-collision environment. In fact, thediscovery of abundant yields of strange antimatter that sometimes exceedchemical equilibrium in heavy-ion collisions is proof that collective for-mation mechanisms akin to those operating in quark-soup hadronizationhave been operational.Somewhat surprised by the high yield of (strange) antibaryons observedin heavy-ion collisions, some authors have proposed to use detailed bal-ance considerations to infer an effective rate of five-hadron collisionalproduction of a baryon–antibaryon pair, using as reference in detailed-balance analysis the reverse reaction N + N̄ → 5π. In our opinion,use of detailed-balance arguments to infer a five-body collisional reac-tion rate is wrong physics. Certainly five-particle collisions occur morerarely than do four-particle collisions, which occur more rarely than dothree-particle collisions, etc. One has first to establish the dominant for-ward and backward channels (if such exist, for equilibrium could be theresult of multistep processes) when one is using the detailed-balance ar-gument to estimate a rate. For the case of baryon–antibaryon formationand annihilation in HG, depending on temperature and density, two- andthree-body reactions among mesons should dominate higher many-bodyprocesses.We shall now review a few properties of the solutions of the masterequations Eq. (18.11). To maximize the yield of strangeness that canbe obtained, and to maximize the production of strange antibaryons, weassume that, after τ0 1 fm, the densities both of pions and of antinu-cleons should have reached approximately their chemical-equilibrium val-ues, and, at this point in time, the thermal processes for production ofstrangeness are turned on in HG. First, we turn our attention to the evo-lution of the total yield of strangeness in the HG phase assuming thatthe initial densities of strange particles are zero. Then, we consider theevolution of the yields of individual strange hadrons, including strangeantibaryons.We study the total strangeness in HG, to determine the time requiredfor the total abundance of strangeness to reach the chemical equilibrium.In Fig. 18.6, the evolution of the total abundance of strangeness is shown,and we see that the production of strangeness in HG is roughly 100 timesslower than the time for chemical relaxation in QGP, see Fig. 17.11, forthe example of temperature T = 160MeV. The considered range of thebaryo-chemical potential µb ∈ [0, 450]MeV covers all regions of interestand, as we see, has a negligible impact on this result.18 The strangeness background 351b( = 450)µb( = 0)µ( ) [fm ] t 3−sρ1010101010 10 10 1− 2− 3− 4−1 [fm]tsρsρequequFig. 18.6. The thermal production of strangeness as a function of time in aconfined hadron gas at T = 160MeV. Results for two values of baryo-chemicalpotential (µb = 0 and 450MeV) are shown [164].The yield of strangeness observed in experiments at the SPS and RHICcan not be produced by thermal HG processes. This becomes even cleareron considering individual particles with higher strangeness content suchas (multi)strange antibaryons, shown in Fig. 18.7. For reasons we dis-cussed above, strange antibaryons remain more distant from the equi-librium distribution than do strange mesons and strange baryons. Infact, to arrive within the time scale shown at a measurable yield, theresult presented benefits from the rather optimistic assumptions of equi-librium abundances for the non-strange antibaryons. As expected, wesee in Fig. 18.7 that kaons, K = s̄q, antikaons, K̄ = sq̄, and hyperons,Y = qqs, are the first to reach equilibrium abundance – the other s̄ car-riers are delayed by another factor of 3–5 in getting to their HG limits,with Ω = s̄s̄s̄ trailing far behind. We expect that the HG-based produc-tion of multistrange antibaryons generates negligible background. This isone of the important reasons allowing multistrange antibaryons to signalthe formation and hadronization of a deconfined quark–gluon phase inheavy-ion collisions.352 Strangeness10 4−10 6−10 8−10 2−1021031 10ρKρΞ ρΩρYρKeqρeqYρeqΞρeqΩ(a)1021031 1010101010 1− 3− 5− 7− 3 −ρ( ) [fm ] t i 3 −ρ( ) [fm ] t iYρYρΞ ρΩρeqΩρeqΞρKρKeq(b)Fig. 18.7. The approach to chemical equilibrium by strange hadrons in hothadronic matter at temperature T = 160, µb = 450MeV – s̄ hadrons are shownin (a), whereas s hadrons are shown in (b) [164].19 Hadron-freeze-out analysis19.1 Chemical nonequilibrium in hadronizationIn the final state, we invariably see many hadronic particles, and naturallywe observe their spectra and yields only in restricted domains of phasespace. An extra reaction step of ‘hadronization’ is required in order toconnect the properties of the fireball of deconfined quark–gluon matter,and the experimental apparatus. In this process, the quark and gluoncontent of the fireball is transferred into ultimately free-flowing hadronicparticles. In hadronization, gluons fragment into quarks, and quarks co-alesce into hadrons.Hadronization of course occurs in all reactions in which final-state had-rons are observed: for example, in high-energy e+e− → qq̄ reactions, wesee jets of final-state hadrons carrying the energy and momentum of thetwo quarks produced. It is not yet clear whether there is a fundamentaldifference between the hadronization of a thermal fireball and that of a19 Hadron-freeze-out analysis 353fast-moving quark. We refer to the process in which a thermal fireball ofquarks and gluons turns into hadrons as statistical hadronization. Thisis an area of nonperturbative strong-interaction physics in rapid develop-ment, and this interesting topic could nearly fill a review of the size ofthis book.The question we address is that of whether chemical-equilibrium hadronphase space (the so-called ‘hadronic gas’) can be used consistently todescribe the physics of thermal hadronization. We first note that theproduction of entropy is small, or even null, in hadronization of thermalQGP. Color ‘freezes’, and the excess entropy of QGP has to find a wayto get away, so any additional production is hindered. We consider theGibbs–Duham relation for a unit volume, Eq. (10.26), and combine itwith the instability condition of dynamic expansion, Eq. (3.31), cast inthe form0 = P |h + (P |h + ε|h)κv2c1− v2c. (19.1)The result isεσ∣∣∣h=(T |h +µbνbσ∣∣∣h)(1 +κv2c1− v2c). (19.2)Using the energy, E, the entropy S, and the baryon number b as variables,we obtain [222]ES∣∣∣∣h= (T |h + δT |h)(1 +κv2c1− v2c), δT = µbνbσ=µbS/b. (19.3)For the RHIC, we have δT |h < 0.4 MeV, considering that µb < 40 MeVand S/b > 100, whereas at the top SPS energy we have µb 200–250MeV and S/b 25–45, and thus δT |h 5–8 MeV. Eq. (19.3) shows thatES∣∣∣∣h> T |h (19.4)when super-cooling occurs prior to hadronization. Since all three quanti-ties, E|h , S|h , T |h can be obtained from particle abundances, this condi-tion can be verified. More generally, we can transcribe the Gibbs–Duhamrelation, Eq. (10.26), to obtain the relationES+PVS= T + δT > T. (19.5)Solving for P/ε, we obtainPε=T + δT − E/SE/S. (19.6)354 StrangenessThe study of hadron production, which will follow, allows an evalu-ation of both the freeze-out temperature and the energy per entropyE/S. The Gibbs–Duham relation in the form Eq. (19.6) determines akey property of the equation of state, the ratio of the pressure to theenergy density. We expect to find P/ε negative and small in magnitude,in nonequilibrium-sudden hadronization. In an equilibrium transforma-tion, the range 13 < P/ε <17 , see Fig. 11.3 on page 211, is expected,which spans the domain of highly relativistic matter and realistic hadronicgas.Chemical nonequilibrium is naturally connected to the sudden (statis-tical) hadronization, which is the favored reaction mechanism in view ofthe results seen at the RHIC, and also has been applied successfully to ex-plain the results obtained at the top energy at the SPS. The reader shouldkeep in mind that, as the collision energy is reduced, the transverse ex-pansion of the fireball of dense matter diminishes, and, at a sufficientlylow collision energy, a more adiabatic hadronization has to occur. In this‘AGS limit’, some of the signatures of the QGP phase we are discussingin this chapter may be erased.The word sudden refers to the time hadrons have available, followingtheir formation, to rescatter from other hadrons. The observed high abun-dances of short-lived hadronic resonances such as K∗(892) [279], whichhas a natural half life of τK∗ ln 2/ΓK∗ = 2.8 fm, implies that the decayoccurs mostly outside of the hadronic environment. Had the decay prod-ucts undergone rescattering, reconstruction of the K∗(892) would in mostcases be impossible. Conversely, reconstruction of abundant K∗(892) im-plies that the two decay products (π and K) did not propagate throughdense hadron matter [224, 260]. The HBT results, section 9.3, also placea very severe constraint on the size and life span of the pion source. Thesource is smaller than one would expect if the prolonged expanding HGphase were to exist. Finally, the hadron spectra are described in terms ofa source breaking up at T = 165±7 MeV and expanding with v⊥ = 0.52c[84], with chemical and thermal freeze-outs coinciding.It has been proposed that a mechanical instability associated withsuper-cooling of QGP is at the origin of the sudden-breakup mechanism.In section 3.5, we have seen that the motion of the quark–gluon fluidadds to the pressure exercised against the vacuum. Equation (3.31) de-scribes the balance condition when the dynamic expansion has run outof ‘speed’. The normal stable case is that vc = 0. However, if the initialcondition generated by the great collisional compression produces a fastexpansion with a finite velocity vc = 0 when the condition Eq. (3.31) issatisfied, this means that the outward flowing QGP matter is at a pres-sure P = Pp − B < 0, a highly unstable situation, in the (locally atrest) frame of reference. The fireball matter reaching this super-cooled19 Hadron-freeze-out analysis 355condition breaks up into smaller clusters. These drops of QGP separateand hadronize into free-streaming hadrons. A HG phase is never formed.However, production of particles occurs as dictated by the phase spaceavailable. Their yields are controlled by abundances of quarks in thehadronizing phase.19.2 Phase space and parametersBefore proceeding with this section, the reader should refresh his memoryabout the role chemical potentials play in counting hadrons (sections 11.2and 11.4). The approach we present is in its spirit a generalization ofFermi’s statistical model of hadron production [121, 122], in that the yieldof hadrons is solely dictated by the study of the magnitude of the phasespace available.The relative number of final-state hadronic particles freezing out from,e.g., a thermal quark–gluon source, is obtained by noting that the fugacityfi of the ith emitted composite hadronic particle containing k componentsis derived from fugacities λk and phase-space occupancies γk:Ni ∝ e−Ei/Tffi = e−Ei/Tf∏k∈iγkλk. (19.7)In most cases, we study chemical properties of the light quarks u andd jointly, though, on occasion, we will introduce the isospin asymmetry,Eq. (11.11). As seen in Eq. (19.7), we study particle production in terms offive statistical parameters, T, λq, λs, γq, and γs. In addition, to describethe shape of spectra, one needs matter-flow-velocity parameters; thesebecome irrelevant when only total abundances of particles are studied;these are obtained by integrating over all of phase space, or equivalentlyin the presence of strong longitudinal flow, when we are looking at a yieldper unit rapidity.Assuming a QGP source, several of the statistical parameters have nat-ural values.1. λs. The fugacity of the strange quark λs can be obtained from therequirement that strangeness balances, 〈ns〉 − 〈ns̄〉 = 0, which, for asource in which all s and s̄ quarks are unbound and have symmetricphase space, Eq. (11.13), implies λs = 1. However, the Coulomb dis-tortion of the strange-quark phase space plays an important role in theunderstanding of this constraint for Pb–Pb collisions, leading to theCoulomb-deformed value λs 1.1; see section 11.3.2. γs. The strange-quark phase-space occupancy γs can be computed, sec-tion 17.5, within the framework of kinetic theory and γs 1. Recallthat the difference between the two different types of chemical param-eters λi and γi is that the phase-space-occupancy factor γi regulates356 Strangenessthe number of pairs of flavor ‘i’, and hence applies in the same mannerto particles and antiparticles, whereas the fugacity λi applies only toparticles, while λ−1i is the antiparticle fugacity.3. λq. The light-quark fugacity λq, or, equivalently, the baryo-chemicalpotential, Eq. (11.2a), regulates the baryon density of the fireball andhadron freeze-out. This density can vary depending on the energy andsize of colliding nuclei, and the value of λq is not easily predicted. How-ever, it turns out that this is the most precisely measurable parameter,with everybody obtaining the same model-independent answer, for itdirectly enters all highly abundant hadrons. Since T differs dependingon the strategy of analysis, the value of µb is not so well determined.4. γq. The equilibrium phase-space occupancy of light quarks γq is ex-pected to significantly exceed unity in order to accommodate the ex-cess entropy content in the plasma phase. There is an upper limit,Eq. (7.20). We addressed this in section 7.5.5. Tf . The freeze-out temperature Tf is expected to be within 10% ofthe Hagedorn temperature TH 160MeV, which characterized theproduction of particles in proton–proton reactions; see chapter 12.6. vc. The collective-expansion velocity vc is expected to remain near tothe relativistic velocity of sound,vc ≤ 1/√3,the natural speed of flow of information in the QGP phase. There isa longitudinal velocity, which is needed in order to describe rapidityspectra, section 8.3, and there is a motion of the hadronization surface,aside from many further parameters one may wish to use to model thevelocity profile of flowing matter.The resulting yields of final-state hadronic particles are most conve-niently characterized by taking the Laplace transform of the accessiblephase space. This approach generates a function that, in terms of itsmathematical properties, is identical to the partition function. For exam-ple, for the open-strangeness sector, we find (with no flow)L(e−Ei/Tf∏k∈iγkλk)∝ lnZHGs , (19.8)with lnZHGs given in Eq. (11.19).It is important to keep in mind that• Eq. (19.8) does not require formation of a phase comprising a gas ofhadrons, but is not inconsistent with such a step in the evolution of the19 Hadron-freeze-out analysis 357matter; Eq. (19.8) describes not a partition function, but just a look-alike object arising from the Laplace transform of the accessible phasespace;• the final abundances of particles measured in an experiment are ob-tained after all unstable hadronic resonances ‘j’ have been allowed todisintegrate, contributing to the yields of stable hadrons; and• in some experimental data, it is important to distinguish between thetwo light-quark flavors, for example experiments are sensitive only toΞ−, not to Ξ0, and averaging over isospin does not occur.The unnormalized particle multiplicities arising are obtained by differ-entiating Eq. (19.8) with respect to the particle fugacity. The relativeparticle yields are simply given by ratios of corresponding chemical fac-tors, weighted by the size of the momentum phase space accepted bythe experiment. The ratios of strange antibaryons to strange baryons ofthe same type of particles are, in our approach, simple functions of thequark fugacities; see Eqs. (11.21a)–(11.21e). When particles of unequalmass are considered, and are fed by decay of other hadrons, considerablenumerical effort is required to evaluate yield ratios for particles, in partic-ular, if these are available in a fraction of phase space only. To the bestof our knowledge, the numerical results of various groups working withthe statistical-hadronization method are consistent, though the physicscontent can vary widely, depending on assumptions introduced.19.3 SPS hadron yieldsWe have argued that, in general, we must expect γq = 1, section 19.1,i.e., chemical nonequilibrium at hadron freeze-out is an expected ingre-dient in a precise interpretation of the experimental results on particleratios obtained in the SPS energy range. For strangeness, it has beenexpected and was seen early on in experimental data [216]. Full chemicalnonequilibrium was first noted in the study of the S–Au/W/Pb collisionsat 200A GeV [176]. On fitting the yields of hadrons observed, it wasnoted that the statistical significance increased when chemical nonequi-librium was introduced. The statistical significance is derived from thetotal statistical error:χ2 ≡∑j(Rjth −Rjexp)2(∆Rjexp)2. (19.9)It is common to normalize χ2 by the difference between the number ofdata points and parameters used, the so-called ‘dof’ (degrees-of-freedom)quantity.358 StrangenessTable 19.1. Statistical parameters obtained from fits of data for S–Au/W/Pbcollisions at 200A GeV, without enforcing conservation of strangeness [176]Tf [MeV] λq λs γs γq χ2/dof145 ± 3 1.52 ± 0.02 1∗ 1∗ 1∗ 17144 ± 2 1.52 ± 0.02 0.97 ± 0.02 1∗ 1∗ 18147 ± 2 1.48 ± 0.02 1.01 ± 0.02 0.62 ± 0.02 1∗ 2.4144 ± 3 1.49 ± 0.02 1.00 ± 0.02 0.73 ± 0.02 1.22 ± 0.06 0.90∗ denotes fixed (input) valuesWe show the resulting statistical parameters obtained in hadron-yieldfits in table 19.1, for S–Au/W/Pb collisions at 200A GeV. Asterisks (∗)mark fixed (input) values, thus the first column assumes not only chemicalequilibrium, but also the symmetric value λs = 1 for the QGP phasespace. Interestingly, little is gained by allowing λs to vary, and all differentfitting strategies point to λs = 1. However, allowing for strange γs = 1and then light-quark γq = 1, nonequilibrium brings the result of the fitprogressively to statistical significance. For systems we study, with a fewdegrees of freedom (typically 5–15), a statistically significant fit requiresthat χ2/dof < 1. For just a few ‘dof’, the error should be as small asχ2/dof < 0.5. The usual requirement χ2 → 1 applies only for infinitelylarge ‘dof’. We learn from these results that the chemical nonequilibriumfactor γi for both strange and light quarks is a required ingredient in thestatistical hadronization model.Turning now to the Pb–Pb system at collision energy 158A GeV, weconsider the particles listed in the top section of table 19.2 from the exper-iment WA97, for p⊥ > 0.7GeV, within a narrow central-rapidity window∆y = 0.5. Further below are shown results from the large-acceptanceexperiment NA49, extrapolated by the collaboration to full 4π coverageof phase space. The total error χ2 for the two columns of results is shownat the bottom of this table along with the number of data points ‘N ’and parameters ‘p’ used, and the number of (algebraic) redundancies ‘r’connecting the experimental results. For r = 0, it is more appropriateto quote the total χ2, since the condition for statistical relevance is moredifficult to establish given the constraints, but since χ2/(N−p−r) < 0.5,we are certain to have a valid description of hadron multiplicities. Wewill return to discuss the yields of Ω and Ω at the end of this section.In table 19.2 second from last column, the superscript ‘s’ means thatλs is fixed by strangeness balance. The superscript ‘γq’, in the two lastcolumns means that γq = γcq = emπ/(2Tf) is fixed in such a way as tomaximize the entropy content in the hadronic phase space. The fits pre-sented were obtained with the updated NA49 experimental results, i.e.,they include the updated h−/b, newly published yield of φ [43], and pre-19 Hadron-freeze-out analysis 359Table 19.2. WA97 (top) and NA49 (bottom) Pb–Pb 158A-GeV-collision hadronratios compared with phase-space fitsRatios Reference Experimental data Pb|s,γq Pb|γqΞ/Λ [171] 0.099 ± 0.008 0.096 0.095Ξ/Λ̄ [171] 0.203 ± 0.024 0.197 0.199Λ̄/Λ [171] 0.124 ± 0.013 0.123 0.122Ξ/Ξ [171] 0.255 ± 0.025 0.251 0.255K+/K− [79] 1.800 ± 0.100 1.746 1.771K−/π− [248] 0.082 ± 0.012 0.082 0.080K0s/b [152] 0.183 ± 0.027 0.192 0.195h−/b [43] 1.970 ± 0.100 1.786 1.818φ/K− [21] 0.145 ± 0.024 0.164 0.163Λ̄/p̄ y = 0 0.565 0.568p̄/π− All y 0.017 0.016χ2 1.6 1.15N ; p; r 9; 4; 1 9; 5; 1dict the Λ̄/p̄ ratio. b is here the number of baryon participants, andh− = π−+K−+ p̄ is the yield of stable negative hadrons, which includespions, kaons, and antiprotons. We see, on comparing the two columns,that conservation of strangeness (which is enforced in the second fromlast column) is consistent with the experimental data shown; enforcing itdoes not change much the results for particle multiplicities.The six parameters (T, vc, λq, λs, γq and γs) describing the abundancesof particles are shown in the top section of table 19.3. Since the resultsof the WA97 experiment do not cover the full phase space, one findsa reasonably precise value for one velocity parameter, taken to be thespherical surface-flow velocity vc of the fireball hadron source.As in S-induced reactions in which λs = 1 [176], so also in Pb-inducedreactions, a value λPbs 1.1 characteristic of a source of freely movablestrange quarks with balancing strangeness, i.e., with λ̃s = 1, is obtained;see Eq. (11.17). Since all chemical-nonequilibrium studies of the Pb–Pbsystem converge to the case of maximum entropy, see Fig. 7.7 on page 125,we have presented the results with fixed γq = γcq = emπ/(2Tf). The largevalues of γq > 1 seen in table 19.3 confirm the need to hadronize theexcess entropy of the QGP that possibly is formed. This value is derivedboth from the specific abundance of negative hadrons h−/b and from therelative yields of strange hadrons.The results shown in table 19.3 allow us to evaluate P/ε, section 19.1.Using Eq. (19.6) with δT = 8 MeV (see Eq. (19.3)), T = 148 MeV, and360 StrangenessTable 19.3. Upper section; the statistical model parameters which best describethe experimental results for Pb–Pb data seen in table 19.2; bottom section, en-ergy per entropy, antistrangeness, and net strangeness of the full hadron phasespace characterized by these statistical parameters. In column two, we fix λs bythe requirement of conservation of strangenessPb|s,γqv Pb|γqvT [MeV] 151 ± 3 147.7 ± 5.6vc 0.55 ± 0.05 0.52 ± 0.29λq 1.617 ± 0.028 1.624 ± 0.029λs 1.10∗ 1.094 ± 0.02γq γcq∗ = emπ/(2Tf) = 1.6 γcq∗ = emπ/(2Tf) = 1.6γs/γq 1.00 ± 0.06 1.00 ± 0.06E/b [GeV] 4.0 4.1s/b 0.70 ± 0.05 0.71 ± 0.05E/S [MeV] 163 ± 1 160 ± 1(s̄− s)/b 0∗ 0.04 ± 0.05∗ indicates values resulting from constraints.E/S = 160 MeV, we obtain P/ε ∼ −1/40. This negative and small valueis consistent with the super-cooling hypothesis. One other interestingquantitative results of this analysis is shown in the bottom section oftable 19.3: the yield of strangeness per baryon, s/b 0.7. We see, inthe lower portion of table 19.3, that near strangeness balance is obtainedwithout constraint.The most rarely produced hadrons are the triply strange Ω(sss) andΩ(s̄s̄s̄), which are the heaviest stable hadrons, MΩ = 1672MeV. Thephase space for Ω is ten times smaller than that for Ξ under the conditionsof chemical freeze-out we have obtained, and any contribution from non-statistical hadronization would be visible first in the pattern of productionof Ω and Ω . For the parameters in table 19.3, the yields of Ω reportedfor the experiment WA97 are underpredicted by nearly a factor of two.This excess yield originates at the lowest m⊥, as can be seen in Fig. 8.11on page 155. The ‘failure’ of a statistical-hadronization model to describeyields of Ω (and, by 30%, Ω) has several possible explanations.One is the possibility that an enhancement in production of Ω and Ωis caused by pre-clustering of strangeness in the deconfined phase [215].This would enhance the production of all multistrange hadrons, but mostprominently the phase-space-suppressed yields of Ω and Ω. This mech-anism would work only if pairing of strange quarks near to the phasetransition were significant. Current models of ‘color superconductivity’support such a clustering mechanism [30, 185, 206, 237]. We have men-19 Hadron-freeze-out analysis 361tioned, at the end of section 11.5, the possibility that distillation of stran-geness followed by breakup of strangelets could contribute to productionof Ω and Ω. The decay of disoriented chiral condensates has also beenproposed as a source of soft Ω and Ω [158].A more conventional explanation for the excess production of Ω and Ωis obtained by noting that, due to the low reaction cross section, Ω and Ωcould decouple from the HG background somewhat sooner than do all theother hadrons [55, 263]. To augment the yields by factor k, it is sufficientto take an incrementally δTΩ higher freeze-out temperature, determinedfrom studying the Ω phase space:δTΩ Tln kMΩ/T. (19.10)In order to increase the yield by a factor of two, the freeze-out of Ωwould need to occur at TΩ = 160MeV rather than at T = 150MeV.Since the temperature drops as the expansion of the fireball develops, ahigher freeze-out temperature means freeze-out occurring slightly earlierin time. Even though the required staging in time of hadron productionis apparently small, a consistent picture requires fine-tuning and it seemsunnatural, considering that yields of all the other particles are perfectlyconsistent with just one sudden chemical-freeze-out condition.In view of these pre- and post-dictions of the anomalous yield of Ω andΩ, one should abstain from introducing these particles into statistical-hadronization-model fits. We note that the early statistical descriptionsof yields of Ω and Ω have not been sensitive to the problems we described[61, 180]. In fact, as long as the parameter γq is not considered, it is notpossible to describe the experimental data at the level of precision thatwould allow recognition of the excess yield of Ω and Ω within statisti-cal hadronization model. For example, a chemical-equilibrium fit, whichincludes the yield of Ω and Ω has for 18 fitted data points with two pa-rameters a χ2/dof = 37.8/16 [80]. Such a fit is quite unlikely to containall the physics even if its appearance to the untrained eye suggests thata very good description of experimental data as been achieved.19.4 Strangeness as a signature of deconfinementWe have found that the rate of production of strangeness in QGP issensitive to the temperature achieved at the time gluons reach chemi-cal and thermal equilibrium. There is considerable uncertainty regardinghow short the time required to relax the strangeness flavor in the thermalgluon medium is, with the upper limit being the hatched area in Fig. 17.11.Consideration of the small mass for strangeness found in lattice studiesof strange hadrons has yielded much smaller τs (the thin-dotted line in362 StrangenessFig. 17.11). There is also the probable further reduction due to the next-to-leading-order effects (the K-factor). In view of this, we now establisha benchmark yield of strangeness, assuming that the equilibration pro-cess leads to near-chemical-equilibrium conditions for hadronizing QGP.Specifically, the abundance of light quarks in the QGP phase may beconsidered near to the equilibrium yield, γQGPq → 1, whereas the yield ofstrange quarks characterized by the QGP phase-space occupancy beforehadronization, γQGPs , may differ appreciably from equilibrium.We consider the ratio of the equilibrium density of strangeness, arisingin the Boltzmann-gas limit, Eq. (10.51), to the baryon density in a fireballof QGP given in Eq. (10.75):ρsρb=sb=sq/3=γQGPsγQGPq(3/π2)T 3W (ms/T )23(µqT 2 + µ3q/π2) . (19.11)W is as defined in Eq. (10.50a) and shown in Fig. 10.1. The equilibriumstrange-quark density Eq. (10.51), with gs = 6, is the first term in theexpansion Eq. (10.63). Higher-order quantum-statistics correction termsare negligible, givenms/T = O(1). To a first approximation, perturbativethermal QCD corrections, see Eq. (16.24), cancel out in the ratio. Forms = 200 MeV and T = 150 MeV, we havesb γQGPsγQGPq0.7lnλq + (lnλq)3/π2. (19.12)The relative yield s/b is mainly dependent on the value of λq. In theapproximation considered, it is nearly temperature-independent. Thislight-quark fugacity pertinent to the final-state hadrons is well determinedand does not vary depending on the strategy used for analysis of data.We show in Fig. 19.1, as a function of λq − 1 (the variable chosen toenlarge the interesting region λq → 1), the expected relative yield perbaryon originating from the QGP, defined in Eq. (19.12) with γQGPs =γQGPq = 1. At the top SPS energy, we see that the equilibrium yieldis 1.5 pairs of strange quarks per participating baryon (for λq 1.5–1.6). Considering the experimental yield in table 19.3, γQGPs ∼ 0.5. Theexplanation of this is that, if formed, the QGP system did not become hotenough for long enough. In p–p collisions at the corresponding energy,the yield is below 0.3 pairs of strange quarks per participant [277], whichis 40% of the Pb–Pb yield.For the RHIC 130-GeV run, the value λq = 1.09 allows one to under-stand many particle yields at central rapidity. We see, in Fig. 19.1, thatthe specific yield of strangeness in a fireball of QGP at equilibrium is anorder of magnitude greater than that currently observed at the SPS top19 Hadron-freeze-out analysis 363Fig. 19.1. The yield of strangeness per baryon as a function of λq in equilibratedQGP.energy. This remarkable increase is due to the expected chemical equili-bration (γs → 1) at the RHIC, see section 17.5, as well as to a substantialreduction of baryon density at central rapidity. In comparison with thegeneral hadron multiplicity, only a modest enhancement of productionof strangeness at most can be expected at the RHIC, compared to SPS;the remarkable feature of the RHIC situation is that this enhancementis found in the abundance of (multistrange) baryons. Given the largestrangeness-per-baryon ratio, baryons and antibaryons produced at theRHIC are mostly strange [221]. We are not aware of any reaction modelother than formation of QGP and its hadronization that could producethis type of yield anomaly.While the specific yield of strangeness s/b is a clear indicator for theextreme conditions reached in heavy-ion collisions, a more directly acces-sible observable is the occupancy of the hadron-strangeness phase space,γHGs . Due to the need to hadronize into a strangeness-poor phase, γHGscan be appreciably greater than unity. To understand this, we comparethe phase space of strangeness in QGP with that of the resulting HG.The absolute yields must be the same in both phases. This hadroniza-tion condition allows us to relate the two phase-space occupancies in HGand QGP, by equating the strangeness content in the two phases. Oncanceling out the common normalization factor T 3/(2π2), Eq. (11.26), weobtainγQGPs VQGPgsW( msTQGP) γHGs V HG(γqλqλsFK+γ2qλ2qλsFY). (19.13)364 StrangenessFig. 19.2. The HG/QGP strangeness-occupancy γs ratio in sudden hadroniza-tion as a function of λq. Solid lines, γHGq = 1; long-dashed lines, γHGq = 1.3; andshort-dashed lines, γHGq = 1.6. Thin lines are for T = 170 and thick lines forT = 150 MeV, for both phases.Here we have, without loss of generality, followed the s̄-carrying hadrons inthe HG phase space, and we have omitted the contribution of multistrangeantibaryons for simplicity. We now use the condition that strangeness isconserved, Eq. (11.28), to eliminate λs from Eq. (19.13), and obtainγHGsγQGPsV HGV QGP=gsW (ms/TQGP)√(γqFK + γ2qλ−3q FY)(γqFK + γ2qλ3qFY). (19.14)In sudden hadronization, V HG/V QGP 1, the growth of volume is neg-ligible, TQGP THG, the temperature is maintained across the hadroniza-tion front, and the chemical occupancy factors in both states of matteraccommodate the different magnitude of the particle phase space. In thiscase, the ‘squeezing’ of the strangeness of the QGP into the smaller HGphase space results in γHGs /γQGPs > 1. We show, in Fig. 19.2, the enhance-ment of phase-space occupancy expected in sudden hadronization of theQGP. The temperature ranges T = 150 MeV (thick lines) and T = 170MeV (thin lines) span the ranges being considered today at the SPS andRHIC. The value of γq is in the range of the chemical equilibrium in HG,γq = 1 (solid lines), to the expected excess in sudden hadronization, seesection 7.5, γq = 1.6 (short-dashed lines), with the intermediate valueγq = 1.3 (long-dashed lines).19 Hadron-freeze-out analysis 365Fig. 19.3. The HG/QGP strangeness-occupancy γs ratio in sudden hadroniza-tion, as a function of T for λq = 1. Solid line, γHGq = 1; long-dashed line,γHGq = 1.3; and short-dashed line, γHGq = 1.6.We note that, for the top SPS energy range, for which λq = 1.5–1.6,sudden-hadronization analysis of data implies T 150MeV and γq 1.6,and the value of γs increases across hadronization by factor 2.7. Since theyield of strangeness seen at the SPS implies that γQGPs 0.6, this in turnimplies that γHGs 1.6 γHGq , as is indeed found in hadron-productionanalysis in the sudden-hadronization picture; see table 19.3.Because γHGs /γHGq 1, one can also model the hadronization at theSPS energy in terms of an equilibrium-hadronization model. The en-hancement in production of pions associated with the high-entropy phasecan be accommodated by use of two temperatures, one for the determi-nation of absolute yields of particles, and another for determination ofthe spectral shape. Such an approach has a similar number of param-eters, and comparable predictive power. However, the SPS condition,γHGs /γHGq 1, does not hold for the RHIC energy range. We thereforeexpect that a much clearer picture about the dynamics of hadronizationof QGP should emerge from the study of yields of strange particles at theRHIC.As can be seen in Fig. 19.2, near λq → 1 (the condition at the RHIC)there is practically no variation in the ratio γHGs /γHGq . Therefore, weconsider, in Fig. 19.3 for λq = 1, the dependence on the hadronizationtemperature of γHGs /γHGq , with the three cases shown: solid line, γq = 1;long-dashed line, γq = 1.3; and short-dashed line, γq = 1.6. We nowcompare the hadronization conditions for the RHIC in the range between366 StrangenessT = 180 MeV and γq = 1 and T = 155 MeV and γq = 1.6. Across thisdomain, we see that the phase space of strangeness in the HG is threetimes smaller than that in QGP, or, as shown in Fig. 19.3, there is thislarge increase in occupancy of the strange-quark phase space.In the likely event that the QGP formed at the RHIC evolves towardthe chemical-equilibrium abundance of strangeness, or possibly even ex-ceeds it, section 17.5, we should expect a noticeable over-occupancy ofstrangeness to be measured in terms of the chemical-equilibrium abun-dance of final-state hadrons. This high phase-space occupancy is oneof the requirements for the enhancement of production of multistrange(anti)baryons, which is an important hadronic signal of QGP phenomena[213–215, 226]. In particular, we hope that hadrons produced in phasespace with a small probability, such as Ω and Ω, will be observed withyields above expectations, continuing the trend seen at the SPS. Becausemuch of the strangeness is in the baryonic degrees of freedom, the kaon-to-pion ratio appears suppressed, in Fig. 1.5 on page 17, relative to SPSresults, however we will show below that this is not the case.Many results from the RHIC run at√sNN = 130 GeV are still prelim-inary and the following quantitative discussion is probably not the finalword on this matter. However, the results we find are very interesting,and in qualitative agreement with the sudden-breakup reaction picture ofQGP predominantly presented in this book. The data is mainly obtainedin the central-rapidity region, where, due to approximate longitudinalscaling, the effects of flow cancel out and we can evaluate the full phase-space yields in order to obtain particle ratios. We do not explore trivialresults such as π+/π− = 1, since the large hadron yield combined withthe flow of baryon isospin asymmetry toward the fragmentation rapidityregion assures us that this result will occur to within a great precision.We also do not use the results for K∗ and K̄∗ since these yields dependon the degree of rescattering of resonance decay products [224, 259]. Thedata we use has been reported in conference reports of the STAR collabo-ration of summer 2001, which has been combined with data of PHENIX,BRAHMS, and PHOBOS; for more discussion of the origin of data, see[81]. We assume, in our fit in table 19.4, that the multistrange weak-interaction cascading Ξ → Λ, in the STAR result we consider, is cut byvertex discrimination and thus we use these uncorrected yields.We first look at the last column in table 19.4, the chemical-equilibriumfit. Its large χ2 originates from the inability to account for production ofmultistrange Ξ and Ξ. Similar results are presented in [81], in an equi-librium fit that does not include multistrange hadrons. The equilibriumfit yields E/S = 159MeV < T = 183 MeV. With a negligible contribu-tion from δT , Eq. (19.6) implies that P/ε 1/6.6 as is expected for thehigh-freeze-out temperature.19 Hadron-freeze-out analysis 367Table 19.4. Fits of central-rapidity hadron ratios for the RHIC run at√sNN =130 GeV. Top section: experimental results, followed by chemical parameters,physical properties of the phase space, and the fitting error. Columns: data,the full nonequilibrium fit, the nonequilibrium fit constrained by conservation ofstrangeness and supersaturation of pion phase space, and, in the last column, theequilibrium fit constrained by conservation of strangeness, superscript ∗ indicatesquantities fixed by these considerationsFit FiteqData Fit s− s̄ = 0 s− s̄ = 0p̄/p 0.64 ± 0.07 0.637 0.640 0.587p̄/h− 0.068 0.068 0.075Λ/Λ 0.77 ± 0.07 0.719 0.718 0.679Λ/h− 0.059 ± 0.001 0.059 0.059 0.059Λ/h− 0.042 ± 0.001 0.042 0.042 0.040Ξ/Ξ 0.83 ± 0.08 0.817 0.813 0.790Ξ−/Λ 0.195 ± 0.015 0.176 0.176 0.130Ξ−/Λ 0.210 ± 0.015 0.200 0.200 0.152K−/K+ 0.88 ± 0.05 0.896 0.900 0.891K−/π− 0.149 ± 0.020 0.152 0.152 0.145KS/h− 0.130 ± 0.001 0.130 0.130 0.124Ω/Ξ− 0.222 0.223 0.208Ω/Ξ− 0.257 0.256 0.247Ω/Ω 0.943 0.934 0.935T 158 ± 1 158 ± 1 183 ± 1γq 1.55 ± 0.01 1.58 ± 0.08 1∗λq 1.082 ± 0.010 1.081 ± 0.006 1.097 ± 0.006γs 2.09 ± 0.03 2.1 ± 0.1 1∗λs 1.0097 ± 0.015 1.0114∗ 1.011∗E/b [GeV] 24.6 24.7 21s/b 6.1 6.2 4.2S/b 151 152 131E/S [MeV] 163 163 159χ2/dof 2.95/(10− 5) 2.96/(10− 4) 73/(10− 2)The chemical-nonequilibrium fit appears more internally consistent.The value of the hadronization temperature T = 158 MeV is below thecentral expected equilibrium phase-transition temperature for the case of2 + 1 flavors, section 15.5. This is also near to the P = 0 condition for thequark–gluon-liquid model developed in chapter 16, see Fig. 4.2 on page 70,and, as is appropriate, a little above the temperatures seen for the SPSbaryon-rich freeze-out; see table 19.3. We find, Eq. (19.6) P/ε −1/33, avalue consistent with super-cooling and sudden QGP hadronization. Thisreaction picture is in agreement with the relatively large γs and γq > 1368 Strangenessobtained. Comparing the two types of results, we conclude that it is theinclusion of the yields of the multistrange antibaryons in the RHIC dataanalysis, along with the hypothesis of chemical nonequilibrium, whichallows us to discriminate between the different scenarios of reaction.We look next at the strangeness content, s/b = 6, in table 19.4: thefull QGP phase space would have yielded 8.6 pairs of strange quarks perbaryon at λq = 1.085, as is seen in Fig. 19.1, and γQGPs = 6/8.6 =0.7. With this value, and using the fitted value γHGs = 2.1, we computeγHGs /γQGPs = 2.1/0.7 = 3 and, as we see in Fig. 19.3, this is, for T = 158MeV and γq = 1.55, the expected condition for hadronization of QGP.The fact that the inferred strangeness phase space in QGP is not fullysaturated is, on a second careful look, in qualitative agreement with pre-dictions of kinetic theory, Fig. 17.13. Namely, particle multiplicities ob-served and the shape of particle spectra suggest that the initial conditionsare near T = 300 MeV in the RHIC run at√sNN = 130 GeV, whichpushes the value of γQGPs , shown in Fig. 17.13, toward 0.9.A further reduction in this prediction, more applicable to the highestRHIC energy range, arises since the initial volume of the fireball assumedin order to obtain the result in Fig. 17.13 is up to twice as large as thatimplied by the RHIC 130A-GeV results. This is understood by consid-ering the total experimental hadron multiplicity entropy, which we canderive from the fitted entropy per baryon, S/b 150. We cannot multiplythis by the total number of participants, since many of the 350 partic-ipating nucleons are found in the fragmentation regions. If we assumethat the central-rapidity fraction is 100 baryons, the central entropy con-tent is 15 000, while the calculations for Fig. 17.13 are based on 38 000entropy units. Thus, the initial volume of the fireball populating the par-ticle yields in the central-rapidity region is about half that used to obtainthe results in Fig. 17.13. A smaller system lives for a shorter time, and,since we are not yet in the regime of full equilibration of strangeness, asmaller value of γQGPs than that seen in Fig. 17.13 is expected. Also, thesmaller volume of the system is more consistent with the HBT results.The value of the thermal energy content, E/b = 25 GeV, seen in thebottom portion of table 19.4 is also in very good agreement with expec-tations once we allow for the kinetic-energy content associated with lon-gitudinal and transverse motion. The energy of each particle is ‘boosted’by the factor γv⊥ cosh y‖, see, e.g., Eq. (8.39). For v⊥ = c/√3, we haveγv⊥ = 1.22. The range of longitudinal flow is about ±2.3 rapidity units; seeFig. 9.19. To obtain the the increase in energy due to longitudinal flow, wehave to multiply by the average,∫dy‖ cosh y‖/y‖ → sinh(2.3)/2.3 = 2.15,for a total average increase in energy by factor 2.62, which takes the fullenergy content to Ev/b 65 GeV as expected.19 Hadron-freeze-out analysis 369We now consider what these results imply about the total yield of stran-geness in the RHIC fireball. First, we sum up the yield of strange quarkscontained in hyperons. Recalling Fig. 2.6 on page 32, we have in singlystrange hyperons 1.5 times the yield observed in Λ. Also, accounting forthe doubly strange Ξ−, which are half of all the Ξ, we have〈s〉Yh−= 1.5× 0.059 + 2× 2× 0.195× 0.059 = 0.133.Allowing for the unobserved Ω at the theoretical rate, this number in-creases to 〈s〉Y/h− = 0.14. Repeating the same argument for antihyper-ons, the result is 0.10. The s and s̄ content in kaons is four times that inKS and thus〈s + s̄〉h−= 0.76,with 32% of this yield contained in hyperons and antihyperons. Up tothis point, the analysis is practically solely based on direct measurementsand established yields of particles.We now estimate the increase in the ‘strangeness-suppression’ factorWs, Eq. (18.1). Correcting for the presence of K− among negativelycharged hadrons, and assuming that all three pions are equally abundant,we find〈s + s̄〉π− + π+ + π0 0.30.The total number of pions produced comprises pions arising from reso-nance decays and from fragments of the projectile and target. Thus, asfew as half of the pions originate from the newly made qq̄ pairs. In theRHIC run at√sNN = 130 GeV, we estimateWs 0.6. The increase com-pared with the SPS energy is largely due to the strangeness content inhyperons. Considering that γQGPs 0.7 at√sNN = 130 GeV, there is stillspace for a further rise in yield of strangeness at the highest RHIC energy,and we hope and expect that Ws → 1 when the initial temperature risesto well above the mass of the strange quark for a sufficient length of time.We have learned to appreciate, in this last part, how the deconfinedthermal phase, through its gluon content, manifests itself as a strange-ness signature of QGP. The presence of gluons is essential for rapid ther-mal production of strangeness. The SPS strangeness results decisivelyshow interesting new physics, with a significant excess of strangeness andstrange antibaryons. We see, at the SPS and at the RHIC, considerableconvergence of the understanding of the production of hadrons aroundproperties of suddenly hadronizing entropy and strangeness-rich QGP.We see hadronization into pions, at γq → γcq = emπ/(2Tf) 1.6, which is370 Strangenessan effective way to convert the excess of entropy in the deconfined stateinto hadrons. We have seen that strange-particle signatures of hadroniza-tion of QGP become more extreme and clearer at the RHIC.It is important to remember that it is not only at the RHIC, and inthe near future at the LHC, that QGP can be studied. An alternativeenergy domain for investigating QGP is the phase-transition region athigh baryon density. It is very probable that the onset of deconfinementoccurs at modest collision energies, perhaps in collisions of 20–40A-GeVPb projectiles with a laboratory target. The formation of the QGP phaseis an endothermic process. In experiments near to the condition for phasetransformation, energy balance, lack of explosive flow, and the onset ofabundant production of entropy and strangeness as a function of the en-ergy and reaction volume should provide good global signatures of newphysics. Anomalous production of multistrange baryons and antibaryonsshould help to pinpoint the deconfined phase.We hear sometimes the following question: “Where is the ‘smokinggun’ signature of QGP?”. The disappearance of the suppression of theproduction of strangeness is surely one such observation. However, wemust remember that the discovery of QGP, unlike the discovery of anew particle, requires a change in our understanding of the fundamentalhadronic degrees of freedom. This is a deductive process, and requires aglobal cross check of consistency at each step in its development. It isunlikely that our detailed arguments will persuade the skeptic, however,we hope to show the new way to the uninitiated.QGP discovery is similar to the slow and painful path to the under-standing of electricity. The reader is invited to think through how he/shewould proceed without the plug in the wall, the instrument in the lab-oratory, and the battery in the drawer, and without comprehension ofthe principles of conductivity, to introduce in a lecture the discovery ofthe existence of electricity. The challenge of understanding the ‘ionized’quark–gluon form of matter is certainly more complex than the unravelingof electricity.References[1] S. Abatzis et al., WA85 collaboration, 1990. Λ and Λ production insulphur–tungsten interactions at 200 GeV/c per nucleon. Phys. Lett. B,244, 130.[2] S. Abatzis et al., WA85 collaboration, 1991. 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In Proceedings of the Strangeness’96 Workshop, Budapest, page361, 1996.Indexαsbag model fit, 262lattice, 298, 299running first order, 278running second order, 281, 282at low energy, 284, 285scale dependence, 280, 281thermal, 286approximant, 286antimatterin the Universe, 6matter symmetry, 19, 153antiproton-to-proton ratio, 183from QGP, 16heavy ion enhancement, 19bag model, 258action, 263bag constant, 38, 69, 261, 306boundary conditions, 263static-cavity solutions, 264strange-quark mass, 261baryon, 25, 26barys , 1densityhydrodynamic expansion,108quark–gluon liquid, 310, 312,313Bessel functionIn, 140Kν , 195x2K2, 197non-relativistic limit, 196relativistic limit, 196big-bang, 1differences from micro-bang, 4bosoncondensate, 193distribution function, 194entropy per particle, 207calorimeter, 177canonicalconservation of strangeness, 223ensemble, 192multistrange particles, 226particle enhancement, 231particle suppression, 225, 232hadron yield enhancement, 233partition function, 224cascade(ssq), 33Ξ∗(1530) decay, 34center of momentumaccelerator energies, 72coordinate system, 82rapidity in asymmetric collisions,84charged-hadron multiplicityenergy dependence, 180, 181ratio, 123charmcanonical suppression, 234, 235open, 23389390 Indexthermal production, 329, 333, 334LHC energy, 336charmonium, 36enhancement, 22suppression, 21, 22chemicalentropy equilibration, 115equilibrationthermal rate, 99equilibrium, 90elementary interactions, 234failure, 171strangeness in HG, 352gluon equilibration, 324nonequilibriumentropy, 115heavy quarks, 98SPS results, 357relaxation time for strangeness,335chemical parametersPb–Pb at√sNN = 130 GeV, 367Pb–Pb system, 359, 360S–Au/W/Pb system, 358chemical potential, 57, 212antiparticle, 60isospin asymmetry, 213local, 90chiral condensate, 297disoriented, 269chiral symmetry, 44bag model, 258breaking, 44color, 7confinement, 38current, 273dynamic charge, 38flux tube, 41hyperfine interaction, 259superconductive phase, 47confinement, 6, 38boundary condition, 264origin, 38correlationsBose–Einstein, 171cross sectiongeometric, 78momentum average, 317thermal average, 318current algebra, 45decoherence, 15deconfinement at the AGS and SPS,95degeneracyeffective, 66quantum gas, 61degrees of freedom, 66, 67electro-weak particles, 9QGP, 52detailed balance, 316antibaryon production, 350dileptons, 24Dirac equation, 262Doppler factor, 149energy density, 195at the RHIC, 185hadronic gas, 209, 211lattice QCD, 300QGP, 53, 310quantum gas, 69quark–gluon liquid, 310, 312–314ensemblecanonical, 192grand-canonical, 192micro-canonical, 191statistical, 191enthalpy, 191entropyBoltzmann gas, 114chemical equilibration, 115classical gas, 205, 206conservationideal flow, 106scaling solution, 128contentphase-space distribution, 113Fermi gas, 113glue fireball, 116, 117hadronization, 126initial state, 129isolated system, 114Index 391measurement, 121, 122nonequilibrium, 115particle production, 112per baryon, 206per particle, 114, 207pion gassuper-saturated, 124entropy density, 187quark–gluon liquid, 310, 312–314quark–gluon plasma, 315equation of stateeffect of particle mass, 199finite-volume correction, 209quark–gluon plasma, 303relativistic ideal gas, 199equilibriumabsolute chemical, 90chemical, 90local, 90, 95kinetic, 96relative chemical, 90thermal, 90explanation, 97local, 96eta function, 201Euclidian space, 288expansioncooling, 112decrease in temperature, 11Hubble, 11exponential mass spectrumphase transition, 238extensive variables, 187Fadeev–Popov ghosts, 275fermiondegeneracy factor, 201distribution function, 194, 203domain wall, 292entropy per particle, 207ideal-gas quark density, 204lattice doubling, 291fireball, 2, 95entropy, 115expanding, 137thermal particle spectra, 141expansion, 50explosion, 51flow of matter, 138life span, 91mass, 80stages of evolution, 93super-cooling, 51fireball staticparticle spectra, 131first law of thermodynamics, 187flavorconservation, 214symmetry, 269free energy, 187, 188lowest order in αs, 304perturbative expansion, 303freeze-outchemical and thermal, 158Cooper–Frye formula, 141surface1 + 1 dimensions, 109, 110velocity, 142FRITIOF model, 103fugacity, 56, 57, 213antiparticle, 60time dependence, 214valence quark, 212gauge invariance, 268covariant derivative, 272minimal coupling, 268Gibbs conditionchemical potential, 50temperature, 48Gibbs’ conditionpressure, 48Gibbs–Duham relation, 107, 190gluon, 6, 24, 38current, 273degeneracy, 53density, 63equilibration, 324field, 271field condensate, 39field correlator, 39spectra, 120392 Indexthermal mass, 305yield of strangeness, 340Goldstone boson, 44grand-canonical ensemble, 57hadronabundances, 169RHIC, 366theoretical yield error, 240finite-volume cluster, 247in modification of a medium, 345massbag model, 259multiplicity of h− at SPS, 166ratios in A–A collisions at 14AGeV, 170ratios in A–A collisions at 200AGeV, 169size, 260hadronichadros , 1cascade, 102mass spectrum, 217exponential growth, 235–237resonance interactions, 243hadronic gas, 48ε/P , 210asymmetry of strangeness, 215energy density, 210excluded volume, 251consistency, 250correction, 248finite size EoS, 209overheated, 51phase space for strangeness, 217pressure, 210properties, 65relativistic limit, 66relaxation time for strangeness,350scattering phase shifts, 243hadronizationdeconfined matter, 350in a volume, 141statistical, 352sudden, 126entropy content, 126surface emission, 142Hagedorn gas, 236critical temperature, 240Hagedorn temperature, 53HBT, 171, 172correlations, 174, 175kaon, 174resonances, 174transverse mass, 176heat function, 191heavy ionbaryon stopping, 74collisionaxis, 81event generators, 102interaction vertex, 81participants, 78spectators, 79systems, 72transport models, 102experimental programBNL–RHIC, 76CERN–SPS, 75rapidity gap, 73hydrodynamics, 104equations of motion, 105Euler relation, 104flow forces, 210one-dimensional solution, 111hyperon, 28lambda decay, 28lambda resonances, 28number, 25resonance, 31sigma–baryon, 30yield, 32, 33ideal gasclusters in bootstrap model, 256energy per particle, 201, 202entropy, 205entropy per baryon, 206quark partition function, 203impact parameter, 79isospin, 25Index 393particle counting, 214quark current, 43jet quenching, 23kaon, 35latent heat, 38, 69, 258, 261latticecell size, 294continuum limit, 294critical temperature, 301domain-wall fermions, 292dynamic quarks, 289energy density, 300infrared cutoff, 289mass of strange hadrons, 299mass of strange quarks, 299naive quark action, 291plaquette, 290pressure, 301with staggered fermions, 302procedure for simulations, 294,295QCD action, 289quark mass, 298quenched quarks, 289, 296running coupling constant, 298, 299scaling violation, 294staggered quark action, 292ultraviolet cutoff, 289Wilson action, 290, 291lepton, 6leptos , 1level densityN -particle, 241scattering phase shift, 242single-particle, 61LHCcharge multiplicity, 181Lorentzboosts, 83contraction, 82covariant gauge, 275invariant spectra, 132, 139Mandelstam variables, 319mass thermal, 305matter–antimatter, 182symmetry, 4Maxwell construction, 49meson, 26, 27φ, 36mesos , 1strange, 25omega, 34chemical equilibration, 349freeze-out temperature, 361production of decay in strangelets,223OSCAR, 100pair productionperturbative, 326Schwinger mechanism, 40particledensity, 194glue fireball, 119phase-space distribution, 317energy per particle, 198ensemble, 192indistinguishable, 55momentum, 81production, 95∝ e−2m/T , ∝ e−m/T , 228secondaries, 81spectra, 62pseudorapidity, 137scaling solution, 128thermal, 135surface emission, 143temperature, 152particle ratiosantiproton to proton, 182chemical fugacities, 218in A–A collisions at 14AGeV, 170in A–A collisions at 200A GeV,169in Pb–Pb collisions at√sNN =130 GeV, 367in Pb–Pb collisions at 158A GeV,359394 Indexstrange baryon–antibaryon, 160,164partition functionBoltzmann, 61canonical, 56, 224generating function, 58grand-canonical, 57quantum, 61multicomponent system, 193pressure ensemble, 190quantum, 59strange particles, 217vacuum, 69partoncascade, 102thermalization, 23path integral, 288Fermi determinant, 293phasecrossover, 53diagram, 46, 47, 49, 50metastable, 51mixed, 49transition, 53change in g, 9quark-mass dependence, 53, 54phase spaceN -particle volume element, 242Coulomb distortion, 215entropy contentsingle-particle, 113integral, 197Lorentz-invariant, 139occupancykinetic evolution, 323quantum particle, 193phase transitionearly Universe, 9, 14finite-volume, 238fluctuations, 126photondensity, 63direct from QGP, 23production, 24pionexcess, 167, 168gas, 124properties, 125–127productionenhancement, 127suppression, 167yieldcharge asymmetry, 90plasmaelectron–ion, 54Podolanski–Armenteros analysis, 29,30pressure, 188criticalearly Universe, 10effect of particle mass, 68, 199hadronic gas, 209, 211thermal, 306pseudorapidity, 85error, 87, 89particle emission angle, 85particle energy and momentum,136rapidity, 86–88QCD, 267K-factorflavor production, 330Λ parameter, 283asymptotic freedom, 278charge definition, 279color-magnetic instability, 40critical temperature, 301Feynman diagrams, 277, 326Lagrangian, 273lattice action, 289lattice formulation, 287lattice pressure, 301perturbative, 38, 274Polyakov loop, 296, 297renormalization, 278group, 280particle spin, 279running, 282β and γ functions, 281running αsinitial conditions, 284Index 395sum rules, 39, 45temperature dependence of αs,286thermal Feynman diagrams, 304transmutation of scales, 279two-loop αs, 286value of Λ, 283QED instability, 40QGPenergy density, 315entropy density, 315pressure, 307quark, 6, 24, 38bag model, 38, 262cavity state, 260charge, 6chiral condensate, 297chiral symmetry, 44cold quark matter, 47confinement, 38degeneracy, 53energy in the bag, 266flavors, 6free, 53, 54ideal-gas density, 204Lagrangian, 273lattice action, 291mass, 7massless limit, 45phase structure, 53, 54running, 282, 283, 328sum rules, 46pairing, 47productionrunning threshold, 284strange, 8sum rules, 45thermal mass, 305quark densitystatistical equilibrium density,325quark–gluon liquid, 70, 306quark–gluon plasmaBc formation, 37comparison of signatures, 24critical temperature, 10degeneracy, 53degrees of freedom, 52energy density, 53, 310equation of state, 303equations of state, 310, 312, 313evidence, 162phase-space enhancement,363–365formation, 153hadronization at the RHIC, 367in the early Universe, 5negative pressure, 52observability, 15, 16partition function, 70, 304perturbative QCD interactions,70, 304phase diagram, 48strange-antibaryon signature, 351strange-particle signature, 18sudden hadronization, 52super-coolingmechanical instability, 52thermodynamic potential, 310undercooled, 51yield of strangeness, 362, 363quasi-particle, 59, 305quasirapidity distributionprotons and kaons, 89rapidity, 82Λ and Λ spectra in S–S collisions,144asymmetric systems, 84baryon-poor region, 184CM reflection, 144fragmentation region, 145gap, 72, 73negative-hadron distribution, 165particle spectra, 83scaling solution, 128pseudorapidity, 86–88velocity relation, 82reference framecenter of momentum, 79relaxation time, 98electro-weak interactions, 92396 Indexproduction of strangeness, 336thermal production of charm, 337RHICcharged-hadron production, 180,182first results, 178Riemann eta and zeta functions, 200search for strangelets, 223spectrainverse transverse slopesystem size, 152Lorentz-invariant, 139pseudorapidity, 136, 138rapidity‘net’ baryons, 146massless QGP quanta, 145schematic representation, 147three-fluid model, 146rapidity window, 133strange hadroninverse transverse slope, 153thermal, 135thermal fitstatistical parameters, 157transverse mass, 148Λ, Ω + Ω, 155π0 and η, 151strange particles, 149, 150strange-particle analysis, 154,155spin–spin interaction, 40statistical bootstrap, 244cluster formation, 256critical behavior, 254critical curve, 255hypothesis, 243idea, 244model, 247, 252physical interpretation, 257singularity, 246statistical ensemble, 191statistical hadronization, 352enhancement of occupancy of phasespace, 363excess of omega, 360RHIC, 366statistical mechanicscovariant formulation, 248statistical significance, 358Stefan–Boltzmann law, 52, 69strange antibaryonssignature of deconfinement, 160,164, 351strange hadroninverse transverse slope, 153strange particlenon-statistical yield, 157spectra, 19strangenessφ, 153abundance at the RHIC, 184chemical equilibrium, 98chemical-relaxation time, 335,336conservation, 222canonical, 223canonical QGP and HG, 229distillation, 222enhancement, 19, 160, 163, 164exchange reaction, 344, 345excitation function, 16, 17hyperon yield, 31, 32in baryons and mesons, 25kinetic evolution at the RHIC,338lattice quark mass, 299negative chemical potential, 221,222observables, 171Okubo–Zweig–Izuka rule, 344particle decays, 81partition function, 217phase-space occupancy, 339, 364,365enhancement, 363production, 299, 326, 327glue equilibration, 340SPS rapidity distribution, 159production in HG, 343, 344, 348,351QGP signature, 18Index 397quark mass, 8, 258relative equilibrium, 91signature of QGP, 15, 24, 160, 164symmetry in QGP, 214thermal production, 332, 334Wróblewski factor, 340, 341yield at SPS, 159yield at the RHIC, 369yield in QGP, 98, 362, 363streamer chamber, 81SU(3)adjoint representation, 270Gell-Mann representation, 269temperature, 188evolution, 94glue fireball, 118Hagedorn, 53inverse slope, 56local, 90transverse slope, 152thermalcollision frequency, 320, 331equilibration, 93equilibrium, 90Feynman diagrams, 304particle spectra, 135expanding fireball, 141pressure, 306QCD energy scale, 285reaction rate, 318reactivity, 318thermal mass, 305, 308thermalizationtransport models, 104thermodynamic potentialquark–gluon liquid, 310three-body reactions, 349transverse energy, 177distribution, 177, 178per charged particle, 179pseudorapidity density, 180scaling with A, 178transverse mass, 82, 148transverse-momentum acceptance,134upsilonium, 37vacuumenergy, 41density, 42zero-point, 41instability, 41latent heat, 38, 261partition function, 69perturbative, 38polarization, 277restoration of symmetry, 46structure, 258true, 38velocitycylindrical representation, 139relative, 319sound, 11, 107Wall Street, 63Wilson action, 291Yang–Mills fields, 271zeta function, 200

Melting Hadrons, Boiling Quarks

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Melting Hadrons, Boiling Quarks

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