Effective quantum theories with short- and long-range forces Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakulta¨t der Rheinischen Friedrich-Wilhelms-Universita¨t Bonn vorgelegt von Sebastian Ko¨nig aus Northeim Bonn, August 2013 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakulta¨t der Rheinischen Friedrich-Wilhelms-Universita¨t Bonn 1. Gutachter: Prof. Dr. Hans-Werner Hammer 2. Gutachter: Prof. Dr. Ulf-G. Meißner Tag der Promotion: 23. Oktober 2013 Erscheinungsjahr: 2013 Abstract At low energies, nonrelativistic quantum systems are essentially governed by their wave functions at large distances. For this reason, it is possible to describe a wide range of phenomena with short- or even finite-range interactions. In this thesis, we discuss several topics in connection with such an effective description and consider, in particular, modifications introduced by the presence of additional long-range potentials. In the first part we derive general results for the mass (binding energy) shift of bound states with angular momentum ` ≥ 1 in a periodic cubic box in two and three spatial dimensions. Our results have applications to lattice simulations of hadronic molecules, halo nuclei, and Feshbach molecules. The sign of the mass shift can be related to the symmetry properties of the state under consideration. We verify our analytical results with explicit numerical calculations. Moreover, we discuss the case of twisted boundary conditions that arise when one considers moving bound states in finite boxes. The corresponding finite-volume shifts in the binding energies play an important role in the study of composite-particle scattering on the lattice, where they give rise to topological correction factors. While the above results are derived under the assumption of a pure finite-range inter- action—and are still true up to exponentially small correction in the short-range case—in the second part we consider primarily systems of charged particles, where the Coulomb force determines the long-range part of the potential. In quantum systems with short-range interactions, causality imposes nontrivial constraints on low-energy scattering parameters. We investigate these causality constraints for sys- tems where a long-range Coulomb potential is present in addition to a short-range in- teraction. The main result is an upper bound for the Coulomb-modified effective range parameter. We discuss the implications of this bound to the effective field theory (EFT) for nuclear halo systems. In particular, we consider several examples of proton–nucleus and nucleus–nucleus scattering. For the bound-state regime, we find relations for the asymptotic normalization coefficients (ANCs) of nuclear halo states. Moreover, we also consider the case of other singular inverse-power-law potentials and in particular discuss the case of an asymptotic van der Waals tail, which plays an important role in atomic physics. Finally, we consider the low-energy proton–deuteron system in pionless effective field the- ory. Amending our previous work, we focus on the doublet-channel spin configuration and the 3He bound state. In particular, we study the situation at next-to-leading order in the EFT power counting and provide numerical evidence that a charge-dependent counterterm is necessary for correct renormalization of the theory at this order. We furthermore argue that the previously employed power counting for the inclusion of Coulomb contributions should be given up in favor of a scheme that is consistent throughout the bound-state and the scattering regime. In order to probe the importance of Coulomb effects directly at the zero-energy threshold, we also present a first calculation of proton–deuteron scattering lengths in pionless effective field theory. Parts of this thesis have been published in the following articles: • S. Ko¨nig and H.-W. Hammer, Low-energy p-d scattering and He-3 in pionless EFT, Phys. Rev. C 83, 064001 (2011) • S. Ko¨nig, D. Lee, and H.-W. Hammer, Volume Dependence of Bound States with Angular Momentum, Phys. Rev. Lett. 107, 112001 (2011) • S. Bour, S. Ko¨nig, D. Lee, H.-W. Hammer, and U.-G. Meißner, Topological phases for bound states moving in a finite volume, Phys. Rev. D 84, 091503(R) (2011) • S. Ko¨nig, D. Lee, and H.-W. Hammer, Non-relativistic bound states in a finite volume, Annals Phys. 327, 1450–1471 (2012) • S. Ko¨nig, D. Lee, and H.-W. Hammer, Causality constraints for charged parti- cles, J. Phys. G: Nucl. Part. Phys. 40, 045106 (2013) • S. Ko¨nig and H.-W. Hammer, The Low-Energy p–d System in Pionless EFT, Few-Body Syst. 54, 231–234 (2013) • S. Elhatisari, S. Ko¨nig, D. Lee, and H.-W. Hammer, Causality, universality, and effective field theory for van der Waals interactions, Phys. Rev. A 87, 052705 (2013) fu¨r meinen Opa Contents 1 Introduction 1 2 General concepts 3 2.1 Finite-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Non-local interactions . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Solutions of the free equation . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Asymptotic form of the wave function . . . . . . . . . . . . . . . . . 6 2.1.4 The effective range expansion . . . . . . . . . . . . . . . . . . . . . 6 2.1.5 Bound states and asymptotic normalization constants . . . . . . . . 7 2.2 Some formal scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 The Lippmann–Schwinger equation . . . . . . . . . . . . . . . . . . 9 2.2.2 The T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Hierarchy of partial waves . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Universality for large scattering length . . . . . . . . . . . . . . . . 14 2.3.3 The theorist’s point of view . . . . . . . . . . . . . . . . . . . . . . 15 2.3.4 From potential models to a modern perspective . . . . . . . . . . . 16 2.4 Effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Pionless effective field theory . . . . . . . . . . . . . . . . . . . . . . 18 2.4.3 Effective field theory for nuclear halo states . . . . . . . . . . . . . 19 2.5 Lattice calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.1 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.2 Nuclear lattice simulations . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.3 Numerical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 i 3 Finite-volume calculations 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Bound states in a finite volume . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Infinite volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Finite volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 S-wave result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Extension to higher partial waves . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 Sign of the mass shift . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.3 Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.1 Lattice discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Two-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Twisted boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.1 Generalized derivation . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7.2 Topological volume factors . . . . . . . . . . . . . . . . . . . . . . . 48 3.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 The Coulomb force 53 4.1 Coulomb wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.1 The Gamow factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.2 Analytic wave functions . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Modified effective range expansion . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Bound-state regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.1 Asymptotic wave function . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.2 Bound-state condition and ANC . . . . . . . . . . . . . . . . . . . . 59 4.4 The Coulomb T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.1 Yukawa screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 Expression in terms of hypergeometric functions . . . . . . . . . . . 64 5 Causality bounds for charged particles 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ii 5.2 Setup and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Derivation of the causality bound . . . . . . . . . . . . . . . . . . . . . . . 70 5.3.1 Wronskian identities . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3.2 Rewriting the wave functions . . . . . . . . . . . . . . . . . . . . . 71 5.3.3 The causality-bound function . . . . . . . . . . . . . . . . . . . . . 72 5.3.4 Calculating the Wronskians . . . . . . . . . . . . . . . . . . . . . . 73 5.3.5 Constant terms in the causality-bound function . . . . . . . . . . . 76 5.4 The causal range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4.1 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . 78 5.5 Examples and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.5.1 Proton–proton scattering . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5.2 Proton–deuteron scattering . . . . . . . . . . . . . . . . . . . . . . 81 5.5.3 Proton–helion scattering . . . . . . . . . . . . . . . . . . . . . . . . 82 5.5.4 Alpha–alpha scattering . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.6 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.7 Relation for asymptotic normalization constants . . . . . . . . . . . . . . . 86 5.7.1 Derivation for the neutral system . . . . . . . . . . . . . . . . . . . 86 5.7.2 Derivation for charged particles . . . . . . . . . . . . . . . . . . . . 88 5.7.3 Application to the oxygen-16 system . . . . . . . . . . . . . . . . . 91 5.8 Other long-range forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8.1 Singular potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8.2 Causality bounds for van der Waals interactions . . . . . . . . . . . 95 5.9 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6 The proton–deuteron system revisited 103 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Formalism and building blocks . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.1 Full dibaryon propagators . . . . . . . . . . . . . . . . . . . . . . . 106 6.2.2 Coulomb contributions in the proton–proton system . . . . . . . . . 108 6.2.3 Power counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2.4 Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2.5 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 118 6.3 Scattering phase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 iii 6.3.1 Quartet channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.2 Doublet channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4 Trinucleon wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.4.1 Homogeneous equation . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4.2 Normalization condition . . . . . . . . . . . . . . . . . . . . . . . . 122 6.5 Helium-3 properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.5.1 Energy shift in perturbation theory . . . . . . . . . . . . . . . . . . 126 6.5.2 Nonperturbative calculation . . . . . . . . . . . . . . . . . . . . . . 127 6.5.3 Leading-order results . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.6 The Coulomb problem at next-to-leading order . . . . . . . . . . . . . . . . 132 6.6.1 Scaling of the dibaryon propagators . . . . . . . . . . . . . . . . . . 133 6.6.2 Ultraviolet behavior of the amplitude . . . . . . . . . . . . . . . . . 133 6.6.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.6.4 Nonperturbative calculation . . . . . . . . . . . . . . . . . . . . . . 136 6.6.5 Back to the scattering regime . . . . . . . . . . . . . . . . . . . . . 137 6.7 Proton–deuteron scattering lengths . . . . . . . . . . . . . . . . . . . . . . 142 6.7.1 Numerical calculation of the Gamow factor . . . . . . . . . . . . . . 143 6.7.2 Bubble diagram with full off-shell Coulomb T-matrix . . . . . . . . 145 6.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7 Concluding remarks 153 A The Coulomb wave functions of Bolle´ and Gesztesy 157 B Explicit expressions for the causality bound function 159 B.1 Repulsive case, γ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B.2 Attractive case, γ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 C Equivalence of ANC relations 161 D Bound states in nonrelativistic effective field theory 163 D.1 Simplified nucleon–deuteron system . . . . . . . . . . . . . . . . . . . . . . 163 D.2 Bethe–Salpeter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 D.2.1 Momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 iv D.2.2 Bound state contribution . . . . . . . . . . . . . . . . . . . . . . . . 167 D.2.3 Homogeneous equation . . . . . . . . . . . . . . . . . . . . . . . . . 168 D.2.4 Three-dimensional reduction . . . . . . . . . . . . . . . . . . . . . . 168 D.3 Operator formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 D.3.1 Lippmann–Schwinger equation . . . . . . . . . . . . . . . . . . . . . 169 D.4 Normalization condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 D.4.1 General derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 D.4.2 Explicit form in three dimensions . . . . . . . . . . . . . . . . . . . 171 D.4.3 Factorization of the T-matrix . . . . . . . . . . . . . . . . . . . . . 173 D.5 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 v vi Chapter 1 Introduction A lot of interesting physics can be done with short or even finite-range interactions. In many situations physicists encounter “universality” in the sense that observables are independent, or at least approximately so, of the short-range details of a given system. The simple—yet very powerful—concept behind this observation is that low-energy phenomena do not probe physics at short distances. This separation of scales is also the foundation of effective field theory. However, such universal aspects are usually just the starting point for a deeper under- standing. In general, universal behavior is not exactly realized in nature and as soon as one wants to go beyond a certain basic level of precision, it is necessary to take into account corrections. One such correction that plays indeed a very prominent role at low energies is the Coulomb interaction. It is particularly important in nuclear physics because most processes in that field involve charged particles. The above two paragraphs already mention the most important aspects of the topics to be discussed in this thesis. It summarizes the work done in a little more than three years of doctoral studies, divided into several projects that will be presented in separate chapters. Starting with a quick tour through quantum mechanics with short- and finite- range interactions, we will discuss low-energy universality and its connection to effective field theory (EFT) in Chapter 2. With a focus on applications in nuclear physics, we will present there the general concepts that will be important throughout this work, both as theoretical foundations and to put the results to be derived in a broader context. In Chapter 3 we consider how calculations in finite volumes with periodic boundary con- ditions, as they appear frequently in numerical lattice calculations, affect the properties of nonrelativistic bound states. When the latter arise from a finite-range interaction, it is possible to derive an analytic formula for their leading-order volume dependence. Gener- alizing a result for S-waves that has been obtained by Lu¨scher in the 1980s [8], we derive the finite-volume mass shift for states with arbitrary orbital angular momentum. For the rest of this work we will then mainly be concerned with systems of charged particles, where the Coulomb force plays an important role, and particularly so at low energies. Chapter 4 is dedicated to collecting and reviewing several results concerning this topic. 1 2 Chapter 1. Introduction Subsequently, in Chapter 5, we derive a generalization of the so-called Wigner causality bound to interactions with Coulomb tails. Results of this kind are both interesting theo- retically and as a guide to improve the convergence of effective-field-theory calculations. We also consider other long-range forces and in particular the van der Waals interaction, which plays an important role in low-energy atomic physics. While in the derivation of the causality bounds the Coulomb force enters mainly as a theoretical challenge in a configuration-space treatment, we will look at the difficulties it creates from a more practical point of view in Chapter 6, where we consider the low- energy proton–deuteron system in pionless effective field theory. Performing calculations in both the scattering and the bound-state regime, we obtain results for phase shifts, scattering lengths, and the Helium-3 binding energy. In all cases, Coulomb effects raise the interesting question of how to treat them consistently in an EFT power counting scheme. At the same time, they also constitute a challenge for numerical calculations in momentum space. Finally, we close with some concluding remarks in Chapter 7. Chapter 2 General concepts Overview In the first part of this chapter, we review several well-known results from nonrelativistic quantum mechanics. Focussing first on solutions of the radial Schro¨dinger equation and the effective range expansion in Section 2.1 , we subsequently give an overview of scattering theory from a more formal point of view in Section 2.2. Moving on, we then discuss low- energy universality and its connection to effective field theories Sections 2.3 and 2.4. Finally, in Section 2.5, we briefly discuss lattice calculations as a prelude to the following Chapter 3. 2.1 Finite-range interactions Consider a two-particle system with reduced mass µ interacting via a spherically-symmetric potential V (r), where r = |r1 − r2| is the relative distance of the two particles in their center-of-mass frame. It is a well-known fact in quantum mechanics that a solution ψ(r) of the time-independent Schro¨dinger equation1,2[ −∆r 2µ + V (r)− E ] ψ(r) = 0 , (2.1) describing a state with energy E and angular momentum quantum numbers (`,m), can be separated as ψ(r) = ψ`m(r) = R`(r)Y m ` (θ, φ) = u`(r) r Y m` (θ, φ) (2.2) into a known angular part, given by the spherical harmonics Y m` (θ, φ), and a (reduced) radial wave function u`(r), which is a solution of[ − d 2 dr2 + `(`+ 1) r2 + 2µ [ V (r)− E]]u`(r) = 0 . (2.3) 1∆r denotes the Laplacian. Throughout this work, we always indicate the coordinate that a differ- ential operator acts on with an explicit subscript. 2Here and in the following we work in natural units where ~ = c = 1. 3 4 Chapter 2. General concepts This is known as the radial Schro¨dinger equation. In the following, we are primarily interested in interactions with a finite range R. For the potential V (r) this means that it fulfills the condition V (r) = 0 for r > R . (2.4) Moreover, throughout this work we employ the following canonical classification scheme for potentials: • if for r →∞ the potential falls off faster than any power law, e.g., like an exponen- tial, we call it short-ranged. • if, on the other hand, the potential has a power-law form r−α with α ≥ 1 at large r, we call it a long-range potential. A very prominent member of this class is the Coulomb potential VC(r) ∼ 1/r, which will be discussed in more detail in Chapter 4. We do not consider potentials that do not fall off at large distances. 2.1.1 Non-local interactions The interaction of a quantum system can be described in more general terms by a real symmetric operator V (r, r′) = 〈r|Vˆ |r′〉 . (2.5) Assuming, as we will do throughout this work whenever we consider such non-local inter- actions, that V (r, r′) allows an expansion in Legendre polynomials (partial-wave decom- position), V (r, r′) = ∞∑ `=0 (2`+ 1)V`(r, r ′)P`(cos θ) , cos θ = r · r′ rr′ , (2.6) a generalized form of the radial Schro¨dinger equation (2.3) can be written as p2u`(r) = − d 2 dr2 u`(r) + `(`+ 1) r2 u`(r) + 2µ ∫ ∞ 0 dr′ 4pir′ V`(r, r′)u`(r′) , (2.7) where we have furthermore introduced the momentum scale p2 = 2µE. From this one directly sees that the interaction no longer depends on the wave function at just a single point but over some extended range and is thus “smeared out” or “non-local” in that sense. For the special case where V (r, r′) = V (r)× δ(3)(r− r′) , (2.8) we simply get back Eq. (2.3). Note that the partial-wave decomposition (2.6) is in particular possible for non-local potentials which are a function of the relative distance only, V (r, r′) = V (|r− r′|) , (2.9) 2.1. Finite-range interactions 5 which is the straightforward generalization of spherical symmetry to the case of non-local interactions. This condition, however, is only a sufficient and not a necessary one, and we do not make such explicit assumptions on the interaction in the following. Due to the symmetry of V (r, r′), which ensures that the full Hamiltonian is a symmetric operator, the generalized condition for the interaction to have a finite range now reads V (r, r′) = 0 if r = |r| > R or r′ = |r′| > R , (2.10) which implies the same for all partial-wave components V`(r, r ′). The integral in Eq. (2.7) does then not extend all the way to infinity but is cut off at r′ = R. Since in this work we are not concerned with the exact form of the interaction, we will from now on absorb the prefactor 4pir′ in Eq. (2.7) into the potential and also omit the subscript ` in writing down the radial Schro¨dinger equation. 2.1.2 Solutions of the free equation In the absence of any interaction (in particular, outside the range of a finite-range poten- tial), one is left with the free radial Schro¨dinger equation, p2u`(r) = [ − d 2 dr2 + `(`+ 1) r2 ] u`(r) , (2.11) the solutions of which are well known. Commonly (see, e.g., Ref. [9]) the Riccati–Bessel and Riccati–Neumann functions S`(pr) and C`(pr) are chosen as a base pair of linearly- independent solutions.3 They are defined as S`(z) = √ piz 2 J`+1/2(z) , (2.12a) C`(z) = (−1)` √ piz 2 J−`−1/2(z) , (2.12b) where Jn(z) denotes the ordinary Bessel function. Any solution of Eq. (2.11) can be written as a linear combination of S`(pr) and C`(pr). Their asymptotic behavior for large arguments is determined by S`(z) |z|→∞∼ sin (z − `pi/2) , C`(z) |z|→∞∼ cos (z − `pi/2) , (2.13) whereas close to the origin (in the limit z → 0) S` and C` scale like z`+1 and z−`, respectively. Due to this behavior S`(pr) and C`(pr) are also called the regular and irregular solutions of (2.11). In some situations it is more convenient to work instead with the Riccati–Hankel functions Hˆ±` (z) = C`(z)± iS`(z) , (2.14) which have the asymptotic form e±iz as |z| → ∞ and thus, for z = pr, correspond to outgoing and incoming waves for z = pr, respectively. 3Ref. [9] actually uses the notation S`(z) = ˆ`(z) and C`(z) = nˆ`(z). The convention we employ here is the same as in Ref. [10] with L = ` and d = 3. 6 Chapter 2. General concepts 2.1.3 Asymptotic form of the wave function Going back to Eq. (2.7), we now consider an arbitrary interaction that fulfills the finite- range condition (2.10), and which we assume to be sufficiently well-behaved at the origin to allow for a solution that is regular there (i.e., u`(r)→ 0 as r → 0 with finite derivative). We denote such a solution for a given—possibly complex, to also cover the bound-state regime—momentum parameter p with an explicit superscript “(p).” Outside the range of the interaction, it can be written in the form u (p) ` (r) = p ` [cot δ`(p)S`(pr) + C`(pr)] (r > R) , (2.15) where he have adopted the momentum-dependent normalization convention from Ref. [10]. In the following, whenever we write u (p) ` (r), it is also implied that the wave function is normalized exactly as in Eq. (2.15). The scattering phase shift δ`(p) can be interpreted as the additional phase (compared to the regular solution S`(pr) of the free equation) that the wave function picks up due to the interaction. It is related to the partial-wave S-matrix s`(p) via s`(p) = e 2iδ`(p) , (2.16) such that up to an overall change in the normalization, the asymptotic form of u (p) ` (r) can also be written as u (p) ` (r) ∝ H−` (pr)− s`(p)H+` (pr) (r > R) (2.17) in terms of Riccati–Hankel functions. Alternatively, by factoring out an overall eiδ`(p)/ sin δ`(p), one finds that u (p) ` (r) ∝ [ cot δ`(p)− i ] H−` (pr)− [ cot δ`(p) + i ] H+` (pr) (r > R) . (2.18) The latter form can also be derived directly by inverting Eq. (2.14) and inserting the result into Eq. (2.15). For interactions with a short but not strictly finite range, e.g., a potential with an exponential or Gaussian tail, the relations above do not hold exactly, but are fulfilled asymptotically as r →∞. 2.1.4 The effective range expansion For systems with short- or finite-range interactions, the scattering phase shift δ`(p) can be expressed in terms of the well-known effective range expansion p2`+1 cot δ`(p) = − 1 a` + 1 2 r` p 2 + · · · , (2.19) where a` and r` are the scattering and effective range parameters, respectively, and the ellipses stand for higher-order shape parameters (∝ p4, p6, . . .) that we have not written out here. Eq. (2.19) states that the cotangent of the scattering phase shift, multiplied by appropriate powers of the momentum, is an analytic function of p2 and thus of the energy. Historically, this expansion was first used to describe and interpret nucleon–nucleon scat- tering at low-energies. This is discussed, for example, in the review by Jackson and 2.1. Finite-range interactions 7 Blatt [11]. As noted there (and already in an earlier publication by the same authors [12]), the S-wave (` = 0) version of Eq. (2.19) was first derived by Schwinger, but only published in lecture notes. A derivation based on a direct analysis of the radial wave functions was later given by Bethe in Ref. [13]. An extension of Bethe’s formalism to also treat higher partial waves can be found in the textbook of Goldberger and Watson [14]. For a modern derivation of Eq. (2.19) for arbitrary `, based directly on the analytic properties of the partial-wave S-matrix and scattering amplitude (both of which will be discussed in more detail in the following sections), see, for example, Ref. [9]. In general, an expansion as in Eq. (2.19), which is essentially a Taylor series of the left- hand side around p2 = 0, only has a finite radius of convergence. For example, if the interaction is given by a short-range potential of the Yukawa form exp(−µr)/r (as it arises from the exchange of a particle with mass µ), p2`+1 cot δ`(p) has a cut starting at −µ2/4 in the complex p2 plane (see, for example, Ref. [15]), which naturally limits the radius of convergence of the expansion. Within its region of analyticity, however, p2`+1 cot δ`(p) can also be expanded around points other than p = 0. For example, in the 3S1 channel of neutron–proton scattering it is customary to work with an effective range expansion around the deuteron pole [16]. In the presence of long-range potentials p2`+1 cot δ`(p) is in general no longer analytic in p2 and the ordinary effective range expansion as given above thus not valid for systems where the interaction has a power-law tail. It is, however, often possible to modify the left-hand side of Eq. (2.19) in such a way that analyticity is restored. Essentially, this means that known non-analytic terms due to the long-range component of the potential can be taken into account explicitly. Most notably, for the scattering of charged particles one has the so-called Coulomb-modified effective range expansion that will be discussed in Chapter 4. Note furthermore that a general modified effective range function for the case where the interaction can be written as the sum of a short-range and a long-range potential has been derived by van Haeringen and Kok in Ref. [17]. The methods just mentioned no longer work if the long-range part of the potential has a power-law form that is more singular than 1/r2 at the origin. For such cases, an extension of a formalism called quantum defect theory [18–20] has been developed by Gao [21]. This is particularly interesting for applications in atomic physics, where a long-range van der Waals tail (∼ 1/r6) arises due to the mutual polarization of the atoms, a case we will come back to in Chapter 5. 2.1.5 Bound states and asymptotic normalization constants Going back to the case of pure finite-range interactions, we now consider bound states. Their wave functions correspond to solutions of Eq. (5.2) for negative energies, p2 = 2µE < 0 . (2.20) More precisely, in the complex p-plane, the bound states are located on the positive imaginary axis, whereas the negative imaginary axis is the location of virtual states [9]. We write p = iκ in the bound-state regime and call κ > 0 the binding momentum. 8 Chapter 2. General concepts A bound state with angular momentum ` corresponds to a (simple) pole in the partial- wave S-matrix s`(p) at p = iκ. From Eq. (2.17) it can be seen that this means that there is no incoming component because s`(p), the ratio of the outgoing wave compared to the incoming one, becomes infinite. Alternatively one can choose a normalization with an overall coefficient s`(p) factored out in Eq. (2.17), leaving s`(p) −1 in front of the incoming component H−` (pr) that has to vanish for p = iκ. From Eq. (2.18) one furthermore sees that cot δ`(p = iκ) = i (2.21) is the bound-state condition for the scattering phase shift. The same result can be found more directly by writing the S-matrix as s`(p) = 1 + 2ikf`(p) (2.22) with the partial-wave scattering amplitude f`(p) = p2` p2`+1 [cot δ`(p)− i] . (2.23) Since the pole in s`(p) has to come from the second term in (2.22) we again obtain the condition stated in Eq. (2.21). On the bound-state wave function we impose the usual normalization condition∫ ∞ 0 dr ∣∣u`(r)∣∣2 = 1 . (2.24) From the above discussion it follows immediately that u (iκ) ` (r) ∝ H+` (iκr) ∼ e−κr (2.25) as r →∞, which together with our regularity assumption on the interaction ensures that the wave function is indeed normalizable. Note, however, that the bound-state normalization condition (2.24) is not directly com- patible with the asymptotically fixed form as given in Eq. (2.15). In order to discuss the precise asymptotic form of the bound-state wave functions we closely follow Ref. [9] and define solutions χ±`,p(r) of (2.7) normalized such that they exactly fulfill the condition χ±`,p(r) r→∞∼ H±` (pr) . (2.26) The bound-state solution normalized according to (2.24), which in the following we denote as u (iκ) A,` (r), can then be written as u (iκ) A,` (r) = i `Aκχ + `,iκ(r) , (2.27) where Aκ = (∫ ∞ 0 dr |χ+`,iκ(r)|2 )−1/2 (2.28) 2.2. Some formal scattering theory 9 is the so-called asymptotic normalization constant/coefficient (ANC). The factor i` con- veniently adjusts the phase of u (iκ) ` (r) such that it is a real function. Outside the range of the interaction one then has the exact identity u (iκ) A,` (r) = i `AκH + ` (iκr) for r > R . (2.29) The wave functions u (iκ) ` (r) with the asymptotic behavior determined by Eq. (5.3), on the other hand, have the form u (iκ) ` (r) = i `κ`H+` (iκr) for r > R . (2.30) We use the different notations given in Eqs. (2.29) and (2.30) to indicate which convention is used. Asymptotic normalization constants are interesting quantities because for shallow bound states (i.e., states with small binding energy/momentum) they are closely related to scattering processes. For example, they are directly connected to zero-energy capture reactions [22], which play an important role in nuclear astrophysics. In Chapter 5 (Sec- tion 5.7) we will discuss how the ANC can be expressed directly in terms of the parameters that appear in the effective range expansion (2.19). 2.2 Some formal scattering theory In the preceding sections, we have established that for finite-range interactions the asymp- totic radial wave functions of two-particle systems have an explicitly-known analytic form. Since low-energy physics is governed by large-distance scales, this fact will be very useful in the following chapters to derive relations based on just the properties of the wave-function tails. Since the focus so far was on consequences of the finite-range assumption, we have worked directly with radial Schro¨dinger equations and wave functions in configuration space, and only introduced scattering concepts like the phase shift and the S-matrix as they appear in that context. In the following, we discuss quantum-mechanical scattering theory from a more formal point of view, establishing the connection with the relations given in Section 2.1 along the way and focussing on results that will be relevant later in this work. Unless otherwise indicated with an explicit citation, what follows is mostly taken from Sakurai’s textbook [23], but uses slightly different conventions in some places. Until further notice, we also lift the finite-range assumption on the interaction. 2.2.1 The Lippmann–Schwinger equation In its abstract operator form, the time-independent Schro¨dinger equation (2.1) reads Hˆ|ψ〉 = E|ψ〉 (2.31) with the Hamilton operator Hˆ = Hˆ0 + Vˆ given by 〈r|Hˆ|ψ〉 = −∆r 2µ ψ(r) + ∫ d3r′ V (r, r′)ψ(r′) (2.32) 10 Chapter 2. General concepts in configuration space, where we have allowed the interaction to be non-local. What we have ignored so far is that the Schro¨dinger equation alone does not specify the boundary condition for the solutions. In the time-independent scattering formalism we are consid- ering here, one is primarily interested in solutions that reduce to a plane-wave state |p〉 when the interaction is “switched off,” Vˆ ≡ 0. This behavior can be enforced by making the ansatz |ψ(+)p 〉 = |p〉+ (Ep − Hˆ0 + iε)−1Vˆ |ψ(+)p〉 , Ep = p2 2µ , (2.33) where the small imaginary part iε has been introduced to make the otherwise singular operator E − Hˆ0 invertible. More precisely, the zero modes of E − Hˆ0 are given by the plane-wave states |p〉, (Ep − Hˆ0)|p〉 = 0 for all |p〉 . (2.34) Acting on both sides of Eq. (2.33) with E − Hˆ0 + iε and sending ε → 0 (which in the following is always implicitly understood to be done at the end of all manipulations) gives back the Schro¨dinger equation in the form (3.3). Equation (2.33) is called the Lippmann-Schwinger equation for the scattering states. It can be interpreted as an integral formulation of the Schro¨dinger equation with the bound- ary conditions determined by the inhomogeneous term |p〉 and the iε-prescription. We have chosen a positive sign for the imaginary part and indicated this by writing |ψ(+)p 〉 for the scattering state. This gives the physically most relevant solution that corresponds to an incoming plane wave and an outgoing scattered wave when one goes over to a time-dependent framework. By choosing a configuration-space basis {|r〉 : r ∈ R3} we get the ordinary wave functions ψ(+)p (r) = 〈r|ψ(+)p 〉 , (2.35) and the plane-wave states are just 〈r|p〉 = exp(ip·r). In general, the wave function ψ(+)p (r) is not characterized by a single pair of angular-momentum quantum numbers (`,m) but rather has an expansion in spherical harmonics or Legendre polynomials [24], ψ(+)p (r) = ( 2µ pip )1/2 ∞∑ `=0 ∑` m=−` Y m` (rˆ)Y m∗ ` (pˆ) u`(r) r (2.36) = ( 2µ pip )1/2 1 4pi ∞∑ `=0 (2`+ 1)i`P`(cos θ) u`(r) r , cos θ = pˆ · rˆ , (2.37) with the unit vectors pˆ = p/p and rˆ = r/r. The wave functions u`(r) are then solutions of the radial Schro¨dinger equation (5.2), and the normalization in Eq. (2.37) is chosen such that the latter can be rewritten as an integral equation with the inhomogeneous term given by the Riccati–Bessel function S`(pr). 2.2.2 The T-matrix Defining the Green’s function operator (the free resolvent) Gˆ (+) 0 (E) = (E − Hˆ0 + iε)−1 , (2.38) 2.2. Some formal scattering theory 11 we can write the Lippmann-Schwinger equation (2.33) as |ψ(+)p 〉 = |p〉+ Gˆ(+)0 (Ep) Vˆ |ψ(+)p 〉 . (2.39) For the free Hamiltonian defined implicitly in Eq. (2.32), the Green’s function in config- uration space is given by the expression G (+) 0 (E; r, r ′) = 〈r|Gˆ(+)0 (E)|r′〉 = − µ 2pi eik|r−r ′| |r− r′| , (2.40) where k is a momentum scale defined as k = √ 2µ(E + iε). It satisfies the equation 1 2µ (∆r + k 2)G (+) 0 (E; r, r ′) = δ(3)(r− r′) . (2.41) Note that other conventions exist in the literature where the factor 1/(2µ) is absorbed into the definition of G (+) 0 . Our choice here has the advantage of staying close to the operator notation, such that from Eq. (2.38) one can directly read off the momentum- space expression G (+) 0 (E; q,q ′) = 〈q|Gˆ(+)0 (E)|q′〉 = (2pi)3δ(3)(q− q′) E − q2/(2µ) + iε . (2.42) Introducing now the operator Tˆ , which in the following we simply refer to as the T-matrix, via the implicit definition Vˆ |ψ(+)p 〉 = Tˆ |p〉 , (2.43) we obtain the formal solution |ψ(+)p 〉 = ( 1 + Gˆ (+) 0 (Ep) Tˆ ) |p〉 (2.44) for the scattering state. The original problem of finding the solution for the scattering wave function is thus modified to the question of determining the T-matrix. In order to proceed in this direction, one can act with the potential operator Vˆ on both sides of Eq. (2.44) from the left and then use Eq. (2.43) once again to find Tˆ |p〉 = ( Vˆ + Vˆ Gˆ (+) 0 (Ep) Tˆ ) |p〉 . (2.45) Demanding further that this holds for all states {|p〉 : p ∈ R3} yields an operator equation for Tˆ . In fact, at this point one can furthermore lift the restriction that the Green’s function operator is evaluated at the on-shell energy E = Ep but rather allow this value to be an arbitrary complex parameter. The resulting equation reads Tˆ (E) = Vˆ + Vˆ Gˆ (+) 0 (E) Tˆ (E) (2.46) and is called the Lippmann–Schwinger equation for the T-matrix. The momentum-space representation T (E; q,p) = 〈q|Tˆ (E)|p〉 (2.47) 12 Chapter 2. General concepts with no imposed connection between the variables E, p, and q is commonly referred to as the full off-shell T-matrix, whereas the quantity T (q,p) = 〈q|Tˆ (E = Ep)|p〉 (2.48) is called the half off-shell (or simply half-shell) T-matrix. It has the useful property of being directly related to the scattering wave function in momentum space via ψ(+)p (q) = 〈q|ψ(+)p 〉 = (2pi)3δ(3)(q− p) + 2µT (q,p) p2 − q2 + iε , (2.49) according to Eq. (2.44). The scattering amplitude From the Green’s function in configuration space, Eq. (2.40), one can deduce that at asymptotically large distances the scattering wave function ψ (+) p (r) can be written as the sum a plane wave (given by the state |p〉 and interpreted as the originally incoming particle flux) and an outgoing spherical wave describing the effect of the scattering process, ψ(+)p (r) r→∞∼ eip·r + e ipr r f(p′,p) , (2.50) where p′ = prˆ and f(p′,p) = − µ 2pi 〈p′|Vˆ |ψ(+)p 〉 = − µ 2pi 〈p′|Tˆ |p〉 (2.51) is the scattering amplitude. From the definitions above it is clear that f(p′,p) only depends on the magnitude p of p and the angle θ between p and r. It thus has an expansion in partial waves, f(p′,p) = ∞∑ `=0 (2`+ 1)f`(p)P`(cos θ) , (2.52) and it is precisely the partial-wave amplitudes f`(p) appearing in Eq. (2.52) that were already mentioned in Section 2.1.5. Assuming that the potential can be expanded in Legendre polynomials (cf. Section 2.1.1), an analogous expansion also exists for the T- matrix, T (E; q,p) = ∞∑ `=0 (2`+ 1)T`(E; q, p)P`(cos θ) , (2.53) where θ is now the angle between the momentum vectors q and p. It then follows that the partial-wave scattering amplitude—and thus also the scattering phase shift δ`(p)—is determined by the partial-wave T-matrix at the on-shell point: f`(p) = e2iδ`(p) − 1 2ip = − µ 2pi T`(Ep; p, p) . (2.54) 2.3. Universality 13 2.3 Universality We now turn back our attention to short- and finite-range interactions. As shown in Section 2.1, the asymptotic form of the radial wave function for a given two-particle scattering system is known analytically and can be parametrized in terms of the scattering phase shift δ`(p). The relations for the wave functions in Eqs. (2.15) and (2.29) hold rigorously for pure finite-range potentials and still up to exponentially small (as r →∞) corrections if the interaction is short-ranged. In the latter situation, the interaction range is limited by the typical fall-off scale of the potential. In either case, the effective-range expansion (2.19) furthermore provides a method to express the phase shifts in terms of just a few low-energy parameters. In many situations, already the scattering length and effective range parameter a` and r` are sufficient for a reasonably accurate description of the experimental data. The physics behind all this is that at sufficiently low energies (and thus small scattering momenta p) the details of the interaction cannot be resolved because the de Broglie wavelength λp ∝ 1/p corresponding to the incoming particle flux is too large compared to the spatial extent of the scattering center. More quantitatively, the criterion for not resolving the details of an interaction with range R is p R−1 ⇐⇒ pR 1 . (2.55) This simple principle is very powerful because it means that—provided the underlying interaction has a finite (or short) range—low-energy quantum scattering can be described in a universal way by the parameters appearing in the effective range expansion. 2.3.1 Hierarchy of partial waves In the formal discussion so far we have mostly considered some fixed but arbitrary angular momentum `. The really important quantity, however, is the scattering amplitude f(p′,p) defined in Section 2.2.2 since it is directly related to the differential cross section that is determined in experiments, dσ dΩ = |f(p′,p)|2 . (2.56) This means that all physically significant information about the process is contained in the scattering amplitude. Combining Eqs. (2.52) and (2.54), we can express the scattering amplitude in terms of the phase shifts δ`(p) as f(p′,p) = ∞∑ `=0 (2`+ 1) e2iδ`(p) − 1 2ip P`(cos θ) . (2.57) If all terms in this sum were equally important, the universal parametrization mentioned above would not be very useful because describing the physical system would still require an infinite number of parameters. Fortunately, the phase shifts themselves have the low- 14 Chapter 2. General concepts energy behavior4 δ`(p) ∝ p2`+1 , (2.58) which means that at low energies only a few partial waves are important. Usually, the dominant term is the S-wave (` = 0), but this is not necessarily always the case.5 The physical origin behind the behavior (2.58) is of course the centrifugal barrier ∼ `(`+1)/r2 that the scattering particles cannot penetrate appreciably at low energies. 2.3.2 Universality for large scattering length A particularly interesting situation occurs if the S-wave scattering length of a system is unnaturally large. By “unnaturally large” we mean in this context that it is much larger than the range R of the potential, a0 R, because na¨ıvely one would expect that all length scales in the system are of the “natural” order of magnitude. Usually, one does of course not know the exact range R of the underlying interaction, nor is the notion of a strict finite-range interaction a very realistic picture. In general, if the interaction is assumed to be mediated by some kind of particle, the corresponding potential has an exponential (Yukawa) tail, and the inverse mass of the exchange particle provides a good estimate for the interaction range. In low-energy nuclear physics, for example, the typical length scale is set by the inverse pion mass, m−1pi ≈ 1.4 fm. Typically, the S-wave effective range is found to be of the order of magnitude estimated for the underlying interaction, such that, in the absence of more direct information, a0 r0 can be used as a criterion for asserting that the scattering length is unnaturally large. Since the total cross section for two-particle scattering at zero energy is given by6 σtot(0) = 4pia 2 0 , (2.59) a large S-wave scattering length means that the particles interact strongly. In fact, one can say that the low-energy physics of such a system is completely governed by the large S-wave scattering length. For example, if the interaction supports a two-particle bound state (simply called a dimer in the following) at p = iκ, the combination of Eqs. (2.21) and (2.19) tells us that κ ≈ 1 a0 + r0 2a20 = 1 a0 ( 1 +O(a0/r0) ) , (2.60) where the correction is negligible if r0 a0. This means that at leading order the dimer binding energy is just Ed = 1 2µa20 . (2.61) 4This can be seen, for example, from the effective range expansion (2.19). Since cot(x) = 1 +O(x), one has p2`+1/δ`(p) ≈ const. as p→ 0. 5For example, in the scattering of two identical fermions the Pauli principle only allows odd `, such that the leading low-energy contribution is given by the P-wave (` = 1). 6This follows from the discussion in the preceding sections by noting that limp→0 f0(p) = −a0. 2.3. Universality 15 Whether or not a bound state actually exists can furthermore be related to the sign of the scattering length. In the conventions that we are using here, a large positive scattering length implies the existence of a shallow dimer state.7 The Efimov effect A very intriguing phenomenon occurs in the three-body sector of a system of particles that have a large two-body scattering length and in that sense a strong pairwise interaction. It can be shown that the bound-state spectrum of such a system exhibits a tower of (approximately) geometrically-spaced three-body “trimer” states, i.e., a series of bound- state energies fulfilling E (n+1) trimer /E (n) trimer ≈ const. . (2.62) In the limit where the magnitude of a0 goes to infinity, 8 so does the number of trimer states, and the geometrical spacing becomes exact with a universal scaling factor determined only by the mass ratio of the particles. This effect was first proposed by (and named after) V. Efimov [25] and later proven by Amado and Nobel [26, 27]. For a detailed discussion of quantum systems with a large scattering length and the Efimov effect, we refer here to the review by Braaten and Hammer [28]. To conclude this subsection, we summarize that we speak of low-energy universality in the general sense whenever a description with finite-range interaction is a good approximation to describe the physics of a given (two-particle) system. In a manner of speaking, physics at low energies is governed by the tails of the (radial) wave functions, which have a universal analytic form, and the essential features of the system can be well described by just a few parameters, namely the scattering lengths and effective ranges for a small number of partial waves. Since the effective range expansion is valid in the complex p- plane, this includes the description of bound states with small binding energies. Moreover, if the (S-wave) scattering length is unnaturally large, this parameter governs the whole system and one finds relations that are “even more universal,” like the binding energies of shallow dimer states or the Efimov effect in the three-body sector. Finally, it is important to point out again that the above statements still hold (up to corrections that are often negligible) in the more realistic setup where the interaction has no strictly finite range but is short-ranged and falls off rapidly. 2.3.3 The theorist’s point of view Having read the preceding sections, the caption above this sentence might seem confusing at first. By no means do we want to say that we now switch gears to look at physics from the theoretical side, but rather that this is what we have been doing all along so far. Here, we want to emphasize that finite-range potentials are not an experimental concept but, since potentials are even not observable quantities, really just a convenient theoretical 7For a derivation of this statement see, for example, Ref. [23]. 8More precisely, the relevant limit is |a0/R| → ∞, which means that the effect can also be found for zero-range interactions. 16 Chapter 2. General concepts tool. The same should be said in fact, at least to a good extent, about partial-wave scattering phase shifts. What is directly accessible in scattering experiments (via count rates and their angular distribution) are (differential) cross sections and thus, according to Eq. (2.56), the absolute value squared of the scattering amplitude. Phase shifts are then only obtained from an inversion of Eq. (2.52), necessarily truncated at some ` and based on data in only a limited energy (and usually angular) regime. This procedure can be very delicate in practice and that the result, due to the limited amount of data, is not truly unique. Potentials are still a very useful tool for the theoretical description of physical processes, and scattering phase shifts provide a convenient “interface” to compare the results of cal- culations with experiments, but it is important to keep in mind the limitations described above. 2.3.4 From potential models to a modern perspective Although low-energy universality arises naturally in the the theory of finite- or short- range interactions, it was first found as an experimental phenomenon. It was observed9 that low-energy S-wave scattering phase shifts in the two-nucleon system could be well described with just two parameters since the data points could be fitted by a straight line in a suitable representation – the left-hand side of the effective range expansion (2.19) or its Coulomb-modified analog that will be discussed in Chapter 4. Different forms of potentials could thus be used equally well to model the system as long as they had two parameters that could be adjusted to reproduce that line. At the time, it was concluded that measurements at higher energies were needed to really determine the shape of the nuclear potentials. This led, subsequently to the construction of very sophisticated potential models, many of which, like Nijmegen I,II [29], AV18 [30], and CD-Bonn [31], are still well known and often used today (a comparison of the different models and an historical overview of their development can be found, for example, in Ref. [32]). All of these potentials describe the nucleon–nucleon phase shifts (and deuteron properties) very well, but differ quite substantially in their details. Of course, this ambiguity simply illustrates explicitly the fact that potentials are not observable. This, together with the difficulty of these approaches to consistently describe and/or implement the physics of more than two particles (see, for example, Ref. [33]), as well as the essentially unsystem- atic way they are constructed, has led to the development of a more modern perspective. Rather than continuing the ultimately futile endeavor of trying to find the nuclear poten- tial, one simply constructs so-called effective potentials as a systematic expansion, where new operators are added to describe physics at subsequently higher energy scales (and/or of an increasing number of nucleons). This approach explicitly incorporates the observed low-energy universality, or can, in fact, be characterized as being based on it. More gen- erally, the underlying concept is that of effective field theory (EFT), which we now turn to discuss in some more detail. 9See, for example the review of proton–proton scattering by Jackson and Blatt [11] and original references therein. 2.4. Effective field theory 17 2.4 Effective field theory The main concepts of effective field theories can be summarized as follows: • First, choose degrees of freedom that are appropriate for the energy scale under consideration (nucleons instead of quarks, for example). • Then, construct a Lagrangian out of fields corresponding to these degrees of free- dom, including all terms allowed by general principles like unitarity, analyticity and whatever other symmetries relevant for the system at hand (like isospin, in the example above). • Finally, since such a Lagrangian contains in principle infinitely many terms, a further key ingredient is an ordering scheme (“power counting”) that serves to estimate the relative importance of the individual terms. The connection to low-energy universality is that the ordering in the final step is often (but not necessarily always or exclusively) based on the relative momenta in the system one wishes to describe. We will illustrate this in the following by looking at a few explicit examples. First, it is important to point out that the coefficients of operators in a Lagrangian constructed in the way laid out above are not normally dictated by general concepts. Instead, these so-called low-energy constants have to be determined by matching the results of calculations to known observables.10 The predictive power of the theory then lies in the fact that usually the same coefficients (or combinations thereof) appear in different observables. Having measured some of them, it is possible to make predictions for others. 2.4.1 Historical overview The approach laid out above was pioneered by Weinberg in Ref. [34], where (in the context of pion physics) he formulated the idea that a quantum field theory has, ultimately, no other content than analyticity, unitarity, cluster decomposition, and symmetry. Such “phenomenological Lagrangians” had been used before, but were usually based on current- algebra concepts (see, e.g., Ref. [35]). The big step that Weinberg took was to promote them to the starting point of calculations, thereby abandoning their unsystematic heritage. A key role in Weinberg’s original application to hadron physics at low energies is played by the approximate chiral SU(2)L × SU(2)R symmetry of the strong interaction, the spontaneous breaking of which is responsible for the pion mass being so small.11 This mass and the momenta of low-energy pions are used as small scales in the power counting. 10Alternatively, if an underlying more fundamental theory is known, they can sometimes also be calculated from that. 11If there was no explicit chiral-symmetry breaking, they would be massless Goldstone bosons. 18 Chapter 2. General concepts Weinberg’s approach, which became famous as chiral perturbation theory, was worked out further by Gasser and Leutwyler [36, 37] and has a vast number of applications and exten- sions today. In particular, the formalism has been extended to include also nucleons and other baryons (see, for example, Refs. [38–40]). This important step made it possible to construct the effective nuclear potentials mentioned at the end of Section 2.3.4. In an idea that again goes back to Weinberg [41, 42], a low-energy expansion of the effective interac- tion between nucleons is built out of diagrams derived from the chiral Lagrangian. This means that rather than doing plain perturbation theory by summing Feynman diagrams up to a given order, the power counting is applied to derive an effective potential, which can subsequently used in calculations based on Schro¨dinger (or Lippmann–Schwinger) equations. Of course, with the above summary we have only scratched the surface of a very complex field. Much more detailed discussions of the points mentioned here (and many more) can be found in the reviews by Epelbaum et al. [43], or Machleidt and Entem [44], which we just refer to here for simplicity. Furthermore, it is important to point out that although the origins of effective field theory lie in hadron and nuclear physics, it is by no means limited to those applications. In fact, it is used in many areas of modern theoretical physics. Since the key ingredient is merely a separation of scales and not that the total energy of the system one wishes to describe is a small scale, there are even applications in high-energy physics, like the soft-collinear effective theory started in Refs. [45, 46]. We do not make here the futile attempt to give a comprehensive list of current EFT applications. Rather, we focus on two examples that will play a role later in this work. 2.4.2 Pionless effective field theory At very low energies in nuclear physics, even the pions can be “integrated out,” which then leaves only nucleons as effective degrees of freedom. The interactions between them are simple contact terms, corresponding to delta-peak potentials and derivatives thereof in configuration space, and since in the low-energy regime all relative momenta are small, a nonrelativistic description is appropriate. The latter means that all particles only prop- agate forward in time and that there is no pair creation. In fact, this kind of EFT can be thought of as a convenient reformulation of quantum mechanics. Naturally, this pionless effective field theory is limited to energy (momentum) scales below the pion mass, but nevertheless it is very interesting and exhibits a rich set of features. The reason for this is that the S-wave nucleon–nucleon system is an important example for the case of unnaturally large scattering lengths. Both ad ≈ 5.42 fm in the 3S1 (isospin 0) and at ≈ −23.71 fm in the 1S0 (isospin 1) channel are significantly larger than the typical scale of about 1.4 fm set by the inverse pion mass (cf. Section 2.3.2). The corresponding effective ranges, on the other hand, have the values 1.75 and 2.73 fm, respectively, and are thus indeed of the expected natural order of magnitude.12 12The numbers quoted here for the scattering lengths and effective ranges are quoted from Refs. [16] and [47]. 2.4. Effective field theory 19 The deuteron appears in this picture as a nearly universal shallow dimer state correspond- ing to the large positive scattering length ad. From Eq. (2.61) one obtains Ed ≈ 1.4 MeV as leading-order result for its binding energy, which is not far from the experimental value Eexpd = 2.225 MeV [48]. The agreement here is rather coarse because the effective range in this channel is only about a factor of three smaller than the scattering length, which makes the range corrections to the leading-order expression quite significant. Indeed, if Eq. (2.60) is used to calculate the binding momentum including the effective-range term, the result is κ ≈ 45.6 MeV, corresponding to a binding energy of about 2.21 MeV, much closer to the experimental value. The presence of the shallow deuteron state has a significant impact on the construction of pionless effective field theory. Since perturbation theory cannot produce bound states, the applicability of the approach would na¨ıvely be limited to momentum scales Q below the deuteron pole, Q < κd ∼ 1/ad. If one also takes into account the pole corresponding the virtual bound state in the 1S0 channel, where the scattering length is larger yet, the range of applicability is even narrowed down to Q < 1/at. In either case, this is much smaller than the natural breakdown scale set by the pion mass. The solution, introduced in Refs. [49–52], is to include certain contributions up to all orders in the perturbative expansion and thus generate the shallow states corresponding to the large scattering lengths in a nonperturbative manner. The theory with this scheme applied and its low- energy constants fixed by matching two-body amplitudes to the effective range expansion is then valid for low-energy scales of the order Q ∼ 1/ad,t and with the natural breakdown scale Λ ∼ m−1pi , corresponding to an EFT expansion parameter Q/Λ ∼ 1/3.13 In Refs. [53– 55], the formalism has been extended to the three-nucleon sector. The situation there is particularly interesting because the triton can be interpreted as an approximate Efimov state. Since the physics it describes are to a significant extent governed by the large S-wave scattering lengths, pionless effective field theory has a lot in common with a simpler EFT that describes identical bosons with a large two-body scattering length. The key features mentioned above (nonperturbative resummation to reproduce shallow dimer bound states and the Efimov effect in the three-body sector) can be studied there without complications due to different spin and isospin channels. A comprehensive review of this EFT and a broader discussion of universality in systems with large scattering length can be found in the review by Braaten and Hammer [28]. For applications to cold-atomic systems with large scattering lengths, see also K. Helfrich’s doctoral thesis [56] and further references therein. In Chapter 6, we will discuss pionless effective theory in more detail and use it to analyze the proton–deuteron system. 2.4.3 Effective field theory for nuclear halo states Almost the same formalism as for the pionless EFT discussed above can also be used to construct an effective field theory that is useful to calculate properties of nuclear halo systems. Such states, also called halo nuclei, can be thought of as a tightly bound core 13Note that if the size of corrections is estimated directly in terms of the effective range parameters, one gets the same result: 1.75/5.42 ≈ 0.32. 20 Chapter 2. General concepts nucleus with one or more weakly-bound valence nucleons (for reviews of such states see, for example, Refs. [57, 58]). The separation of scales, which is the crucial ingredient for the construction of an EFT, is given for such systems by the small separation energy of the valence nucleons compared to the binding energy of the core.14 In an effective two-body picture that neglects the internal structure of the core (which cannot be resolved at low energies), a one-nucleon halo nucleus can be thought of as a shallow dimer state that occurs due to a large scattering length in the corresponding nucleon–nucleus scattering system. This is exactly the same situation as with the deuteron in the few-nucleon sector, and it is thus natural to adopt the concepts of the pionless effective field theory described in the previous section. The resulting effective Lagrangian contains, in addition to the nucleon terms, an additional field to describe the core as a whole, interacting with the nucleons via contact terms and/or derivatives thereof. This so-called halo EFT was first introduced in Refs. [60, 61] to study neutron–alpha and has since then been extended to describe a number of other phenomena like alpha–alpha scattering [62], bound single-neutron halo states such as 11Be [59] and 8Li [63, 64], and various two-neutron halo systems [65–67]. Recently, is has also been used to calculate charge form factors of two-neutron halo nuclei [68]. 2.5 Lattice calculations In the preceding sections we have repeatedly touched the subject of nuclear physics, but almost exclusively discussed it directly from an effective point of view. It is thus due time to mention quantum chromodynamics (QCD) as the widely accepted underlying15 theory of the strong interaction which, up to electromagnetic and weak effects, ultimately governs the properties of nucleons and nuclei.16 It is defined by the Lagrangian LQCD = ∑ f ψ¯f (iDµγ µ −mf )ψf − 1 4 ∑ a F aµνF µν,a , Dµ = ∂µ + igA a µt a , F aµν = ∂µA a ν − ∂νAaµ + igfabcAbµAcν , (2.63) which is an essential component of the Standard Model of particle physics. In Eq. (2.63), the sum runs over all quark flavors f represented by the Dirac fermion field ψf and A a µ(x) are the gluon gauge fields. Furthermore, ta and fabc are the SU(3) group generators and structure constants, and a, b, c = 1, . . . , 8 are the corresponding color indices. In the limit of vanishing quark masses, mf = 0 for all f , LQCD exhibits an exact chiral symmetry, which means that right and left-handed quark fields ψf,R/L = (1 ± γ5)/2ψf decouple. If one only considers two light quark flavors f = u, d, which is usually sufficient 14Alternatively, as it is done for example in Ref. [59], one can compare the large matter radius of the halo nucleus as a whole to the small radius of the core alone. 15We write “underlying” here to avoid the question of whether QCD can also be regarded as a funda- mental theory. It is of course possible, if not likely, that QCD and the standard model in general are also just effective theories of some sort, and the author takes the stance that it is not even logically possible to assert any given theory as truly “fundamental.” 16Note that in nuclear physics electromagnetic effects are not a small correction, but play an important role for the description of all nuclei heavier than the deuteron. 2.5. Lattice calculations 21 for nuclear-physics applications, this is broken down spontaneously to the approximate isospin symmetry (exact in the limit mu = md) that is still visible in the hadronic spec- trum. We assume all this to be well-known and will not go into further detail here. The only aspect really important for the present discussion is that the running of the strong coupling constant αs = g 2/(4pi) that makes QCD perturbative at high energy scales and leads to the famous “asymptotic freedom” at the same time renders the theory strongly-interacting and thus non-perturbative at low energies. This is directly reflected by the fact that quarks and gluons are not physically observable degrees of freedom, but only exist confined into hadrons. One way to circumvent the breakdown of perturbation theory are the effective field theo- ries introduced in Section 2.4. Yet, as powerful as they are, it is nevertheless desirable to also deduce the properties of hadrons, their interactions, and ultimately those of nuclei, directly from LQCD. A way to achieve this is to “solve” the theory numerically by putting it on a discretized space-time lattice. 2.5.1 Lattice QCD The idea just mentioned was initially conceived by Wilson [69] in 1974 as an attempt to explain confinement. Although the latter is still an unsolved problem, with the rapid advance of computer power during the last two decades, the lattice formulation of QCD has evolved into a very successful tool. In the following few paragraphs, we will briefly review its main concepts. Much more thorough discussions of the topic can be found in introductory lattice QCD texts such as Refs. [70, 71]. As Davies puts it in Ref. [70], “Lattice QCD is just QCD, no more and no less.” Although in practice there are a number of technical issues—in particular with the implementation of fermions and chiral symmetry—this statement provides a good summary of the approach. Its starting point is the Feynman path-integral formulation of field theory, which encodes all physical information in the partition function Z = ∫ DψDψ¯DAaµ eiSQCD[ψ,ψ¯,A a µ] . (2.64) Here, SQCD = ∫ d4xLQCD (2.65) denotes the QCD action and we have omitted all flavor indices f for simplicity. Phys- ical observables can be calculated by considering vacuum expectation values of suitable operators O given by 〈O〉 = 1 Z ∫ DψDψ¯DAaµO[ψ, ψ¯, Aaµ] eiSQCD[ψ,ψ¯,A a µ] . (2.66) Under a Wick rotation that yields a description in Euclidean spacetime, the exponential term in the above expression becomes eiSQCD[ψ,ψ¯,A a µ] −→ e−SEQCD[ψ,ψ¯,Aaµ] , (2.67) 22 Chapter 2. General concepts where the superscript E indicates the Euclidean action. With this, complex phase os- cillations in Eqs. (2.64) and (2.66) go over into exponential suppression factors of terms away from the classical (i.e., minimal) action. The resulting expressions are then nu- merically well-behaved and one can calculate the functional integrals over the symbolic measure DψDψ¯DAaµ by sampling all fields on a discrete space-time mesh, giving an ordi- nary four-dimensional integral at each point.17 This integral is, of course, a tremendously high-dimensional one and cannot be performed with straightforward numerical quadra- ture rules. Instead, one has to resort to Monte Carlo methods to make the calculation feasible at all. Even with that, the procedure in general still requires supercomputing power, and in practice one furthermore has to cope with a number of problems related to the numerical treatment, some of which we will come back to shortly in Section 2.5.3. Still, over the last few years lattice QCD has been very successful in calculating hadronic properties “from first principles.” With steady improvements in both algorithms and computational facilities, hadron spectroscopy, as one of lattice QCD’s prime disciplines, is moving towards precision calculations [72, 73]. Beyond that, it is also possible to extract resonance properties [74] and pion scattering parameters [75–77], to name just a few examples. There are furthermore promising efforts to calculate nuclear physics processes directly with lattice QCD (see, for example, Ref. [78] and further references therein). Despite the successes just mentioned, lattice QCD is still far from replacing effective field theory in low-energy hadron and nuclear physics. Instead, the two methods are to a good extent complementary. For example, due to computational limitations lattice calculations are usually performed at unphysically large quark masses, and results from chiral perturbation theory are then needed to perform extrapolations of the results back to the point of physical quark masses. On the other hand, lattice QCD can be used to calculate low-energy constants of chiral perturbation theory that would otherwise have to be fixed from experiments [79]. In that sense, it is possible to “close the gap” between the EFT and the underlying theory. 2.5.2 Nuclear lattice simulations One can go even further and adopt the lattice approach directly to perform EFT calcula- tions. This rather new idea opens the door for efficient calculations of few- and many-body systems, ranging from nuclear physics to condensed matter and atomic physics (overviews of computational methods and applications can be found in Refs. [80, 81]). For nuclear physics, the EFT approach is interesting because in lattice QCD it is still very challenging to extract the properties of systems with more than two nucleons. The reason for this is that in such calculations most of the computational effort has to be spent for generating the correct degrees of freedom rather than the interactions between them. An overview of nuclear lattice simulations based on chiral effective theory and applied to selected nuclei with mass number up to A = 12 can be found in Refs. [82, 83]. Unlike 17In practice, the integral over the fermion fields is typically performed analytically with the help of Grassmann variables, yielding a modified action for the gluon fields that involves the determinant of a large matrix. 2.5. Lattice calculations 23 other nuclear many-body approaches like Green’s Function Monte Carlo (GFMC) [84] or the No-Core Shell Model (NCSM) [85], the nuclear lattice approach is particularly suited to study nuclei with a pronounced cluster substructure, such as the famous Hoyle state in carbon-12 [86, 87]. 2.5.3 Numerical aspects All methods mentioned in the preceding sections have in common that by putting the physical system on a discrete space-time mesh one necessarily introduces numerical ap- proximations. Inherent to the approach is that the degrees of freedom no longer reside in a continuous space and time, but only on fixed sites18 which are separated by a lattice spacing a (we focus on the spatial separation here and for simplicity ignore that one can choose a different lattice spacing in the time direction). Most prominently, this quantity enters in the definition of derivatives as finite differences on the lattice, which only gives the continuum result up to higher orders in a. In practice, a has thus to be kept small to avoid large discretization artifacts, and ideally calculations have to be performed at a number of different lattice spacings in order to extrapolate the results to their continuum values. Furthermore, since computing power is a finite resource, so is the space and time that can be simulated in a lattice calculation. Typically, one chooses a cube of box length L with periodic boundary conditions for the spatial simulation volume, which is used to sample the path integral for a number of time steps Lt. Naturally, the volume must not be too small compared to the typical length scale of the system one wants to simulate. Both, the need to increase the simulation volume and decrease the lattice spacing drive up the computer time (and memory) required for a given calculation. However, while the continuum extrapolation is something that simply has to be done, the volume dependence of physical observables can actually be used as a tool. This idea, pioneered by Lu¨scher in the 1980s [8, 88], will play an important role in the following chapter. 18Strictly speaking, gauge fields like the gluons in lattice QCD are actually defined not on the lattice sites but on the links between them. 24 Chapter 2. General concepts Chapter 3 Finite-volume calculations Overview In this chapter we derive general results for the mass (binding energy) shift of bound states with angular momentum ` ≥ 1 in a periodic finite volume. Most of the following content is the same as published in Ref. [4], some results of which were first summarized in a letter [2]. Section 3.7 summarizes results from Ref. [3] after giving a detailed derivation of the present author’s main contribution to that publication. 3.1 Introduction As already mentioned at the end of the previous chapter, lattice simulations are used in many areas of quantum physics, ranging from nuclear and particle physics to atomic and condensed matter physics [78, 80, 81, 89]. In such calculations, the system is solved numerically using a discrete space-time lattice over a finite volume. In practice, this finite volume is usually taken to be a cubic box with periodic boundary conditions. When simulating composite objects such as bound states, these boundaries of the box modify quantum wave functions, leading to finite-volume shifts in the binding energies of the states. A detailed knowledge of such effects is necessary in order to improve high-precision lattice calculations. In Ref. [8], Lu¨scher derived a formula for the finite-volume mass shift of S-wave bound states of two particles with reduced mass µ interacting via a potential with finite range R. When such a state with energy E = −EB is put in a periodic cubic box of length L, its mass (energy) in the rest frame1 is shifted by an amount ∆mB = −3|Aκ|2 e −κL µL +O(e−√2κL) , (3.1) where κ = √ 2µEB is the binding momentum and Aκ is the asymptotic wave function normalization defined by ψB(r) = Aκ e −κr/( √ 4pir) for r > R; cf. Section 2.1.5. For 1Lu¨scher uses the term “mass shift” because he was more interested in a relativistic setup. We adopt this convention here, but sometimes also use the term “binding energy shift” synonymously. 25 26 Chapter 3. Finite-volume calculations potentials with exponential fall-off, V (r) ∼ exp(−r/R) for large r, the formula is modified by exponentially small corrections provided that the binding momentum κ is smaller than 1/R. The generalization of Lu¨scher’s formula (3.1) for the finite-volume mass shift to bound states in higher partial waves was briefly discussed in Ref. [2]. In this chapter we present the full derivation of these results as it appeared in Ref. [4]. We give explicit results for the mass shift of states with angular momentum up to ` = 3 and discuss how, in general, the mass shift for a given state depends on its transformation properties with respect to the symmetry group of the cubic box. In addition to reducing finite-volume effects in precision lattice calculations, our results can also be used as a diagnostic tool to probe the angular momentum and radial structure of the bound-state wave function. Furthermore, we discuss how the mass-shift formula can be generalized to two-dimensional systems and different (“twisted“) boundary conditions (in three dimensions). The latter result is a key ingredient for studying bound states that are moving in a finite periodic volume, which have a topological phase correction to the energy [3, 90]. This factor contains information about the number and mass of the constituents of the bound states, and it must be included when determining scattering phase shifts for composite objects in a finite volume. For a discussion of how scattering phase shifts in S- and higher partial waves can be extracted from finite-volume energy levels, see Refs. [88, 91]. Our results are universal and can be applied to a wide range of systems. In particle physics, for example, there is some interest in hadronic molecules with angular momentum [92–94]. In the case of S-waves, the deuteron and some exotic weakly-bound states such as the H-dibaryon were recently studied in lattice QCD [95]. Similar investigations for exotic bound states with angular momentum appear feasible in the future. In atomic physics, several experiments have investigated strongly-interacting P-wave Feshbach resonances in 6Li and 40K [96–98], which can be tuned to produce bound P-wave dimers. If such systems are simulated in a finite volume, our results can be used to describe the volume dependence of the dimer binding energies. Other systems that are relevant in this context are the halo nuclei introduced in Sec- tion 2.4.3. Among these weakly-bound nuclei with molecular character there are some systems with nonzero orbital angular momentum. A well-known example of a P-wave halo state is the JP = 1/2− excited state in 11Be. The electromagnetic properties of the low-lying states in this nucleus can be well described in a two-body halo picture of a 10Be core and a neutron [59, 99]. In order to study such a system in, for example, a nuclear lattice simulation as discussed in Section 2.5.2, it is crucial to understand the volume dependence. A related class of systems is given by nuclei with an α-cluster structure such as 8Be and excited states of 12C [86, 100, 101]. Finally, we point out that the asymptotic normalization coefficient (ANC) of the bound- state wave function appears in the mass-shift formula. Our results can hence be used to extract this quantity from lattice calculations at finite volumes. The chapter is organized as follows. Based largely on the prerequisites given in Chapter 2, we start with a general discussion of the finite-volume mass shift in Section 3.2. Lu¨scher’s result for S-waves is recovered in Section 3.3, while our extension to higher partial waves is given in Section 3.4. In particular, we discuss the mass shift for the irreducible repre- 3.2. Bound states in a finite volume 27 sentations of the cubic group, relate the sign of the shift to the leading parity, and derive a trace formula for the multiplet-averaged mass shift of states with arbitrary angular mo- mentum `. In Section 3.5, we verify our results numerically for two model systems. The case of two spatial dimensions is treated in Section 3.6, while in Section 3.7 we discuss the mass shift for twisted boundary conditions and how it leads to topological phase factors in the finite-volume calculations of composite-particle scattering. Finally, a brief summary and outlook is given in Section 3.8. 3.2 Bound states in a finite volume As a starting point, we first review several results from Ref. [8]. We closely follow Lu¨scher’s derivation, but consider a more general system with arbitrary angular momentum and non-local interactions. 3.2.1 Infinite volume Before we discuss the finite volume, we briefly review the infinite-volume case. Our basic setup is the one discussed in Section 2.1, i.e., we consider a system of two spinless particles with reduced mass µ and zero total momentum with a rotationally-symmetric finite- range interaction described by the symmetric operator V (r, r′) in configuration space. We furthermore assume that the interaction is such that it supports a bound state |ψB〉 with energy E = −EB = −κ2/(2µ) and angular-momentum quantum numbers (`,m). We consider the full three-dimensional wave function in this chapter to capture the angular dependence of the state. Recall that the finite-range assumption on the potential implies that V (r, r′) = 0 if |r| > R or |r′| > R . (3.2) The Schro¨dinger equation Hˆ|ψB〉 = −EB|ψB〉 , (3.3) can be written as − 1 2µ ∆r ψB(r) + ∫ d3r′ V (r, r′)ψB(r′) = −EB ψB(r) (3.4) in configuration space and for a local potential, V (r, r′) = V (r) δ(3)(r− r′) , (3.5) it reduces to the familiar form[ − 1 2µ ∆r + V (r) ] ψB(r) = −EB ψB(r) . (3.6) According to Eqs. (2.2) and (2.29), the asymptotic form of the wave function ψB(r) is determined by the Riccati–Hankel function H+` , ψB(r) = i `Aκ Y m ` (r/r) H+` (iκr) r (r > R) , (3.7) 28 Chapter 3. Finite-volume calculations regardless of the locality of the interaction, and where we have written the spherical harmonics as a function of the unit vector r/r instead of the angles θ and φ. For future reference, we give here the explicit expressions for H+` (z) for ` = 0, 1, 2: H+0 (z) = e iz , (3.8a) H+1 (z) = ( 1 + i z ) ei(z−pi/2) , (3.8b) H+2 (z) = ( 1 + 3i z − 3 z2 ) ei(z−pi) . (3.8c) 3.2.2 Finite volume We now consider what happens when the two-body system is put into a periodic cubic box with edge length L R. For this problem it is convenient to define a periodic extension of the potential VL(r, r ′) = ∑ n∈Z3 V (r + nL, r′ + nL) . (3.9) We take |ψ〉 to be an exact periodic solution of the finite-volume Schro¨dinger equation, HˆL|ψ〉 = −EB(L)|ψ〉 , (3.10) with the finite-volume Hamiltonian HˆL = Hˆ0 + VˆL and the volume-dependent binding energy EB(L). The periodic boundary conditions that we impose require that ψ(r + nL) = ψ(r) (3.11) for all integer vectors n ∈ Z3. It is clear that EB(L) approaches the infinite-volume eigenvalue EB and that |ψ〉 → |ψB〉 as L→∞. We now derive a formula for the finite-volume mass (energy) shift, ∆mB ≡ EB(∞)− EB(L) . (3.12) To proceed, we define a state |ψ0〉 by adding together periodic copies of the infinite-volume wave function in (3.4), 〈r|ψ0〉 = ψ0(r) = ∑ n ψB(r + nL) . (3.13) This clearly satisfies the periodicity condition (3.11). Acting upon this state with the 3.2. Bound states in a finite volume 29 finite-volume Hamiltonian, we get HLψ0(r) = H0 ∑ n′ ψB(r + n ′L) + ∑ n′ ∑ n ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) = ∑ n′ { H0 ψB(r + n ′L) + ∫ d3r′ V (r + n′L, r′ + n′L)ψB(r′ + n′L) + ∑ n6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) } = −EB(∞) ∑ n′ ψB(r + n ′L) + ∑ n′ ∑ n6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r + n′L) . (3.14) The final result can be written as HˆL|ψ0〉 = −EB(∞)|ψ0〉+ |η〉 , (3.15) where we have defined |η〉 as η(r) = ∑ n′ ∑ n 6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) . (3.16) With the substitution r′ → r′ − nL for each term in the sum, this can be rewritten as η(r) = ∑ n′ ∑ n6=n′ ∫ d3r′ V (r + nL, r′)ψB ( r′ + (n′ − n)L) . (3.17) Due to the finite range of the potential we only get contributions from the domain |r′| < R. We note that |r′ + (n′ − n)L| > R when n 6= n′ and R L. Therefore, we can use the asymptotic form of the wave function and find that |η〉 = O(e−κL). This means that |ψ0〉 is an approximate solution of the finite-volume Schro¨dinger equation (3.10) for large L. Motivated by this, we write the exact finite- volume solution |ψ〉 explicitly as |ψ〉 = α|ψ0〉+ |ψ′〉 with |ψ′〉 = O ( e−κL ) . (3.18) We take |ψ〉 to be unit-normalized per volume L3. The same is true of |ψ0〉 up to correc- tions of order e−κL. We will choose α in Eq. (3.18) such that 〈ψ0|ψ′〉 = 0 . (3.19) Consider now the matrix element 〈ψ|HˆL|ψ0〉. Acting with HˆL on |ψ0〉, we get 〈ψ|HˆL|ψ0〉 = −EB(∞)〈ψ|ψ0〉+ 〈ψ|η〉 = −EB(∞)〈ψ0|ψ0〉 · α + 〈ψ|η〉 (3.20) according to (3.15) and (3.18). On the other hand, acting with HˆL on 〈ψ| yields 〈ψ|HˆL|ψ0〉 = −EB(L)〈ψ|ψ0〉 = −EB(L)〈ψ0|ψ0〉 · α . (3.21) 30 Chapter 3. Finite-volume calculations Combining these two results we find EB(∞)− EB(L) = ∆mB = 〈ψ|η〉 α〈ψ0|ψ0〉 . (3.22) We first consider the numerator in this expression. Obviously, 〈ψ|η〉 = α〈ψ0|η〉+ 〈ψ′|η〉 = α〈ψ0|η〉+O ( e−2κL ) . (3.23) Note that the factor of α here will cancel the α in the denominator of Eq. (3.22). We can now simplify further, starting with 〈ψ0|η〉 = ∑ n′′ ∑ n′ ∑ n 6=n′ ∫ d3r ∫ d3r′ ψ∗B(r + n ′′L)V (r + nL, r′ + nL)ψB(r′ + n′L) . (3.24) For each n we can make the substitutions r→ r− nL and r′ → r′ − nL. These leave the integrals invariant, and we get 〈ψ0|η〉 = ∑ n′′ ∑ n′ ∑ n 6=n′ ∫ d3r ∫ d3r′ ψ∗B ( r + (n′′−n)L)V (r, r′)ψB(r′+ (n′−n)L) . (3.25) Setting m = n′ − n and m′ = n′′ − n yields 〈ψ0|η〉 = C · ∑ m′ ∑ m 6=0 ∫ d3r ∫ d3r′ ψ∗B(r + m ′L)V (r, r′)ψB(r′ + mL) , (3.26) where C counts the number of repeated periodic copies. The fact that C diverges simply reflects the fact that we are working with periodic wave functions with normalization measured per volume L3, and C will cancel in the final result. For the integral to be non-zero, both r and r′ have to be close to 0 due to the finite range of the potential. From the assumption L R it then follows that all terms with m′ 6= 0 are suppressed by at least a factor of e−2κL, and we have 〈ψ0|η〉 = C · ∑ m 6=0 ∫ d3r ∫ d3r′ ψ∗B(r)V (r, r ′)ψB(r′ + mL) +O ( e−2κL ) . (3.27) The possible nonvanishing values of |m| are 1,√2,√3, . . . . We therefore arrive at 〈ψ0|η〉 = C · ∑ |m|=1 ∫ d3r ∫ d3r′ ψ∗B(r)V (r, r ′)ψB(r′ + mL) +O ( e− √ 2κL ) . (3.28) For the denominator in (3.22), an analogous procedure yields 〈ψ0|ψ0〉 = C · ∑ m ∫ d3r ψ∗B(r)ψB(r + mL) = C · [ 1 +O(e−κL)] (3.29) with the same constant C as above. Combining (3.28) and (3.29), the constant cancels and we get ∆mB = ∑ |n|=1 ∫ d3r ∫ d3r′ ψ∗B(r)V (r, r ′)ψB(r′ + nL) +O ( e− √ 2κL ) , (3.30) 3.3. S-wave result 31 where we have renamed m back to n. Eq. (3.30) is a general result valid for any angular momentum. The dependence of the mass shift on quantum numbers (`,m) will emerge from the wave function ψB and the resulting overlap integrals in (3.30). In the following, we explore this dependence in detail and denote the mass shift as ∆m (`,m) B . 3.3 S-wave result For ` = 0 the asymptotic wave function (3.7) is simply given as ψB(r) = ψB(|r|) = √ 1 4pi u0(r) r (3.31a) with u0(r) = AκH + 0 (iκr) = Aκ e −κr for r > R . (3.31b) Due to the finite range R L of the potential we only have contributions with |r′+nL| > R in Eq. (3.30). Hence we can insert the asymptotic form for ψB(r ′ + nL) and get ∆m (0,0) B = Aκ√ 4pi ∑ |n|=1 ∫ d3r ∫ d3r′ ψ∗B (|r|)V (r, r′) e−κ|r′+nL||r′ + nL| +O(e−√2κL) . (3.32) We can furthermore use the Schro¨dinger equation (3.4) to eliminate the potential. Doing this and then renaming r′ → r, we get ∆m (0,0) B = Aκ√ 4pi ∑ |n|=1 ∫ d3r {[ ∆r 2µ − EB ] ψ∗B (|r|)} e−κ|r+nL||r + nL| +O(e−√2κL) = Aκ√ 4pi ∑ |n|=1 ∫ d3r ψ∗B (|r− nL|) 1 2µ [ ∆r − κ2 ] e−κr r +O(e−√2κL) . (3.33) In the second line we have shifted the integration variable and used partial integration to let the Laplacian act on exp(−κr)/r. Finally, we use the fact that exp(−κr)/(4pir) is a Green’s function for the operator ∆r − κ2,[ ∆r − κ2 ] e−κr 4pir = −δ(3)(r) , (3.34) cf. Eq.(2.41). This allows us to trivially solve the integral and arrive at ∆m (0,0) B = − √ piAκ µ ∑ |n|=1 ψ∗B (|nL|)+O(e−√2κL) = −3|Aκ|2 e −κL µL +O(e−√2κL). (3.35) In the last step we have inserted the asymptotic form of the wave function for ψ∗B (|nL|) = ψ∗B(L), and the sum yields a factor of six. This is just Lu¨scher’s result (3.1) as given in the introduction. 32 Chapter 3. Finite-volume calculations 3.4 Extension to higher partial waves We now discuss the generalization of the mass-shift formula to arbitrary angular momen- tum. The general form for the asymptotic wave function is ψB,(`,m)(r) = Y m ` (θ, φ) i`AκH + ` (iκr) r . (3.36) Inserting this into Eq. (3.30) and performing steps analogous to those for the S-wave case, we find ∆mB = ∑ |n|=1 ∫ d3r { 1 2µ [ ∆r − κ2 ] ψ∗B(r− nL) } Y m` (θ, φ) i`AκH + ` (iκr) r +O(e−√2κL) . (3.37) The crucial ingredient is now the relation Y m` (θ, φ) H+` (iκr) r = (−i)`Rm` ( −1 κ ∇r )[ e−κr r ] , (3.38) where Rm` are the solid harmonics defined via R m ` (x, y, z) = R m ` (r) = r `Y m` (θ, φ). The result (3.38) follows from Lemma B.1 in Ref. [102], which proves that Rm` (∇r)f(r) = Rm` (r) ( 1 r d dr )` f(r) (3.39) for any smooth function f(r). From this we obtain Eq. (3.38) by using the relation2,( 1 z d dz )` h (1) 0 (z) = (−1)` z−` h(1)` (z) (3.40) and noting that e−κr = H+0 (iκr) and H + ` (z) = iz h (1) ` (z), where h (1) ` (z) is a spherical Hankel function of the first kind. We can illustrate Eq. (3.38) with an example. For the case ` = 1 and m = 0 we have H+1 (iκr) ∼ ( 1 + 1 κr ) e−κr r (3.41) and Y 01 (θ, φ) ∝ cos θ. A straightforward calculation using cos θ = z/r shows that indeed cos θ ( 1 + 1 κr ) e−κr r = −1 κ ∂ ∂z [ e−κr r ] . (3.42) Rewriting Eq. (3.37) with the help of Eq. (3.38), we get ∆mB = Aκ 2µ ∑ |n|=1 ∫ d3r {[ ∆r − κ2 ] ψ∗B(r− nL) }{ Rm` ( −1 κ ∇r )[ e−κr r ]} +O(e−√2κL) . (3.43) 2The relation (3.40) is just a special case of Eq. (10.1.24) in [103], which also holds for other spherical Bessel functions. 3.4. Extension to higher partial waves 33 We now integrate by parts and pass the Laplacian through the differential operator Rm` (−∇r/κ). Since the operators both consist of partial derivatives, this is not a problem when the wave function is smooth. We assume that this is the case, with the possible exception of a measure-zero region that can be omitted from the integral. The partial integrations give a factor (−1)`. We can now proceed in exactly the same way as for S-waves. Performing one more integration by parts so that the Laplacian acts on exp(−κr)/r yields a delta function times a factor of −4pi. The final result is then ∆m (`,m) B = (−1)`+1 · 2piAκ µ ∑ |n|=1 Rm` ( −1 κ ∇r ) ψ∗B,(`,m)(r− nL) ∣∣∣∣∣ r=0 +O(e−√2κL) . (3.44) For ψ∗B,(`,m) we can insert the asymptotic form (3.36) since it is evaluated in the asymptotic region. 3.4.1 Results For ` = 1, we find the same result for all three P-wave states: ∆m (1,0) B = ∆m (1,±1) B = 3|Aκ|2 e−κL µL +O(e−√2κL) . (3.45) Compared to the S-wave case, the sign of the P-wave mass shift is reversed while the magnitude is exactly the same. Qualitatively, this means that S-wave bound states are more deeply bound when put in a finite volume while P-wave bound states are less bound (compared to the infinite-volume result). This behavior will be analyzed in more detail later. We next discuss the results for ` = 2. From Eq. (3.44) we find ∆m (2,0) B = −15|Aκ|2 e−κL µL · F 02 ( 1 κL ) +O(e−√2κL) , (3.46) ∆m (2,±1) B = +15|Aκ|2 e−κL µL · F 12 ( 1 κL ) +O(e−√2κL) , (3.47) ∆m (2,±2) B = −15|Aκ|2 e−κL µL · F 22 ( 1 κL ) +O(e−√2κL) , (3.48) where F 02 (x) = 1 2 + 3x+ 27 2 x2 + 63 2 x3 + 63 2 x4 , (3.49) F 12 (x) = 2x+ 9x 2 + 21x3 + 21x4 , (3.50) F 22 (x) = 1 4 + 1 2 x+ 9 4 x2 + 21 4 x3 + 21 4 x4 . (3.51) We note that here the size and even the sign of the mass shift both depend on the quantum number m. To understand this effects we need to take into account that our cubic finite volume breaks the rotational symmetry group down to a cubic subgroup. 34 Chapter 3. Finite-volume calculations Representations of the cubic group The cubic symmetry group O is a finite subgroup of SO(3) with 24 elements. There are five irreducible representations of O, conventionally called A1, A2, E, T1, and T2. Their dimensionalities are 1, 1, 2, 3, and 3, respectively. Irreducible representations D` of the rotation group SO(3) are reducible with respect to O for ` > 1. For further details about the decomposition see, for example, Ref. [104]. In our discussion we assume that the infinite-volume system has no partial wave mixing, such that orbital angular momentum ` is a good quantum number. We also assume that there are no accidental degeneracies in the bound state spectrum, so we can use ` as a label for the family of cubic representations split apart at finite volume. Parity invariance remains unbroken by the cubic volume, and we have P = (−1)` just as in the infinite- volume case. For clarity, however, we will indicate parity explicitly with ± superscripts in the following. With our assumptions, an S-wave state in infinite volume will map onto an A+1 state at finite volume. Also a P-wave triplet will simply map onto the three elements of the T−1 representation at finite volume. For ` = 2, however, the five D-wave states are split into a T+2 triplet and an E + doublet, D2 = T+2 ⊕ E+ . (3.52) In the following we use the notation |Γ, `;α〉, α = 1, . . . , dim(Γ) for the basis vectors of the irreducible cubic representations. We can rewrite the finite volume mass shift in Eq. (3.30) as ∆m (Γ,`,α) B ≡ 〈 Γ, `;α ∣∣Vˆ ∑ |n|=1 Tˆ (nL) ∣∣Γ, `;α〉 , (3.53) where Tˆ (x) is the translation operator for displacement by a vector x. We can also calculate the matrix elements of ∆mB in the (`,m) basis. In this case there will be off- diagonal matrix elements connecting (`,m) and (`,m′) when m and m′ are equivalent modulo 4. According to Ref. [104], the unitary transformation between the two basis sets for the five D-wave states is |T+2 , 2; 1〉 = − 1√2 (|2,−1〉+ |2, 1〉) , (3.54a) |T+2 , 2; 2〉 = i√2 (|2,−1〉 − |2, 1〉) , (3.54b) |T+2 , 2; 3〉 = − 1√2 (|2,−2〉 − |2, 2〉) (3.54c) and |E+, 2; 1〉 = |2, 0〉 , (3.55a) |E+, 2; 2〉 = 1√ 2 (|2,−2〉+ |2, 2〉) . (3.55b) So, for example, we have ∆m (T+2 ,2;1) B = 1 2 ( ∆m (2,−1,−1) B + 2∆m (2,−1,1) B + ∆m (2,1,1) B ) = −15|Aκ|2 e −κL µL · ( 2 κL + 9 κ2L2 + 21 κ3L3 + 21 κ4L4 ) +O(e−√2κL) , (3.56) 3.4. Extension to higher partial waves 35 where we have defined ∆m (`,m1,m2) B = (−1)`+1 · 2piAκ µ ∑ |n|=1 Rm1` ( −1 κ ∇r ) ψ∗B,(`,m2)(r− nL) ∣∣∣∣∣ r=0 +O(e−√2κL) (3.57) as a straightforward generalization of Eq. (3.44). As expected from cubic symmetry, the mass shift is the same for all three T+2 states and also within the E+ doublet. To summarize our results, we write the mass shift for a state belonging to irreducible representation Γ with angular momentum ` as ∆m (`,Γ) B = α ( 1 κL ) · |Aκ|2 e−κL µL .+O(e−√2κL) (3.58) We list the coefficients α ( 1 κL ) for ` = 0, . . . , 3 in Table 3.1. ` Γ α(x) 0 A+1 −3 1 T−1 +3 2 T+2 30x+ 135x 2 + 315x3 + 315x4 2 E+ −1/2 (15 + 90x+ 405x2 + 945x3 + 945x4) 3 A−2 315x 2 + 2835x3 + 12285x4 + 28350x5 + 28350x6 3 T−2 −1/2 (105x+ 945x2 + 5355x3 + 19530x4 + 42525x5 + 42525x6) 3 T−1 1/2 (14 + 105x+ 735x 2 + 3465x3 + 11340x4 + 23625x5 + 23625x6) Table 3.1: Coefficient α(x) in the expression for the finite-volume mass shifts for ` = 0, . . . , 3. Γ indicates the corresponding representation of the cubic group. 3.4.2 Sign of the mass shift The sign of the finite-volume mass shift can be understood in terms of the parity of the wave function. In infinite volume the tail of each bound state wave function must vanish at infinity. In the finite volume, however, the bound state wave functions with even parity along a given axis can remain nonzero everywhere. Only the derivative needs to vanish, and the kinetic energy is lowered by broadening the wave function profile. On the other hand, a wave function with odd parity along a given axis must change sign across the boundary. In this case the wave function profile is compressed and the kinetic energy thus increased. We have illustrated both cases for a one-dimensional square-well potential in Fig. 3.1. In three dimensions, the situation is slightly more complicated, which can be seen from the fact that for ` = 2 the sign of the mass shift depend on the representation of the cubic group even though the parity is just (−1)2 = +1 for all states. In order to understand this we consider the basis polynomials for the cubic representations. These basis polynomials are obtained by decomposing the cubic basis vectors in terms of solid harmonics which 36 Chapter 3. Finite-volume calculations x ψeven ψodd DˆB Figure 3.1: Wave functions with even (bottom) and odd parity (top) for a one- dimensional square well potential in a box with periodic boundary conditions. The dashed lines give the infinite volume solutions for comparison. are homogeneous polynomials in x, y and z. For ` = 0, . . . , 4 the basis polynomials are given explicitly in [102]. For a given polynomial P (x, y, z), we define its leading parity as lpP = (−1)dmax , (3.59) where dmax = max{degx P, degy P, degz P} (3.60) is the maximum degree of P with respect to any one of the three variables. It is this leading parity that determines the asymptotic behavior of the mass shift as κL → ∞. More precisely, we have α ( 1 κL ) ∼ (−1)dmax+1 ( 1 κL )`−dmax as κL→∞ (3.61) for the α ( 1 κL ) in Eq. (3.58). It can easily be checked that this relation holds for all results presented in Table 3.1. For ` = 2, for example, we have the basis polynomials P2,T+2 ∼ xy , yz , zx , (3.62a) P2,E+ ∼ x2 − y2 , y2 − z2 , (3.62b) and hence dmax = 1 for the T + 2 representation and dmax = 2 for the E + representation. 3.4. Extension to higher partial waves 37 3.4.3 Trace formula The expressions for the finite-volume mass shift become simpler when we sum over all m for a given `. We can rewrite Eq. (3.44) as ∆m (`,m) B = (−1)`+1 · 2piAκ µ ∑ |n|=1 Rm` ( −1 κ ∇r ) ψ∗B,(`,m)(r) ∣∣∣∣∣ r=nL +O(e−√2κL) . (3.63) Inserting the asymptotic form of the wave function, ψ∗B,(`,m)(r) ∣∣∣ r=nL = [ Y m` (θ, φ) i`AκH + ` (iκr) r ]∗ ∣∣∣∣∣ r=nL , (3.64) and using Eq. (3.38) a second time yields ∆m (`,m) B = (−1)`+1 · 2pi|Aκ|2 µ ∑ |n|=1 Rm` ( −1 κ ∇r ) R∗m` ( −1 κ ∇r )[ e−κr r ] ∣∣∣∣∣ r=nL +O(e−√2κL) . (3.65) Now, from the well-known relation ∑` m=−` Y m` (θ, φ)Y ∗m ` (θ, φ) = 2`+ 1 4pi (3.66) and Rm` (r) = r `Y m` (θ, φ) we get an analogous expression for the solid harmonics, which then carries over to∑` m=−` Rm` ( −1 κ ∇r ) R∗m` ( −1 κ ∇r ) f(r) = 1 κ2` · 2`+ 1 4pi (∆r) `f(r) (3.67) for any sufficiently smooth function f(r). Finally, we have (∆r) ` e −κr r = κ2` e−κr r (r 6= 0) , (3.68) which follows from Eq. (3.34). Putting everything together, we arrive at ∑` m=−` ∆m (`,m) B = (−1)`+1 · 2pi|Aκ|2 µ · 2`+ 1 4pi ∑ |n|=1 1 κ2` (∆r) ` [ e−κr r ] ∣∣∣∣∣ r=nL +O(e−√2κL) = (−1)`+1(2`+ 1) · 3|Aκ|2 e −κL µL +O(e−√2κL) , (3.69) where the sum just yields a factor of six. Dividing by 2`+ 1, we obtain the average mass shift for states with angular momentum `, ∆m (`) B = (−1)`+1 · 3|Aκ|2 e−κL µL +O(e−√2κL) . (3.70) 38 Chapter 3. Finite-volume calculations Apart from the alternating sign, this average shift is independent of `. Eq. (3.70) can be verified explicitly for the the results presented in Section 3.4.1 (cf. Ta- ble 3.1). For ` = 2, for example, one has to average over the three-dimensional represen- tation T+2 and the two-dimensional representation E +.3 3.5 Numerical tests In order to verify our predictions numerically, we put the Schro¨dinger equation (3.10) on a discrete spatial lattice such that the Hamiltonian becomes an ordinary matrix. We then calculate the corresponding energy eigenvalues and eigenvectors. 3.5.1 Lattice discretization We use a hat symbol to denote dimensionless lattice units. For example, we have Lˆ = L/a and EˆB = EB · a , (3.71) where a denotes the lattice spacing. The free lattice Hamiltonian is given by Hˆ0 = ∑ nˆ [ 3 µˆ a†(nˆ)a(nˆ)− 1 2µˆ ∑ l=1,2,3 ( a†(nˆ)a(nˆ + eˆl) + a†(nˆ)a(nˆ− eˆl) )] (3.72) where a†(nˆ) and a(nˆ) are creation and annihilation operators for a lattice site nˆ and eˆl is a unit vector in the l-direction. The corresponding lattice dispersion relation is Eˆ(qˆ) = Q2(qˆ) 2µˆ (3.73) with the lattice function Q2(qˆ) = 2× ∑ l=1,2,3 (1− cos qˆi) = ∑ l=1,2,3 qˆ2l [ 1 +O(qˆ2l ) ] (3.74) and the lattice momenta qˆ = 2pinˆ/Lˆ . (3.75) The binding momentum for a bound state with energy −EˆB is determined by −µˆEˆB = 1− cos(−iκˆ) = 1− cosh(κˆ) . (3.76) The lattice Green’s function for the Hamiltonian (3.72) is Gˆ(nˆ, Eˆ) = 1 L3 ∑ qˆ e−iqˆ·nˆ Q2(qˆ) + 2µˆEˆ . (3.77) 3Note that the mapping from the angular momentum eigenstates to the cubic group states is a unitary transformation. 3.5. Numerical tests 39 We impose periodic boundary conditions by defining the distance rˆ to the origin as rˆ (nˆ) = √√√√ ∑ l=1,2,3 min { nˆ2l , ( Lˆ− nˆl )2} . (3.78) 3.5.2 Methods We calculate the mass shift using three different methods: 1. As a direct difference in energies, Eq. (3.12), where we use a very large volume (L∞) to approximate the infinite-volume result. 2. From the overlap formula (3.30). 3. Using discretized versions of Eqs. (3.35) and (3.45), which we obtain by replacing exp(−κr)/r with the lattice Green’s function. More precisely, we write the asymp- totic bound-state wave function (3.7) as ψB(r) = i `Aκ Y m ` (r/r)H + ` (iκr) e κr · 4piGκ(r) for r > R (3.79) and replace the continuum Green’s function Gκ(r) = e−κr 4pir (3.80) with the lattice version Gˆκˆ(nˆ) ≡ Gˆ ( nˆ, −κˆ2 2µˆ ) . (3.81) Effectively, this amounts to the replacement e−κˆLˆ/Lˆ −→ 4piGˆκˆ(Lˆ, 0, 0) (3.82) in the mass-shift formula. The lattice Green’s function is also used to calculate the asymptotic normalization Aκ from the lattice data. This procedure has the advantage of avoiding large discretization errors. 3.5.3 Results In the following we report physical quantities in units where the reduced mass µ is set to one. 40 Chapter 3. Finite-volume calculations Gaussian potential We first use a Gaussian potential, VGauss(r) = −V0 exp (−r2/(2R2)) , (3.83) with R = 1 and V0 = 6. This potential does not have a finite range in a strict mathematical sense, but the range corrections can be entirely neglected in comparison with other errors in our numerical calculation. The smoothness of the Gaussian potential helps to minimize lattice discretization artifacts. In Fig. 3.2 we show the S- and P-wave mass shifts obtained with the three methods described in Section 3.5.2. The results from the three different methods described above agree well for both S- and P-waves. In order to compare the dependence on the box size L with the predicted behavior we have plotted log(L · |∆mB|) against L (we use the absolute value of ∆mB since the S-wave mass shift is negative). For both S- and P-waves, the expected linear dependence is clearly visible. 3 4 5 6 7 8 9 10 11 12 13 -20 -15 -10 -5 0 V = VGauss S-wave P-wave lo g (L ·|∆ m B |) L direct difference overlap integral Green’s function Figure 3.2: S-wave and P-wave mass shifts log(L · |∆mB|) as functions of the box size L (in lattice units) for a Gaussian potential. We show the results obtained from the direct difference Eq. (3.12) (crosses), evaluation of the overlap integral Eq. (3.30) (squares), and discretized versions of Eqs. (3.35), (3.45) (circles). The dashed lines show linear fits to the overlap integral results. When we perform a linear fit to the overlap integral data (dashed lines in Fig. 3.2) we obtain κ = 2.198± 0.005, |Aκ| = 11.5± 0.2 for the S-wave results and κ = 1.501± 0.004, |Aκ| = 7.0 ± 0.1 for the P-wave results. The values for the asymptotic normalization constants are in good agreement with the results |Aκ| ∼ 11.5 (S-wave) and |Aκ| ∼ 7.2 (P-wave) that are obtained directly from the L∞ = 40 data. Inserting the corresponding energy eigenvalue into the lattice dispersion relation (3.76), we find κ ∼ 2.211 (S-wave) and κ ∼ 1.501 (P-wave), again in very good agreement with the fit results. The re- maining small discrepancies can be attributed to the mixing with higher partial waves induced by the lattice discretization and the fact that we have not performed a continuum extrapolation to vanishing lattice spacing. 3.5. Numerical tests 41 Simple step potential For a simple step potential, Vstep(r) = −V0 θ(R− r) , (3.84) which we use with R = 2 and V0 = 3, the numerical calculation becomes more difficult because the discontinuous shape introduces considerable lattice artifacts. Still, we discuss it here due to its strict finite range and since we find that for a small lattice spacing of a = 0.2 the results are satisfactory. In Fig. 3.3 we show a plot analogous to the one presented for the Gaussian potential. Again, the results from the different methods agree well and the expected linear behavior is clearly visible. Furthermore, the results from the three methods agree well with each other already for smaller L (compared to the results for the Gaussian potential), as expected from the fact that the step potential does not have a tail. 4 5 6 7 8 9 10 11 12 -15 -10 -5 0 V = Vstep S-wave P-wave lo g (L ·|∆ m B |) L direct difference overlap integral Green’s function Figure 3.3: S-wave and P-wave mass shifts log(L · |∆mB|) as functions of the box size L (in lattice units) for a simple step potential. The symbols are as in Fig. 3.2. From fitting to the overlap integral data (dashed lines in Fig. 3.3) we obtain κ = 2.0636± 0.0005, |Aκ| = 29.17 ± 0.06 for the S-wave results and κ = 1.6192 ± 0.0009, |Aκ| = 12.48±0.05 for the P-wave results. From the L∞ = 40 data we find κ ∼ 2.0666, |Aκ| ∼ 29.6 (S-wave) and κ ∼ 1.6242, |Aκ| ∼ 12.8 (P-wave). Given that we do not have error estimates for the L∞ = 40 results, the overall agreement is quite good. Finally, we also check our result for the D-wave mass splittings, using again the step potential with a = 0.2. In Fig. 3.4 we show the mass shift for the D-wave states in both the T+2 and the E + representation. Due to the polynomial coefficients α ( 1 κL ) —see Eq. (3.58) and Table 3.1—one does not expect a linear dependence on L for log(L · |∆mB|). Hence, we simply plot ∆mB as a function of L directly and do not perform a fit. Nevertheless, we see that (except for very small L, where obviously the condition L R is not satisfied) the agreement between the three methods to calculate ∆mB is very good and hence conclude that our mass-shift formula indeed gives the right result also for ` = 2. 42 Chapter 3. Finite-volume calculations 4 5 6 7 8 9 10 11 12 0 0.1 0.2 0.3 0.4 V = Vstep, D-wave, T2 rep. ∆ m B L direct difference overlap integral Green’s function · 4 5 6 7 8 9 10 11 12 -0.5 -0.4 -0.3 -0.2 -0.1 0 V = Vstep, D-wave, E rep. ∆ m B L direct difference overlap integral Green’s function · Figure 3.4: D-wave, mass shift ∆mB for T + 2 rep. (left panel) and E + rep. (right panel) as a function of the box size L (in lattice units) for a simple step potential. The symbols are as in Fig. 3.2. 3.6 Two-dimensional systems In this section we derive a formula for the finite-volume (or rather finite-area) mass shift of bound states in two-dimensional systems. The results can be used, for example, in lattice simulations of cold atomic systems, which can be prepared experimentally to be effectively two-dimensional [105, 106]. We note that the S-wave case in two dimensions was previously investigated in Ref. [107]. In two dimensions, the Schro¨dinger equation is − 1 2µ ∆2Dr ψB(r) + ∫ d2r′ V (r, r′)ψB(r′) = −EB ψB(r) (3.85) with ∆2Dr ψB(r) = [ 1 r ∂ ∂r + ∂2 ∂r2 + 1 r2 ∂2 ∂θ2 ] ψB(r) (3.86) in polar coordinates. States are described by a single angular momentum quantum number m = 0,±1,±2, . . ., and for the wave function we have the separation ψB(r) = um(r)Ym(θ) (3.87) with Ym(θ) = eimθ√ 2pi . (3.88) The two linearly independent solutions of the free radial equation( d2 dr2 + 1 r d dr − m 2 r2 + p2 ) um(r) = 0 , (3.89) are just the Bessel and Neumann functions Jm(pr) and Nm(pr). For a bound state, we have p2 = −κ2 = −2µEB, and the wave function has the asymptotic form um(r) = AκKm(κr) for r > R , (3.90) 3.6. Two-dimensional systems 43 where Km is the modified Bessel function of the second kind. It is related to the Hankel function of the first kind, H(1)m (z) = Jm(z) + iNm(z) , (3.91) via Km(x) = pi 2 im+1H(1)m (ix) . (3.92) As in the three-dimensional case, Aκ is the asymptotic normalization constant. Inserting Eq. (3.92) into Eq. (3.90) yields a form which is more similar to the three-dimensional expression. To render the analogy to the calculations in Section 3.4 as explicit as possible, we will use the Hankel function in the following intermediate steps and only express the final results in terms of the modified Bessel function. Nearly all of the three-dimensional calculation carries over if we just replace all exponential terms with Hankel functions. The overlap integral for the mass shift is now ∆m (m) B = ∑ |n|=1 ∫ d2r ∫ d2r′ ψ∗B,m(r)V (r, r ′)ψB,m(r′ + nL) +O ( iH(1)m (√ 2κL )) . (3.93) From the asymptotic form of the Hankel function, H(1)m (z) ∼ √ 2 piz ei(z− m 2 pi−pi 4 ) as |z| → ∞ , (3.94) it is clear that in principle we still have an exponential behavior. In deriving Eq. (3.93) we have used this to write O ( H(1)m ( iκL )2) ∼ O (H(1)m (2iκL)) O (H(1)m (√2iκL)) . (3.95) In the following we will simply write the correction terms as O(e−√2κL), as in the three- dimensional case. The two-dimensional analog of the relation (3.38) is Ym(θ)H (1) m (iκr) = (−i)mRm ( −1 κ ∇2Dr ) H (1) 0 (iκr) , (3.96) where Rm(r, θ) = r mYm(θ). This follows from Rm(∇2Dr )f(r) = Rm(r) ( 1 r d dr )m f(r) (3.97) and ( 1 z d dz )m H (1) 0 (z) = (−1)m z−mH(1)m (z) . (3.98) The derivation of Eq. (3.97) can be carried out in the same manner as the three-dimensional proof of Lemma B.1 in [102], using the expansion of eip·r (2D vectors) in terms of Bessel functions. As the final ingredient we have[ ∆2Dr − κ2 ] i 4 H (1) 0 (iκr) = −δ(2)(r) . (3.99) 44 Chapter 3. Finite-volume calculations Using all this in steps completely analogous to those in three dimensions, we get ∆m (m) B = (−1)m+1 · piAκ µ ∑ |n|=1 Rm ( −1 κ ∇2Dr ) ψ∗B,m(r− nL) ∣∣∣∣∣ r=0 +O(e−√2κL) . (3.100) For m = 0 (two-dimensional S-waves), this directly yields ∆m (0) B = −2 |Aκ|2 µ K0(κL) +O ( e− √ 2κL ) . (3.101) In fact, Eq. (3.100) can be simplified further. Inserting the asymptotic form for the wave function for ψ∗B,m and using (3.96) a second time gives ∆m (m) B = (−1)m+1 · pi|Aκ|2 µ ∑ |n|=1 Rm ( −1 κ ∇2Dr ) R∗m ( −1 κ ∇2Dr )[ i pi 2 H (1) 0 (iκr) ] ∣∣∣∣∣ r=nL +O(e−√2κL) . (3.102) From Eq. (3.88) and Rm(r) = r mYm(θ) it is clear that Rm(θ)R ∗ m(θ) = (r2)m 2pi , (3.103) which then yields Rm ( −1 κ ∇2Dr ) R∗m ( −1 κ ∇2Dr ) f(r) = 1 κ2m · 1 2pi ( ∆2Dr )m f(r) (3.104) for any sufficiently smooth f(r). This is essentially the same relation that we used to derive the trace formula in the three-dimensional case, only that here we do not have to sum over different m. Together with the two-dimensional analog of Eq. (3.68),( ∆2Dr )m H (1) 0 (iκr) = κ 2mH (1) 0 (iκr) (r 6= 0) , (3.105) we then get ∆m (m) B = (−1)m+1 · Aκ|2 2µ ∑ |n|=1 1 κ2m ( ∆2Dr )m [ i pi 2 H (1) 0 (iκr) ] ∣∣∣∣∣ r=nL +O(e−√2κL) = (−1)m+1 · 2|Aκ| 2 µ K0(κL) +O ( e− √ 2κL ) . (3.106) As we shall see in the following, this is the final result for m = 0 and any odd m, whereas for even m 6= 0 things become slightly more complicated. In general, we have to take into account that the finite volume breaks the original planar rotational symmetry of the system down to the symmetry group of a square. We find 3.7. Twisted boundary conditions 45 that states with the same absolute value of m may mix to form good eigenstates in the finite volume. More precisely, we have the symmetric and antisymmetric combinations |m,±〉 = 1√ 2 (|m〉 ± |−m〉) (3.107) for m 6= 0. When we calculate the mass shift for these states (in the same way as described in Section 3.4.1), we get mixing terms of the form ∆m (m,mixed) B = (−1)m+1 · piAκ µ ∑ |n|=1 Rm ( −1 κ ∇2Dr ) ψ∗B,−m(r) ∣∣∣∣∣ r=nL +O(e−√2κL) . (3.108) Since the condition for the mixing of states is 2m ≡ 0 mod 4 , (3.109) they do not play a role for odd m (in fact, they vanish in this case). For even m, however, we have to take them into account and find ∆m (m,±) B = 1 2 ( ∆m (m) B ± 2∆m(m,mixed)B + ∆m(−m)B ) (3.110) as our final result. As an illustration, we give the explicit results for |m| = 2: ∆m (2,+) B = −4 |Aκ|2 µ [( 1 + 12 κ2L2 ) K0(κL) + ( 4 κL + 24 κ3L3 ) K1(κL) ] +O(e−√2κL) , (3.111a) ∆m (2,−) B = 16 |Aκ|2 µ [ 3 κ2L2 K0(κL) + ( 1 κL + 6 κ3L3 ) K1(κL) ] +O(e−√2κL) . (3.111b) 3.7 Twisted boundary conditions In this section, we go back to the three-dimensional case and discuss a generalization of the mass-shift formula obtained by changing the boundary condition imposed on the finite-volume wave function. Instead of the periodicity (3.11) we now require that ψ(r + nL) = ψ(r) e−iθ·n (3.112) for all n ∈ Z3, where θ is an arbitrary vector of phases. It is clear that setting θ = 0 in this so-called twisted boundary condition gives back Eq. (3.11). As we will discuss in more detail shortly in Section 3.7.2, a boundary condition of this form arises when one considers a system of more than two particles, two of which form the bound state whose mass shift we are interested in. In such a setup one has to consider the full two-particle wave function Ψ(r1, r2) = Ψ(R, r) = e iP·R ψ(r) , (3.113) where r = r1 − r2 and R = η1r1 + η2r2 (3.114) 46 Chapter 3. Finite-volume calculations are the relative and center-of-mass coordinates, respectively, and η1,2 are the mass ratios η1 = m1 m1 +m2 , η2 = 1− η1 . (3.115) If one demands that Ψ(r1, r2) is periodic in both coordinates, e.g., Ψ(r1 + nL, r2) = e iP·R eiη1LP·n ψ(r + nL) = Ψ(r1, r2) , (3.116) one finds that the behavior of ψ(r + nL) has to cancel the additional phase, ψ(r + nL) = e−iη1LP·nψ(r) . (3.117) This is just Eq. (3.112) with θ = η1LP. More generally, one can of course write the boundary condition in the form Ψ(r1 + n1L, r2 + n2L) = eiP·R eiη1LP·n1 eiη2LP·n2 ψ(r + (n1 − n2)L) = Ψ(r1, r2) , (3.118) but this again gives Eq. (3.117) since eiη1LP·n1eiη2LP·n2 = eiη1LP·n1ei(1−η1)LP·n2 = eiη1LP·(n1−n2) , (3.119) where the last equality follows by noting that the total momentum for the center-of-mass movement P is quantized in the finite volume, P = 2pi L K , K ∈ Z3 . (3.120) Keeping in mind these considerations as a motivation, we now derive the the finite-volume mass shift for a the two-particle system with twisted boundary conditions—without writ- ing θ in terms of the momentum P, however, but rather keeping it as an arbitrary pa- rameter. 3.7.1 Generalized derivation We carry out the derivation along the lines laid out in Section 3.2 and start by making an ansatz |ψθ〉 for the finite-volume wave function of the form 〈r|ψθ〉 = ψθ(r) = ∑ n ψB(r + nL) e iθ·n . (3.121) Like the |ψ0〉 defined in Eq. (3.13), this has at least the correct boundary behavior: ψθ(r + nL) = ∑ n′ ψB(r + n ′L+ nL) eiθ·n ′ = ∑ n′′=n′+n ψB(r + n ′′L) eiθ·(n ′′−n) = ψθ(r) e−iθ·n . (3.122) 3.7. Twisted boundary conditions 47 Acting on this state with the finite-volume Hamiltonian, we get HLψθ(r) = H0 ∑ n′ ψB(r + n ′L) eiθ·n ′ + ∑ n′,n ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) eiθ·n ′ = ∑ n′ {[ H0 ψB(r + n ′L) + ∫ d3r′ V (r + n′L, r′ + n′L)ψB(r′ + n′L) ] eiθ·n ′ + ∑ n6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) eiθ·n ′ } = −EB(∞) ∑ n′ ψB(r + n ′L) eiθ·n ′ + ∑ n′ ∑ n6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) eiθ·n ′ , (3.123) which analogously to Eq. (3.15) we write as HˆL|ψθ〉 = −EB(∞)|ψθ〉+ |ηθ〉 (3.124) with ηθ(r) = ∑ n′ ∑ n6=n′ ∫ d3r′ V (r + nL, r′ + nL)ψB(r′ + n′L) eiθ·n ′ . (3.125) We see that in the end the phase factor eiθ·n ′ is simply carried through the whole cal- culation. Since furthermore it does not depend on the variable r, we can now conclude that all steps carried out in Section 3.2.2 remain valid in spite of its presence and directly arrive at the the integral formula for the finite-volume mass shift with twisted boundary conditions: ∆mB(θ) = ∑ |n|=1 ∫ d3r ψ∗B(r)V (r)ψB(r + nL) e iθ·n +O(e−√2κL) . (3.126) For S-wave states, the final result then is ∆m (0) B (θ) = −|Aκ|2 e−κL µL × ∑ n=eˆx,eˆy ,eˆz cos(θ · n) +O(e−√2κL) , (3.127) i.e., the phases from pairs of opposite directions combine to give cosine factors. Again, it is clear that setting θ = 0 gives back the old result (3.35) for periodic boundary conditions. For P-wave states, there is now a dependence on the quantum number m: ∆m (1,0) B (θ) = |Aκ|2 e−κL µL × −(1 + κL)(cos θx + cos θy) + (2 + κL(2 + κL)) cos θz κ2L2 , (3.128a) ∆m (1,±1) B (θ) = |Aκ|2 e−κL µL × (1 + κL(1 + κL))(cos θx + cos θy)− 2(1 + κL) cos θz 2κ2L2 . (3.128b) 48 Chapter 3. Finite-volume calculations However, this goes away as it should if one sends θ → 0. For m = 0, for example, one finds that − (1 + κL)(cos θx + cos θy) + (2 + κL(2 + κL)) cos θz = −2(1 + κL)2 + (2 + κL(2 + κL)) = κ2L2 , (3.129) cancelling the additional factor in the denominator and thus giving back the old result. Improved volume dependence Already in the two-body sector the results given above are actually quite interesting. From Eq. (3.127), for example, one can see that for a bound state with equal mass constituents (such that the mass ratios are η1 = η2 = 1/2), the leading finite-volume mass shift can be made to vanish if one chooses to perform the calculation in a “boosted” frame with K = (1, 1, 1)/2 rather than in the rest frame of the bound state, which would be the na¨ıve choice. In Ref. [3] it was argued and demonstrated numerically that this procedure can be gener- alized to bound states of more than two particles if one assumes that there is no cluster substructure. Furthermore, in Ref. [90] Davoudi and Savage discuss a general method for reducing finite-volume corrections in two-body calculations by choosing appropriate combinations of boosted frames. 3.7.2 Topological volume factors As mentioned at the outset, the mass shift for twisted boundary conditions is important for systems of more that two particles in a finite volume, two of which form a dimer bound state. In the following, we will discuss how these shifts yield correction factors that have to be taken into account in finite-volume determinations of scattering phase shifts where one or more of the particles is composite.4 To this end, consider the scattering at low energies of two particle labeled A and B, both of which can be either elementary (pointlike) or composite bound states. For the sake of definiteness we take A to be a pointlike “atom” and B to be a dimer bound state. In order to extract the S-wave scattering phase shift δ0(p) from finite-volume calculations, a popular strategy is to apply Lu¨scher’s formula p cot δ0(p) = 1 piL S(η) , η = ( Lp 2pi )2 , (3.130) where p is the center-of-mass momentum of the A–B system and S(η) = √ 4pi · Z00(1, η) = lims→1 ∑ n∈Z3 1 (n2 − η)s (3.131) 4The relations presented in the following, published in Ref. [3], were derived by D. Lee [108]. The present author’s main contribution to Ref. [3] was the derivation of the twisted-boundary mass-shift formula. 3.7. Twisted boundary conditions 49 is the Lu¨scher Zeta-function (see, e.g., Ref. [102]).5 The crucial point here is that the momentum p is determined from energy levels of the A–B system in the finite volume which, in turn, are functions of the box size L, i.e. p = p ( EAB(L) ) . (3.132) If now at least one of the particles A and B is a composite bound state, there will be a contribution to the volume dependence of the total energy that is solely due to the mass shift of the bound state. In order to apply Lu¨scher’s formula (3.130), this contribution has to be determined and subtracted. Turning back to our atom–dimer example, we assume that the underlying interactions between the atoms has a finite range R and impose periodic boundary conditions. We denote an A–B scattering state with momentum p = |p| in the center-of-mass frame by Ψp and write it as 〈r|Ψp〉 = c · ∑ k∈Z3 ei 2pik L ·r( 2pik/L )2 − p2 (3.133) in configuration space, with some normalization constant c. In writing this expression, which corresponds to a pure S-wave, we have assumed that contributions from higher partial waves can be neglected. It is then valid as an approximation for all r outside shifted copies of the finite-range interaction if we neglect exponentially-suppressed contri- butions in the effective interaction between composite particles that are introduced by the bound-state wave functions. As discussed in Ref. [3], this is valid because for the current derivation we are only interested in the contributions due to the shifts in the binding energies. According to the formulas derived above, these are also exponentially small, but their contributions to the scattering are suppressed further by inverse powers of L. Note that upon discretization the expression in Eq. (3.133) essentially goes over into the lattice Green’s function defined in Section 3.5.1. Let Hˆ denote the Hamiltonian for the system we are considering. Acting with this on the state |Ψp〉 then gives 〈r|Hˆ|Ψp〉 = c · ( L 2pi )2 ∑ k∈Z3 ei 2pik L ·r [ p2 2µAB + E 2pik/L A (L) + E −2pik/L B (L) ] k2 − η (3.134) in configuration space [108], where µAB is the reduced mass of the A–B system and with η as defined in Eq. (3.130). In this expression, E ±2pik/L A,B (L) are the energy contributions (at volume L) due to the binding of the states A and B, moving with momentum ±2pik/L. By definition, they are zero for point particles. In particular, for our specific example here we have E 2pik/L A (L) = 0 for all L, but we keep it in the equations in order to give a more general result. The total energy of the A–B system is then EAB(p, L) = 〈Ψp|Hˆ|Ψp〉〈Ψp|Ψp〉 = 1 N ∑ k∈Z3 p2 2µAB + E 2pik/L A (L) + E −2pik/L B (L) (k2 − η)2 (3.135) 5As discussed in Ref. [102], Z00(s, η) is initially defined by the sum in Eq. (3.131) for Re(s) > 3/2 and then extended to the whole complex plane by analytic continuation; this is what we express with the limit in Eq. (3.131). 50 Chapter 3. Finite-volume calculations with the normalization factor N = ∑ k ( k2 − η)−2 . (3.136) The volume dependence of E −2pik/L B (L) is known from the derivation in Section 3.7.1. According to Eq. (3.127), it is given as ∆EB(k, L) ≡ E−2pik/LB (L)− E−2pik/LB (∞) = ∆m (0) B ( θ = −2piηBk ) = −|Aκ|2 e −κL µBL × ∑ i=x,y,z cos (2piηBki) +O ( e− √ 2κL ) , (3.137) where µB and ηB are the reduced mass and mass ratio of the constituents comprising the state B. If the state A is composite as well, one has an analogous expression for the contribution E 2pik/L A (L). The total volume dependence of the energy EAB can then be written as EAB(p, L)− EAB(p,∞) = τA(η) ·∆EA(0, L) + τB(η) ·∆EB(0, L) (3.138) with the topological volume factors τA,B(η) = 1 N ∑ k ∑ i=1,2,3 cos (2piηA.Bki) 3 (k2 − η)2 . (3.139) They are obtained by inserting the expressions for ∆EA,B(k, L) into the sum over all k in Eq. (3.135) and factoring out the volume dependence of the states A and B at rest. Eq. (3.138) is the desired contribution to the volume dependence of the energy levels EAB (determined in numerical calculations) that should be subtracted before applying Lu¨scher’s formula (3.130) for an extraction of the scattering phase shift. Figure 3.5 demonstrates the importance of this procedure. For the atom–dimer (more generally called “fermion–dimer” here) system it shows lattice results for the scattering length and effective range calculated from S-wave phase shifts that were extracted using Lu¨scher’s finite-volume formula (3.130). The results shown in the plot were obtained in a calculation by Bour et al., the full details of which can be found in Ref. [109]. They are shown here with the kind permission of the authors. Using dimensionless units obtained by rescaling all quantities with the dimer binding momentum κD, both the scattering length aFD and the effective range rFD are plotted against different values of the lattice spacing alatt on the x-axis, such that a continuum extrapolation can be performed. The two data sets shown in the plots were obtained using two different lattice Hamiltonians that yield the same continuum limit. One clearly sees that the correct values (indicated by the blue triangles in Fig. 3.5) are only reached when the topological corrections factors are included in the calculation. The effect is particularly prominent for the effective range, where otherwise the result is completely off. It should be pointed out here that the effect of the topological volume contributions is so strong in the chosen example because the dimer was tuned to be very shallow and 3.8. Summary and outlook 51 the volumes used in the numerical calculation were not very large. Together, these two factors enhance the relative importance of the binding-energy shifts. In a calculation with deeper dimers and/or larger volumes, the effect can be much weaker or even negligible. However, in general it is always there and should be taken into account, especially when the exact situation is a priori unknown. -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 1.16 1.18 1.2 1.22 1.24 1.26 1.28 κ D a F D κ D r F D κDalatt H1, full H2, full H1, τ(η) = 1 H2, τ(η) = 1 STM equation Figure 3.5: Lattice results and continuum extrapolation with error estimates for the fermion–dimer scattering length (top) and effective range parameter (bottom). For com- parison we show the continuum results obtained via the Skorniakov–Ter-Martirosian equa- tion. 3.8 Summary and outlook In this chapter we have derived explicit formulae for the mass shift of P- and higher-wave bound states in a finite volume and discussed their decomposition into states transforming according to the representations of the cubic group. We have compared our numerical 52 Chapter 3. Finite-volume calculations results for ` ≤ 2 with numerical calculations of the finite-volume dependence for lattice Gaussian and step potentials and found good agreement with the predictions. For ` ≥ 2, the mass shift of a given state (`,m) depends on the angular momentum projection m due to the breaking of rotational symmetry. Averaged over all m in a multiplet, however, the absolute value of the mass shift is even independent of `. The mass shift for states in representations of the cubic group is the same for all states, and its sign can be understood from the leading parity of the representations. We have furthermore derived corresponding expressions for the finite-volume mass shift in two-dimensional systems. With the known volume dependence, lattice calculations provide a method to extract asymptotic normalization coefficients of bound state wave functions, which are of interest, for example, in low-energy astrophysical capture reactions. Using a relation that will be discussed in more detail in Chapter 5, the asymptotic normalization and binding momentum of a shallow bound state can be used to extract the effective range from a simulation. Furthermore, we have shown how twisted boundary conditions arise if one studies dimer states in moving frames and how the mass shift for this case leads to topological corrections factors that have to be taken into account in finite-volume calculations of composite- particle scattering. A precise knowledge of these corrections is particularly important for processes involving shallow dimer states in volumes that are not very large. Our work provides a general framework for future lattice studies of molecular states with angular momentum in systems with short-range interactions. Applications to nuclear halo systems and molecular states in atomic and hadronic physics appear promising. An important next step would be to include Coulomb effects into the framework in order to investigate the volume dependence of bound states of charged particles, which are much easier to treat experimentally. In particular, this extension of the formalism is important if one wants to describe proton-halo nuclei. Another interesting direction would be to analyze the volume dependence of resonances along the lines of Refs. [104, 110]. Chapter 4 The Coulomb force Overview The Coulomb potential, although notoriously difficult to handle due to its long-range nature, is one of the most important and probably most thoroughly investigated inter- actions in quantum mechanics. We review here results from the vast literature on the subject with a focus on aspects that will be important for the following two chapters of this thesis. The first part, where we introduce the Coulomb wave functions and the mod- ified effective range expansion, is based largely on the introductory section of Ref. [5], but provides some additional details. In the second part we discuss the full off-shell Coulomb T-matrix. In particular, we give an approximate expression for this function in the case of a Yukawa-screened Coulomb potential, originally derived by Gorshkov [111]. This section contains, at least to the author’s best knowledge, some results that have not previously been published. 4.1 Coulomb wave functions For a pure Coulomb interaction the radial Schro¨dinger equation for two particles carrying electromagnetic charges Z1e and Z2e reads p2w`(r) = − d 2 dr2 w`(r) + `(`+ 1) r2 w`(r) + γ r w`(r) (4.1) with the Coulomb parameter γ = 2µαZ1Z2 , (4.2) and the fine-structure constant α = e2 4pi ≈ 1 137 . (4.3) We use here the letter w to denote the wave functions in order to distinguish them from the u`(r) describing the asymptotically non-interacting “neutral” system discussed in Chapter 2. 53 54 Chapter 4. The Coulomb force The solutions of (5.29) are the so-called Coulomb wave functions, and their properties are well-known. We use here the conventions introduced by Yost, Breit and Wheeler in Ref. [112] and summarize some important relations in the following. For a more compre- hensive discussion we refer to the review article by Hull and Breit [113]. Explicitly, we write the wave functions as [114, 115] F (p) ` (r) = 1 2 ∣∣∣∣∣ei pi 2 k Γ ( 1 2 +m− k) Γ(2m+ 1) ∣∣∣∣∣ e−ipi2 ( 12+m)Mk,m(z) (4.4a) G (p) ` (r) = Γ ( 1 2 +m− k)∣∣Γ(1 2 +m− k)∣∣e−ipi2 ( 12−m+k)Wk,m(z) + iF (p)` (r) , (4.4b) where ρ = pr , η = γ 2p , (4.5) and z = 2iρ , k = iη , m = `+ 1 2 . (4.6) The functions Mk,m and Wk,m are Whittaker functions, which can be expressed in terms of hypergeometric functions as Mk,m(z) = e − 1 2 zz 1 2 +m 1F1 ( 1 2 +m− k, 1 + 2m; z) , (4.7) Wk,m(z) = e − 1 2 zz 1 2 +mU ( 1 2 +m− k, 1 + 2m; z) . (4.8) 1F1(a, b; z) is Kummer’s function of the first kind, 1F1(a, b; z) = ∞∑ n=0 a(n)zn b(n)n! , a(n) = a(a+ 1) · · · (a+ n− 1) , (4.9) and U(a, b; z) = Γ(1− b) Γ(1 + a− b)1F1(a, b; z) + Γ(b− 1) Γ(a) z1−b1F1(a− b+ 1, 2− b; z) . (4.10) Due to their behavior in the limit ρ = pr → 0, F (p) ` (r) ∼ Cη,` ρ`+1 , (4.11a) G (p) ` (r) ∼ ρ−` Cη,`(2`+ 1) , (4.11b) F (p) ` and G (p) ` are commonly called the regular and irregular Coulomb wave functions, respectively. The factor Cη,`, which in the following we refer to as the Gamow factor, 1 is given by C2η,0 = 2piη e2piη − 1 , (4.12) 1Note that there is no general agreement about the name of Cη,` in the literature. It is sometimes also referred to as the Sommerfeld factor or, perhaps introduced as a sort of compromise, Gamow–Sommerfeld factor. 4.1. Coulomb wave functions 55 and C2η,` = 22` [(2`+ 1)!]2 ∏` s=1 (s2 + η2) · C2η,0 . (4.13) Sometimes it is convenient to write it in the more general form [103] Cη,` = 2` e− piη 2 [Γ(`+ 1 + iη)Γ(`+ 1− iη)] 12 Γ(2`+ 2) . (4.14) For asymptotically large ρ = pr, on the other hand the Coulomb wave functions behave as F (p) ` (r) ∼ sin(ρ− `pi/2− η log(2ρ) + σ`) , (4.15a) G (p) ` (r) ∼ cos(ρ− `pi/2− η log(2ρ) + σ`) (4.15b) with the Coulomb phase shift σ` = arg Γ(`+ 1 + iη) . (4.16) From Eqs. (4.11) and (4.15) it is clear that F (p) ` (r) and G (p) ` (r) are the direct analogues of the Riccati–Bessel functions S`(pr) and C`(pr) that solve the free radial Schro¨dinger equa- tion. The key differences to point out are the presence of the factors of Cη,` in (4.11) and of the additional phases η log(2pr) σ` in (4.15). The latter—in particular the logarithmically- divergent term—reflect the inherent long-range nature of the Coulomb potential: no matter how large the separation r of two charged particles becomes, the phase of their relative-motion wave function (determined solely by the regular function F (p) ` (r) for a pure Coulomb interaction) never goes to a constant, and the particles are thus never free [116]). 4.1.1 The Gamow factor The factor C2η,0 has a direct physical interpretation. To see this, note first that the full three-dimensional continuum Coulomb wave function (with outgoing asymptotics for the spherical wave and normalized such that the incoming plane-wave component ∼ exp(ip·r) has unit amplitude) can be written in terms of a confluent hypergeometric function as [117] ψ(+)p (r) = e −piη 2 Γ(1 + iη) eip·r1F1(iη, 1; ipr − ip · r) . (4.17) Its partial-wave expansion is given in terms of the regular Coulomb wave functions as (see Refs. [118, 119] and cf. also Ref. [120]) ψ(+)p (r) = ∞∑ `=0 (2`+ 1)i`eiσ` F (p) ` (r) pr P`(cos θ) . (4.18) It is clear that only the S-wave term contributes in the limit r → 0, such that by inserting the threshold behavior of F (p) ` (r) from Eq. (4.11a) we find that lim|r|→0 ψ(+)p (r) = e iσ`Cη,0 . (4.19) 56 Chapter 4. The Coulomb force Alternatively, one can derive directly from Eq. (4.17) that |ψ(+)p (0)|2 = e−piηΓ(1 + iη)Γ(1− iη) = C2η,0 , (4.20) where the last equality follows from Eq. (4.14). As pointed out in Ref. [117] (for example) this means that the S-wave Gamow factor (squared) is the probability of two charged particles to be found at zero separation.2 4.1.2 Analytic wave functions Of all possible linearly independent pairs of solutions for the Schro¨dinger equation (5.29), the Coulomb wave functions F (p) ` (r) and G (p) ` (r) are convenient to use because with their behavior at the origin and for asymptotically large distances, given in Eqs. (4.11) and (4.15), respectively, they correspond most directly to the solutions of the free Schro¨dinger equation discussed in Chapter 2. In contrast to those functions, however, F (p) ` (r) and G (p) ` (r) cannot directly be used in order to obtain expressions that are analytic in p 2. Rather, the fact that they are expressed in terms of the variables ρ = pr and η ∼ 1/p yields series expansions [112, 113] with a complicated “entanglement” of terms in p and r. As first done by Lambert in Ref. [121], it is possible to define Coulomb wave functions that are directly analytic in p2. Lambert’s result was later generalized by Bolle´ and Gesztesy in Ref. [122]. Their pair of analytic wave functions, which we denote by F (0) n (p, r) and G˜ (0) n (p, r) and discuss in more detail in Appendix A, will be very useful in the following Chapter 5. Here, we note for completeness that Coulomb wave functions analytic in the energy are also discussed in a more recent publication by Seaton [123]. 4.2 Modified effective range expansion Analogous to the discussion in Section 2.1.4 we now consider the case where a finite-range interaction V (r, r′) is present in addition to the Coulomb potential. We again assume that the interaction allows for a solution that is regular at the origin, and the finite-range condition (5.1) implies that V (r, r′) vanishes if r > R or r′ > R for some fixed but arbitrary range R. The radial Schro¨dinger equation now reads p2w`(r) = − d 2 dr2 w`(r) + `(`+ 1) r2 w`(r) + 2µ ∫ R 0 dr′ V (r, r′)w`(r′) + γ r w`(r) . (4.21) A general solution of Eq. (4.21) for momentum p can be written as a linear combination of the regular and irregular Coulomb wave functions F (p) ` and G (p) ` defined in the previous section, w (p) ` (r) ∝ [ cot δ˜`(p)F (p) ` (r) +G (p) ` (r) ] for r > R , (4.22) 2In particular, the wave function at the origin stays finite although the Coulomb potential diverges for r → 0. The latter is only a mathematical pathology that is always regulated by screening effects in a real physical system. 4.2. Modified effective range expansion 57 with an arbitrary overall normalization, and where δ˜` is the phase shift of the full solution w (p) ` compared to the regular Coulomb function F (p) ` [13]. Since it can be related to the partial-wave expansion of a function TSC defined by [124, 125] T = TSC + TC ⇐⇒ TSC = T − TC , (4.23) where T is the full T-matrix corresponding to the combined interaction Vˆ + VˆC and TC is the pure Coulomb T-matrix (see Section 4.4), δ˜` is called the “Coulomb-modified” or “Coulomb-subtracted” scattering phase shift. Instead of the ordinary effective range expansion (2.19) we now have the more complicated expression C2η,` p 2`+1 cot δ˜`(p) + γ h`(p) = − 1 aC` + 1 2 rC` p 2 + · · · , (4.24) where h`(p) = p 2` C2η,` C2η,0 h(η) , (4.25) h(η) = Reψ(iη)− log |η| , (4.26) and the digamma function ψ(z) = Γ′(z)/Γ(z) is the logarithmic derivative of the Gamma function. It means that in order to get an expression that is analytic in p2, one has to multiply p2`+1 cot δ˜`(p) by the Gamow factor squared and add an additional function that cancels the remaining non-analytic terms. Eq. (4.24) is called the Coulomb-modified effective range expansion. For ` = 0, it simplifies to C2η,0 p cot δ˜0(p) + γ h(η) = − 1 aC0 + 1 2 rC0 p 2 + · · · . (4.27) A derivation of Eq. (4.27) for the case of proton–proton scattering can be found, for example, in Ref. [13].3 See also Ref. [11] for a detailed discussion. The analytic properties of the ` = 0 modified effective range function are investigated in Ref. [126]. In Ref. [122], Bolle´ and Gesztesy derived a very general form of the Coulomb-modified effective range expansion for an arbitrary number of spatial dimensions. Specializing their result to the three-dimensional case, a version of Eq. (4.24) can be written as C2η,` p 2`+1 ( cot δ˜`(p)− i ) + γ h˜`(p) = − 1 aC` + 1 2 rC` p 2 + · · · (4.28) with4 h˜`(p) = (2p)2` Γ(2`+ 2)2 |Γ(`+ 1 + iη)|2 |Γ(1 + iη)|2 ( ψ(iη) + 1 2iη − log(iη) ) . (4.29) 3As a remark we note that on first sight the expansion given in Eq. (51) of Bethe’s paper [13] seems to be different from the one given here in Eq. (4.27), which is the same as given in later publications referring to Bethe’s result. The η-dependent function on the left hand side of Bethe’s expansion appears to differ from our h(η) by two times the Euler-Mascheroni constant γE . This apparent conflict can be resolved by noting that the g(η) in Eq. (51) of Ref. [13] is not the function defined in Eq. (47a) of the same paper, but rather given by limη1→∞[g(η)− g(η1)], where in this latter expression the g from Eq. (47a) is meant. The limiting process then yields exactly the term −2γE . 4This definition essentially comes from combining Eqs. (4.1) and (4.2) of Ref. [122], with the correction that the exponent in Eq. (4.2) should be −2 rather than 2. 58 Chapter 4. The Coulomb force The latter function can be rewritten using C2η,` = 22` Γ(2`+ 2)2 |Γ(`+ 1 + iη)|2 |Γ(1 + iη)|2 C 2 η,0 , (4.30) with C2η,0 as defined in Eq. (4.12). The expressions given here reproduce Eqs. (4.24) and (4.13) when one explicitly assumes that the momentum p is real. In fact, one has to rewrite Eq. (4.29) in this manner in order to get an effective range function that is analytic in p2 around threshold. The form of the Coulomb-modified effective range expansion for general ` that we have given in Eqs. (4.24) and (4.28) is the same as in Ref. [122]. Note that sometimes another convention, differing from ours by an overall momentum-independent factor, is used in the literature. The effective range expansion given in Refs. [124, 127, 128] can be written as ( Γ(2`+ 2) 2` Γ(`+ 1) )2 [ C2η,` p 2`+1 ( cot δ˜`(p)− i ) + γ h˜`(p) ] = − 1 a˜C` + 1 2 r˜C` p 2 + · · · . (4.31) This expression has the advantage of having a more direct connection to the ordinary effective range expansion without Coulomb effects. For ` = 0, both our choice and the form in Eq. (4.31) give the same expression. In this work we will, because of its simpler form, primarily use the convention of Eqs. (4.24) and (4.28). 4.3 Bound-state regime In order to discuss the bound-state regime for systems of charged particles we need to know the solutions of Eq. (5.29) with the appropriate (exponentially decaying) behavior. In other words, we need the Coulomb analogs of the Hankel functions H±` . 4.3.1 Asymptotic wave function Essentially, this role is played by the Whittaker functions W∓iη,`+ 1 2 . From the defini- tions (4.4) one directly sees that G (p) ` (r)− iF (p)` (r) = e−iσ`ei pi 2 (`−iη)Wiη,`+ 1 2 (2ipr) (4.32) where the prefactor is found by noting that with m and k as in Eq. (4.6) we have Γ ( 1 2 +m− k)∣∣Γ(1 2 +m− k)∣∣ = ei arg Γ(`+1−iη) = e−iσ` (4.33) due to the property Γ(z) = Γ(z) of the Gamma function. Using this, one can also write the Coulomb phase shift in the form eiσ` = ( Γ(`+ 1 + iη) Γ(`+ 1− iη) )1 2 , (4.34) 4.3. Bound-state regime 59 which is useful for showing that furthermore G (p) ` (r) + iF (p) ` (r) = e iσ`e−i pi 2 (`+iη) W−iη,`+ 1 2 (−2ipr) . (4.35) More precisely, this follows from the definitions (4.4) after a short calculation involving the relation [114] Mk,m(z) = e −ipik Γ(2m+ 1) Γ(1 2 +m− k)W−k,m(−z)− e ipi(m− 1 2 +k) Γ(2m+ 1) Γ(1 2 +m+ k) Wk,m(z) , (4.36) the second term of which cancels the Wk,m(z) from G (p) ` (r) after simplifying the prefactors with the help of (4.34). These and further relations for Coulomb wave functions with complex arguments and/or parameters can also be found in the articles by Humblet [129] and Dzieciol et al. [130]. According to Hull and Breit [113], the asymptotic behavior of the Whittaker functions for large |z| is Wk,m(z) ∼ e−z/2 zk , W−k,m(−z) ∼ ez/2 (−z)−k , (4.37) such that the normalizable bound-state solution is given by W−iη,`+ 1 2 (−2ipr) ∝ e −κr r|η| as |z| → ∞ (4.38) for bound-state momenta p = iκ, κ > 0. 4.3.2 Bound-state condition and ANC We now go back to the case where a finite-range interaction is present in addition to the Coulomb tail and consider solutions w (p) ` (r) of Eq. (4.21) with the asymptotic from as given in Eq. (4.22). Inverting Eqs. (4.32) and (4.35) in order to express W±iη,`+ 1 2 in terms of F (p) ` and G (p) ` and inserting the result into Eq. (4.22) gives w (p) ` (r) ∝ [ cot δ˜`(p)− i ] Wiη,`+ 1 2 (2ipr)− [ cot δ˜`(p) + i]e2iσ`e−ipi`W−iη,`+ 1 2 (−2ipr) (4.39) for r > R, in direct analogy to Eq. (2.18) that describes the case without Coulomb inter- action (see Section 2.1.3). Repeating the argument that for a bound state the component representing the incoming wave—given by the Wiη,`+1/2—has to vanish, one finds that the condition for the existence of a bound state with binding momentum κ is cot δ˜`(p = iκ) = i . (4.40) In other words, one simply has to replace the scattering phase shift δ`(p) in Eq. (2.21) with its Coulomb-modified analog δ˜`(p). Furthermore, we define a bound-state solution w (iκ) A,` (r) that behaves exactly like the Whittaker function, w (iκ) A,` (r) = A˜κ ·W−iη,`+ 12 (2κr) for r > R , (4.41) where A˜κ denotes the asymptotic normalization constant (ANC) for a bound state of charged particles. 60 Chapter 4. The Coulomb force 4.4 The Coulomb T-matrix For the pure Coulomb interaction it is possible to write down a closed expression for the full off-shell T-matrix (see Section 2.2.2). If we write the Coulomb interaction as an operator VˆC with 〈r|VˆC |r′〉 = δ(3)(r− r′)VC(r) with VC(r) = VC(r) ≡ γ 2µr , (4.42) 〈p|VˆC |q〉 = 2piγ µ 1 (p− q)2 ≡ VC(p,q) , (4.43) the Lippmann-Schwinger equation (2.46) for the Coulomb T-matrix TC reads TˆC(E) = VˆC + VˆC Gˆ (+) 0 (E) TˆC(E) , (4.44) where the energy E is a free (complex) parameter. Slightly altering our notation for the rest of this chapter, we introduce the center-of-mass momentum scale k (instead of denoting it by p as done so far) and write E = k2 2µ , η = γ 2k . (4.45) With this, a solution of Eq. (4.44) in momentum space can be written in the Hostler form5 [131–133] TC(k; p,q) = VC(p,q) { 1− 2iη ∫ ∞ 1 ( s+ 1 s− 1 )−iη ds s2 − 1− } , (4.46) where = (p2 − k2)(q2 − k2) k2(p− q)2 . (4.47) Alternatively, it can be recast in terms of hypergeometric functions as TC(k; p,q) = VC(p,q) { 1−∆−1 [ 2F1 ( 1, iη, 1 + iη; ∆− 1 ∆ + 1 ) − 2F1 ( 1, iη, 1 + iη; ∆ + 1 ∆− 1 )]} , (4.48) with the new variable ∆ defined via ∆2 = 1 + . (4.49) For future reference we note that this can be shown by using the integral representa- tion [103] 2F1(a, b; c; z) = Γ(c) Γ(b)Γ(c− b) ∫ 1 0 tb−1(1− t)c−b−1(1− tz)−a dt (4.50) 5Up to a prefactor (e2piη−1)−1 in front of the integral, this is the form given in Eq. (90) of Ref. [131]. Note, however, that the additional factor should actually not be there, which can be seen by starting from Eq. (86) in the same reference. 4.4. The Coulomb T-matrix 61 for the hypergeometric function to obtain first 2F1 ( 1, iη; 1 + iη; ∆− 1 ∆ + 1 ) = Γ(1 + iη) Γ(iη) ∫ 1 0 tiη−1 dt 1− t∆−1 ∆+1 = iη ∫ ∞ 1 t−iη dt t− ∆−1 ∆+1 , (4.51) and subsequently, using the transformation t = s+ 1 s− 1 , dt = 2 ds (s− 1)2 (4.52) to arrive at 2F1 ( 1, iη; 1 + iη; ∆− 1 ∆ + 1 ) − 2F1 ( 1, iη; 1 + iη; ∆ + 1 ∆− 1 ) = iη ∫ ∞ 1 ( s+ 1 s− 1 )−iη 2∆ ds s2 −∆2 . (4.53) This gives back the integral form (4.46) of TC(k; p,q) when inserted into Eq. (4.48). 4.4.1 Yukawa screening In many situations, most notably in numerical calculations, it is necessary to suppress the long range of the Coulomb potential by screening it a large distances. A popular choice to implement this is to replace the plain Coulomb interaction with a Yukawa potential, VC(r) −→ VC,λ(r) = γ 2µ e−λr r , (4.54) where the screening parameter λ can be interpreted as a photon mass. In momentum space, the Yukawa potential is given by VC,λ(p,q) = 2piγ µ 1 (p− q)2 + λ2 . (4.55) In fact, the Coulomb potential in momentum space is usually defined by taking the limit λ → 0 in this expression because the Fourier transform of 1/r is not immediately well- defined. From the above expression it is clear that the photon mass regulates the singu- larity that otherwise occurs in forward direction (q = p, i.e., for vanishing momentum transfer). Ref. [131] gives an expression for what we in the following call the “partially screened” Coulomb T-matrix TˆC,λ, originally derived by Gorshkov [111]. It is defined by the relation TˆC,λ = VˆC,λ + VˆC,λ Gˆ (+) 0 TˆC , (4.56) where we have not written out the energy dependence of the functions for notational convenience. Note that this is not a Lippmann–Schwinger equation because the operator that appears on the right-hand side is the unscreened Coulomb T-matrix TˆC . Still, TˆC,λ is an interesting 62 Chapter 4. The Coulomb force object to study because it can be written down as a closed expression that converges to the unscreened Coulomb T-matrix TˆC in the limit λ→ 0. Due to this property it is useful in numerical calculations where the the Coulomb interaction—with its pole at vanishing momentum transfer that would otherwise create problems—has to be regulated. Ideally, one would of course like to use an expression for the exact Yukawa T-matrix in such an approach, but no closed solution for that quantity is known so far. We thus propose here to use TˆC,λ as a pragmatic alternative and will discuss its application to the low-energy proton–deuteron system in Chapter 6. Since it has the right behavior in the limit λ→ 0, we expect it to adequately describe most of the nonperturbative Coulomb effects. Unfortunately, the expression given for TC,λ(k; p,q) in Eqs. (246) and (247) of Ref. [131] is no fully correct.6 Since in the original paper by Gorshkov [111] the limit λ→ 0 is taken without first giving the explicit form of the partially screened T-matrix, we will derive it here in the following. To this end we start from Eq. (244) of Ref. [131], which in our notation reads TC,λ(k; p,q) = VC,λ(p,q)− iη ∫ 1 0 dx Λ0(x) VC,λ−ikΛ0(x)(xp,q)× exp { −iη ∫ 1 x dx1 x1Λ0(x1) } , (4.57) where Λ0(x) is defined as the positive root of Λ20(x) = [ 1− (p/k)2x] (1− x) , (4.58) and VC,λ−ikΛ0(x) is just the Yukawa potential (4.55) with the substitution λ −→ λ−ikΛ0(x). We now consider the integral in Eq. (4.57). With the substitution [111] x = s2 − 1 s2 − (p/k)2 , dx ds = 2s ( 1− (p/k)2)( s2 − (p/k)2)2 (4.59) one finds that Λ0(x) = s(1− x) , (4.60) and since furthermore 1− x = 1− (p/k) 2 s2 − (p/k)2 , (4.61) the integral in the exponent is just∫ 1 x dx1 x1Λ0(x1) = ∫ ∞ s 2 ds1 s21 − 1 = log (s+ 1 s− 1 ) . (4.62) For the potential term under the integral we find VC,λ−ikΛ0(x)(xp,q) = 2piγ µ × (s 2 − (p/k)2) (s2 − 1)(q− p)2 + 1 k2 [ λ2(k2s2 − p2)− 2iλks(k2 − p2)− (k2 − q2)(k2 − p2) ] (4.63) 6This can be seen by a straightforward dimensional analysis of Eq. (247) in Ref. [131]. Furthermore, the prefactor in Eq. (246) is written in terms of the unscreened Coulomb potential, which is clearly not correct. 4.4. The Coulomb T-matrix 63 after a lengthy but straightforward calculation. Adding 0 = λ2(s2 − 1)− λ2s2 + λ2 (4.64) in the denominator, we can rewrite this as VC,λ−ikΛ0(x)(xp,q) = 2piγ µ × (s 2 − (p/k)2) (s2 − 1)[(q− p)2 + λ2]− 1 k2 [ (k2 − q2)(k2 − p2) ] + 1 k2 [ (k2 − p2)(λ2 − 2iλks) ] . (4.65) Finally, noting that the term in the numerator cancels against the same factor in dx Λ0(x) = 2 ds s2 − (p/k)2 , (4.66) and factoring out the Yukawa potential, we arrive at TC,λ(k; p,q) = VC,λ(p,q) { 1− 2iη ∫ ∞ 1 ( s+ 1 s− 1 )−iη ds s2 − 1− λ + ζλ(s) } (4.67) with λ = (k2 − p2)(k2 − q2) k2 [(q− p)2 + λ2] (4.68) and ζλ(s) = (k2 − p2)(λ2 − 2iλks) k2 [(q− p)2 + λ2] . (4.69) This expression is very similar to the integral form of the unscreened Coulomb T-matrix. The only differences are given by the new term ζλ(s) in the denominator and the fact that all singularities, both in the overall prefactor and under the integral, are now regulated by adding λ2. In fact, one directly sees that in the limit λ → 0, Eq. (4.67) converges to the unscreened expression given in Eq. (4.46). 4.4.2 Expression in terms of hypergeometric functions Something that is not noted in Refs. [131] and [111] is that—just like the unscreened Coulomb T-matrix—TC,λ(k; p,q) can also be expressed in terms of hypergeometric func- tions. To obtain this expression we first note that the denominator in Eq. (4.67) can be written as s2 − 1− λ + ζλ(s) = s2 − (1 + d1 ·D2)− (d2 ·D2)s (4.70) with d1 = k 2 − q2 − λ2 , d2 = 2iλk (4.71) and D2 = k2 − p2 k2 [(q− p)2 + λ2] . (4.72) 64 Chapter 4. The Coulomb force After making the transformation s = t+ 1 t− 1 , ∫ ∞ 1 ds = 2 ∫ ∞ 1 dt (t− 1)2 (4.73) we get TC,λ(k; p,q) = VC,λ(p,q) × { 1− 4iη ∫ ∞ 1 t−iη dt −D2(d1 + d2) t2 + (4 + 2D2 · d1) t−D2(d1 − d2) } . (4.74) To proceed further, we use the indefinite integral7∫ tν dt x2t2 + x1t+ x0 = 1 X1 2−ν tν ν { 2F1 (− ν,−ν; 1− ν;X+2 (t)) ·X+3 (t)−ν − 2F1 (− ν,−ν; 1− ν;X−2 (t)) ·X−3 (t)−ν } , (4.75) where X1 = √ x21 − 4x0x2 , X±2 (t) = x1 ∓X1 x1 + 2tx2 ∓X1 , X ± 3 (t) = tx2 x1 + 2tx2 ∓X1 . (4.76) Evaluating this at t = 1 is straightforward, but considering t → ∞ requires a little more care. From Eq. (4.76) one sees that X±2 (t) goes to zero like 1/t as t → ∞, such that the hypergeometric functions simply yield one in this limit. Since the potentially problematic (because ν = −iη) prefactor tν is cancelled by the numerator of X±3 (t)−ν with the remainder then going to zero as t → ∞, we can conclude that there is actually no contribution to the integral from the upper boundary in Eq. (4.74) and that its value is hence given by the right-hand side of Eq. (4.82) with t = 1. Before inserting this into Eq. (4.74), we subsequently apply the identities [103] 2F1(a, b; c; z) = (1− z)c−a−b 2F1(c− a, c− b; c; z) (4.77) and 2F1(a, b; c; z) = (1− z)−a 2F1 ( a, c− b; c; z z − 1 ) (4.78) to rewrite 2F1(−ν,−ν; 1− ν; z) = (1− z)1+ν 2F1(1, 1; 1− ν; z) = (1− z)ν 2F1 ( 1,−ν; 1− ν; z z − 1 ) . (4.79) This is useful because from Eq (4.76) one finds that for z = X±2 (t), (1− z)ν = 2νX±3 (t)ν , (4.80) 7This result has been obtained with the help of computer algebra software (Wolfram Mathematica). 4.4. The Coulomb T-matrix 65 canceling the inverse factors of this in Eq. (4.82). Moreover, the arguments simplify to z z − 1 = − 1 2 X±2 (t) X±3 (t) = −x1 ∓X1 2t x2 . (4.81) With this, we then have ∫ ∞ 1 tν dt x2t2 + x1t+ x0 = 1 νX1 { 2F1 ( 1,−ν; 1− ν;−x1 +X1 2x2 ) − 2F1 ( 1,−ν; 1− ν;−x1 −X1 2x2 )} . (4.82) Finally, applying the above result to Eq. (4.74), we can write the partially-screened Coulomb T-matrix as TC,λ(k; p,q) = VC(p,q) { 1−∆−1λ [ 2F1 ( 1, iη, 1 + iη;X−λ )− 2F1(1, iη, 1 + iη;X+λ ) ]} , (4.83) with ∆2λ = 1 + (k2 − p2)(k2 − q2 − λ2) k2 [(q− p)2 + λ2] − λ2(k2 − p2)2 k2 [(q− p)2 + λ2]2 (4.84) and X±λ = 2k2 [(q− p)2 + λ2] (1±∆λ) + (k2 − p2)(k2 − q2 − λ2) (k2 − p2) [(k + iλ)2 − q2] . (4.85) As it should, this reduces to the hypergeometric expression (4.48) for the unscreened Coulomb T-matrix in the limit λ→ 0. It is directly clear from Eqs. (4.84) and (4.49) that limλ→0 ∆2λ = ∆ 2 , (4.86) and a straightforward calculation then furthermore shows that limλ→0X±λ = ∆± 1 ∆∓ 1 . (4.87) 66 Chapter 4. The Coulomb force Chapter 5 Causality bounds for charged particles Overview In this chapter, we derive a generalization of the so-called Wigner causality bound for a system of charged particles, where the Coulomb force determines the long-range in- teractions. The majority of the material presented in the following sections has been published in Ref. [5]. The review of Coulomb wave functions and the modified effective range expansion from that reference have already been given in the preceding chapter; the remaining parts are included here in a slightly re-arranged and amended form. A part of Section 5.7 is based on results from Ref. [4] that have been omitted in Chapter 3 to put them in a more suitable context here. Finally, the discussion of causality bounds for van der Waals tails in Section 5.8 is summarized from Ref. [7], which was mainly worked out by S. Elhatisari. 5.1 Introduction The constraints of causality for two-body scattering with finite-range interactions were first derived by Wigner [134]. The causality bound can be understood as a lower bound on the time delay ∆t between the incoming and outgoing wave packets. When ∆t is negative, the outgoing wave packet departs earlier than for the non-interacting system. However, the incoming wave must first reach the interaction region before the outgoing wave can leave. In low-energy scattering this manifests itself as an upper bound on the effective range parameter. In Ref. [135], Phillips and Cohen derived this bound for S-wave scattering with finite-range interactions. Some constraints on nucleon–nucleon scattering and the chiral two-pion exchange potential were considered in Ref. [136], and relations between the scattering length and effective range have been explored for one- boson exchange potentials [137] and van der Waals potentials [138]. In Refs. [10, 139] the causality bounds for finite-range interactions were extended to an arbitrary number of space-time dimensions and arbitrary angular momentum. The extension to systems with 67 68 Chapter 5. Causality bounds for charged particles partial-wave mixing was first studied in Ref. [140]. Here, we consider the causality constraints for the scattering of two charged particles with an arbitrary finite-range interaction. This analysis, published in Ref. [4], is the first study of causality bounds that takes into account the long-range Coulomb force. The results presented here are relevant to studies of low-energy scattering of nuclei and nucleons using effective field theory (EFT), in particular for the application of effective field theory to the nuclear halo systems discussed in Section 2.4.3. There is an important connection between causality bounds and the convergence of effective-field-theory calculations with increasing order [140]. For local contact inter- actions, the range of the effective interaction is controlled by the momentum cutoff scale of the effective theory. In effective theories with non-perturbative renormalization, which typically occur in nuclear physics, exact cutoff-independence can generally not be achieved. There is a “natural” value of the cutoff at which all higher-order corrections scale as expected from dimensional analysis. If the cutoff is taken larger, “new physics” intervenes, the corrections scale unnaturally, and unitarity violations may occur. This is different from what one encounters in high-energy particle physics where the renormal- ization is typically perturbative and cutoff momenta can be chosen arbitrarily large. For calculations using dimensional regularization, the renormalization scale plays a similar role in regulating ultraviolet physics. The term “new physics,” in the above context, refers to details left out (integrated out) in the effective theory. In the case of halo EFT, these details are the finite size of the core nucleus and its internal excitations as well as the exponential tail of the pion-exchange interaction. Problems with convergence of the effective theory can occur if the cutoff scale is set higher than the scale of the new physics. It is desirable to have a more quantitative measure of when problems may appear, and this is where the causality bound provides a useful diagnostic tool. For each scattering channel we use the physical scattering pa- rameters to compute a quantity called the causal range, Rc. It is the minimum range for finite-range interactions consistent with the requirements of causality and unitarity. For any fixed cutoff scale, the causality bound marks a branch cut of the effective theory when viewed as a function of physical scattering parameters [140]. The coupling constants of the effective theory become complex when scattering parameters violating the causality bound are enforced. These branch cuts do not appear in perturbation theory; however, a nearby branch point can spoil the absolute convergence of the perturbative expansion. Our results can be viewed as a guide for improving the convergence of halo-EFT calcu- lations. In particular, if the cutoff momentum used in a calculation is too high, then problems with convergence may appear in some observables. Consequently, the causal range can be used to estimate the “natural” ultraviolet cutoff Λ of the effective theory as R−1c . The natural cutoff is optimal in the sense that no known infrared physics is left out of the theory and that all corrections involving the ultraviolet cutoff scale naturally [141, 142]. Increasing the cutoff beyond the natural value will not improve the accuracy of the calculation. The causality bounds also have an impact in the regime of bound states. For two-body halo states—or more generally whenever there is a shallow two-body bound state close to threshold—the same integral identity that yields the causality bound for the effective 5.2. Setup and preliminaries 69 range can be used to derive a relation between the asymptotic normalization constant (ANC) of the bound-state wave function, the binding momentum, and the effective range for the scattering of the two halo constituents. This relation can be shown to be equivalent to a result previously derived by Sparenberg et al. [143]. Its significance lies in the fact that the ANC is an important input parameter for the calculation of near-threshold radiative capture and photodissociation reactions. The causality bounds also constrain the range of model potentials that are fitted to scattering data in order to extract ANCs. The organization of this chapter is as follows. After briefly reviewing in Section 5.2 the theoretical setup for two charged particles with additional short-range interactions (dis- cussed in detail in Chapter 4), we derive the charged-particle causality bounds for arbitrary values of the orbital angular momentum in Section 5.3. This analysis includes both at- tractive and repulsive Coulomb forces. In Section 5.4 we define the causal range and then extract and discuss this quantity in Section 5.5 for several nuclear scattering processes including proton–proton, proton–deuteron, proton–3He, proton–alpha, and alpha–alpha scattering. Some numerical calculations are given in Section 5.6. In Section 5.7, we eluci- date the relation for asymptotic normalization constants mentioned above and extract, as an application, the ANCs of the excited 2+ and 1− states in 16O from α−12C scattering data. Before briefly touching the subject of causality bounds for other long-range forces (in particular, for a van der Waals potential) in Section 5.8, we then conclude with a summary of the main results and provide an outlook. 5.2 Setup and preliminaries We consider a two-particle system with reduced mass µ interacting via a finite-range potential with range R. As already done in the previous chapters, we write the interaction as a real symmetric operator with kernel V (r, r′) satisfying the finite-range condition, V (r, r′) = 0 if r > R or r′ > R . (5.1) In particular, we assume that the interaction is energy-independent. After giving a de- tailed formal derivation of the causality bounds in the following sections, we will come back to the question what the above assumptions mean for the application to (halo) EFT calculations in Section 5.5. In the absence of Coulomb interactions the system with (fixed but arbitrary) angular momentum ` is described by the radial Schro¨dinger equation, p2u (p) ` (r) = − d2 dr2 u (p) ` (r) + `(`+ 1) r2 u (p) ` (r) + 2µ ∫ R 0 dr′ V (r, r′)u(p)` (r ′) . (5.2) As done in Chapter 2, we adopt the conventions of Ref. [10] and choose the normalization of u (p) ` that for r ≥ R we have u (p) ` (r) = p ` [cot δ`(p)S`(pr) + C`(pr)] , (5.3) where S` and C` are the Riccati-Bessel functions and δ`(p) is the scattering phase shift. 70 Chapter 5. Causality bounds for charged particles If the particles carry electromagnetic charges Z1e and Z2e, respectively, there is a Coulomb potential in addition to the finite-range interaction. As done in Chapter 4, we write this as VC(r) = γ 2µr = αZ1Z2 r , (5.4) such that in the radial Schro¨dinger equation we simply get a term ∼ γ/r because the factor of 2µ in the denominator cancels out. From Section 4.2 we quote it in the form p2w (p) ` (r) = − d2 dr2 w (p) ` (r) + `(`+ 1) r2 w (p) ` (r) + 2µ ∫ R 0 dr′ V (r, r′)w(p)` (r ′) + γ r w (p) ` (r) , (5.5) where again we use the superscript “(p)” to denote the solution for a given center-of-mass momentum p. We choose the normalization of w (p) ` such that for r ≥ R we have1 w (p) ` (r) = p `Cη,` [ cot δ˜`(p)F (p) ` (r) +G (p) ` (r) ] , (5.6) with the Coulomb-subtracted phase shift δ˜` and the regular and irregular Coulomb wave functions F (p) ` and G (p) ` as defined in Chapter 4. The inclusion of the Gamow factor in the normalization Cη,` will be convenient later, when we rewrite Eq. (5.6) in terms of a different pair of functions and relate it to the Coulomb-modified effective range expansion (4.24), rearranged in the form C2η,` p 2`+1 cot δ˜`(p) = −γ h`(p)− 1 aC` + 1 2 rC` p 2 + · · · , (5.7) with h`(p) as defined in Chapter 4. 5.3 Derivation of the causality bound With the Coulomb wave functions and Coulomb-modified effective range expansion at our hands, we can now closely follow the derivation presented in Ref. [10] for scattering in the absence of Coulomb interactions. 5.3.1 Wronskian identities We consider solutions of the radial Schro¨dinger equation (4.21) for two different momenta pA and pB. Introducing the short-hand notation wA,B(r) = w (pA,B) ` (r) , (5.8) i.e., suppressing the angular-momentum subscript ` for convenience, we get (p2B − p2A) ∫ r dr′wA(r′)wB(r′) = (wBw′A − wAw′B) ∣∣r − 2µ ∫ r dr′ ∫ R 0 dr [wB(r)V (r, r ′)wA(r′)− wA(r)V (r, r′)wB(r′)] (5.9) 1Note that for ` = 0 our normalization is the same as chosen in Ref. [13], i.e., for r ≥ R our solution w (p) 0 coincides with the function ϕ defined in Eq. (42) of that paper. 5.3. Derivation of the causality bound 71 by subtracting wA times the equation for wB from that for wB multiplied by wA, as it is done in Ref. [10], and integrating from some small radius to r. We assume that our interaction V (r, r′) is such that it alone (without the additional Coulomb potential) permits a solution that is sufficiently regular at the origin, i.e., u`(0) = 0 and ∂ru` stays finite as r → 0, where u` is a solution of Eq. (5.2). As boundary condition for the solutions wA,B of the full radial Schro¨dinger equation we can then demand as well that they vanish with finite derivative at the origin. If we only had the Coulomb potential and no additional interaction, this is fulfilled by the regular Coulomb function F (p) ` (r), cf. Eq. (4.11a). We can thus take the limit → 0 in Eq. (5.9) and get the relation W [wB, wA](r) = (p 2 B − p2A) ∫ r 0 dr′wA(r′)wB(r′) , (5.10) where the Wronskian W [wB, wA] is defined as W [wB, wA](r) = wB(r)w ′ A(r)− wA(r)w′B(r) . (5.11) 5.3.2 Rewriting the wave functions Following further the derivation presented in Ref. [10], we re-express the solutions w (p) ` (r) in terms of functions f(p, r) and g(p, r) such that w (p) ` (r) = p 2`+1C2η,` cot δ˜`(p) f(p, r) + g(p, r) (5.12) for r ≥ R, with f(p, r) analytic in p2, f(p, r) = f0(r) + f2(r) p 2 +O(p4) , (5.13) and g(p, r) = g˜(p, r) + φ(p) · f(p, r) . (5.14a) The g(p, r) contains a term which is non-analytic in p2 and is proportional to f(p, r). The remainder g˜(p, r), however, is analytic in p2, g˜(p, r) = g0(r) + g2(r) p 2 +O(p4) . (5.14b) Combining Eqs. (5.6) and (5.12), we find f(p, r) = 1 p`+1Cη,` F (p) ` (r) (5.15a) and g(p, r) = p`Cη,`G (p) ` (r) . (5.15b) These functions are directly related to the analytic Coulomb wave functions of Bolle´ and Gesztesy [122] mentioned in Section 4.1.2 and discussed further in Appendix A. In fact, one simply has that f(p, r) is exactly the F (0) n (p, r) defined in Eq. (A.3a), whereas comparison of Eqs. (5.15b) and (A.3b) shows that g(p, r) = G˜(0)n (p, r) + ( γ h˜`(p)− ip2`+1C2η,` ) · F (0)n (p, r) (5.16) 72 Chapter 5. Causality bounds for charged particles with G˜ (0) n (p, r) as defined in Eq. (A.4). This implies that g˜(p, r) = G˜(0)n (p, r) (5.17) and φ(p) = γ h˜`(p)− ip2`+1C2η,` = γ h`(p) , (5.18) where the last step follows from the combination of Eqs. (4.29) and (A.4) after a short calculation. When we insert now the modified effective range expansion (4.24) into the asymptotic Coulomb wave function (5.12), the non-analytic term involving h`(p) conveniently drops out and we are left with w (p) ` (r) = ( − 1 aC` + 1 2 rC` p 2 + · · · ) f(p, r) + g˜(p, r) for r ≥ R . (5.19) Thus, it is possible to choose a normalization such that w (p) ` (r) is analytic in p 2. Combining this with the expansions (5.13) and (5.14), we arrive at w (p) ` (r) = − 1 aC` f0(r) + g0(r) + p 2 [ 1 2 rC` f0(r)− 1 aC` f2(r) + g2(r) ] +O(p4) . (5.20) 5.3.3 The causality-bound function From here we can proceed exactly as in Ref. [10]. For the Wronskian of two solutions wA and wB for r ≥ R we find W [wB, wA](r) = (p 2 B − p2A) { 1 2 rC` W [f0, g0](r) + ( 1 aC` )2 W [f2, f0](r) − 1 aC` [W [f2, g0](r)−W [g2, f0](r)] +W [g2, g0](r) } +O(p4A,B) . (5.21) Note that in the O(p4A,B) we have also included terms of the form p2Ap2B. We set pA = 0 in Eq. (5.10) and furthermore take the limit pB → 0. Using the expansion (5.21), we get −rC` W [f0, g0](r) = bC` (r)− 2 ∫ r 0 dr′ [ w (0) ` (r ′) ]2 (5.22) for r ≥ R, with w(0)` (r) = limp→0w(p)` (r) and the causality-bound function bC` (r) = 2W [g2, g0](r)− 2 aC` {W [f2, g0](r) +W [g2, f0](r)}+ 2 (aC` ) 2W [f2, f0](r) . (5.23) Written as a function of ρ = p · r, the Wronskian of the Coulomb wave functions is2 W [F (p) ` , G (p) ` ](ρ) = −W [G(p)` , F (p)` ](ρ) = −1 . (5.24) 2See, for example, Eq. (14.2.4) in Ref. [103]. 5.3. Derivation of the causality bound 73 Since d/dr = p · d/dρ and W [f, f ] ≡ 0, we also have W [f, g](r) = W [f, g˜](r) = −1 . (5.25) Plugging in the expansions (5.13) and (5.14), we see that W [f0, g0](r) = −1 for the leading-order functions, and W [f2, g0](r) = W [g2, f0](r) for the terms at O(p2). Inserting these relations into Eq. (5.22), we get rC` = b C ` (r)− 2 ∫ r 0 dr′ [ w (0) ` (r ′) ]2 , (5.26) where bC` (r) has been simplified to bC` (r) = 2W [g2, g0](r)− 4 aC` W [f2, g0](r) + 2 (aC` ) 2W [f2, f0](r) . (5.27) Since the integral in Eq. (5.26) is positive definite, the resulting causality bound is rC` ≤ bC` (r) , ∀ r ≥ R . (5.28) 5.3.4 Calculating the Wronskians We now derive the explicit form of the function bC` (r). To do this, we need expressions for the Wronskians that appear in Eq. (5.27). We can obtain them by first noting that f(p, r) and g˜(p, r), being linear combinations of Coulomb wave functions (with p-dependent co- efficients), are solutions of the Coulomb Schro¨dinger equation,[ − d 2 dr2 + `(`+ 1) r2 + γ r − p2 ] x(p, r) = 0 , (5.29) which, of course, corresponds to setting V (r, r′) = 0 in Eq. (4.21). Here and in the following, x stands for either f or g˜. Inserting the expansion x(p, r) = x0(r) + p 2 x2(r) +O(p4) (5.30) into Eq. (5.29) and comparing orders in p2, we find that[ − d 2 dr2 + `(`+ 1) r2 + γ r ] x0(r) = 0 , (5.31) i.e., x0 is a solution of the zero-energy Coulomb Schro¨dinger equation, and[ − d 2 dr2 + `(`+ 1) r2 + γ r ] x2(r) = x0(r) . (5.32) From this we readily obtain the differential equations d dr W [f2, f0](r) = [f0(r)] 2 , (5.33a) d dr W [g2, g0](r) = [g0(r)] 2 , (5.33b) d dr W [f2, g0](r) = f0(r)g0(r) (5.33c) 74 Chapter 5. Causality bounds for charged particles for the desired Wronskians. Put together, this yields a simple first-order differential equation for bC` (r), d dr bC` (r) = 2 ( g0(r)− 1 aC` f0(r) )2 . (5.34) From Eqs. (A.7) and (A.8) in Ref. [122] we have the explicit expressions f0(r) = (2l + 1)!√ γ2l+1 √ r I2`+1(2 √ γr) , (5.35a) g0(r) = 2 √ γ2l+1 (2l + 1)! √ r K2`+1(2 √ γr) (5.35b) for γ > 0, where Iα and Kα are modified Bessel functions, and f0(r) = (2l + 1)!√ (−γ)2l+1 √ r J2`+1(2 √−γr) , (5.36a) g0(r) = −pi√(−γ)2l+1 (2l + 1)! √ r N2`+1(2 √−γr) (5.36b) for γ < 0, where Jα and Yα are the ordinary Bessel functions. 3,4 Using these expressions for f0 and g0 and Eq. (5.34) we can determine b C ` (r) up to an integration constant. In order to fix this constant, we must work directly with the Wron- skians in Eq. (5.23). Before we do that, however, we first discuss the general form of bC` (r). We break apart the function as a sum of two functions, X`(r) and Y`(r), and a constant term Z`, bC` (r) = X`(r) + Y`(r) + Z` . (5.37) We take X`(r) to be a function consisting entirely of a sum of terms that have a pole at r = 0, ranging from order 1 to `, X`(r) = ∑` m=1 X`,mr −m . (5.38) By furthermore requiring the function Y`(r) to vanish at r = 0, the decomposition in Eq. (5.37) is unique. Where exactly the contributions to the three terms in the decomposition originate from can be inferred from the behavior of f(p, r) and g(p, r) at the origin. From Eq. (4.11a) combined with Eq. (5.15) we find that f(p, r) ∼ r`+1 as r → 0 . (5.39) 3From Eq. (9.1.50) and the remark above Eq. (9.6.41) in Ref. [103] it is clear that these f0 and g0 are indeed solutions of Eq. (5.31). 4Bolle´ and Gesztesy actually give an expression for g0 in the attractive case (γ < 0) that involves the Hankel function H (2) α times i instead of the Neumann function Nα (also called Bessel function of the second kind and denoted then by Yα). With that, however, g0 would not be real, which it should be. Our g0 as in Eq. (5.36b) is taken from the results of Lambert [121]. 5.3. Derivation of the causality bound 75 This implies that every term in the expansion of f(p, r) is O(r`+1). Therefore, limr→0W [f2, f0](r) = 0 (5.40) for all `, which means that this Wronskian only yields contributions to the Y`(r). Furthermore, from Eq. (17) in Ref. [112] we know that the irregular Coulomb wave func- tion has the asymptotic behavior G (p) ` ∼ Dη,` ρ−` as ρ→ 0 , (5.41) with Dη,` fulfilling Cη,`Dη,` = 2`+ 1. Using Eq. (5.15b) then yields g(p, r) ∼ r −` 2`+ 1 as r → 0 . (5.42) We note that g0(r) has exactly the same behavior near r = 0 and can thus show that g2(r) is subleading as r → 0, g2(r) ∼ r−`+c for c > 0. From this we infer that limr→0W [f0, g2](r) = 0 (5.43) for all `, so also from this Wronskian we only get contributions to Y`(r). Both the singular X`(r) and the constant Z`, therefore, only come from the Wronskian W [g2, g0](r). For ` = 0 the situation is still simple because the above analysis also tells us that limr→0W [g2, g0](r) = 0 for ` = 0 , (5.44) i.e., bC0 (r) is given entirely by Y0(r). With the knowledge that it vanishes at the origin, it is actually straightforward to give an explicit expression for bC0 (r) in terms of antiderivatives of the right hand side of Eq. (5.34), where one has to insert the f0(r) and g0(r) from Eqs. (5.35) and (5.36). The result, obtained by integrating from 0 to r, is bC0 (r) = 2r3 3 ( aC0 )−2 1F2 ( 3 2 ; 2, 4; 4γr ) − 4r 2 √ pi ( aC0 )−1 G2,11,3 ( 4γr ∣∣∣∣ 120, 1,−2 ) + 4 √ piγr2G3,12,4 ( 4γr ∣∣∣∣ −1, 12−1, 0, 1,−2 ) (5.45) for the repulsive case, and bC0 (r) = 2r3 3 ( aC0 )−2 1F2 ( 3 2 ; 2, 4; 4γr ) + 4 √ pir2 ( aC0 )−1 G2,12,4 ( −4γr ∣∣∣∣ 12 ,−120, 1,−2,−1 2 ) + 2pi2 [ γ2r3 3 1F2 ( 3 2 ; 2, 4; 4γr ) − 2γr 2 √ pi G3,13,5 ( −4γr ∣∣∣∣ −1, 12 ,−12−1, 0, 1,−2,−1 2 )] (5.46) for an attractive Coulomb interaction. In the above equations, pFq and G m,n p,q denote the (generalized) hypergeometric and Meijer G-functions, respectively. For general ` ≥ 1, W [g2, g0](r) is singular at r = 0 and the analysis becomes more complicated. For practical purposes one can simply use power-series expansions for the Bessel functions that appear in the expressions for the zero-energy functions and integrate these term by term until a desired precision is reached. The only additional ingredients needed are the values for the constant terms Z` because these are obviously not generated by the integration. 76 Chapter 5. Causality bounds for charged particles 5.3.5 Constant terms in the causality-bound function Obtaining these constants turns out to be the central difficulty in the derivation of the causality bound for the charged-particle system, which so far—apart from having to cope with more complicated expressions—very closely followed the path laid out for the neutral system in Ref. [10]. In order to determine the Z` we consider the explicit form of g˜(p, r). From the results of Bolle´ and Gesztesy [122], we have5 g˜(p, r) = N`(p) · γ log (|γ|r) · f(p, r) + γ Re { N`(p) · r`+1 e−ipr · ∞∑ n=0 [a`,n(p) + b`,n(p)] r n } + Re { r−` 2`+ 1 e−ipr · 2∑` n=0 d`,n(p) r n } , (5.47) where N`(p) = (2p) 2` Γ(2`+ 2)2 ∏` s=1 (s2 + η2) , (5.48) a`,n(p) = −Γ(2l + 2) Γ(n+ 1)Γ(n+ 2`+ 2) (2ip)n n∏ s=1 (s+ `− iη) · [ψ(n+1)+ψ(n+2`+2)] , (5.49a) b`,n(p) = Γ(2l + 2) Γ(n+ 1)Γ(n+ 2`+ 2) (2ip)n n∏ s=1 (s+ `− iη) · n+∑` j=1 1 j − iη , (5.49b) and d`,n(p) = 1 Γ(n+ 1) (2ip)n n∏ s=1 ( s− `− 1− iη s− 2`− 1 ) . (5.50) With this result and the appropriate expression for g0(r) from Eq. (5.35) or (5.36) one can use the following procedure to calculate the Z`, i.e., the terms of order r 0 in the Wronskian W [g2, g0](r). 1. Note that W [g˜, g0](r) = p 2W [g2, g0](r) +O(p4) (5.51) and calculate this Wronskian using a truncated version of g˜(p, r) as given in Eq. (5.47). Including terms up to the order 2`+ 1 in r is sufficient. 2. From the result, extract the terms that are of the order r0. 3. From that expression then extract the terms that are of the order p2. They consti- tute the O(r0) contributions in a series expansion of W [g2, g0](r) which cannot be obtained from a term-by-term integration of g0(r) 2. 5One gets this form from Eq. (A.5) in Appendix A by inserting n = 2`+ 3, η = γ/(2p) and assuming that the momentum p is real and positive. 5.4. The causal range 77 With the help of computer algebra software, this prescription is straightforward to imple- ment. The results for ` = 0, . . . , 2 are shown in Table 5.1. ` 0 1 2 Z` 0 γ ( 1 6 − 2γE 9 ) γ3 ( 79 21600 − γE 360 ) Table 5.1: Constant term Z` in Eq. (5.37) for ` = 0, 1, 2. γE = 0.577216 . . . is the Euler-Mascheroni constant. The values are the same for repulsive (γ > 0) and attractive (γ < 0) Coulomb potentials. At this point we remark that, in principle, it is also possible to use the wave functions defined by Seaton [123] to get explicit expressions for f2(r) and g2(r) (in addition to the already known zero-energy functions), and then simply calculate all the Wronskians directly. However, the analytic irregular Coulomb function defined by Seaton is slightly different from our g˜(p, r). More importantly, not all coefficients needed for the expansions are given explicitly. Finally, writing everything in terms of the wave functions of Bolle´ and Gesztesy paves the way for a generalization of the results presented here to an arbitrary number of spatial dimensions. Knowing now what the constant terms Z`, it is possible to write down explicit expressions for the causality-bound functions bC` (r) also for ` > 0. To do that we use that the antiderivative of the right-hand side of Eq. (5.34) can be expressed in terms of (generalized) hypergeometric functions pFq and Meijer G-functions G m,n p,q . The only additional point to be taken into account is that the antiderivative of g0(r) 2 in general includes a constant term that is different from the desired Z`. Hence, one has to determine this term and add another constant such that their sum is exactly equal to the Z` given in Table 5.1. We have carried out this procedure explicitly for ` = 1 and ` = 2. Since the results are rather lengthy, we give the complete expressions for bC1 (r) and b C 2 (r) in Appendix B. 5.4 The causal range The causality-bound equation (5.28) can be rewritten as bC` (r)− rC` ≥ 0 ∀ r ≥ R . (5.52) In cases where the details of the interaction (in particular its range, assuming that a description with finite-range potentials is applicable) is not known, one can use Eq. (5.52) to define the causal range Rc of a scattering system as that value of r for which the bound is just satisfied, i.e., bC` (Rc)− rC` = 0 . (5.53) We note from Eq. (5.34) that the derivative of bC` (r) is non-negative, d dr bC` (r) = 2 ( g0(r)− 1 aC` f0(r) )2 ≥ 0 . (5.54) 78 Chapter 5. Causality bounds for charged particles Hence, bC` is an increasing function of r and the causal range is defined uniquely. For the case that Eq. (5.53) does not have a solution (i.e., if bC` (r) is positive already for r = 0) we define the causal range to be zero. Note that the causal range is a function only of the scattering length and the effective range. It can thus be calculated from observables in a well-defined way. The importance of the causal range is given by the fact that it can be interpreted as the minimum range a potential is allowed to have to be consistent with causality. If, for a given system, the values in individual partial waves differ significantly, the maximum value should be taken as the causal range of the underlying potential. Alternatively, one can model the interaction with an `-dependent potential. For effective field theories with short-range interactions such as halo EFT, the causal range constrains the allowed values of the momentum-space cutoff or the lattice spacing used in numerical calculations. 5.4.1 Practical considerations At this point we recall that our derivation of the causality bounds was based on the assumption that the concrete system under consideration is described by a finite-range (though possibly non-local) two-body interaction which is energy-independent. In EFT calculations one frequently obtains effective interactions that explicitly depend on the energy. We note that this energy dependence can be traded for momentum dependence at any given order in the power counting (EFT expansion) by using the equations of mo- tion obtained from the effective Lagrangian. However, the energy dependence introduces another length scale into the system, and so the conversion to momentum-dependent in- teractions could produce an interaction range so large that the causality bounds may not be useful in practice. There are also other theoretical frameworks, e.g., Feshbach reaction theory, that explicitly use energy-dependent interactions. Here again the energy dependence introduces a length scale which acts as an interaction range. This can be seen from the time delay of the scattered wavepacket, which is proportional to the derivative of the phase shift with respect to energy. By setting up a very strong energy dependence for the interactions it is possible to produce a time delay which is arbitrarily large and negative. This has the same effect as interactions at arbitrarily large separations. Furthermore, the assumption of a strict finite range certainly is an idealization that is only applicable to a varying degree of validity to concrete physical systems. For example, there can be exchange forces arising from the Pauli principle. Consider, for example, nuclear halo systems with a tightly bound core and a halo nucleon which is only weakly bound to the core. The exchange of a nucleon from the core and the halo nucleon that is necessary to anti-symmetrize the system can only give a sizable contribution if there is spatial overlap between the wave function of the core and the wave function of the halo nucleon. This yields a short-range exponential tail that, within the domain of validity of the effective theory, can be subsumed in the effective range parameters of the halo–core interaction. The same analysis would apply to low-energy nucleon–nucleus scattering upon the core nucleus. Another more prominent effect is given by exponential tails generated by simple pion-exchange contributions; cf. Ref. [140] and the discussion in the following section. 5.5. Examples and results 79 5.5 Examples and results We now calculate explicit values for causal ranges in few-nucleon systems. In Fig. 5.1 we plot the left-hand side of Eq. (5.52) as a function of r for the case of proton–proton S-wave scattering. The causal range can then be read off as the point where the function becomes zero. Fig. 5.2 shows analogous plots for α–α S- and D-wave scattering. In this system, there are visible error bands due to the larger uncertainties in the effective range parameters. 0 0.5 1 1.5 2 2.5 3-3 -2 -1 0 1 2 3 4 r (fm) bC (r ) − rC (f m ) B Figure 5.1: Causal range plot for S-wave proton–proton scattering. In Table 5.2 we give a summary of the causal ranges that one finds for various two-body systems of light nuclei where low-energy scattering parameters and/or phase shifts are available from experiments. The results are briefly discussed in the following subsections. 5.5.1 Proton–proton scattering For p–p S-wave scattering one finds a causal range of about 1.38 fm. This value is very close to the range estimate obtained by assuming that the typical length scale of the N–N interaction is set by the inverse pion mass, ~c/Mpi ≈ 1.4 fm. The value one finds in the 3P0 channel is somewhat larger (Rc ≈ 2.3 fm), whereas the 3P1 effective range parameters impose almost no constraint on the range of the nuclear potential in this channel. As we will discuss in more detail below, this suggests some significant differences in the radial dependence of the interactions for the 3P1 channel. For effective-field-theory calculations with purely local interactions (e.g., pionless effective field theory), our results suggest to keep the cutoff momentum smaller than Mpi for the 1S0 and 3P0 channel. However, there is more freedom to take a higher cutoff for the 3P1 channel. In Ref. [140], causality bounds were investigated for neutron–proton scattering. The 80 Chapter 5. Causality bounds for charged particles -0.3 -0.2 -0.1 0 0.1 1 1.5 2 2.5 3 3.5-0.02 -0.015 -0.01 -0.005 0 0.005 r (fm) bC (r ) − rC (f m ) D-wave S-wave DˆB Figure 5.2: Causal range plot for S-wave α–α scattering. System Reference Channel aC/ fm2`+1 rC/ fm−2`+1 Rc / fm p–p [144] 1S0 −7.828± 0.008 2.80± 0.02 1.38± 0.01 p–p [145] 3P0 −3.03± 0.11 4.22± 0.11 2.33± 0.05 p–p [145] 3P1 2.013± 0.053 −7.92± 0.17 ≈ 0.03 p–d [146] 2S1/2 2.73± 0.10 2.27± 0.12 3.90± 0.15 p–d [147] 2S1/2 4 −2.8 0 p–d [146] 4S3/2 11.88± 0.40 2.63± 0.02 2.20+0.07−0.06 p–d [147] 4S3/2 11.11 2.64 2.29 p–3He [148] 1S0 11.1± 0.4 1.58± 0.12 1.32+0.21−0.17 p–3He [148] 3S1 9.04± 0.14 1.50± 0.06 1.27+0.10−0.09 p–α [149] S1/2 4.97± 0.12 1.295± 0.082 1.32+0.40−0.21 p–α [149] P1/2 −19.36± 0.50 0.349± 0.021 2.65± 0.07 p–α [149] P3/2 −44.83± 0.51 −0.365± 0.113 0.49+0.17−0.10 α–α [62] S (−1.65± 0.17) · 103 1.084± 0.011 2.58+0.14−0.13 α–α [150] D (−7.23± 0.61) · 103 (−1.31± 0.22) · 10−3 2.03+0.09−0.07 Table 5.2: Summary of causal-range results obtained from experimental input for various few-nucleon systems. 5.5. Examples and results 81 results (Rc = 1.27 fm for 1S0, Rc = 3.07 fm for 3P0, and Rc = 0.23 fm for 3P1) are qualitatively very similar to what we find for the p–p system. This would indicate only a moderate amount of isospin breaking. In the same publication [140], the influence of the shape of the potential upon the neutron– proton causal range was also studied numerically. When the potential is repulsive at shorter distances (less than ∼ 1 fm) and attractive at larger distances (greater than ∼ 1 fm), the causal range comes out on the larger side, about 2 fm or more. When the potential is attractive at intermediate distances and repulsive at larger distances, then the causal range is smaller, about 1 fm or less. The pion tail determines the sign of the potential at larger distances. For both the n–p and the p–p interaction, the one-pion exchange tail is repulsive in the 3P1-channel while it is attractive in the 3P0-channel. Note that causality bounds in the presence of pion-exchange contributions were also dis- cussed by Phillips and Cohen in Ref. [135]. Ideally, one would account for the one-pion- exchange tail explicitly in the calculation of the causal range, as it was done in this work for the long-range Coulomb potential. Without knowing analytical solutions of the Schro¨dinger equation involving a Yukawa-like potential (plus a Coulomb part, in the p–p case), however, such a procedure can at best be implemented numerically. For an example, see Ref. [151]. 5.5.2 Proton–deuteron scattering There are several experimental determinations of p–d effective range parameters. In Ta- ble 5.2 we have included results from Arvieux [146] and Huttel et al. [147]. While for the quartet-channel there is a good agreement between the scattering lengths and effective ranges (and, of course, of the resulting causal ranges, which come out as 2.2–2.3 fm), there is a large discrepancy for the doublet-channel results. The difficulty of determining the proton–deuteron doublet-channel scattering length has previously been discussed by Orlov and Orevkov [152]. Comparing different models, the authors conclude that 0.024 fm is currently the best theoretical estimate for the doublet-channel p–d scattering length. From Table 3 in Ref. [152] one reads off that the corresponding value for the Coulomb-modified effective range is as huge as 8.23 · 105 fm. Inserting these numbers into the causal-range calculation one gets a very large value of Rc ≈ 8.15 fm. As a consequence, no definite conclusion can be reached with the currently available data. Note that three-body forces have been found to be very important for theoretical calcula- tions of the p–d (and n–d) threshold scattering parameters. Our analysis here, however, is independent of the microscopic origin of the effective interaction between proton and deuteron. In a detailed picture, the force might arise from two-nucleon forces or three- nucleon forces, but the result is always some effective two-body interaction between the proton and the deuteron. The causal range we calculate is the minimum range that this effective interaction has to have in order to be able to reproduce the experimentally determined scattering parameters. 82 Chapter 5. Causality bounds for charged particles 5.5.3 Proton–helion scattering For the scattering of protons off a helium nucleus we were able to find data for both p–3He and p–α scattering. In the first case, there was only enough data available to calculate the causal range for the S-wave channels. Since both the scattering lengths and effective ranges are very similar for the singlet and the triplet channel, so are the resulting causal ranges, which come out as approximately 1.3 fm. Incidentally, one finds almost the same value for the S-wave in p–α scattering. For this sys- tem, it is interesting to compare to the neutron–alpha system, where there is no Coulomb repulsion in the scattering process. Results for the n–α causal ranges can be read off from Fig. 5 in Ref. [10] (obtained using effective range parameters from Ref. [149]). Even though from the plot one only gets quite rough estimates, one clearly sees that the results for the S1/2 and P1/2 channels agree very well between p–α and n–α scattering, which as in the nucleon–nucleon case discussed above could be interpreted as only a moderate amount of isospin breaking. However, the causal ranges for the P3/2 channels are very dif- ferent (∼ 0.5 fm for p–α, ∼ 2 fm for n–α). It is an interesting question if this discrepancy hints at an error in the extraction of the effective range parameters (either for one of the systems or possibly both), or if there actually is a physical reason behind the difference in the causal ranges. 5.5.4 Alpha–alpha scattering For α–α S-wave scattering we use the values given by Higa et al. (see Ref. [62] and experimental references therein), aC0 = (−1.65±0.17)·103 fm and rC0 = (−1.084±0.011) fm to find a causal range of about 2.58 fm. For the ` = 2 channel, no effective range parameters could be found in the literature. We have thus used the phase-shift data collected in the review article by Afzal et al. [150] to perform the fit to the effective range expansion (Eq. (4.24) with ` = 2) ourselves. By including the phase-shift data up to Elab ≈ 6.5 MeV we find aC2 = (−7.23±0.61) ·103 fm5 and rC2 = (−1.31± 0.22) · 10−3 fm−3. However, the fit is strongly dominated by the O(p4) shape parameter, so the actual uncertainties of aC2 and r C 2 should probably be somewhat larger. For the causal range in this channel we find a value of about 2 fm, which is just slightly smaller than the S-wave result. 5.6 Numerical calculations In order to check our relations and to get a better understanding of the values for the causal range, we now present some explicit numerical calculations. By cutting off the singular parts of the potential (i.e., the Coulomb potential and the angular momentum term for ` ≥ 0) at very small distances, it is a simple task to numeri- cally solve the radial Schro¨dinger equation (4.21) in configuration space. From the radial wave functions one can extract the Coulomb-modified phase shifts by looking for a zero 5.6. Numerical calculations 83 at some large (i.e., much larger than the range R of the short-range potential) distance, w (p) ` (r0) = 0 , r0 R , (5.55) and then calculating cot δ˜`(p) = −G`(p)(r0) F`(p)(r0) . (5.56) For the simplest case of a local step potential, V (r, r′) = Vstep(r) · δ(r − r′) ≡ V0 θ(R− r) · δ(r − r′) , (5.57) one can of course also obtain the phase shift directly by matching the wave functions at r = R. The effective range parameters are then obtained by repeating the calculation for several (small) momenta and fitting Eq. (4.24) to the results. In order to test Eq. (5.26) directly one needs the wave function to calculate the integral∫ r 0 dr′ [ w (0) ` (r ′) ]2 = limp→0 ∫ r 0 dr′ [ w (p) ` (r ′) ]2 . (5.58) Even if we do not actually take the limit p→ 0 but rather just insert some small p0 = 0.1 (in units of an arbitrary inverse length scale), we find that the relation rC` ≈ bC` (R)− 2 ∫ R 0 dr′ [ w (p0) ` (r ′) ]2 (5.59) is typically fulfilled to better than one-percent accuracy for the simple step potential defined above. For illustration, in the following we choose units where the radial distance is measured in fm. The potential range is set to 1 fm and its strength is measured in MeV.6 Furthermore, the reduced mass and Coulomb parameter are set to the values for the proton–proton system, i.e., 2µ = mN ≈ 940 MeV and γ = γp–p ≈ 0.035 fm−1. In Figs. 5.3 and 5.4 we show the results (for ` = 0, 1, 2) for both repulsive and attractive step potentials. Quite interestingly, the ` = 0 causal range for the repulsive potential stays at zero (meaning that one could reproduce the same values of the scattering length and the effective range also with a contact interaction) until a potential strength of about 100 MeV. For higher partial waves the causal range takes a nonzero value for much weaker potentials, but the rise is less steep. In general, it is remarkable that the causal range is typically considerably smaller than the actual potential range (R = 1 fm). For attractive potentials the causal range grows much faster as the potential strength (now negative) increases. In contrast to what one might expect, no special features are seen in the causal ranges as the potential becomes strong enough to support a new bound state close to threshold, i.e., when there is a pole in the scattering length parameter. To conclude this section, we show the general dependence of the causal range on both the scattering length and the effective range, which has the advantage of not depending on a 6Note that with these conventions, the quantity that is used in the numerical calculation is v0 ≡ 2µV0/(~c)2, where ~c ≈ 197.33 MeV · fm is used for the unit conversion. 84 Chapter 5. Causality bounds for charged particles 0.1 1 10 100 1000 10000 0 0.2 0.4 0.6 0.8 1 V0 (MeV) R c (f m ) ℓ = 0 ℓ = 1 ℓ = 2 · Figure 5.3: Causal range for a repulsive step potential and γ = γp–p. -4 -2 0 2 4 -250 -200 -150 -100 -50 0 0 0.2 0.4 0.6 0.8 V0 (MeV) R c (f m ) a C (f m 2 ℓ+ 1 ) ℓ = 0 ℓ = 1 ℓ = 2 · · Figure 5.4: Causal range (thick lines) for an attractive step potential and γ = γp–p. The thin lines show the corresponding scattering lengths. 5.7. Relation for asymptotic normalization constants 85 certain model potential. For illustration, we again measure distances in fm and set the Coulomb parameter to the value of the proton–proton system. In Figs. 5.5 and 5.6 we show the results for ` = 0 and ` = 1. For negative rC , the causal range stays essentially zero. For positive effective ranges, it increases as the absolute value of aC becomes larger. If one gradually turns off the Coulomb interaction by letting γ → 0, the ` = 1 plot stays almost unchanged, whereas the ` = 0 result remains qualitatively the same, but with a much steeper rise in the quadrant where aC > 0 and rC > 0. Figure 5.5: Causal range for γ = γp–p and ` = 0 in dependence of a C 0 and r C 0 , both measured in fm. 5.7 Relation for asymptotic normalization constants The formalism that has been used so far in this chapter in order to derive the causality bound for the effective range can be extended to the bound-state regime to derive a relation between the asymptotic normalization of the bound state wave function and the effective range of the corresponding two-particle scattering process. It is well-known [9] and was, as noted already in Chapter 3, also pointed out by Lu¨scher [8] that the asymptotic normalization constant (ANC) of the bound-state wave function is related to scattering parameters. More precisely, the residue of the bound-state pole in the analytically-continued elastic scattering amplitude (the f`(p) defined in Eqs. (2.22) and (2.23) in Chapter 2) is proportional to |Aκ|2. In the limit of shallow bound states, κ→ 0, it is possible to make a more direct connection to the effective range in the corre- sponding scattering channel. Ultimately, however, it can be viewed again as a manifesta- tion of the analyticity of the scattering amplitude as a function defined on the complex energy plane. Before we discuss the relation for our system of charged particles, we first present its 86 Chapter 5. Causality bounds for charged particles Figure 5.6: Causal range for γ = γp–p and ` = 1 in dependence of a C 1 and r C 1 , measured in fm3 and fm−1, respectively. derivation for the simpler case without Coulomb interaction. 5.7.1 Derivation for the neutral system As a starting point we quote the neutral version of our “master equation” (5.26) from Ref. [10], r` = b`(R)− ∫ R 0 dr [ u (0) ` (r) ]2 , (5.60) where r` is the effective range as defined by Eq. (2.19) and u (0) ` (r) = limp→0 u (p) ` (r) (5.61) is the zero-energy limit of the scattering wave function for the uncharged system, normal- ized as in Eq. (5.3). The neutral version of the causality-bound function, b`(R), can be given in closed form for arbitrary angular momentum. Specializing the general formula given in Ref. [10] to the three-dimensional case one finds b`(R) = − 2Γ ( `− 1 2 ) Γ ( `+ 1 2 ) pi ( R 2 )−2`+1 − 4 `+ 1 2 1 a` ( R 2 )2 + 2pi Γ ( `+ 3 2 ) Γ ( `+ 5 2 ) 1 a2` ( R 2 )2`+3 = −2Γ ( `− 1 2 ) Γ ( `+ 1 2 ) pi ( R 2 )−2`+1 +O(a−1` ) . (5.62) For ` = 0, this simply gives b0 = 2R +O(a−1` ) . (5.63) 5.7. Relation for asymptotic normalization constants 87 Note that a−1` → 0 as κ → 0, and since we only consider finite-range potentials, we can in fact write O(a−1` ) = O(κ). Recall now that following the conventions of Section 2.1.3 we use the explicitly momentum- dependent normalization condition (5.3) for the scattering wave functions u (p) ` (r). Upon going to the bound-state pole, p→ iκ , cot δ`(p→ iκ) = i , (5.64) we get the wave function u (iκ) ` (r) with the asymptotic form as given in Eq. (2.30). By comparison with Eq. (2.29) one directly finds that its relation to the unit-normalized u (iκ) A,` (r) is simply u (iκ) ` (r) = κ` Aκ u (iκ) A,` (r) , (5.65) where Aκ is the ANC. Writing the normalization condition for u (iκ) A,` (r) in the form 1 = (∫ R 0 dr + ∫ ∞ R dr ) |u(iκ)A,` (r)|2 (5.66) and using the relation (5.65), we can rewrite Eq. (5.60) as r` = b`(R)− 2 limκ→0 { κ2` A2κ − ∫ ∞ R dr [ u (iκ) ` (r) ]2} . (5.67) According to Eq. (2.30), the remaining integral is ∫ ∞ R dr [ u (iκ) ` (r) ]2 = κ2` ∞∫ R dr [ i`H+` (iκr) ]2 . (5.68) For ` = 0, we get∫ ∞ R dr [ u (iκ) ` (r) ]2 = e−2κR 2κ = 1 2κ −R +O(κ) as κ→ 0 . (5.69) Together with Eq. (5.63) this yields r0 + 2 A2κ − 1 κ = O(κ) , (5.70) which is, up to the given order, equivalent to the relation A2κ = 2κ 1− κr0 (5.71) from Ref. [14]. For ` ≥ 1, the integral is [153]∫ ∞ R dr [ u (iκ) ` (r) ]2 = Γ ( `− 1 2 ) Γ ( `+ 1 2 ) pi ( R 2 )−2`+1 +O(κ) as κ→ 0 . (5.72) 88 Chapter 5. Causality bounds for charged particles We see that the leading term exactly cancels the one in Eq. (5.62) and thus arrive at r` + 2κ2` A2κ = O(κ) . (5.73) There are two things to point out about the cancellation of terms encountered above. Firstly, it is clear that Eq. (5.67) can be rigorously valid only for zero-energy bound states, i.e., if one strictly considers the limit κ → 0. Since the left-hand side is a constant, the R-dependence of b`(R) has to cancel that of the integral. In the scattering regime this is ensured by taking the zero-energy limit, i.e., considering scattering directly at threshold. In the bound-state regime, on the other hand, one is of course interested in finding a relation for the case of a bound state close to threshold, where κ is small but finite. It is obvious that in principle this poses a problem because the b`(R) in Eq. (5.67) is a strictly increasing function of R, whereas the integral is always bounded and in fact becomes smaller with increasing R. The key observation is that for finite binding momenta, the cancellation of the R-dependence is valid only up to higher orders in κ. In other words, the terms subsumed as O(κ) in Eqs. (5.70) and (5.73) are still functions of R. Secondly, however, the cancellation of the leading R-dependent terms is by no means a coincidence, but can—as we shall do in the next section when we derive the relation for a system of charged particles—be proven on more general grounds. Upon doing that one finds that Eq. (5.73) is in fact true in a stronger form with only terms of the order κ2 on the right-hand side. 5.7.2 Derivation for charged particles We now derive the analogous relation for a system of charged particles. The appropriate starting point is Eq. (5.26) derived in Section 5.3, from which we already obtained the causality bounds for the effective range parameter. For bound-state momenta, we write it as rC` = b C ` (r)− 2 limκ→0 ∫ r 0 dr′ [ w (iκ) ` (r ′) ]2 for all r > R , (5.74) with bC` (r) now as defined in Eq. (5.27). In order to proceed we need again the precise connection between the radial wave functions w (iκ) ` appearing in Eq. (5.74) and the unit- normalized proper bound-state wave functions w (iκ) A,` (r) defined in Eq. (4.41). Formally, we can write w (iκ) ` (r) = (iκ) `Cη,` [ cot δ˜`(iκ)F (iκ) ` (r) +G (iκ) ` (r) ] for r > R , (5.75) but we need to be careful with the way the analytic continuation to the bound-state regime is done. In Section 4.3 we showed that in complete analogy to the neutral case, cot δ˜`(iκ) = i for a bound state and that the resulting linear combination G (p) ` (r)+iF (p) ` (r) of Coulomb wave functions is proportional to the Whittaker function W−iη,`+ 1 2 (−2ipr). More precisely, from Eq. (4.35) we have that Cη,` [ G (p) ` (r) + iF (p) ` (r) ] = Cη,` e iσ`e−i pi 2 (`+iη) W−iη,`+ 1 2 (−2ipr) . (5.76) 5.7. Relation for asymptotic normalization constants 89 The combination of prefactors in this turns out to be crucial to get a well-behaved ex- pression in the limit p→ iκ. Combining Eqs. (4.14) and (4.34), we have Cη,` e iσ` = 2` e− piη 2 [Γ(`+ 1 + iη)Γ(`+ 1− iη)] 12 Γ(2`+ 2) ( Γ(`+ 1 + iη) Γ(`+ 1− iη) )1 2 . (5.77) Note that iη will be real for bound-state momenta, and in the following, where we only consider the case of a repulsive Coulomb potential7 (i.e., γ > 0), it is in fact a positive number. Conveniently, the term Γ(` + 1 − iη), which might hit a pole in the Gamma function, cancels in Eq. (5.77). Noting that also the phase exp(piη/2) drops out, we can finally write w (iκ) ` (r) = κ `C˜η,`W−iη,`+ 1 2 (2κr) for r > R , (5.78) where we have defined C˜η,` = 2` Γ(`+ 1 + iη) Γ(2`+ 2) . (5.79) Comparing this with Eq. (4.41) we readily infer that w (iκ) ` (r) = κ` A˜κ C˜η,` · w(iκ)A,` (r) (5.80) with the asymptotic normalization constant A˜κ. Following now the same steps as in Section 5.7.1, we can rewrite Eq. (5.74) as rC` = b C ` (r)− 2 limκ→0 { κ2` A˜2κ C˜2η,` − ∫ ∞ r dr′ [ w (iκ) ` (r ′) ]2} for r > R . (5.81) Canceling the r-dependence We now show that the r-dependence in Eq. (5.81) drops out up to corrections of higher order in κ. When we derived the relation for the neutral case we established the cancel- lation of the leading term by analytically carrying out the integral over the asymptotic wave function (which was just a simple Riccati–Hankel function in that case) and then expanding the result in powers of κ. Since the asymptotic wave function is now given by a Whittaker function, the situation for the charged case is much more complicated. However, as already mentioned at the end of Section 5.7.1, the cancellation can in fact be established in a more general way. To this end we recall from Eq. (5.20) that w (p) ` (r) = − 1 aC` f0(r) + g0(r) +O(p2) . (5.82) It is straightforward to assume that this is valid also in the bound-state regime since we are working with a wave function explicitly analytic in p2. Hence,[ w (iκ) ` (r) ]2 = ( − 1 aC` f0(r) + g0(r) )2 +O(κ2) = 1 2 d dr bC` (r) +O(κ2) , (5.83) 7Considering attractive Coulomb potentials does not make much sense in the present context because it would create an infinite tower of bound states arbitrarily close to threshold. 90 Chapter 5. Causality bounds for charged particles where in the last step we have inserted Eq. (5.34). This directly tells us that bC` (r) in Eq. (5.81) cancels with the integral up to higher-order terms in κ (which are still functions of r) and, possibly, an integration constant. This situation is already familiar from the calculation of the Wronskian W [g2, g0](r). Again the constant term can only come from the integral over g0(r) 2. To determine it, we note that∫ ∞ r dr′ [g0(r′)] 2 <∞ if r > 0 , (5.84) i.e., the integral is actually convergent and gives a Meijer G-function, as already encoun- tered in Section 5.3.4. The constant term coming from this function has to be combined with the already known Z` in b C ` (r). We define ∆Z` = W [g2, g0](r) + ∫ ∞ r dr′ [g0(r′)] 2 , (5.85) which is explicitly r-independent, to finally arrive at rC` + 2κ2` A˜2κ C˜2η,` + 2 ∆Z` = O(κ2) (5.86) or, equivalently,8 |A˜κ| = κ`C˜η,` ( −r C ` 2 −∆Z` +O(κ2) )−1/2 , (5.87) both valid in the limit where κ→ 0. In Table 5.3 we give the resulting values for the ∆Z` for ` = 0, 1, 2. Note that up to a minus sign, these are exactly the correction terms mentioned at the end of Section 5.3.5. This means that the antiderivative used to obtain the explicit expressions for the ` = 1 and ` = 2 causality-bound functions given in Appendix B corresponds essentially to the definite integral in Eq. (5.84), at least for the case of a repulsive Coulomb interaction, where this is convergent. ` 0 1 2 ∆Z`, −1/(3γ) −γ/108 −17γ3/10800 Table 5.3: Integration constant ∆Z` in Eqs. (5.86) and (5.87)) for ` = 0, 1, 2. Finally, we note that Sparenberg et al. [143] have previously derived an ANC relation equivalent to ours, but written in terms of the scattering length instead of the effective range parameter. The equivalence of the two relations up to the contributions of higher- order shape parameters is shown in Appendix C. 5.7.3 Application to the oxygen-16 system The 16O nucleus has two excited states lying just below the α–12C threshold, a 2+ at about −245 keV and an even more shallow 1− at only −45 keV. The properties of these 8Note that in Ref. [5] there was a typo that accidentally put the C˜η,` in Eq. (5.87) in the denominator. 5.7. Relation for asymptotic normalization constants 91 states play an important role in astrophysical helium burning processes [154–156]. In the following, we calculate asymptotic normalization constants for both states under the assumption that they can be described in a α + 12C halo picture. We use the recent data obtained by Tischhauser et al. [157] (for the actual phase shifts see their Ref. [32]) in order to extract the Coulomb-modified effective range for the α–12C P- and D-wave channels. Focussing first on the D-wave, we note that the combination of the strong Coulomb repulsion and the ` = 2 centrifugal barrier makes the low-energy phase shifts very small over a wide energy range. Moreover, there is a narrow resonance at a center-of-mass energy of about 2.7 MeV, which strongly constrains the energy region for a straightforward fit to the effective range function. From a simple fit to the data up to the narrow resonance one only obtains an effective range parameter with an uncertainty too large (50%) to allow an extraction of the ANC because the latter depends on rC very sensitively. To mitigate this problem, we use the position of the 2+ oxygen state as an additional input parameter for a self-consistent extraction of the effective range. In the following, we describe this procedure in more detail. At the position of the bound state, where cot δ˜`(iκ) = i, the Coulomb-modified effective range expansion (4.28) reduces to γ h˜`(iκ) = − 1 aC` − 1 2 rC` κ 2 + · · · , (5.88) where now h˜`(iκ) = (2iκ)2` (2`+ 1)!2 ∏` s=1 (s2 + η2)× ( ψ(iη) + 1 2iη − log(iη) ) , η = γ 2iκ . (5.89) A straightforward calculation shows that for the prefactor we have (2iκ)2` (2`+ 1)!2 ∏` s=1 (s2 + η2) = γ2` (2`+ 1)!2 +O(κ2) . (5.90) Furthermore, for the digamma function we have the asymptotic expansion (Eq. (6.3.18) in Ref. [103]) ψ(z) = log z − 1 2z − ∞∑ n=1 B2n 2n z2n (5.91) for |z| → ∞, where the B2n are the Bernoulli numbers, B0 = 1, B1 = −1 2 , B2 = 1 6 , . . . . (5.92) Noting that the sum in Eq. (5.91) only starts at n = 1, we find that γ h˜`(iκ) only starts at order κ2, and inserting the precise relation into Eq. (5.88), we arrive at 1 aC` = ( −1 2 rC` + γ2`−1 3(2`+ 1)2 ) κ2 +O(κ4) . (5.93) 92 Chapter 5. Causality bounds for charged particles Neglecting the effective-range contribution at leading order, we insert the binding momen- tum κ ≈ 0.187 fm−1 of the 2+ state to get a first approximation for the scattering length parameter. This is then used to constrain a subsequent fit to the phase-shift data up to about 2.6 MeV, i.e., just below the narrow resonance. We include a single O(p4) shape parameter in the fit since some curvature is clearly necessary. The rC2 obtained from this is then used in Eq. (5.93) to get a better result for 1/aC2 , which, in turn, is fed back into the fit. Iterating this procedure a couple of times yields a well-converged self-consistent result for rC2 . After eight iterations, we find rC2 = (5.94± 0.35) · 10−4 fm−3 (5.94) for the α–12C D-wave. Including a second shape parameter in the fitting procedure only changes this result within the given uncertainty, so we conclude that for the energy range we have been fitting, a single shape parameter really is sufficient to account for the curvature. Inserting the fit result into Eq. (5.87) yields |A(2+)| = (2.41± 0.38) · 104 fm−1/2 (5.95) for the 2+ state in 16O. Here and in the following, we simply write A instead of A˜κ to keep the notation simple. Including in Eq. (5.86) an O(κ4) term of the order of the shape parameter gives a consistent result for |A(2+)| within the error given above. When we apply the same same procedure to the 1− state just 45 keV below the α–12C threshold, we also see a nice convergence and obtain the results rC1 = (4.546± 0.002) · 10−2 fm−1 (5.96) and |A(1−)| = (1.188± 0.024) · 1014 fm−1/2 . (5.97) The overall picture for the 1− state is somewhat more complicated, however. On the one hand, the extraction of the effective range parameters is easier in this case because there is no narrow resonance limiting the fit range (we have used the data up to Ecm = 3.75 MeV). By doing a simple fit without the self-consistent iteration we get values for the effective range and the ANC that are slightly smaller but overlap with the results given above when the respective uncertainties are taken into account. On the other hand, allowing for a second shape parameter changes the value of the effective range quite dramatically to 0.046 fm−1, which leads to an imaginary ANC. This could indicate that the effective range parameters from the simple fit violate the causality bound. The fact that rC has to be such that Eq. (5.86) yields a real value for the ANC can be interpreted as a weaker remnant of the original causality-bound relation. Alternatively, the cluster picture might not be applicable for the shallow 1− state in 16O. Our result for the ANC of the 2+ state is about a factor five smaller than the value |A(2+)| = (1.11 ± 0.11) · 105 fm−1/2 obtained by Brune et al. [154], while the value for |A(1−)| is about a factor two smaller. Other, more recent determinations [158, 159] have found even larger values. For the interpretation of this discrepancy, note that our calculations are predictions of the ANCs based only on alpha–carbon elastic scattering data and the assumption that the system can be approximately described in an effective 5.8. Other long-range forces 93 two-body picture with a finite-range interaction. In the references mentioned above, the ANCs are extracted from alpha–carbon transfer measurements. Note furthermore that a comparison of the experimental extractions in Ref. [158] exhibits quite some discrepancy (factors of two up to roughly an order of magnitude) also between the cited individual experimental determinations of the ANCs. Sparenberg et al. [160] have carried out a similar analysis for the 2+ state based on their ANC relation (cf. Ref. [143]). They subtracted the narrow resonance from the α–12C D-wave phase shift data in order to extract a set of higher-order shape parameters and concluded that present-day data are not sufficient to constrain the ANC strongly. As discussed above, our approach of performing a self-consistent fit to the data below the resonance constrained by the separation energy and using the effective range instead of the scattering length as input in the ANC relation improves the stability of the extraction. However, compared to other determinations our ANC values are generally smaller by a factor two to five. We thus conclude that this issue requires further study. 5.8 Other long-range forces The causality bounds derived above for the system with Coulomb interactions are, despite being quite involved due to the complicated nature of the Coulomb wave functions, not fundamentally different from the result that one obtains in the purely finite-range case. In Section 5.7.1 we have already quoted the causality-bound function for this case from Ref. [10]. According to Eq. (5.62), its behavior is b`(r) = − 2Γ ( `− 1 2 ) Γ ( `+ 1 2 ) pi (r 2 )−2`+1 +O(r2) . (5.98) From the explicit results for the Coulomb causality-bound functions, given in Eqs. (5.45) and (5.46) for ` = 0, and in Appendix B for ` = 1, 2, one finds that bC` (r) = − ( 2` Γ(`+ 1) Γ(2`+ 2) )2 × 2Γ ( `− 1 2 ) Γ ( `+ 1 2 ) pi (r 2 )−2`+1 +O(r−2`+2) . (5.99) Up to an additional prefactor and the subleading pole terms (if ` ≥ 2) in Eq. (5.99), the leading behavior is exactly the same as for pure finite-range interactions.9 The overall prefactor goes away if one uses the conventions of Eq. (4.31) for the Coulomb-modified effective range expansion. In both cases, the S-wave causality-bound function vanishes at the origin, whereas for all ` ≥ 1 it has a pole. It is this pole that requires the effective range parameter to tend towards negative infinity in the limit where the range of the potential goes to zero and therefore prevents one from tuning the effective range parameter for P- and higher partial wave to zero in a calculations with exact zero-range interactions [10, 139]. 9Strictly, we have explicitly derived this only for ` = 0, 1, 2. We assume here, however, that Eq. (5.99) is true in general. 94 Chapter 5. Causality bounds for charged particles 5.8.1 Singular potentials The above comparison has been given in Ref. [7], where we have just corrected a mi- nor typo10 here. As furthermore discussed there, the situation changes significantly for interactions with long-range tails of the form Vα(r) = g rα with α > 2 , (5.100) where g denotes the coupling strength. The properties of such so-called singular potentials have been reviewed in detail by Frank et al. in Ref. [161]. They are particularly peculiar if they are attractive (g < 0) because in that case there always exists an infinite spectrum of bound states that is not bounded from below. Obviously, this situation is unphysical. Still, it is interesting to study such singular potentials if one assumes that they only determine the long-range tail of a given interaction and are, by some mechanism, cut off at small distances. For an analysis of systems with this kind of behavior, it is important to study solutions of the Schro¨dinger equation with pure inverse-power-law interactions. For the 1/r4 potential, for example, such solutions have been known for a long time and can be written in terms of Mathieu functions [162]. At zero energy, it is possible to write down the general solution of the radial Schro¨dinger equation with a potential of the form (5.100). From Ref. [161] we quote the result that the physical solution (vanishing at the origin) is given by uα,`(r) ∝ r1/2Kn(βrσ) with n = 2`+ 1 α− 2 , (5.101) where Kn(z) is a modified Bessel function, and β = 2g1/2 α− 2 , σ = 1− α 2 . (5.102) A second linearly-independent solution is given by inserting the modified Bessel function In(z) in Eq. (5.101). From this one already sees that the highly singular nature of the potential leads to a peculiar behavior of the wave function. For an attractive potential (g < 0), Eq. (5.102) implies that the parameter β becomes purely imaginary. Although in that case the amplitude of uα,`(r) still goes to zero in the limit r → 0, at the same time it oscillates more and more rapidly and in fact has an essential singularity at the origin. We will come back to this point and discuss its implication for the causality bound in the next section. Of particular physical importance is the attractive van der Waals potential, VvdW(r) = −C6 r6 , C6 > 0 , (5.103) because it describes the long-range interaction of neutral atoms (or molecules) due to their mutual polarization [28]. For example, it determines the attractive part of the phenomenological Lennard–Jones potential [161]. 10The subleading pole terms were given in Ref. [7] as O(r−2`) instead of O(r−2`+2), which obviously made them not subleading but rather more singular than the first term. 5.8. Other long-range forces 95 Based on explicit solutions derived by Gao [163], causality bounds for this kind of potential have been studied in Ref. [7]. In the following we summarize the central results of that publication. 5.8.2 Causality bounds for van der Waals interactions Introducing the van der Waals length scale β6 = (2µC6) 1/4 , (5.104) the Schro¨dinger equation for a two-particle system with a (non-local) finite-range inter- action V (r, r′) and an additional van der Waals tail of the form (5.103) can be written as p2v (p) ` (r) = − d2 dr2 v (p) ` (r) + `(`+ 1) r2 v (p) ` (r) + 2µ ∫ R 0 dr′ V (r, r′) v(p)` (r ′) + θ(r −R)β 4 6 r6 v (p) ` (r) . (5.105) We use the letter v here to denote the radial wave functions in order to distinguish them from those for the asymptotically non-interacting (u) and the Coulomb (w) case. As usual, we denote a solution for momentum p with a corresponding superscript. Due to its highly singular behavior, the van der Waals potential has to be cut off at small distances. In Eq. (5.105) we have conveniently done this with the step function θ(r−R). In principle, other methods (and cut-off radii) can be used, but it is always possible to absorb the difference into V (r, r′), which we have not specified explicitly. The only important point here is that for r > R the interaction is given exactly and explicitly by the van der Waals tail because that is the setup we are interested in. Asymptotic wave functions It is convenient to introduce the rescaled variables rs = r/β6 and ps = pβ6 . (5.106) In the outer region (r > R), the Schro¨dinger equation (5.105) then reduces to[ d2 dr2s − `(`+ 1) r2s + 1 r6s + p2s ] v (p) ` (r) = 0 . (5.107) The exact solutions of this equation (valid for all rs > 0) have been derived by Gao in Ref. [163] and are given in terms of infinite sums of Bessel functions. Up to a minor 96 Chapter 5. Causality bounds for charged particles change in notation,11 we quote them here in the form used in Ref. [7], F vdW` (p, r) = r 1/2 s X2` (ps) + Y 2 ` (ps) [ X`(ps) ∞∑ m=−∞ bm(ps) Jν+m (ρs) − Y`(ps) ∞∑ m=−∞ bm(ps)Nν+m (ρs) ] , (5.108a) GvdW` (p, r) = r 1/2 s X2` (ps) + Y 2 ` (ks) [ X`(ps) ∞∑ m=−∞ bm(ps)Nν+m (ρs) + Y`(ps) ∞∑ m=−∞ bm(ps) Jν+m (ρs) ] , (5.108b) with the radial variable ρs = 1 2r2s , (5.109) and with the momentum-dependent coefficients X`(ps), Y`(ps)—not to be confused with the functions introduced in Section 5.3.4—and bm(ps) as defined in the appendix of Ref. [7]. The variable ν that appears in the indices of the Bessel functions is determined as the root of a (transcendental) equation that again involves the momentum ps. As such, it is also a momentum-dependent function, but for simplicity we have not indicated this explicitly in Eqs. (5.108). Both van der Waals wave functions defined above have an essential singularity at the origin which lets their amplitude vanish in the limit r → 0 with infinitely-rapid oscillations. This is the same behavior that was mentioned above for the general zero-energy solutions of attractive singular power-law potentials. An important consequence is that there is no singled-out regular solution defined by its vanishing at the origin and thus no obvious choice for the definition of a scattering phase shift. Rather, one has an infinite number of linearly-independent pairs of solutions that can, in principle, all be used to express the asymptotic wave function of the system. As already mentioned in Section 2.1.4 in Chapter 2, Gao has extended the formal- ism of quantum defect theory to describe systems with tails of the singular power-law type (5.100) in Ref. [21]. The application to the special case of a van der Waals potential is discussed in Ref. [164]. By writing the general solution of Eq. (5.105) for r > R as v (p) ` (r) = F vdW ` (p, r)− tan δ(short)` (p)GvdW` (p, r) , (5.110) we define the short-range K-matrix 12 tan δ (short) ` (p) = Kˆ`(p) . (5.111) 11In Ref. [7], these solutions are called FL and GL, where L denotes the angular momentum. They are related to the functions f c0l and g c0 l defined in Ref. [21] by the normalization factors F vdW ` = f c0 l / √ 2 and GvdW` = −gc0l / √ 2, with l = `. 12Note that our Kˆ` is called K 0 l in Refs. [163, 165] and corresponds to −Kc0l of Refs. [21, 164]. 5.8. Other long-range forces 97 Other choices for the basis pair of van der Waals wave functions correspond to different K-matrices. The Kˆ`(p) defined above in terms of F vdW ` (p, r) and G vdW ` (p, r) is the most convenient choice to describe the system near the zero-energy threshold [21]. In particular, it is an analytic function of p2 [165], Kˆ`(p) = ∞∑ n=0 K`,2n p 2n . (5.112) The parameters appearing in this expansion characterize the low-energy physics of the system, which means that Eq. (5.112) can be seen as analog of the ordinary effective range expansion. K`,0 is, essentially, an inverse scattering length parameter, and K`,2 plays the role of the effective range. Note, however, that this is just an approximate cor- respondence, which can already be seen from looking at the dimensions of these quantities (K`,0 is dimensionless whereas K`,2 has the dimension of an inverse momentum squared, corresponding to an area). The short-range K-matrix Kˆ` has been used by Gao to define so-called generalized scattering lengths and effective ranges [164]. An important quantity that enters in these relations is the van der Waals length scale β6 defined in Eq. (5.104). In fact, as discussed in Ref. [7], for almost all scattering systems of alkali atoms that have been studied experimentally so far one finds that β26 dominates the generalized effective range compared to the parameter K`,2. Despite their complicated nature, the van der Waals wave functions are analytic functions of p2 [163] and thus possess expansions F vdW` (p, r) = f vdW `,0 (r) + f vdW `,2 (r) p 2 +O(p4) , (5.113a) GvdW` (p, r) = g vdW `,0 (r) + g vdW `,2 (r) p 2 +O(p4) . (5.113b) With ν0 = (2`+ 1)/4 , (5.114) the leading-order functions are simply fvdW`,0 (r) = r 1/2 s Jν0(ρs) , (5.115a) gvdW`,0 (r) = r 1/2 s Nν0(ρs) , (5.115b) where Jn(z) and Nn(z) are Bessel functions of the first and second kind, respectively. Of course, these are just linear combinations of the general zero-energy solutions mentioned in Section 5.8.1. The quadratic terms in Eqs. (5.113) are somewhat more involved, but can also be written down explicitly [7]. The van der Waals causality bound With the ingredients given in the previous section one has everything at hand to derive the van der Waals causality bound exactly as in Ref. [10] and done for the Coulomb potential in this chapter. In fact, the derivation here is almost as simple as in the purely finite- range case because the quadratic radial functions f`,2 and g`,2—and thus the Wronskians appearing in the causality-bound function—are known explicitly. The result is K`,2 = b vdW ` (r)− pi 4 ∫ r 0 dr′ [ v (0) ` (r ′) ]2 , (5.116) 98 Chapter 5. Causality bounds for charged particles with bvdW` (r) = pi 4 { W [fvdW`,2 , f vdW `,0 ](r) +K 2 `,0W [g vdW `,2 , g vdW `,0 ](r) −K`,0 ( W [gvdW`,2 , f vdW `,0 ](r) +W [f vdW `,2 , g vdW `,0 ](r) )} . (5.117) In the well-known fashion, the positive-definiteness of the integral term in Eq. (5.116) yields a causality bound of the form K`,2 ≤ bvdW` (r) for all r > R , (5.118) and the van der Waals causal range RvdWc is defined by K`,2 = b vdW ` (R vdW c ) . (5.119) The Wronskians in Eq. (5.117) are given explicitly in Ref. [7]. They have the important property that for all angular momenta ` they vanish in the limit r → 0, which implies the same for the van der Waals causality-bound function bvdW` (r). This stands in clear contrast to the behavior found for the asymptotically non-interacting system and for interactions with Coulomb tails. As summarized in Eqs. (5.98) and (5.99), in both these cases the causality-bound function for ` ≥ 1 diverges at the origin. This important difference can be traced back to the behavior of the wave functions. Since both van der Waals wave functions vanish as r → 0 (faster than a power law), the same is true for the coefficient functions in Eqs. (5.113) and, in turn, for the Wronskians. At the same time this means that the latter can be calculated as straightforward integrals from 0 to r over products of the zero-energy solutions (5.115), cf. Section 5.3.4. Since all functions vanish in the limit r → 0, there are no peculiar integration constants as encountered for Coulomb tails. In Ref. [7] it is argued further that this should also be true for other singular-potential tails that lead to the same behavior of the wave functions. The fact that the van der Waals causality-bound function bvdW` (r) does not diverge at r = 0 is quite significant. It means that as long as the short-range K-matrix parameter K`,2 is less than or equal to zero, there is no real constraint on the interaction range from the causality-bound relation (5.118). As a consequence, in an effective field with contact interactions that incorporates a van der Waals tail of the interaction, the causality bound does not induce convergence problems when the cutoff is taken to infinity, provided that K`,2 ≤ 0 holds for the angular-momentum channel one is interested in. However, the van der Waals length scale apparently plays quite an important role. As already pointed out, it is found to be the dominant scale when K`,2 is expressed in terms of Gao’s generalized effective range parameters [7, 164]. Furthermore, causal ranges that were calculated in Ref. [7] for systems of alkali atoms where scattering parameters could be found in the literature13 all came out significantly smaller than β6. 13Unfortunately, only S-wave parameters could be found. It would be very interesting to see if the reported result persists for higher partial waves. 5.9. Summary and outlook 99 Causal range near an S-wave magnetic Feshbach resonance Since their first experimental observation in 1998 [166], Feshbach resonances have become a popular tool to tune the interactions in cold atomic systems by varying an externally applied magnetic field [167] (see also K. Helfrich’s doctoral thesis [56] and references therein). It is thus interesting to study the impact of the van der Waals causality bound in this context. In Ref. [168] the multi-channel problem of scattering around a magnetic Feshbach reso- nance is reduced to a description by an effective single-channel K-matrix that depends on the applied magnetic field B. The behavior around the resonance is described by several parameters. B0,` is the position of the Feshbach resonance, while gres parametrizes its width. Kbg` is a background value for the K-matrix, and a scale dB,` is introduced in order to define a dimensionless magnetic field. With these parameters, we write the effective single-channel K-matrix as Kˆeff` (p,B) = −Kbg` [ 1 + gres p2β26 − gres (Bs + 1) ] , (5.120) with Bs = (B −B0,`) dB,` . (5.121) The parametrization given above corresponds to Eq. (18) in Ref. [168]. Note that we have slightly changed the notation and are using a different sign convention for the K-matrix. By expanding the right-hand side of Eq. (5.120) in p2 it is straightforward to determine the K-matrix expansion parameters K`,0 and K`,2. A short calculation yields that Keff`,0 = −Kbg` ( 1 + 1 Bs + 1 ) (5.122) and Keff`,2 = β26K bg ` gres(Bs + 1)2 . (5.123) As noted in Ref. [168], the parameters Kbg` and gres are constrained by the condition Kbg` gres < 0 . (5.124) This implies that Keff`,2 as given by Eq. (5.123) is always negative. From the causality bound in Eq. (5.118) it then follows that wherever the effective single-channel description considered here is applicable and correctly captures the entire energy-dependence of the short-range K-matrix, the causal range will be zero when the interaction is tuned close to a Feshbach resonance. 5.9 Summary and outlook In this chapter, we have investigated the constraints imposed by causality on the low- energy scattering parameters of charged particles interacting via a short-range interaction 100 Chapter 5. Causality bounds for charged particles and a long-range Coulomb potential. Similar to the case of neutral particles without Coulomb interaction [10, 135–140], our considerations yield a constraint on the maximum value of the Coulomb-modified effective range. While conceptually straightforward, the calculation of the Wronskians required for the derivation of the bound function is intricate. We have calculated them through term- by-term integration of the power-series expansion of the zero-energy wave functions and additionally determining the integration constants that are not generated by this process. We define the causal range as the minimum value of the interaction range consistent with the causality bound. In effective field theories with contact interactions such as halo EFT, the natural momentum cutoff is of the order of the inverse of the causal range. If the natural cutoff is not known from other considerations, its size can be estimated from the causal range. If the momentum cutoff used in a calculation is too high, then problems with convergence of higher order corrections can appear. For example, the convergence pattern might be such that an improvement in higher orders of the EFT can only be sustained through large cancellations between individual terms. Such an unnatural pattern would be especially undesirable for the stability of numerical (lattice) calculations. Our results can thus be viewed as a guide for improving the convergence pattern of EFT calculations with contact interactions. In lattice simulations of halo EFT, the lattice spacing should not be taken smaller than the causal range. We have analyzed the causal ranges for a variety of systems ranging from proton–proton scattering to alpha–alpha scattering. Our results for causal ranges in different partial waves in these systems typically vary by factors of 2–3. The precise values are quite sensitive to small uncertainties in the effective range parameters. In channels with a large negative effective range the causal range is very close to zero, which implies that causality provides almost no constraints on the range of the interaction in this case. Thus, the causal range provides a good order-of-magnitude estimate of the range of interaction, but drawing more quantitative conclusions about the structure of the underlying potentials is difficult. In order to illustrate the dependence of the causal range on scattering parameters, we have performed explicit numerical calculations for a simple step potential with a Coulomb tail. After an analytic continuation to the bound state regime, the integral relations for the causality bound can also be used to derive a model-independent expression for the ANC of shallow bound states. If the state is a two-body halo state, the relation can be used to extract the ANC from low-energy scattering parameters. Up to higher-order shape parameters, our relation is equivalent to the one previously derived by Sparenberg et al. in Ref. [143] (see Appendix C for details). One difference is that we express the ANC in terms of the binding momentum and the effective range rather than the binding momentum and the scattering length. We find this form more suitable for the extraction of ANCs from scattering data since the effective range is typically more precisely determined than the scattering length for shallow states. Moreover, extracting the effective range in a self-consistent fit from the scattering data that reproduces the correct separation energy improves the stability of the extraction. We have illustrated our relation by extracting the ANCs of the excited 2+ and 1− states in 16O from α−12C scattering data. Compared to previous extractions [154, 158, 159], our values are generally smaller. Whether this difference is physically significant requires 5.9. Summary and outlook 101 further study. The application of our relation to other shallow cluster states and a bench- mark against model calculations would also be very interesting. Finally, we have discussed causality bounds for systems where the long-range interaction is given by a singular potential. In particular, we have reviewed the case of an attractive van der Waals tail as it appears in the scattering of neutral atoms. In this case, the situation is fundamentally different from the systems of charged or asymptotically non- interacting particles. Due to the behavior of the singular-potential wave functions, the causality-bound function never has a pole at the origin, which implies that there is no constraint on the interaction range as long as the short-range parameter K`,2 is negative or zero. We have shown that this is always the case for scattering around a magnetic Feshbach resonance provided that the effective single-channel description of Ref. [164] is applicable. These results can be useful in developing an EFT for systems zero-range interactions and an attractive van der Waals tail. 102 Chapter 5. Causality bounds for charged particles Chapter 6 The proton–deuteron system revisited Overview In this chapter we consider the S-wave proton–deuteron system in pionless effective field theory, a project already started in the present author’s diploma thesis [169]. The first part of what follows is largely based on a subsequent publication [1] (a summary of which appeared in the APFB2011 conference proceedings [6]), with a few details changed, added, and corrected. The material presented in the second part (Section 6.5.2 and beyond), however, is new and has not previously been published. 6.1 Introduction In Section 2.4 we introduced effective field theories as a powerful tool that can be used to carry out calculations in a formalism involving directly the “correct” degrees of freedom for the physical system under consideration. In particular, in nuclear systems at very low energies and momenta, pion-exchange effects cannot be resolved and one can hence use the pionless effective field theory introduced in Section 2.4.2. This approach only includes short-range contact interactions between nucleons [49, 52] and is constructed to reproduce the effective range expansion [13] in the two-body system. It furthermore recovers Efimov’s universal approach to the three-nucleon problem [170, 171]. The extension of pionless EFT to include the long-range Coulomb interaction was first discussed by Kong and Ravndal for the proton–proton channel [117, 172]. In Ref. [173], this analysis was extended to next-to-next-to-leading order. A renormalization-group analysis of proton–proton scattering in a distorted wave basis was carried out in Refs. [174, 175]. Moreover, the theory was applied to proton–proton fusion in Refs. [176, 177]. Here, we are interested in the nuclear three-body system with two charged particles, i.e., protons. Although the Coulomb interaction can be treated as a perturbative correction for intermediate and higher energies, it becomes strong close to threshold and has to 103 104 Chapter 6. The proton–deuteron system revisited be treated nonperturbatively there. In Ref. [178], Rupak and Kong have formulated a power counting that takes into account such strong Coulomb contributions in the quartet channel of proton–deuteron (p–d) scattering. They calculated the phase shifts to next-to- next-to-leading order (N2LO) in the pionless EFT and included Coulomb effects to next- to-leading order (NLO) [178]. However, they were not able to extend their calculation to the threshold region below center-of-mass momenta of 20 MeV. Furthermore, they did not consider the doublet channel and the 3He bound state. A leading order calculation of the 3He nucleus including nonperturbative Coulomb inter- actions has been carried out by Ando and Birse [179]. Including isospin breaking effects in the nucleon–nucleon scattering lengths, they obtained a good description of the 3He–3H binding energy difference, but they did not consider scattering observables. A similar study at next-to-leading order in the pionless EFT was carried out using the resonating group method [180]. Those results do not include isospin breaking and are consistent with other determinations of the 3He–3H binding energy difference. In this chapter, we apply the power counting of Rupak and Kong to study the S-wave p–d system in both the quartet and the doublet channel. Focussing first on results al- ready published in Ref. [1], we show how by using a special integration mesh we are able to calculate scattering phase shifts in both channels down to very small momenta of the order 3 MeV. After reviewing the formalism of pionless EFT in Section 6.2, we discuss these results in Section 6.3. Amending then our previous work, we consider in Sections. 6.4 and 6.5 both a perturbative evaluation of the Coulomb contribution to the 3He–3H binding-energy difference as well as a nonperturbative calculation of the 3He bind- ing energy and wave function. Although our calculation only includes Coulomb photons, it gives almost the same result as found by Ando and Birse [179]. From the nonperturbative bound-state calculation we find that the doublet-channel sys- tem beyond leading order does not seem to be renormalized correctly when Coulomb contributions are taken into account. We discuss this in some detail in Section 6.6 and also critically review the situation in the scattering regime. As a further new result we present in Section 6.7 a first preliminary calculation of proton–deuteron scattering lengths in pionless EFT. We then conclude with a brief summary and an outlook. 6.2 Formalism and building blocks We write the effective Lagrangian in the form L = N † ( iD0 + D2 2MN ) N −di† [ σd + ( iD0 + D2 4MN )] di− tA† [ σt + ( iD0 + D2 4MN )] tA + yd [ di† ( NTP idN ) + h.c. ] + yt [ tA† ( NTPAt N ) + h.c. ] + Lphoton + L3 , (6.1) with the nucleon field N and two dibaryon fields di (with spin 1 and isospin 0) and tA (with spin 0 and isospin 1), corresponding to the deuteron and the spin-singlet isospin- triplet virtual bound state in S-wave nucleon–nucleon scattering. Both dibaryon fields are formally ghosts since their kinetic terms have a negative sign. This is required to 6.2. Formalism and building blocks 105 reproduce the positive values of the effective ranges with short-range interactions [135]. Despite these “wrong” signs, the Lagrangian (6.1) can be shown to be equivalent to the most general version including only nucleon fields (see, for example, Ref. [181]). One can interpret the choice of signs in Eq. (6.1) as “avoiding” the Wigner bound (cf. Chapter 5), but since the effective N–N interactions (obtained by eliminating the dibaryon fields with the kinetic energy terms included) become energy-dependent, this is not a rigorous statement. Spin and isospin degrees of freedom are included by treating the field N as a doublet in both spaces, but for notational convenience we usually suppress the spin and isospin indices of N . The operators P id = 1√ 8 σ2σiτ 2 , PAt = 1√ 8 σ2τ 2τA , (6.2) with the Pauli matrices ~σ and ~τ operating in spin and isospin space, respectively, project out the 3S1 and 1S0 nucleon–nucleon partial waves. The covariant derivative Dµ = ∂µ + ieAµQˆ , (6.3) where Qˆ is the charge operator, includes the coupling to the electromagnetic field. Fur- thermore, we have the kinetic and gauge fixing terms for the photons, Lphoton = −1 4 FµνF µν − 1 2ξ (∂µA µ − ηµην∂νAµ)2 , (6.4) of which we only keep contributions from Coulomb photons. These correspond to a static Coulomb potential between charged particles, but for convenience we introduce Feynman rules for a Coulomb-photon propagator, ∆Coulomb(k) = i k2 + λ2 , (6.5) which we draw as a wavy line, and factors (±ie Qˆ) for the vertices.1 Following Ref. [178], we have regulated the singularity of the Coulomb-photon propagator at zero momentum transfer by introducing a photon mass λ in Eq. (6.5). This corresponds to a screening of the Coulomb interaction in configuration space by writing it as a Yukawa potential ∼ e−λr/r. In the numerical calculations that will be discussed later on, λ is always taken to be small (typically well below 1 MeV). In fact, by choosing a mesh-point distribution dense around the Coulomb peak it is possible to numerically take the zero-screening limit λ→ 0 [169]. S-wave nucleon–deuteron scattering can take place in either a spin-3/2 (quartet channel) or spin-1/2 (doublet channel) configuration. In the doublet channel, a three-body contact interaction is required for renormalization already at leading order in the EFT [54]. We 1Due to the sign convention chosen in the Lagrangian (6.1), dibaryon–photon vertices get an additional minus sign. 106 Chapter 6. The proton–deuteron system revisited write it here in the form given by Ando and Birse [179], L3 = MNH(Λ) 3Λ2 ( y2dN †(~d · ~σ)†(~d · ~σ)N + y2tN †(~t · ~τ)†(~t · ~τ)N − ydyt [ N †(~d · ~σ)†(~t · ~τ)N + h.c. ]) , (6.6) where Λ is a momentum cutoff applied in the three-body equations discussed below and H(Λ) a known log-periodic function of the cutoff that depends on a three-body parameter Λ∗. The Lagrangian in Eq. (6.6) differs slightly from the one quoted in Refs. [1, 169], which was taken over from Ref. [55]. The version given above corrects a mistake in the old expression and has also been used in Ref. [182]. Fortunately, the results in the older publications were not actually affected by this problem (see Section 6.2.4 for details). 6.2.1 Full dibaryon propagators In the strong sector, we adopt the standard power counting for large S-wave scattering length [49, 52]. A nucleon bubble together with a bare dibaryon propagator scales as O(1). The bare dibaryon propagators therefore are dressed by nucleon bubbles to all orders, giving the geometric series for the full propagators that are shown diagrammatically in Fig. 6.1. (a) = + + + · · · (b) = + + + · · · Figure 6.1: Full dibaryon propagators in (a) the 3S1 state (i.e. the deuteron) and (b) the 1S0 state. For convenience, we also resum the effective range corrections that arise when the dibaryons in the theory are promoted to dynamical fields by including their kinetic terms. We do not go into the details of the calculations here and simply quote the results for the renor- malized propagators. They are obtained by demanding that the (S-wave) effective range expansions k cot δd = −κd + ρd 2 (k2 + κ2d) + · · · (6.7) around the deuteron pole2 at k = iκd = i √ MNEd, and k cot δt = − 1 at + ρt 2 k2 + · · · (6.8) 2The notation ρd is used in the quadratic term of Eq. (6.7) because the expansion is not around zero momentum. The difference of ρd to the effective range in a standard expansion around zero momentum is smaller than 1% [16]. Note furthermore that in Ref. [1] we used the more common (at least in the context of pionless EFT) notation γd for the deuteron binding momentum. We changed it here to instead of κd in order to be consistent with the conventions used throughout the present work. 6.2. Formalism and building blocks 107 for the singlet channel are reproduced. The expansion for the singlet channel is around zero momentum; alternatively one could also expand here around the position of the virtual bound state. In writing Eq. (6.8), however, we have used that ρt = r0t to the order we are working at and will in the following also identify κt ≡ 1/at to make the notation more symmetric. After renormalization (using the PDS scheme [50]), the fully resummed propagators are ∆ijd (p) ≡ δij∆d(p) = − 4pii MNy2d · δ ij −κd + √ p2 4 −MNp0 − iε− ρd2 ( p2 4 −MNp0 − κ2d ) (6.9) and ∆ABt (p) ≡ δAB∆t(p) = − 4pii MNy2t · δ AB −κt + √ p2 4 −MNp0 − iε− ρt2 ( p2 4 −MNp0 ) , (6.10) which means that we have fixed the parameters appearing in the effective Lagrangian according to3 σd,t = 2 MN µ− κd,t ρd,t , y2d,t = 8pi M2N 1 ρd,t . (6.11) These expressions are valid up to N2LO since the resummation of the effective-range contributions only includes a subset of higher-order (N3LO etc.) terms. At leading or- der, range corrections are not included and the dibaryon kinetic terms do not contribute. The corresponding propagators are obtained by setting ρt = 0 and ρd = 0 in Eqs. (6.9) and (6.10), while perturbative expressions for the NLO and N2LO propagators can be obtained by expanding the equations up to linear and quadratic order in ρd and ρt, re- spectively. For each expression, the corresponding deuteron wave function renormalization constant is given by the residue at the bound state pole: Z−10 = i ∂ ∂p0 1 ∆d(p) ∣∣∣∣ p0=− κ 2 d MN ,p=0 . (6.12) At leading order, one simply has ZLO0 = κdρd , (6.13) whereas the result from the fully resummed expression (6.9) is ZN 2LO 0 = κdρd 1− κdρd . (6.14) The expressions for the perturbatively expanded propagators can then simply be read off from the geometric series 1 1− κdρd = 1 + κdρd + (κdρd) 2 + · · · . (6.15) 3Choosing the normal positive signs for the dibaryon kinetic energy terms in the Lagrangian (6.1) would have given a minus sign in the expression for y2d,t and thus made the coupling constants purely imaginary. This situation is avoided with the choice of the negative signs. 108 Chapter 6. The proton–deuteron system revisited Note that since κdρd ≈ 0.4, the above series converges only rather slowly. This fact can be taken into account by choosing an alternative renormalization scheme, called Z- parametrization [183], that is constructed in such a way that it produces the fully re- summed Z0 as given in Eq. (6.14) already at NLO. In Ref. [184] the approach has been applied to the three-nucleon system in pionless EFT and shown to improve the overall convergence of the theory. In the present work, however, where we are primarily inter- ested in the inclusion of Coulomb effects, we use the simpler scheme with the propagators as given in Eqs. (6.9) and (6.10). 6.2.2 Coulomb contributions in the proton–proton system The Coulomb interaction breaks the isospin symmetry that is implicit in the dibaryon propagators considered so far. For the p–p part of the singlet dibaryon we can also have Coulomb-photon exchanges inside the nucleon bubble. These can be resummed to all orders, yielding a dressed nucleon bubble [117, 172] that is subsequently used to calculate the full singlet-dibaryon propagator in the p–p channel. This is shown in Fig. 6.2. The result for the leading order propagator is [179] ∆ABt,pp(p) ≡ δAB∆t,pp(p) = − 4pii MNy2t · δ AB −1/aC − γ h˜0(p′) (6.16) with γ = γp–p = αMN , α = e2 4pi ≈ 1 137 , p′ = i √ p2/4−MNp0 − iε , (6.17) and function h˜`(p) as defined in Eq. (4.29), giving h˜0(p ′) = ψ(iη) + 1 2iη − log(iη) , η = η(p′) = γ 2p′ . (6.18) This means, of course, that the Coulomb-modified effective range expansion (4.28) dis- cussed in Chapter 4 has been used for renormalization. We denote here the p–p S-wave scattering length simply as aC . Corrections due to the corresponding Coulomb-modified effective range rC can be included in the same way as described in the preceding section by first resumming insertions of the kinetic energy operators to all orders and then matching the result to reproduce the modified effective range expansion up to quadratic order. = + + + · · · = + + + · · · Figure 6.2: Dressed nucleon bubble and full singlet dibaryon propagator in the p-p channel. To simplify the equations that we give later on, we introduce here the propagator functions Dd,t(E; q) ≡ (−i) ·∆d,t ( E − q 2 2MN , q ) (6.19) 6.2. Formalism and building blocks 109 and Dppt (E; q) ≡ (−i) ·∆t,pp ( E − q 2 2MN , q ) . (6.20) 6.2.3 Power counting The power counting of pionless effective field theory has been extensively discussed in the literature (see the reviews [43, 47, 185] and references therein), such that we can be rather brief on this subject here. We will, however, elaborate a bit on the power counting for the Coulomb sector of the theory as it was introduced by Rupak and Kong in Ref. [178]. Strong sector Without electromagnetic effects, the most relevant low-energy scale Q of the theory is set by the deuteron binding momentum κd ≈ 45 MeV. We can formally count the external momenta k, p to be of the same order. Since we are working in a setup without explicit pions, the natural ultraviolet cutoff of our theory is of the order of the pion mass, Λ ∼Mpi. Which cutoff is best to use in practice depends on whether one discusses the quartet- channel or the doublet-channel system. In the first case short-range effects are suppressed by the Pauli principle and one finds that already Λ = Mpi ≈ 140 MeV is sufficient for an accurate description. In the doublet channel, on the other hand, the cutoff must be set to a few hundred MeV to get converged results. In either case, the combination of the two scales yields the expansion parameter O(Q/Λ) ∼ 1/3 of pionless EFT. A further relevant scale in our system is the nucleon mass MN . It appears explicitly in kinetic energies, which scale as O(Q2/MN). As a consequence, the nucleon propagator scales as O(MN/Q2) and the loop integration measure d4q = d3q dq0 scales as O(Q5/MN). Note, however, that in each loop the dq0-integration can be carried out by picking up the residue from one nucleon propagator, so loops with one of the involved nucleon propagators canceled simply scale as O(Q3). We assume y2d ∼ y2t ∼ Λ/M2N for the nucleon–dibaryon coupling constants and σd ∼ σt ∼ QΛ/MN for the bare dibaryon-propagator terms. Together with the scaling of loops as discussed in the previous paragraph it then follows that in order to get the strong (n– d) scattering amplitude, one has to iterate the leading one-nucleon exchange diagram to all orders. The result can be written as an integral equation and will be discussed in Section 6.2.4. Including Coulomb photons The power counting has to be adapted when Coulomb effects are included. From the form of the (Yukawa-screened) Coulomb potential in momentum space, Vc,λ(q) ∼ α q2 + λ2 , (6.21) it is clear that Coulomb contributions dominate for small momentum transfers. As noted in Ref. [178], they enter ∼ αMN/q, i.e., proportional to the Coulomb parameter η intro- 110 Chapter 6. The proton–deuteron system revisited duced in Chapter 4. This behavior is not captured by the power counting for the strong sector, which consequently has to be modified in order to perform calculations including Coulomb effects for small external momenta. Most importantly, one can no longer simply assume that the scale of all momenta is set by the deuteron binding momentum, Q ∼ κd. Instead, one has to keep track of the new scale introduced by the external momenta separately. We generically denote this scale by p and assume p Q for the power counting. This means that we make a simultaneous expansion in two small parameters Q/Λ and p/(αMN) [178]. 4 For p & Q, the Coulomb contributions are small and the results in both schemes agree. With this modified counting, it is not straightforward to deduce the scaling of loops anymore. Kinetic energies always scale like Q, so the scaling of dq0 and the nucleon propagator is not modified in the presence of Coulomb effects. However, where before we could simply assume that all loop momenta are dominated by the scale Q ∼ κd, we now have to check first which contribution is picked up (or rather enhanced) after carrying out the dq0-integral. In the following we illustrate this by deducing the scaling of a few selected diagrams. Selected diagrams In Fig. 6.3 we show all diagrams contributing to p–d scattering that involve a single Coulomb-photon exchange and are thus of the order α. The last one, diagram (d) has the most straightforward behavior since it simply scales as α/p2. Diagram (a) has the same factor because also here the Coulomb-photon propagator involves the external momentum scale p, but it is further enhanced a factor Λ/Q from the nucleon bubble, which is easy to count as there are no Coulomb-photon exchanges inside the bubble. This is of course in perfect agreement with the fact that the direct coupling of the photon to the dibary- on—generated by gauging the dibaryon kinetic energy operators—only enters at NLO in the EFT counting. This makes diagram (d) both O(α) and an effective-range correction ∼ ρd,t. (a) (b) (c) (d) Figure 6.3: Leading O(α) diagrams involving Coulomb photons. Since diagram (b) is more complicated to count, we first consider diagram (c), where clearly the loop does not involve the external momentum p but is rather dominated by the deuteron binding momentum flowing into the triangle loop. This yields then that compared to the simple one-nucleon exchange diagram (without a photon) it is suppressed by a factor αMN/Q. Consistent with this one finds by a direct numerical evaluation that at threshold (vanishing p–d center-of-mass momentum) it is a 15% effect. 4For a related approach in the pionful theory, see also Ref. [186]. 6.2. Formalism and building blocks 111 Finally, although it is not straightforward to see due to the non-planar topology of dia- gram (b), one can argue also here that the deuteron binding momentum flowing into the loop sets the scale there and thus take all momenta to scale like Q. Following Rupak and Kong [178] in doing this, we get the same αMN/Q suppression factor as for diagram (c). A direct numerical calculation shows that it is a 7% effect at threshold. We can summarize the findings so far by saying that unless a diagram directly exhibits the (regulated) Coulomb pole, it is not enhanced and thus a small electromagnetic cor- rection to the same diagram topology without Coulomb-photon exchange. The enhanced Coulomb contributions are particularly large in the threshold region (where p is very small) and should thus be iterated to all orders. In our calculation of scattering phase shifts we neglect both diagrams (b) and (c). Their relative size at threshold mentioned above gives an a priori uncertainty of 7–15 percent, from which we conclude that Coulomb effects are included in the calculation at an accuracy corresponding to the typical NLO uncertainty from the EFT expansion. What we have stepped over so far are diagrams like Fig. 6.3a with more than one Coulomb photon attached to the nucleon bubble. For the diagram with two photons (and thus two loops that share a nucleon propagator) one finds a total scaling ∼ α2MNΛ/(Q3p) which means that compared to diagram (b) it is suppressed by a factor αMNp/Q 2. In Ref. [178] it is argued that the diagram with three photons could contribute with a factor ∼ log(p/Q), which is already small for p ≥ 1 MeV, and that the diagrams with n > 3 photons attached to the bubble are even infrared finite and suppressed by factors of αn. Neglecting all these contributions is thus consistent with not including diagrams (b) and (c). To conclude this section we point out again that the power counting established here is specifically designed to account for Coulomb contributions that become strong in the scattering at low momenta. At larger momenta, where the Coulomb parameter αMN/p becomes small, it would suffice to not iterate any Coulomb diagrams but rather include them strictly perturbatively. However, exactly because they become small we assume that it also does not spoil the calculation to iterate them everywhere. The above discussion clearly shows that the counting scheme employed here is rather com- plicated and, except for the simplest diagrams, certainly not unambiguous. As we will discuss in Section 6.5.2, for calculations in the bound-state regime the counting definitely has to be modified because there all loop-momentum scales are set by the binding momen- tum of the bound state. Based on the findings there, we will then argue in Section 6.6.5 that the same counting, which is actually simpler, should also be used in the scattering regime. For the moment, however, we proceed as in Ref. [1] and apply the counting as described above. 6.2.4 Integral equations From the preceding section it follows that certain diagrams have to be resummed to all orders in order to calculate N–d scattering amplitudes in pionless EFT. We now discuss how this is done by constructing and solving integral equations for the corresponding half off-shell T-matrices in momentum space. We start again by first discussing the strong sec- tor (neutron–deuteron scattering) and then subsequently include Coulomb contributions 112 Chapter 6. The proton–deuteron system revisited to get the desired proton–deuteron equations. The integral equations are essentially Lippmann–Schwinger equations written out in mo- mentum space. In the following, we generically use the calligraphic letter T to denote their solutions and use appropriate subscripts to indicate which system we are referring to. This choice of notation indicates that we adopt the standard convention in the field, where the T-matrix differs from the one defined in Chapter 2 by an overall minus sign, and that in order to obtain the physical T-matrix one also has to include the deuteron wave function renormalization factor. The exact relation is T (E; k,p) = −Z0T (E,k,p) . (6.22) Throughout the text, we write “T -matrix” (or simply “amplitude”) to indicate this dis- tinction. Strong sector = + Figure 6.4: Integral equation for the strong scattering amplitude Ts in the quartet channel. Figure 6.4 shows a diagrammatic representation of the quartet-channel n–d amplitude, which is the simplest case one can have. The only interaction occurring here is the one- nucleon exchange diagram with deuteron legs on both sides. Including all spin-, isospin- and symmetry factors, it is given by = − iMNy 2 d 2 · (σjσi)βαδba · 1 k2 + k · p + p2 −MNE − iε . (6.23) Since the spins of all three nucleons taking part in the reaction have to be aligned to produce a total spin-3/2 state, the Pauli principle prohibits here a three-nucleon contact interaction for which the particles have to occupy the same point in space. Furthermore, only the dibaryon field representing the deuteron can appear in the intermediate state. Generically, we define the S-wave projected amplitude as T (E; k, p) = 1 2 ∫ 1 −1 dcos θ T (E; k,p) , θ = θk·p . (6.24) Applying this to the one-nucleon exchange shown in Eq. (6.23) yields the projected inter- action kernel Ks(E; k, p) ≡ 1 kp Q0 ( k2 + p2 −MNE − iε kp ) (6.25) with the Legendre function of the second kind Q0(a) = 1 2 ∫ 1 −1 dx x+ a = 1 2 log ( a+ 1 a− 1 ) . (6.26) 6.2. Formalism and building blocks 113 The unprojected T -matrix has the same spin-isospin indices as the kernel function shown in Eq. (6.23), T = (T ij)βbαa. The n–d quartet channel is chosen by inserting i = (1−i2)/ √ 2 and j = (1 + i2)/ √ 2 for the deuteron spin-1 indices, α = β = 1 for the nucleon spins, and a = b = 2 to select the neutron in isospin space. As done in Refs. [55, 181], we have used here a short-hand notation for the spin-1 indices i and j that includes prefactors to be used in a linear combination. Written out explicitly, the strong quartet-channel amplitude is accordingly given by T qs = 1 2 ( T 11s + i (T 12s − T 21s )+ T 22s )12 12 (6.27) and fulfills the integral equation T qs = −MNy2dKs + T qs ⊗ [ MNy 2 dDdKs ] . (6.28) As in Ref. [1] we have introduced here the short-hand notation A⊗B ≡ 1 2pi2 ∫ Λ 0 dq q2A(. . . , q)B(q, . . .) (6.29) and used the propagator function Dd as defined in Eq. (6.19). As indicated in Eq. (6.29), we regulate all loop integrations with an explicit momentum cutoff Λ. = + + = + + Figure 6.5: Coupled-channel integral equation for the strong scattering amplitude Ts in the doublet channel. The diagrams involving the three-body force have been omitted. The equation for the doublet-channel amplitude is shown in Fig. 6.5. Since the spin-singlet dibaryon is now allowed to appear in the intermediate state, we have a coupled-channel system of two amplitudes that we call T d,as and T d,bs . Of these, the “upper part” T d,as corresponds directly to the n–d scattering process we are interested in, whereas T d,bs only enters as an off-shell quantity because the spin-singlet dibaryon, being only a virtual bound state, cannot appear as a true asymptotic state. Furthermore, with the spins coupled to a total spin 1/2, the Pauli principle no longer prohibits a three-nucleon interaction, and indeed such a term is needed already at leading order to renormalize the system [54]. The corresponding diagrams have, however, been omitted Fig. 6.5 for simplicity. They can be reinstated by supplementing every one- nucleon exchange with a matching N–d contact interaction from Eq. (6.6). With the n–d doublet-channel projection, T d,as = 1 3 (σi)α ′ α (T a,ijs )β ′b α′a(σ j)ββ′ ∣∣∣ a=b=2 α=β=1 , (6.30a) T d,bs = 1 3 (σi)α ′ α (T b,iBs )βb ′ α′a(τ B)bb′ ∣∣∣ a=b=2 α=β=1 , (6.30b) 114 Chapter 6. The proton–deuteron system revisited and introducing furthermore the abbreviations gdd = MNy 2 d 2 , gdt = MNydyt 2 , gtt = MNy 2 t 2 , (6.31) the result can be written as(T d,as T d,bs ) =  gdd ( Ks + 2H(Λ) Λ2 ) −gdt ( 3Ks + 2H(Λ) Λ2 )  + −gddDd ( Ks + 2H(Λ) Λ2 ) gdtDt ( 3Ks + 2H(Λ) Λ2 ) gdtDd ( 3Ks + 2H(Λ) Λ2 ) −gttDt ( Ks + 2H(Λ) Λ2 ) ⊗(T d,asT d,bs ) . (6.32) For a more detailed derivation of the equations above we refer to Refs. [1, 169].5 Although, as pointed out in Section 6.2, a different three-nucleon Lagrangian has been quoted in those works, the doublet-channel equation found there is the same as in Eq. (6.32) due an incorrect derivation of the corresponding Feynman rules that compensated the prob- lem. Presumably, the same was done by the authors of Ref. [55], which the results were compared to. Including Coulomb effects As discussed in Section 6.2.3, the dominant Coulomb contribution is the bubble diagram shown Fig. 6.3a. Its energy and momentum dependence is given by Kbubble(E; k,p) = Ibubble(E; k,p) (k− p)2 + λ2 , (6.33) where Ibubble(E; k,p) = arctan ( 2p2−k2−k·p√ 3k2−4MNE−iε √ (k−p)2 ) + arctan ( 2k2−p2−k·p√ 3p2−4MNE−iε √ (k−p)2 ) √ (k− p)2 (6.34) is the expression for the bubble loop integral. It can be simplified by noting that due to the denominator in Eq. (6.33) the whole expression is dominated by terms with p2 ≈ k2. When the expression appears under the dq integral, we analogously get p2 ≈ k2 and can furthermore assume that q2 ≈ k2 because of the pole at this position in the deuteron propagator. Inserting then the total center-of-mass energy, E = E(k) = 3k2 4MN − κ 2 d MN , k = |k| , (6.35) we get Ibubble(E; k,p) (k− p)2 + λ2 ≈ 1 2 |κd| 1 (k− p)2 + λ2 (6.36a) 5In particular, a comprehensive discussion of the spin-isospin projections can be found in Appendix B of Ref. [169]. 6.2. Formalism and building blocks 115 and ∆d ( E − q 2 2MN ,q ) · Ibubble(E; q,p) (q− p)2 + λ2 ≈ ∆d ( E − q 2 2MN ,q ) · 1 2 |κd| 1 (q− p)2 + λ2 , (6.36b) by using the expansion arctan(x) = x + O(x3) in Eq. (6.34). The same simplifications, which effectively amount to keeping only loop contributions with q ∼ p, have also been used in Ref. [178]. Note that they explicitly rely on the energy being given by Eq. (6.35) and are thus not valid in the bound-state regime, where E < −Ed = −κ2d/MN . For diagrams with the spin-singlet dibaryon in the intermediate state, as they appear in the doublet-channel calculation, the argument based on the deuteron pole is actually not true. However, for energies in the scattering regime, Eq. (6.36) is still a good approxi- mation (numerically, a 15% effect at threshold and thus compatible with neglecting the diagram shown in Fig. 6.3c). A more subtle point is that the above approximation also changes the ultraviolet scaling of the diagram, an effect for which it is difficult to judge a priori how important it is. Since the only true advantage of the approximation is that it (quite significantly) simplifies the calculation, but certainly does not improve it in any physical sense, it is probably best to not use it if possible. After first giving the results obtained with the approximation in Section 6.3 (as reported in Ref. [1]), we will come back to discussing its impact in Section 6.6.5. Three-channel formalism = + + + × ( + + ) + + = + + × ( + + ) + = + + Figure 6.6: Coupled-channel integral equation for the full (i.e. strong + Coulomb) scattering amplitude Tfull in the doublet channel. The diagrams representing the three- nucleon force have been omitted. Due to the fact that the electromagnetic interaction does not couple to isospin eigenstates we now need two different projections for the amplitude T b with the outgoing spin-singlet 116 Chapter 6. The proton–deuteron system revisited dibaryon: T d,b1full = 1 3 (σi)α ′ α (T b,iBfull )βb ′ α′a(1 · δB3)bb′ ∣∣∣ a=b=1 α=β=1 , (6.37a) T d,b2full = 1 3 (σi)α ′ α (T b,iBfull )βb ′ α′a(1 · δB1 + i1 · δB2)bb′ ∣∣∣a=1, b=2 α=β=1 . (6.37b) The latter corresponds to the amplitude with the outgoing spin-singlet dibaryon in a pure p–p state. For the diagrams where this this component appears in the intermediate state we have to insert the p–p propagator (6.16). Diagrammatically, the resulting three-channel integral equation is shown in Fig. 6.6. It is given by  T d,afull T d,b1full T d,b2full  =  gdd ( Ks + 2H(Λ) Λ2 ) −gdt ( Ks + 2H(Λ) 3Λ2 ) −gdt ( 2Ks + 4H(Λ) 3Λ2 ) +  gddK (d) c 0 0  +  −gddDd ( Ks + 2H(Λ) Λ2 ) gdtDt ( 3Ks + 2H(Λ) Λ2 ) 0 gdtDd ( Ks + 2H(Λ) 3Λ2 ) gttDt ( Ks − 2H(Λ)3Λ2 ) 0 gdtDd ( 2Ks + 4H(Λ) 3Λ2 ) −gttDt ( 2Ks + 4H(Λ) 3Λ2 ) 0 ⊗  T d,afull T d,b1full T d,b2full  +  −gddDdK(d)c 0 gdtDppt ( 3Ks + 2H(Λ) Λ2 ) 0 −gttDtK(t)c −gttDppt ( Ks + 2H(Λ) 3Λ2 ) 0 0 −gttDppt × 4H(Λ)3Λ2 ⊗  T d,afull T d,b1full T d,b2full  (6.38) with the Coulomb kernel functions K(d,t)c (E; k, p) = −αMN × [∫ 1 −1 d cos θ Ibubble(E; k,p) (k− p)2 + λ2︸︷︷︸ LO + ρd,t 2kp Q ( −k 2 + p2 + λ2 2kp ) ︸︷︷︸ NLO ] ≈ αMN 2kp Q ( −k 2 + p2 + λ2 2kp )( 1 |κd| − ρd,t ) . (6.39) In writing Eq. (6.38) we have separated the terms in such a way that the Coulomb contributions can be easily identified. Note that the terms involving H(Λ) in Eq. (6.38) differ slightly from the version given in Refs. [1, 169] due to the incorrect Feynman rules used in those works. Still, this turns out not to be a serious problem because both the old expressions and Eq. (6.38) are such that they reduce to the two-channel n–d equation (6.32) when all Coulomb effects are switched off and one considers the sum T d,b = T d,b1 + T d,b2. As a consequence, one finds the same H(Λ) in a numerical fit to the triton binding energy as with the old expression, and all observables are unaffected. 6.2. Formalism and building blocks 117 In order to calculate Coulomb-subtracted phase shifts, we also need the amplitude for pure Coulomb scattering. Since the electromagnetic interaction in our approximation does not couple the different channels, for both quartet-channel and doublet-channel p–d scattering is given by the simple equation Tc = gddK(d)c − Tc ⊗ [ gddDdK (d) c ] . (6.40) Higher-order corrections From Eq. (6.39) one directly sees that the diagram with the photon coupled directly to a dibaryon (Fig. 6.3d and its analog with a spin-singlet dibaryon) is proportional to the effective range (ρd or ρt) and thus a correction that enters at next-to-leading order in the EFT power counting. This has already been mentioned in Section 6.2.3. Apart from that, the order of our calculation is determined by the expressions used for the dibaryon propagators. The fully resummed propagators given in Eqs. (6.9) and (6.10) have spurious deep poles that do not correspond to actual physical bound states. In the quartet channel, the cutoff can be chosen low enough to avoid that pole. Due to the larger cutoff needed in the doublet channel, however, we cannot use the resummed propagators here. Instead, we follow the approach of Ref. [187] and use the perturbative expansions (more appropriately called “partially resummed propagators”) mentioned in Section 6.2.1. This still resums some higher-order effective-range contributions, but removes the unphysical pole.6 More precisely, at next-to-leading order we always use propagators DNLOd,t that include a single insertion of the dibaryon kinetic energy operator and are thus linear in the effective range. At next-to-next-to leading order (N2LO) the propagators DN 2LO d,t for the doublet- channel calculation include corrections quadratic in the effective ranges, whereas in the quartet-channel we use the fully-resummed expression given by Eq. (6.9) together with a cutoff low enough to avoid the unphysical pole. Since the publication of Ref. [1] an agreement has been reached in the literature [187– 189] that in the doublet channel, a second (energy-dependent) three-nucleon interaction is needed at N2LO for consistent renormalization. As in Ref. [1], we do not include such a term here for simplicity and thus obtain only a partial N2LO result. In fact, as we will discuss in Section 6.6.5, with Coulomb effects included, the question of correct renormalization in the double channel might have to be reconsidered already at next-to- leading order. 6.2.5 Numerical implementation The integral equations presented in the previous sections have to be solved numerically. We do so by discretizing the integrals, using Gaussian quadrature, principal value inte- 6Note that the resummation procedure also affects the ultraviolet behavior of the propagators. The fully resummed expressions fall off faster than the leading-order propagators, whereas the perturbative expansions do not go to zero anymore for large p. This point will become important in Section 6.6. 118 Chapter 6. The proton–deuteron system revisited gration to deal with the singularity of the deuteron propagator, and appropriate trans- formations of the integration domain. The latter are especially important to deal with the numerical difficulties caused by the Coulomb-photon propagators. Even though we have regulated the singularity with an artificial photon mass λ, the latter has to be kept small in order to not modify the theory too much. This then yields strongly peaked functions. As shown in Ref. [169], the Coulomb peaks in the inhomogeneous parts the integral equations are the major numerical difficulty. It can be overcome by concentrating the quadrature points around the peak and always putting half of the quadrature points into the low-momentum region. With this procedure is is then possible to (linearly) extrapolate the results back to the physical case λ = 0. The error introduced by this extrapolation can in general be neglected compared to the theoretical uncertainty from the EFT expansion discussed below. We use the experimental input parameters shown in Table 6.1 for the numerical calcula- tion. Parameter Value Ref. Parameter Value Ref. κd 45.701 MeV [48] ρd 1.765 fm [16] at −23.714 fm [47] ρt 2.73 fm [47] aC −7.8063 fm [145] rC 2.794 fm [145] Table 6.1: Parameters used for the numerical calculation. 6.3 Scattering phase shifts From the solutions of the integral equations we can obtain the S-wave scattering phase shifts as δ(k) = 1 2i log ( 1 + 2ikMN 3pi Z0T (Ek; k, k) ) . (6.41) In order to account for the deuteron as a composite asymptotic state, the T -matrix is multiplied by the wave function renormalization Z0 defined in Eq. (6.12). This procedure also removes the dependence of the amplitude on the coupling constant yd, which so far we have have kept in all equations. We call Z0T (Ek; k, k) the renormalized amplitude. For the p–d system, what we compare to experimental data are the Coulomb-subtracted phase shifts already introduced in Chapter 4. We calculate them here as δ˜(k) ≈ δdiff(k) ≡ δfull(k)− δc(k) , (6.42) where δfull(k) is obtained from the full integral equation including both Coulomb and strong interactions and δc(k) is obtained from Eq. (6.40) which only includes the Coulomb interaction. This procedure (as opposed to calculating the pure Coulomb phase shift analytically) has the advantage of properly taking into account of the finite cutoff, the EFT expansion, and the regulating photon mass λ. In the following, we report the results as given in Ref. [1]. 6.3. Scattering phase shifts 119 6.3.1 Quartet channel In Fig. 6.7 we show the phase shift results for both neutron–deuteron and proton–deuteron scattering as functions of the center-of-mass momentum k. The error bands are generated by varying the cutoff within a range of 120 to 160 MeV. Since the cutoff variation is small, we conclude that the calculation is well converged at these cutoffs. This finding is important because it allows us to use the fully resummed deuteron propagator (6.9) at N2LO, which would be problematic numerically for cutoffs Λ & 200 MeV due to the unphysical second pole introduced by the resummation. For the n–d curve in Fig. 6.7 we have used a single cutoff Λ = 140 MeV since we only include this result here for comparison and because its band produced by the cutoff variation would be as narrow as that of the p–d result at N2LO. The fact that the error bands do not overlap is no point of concern since they only give a lower bound on the uncertainty of the calculation. From the expansion parameter Q/Λ ∼ κdρd ≈ 1/3 of the EFT, the error can be estimated as 30%, 10%, and 3% at LO, NLO, and N2LO, respectively. Thus, at LO, the 30% uncertainty from the expansion parameter clearly dominates. At NLO and N2LO, however, the band from the cutoff variation gives a reasonable estimate of the total error in the calculation. Our N2LO band agrees very well with the results of Rupak and Kong [178] and also with the experimental data included in the plot. Due to our improved numerical procedure we were able to extend the calculation of Ref. [178] into the threshold region below k = 20 MeV (indicated by the dotted line in Fig. 6.7), where Rupak and Kong could not obtain convergent results for small photon masses. 6.3.2 Doublet channel The doublet-channel results for the p–d scattering phase shifts as functions of the center- of-mass momentum k are shown in Fig. 6.8. As in the quartet channel, the p–d curve lies above the n–d curve and at least at N2LO it agrees quite well with the experimental data. A more quantitative comparison is, unfortunately, not possible since there are no errors given for the data points. The error bands were generated here by varying the cutoff within a natural range of 200 to 600 MeV (i.e., a few times the pion mass). Note that according to the power counting of Ref. [187] our N2LO calculation is incom- plete since we did not include a subleading three-body force at this order. Figure 6.8 nevertheless shows how the results improve from order to order. Despite the fact that we omitted the subleading three-body force, our partial N2LO result is stable to within about ten percent under the cutoff variation, which is consistent with the 7–15% error es- timate based on the neglected Coulomb diagrams (see Section 6.2.3 and Ref. [178]). This stability suggests that the scattering is relatively insensitive to the subleading three-body interaction. At higher energies, however, there is some room for such a contribution as our partial result consistently lies two to four degrees above the experimental data. We observe that the shift from LO to NLO is of the same order of magnitude as the shift from NLO to N2LO, a behavior that is typical for effective-range corrections in the doublet channel [191]. The smallness of the NLO corrections can be understood as a 120 Chapter 6. The proton–deuteron system revisited 0 20 40 60 80 100 -100 -80 -60 -40 -20 0 k (MeV) δ (d eg ) p–d LO p–d NLO p–d N2LO p–d Exp. p–d Exp. n–d N2LO · Figure 6.7: N–d quartet-channel S-wave scattering phase shifts as functions of the center-of-mass momentum k. Error bands generated by cutoff variation from 120 to 160 MeV. Experimental p–d phase shift data taken from Refs. [146] (diamonds) and [190] (circles). cancellation between two different contributions to this correction, proportional to κρd and κdρd, respectively, where κ is the typical momentum scale of the process. Furthermore, it is known that at LO observables are often described better than expected from the power counting once the exact pole position of the two-body propagator is reproduced [28]. As a consequence, the shifts in observables from LO to NLO can be small and of a size comparable to the corresponding shifts from NLO to N2LO, which more directly reflect the uncertainty due to the EFT expansion. Upon closer examination it turns out that the above results should actually be taken with a grain of salt. This has already been hinted at when we discussed the power counting in Section 6.2.3 and the approximation made in Eq. (6.36). Indeed, a more careful analysis of the bound-state regime that has been carried out since the publication of Ref. [1] also sheds some new light on the scattering calculation. After first giving a detailed discussion of the new bound-state results in the following sections, we will thus revisit this issue in Section 6.6.5. 6.4 Trinucleon wave functions In order to calculate the 3H–3He binding-energy shift in perturbation theory we need the wave functions that describe the trinucleon (triton) bound state. As illustrated diagram- 6.4. Trinucleon wave functions 121 0 20 40 60 80 100 -70 -60 -50 -40 -30 -20 -10 0 k (MeV) δ (d eg ) p–d LO p–d NLO p–d N2LO p–d Exp. n–d N2LO · Figure 6.8: N–d doublet channel S-wave scattering phase shifts as functions of the center-of-mass momentum k. Error bands generated by cutoff variation from 200 to 600 MeV. Experimental p–d phase shift data taken from Ref. [146]. matically in Fig. 6.9, at the bound-state pole the T -matrix factorizes as T (E; k, p) = −B †(k)B(p) E + EB + terms regular at E = −EB , (6.43) where the B(p) are what we call amputated wave functions or vertex factors. For a derivation of this relation, including the sign, see Section D.4.3 in Appendix D. ∼ E + EB + regular terms Figure 6.9: Diagrammatic representation of the factorization of T -matrix at the bound- state pole. 6.4.1 Homogeneous equation For our coupled-channel problem, T (E; k, p) is a matrix in channel-space, and the wave functions will thus be vectors. Since the Coulomb repulsion does not act in the channel where the spin-singlet dibaryon is in a pure p-p or n-n state, we use here the three-channel equation structure introduced for the p–d scattering equations, but with all Coulomb 122 Chapter 6. The proton–deuteron system revisited contributions switched off. Our interaction kernel is then given by Kˆ ≡  −gdd ( Ks + 2H(Λ) Λ2 ) gdt ( 3Ks + 2H(Λ) Λ2 ) gdt ( 3Ks + 2H(Λ) Λ2 ) gdt ( Ks + 2H(Λ) 3Λ2 ) gtt ( Ks − 2H(Λ)3Λ2 ) −gtt ( Ks + 2H(Λ) 3Λ2 ) gdt ( 2Ks + 4H(Λ) 3Λ2 ) −gtt ( 2Ks + 4H(Λ) 3Λ2 ) −gtt × 4H(Λ)3Λ2  , (6.44) which is found from Eq. (6.38) by factoring out all propagators and omitting the K (d,t) c en- tries. Inserting the factorization at the pole (6.43) into the Lippmann-Schwinger equation and taking the limit E → −EB gives a homogeneous equation of the form ~Bs = (KˆDˆ)⊗ ~Bs (6.45) with ~Bs ≡ (Bd,as ,Bd,b1s ,Bd,b2s )T , Dˆ = diag(Dd, Dt, Dt) , (6.46) The inhomogeneous interaction terms have dropped out here since they are regular as E = −EB, and we have canceled the overall B† terms from the factorization. 6.4.2 Normalization condition In order to calculate quantities based on the wave functions Bs it is important to normalize them correctly. To this end one has to take into account that the “potential” derived from the EFT we are using here, i.e., the one-nucleon exchange kernel as given in Eqs. (6.23) and (6.25), is effectively energy-dependent. The normalization condition is( Dˆ ~Bs )T ⊗ d dE ( Iˆ − Kˆ ) ∣∣∣ E=−E 3HB ⊗ ( Dˆ ~Bs ) = 1 , (6.47) where we have defined the matrix of inverse propagators Iˆ = diag(Id, It, It) with Id,t(E, q, q ′) = 2pi2 q2 δ(q − q′)Dd,t(E; q)−1 . (6.48) A detailed derivation of this can be found in Appendix D, in particular in Section D.4. Note, however, that the insight that energy-dependent interactions imply a nontrivial normalization condition for bound-state wave functions is not new at all but has been known for a long time (see, for example, the overview by Agrawala et al. [192] and references therein). Numerical verification The normalization condition derived above can be checked by considering the residue of the T -matrix at the bound-state pole. Following Hagen et al. [68] we define Z = limE→−EB(E + EB) ∫ Λ 0 dq q2 2pi2 ∫ Λ 0 dq′ q′2 2pi2 D(E, q)T (E; q, q′)D(E, q′) (6.49) 6.4. Trinucleon wave functions 123 and call this quantity the Z-factor of the trinucleon state, or simply trimer Z-factor. As we will discuss shortly, for our coupled-channel system we need to sum over all components of the T -matrix. In Eq. (6.49) we have written D(E, q) to denote a generic dibaryon propagator. Inserting the factorization of the T -matrix at the pole as given in Eq. (6.43), we find that Z = ∫ Λ 0 dq q2 2pi2 ∫ Λ 0 dq′ q′2 2pi2 D(−EB, q)B†(q)B(q′)D(−EB, q′) = ∣∣∣∣∫ Λ 0 dq q2 2pi2 D(−EB, q)B(q) ∣∣∣∣2 , (6.50) where the last equality follows by noting that the propagators are real in the bound- state regime. Of course, it is crucial here that the wave functions B(q) are normalized correctly, so numerically calculating Z from both Eqs. (6.49) and (6.50) and showing that the results agree provides the means to verify our normalization condition (6.47) with an explicit calculation. We now carry out this procedure for the triton wave functions in the two-channel for- malism. In order to implement Eq. (6.49) we also need the part of the amplitude that describes the (unphysical) scattering of a spin-singlet dibaryon and a nucleon. The com- plete T -matrix for the system is then given by Tˆ ds = (T d,as T d,cs T d,bs T d,ds ) , (6.51) where the first column is determined by Eq. (6.32) and the second column is a solution of the analogous equation (T d,cs T d,ds ) = −gdt ( 3Ks + 2H(Λ) Λ2 ) gtt ( Ks + 2H(Λ) Λ2 )  + −gddDd ( Ks + 2H(Λ) Λ2 ) gdtDt ( 3Ks + 2H(Λ) Λ2 ) gdtDd ( 3Ks + 2H(Λ) Λ2 ) −gttDt ( Ks + 2H(Λ) Λ2 ) ⊗(T d,csT d,ds ) . (6.52) The expression for the trimer Z-factor in terms of the T -matrix is then ZT = limE→−EB(E + EB) 2∑ i,j=1 [ Dˆii ⊗ (Tˆ ds )ij ⊗ Dˆjj] , (6.53) where we have switched back to the short-hand notation of Section 6.2.4 (with the mod- ification that Dˆ = diag(Dd, Dt) is only a 2×2 matrix here) and inserted a subscript T in order to distinguish it from the expression in terms of the wave functions, which we write as ZB = 2∑ i=1 ∣∣∣Dˆii ⊗ ( ~Bs)i∣∣∣2 . (6.54) 124 Chapter 6. The proton–deuteron system revisited Numerically calculating ZB is straightforward since it is just a simple integral over the normalized wave functions. Implementing the procedure to calculate ZT , on the other hand, is more delicate since in order to numerically obtain the residue one has to rely on cancellations between large numbers. If, however, we approach the bound-state pole exponentially, i.e., set E(x) = −EB + (EB − E0) · exp(−x) , (6.55) where E0 is some energy between the deuteron binding energy Ed and EB, we find that indeed ZT converges to ZB as a function of the parameter x.7 We show this in Figs. 6.10 and 6.11 for two different cutoffs Λ = 1000 MeV and Λ = 377.69 MeV, where the latter value is approximately the first zero of the three-nucleon force H(Λ).8 In both calculations we have used E0 = 1.1×Ed, which means that at x = 20 the distance to the pole (which is always fixed to be exactly at EB = 8.4818 MeV) is only about 10−8 MeV. Only if one goes even closer to the pole (x & 25), numerical difficulties become significant and ZT starts to visibly deviate from ZB. At x = 20, ZT still agrees with ZB to within about 1.5% for Λ = 1000 MeV and with better than 0.05% accuracy for Λ = 377.69 MeV.9 Altogether, we take the above findings as a clear numerical verification underlining the correctness of our normalization condition (6.47). Note that the value of the Z-factor varies significantly for different cutoffs. This can be understood by noting that the way we have defined it in Eqs. (6.49) and (6.50), Z is an unrenormalized quantity. For the current discussion where we are only interested in establishing the agreement of ZT and ZB, however, this is of no importance. 6.5 Helium-3 properties We are now finally in a position to discuss the Helium-3 system. In the first part of this section we will describe how to obtain the 3H–3He binding energy shift in a perturbative approach based the normalized trinucleon wave functions Bs. The results presented here correct and thus supersede those given in Ref. [1]. We will additionally describe how to obtain the 3He binding energy nonperturbatively by locating the pole in the full T -matrix. The results of both calculations will be shown and discussed in Subsec. 6.5.3. 7The author would like to thank P. Hagen for suggesting this approach. 8This selection of cutoffs is mostly arbitrary. The reason for choosing the latter cutoff was to inves- tigate if the three-nucleon force has any significant influence on the numerical convergence. Comparison with Fig. 6.10 shows that this does not seem to be the case. We have checked that also at other cutoffs the overall behavior is the same. 9The better agreement for Λ = 377.69 MeV is most probably due to the absolute value of the Z- factor being much smaller there than at Λ = 1000 MeV, which makes it easier to accurately determine a numerical result originating from the cancellation between large numbers. 6.5. Helium-3 properties 125 2 4 6 8 10 12 14 16 18 20 4000 6000 8000 10000 12000 14000 16000 18000 20000 x tr im er Z -f ac to r ZB ZT · simple Λ = 1000 MeV Z = 6655.21 Figure 6.10: Triton trimer Z factor for cutoff Λ = 1000 MeV. x is the parameter in Eq. (6.55). 2 4 6 8 10 12 14 16 18 20 50 100 150 200 250 300 350 x tr im er Z -f ac to r ZB ZT · Λ = 377.69 MeV Z = 74.543 Figure 6.11: Triton trimer Z factor for cutoff Λ = 377.69 MeV, where H(Λ) ≈ 0. x is the parameter in Eq. (6.55). 126 Chapter 6. The proton–deuteron system revisited 6.5.1 Energy shift in perturbation theory In Ref. [1] the 3H–3He binding-energy difference was calculated using ∆EoldC = ( Dˆ ~Bs )T ⊗ diag(VC , VC , 0)⊗ ( Dˆ ~Bs ) (6.56) with the S-wave projected Coulomb potential VC(E; q, q ′) = −4piα 2qq′ Q ( −q 2 + q′2 + λ2 2qq′ ) (6.57) in momentum space. The prediction for the 3He binding energy is then given by −EB3He = −EB3H + ∆EoldC . (6.58) However, the na¨ıve approach in Eq. (6.56) is actually not correct. By taking the simple matrix elements of the Coulomb potential between the wave functions ~Bs in the nucleon– dibaryon formalism, one effectively couples the photon directly to the dibaryon. However, as discussed in Section 6.2.3, this coupling only enters at next-to-leading order in the EFT power counting. More subtle, but no less important, is to realize that the expression in Eq. (6.56) is not renormalized correctly. By considering the terms that enter in the normalization condition (6.47) we find that formally the wave functions ~Bs are proportional to the EFT coupling constants yd and yt. In the results for physical quantities this dependence has to drop out, just as it does in the scattering calculation where the deuteron wave function renormalization constant cancels the yd-dependence of the T -matrix. In fact, Eq. (6.56) neglects the three-body nature of the problem. To correct this, what should be done is to calculate the diagram shown in Fig. 6.12a at leading order and only add the contribution shown Fig. 6.12b as an NLO-correction. Effectively, this amounts to calculating ∆EnewC = ( Dˆ ~Bs )T ⊗ KˆC ⊗ ( Dˆ ~Bs ) (6.59) with the matrix KˆC = diag(−gddK(d)c ,−gttK(t)c , 0) . (6.60) The kernel functions Kc here are the same that appear in the integral equation for the full scattering amplitude, with the leading-order contribution given by the bubble diagram, Fig. 6.3a, and the direct coupling to the dibaryon only entering at next-to-leading order. As we will discuss in more detail in the next section, it is important here to not use the approximation (6.36) for the bubble diagram but rather include the full expression given in the first line of Eq. (6.39). The new procedure takes into account the EFT expansion and, since KˆC is proportional to y2d,t, ∆E new C is also renormalized correctly. However, the picture is still not complete. As shown in Fig. 6.13, there are also contributions to the energy shift that arise from Coulomb-photon diagrams not taken into account so far. For the scattering calculation we argued that they can be neglected, but we already mentioned at the end of Section 6.2.3 that the power counting has to be modified in the bound-state regime. As we will discuss shortly, these additional diagrams actually are important and should be taken into account in the energy-shift calculation. The complete KˆC then becomes a non-diagonal matrix. 6.5. Helium-3 properties 127 (a) (b) Figure 6.12: Diagrams contributing to the 3H–3He binding energy difference in pertur- bation theory. (a) Leading-order diagram. (b) NLO-correction. (a) (b) Figure 6.13: Additional diagrams contributing to the 3H–3He binding energy difference in perturbation theory. (a) Box diagram. (b) Triangle diagram. 6.5.2 Nonperturbative calculation In Ref. [179], Ando and Birse calculate the 3He binding energy in pionless EFT by using a nonperturbative framework that involves the full off-shell Coulomb T-matrix. As already discussed by Kok et al. in Refs. [193, 194], the latter complication is not actually necessary in the bound-state regime. In this section we carry out a calculation analogous to that of Ando and Birse, but only involving Coulomb photons. The resulting equation structure is still quite complex, but much simpler and in particular easier to handle numerically than that of Ref. [179]. Diagram scaling in the bound-state regime At this point it is important to recall from Section 6.2.3 that the Coulomb diagrams in Figs. 6.3b and 6.3c, which in the following we simply refer to as the box and triangle diagrams, respectively, are not small per se, but rather that the bubble diagram, Fig. 6.3a, is enhanced in the low-energy scattering regime due to the Coulomb singularity at zero momentum transfer. That effect is particularly prominent in the inhomogeneous part of the integral equation, where in the on-shell limit one directly hits the Coulomb pole and the expression is only rendered finite by the regulating photon mass λ. In the bound-state regime this is no longer the case. There are no inhomogeneous terms that could exhibit a Coulomb peak, and since all loops are dominated by the binding momentum of the bound-state under consideration, the dibaryon propagators do not further enhance integration domains of small momentum transfer. The consequence of all this is that in a nonperturbative calculation of the Helium-3 binding energy all O(α) Coulomb diagrams should be treated on an equal footing and included 128 Chapter 6. The proton–deuteron system revisited in the calculation. Furthermore, for the integral in the bubble diagram we have to use the full expression as given by Eq. (6.34) since the argument allowing the approximation given in Eq. (6.36) is not valid in the bound-state regime. Full equation structure The resulting full equation structure is shown diagrammatically in Fig. 6.14 and given by the expression  T d,a full′ T d,b1 full′ T d,b2 full′  =  gdd ( Ks + 2H(Λ) Λ2 ) −gdt ( Ks + 2H(Λ) 3Λ2 ) −gdt ( 2Ks + 4H(Λ) 3Λ2 ) +  gdd ( K (d) c +Kbox ) −gdtKbox −2gdtK(in)tri  +  −gddDd ( Ks + 2H(Λ) Λ2 ) gdtDt ( 3Ks + 2H(Λ) Λ2 ) 0 gdtDd ( Ks + 2H(Λ) 3Λ2 ) gttDt ( Ks − 2H(Λ)3Λ2 ) 0 gdtDd ( 2Ks + 4H(Λ) 3Λ2 ) −gttDt ( 2Ks + 4H(Λ) 3Λ2 ) 0 ⊗  T d,a full′ T d,b1 full′ T d,b2 full′  +  −gddDd ( K (d) c +Kbox ) 3gdtDtKbox gdtD pp t ( 3Ks + 3K (out) tri + 2H(Λ) Λ2 ) gdtDdKbox −gttDt ( K (t) c −Kbox ) −gttDppt ( Ks +K (out) tri + 2H(Λ) 3Λ2 ) 2gdtDdK (in) tri −2gdtDtK(in)tri −gttDppt × 4H(Λ)3Λ2  ⊗  T d,a full′ T d,b1 full′ T d,b2 full′  , (6.61) which, albeit quite complex, is a direct extension of Eq. (6.38). We use a prime in the subscript to indicate the inclusion of the additional Coulomb contributions. The first of these, Kbox, is initially given by a rather complicated expression but can be simplified to [195] Kbox(E; k, p) = −αMN × 1 2 ∫ 1 −1 dcos θ {arctan( 2p2−k2−k·p√ 3k2−4MNE−iε √ (k−p)2 ) + arctan ( 2k2−p2−k·p√ 3p2−4MNE−iε √ (k−p)2 ) (k2 + p2 + k · p−MNE − iε) √ (k− p)2 − λ (k2 + p2 + k · p−MNE − iε)2 } , (6.62) which is valid up to (negligible) corrections of order λ2. For the triangle-diagram contri- butions one furthermore finds K (out) tri (E; k, p) = −αMN × 1 2 ∫ 1 −1 dcos θ Itri(E; k,p) k2 + p2 + k · p−MNE − iε , (6.63a) 6.5. Helium-3 properties 129 K (in) tri (E; k, p) = K (out) tri (E; k, p) , (6.63b) where the superscripts indicate whether the Coulomb-photon exchange is on the incoming (left) or outgoing (right) side of the diagram. The loop function appearing in Eq. (6.63a) is given by Itri(E; k,p) = i 2 √ k2/4 + k · p + p2 × { log ( i(k2/2− k · p− p2 − λ2 −MNE − iε)√ k2/4 + k · p + p2 + 2 √ λ2 + 3k2/4−MNE − iε ) − log ( i(k2 + p2 + k · p− λ2 −MNE − iε)√ k2/4 + k · p + p2 + 2λ )} . (6.64) The spin and isospin projections for both diagrams are completely analogous to those for the simple one-nucleon exchange diagram, with additional projection operators from the photon vertices ensuring that contributions forbidden by charge conservation vanish as they should. = + + + × ( + + ) + × ( + ) + × ( + ) = + + × ( + ) + × ( + + ) + × ( + ) = + + × ( + ) + × ( + ) Figure 6.14: Coupled-channel integral equation for the full scattering amplitude Tfull′ used for the nonperturbative 3He calculation. The diagrams representing the three- nucleon force have been omitted. 6.5.3 Leading-order results The leading-order results for the 3He binding energy are summarized in Fig. 6.15. The lower dashed curve shows the result of the na¨ıve calculation reported in Ref. [1]. The upper dashed curve is the corrected result according to Eq. (6.59), where only the bubble diagram 130 Chapter 6. The proton–deuteron system revisited is taken into account at leading order. The effect of including all O(α) Coulomb diagrams (that are leading order in the EFT counting) is shown by the lower solid curve, whereas the upper solid curve shows the result obtained from the nonperturbative calculation described in Section 6.5.2. For comparison, the experimental 3He and 3H binding energies are indicated as dotted lines and the cutoff dependence of the three-nucleon force H(Λ) is shown as a thin dashed curve. 1000 10000 -5 0 5 6.4 6.8 7.2 7.6 8 8.4 Λ (MeV) E B (M eV ) H (Λ ) from WF (old) from WF (pure Coulomb) from WF (all contrib.) nonpert. calc. Ando+Birse exp. EB( 3He) exp. EB( 3H) Figure 6.15: Leading-order predictions for the 3He binding energy as a function of the cutoff. Lower dashed curve: result from the na¨ıve calculation reported in Ref. [1]. Up- per dashed curve: corrected result according to Eq. (6.59). Lower solid curve: corrected result including all O(α) Coulomb diagrams. Upper solid curve: nonperturbative result according to Section 6.5.2. Red triangle: result from Ref. [179]. Dotted lines: experimen- tal values for the 3H and 3He binding energies. Thin dashed curve: cutoff-dependence of three-nucleon force. The most striking feature is that the new results, both from the perturbative and the nonperturbative calculation, do not show the fall-off at large cutoffs that was reported in Ref. [1] and is shown here by the lower dashed curve. An explanation for why this effect occurs in the na¨ıve calculation will be given below in Section 6.6. In fact, the new results are both essentially stable beyond cutoffs of about 500 MeV. From Fig. 6.15 we furthermore see clearly that it is important to take into account all O(α) Coulomb diagrams in the perturbative calculation. Comparing it with the result from the bubble diagram alone shows that the box and triangle diagrams together give more than half of the total energy shift (with the larger part coming from the triangle contri- butions). The nonperturbative result comes out closer to the experimental value than the perturbative one, as should be expected from that more complete calculation. Within the typical 30% uncertainty of a leading-order calculation in pionless EFT, however, they agree both with one another and with the experimental Helium-3 binding energy of about 6.6. The Coulomb problem at next-to-leading order 131 7.72 MeV.10 Finally, the result obtained by Ando and Birse in Ref. [179] by using the full off-shell Coulomb T-matrix lies almost directly on our curve. The small energy difference of about 0.004 MeV that we obtain at their cutoff Λ = 380.689 MeV is well below the accuracy of the calculation and cannot be resolved in the plot.11 This is in perfect agreement with the conclusion of Kok et al. [193, 194] that a Coulomb-photon approximation should be well justified in the bound-state regime. Wave functions From the new equation structure (6.61) we can also obtain Helium-3 wave functions. In Fig. 6.16 we show a representative plot, calculated at a cutoff Λ = 400 MeV. The solid curves indicate the result obtained from the full equation including all leading Coulomb diagrams, whereas the dashed curves are obtained by calculating simple trinucleon wave functions (without Coulomb effects) for a system with the three-nucleon force was fixed to reproduce the experimental 3He binding energy. Both are normalized according to the condition (6.47) with the interaction Kˆ set to the appropriate form. Comparing the two results one finds that low-momentum modes are suppressed in the wave functions from the full calculation compared to the simple trinucleon result. This is in good agreement with what one na¨ıvely expects from the repulsive Coulomb force: it is particularly strong for small relative momenta of the charged subsystems, thus lowering the probability of the system to be in such a state. For the wave function component where the dibaryon is in a pure p–p state, this does not apply because in that case there is no Coulomb repulsion between the dibaryon and the third nucleon. Indeed, exactly this can also be seen in Fig. 6.16. The wave functions obtained in this manner can be used as nonperturbative input quan- tities for other calculations, e.g., a consistent determination of the Helium-3 photodisin- tegration in pionless EFT, which is currently work in progress. 6.6 The Coulomb problem at next-to-leading order From the promising results at leading order one might to expect that going to next- to-leading order gives both more precision and better agreement with the experimental Helium-3 energy. However, as shown in Fig. 6.17, exactly the opposite is the case. The results from both the perturbative and the nonperturbative calculation are now closer to—or even above—the triton binding energy, which certainly does not make sense phys- ically. Moreover, the results are strongly cutoff-dependent again and rise to even larger 10For the perturbative calculation this statement of course refers to the result where all O(α) diagrams are included. The error estimate should in this case be taken as 30% of the energy shift (since that is what is calculated), which then gives a marginal agreement with the experimental value. 11Ando and Birse use the single cutoff Λ = 380.689 MeV for their calculation because they find that the three-nucleon vanishes there. We find this zero of H(Λ) at Λ ≈ 377.69 MeV instead. This small discrepancy is most likely due to differences in the numerical implementation and negligible here. 132 Chapter 6. The proton–deuteron system revisited 0 100 200 300 400 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0 100 200 300 400 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 100 200 300 400 0 0.0005 0.001 q (MeV) [Dˆ B] (q ) (a .u .) · · simple trinucleon real He-3 deuteron leg n-p singlet leg p-p singlet leg Figure 6.16: Three-component Helium-3 wave functions calculated for a cutoff Λ = 400 MeV. Solid curves: result from full calculation involving all leading Coulomb dia- grams. Dashed curves: result from simple trinucleon calculation with the three-nucleon force fixed to give the experimental Helium-3 binding energy. binding energies as the cutoff is increased. From these findings we suspect that our next- to-leading order calculation is not renormalized properly and that a new counterterm might be needed to renormalize the system at this order when Coulomb effects are in- cluded. In the following we investigate the situation in more detail by analyzing the large- momentum (ultraviolet) behavior of the components that enter in the expression (6.59) for the perturbative energy shift. 6.6.1 Scaling of the dibaryon propagators The ultraviolet scaling of the dibaryon propagators can be directly inferred from looking at their expressions. We consider here the deuteron propagator Dd as a representative example. At leading order, its scaling is DLOd (E; q) ∝ 1 −κd + √ 3 4 q2 −MNE − iε ∼ 1 q as q →∞ . (6.65) Upon going to next-to-leading order, this is multiplied by a factor proportional to the effective range, DNLOd (E; q) = D LO d (E; q) 1 + ρd 2 · 3q 2/4−MNE − κ2d −κd + √ 3 4 q2 −MNE − iε  , (6.66) 6.6. The Coulomb problem at next-to-leading order 133 1000 10000 -5 0 5 5 6 7 8 9 10 11 12 Λ (MeV) E B (M eV ) H (Λ ) from WF (old) from WF (pure Coulomb) from WF (all contrib.) exp. EB( 3He) exp. EB( 3H) simple Figure 6.17: NLO results for the 3He binding energy as a function of the cutoff. The curves are as in Fig. 6.15 such that the asymptotic behavior is changed to DNLOd (E; q) ∼ const. as q →∞ . (6.67) Repeating this procedure in order to get the N2LO and higher-order propagators one even gets functions that are divergent (∼ q, ∼ q2, etc.) in the ultraviolet. 6.6.2 Ultraviolet behavior of the amplitude In order to find the behavior of the wave functions B(p) introduced in Section 6.4, we first go back to the corresponding T -matrix. As discussed, for example, in Ref. [187], for fixed E and k the n–d doublet channel amplitude T ds (E; k, p) has an asymptotic behavior determined by linear combinations of p±is0−1 in the limit p → ∞. It is the imaginary parts with s0 ≈ 1.0064 that give rise to the log-periodic behavior of the three-nucleon force H(Λ) necessary to renormalize the system. What is important for the discussion here is the modulus of the scaling: |T ds (E; k, p)| ∼ 1 p as p→∞ . (6.68) Since the derivation of this is independent of the energy E, the same scaling also applies in the bound-state regime. In particular, it is inherited by the wave functions, such that also |Bs(p)| ∼ 1 p as p→∞ . (6.69) 134 Chapter 6. The proton–deuteron system revisited 1 10 100 1000 10000 1e+05 1e+06 1e+07 1e+08 -6 -4 -2 0 2 4 6 p (MeV) p ·B (p ) (a .u .) deuteron leg singlet leg 1 singlet leg 2 Figure 6.18: Rescaled leading order trinucleon wave function p·Bs(p) (in arbitrary units) as a function of the momentum p. Indeed, as shown in Fig. 6.18, one also finds numerically that a plot of p · Bs(p) against p shows a log-periodic behavior for large p with constant amplitude. The analysis above is true at leading order. Numerically, we find the approximate scaling |Bs(p)| ∼ |T ds (E; k, p)| ∼ 1 p3/2 as p→∞ (NLO) (6.70) for the wave functions (and the corresponding T -matrix) at next-to-leading order. This behavior is illustrated in Fig. 6.19, where we show the NLO wave functions up to large p, rescaled with a factor p3/2. As in the leading-order case, this gives an oscillating function with approximately constant amplitude. Note that the oscillations are more rapid than at leading order and that the exact log-periodicity is broken at NLO. This is consistent with the behavior of the three-nucleon force H(Λ) at next-to-leading order (cf. Fig. 6.17). The p−3/2 fall-off can be seen more clearly in the inlay shown in Fig. 6.19, where the deuteron- leg component of the wave function (without rescaling) is plotted in a double-logarithmic scale. The same scaling behavior as reported here was found —and explained based on analytical considerations—for the scattering amplitude in Ref. [196]. To some extent this faster fall-off at next-to-leading order compensates the scaling of the dibaryon propagators, but, as we shall discuss below, not by enough to render the result convergent. 6.6. The Coulomb problem at next-to-leading order 135 1 10 100 1000 10000 1e+05 -100 0 100 200 0 5 10 -16 -12 -8 -4 0 p (MeV) p3 / 2 ·B (p ) (a .u .) deuteron leg singlet leg 1 singlet leg 2 log(p/MeV) lo g (B ) Figure 6.19: Rescaled NLO trinucleon wave function p3/2 · Bs(p) (in arbitrary units) as a function of the momentum p. The inlay shows the deuteron-leg component (without rescaling) in a double-logarithmic plot; the dashed line included there has a slope of exactly −3/2. For the other two components not included in the double-logarithmic plot, one finds exactly the same behavior. 136 Chapter 6. The proton–deuteron system revisited 6.6.3 Consequences We first consider the situation in the leading-order case. The scaling of the Coulomb kernel in Eq. (6.59) is determined there by the large-momentum behavior of the bubble diagram without the approximation (6.36). From Eqs. (6.39) and (6.34) we then have KLOc (E; k, q) ∼ 1 q3 as q →∞ . (6.71) Denoting in the following all loop momenta generically by q, we find ∼ ( 1 q )2 B × ( q3 )2 loops × ( 1 q )2 D × ( 1 q3 ) kernel ∼ 1 q (LO) (6.72) for the total scaling of the matrix element. Since this is the only contribution at leading order, the result for the energy shift converges as the cutoff is increased. From the same kind of analysis we can understand what led to the cutoff dependence in the result of Ref. [1]: the kernel VC as defined in Eq. (6.57) only falls off like 1/q 2 asymptotically such that the corresponding matrix element is logarithmically divergent.12 Considering finally the situation at next-to-leading order, we find that we can no longer avoid a divergent result. For the contribution of the bubble diagram we now obtain ∼ ( 1 q3/2 )2 B × ( q3 )2 loops × (q0)2 D × ( 1 q3 ) kernel ∼ q0 (NLO) , (6.73) which gives a logarithmic divergence. Even more problematic is the additional contri- bution entering at this order. For the kernel with the photon coupled directly to the dibaryon we have the scaling ∼ ( 1 q3/2 )2 B × ( q3 )2 loops × (q0)2 D × ( 1 q2 ) kernel ∼ q , (6.74) and thus a linear divergence. This again also applies to the old calculation involving VC , which has the same scaling and thus diverges linearly as well. Indeed, by comparing the dashed curves in Figs. 6.15 and 6.17 we see that the NLO result has a much stronger cutoff dependence than at leading order, and can understand this now from the analysis carried out above. Furthermore, we can explain as well why the new perturbative result at NLO looks almost like a mirror image of the old one. If we rewrite VC in Eq. (6.57) by factoring out a y 2 d,t, the result is just the next-to-leading order contribution to K (d,t) c , but with the sign reversed. Due to the faster divergence of this term we barely see the effect of the “leading” bubble diagram at all. 12The same kind of divergence occurs if the bubble diagram is approximated as in Eq. (6.36). However, as already mentioned repeatedly, the approximation is not valid in the bound-state regime, and the divergence that would appear if we were to use it there is just another manifestation of this fact. 6.6. The Coulomb problem at next-to-leading order 137 6.6.4 Nonperturbative calculation Also the result found from the nonperturbative calculation exhibits a strong cutoff de- pendence at next-to-leading order, with the corresponding curve in Fig. 6.17 even rising somewhat faster than the perturbative result. From the analysis above one is led to sus- pect that again the scaling of the involved diagrams is the origin of this effect. In order to show this, we look at what kind of diagrams are generated when the bound-state integral equation is iterated. (a) (b) Figure 6.20: Two diagrams generated by iterating the bound-state integral equation. Figure 6.20a shows a two-loop diagram with a Coulomb-photon exchange that is generated by iterating the equation once. Since the one-nucleon exchange also scales like 1/q2, we find a total scaling ∼ q1/2 for the diagram by applying the same kind of analysis as in the sections above. This indeed indicates a divergence. However, the answer cannot be quite so simple in this case because one directly sees that the diagram with two subsequent nucleon exchanges actually has the same kind of behavior. The important point here is that exactly the same situation also occurs in the system without Coulomb effects, but that every nucleon-exchange is always accompanied by a vertex from the three-nucleon force. Since according to the renormalization prescrip- tion this three-nucleon force is adjusted at each cutoff in order reproduce a three-body experimental input (the triton binding energy, in our case) one effectively also absorbs the divergence into H(Λ). By adding then afterwards the Coulomb contributions into the equation, the problem is reintroduced. At leading order, the situation is different because there the 1/q-scaling of the dibaryon propagators ensures that additional loops generated by iterating the integral equation do not create divergences, neither due to Coulomb photons nor due to the nucleon-exchange interaction. The three-nucleon force is needed in this case only to fix the oscillating behavior of the scattering amplitude [54]. 6.6.5 Back to the scattering regime The above findings also raise some questions concerning the NLO scattering calculation in the doublet channel. If the assertion that the system is not renormalized correctly is true, the effect should also show up in the cutoff-variation of the scattering phase shifts. Indeed, this is what we find. If we perform the phase-shift calculation with cutoffs of the same size as in Fig. 6.17 (up to Λ ≈ 10000 MeV), the result does not seem to converge. Instead, the curves shown for four different cutoffs in the upper panel of Fig. 6.21 move upwards with increasing Λ. 138 Chapter 6. The proton–deuteron system revisited One can of course take the stance, as we have done in Section 6.3 by varying the cutoff only between 200 and 600 MeV, that the cutoff in an EFT calculation should be taken of a natural order of magnitude (defined by the scale of physics left out from the theory). According to Fig. 6.15, the leading-order Helium-3 results are indeed converged at these cutoffs. At next-to-leading order, however, the results are problematic already in that regime. With this in mind, considering cutoffs far beyond the natural size here and in the preceding sections is primarily a tool to expose and analyze this behavior more clearly. Using the full equation structure (6.61) to fit the three-nucleon force such that it repro- duces the experimental 3He energy turns out to remove the strong cutoff dependence of the p–d results (without affecting the results for lower cutoffs very much), just as it does in the n–d system when H(Λ) is fixed to reproduce the triton binding energy. This is shown in the lower panel of Fig. 6.21, where we plot the curves analogous to those in the upper panel obtained by re-fitting the three-nucleon force. For consistency we have used the same equation structure as in the bound-state regime—including the box and triangle diagrams and without the approximation of the bubble diagram—in the calculations of all phase-shift results shown in Fig. 6.21. We interpret these findings as a further indication that in the p–d system there is a new three-body counterterm, which at least numerically can be absorbed by refitting H(Λ). Such a situation could be accounted for by adding to the three-body Lagrangian (6.6) a piece proportional to the charge operator (making it vanish in the triton system). Since it is not forbidden by symmetry, such a term should be there, and the only question is at which order in the EFT counting it enters. Our results strongly suggest that it is necessary for correct renormalization at NLO. This conclusion, however, comes with a caveat. We cannot rule out here that the whole problem might be due to using the partially resummed propagators in the integral equa- tions. If there truly is a new counterterm at NLO in the system with Coulomb effects, it should also show up in a fully perturbative calculation that only includes range cor- rections up to a fixed order instead of resumming a subset of higher-order contributions. For the related system of three uncharged bosons this procedure has been implemented in Ref. [188]. In the charged system, the situation becomes more involved because the photon coupling directly to the dibaryon introduces a new type of range correction that has to be treated on an equal footing with kinetic-energy insertions in the propagators. The new efficient implementation of perturbative range corrections recently presented in Ref. [197] might be a useful tool to do this. One contribution entering in such a calculation at next-to-leading order would be the diagram shown on the left-hand side of Eq. (6.74), but with all propagators and amplitudes given by their leading-order versions. Counting powers of loop momenta indicates a logarithmic divergence for this. However, since in principle there could be cancellations of divergences between different contributions, one cannot draw definite conclusions here without carrying out the calculation. It will certainly be interesting to investigate this issue further. 6.6. The Coulomb problem at next-to-leading order 139 -60 -50 -40 -30 -20 -10 0 10 0 20 40 60 80 100 -60 -50 -40 -30 -20 -10 0 k (MeV) δ (d eg ) δ (d eg ) Λ = 200 MeV Λ = 600 MeV Λ = 2400 MeV Λ = 9600 MeV H(Λ) fit to triton H(Λ) fit to Helium-3 Figure 6.21: NLO p–d scattering phase shifts for large cutoffs obtained using Eq. (6.61). Upper panel: H(Λ) fit to reproduce triton binding energy in the n–d system. Lower panel: H(Λ) fit to reproduce 3He binding energy in the p–d system. 140 Chapter 6. The proton–deuteron system revisited Neglected diagrams and the bubble approximation Although the main motivation for deriving the full equation structure (6.61) with the box and triangle diagrams included was the nonperturbative Helium-3 calculation, with the complete expression at hand we are now also in a position to directly check the approximations made in the scattering regime. Based on the power counting described in Section 6.2.3, we neglected there the contributions from the box and triangle diagrams, and furthermore used the approximation given in Eq. (6.36) for the bubble diagram. To avoid interference with the problematic situation at NLO discussed in the previous section, we go back to the leading-order calculation and show in Fig. 6.22 the old band (as given in Fig. 6.8) together with the result obtained without approximating the bubble diagram (dashed curve) and furthermore what we get from the full equation with also the box and triangle contributions included (solid curve). As before, all error bands were generated by varying the cutoff between 200 and 600 MeV, which is certainly sufficient at leading order. The band from the full calculation overlaps well with the old result. In fact, if one takes the real uncertainty to be the typical 30% of a leading-order pionless EFT calculation, the two bands are almost indistinguishable. It would be tempting to interpreted this as an a posteriori confirmation of both the Coulomb power counting for the scattering regime and the approximation used for the bubble diagram,for if it were not for the result with only the bubble approximation turned off. Since the corresponding band in Fig. 6.22 is broader and consistently shifted upwards (for momenta k below about 70 MeV), it turns out that really the combination of both things—approximating the bubble diagram and neglecting the additional contributions—is important to get the same result as from the full calculation. This indicates that in the latter case there are some substantial cancellations between the effects of the individual Coulomb diagrams. A similar effect occurs in the quartet channel. In this case we have additionally calculated the result with the bubble diagram still approximated, but the box diagram13 already included. Overall, the results from the individual calculations shown in Fig. 6.7 differ now not so much at low center-of-mass momenta, but quite substantially above the deuteron break-up threshold (k & 52 MeV). Since the error bands would be as narrow as in Fig. 6.7 and are not essential for what we want to show here, we have used a single cutoff Λ = 140 MeV to generate the curves. The cancellation between the two effects is particularly striking here: the result obtained with both the bubble-diagram approximation turned off and the box diagram included at the same time (solid curve) is almost identical to the old result (dotted curve). The same pattern is found also in the higher-order calculations. Altogether, these findings cast some doubt on the Coulomb power counting as discussed in Section 6.2.3. In particular, the good agreement of the N2LO quartet-channel phase shifts with experimental data that was found based on this counting scheme already in Ref. [178] appears now to be somewhat accidental, at least at higher energies. Based on the findings obtained here we propose that instead of using the old counting one should— 13Note that the triangle diagram does not appear in the quartet channel since there are never two protons in the intermediate dibaryon. 6.6. The Coulomb problem at next-to-leading order 141 0 20 40 60 80 100 -70 -60 -50 -40 -30 -20 -10 0 k (MeV) δ (d eg ) p–d LO (old) p–d LO (no approx.) p–d LO (no approx. + box and triangle) · Figure 6.22: Leading order p-d doublet channel S-wave scattering phase shifts as func- tions of the center-of-mass momentum k. Dotted curve: old result as shown in Fig. 6.8. Dashed curve: result without approximating the bubble diagram. Solid curve: same as dashed curve with additional Coulomb diagrams (box and triangle) included. Error bands generated by cutoff variation within 200–600 MeV. also in the scattering regime—simply include all O(α) Coulomb diagrams. At the same time, as already alluded to below Eq. (6.36), one should not use the approximation for the bubble diagram because it has no physical justification and turns out to have a more significant impact on the result than na¨ıvely expected. This new scheme14 has the advantage of being much more straightforward and at the same time completely consistent throughout both the scattering and the bound-state regime. The price to be paid for this is that the calculations to be carried out are quite a bit more involved. With suitable adaptive integration routines to carry out the S-wave projections of the Coulomb diagrams numerically, however, the integral equations can still be solved on ordinary desktop computers. Having concluded that the original Coulomb counting scheme should be modified raises the question how important higher-order contributions (in α) really are. Very close to the zero-momentum threshold they could in principle be quite important. As a first step to tackle this problem, we will in the next section discuss the calculation of (Coulomb- modified) proton–deuteron scattering lengths within the current framework. 14We would like to emphasize here that the new scheme does not imply to do strict perturbation theory in α. The included Coulomb diagrams are still iterated nonperturbatively to all orders, which means that the perturbation expansion up to O(α) is applied to the kernels used in the Lippmann– Schwinger equations. 142 Chapter 6. The proton–deuteron system revisited 0 20 40 60 80 100-80 -60 -40 -20 0 k (MeV) δ (d eg ) p–d LO (old) p–d LO (no approx.) p–d LO (no approx. + box) p–d LO (approx. + box) · Figure 6.23: Leading order p-d quartet channel S-wave scattering phase shifts as func- tions of the center-of-mass momentum k. Dotted curve: old result as shown in Fig. 6.7. Dashed curve: result without approximating the bubble diagram. Dash-dotted curve: result with the additional Coulomb box diagram box included (but the bubble diagram still approximated). Solid curve: same as dash-dotted curve, but without approximating the bubble diagram. All curves were calculated at a cutoff Λ = 140 MeV. 6.7. Proton–deuteron scattering lengths 143 6.7 Proton–deuteron scattering lengths The optimized numerical procedure outlined in Section 6.2.5 allows us to calculate the p–d scattering phase shifts down to very low center-of-mass momenta k and at the same time effectively remove the regulating photon mass λ from the calculation. Yet, due to the absence of experimental phase shift data below k . 20 MeV there is no real test for the phase-shift calculation in that regime. It it thus interesting to extend the calculation to the direct extraction of threshold param- eters, i.e., proton–deuteron scattering (S-wave) scattering lengths. They are defined by the Coulomb-modified effective range expansion (4.24), which we write here in the form C2η,0 k · cot δdiff(k) + γ h(η) = − 1 aC0 +O(k2) (6.75) with γ = γp–d = 4αMN/3 and η = γp–d 2k . (6.76) In the following, since we are only considering S-waves, we will omit all ` = 0 indices. For clarity, we furthermore denote the scattering length as ap–d and use superscripts “4” and “2” to indicate the quartet- and doublet channel results, respectively. The first approach that comes to mind for applying Eq. (6.75) is to simply insert Coulomb- subtracted phase shifts δdiff(k) in with the photon mass already extrapolated to zero. However, this is not a very good idea. To see this note that the use of the regulating photon mass is not the only approximation in the calculation. There is also the finite cutoff and, more importantly, the fact that following the EFT power counting we only include a certain set of Coulomb contributions in the calculation. Effectively this means that our calculation does not directly correspond to using the Coulomb potential α/r in configuration space. To solve this problem we propose that it is better to first calculate scattering lengths from Eq. (6.75) for several finite photon masses and then use those results to extrapolate ap–d to λ→ 0. 6.7.1 Numerical calculation of the Gamow factor The Gamow factor Cη,0 entering in Eq. (6.75) is quite strongly affected by the photon- mass screening. Using the analytic formula for the unscreened Coulomb potential given in Eq. (4.12) would result in a values that are far too small. Therefore, in order to carry out the procedure just described we need to extract the correct approximation for the Gamow factor from the numerical calculation in a consistent way. To this end it is useful to recall from Chapter 4 that, if ψ (+) k (r) denotes the exact three-dimensional scattering wave function for the pure Coulomb potential, then C2η,0 = C 2 η,0(k) = |ψ(+)k (r = 0)|2 . (6.77) This means that it should be possible to calculate the appropriate expression for the Gamow directly from the scattering wave function (for a given center-of-mass momentum k) corresponding to our approximate Coulomb potential. 144 Chapter 6. The proton–deuteron system revisited In fact, this is exactly what one should do. In Ref. [17], van Haeringen and Kok give a general version of the modified effective range expansion for systems where the interaction is given by the sum of a long-range potential VL and a short-range interaction VS. It can be written in the form |F`(k)|−2k2`+1 ( cot δM` (k)− i ) +M`(k) , (6.78) where δM` (k) is the subtracted phase shift δM` = δ S ` − δL` . (6.79) This is just what we obtain from the numerical calculation, where VL is given by the approximate Coulomb potential and VS is the short-range EFT interaction. Furthermore, F`(k) in Eq. (6.78) is the Jost function associated with VL, whereas M`(k) is a rather complicated expression that involves derivatives of the corresponding Jost solution.15 The important point is now that for ` = 0 we simply have [24] |F0(k)|−2 = limr→0 |ψ(+)k,0 (r)|2 , (6.80) where ψ (+) k,0 (r) is the S-wave component of the scattering wave function, cf. Eq. (2.37). From Chapter 2, Eq. (2.49), we recall that the momentum-space scattering wave function can be expressed in terms of the T-matrix. In our present conventions (see the remark at the beginning of Section 6.2.4), we have ψ (+) k,0 (p) = 2pi2 k2 δ(k − p)− 2µZ0T (E; p, k) k2 − p2 + iε , E = E(k) , (6.81) where T (E; p, k) is the S-wave projected T -matrix. Fortunately, this is exactly what we already calculated in order to obtain the Coulomb-subtracted phase shifts. From the expansion of the plane wave eikr in terms of Legendre polynomials, eik·r = eikr cos θ = ∞∑ `=0 (2`+ 1) i` j`(kr)P`(cos θ) , (6.82) where j`(z) is the spherical Bessel function of order `, one finds that the partial-wave projected wave function in configuration space can be obtained from the momentum- space version as16 ψ (+) k,` (r) = 1 2pi2 ∫ ∞ 0 dp p2 i` j`(pr)ψ (+) k,` (p) . (6.83) We only need this here for ` = 0, of course. Using limr→0 j0(pr) = 1 and cutting off the momentum-space integral at p = Λ (since we only know the T -matrix up to that value), we arrive at F0(k)−1 = ψ(+)k,0 (r = 0) ≈ 1 + µ pi2 ∫ Λ 0 dp p2 p2 − k2 − iεZ0T (E; p, k) , (6.84) 15The Jost solution f(k, r) is a solution of the radial Schro¨dinger equation involving VL that fulfills the boundary condition limr→∞ eikrf(k, r) = 1. The Jost function F`(k) is then defined as the Wronskian W (f, ϕ), where ϕ is a solution determined by ϕ(0) = 0 and ϕ′(0) = 1. For details see, for example, Ref. [24]. 16The inverse transformation is ψ (+) k,` (r) = 4pi ∫∞ 0 dr r2(−i)`j`(pr)ψ(+)k,` (r). 6.7. Proton–deuteron scattering lengths 145 which is straightforward to evaluate by numerical principal-value integration. Inserting for T the result of the calculation involving only the pure Coulomb diagrams and squaring the right-hand side of Eq. (6.84) gives the desired Gamow factor consistently extracted from the numerical EFT calculation. Summarizing the above results, we write it as C2η,λ = ∣∣∣∣1 + 2MN3pi2 ∫ Λ 0 dp p2 p2 − k2 − iεZ0Tc(E; p, k) ∣∣∣∣2 , (6.85) where we have added the additional subscript “λ” to distinguish it from the expression for the unscreened Coulomb potential. In writing Eq. (6.85) we have inserted the reduced mass µ = 2MN/3 for the p–d system and set T = Tc, where the latter is defined by Eq. (6.40). With this, the final form of the Coulomb-modified effective-range expansion that we use for the numerical calculation reads C2η,λ k · cot δdiff(k) + γ h(η) = − 1 ap–d +O(k2) . (6.86) Note that in principle also the term involving h(η) should be modified due to the screening. Unfortunately, this quantity cannot be extracted from the numerical calculation in the same simple way as the Gamow factor. Lacking a better expression, we will thus use h(η) as defined in Eq. (4.26). Since it vanishes at k = 0, this is not a problem for the extraction of the scattering length, but it means that we cannot consistently extract effective ranges at the same time. 6.7.2 Bubble diagram with full off-shell Coulomb T-matrix Since the Coulomb interaction is strongest directly at threshold, the scattering-length calculation should be a good testing ground for checking the influence of higher-order Coulomb diagrams that we have not taken into account so far. Rather than simply including O(α2) diagrams (with two Coulomb-photon exchanges) to estimate the effect of higher-order terms, we adopt the approach of Ando and Birse [179] and directly include the full off-shell Coulomb T-matrix in the diagrams. Effectively, this resums all subsequent Coulomb-photon exchanges for a given topology. The resulting diagrams are shown in Fig. 6.24. (a) (b) (c) Figure 6.24: Diagrams involving the full off-shell Coulomb T-matrix (indicated by the blob). In Ref. [179], Ando and Birse only consider the bound-state regime. Without going into further details, they argue that the finite extent of the wave functions helps to regularize 146 Chapter 6. The proton–deuteron system revisited the Coulomb singularity and thus use an unscreened interaction. Since we cannot ap- ply this approach in the scattering regime, we use here the partially screened Coulomb T-matrix TC,λ introduced in Section 4.4.1. We assume that this gives an appropriate description for small photon masses λ because in the limit λ → 0, TC,λ converges to the exact (unscreened) Coulomb T-matrix TC . The most important contribution is again given by the “full bubble diagram” shown in Fig. 6.24a. Its spin- and isospin structure is exactly the same as for the leading expression with just a single Coulomb-photon exchange. Only the loop integral, which before could be evaluated analytically, now has the more complicated form I fullbubble(E; k,p) = ∫ ∞ 0 dq q2 TC,λ ( i √ 3q2/4−MNE; k,p ) × ∫ 1 −1 dcos θ′ ∫ 2pi 0 dφ′ a(b+ c · cosφ′) (6.87) with a = k2 + kq cos θ′ + q2 −MNE , (6.88a) b = p2 + pq cos θ cos θ′ + q2 −MNE , (6.88b) c = pq sin θ sin θ′ . (6.88c) Here, as in Eqs. (6.62) and (6.63a), θ is the angle between the vectors k and p, whereas θ′ denotes the angle between k and the loop momentum q, and the azimuthal angle φ′ enters through rewriting the angle between p and q. It can be integrated over analytically with the result ∫ 2pi 0 dφ′ b+ c · cosφ′ = 2pi√ b2 − c2 (6.89) for b > c, which, according to Eqs. (6.88) is certainly fulfilled for the scattering-length cal- culation we are interested in. Only for calculations above the deuteron breakup threshold, where the energy E is positive, one would have to be more careful here. Setting cos θ′ ≡ x ⇒ sin θ′ = √ 1− x2 , (6.90) the remaining angular integral has the form∫ 1 −1 dx (A+B · x)√C +D · x+ E · x2 (6.91) with A = k2 + q2 −MNE , (6.92a) B = kq , (6.92b) C = (p2 + q2 −MNE)2 − p2q2(1− cos2 θ) (6.92c) D = (p2 + q2 −MNE)× 2pq cos θ (6.92d) E = p2q2 . (6.92e) 6.7. Proton–deuteron scattering lengths 147 Using integration by parts, this can be done analytically. The final result for the full- bubble kernel function is then17 K (full) bub (E; k, p) = − MN 2pi2 × 1 2 ∫ 1 −1 dcos θ I fullbubble(E; k,p) (6.93) with I fullbubble(E; k,p) = ∫ ∞ 0 dq q2 TC,λ ( i √ 3q2/4−MNE; k,p ) × 1√ F [ log ( A+B A−B ) − log ( B(2C +D)− A(D + 2E) + 2√F√C + E +D B(2C −D)− A(D − 2E) + 2√F√C + E −D )] , (6.94) where in addition to Eqs. (6.92) we have defined F = B2C + A2E − ABD . (6.95) The remaining momentum integral and the S-wave projection in Eq. (6.93) have to be carried out numerically. To evaluate the Coulomb T-matrix we use the expression in terms of hypergeometric functions derived in Section 4.4.2, which can be implemented efficiently with the fast routines for 2F1 from Ref. [198]. Unfortunately, the above integrals cannot be used to simplify also the expression for the full box and triangle diagrams shown in Figs. 6.24b and c. In those cases, the Coulomb T-matrix also depends on the loop momentum, which means that the angular integrations can no longer be carried out analytically. Finally, as shown in Fig. 6.25, the full Coulomb T-matrix directly between the dibaryons and the proton corresponds to an infinite series of subsequent Coulomb-photon exchanges. By including (at NLO and higher orders) the tree-level diagram in our integration kernel we already generate this term automatically. It is thus important not to include the diagram on the left-hand side in Fig. 6.25 explicitly, as otherwise we would be double- counting some contributions. = + + + · · · Figure 6.25: Full Coulomb T-matrix between deuteron and proton generated by subse- quent Coulomb-photon exchanges. 6.7.3 Results In the following, we give results for proton–deuteron scattering-lengths and study their dependence on the different variants in which the calculation can be carried out (expression for the bubble diagram and inclusion of other contributions). 17Note that the expression in Eq. (6.94) does not directly reduce to Ibubble as defined in Eq. (6.34) when the Coulomb T-matrix is replaced with a single photon exchange. Rather, in that limit, the kernel function K (full) bub (E; k, p) goes over into the LO-part of Eq. (6.39), including the overall prefactor −αMN . 148 Chapter 6. The proton–deuteron system revisited With the expression for the full bubble diagram discussed in the previous section, the calculation already takes almost a day per data point,18 which is just about feasible when run in parallel on a couple of machines. If we also wanted to include the diagrams shown in Figs. 6.24b and c, which require at least one more numerical integration, the current approach would very inefficient. We will thus only include their one-photon exchange equivalents as shown Fig. 6.3 along with the full bubble diagram. Albeit somewhat inconsistent, this should still be a good approximation, the impact of which we will discuss further after showing the results. Due to the problematic situation with the doublet-channel calculation beyond leading order (see Section 6.6.5), we focus here on the quartet channel. Quartet channel We perform the numerical calculations for three different cutoffs Λ = 120, 140, 160 MeV and for photon masses λ between 0.2 MeV and 1.2 MeV. For each extraction of the scat- tering length we calculate the left-hand side of Eq. (6.86) for a center-of-mass momentum range k ∈ [2, 4] MeV. Fig. 6.26 shows a representative plot. To the data points we fit a polynomial of the form f(k) = x0 + x1k 2 + x2k 4 (6.96) in order to extract the scattering length as ap–d = −1/x0. As noted in Section 6.7.1, we cannot also extract the effective range from x1 because the modification of h(η) due to the finite photons mass is not taken into account in our calculation. The k4 term has been included to allow for some curvature. In Fig. 6.27, we show the the photon-mass and cutoff dependence of the results at N2LO (using, as in the phase-shift calculation presented in Section 6.3.1, the fully resummed deuteron propagator since we keep the cutoff below its unphysical second pole). Since the λ-dependence is clearly linear, it is no problem to extrapolate the scattering lengths to the zero-screening limit (λ→ 0). The results, also for the LO- and NLO-calculations, are shown in Table 6.2. Calculation a4p–d / fm (LO) a 4 p–d / fm (NLO) a 4 p–d / fm (N 2LO) bubble approx. 9.31–9.48 10.51–10.67 12.39–12.58 no bubble approx. 8.93–9.01 9.82–9.85 11.52–11.56 bubble + box 8.40–8.48 9.29–9.31 10.89–10.92 full bubble + box 8.32–8.39 9.16–9.17 10.69–10.72 Table 6.2: Quartet-channel results for the p–d scattering length. One can see that the bubble-diagram approximation has quite an impact on the result. Without the approximation, the scattering length is significantly shifted down towards smaller values and that its cutoff dependence becomes much weaker. Unlike the situation encountered for the scattering phase shifts above the deuteron breakup (see Section 6.6.5), 18This statement refers to performing the calculation for a given cutoff and photon mass on a reason- ably modern (as of the writing of this thesis) desktop computer. 6.7. Proton–deuteron scattering lengths 149 0 0.0001 0.0002 0.0003 0.0004 -0.1198 -0.1196 -0.1194 -0.1192 -0.119 -0.1188 -0.1186 simple k2 (fm−2) m o d . E R E fu n ct io n LO, Λ = 140 MeV, λ = 0.3 MeV Figure 6.26: Coulomb-modified effective range plot for quartet-channel p–d scattering. The y-axis shows the left-hand side of Eq. (6.86); the line is the result of a fit to the formula given in Eq. (6.96). 0.2 0.4 0.6 0.8 1 1.2 10.5 11 11.5 12 12.5 13 13.5 14 λ (MeV) p–d N2LO (old) p–d N2LO (no approx.) p–d N2LO (no approx. + box) p–d N2LO (full bubble + box) a 4 p – d (f m ) · Figure 6.27: Photon-mass dependence of the quartet-channel p–d scattering length at N2LO. The bands were generated by varying the cutoff between 120 and 160 MeV. Dotted curve: result with the bubble diagram approximated. Dashed curve: result without approximating the bubble diagram. Solid curve: result with the additional Coulomb box diagram box included (and without approximating the bubble diagram). Long-dashed curve: same as solid curve, but using the full bubble diagram involving the off-shell Coulomb T-matrix. 150 Chapter 6. The proton–deuteron system revisited the additional inclusion of the box diagram here does not compensate the unapproximated bubble diagram to give back the old result. Rather, it moves the scattering length further towards smaller values. Finally, the effect of using the full off-shell Coulomb T-matrix in the bubble diagram is surprisingly small and gives only a 2% change (at N2LO) in the result. The reason for the latter observation might be that for consistency the full bubble diagram is also included in the pure Coulomb equation used to calculate the Coulomb-subtracted phase shifts and the Gamow factor. Since in principle this might give some cancellations, we cannot unambiguously conclude that using the full off-shell Coulomb T-matrix only has a small effect on the scattering length. To investigate this issue further, it might be interesting to also use the full T-matrix in box diagram. Although from the fact that this diagram is not enhanced by the Coulomb pole one should expect it to be only a small correction, it might still be interesting to include it because it only enters in the full (strong + Coulomb) calculation and not in the pure Coulomb equation. To proceed in this direction however, requires more sophisticated numerical methods to deal with the multi-dimensional integrals. Monte Carlo methods might be a useful tool to deal with the problem. Another issue that has to be pointed out is that the changes from NLO to N2LO in Ta- ble 6.2 are somewhat larger than those from LO to NLO. This is not what one would expect from the EFT expansion parameter ∼ 0.3 and certainly has to be investigated fur- ther. With that and the above discussion in mind, the results for the scattering lengths here should clearly be regarded as preliminary. For this reason, we have given the un- certainties only as far as they are created by the cutoff variation and not attempted to also propagate the errors from the fits used in the extraction. At least for a precision calculation at N2LO, where from the EFT expansion we would expect an accuracy of about 3%, a more sophisticated procedure might be advisable. Until the role of the full Coulomb T-matrix in the box diagram is settled, however, we make a conservative error estimate here and use the combination of the two dashed bands in Fig. 6.27 (extrapolated to λ = 0) to summarize our quartet-channel result as a4p–d = 11.1± 0.4 fm . (6.97) Incidentally, this is nicely compatible with the experimental determinations quoted in Table 5.2 in Chapter 5 that give a4p–d between 11 and 12 fm. A more recent analysis and other theoretical calculations based on realistic nuclear potential models, however, have found somewhat larger values around 14 fm [199]. Furthermore, Fig. 6.27 suggests that with the inclusion of the box diagram, our results come out on the smaller side of the range given in Eq. (6.97). It will certainly be interesting to study this issue further. Doublet channel With the currently unclear situation in the doublet channel according to Sections 6.6 and 6.6.5, we only briefly discuss the extraction of scattering lengths for that case here. Since based on the phase-shift calculations we expect a rather strong cutoff dependence in this channel, and because to be consistent we can at best go up to NLO (due to second 6.7. Proton–deuteron scattering lengths 151 0.2 0.4 0.6 0.8 1 1.2 0.3 0.4 0.5 0.6 0.7 λ (MeV) a 2 p – d (f m ) · Λ = 200 MeV Λ = 300 MeV Λ = 400 MeV Λ = 500 MeV Λ = 600 MeV Figure 6.28: Photon-mass dependence of doublet-channel p–d scattering length at NLO. three-nucleon force at N2LO that is not included in our calculation ), it does not make sense at present to investigate here the effect of the full Coulomb T-matrix. Instead, in order to perform a calculation that is at least internally consistent, we follow the approach that gave the lower panel of Fig. 6.21, i.e., we use the same equation structure as for the bound-state calculation and fit the three-nucleon force to reproduce the experimental He-3 binding energy (instead of fitting the equation without Coulomb contributions to the triton system). The NLO-result is shown in Fig. 6.28 for cutoffs between 200 and 600 MeV. Compared to the quartet-channel case, there is now some non-linearity in the photon-mass dependence, but still the curves are essentially flat. At the same time, the cutoff dependence is much stronger, but beyond 300 MeV the result is quite stable around a2p–d ≈ 0.3 fm. At leading order, the situation is almost the same, but the cutoff dependence is somewhat larger (as expected from the EFT expansion). In that case, for larger cutoffs we find a scattering length that is approximately zero or slightly negative, but since the fit formula as in Eq. (6.96) becomes unstable in that regime, it is difficult to make a more quantitative statement here. What we report here, of course, reflects the difficulty in determining the p–d doublet channel scattering lengths that we already mentioned in Section 5.5.2. Our results are at least qualitatively compatible with other determinations in finding mostly very small values for a2p–d. Improving upon this would first require solving the question of range corrections and correct renormalization beyond leading order in pionless EFT calculations with Coulomb effects included. 152 Chapter 6. The proton–deuteron system revisited 6.8 Summary and outlook In this chapter, we have studied several aspects of the proton–deuteron system in pion- less effective field theory. Using an optimized integration mesh we are able to calculate Coulomb-subtracted scattering phase shifts even at very low center-of-mass momenta, were the Coulomb interaction becomes highly non-perturbative. In the bound-state regime, we have corrected our perturbative calculation of the 3He– 3H binding energy difference from Ref. [1] and additionally performed a nonperturbative calculation of the 3He binding energy that agrees very well with the result with the result obtained by Ando and Birse [179] although we have only used Coulomb photons to include electromagnetic effects. Furthermore, at leading order both the perturbative and the nonperturbative calculation give results that are, within their respective uncertainties, in good agreement with the experimental value for the 3He binding energy. At next-to-leading order, the results of both calculations exhibit a strong cutoff depen- dence, indicating that the calculation is not renormalized properly. We have analyzed this situation by studying the ultraviolet behavior of the diagrams entering in the calculation and argued that a new three-body counterterm might be necessary to renormalize the charged doublet-channel system at NLO. To come to a more definite conclusion about this, one should carry out a fully perturbative calculation that does not resum any range corrections. Our findings are supported by a closer examination of the doublet-channel scattering phase shifts, which also turn out not to converge when the cutoff in the calculation is chosen very large. This cutoff dependence however goes away when the three-nucleon force is fitted to reproduce the experimental 3He binding energy, which we interpret as further evidence for a new charge-dependent counterterm. Since the modified fitting of the three-nucleon force requires using the same equation structure in the bound-state and scattering regime, we have critically reviewed the Coulomb power counting of Rupak and Kong [178] and come to the conclusion that it might be better to simply include all Coulomb diagrams of a given order in α. This approach is simpler and more consistent because it treats the scattering and the bound-state regime on an equal footing. First results for proton–deuteron scattering lengths—obtained using a procedure that applies the new counting scheme and consistently extracts the Gamow factor for the Yukawa-screened Coulomb potential from the same calculation that gives the Coulomb- subtracted scattering phase shifts—look quite promising. It will certainly be interesting to follow this approach further and more closely investigate the role of nonperturbative Coulomb effects. To include the full off-shell Coulomb T-matrix in all relevant diagrams, however, requires more sophisticated numerical methods to deal with multi-dimensional integrals. Chapter 7 Concluding remarks In this thesis, we have covered a number of different topics, ranging from the study of finite-volume effects on bound states with angular momentum to formal investigations of causality and unitarity constraints on quantum systems with finite-range interactions and additional long-range tails, and finally to practical calculations of proton–deuteron scattering. All these loosely-connected things are tied together by the concepts of low- energy universality, effective field theory, and/or modifications thereof due to long-range forces. In many parts of this work, the radial Schro¨dinger equation plays a prominent and essential role. Since physical processes at low energies do not probe the short-distance physics of the underlying interaction, it is often possible to reduce their theoretical description to an effective picture with finite-range interactions. This makes it possible to draw conclusions based only on the analytically-known tails of the wave functions and to encode the short- range details in a limited set of parameters given, for example, by the effective range expansion. In Chapter 3 we have shown how this approach directly leads to a universal relation for the volume dependence of two-body binding energies. In this case, the details of the underlying short-range interaction only enter via the physical bound-state properties (the binding energy and the asymptotic normalization constant), allowing their extraction from numerical calculations in periodic boxes of different sizes. We have generalized this result to states with arbitrary orbital angular momentum, to two-dimensional systems, and also to the case of twisted boundary conditions that appear when one considers the scattering of composite particles in finite periodic boxes and then lead to correction factors that have to be taken into account in such calculations. For the remaining part of this thesis the focus has been on systems of charged particles, where the Coulomb force plays a very important role at low energies. A vast number of analytic results have been obtained over the last century, a selection of which we have reviewed in Chapter 4. Despite this broad understanding, the Coulomb potential is notoriously difficult to handle due to its long-range nature. It would certainly be interesting—in particular for applications in nuclear physics—to extend the finite-volume study of Chapter 3 to bound states of charged particles. Having 153 154 Chapter 7. Concluding remarks read Chapter 4 of this thesis one might think that this is just a matter of expressing the asymptotic wave functions in terms of appropriate Whittaker functions. Unfortunately, one runs into problems at a much earlier point of derivation because the very slow large- distance fall-off of the Coulomb potential forbids a straightforward implementation of periodic boundary conditions. To deal with this problem it might be interesting in the future to either adopt techniques from solid-state physics (e.g., Ewald summation or a similar approach) or alternatively try to study screened Coulomb interactions in periodic boxes. The derivation of causality bounds for charged particles given in Chapter 5 is conceptually straightforward and not fundamentally different from the pure finite-range case, but turns out to be quite involved when going into detail due to the complex analytic structure of the Coulomb wave functions with their “entanglement” of radial position and momentum variables. The final result for the case with Coulomb tails is ultimately very similar to what one finds for pure finite-range interactions. Interestingly, the situation is different for systems with van der Waals tails. Although these are still long-ranged, they fall off very fast and should thus be expected to give a result even closer to that for the system without long-range interactions. Yet, the result in this case is fundamentally different, an effect that can be traced back to the highly singular behavior of the van der Waals potential at short distances. It will certainly be interesting to study this behavior further also for other singular power-law potentials or perhaps directly from a more general point of view. While causality bounds—with or without long-range forces—are primarily interesting from a formal point of view, they may also serve as a guide to improve the convergence of calculations in halo effective field theory or similar approaches. It has to be stressed here again, however, that the derivation of the causality bound is based on the assumption of an effective two-body system with energy-independent potentials. As such, its direct applicability is somewhat limited. In Chapter 6 we have critically revised and extended earlier calculations of proton– deuteron and 3He observables in pionless effective field theory and found that the question of how to include the effects of the Coulomb interaction does not seem to be as settled as previously believed. With our numerical methods we are able to study both p–d scat- tering at very low energies as well as the bound-state regime. In the latter case we find good agreement of our leading-order result for the 3He binding energy from both per- turbative and nonperturbative calculations, but encounter problems in calculations at next-to-leading order in the EFT power counting. Upon closer inspection the problem also shows up in the scattering regime and can be avoided by fitting the existing three- nucleon force to the 3He system. We interpret these findings as evidence for a missing charge-dependent counterterm that is needed to correctly renormalize the p–d doublet- channel system at NLO. To come to a definite conclusion about this, however, it will be necessary to carry out a calculation that incorporates range corrections in a strictly perturbative manner. Independent of that question we propose based on the new results that previously applied Coulomb power counting in pionless EFT should be given up in favor of a simpler scheme that includes all Coulomb contributions up to a given order at the same time and thus 155 treats the scattering and bound-state regimes on an equal footing. Finally, we present a calculation of proton–deuteron scattering lengths in pionless EFT. These observables are interesting because the Coulomb interaction is particularly strong directly at the zero-momentum threshold. As a first step to study this, we have per- formed a calculation that involves—at least in the supposedly most important Coulomb diagram—a “partially screened” expression for the full off-shell T-matrix that we dis- cussed previously in Chapter 4. While the results obtained so far look quite promising when compared with older experimental determinations of the scattering lengths, a more recent analysis and other theoretical calculations have found somewhat larger values. It will certainly be interesting to pursue this issue further. 156 Chapter 7. Concluding remarks Appendix A The Coulomb wave functions of Bolle´ and Gesztesy Generalizing results first obtained by Lambert [121], Bolle´ and Gesztesy [122] define the Coulomb wave functions F (0)n (p, r) = r 1 2 +me−ipr1F1 ( 1 2 +m− k, 1 + 2m; z) (A.1a) and G(0)n (p, r) = Γ ( 1 2 +m− k) Γ(2m+ 1) (2ip)2m r 1 2 +me−iprU ( 1 2 +m− k, 1 + 2m; z) , (A.1b) where, in our case, n = 2` + 3 and m, k, z are as defined in Eqs. (4.5) and (4.6). Using Eq. (4.14) in the form1 Cη,` = 2`e−piη/2 |Γ(`+ 1 + iη)| (2`+ 1)! , (A.2) one finds that F (0)n (p, r) = 1 p`+1Cη,` F (p) ` (r) (A.3a) and G(0)n (p, r) = p `Cη,` [ G (p) ` (r)− iF (p)` (r) ] . (A.3b) It is shown in Ref. [122] that F (0) n (p, r) is analytic in p2. Furthermore, from Eqs. (3.16), (3.17) and (4.1) in that paper it follows that G(0)n (p, r) = G˜ (0) n (p, r) + ( γ h˜`(p)− 2ip2`+1C2η,` ) · F (0)n (p, r) , (A.4) where h˜`(p) is the function defined in Eq. (4.29), and where G˜ (0) n (p, r) is analytic in p2. 1This is Eq. (5.1) in Ref. [115]. From Eq. (3.1) in the same reference one directly sees that this is consistent with Eq. (4.30) in Section 4.2. 157 158 Appendix A. The Coulomb wave functions of Bolle´ and Gesztesy In Eq. (4.3) of Ref. [122], Bolle´ and Gesztesy give an explicit expression for G˜ (0) n (p, r). Since there are two typos in their original equation,2 we quote the whole expression for completeness. Slightly altering the notation to match our conventions, we have G˜(0)n (p, r) = [ Γ(n− 1)−2] (2p)n−3 ∣∣∣∣Γ(n− 12 + iγ2p )∣∣∣∣2 ∣∣∣∣Γ(1 + iγ2p )∣∣∣∣−2 γ log (|γ|r)·F (0)n (p, r) − γ Re {[ Γ(n− 1)−2] (2p)n−3 ∣∣∣∣Γ(n− 12 + iγ2p )∣∣∣∣2 ∣∣∣∣Γ(1 + iγ2p )∣∣∣∣−2 r(n−1)/2 e−ipr × ∞∑ k=0 Γ ( n−1 2 + k − iγ 2p ) Γ(n− 1)(2ipr)k Γ ( n−1 2 − iγ 2p ) Γ(n− 1 + k)Γ(k + 1) [ψ(k + 1) + ψ(k + n+ 1)] } + γ Re {[ Γ(n− 1)−2] (2p)n−3 ∣∣∣∣Γ(n− 12 + iγ2p )∣∣∣∣2 ∣∣∣∣Γ(1 + iγ2p )∣∣∣∣−2 r(n−1)/2 e−ipr × ∞∑ k=0 Γ ( n−1 2 + k − iγ 2p ) Γ(n− 1)(2ipr)k Γ ( n−1 2 − iγ 2p ) Γ(n− 1 + k)Γ(k + 1) n−1 2 +k−1∑ s=1 ( s− iγ 2p )−1} + Re { (n− 2)−1(2ip)n−2r(n−1)/2 e−ipr n−3∑ q=0 Γ ( 3−n 2 + q − iγ 2p ) Γ(3− n)(2ipr)q+2−n Γ ( 3−n 2 − iγ 2p ) Γ(q + 3− n)Γ(q + 1) } , (A.5) valid for any odd n ≥ 3 and where the sum over s in the fourth line is defined to give zero for n = 3 and k = 0. 2The iγ/k in the first line should be iγ/(2k) and the (q + 1) in the last line should be Γ(q + 1). Appendix B Explicit expressions for the causality bound function In this appendix, we give explicit expressions for the Coulomb causality bound functions bC1 (r) and b C 2 (r). They were derived with the procedure outlined in Sections 5.3.4 and 5.3.5. B.1 Repulsive case, γ > 0 The ` = 0 result for a repulsive Coulomb potential is given by Eq. (5.45). For ` = 1, we get bC1 (r) = 6r2 5γ3 ( aC1 )−2 [ 3γ2r2 1F2 ( 5 2 ; 3, 6; 4γr ) − 20γr 1F2 ( 3 2 ; 2, 5; 4γr ) +120 1F2 ( 1 2 ; 1, 4; 4γr ) − 120 2F3 ( 1 2 , 2; 1, 3, 4; 4γr )] − 4r 2 √ pi ( aC1 )−1 G2,22,4 ( 4γr ∣∣∣∣ −1, 120, 3,−3,−2 ) − √ piγ3r2 9 G4,02,4 ( 4γr ∣∣∣∣ −1, 12−3,−2, 0, 3 ) + γ 54 , (B.1) and for ` = 2 the result is bC2 (r) = 50r2 63γ5 ( aC2 )−2 [ 7γ4r4 1F2 ( 9 2 ; 5, 10; 4γr ) − 108γ3r3 1F2 ( 7 2 ; 4, 9; 4γr ) +1296γ2r2 1F2 ( 5 2 ; 3, 8; 4γr ) − 12096γr 1F2 ( 3 2 ; 2, 7; 4γr ) +108864 1F2 ( 1 2 ; 1, 6; 4γr ) − 108864 2F3 ( 1 2 , 2; 1, 3, 6; 4γr )] − 4r 2 √ pi ( aC2 )−1 G2,22,4 ( 4γr ∣∣∣∣ −1, 120, 5,−5,−2 ) − √ piγ5r2 3600 G4,02,4 ( 4γr ∣∣∣∣ −1, 12−5,−2, 0, 5 ) + γ3 21600 . (B.2) 159 160 Appendix B. Explicit expressions for the causality bound function B.2 Attractive case, γ < 0 For an attractive Coulomb potential, the ` = 0 result is given by Eq. (5.46). The ` = 1 result reads bC1 (r) = 6r2 5γ3 ( aC1 )−2 [ 3γ2r2 1F2 ( 5 2 ; 3, 6; 4γr ) − 20γr 1F2 ( 3 2 ; 2, 5; 4γr ) +120 1F2 ( 1 2 ; 1, 4; 4γr ) − 120 2F3 ( 1 2 , 2; 1, 3, 4; 4γr )] + 4 √ pir2 ( aC1 )−1 G2,23,5 ( −4γr ∣∣∣∣ −1, 12 ,−120, 3,−3,−2,−1 2 ) + pi3/2γ3r2 18 [ 2G4,03,5 ( −4γr ∣∣∣∣ −52 ,−1, 12−3,−2, 0, 3,−5 2 ) −G1,22,4 ( −4γr ∣∣∣∣ −1, 123,−3,−2, 0 )] + γ 54 , (B.3) and for ` = 2 one finds bC2 (r) = 50r2 63γ5 ( aC2 )−2 [ 7γ4r4 1F2 ( 9 2 ; 5, 10; 4γr ) − 108γ3r3 1F2 ( 7 2 ; 4, 9; 4γr ) +1296γ2r2 1F2 ( 5 2 ; 3, 8; 4γr ) − 12096γr 1F2 ( 3 2 ; 2, 7; 4γr ) +108864 1F2 ( 1 2 ; 1, 6; 4γr ) − 108864 2F3 ( 1 2 , 2; 1, 3, 6; 4γr )] + 4 √ pir2 ( aC2 )−1 G2,23,5 ( −4γr ∣∣∣∣ −1, 12 ,−120, 5,−5,−2,−1 2 ) + pi3/2γ5r2 7200 [ 2G4,03,5 ( −4γr ∣∣∣∣ −92 ,−1, 12−5,−2, 0, 5,−9 2 ) −G1,22,4 ( −4γr ∣∣∣∣ −1, 125,−5,−2, 0 )] + γ3 21600 . (B.4) Appendix C Equivalence of ANC relations In this appendix, we demonstrate the equivalence of our ANC relation derived in Chapter 5 to the one reported by Sparenberg et al. in Ref. [143]. Their ANC relation, translated to our notational convention, reads |A| `! Γ(`+ 1 + ηˆ) ≈ κ`+1 √ a˜C` . (C.1) The tilde on the a˜C` is there to indicate that Sparenberg et al. use the convention for the Coulomb-modified effective range expansion of Refs. [124, 127, 128], which, as noted at the end of Section 4.2, differs from the one used in this work by an overall factor [2``!/(2` + 1)!]2. Combining equations in Ref. [143] and again matching to our notation, one finds that 1 `!2 ∏` s=1 (s2 + η2) [ C2η,0 p 2`+1 cot δ˜`(p) + γ p 2` h(η) ] = − 1 a˜C` + 1 2 r˜C` p 2 + · · · . (C.2) Note that this is just Eq. (4.31) with the prefactors combined and the imaginary parts canceled. More explicitly, we have a˜C` = ( `! 2` (2`+ 1)! )2 aC` , r˜ C ` = ( (2`+ 1)! `! 2` )2 rC` , etc. . (C.3) At the end of their derivation, Sparenberg et al. eliminate the effective range in favor of the scattering length. Without invoking this final step their relation reads |A| 2 `(2`+ 1)! Γ(`+ 1 + ηˆ) ≈ κ` ( −r C ` 2 + γ2`−1 3(2`+ 1)!2 )−1/2 , (C.4) where we are now using our convention for the effective range expansion. With the definition of C˜η,` from Eq. (5.79) and the values for ∆Z` from Table 5.3 one sees that at least for ` = 0, 1, 2 this is exactly equivalent to our Eq. (5.87) with the O(κ2) set to zero. In order to prove the equivalence for arbitrary ` one would need a general expression for ∆Z`. This, in turn, requires knowledge of the constant terms in W [g2, g0](r) for arbitrary 161 162 Appendix C. Equivalence of ANC relations `. It would thus probably be more interesting to turn the argument around, i.e., take the equivalence of the relations for granted and derive from it a general expression for the constant terms in the Wronskians. The only additional ingredient one would need for this procedure is a general series expansion for the Meijer G-functions that arise from the integral of g0(r) 2. Appendix D Bound states in nonrelativistic effective field theory In this appendix appendix we discuss the trinucleon wave functions used in Chapter 6 from a very general point of view on bound states in nonrelativistic field theory. In particular, we give a detailed derivation of the correct normalization condition for the effectively energy-dependent one-nucleon-exchange interaction. The material presented here extends and—in some places—corrects that given in an analogous appendix in the author’s diploma thesis [169]. A considerable overlap with the previous work is accepted in order to make the discussion here self-contained. D.1 Simplified nucleon–deuteron system Although the results presented in the following are of a general nature and can easily also be applied to other nonrelativistic field theories (e.g., the EFT for cold atomic systems with large scattering length [28]), we work here, for the sake of an explicit illustration, with a model set up to resemble the doublet-channel nucleon–deuteron system in pionless EFT. Neglecting the isospin degree of freedom and thus also the virtual spin-singlet state, we write down a model Lagrangian of the form L = N † ( i∂0 + ∇2 2MN ) N − σd di†di + yd [ di† ( NTP idN ) + h.c. ] (D.1) with P id = 1√ 8 σ2σ iτ2 . (D.2) The notation here is the same as in Chapter 6. From there we also quote the projection onto the doublet channel. If (Oi)α is a generic object with spin-1 index i and spin-1/2 index α, then (Od)α = (σi)α′α (Oi)α′ (D.3) 163 164 Appendix D. Bound states in nonrelativistic effective field theory gives the projection onto the spin-doublet channel. The (renormalized) nucleon and deuteron propagators are given by ∆αβN (p) = p α β = iδαβ p0 − p22MN + iε (D.4) and ∆ijd (p) = p i j = − 4pii MNy2d · δ ij −κd + √ p2 4 −MNp0 − iε , (D.5) and the only interaction is given by the doublet-projected one-nucleon-exchange diagram K(k0,k, p0,p;P ) = → E = Ed + EN → k p = −iy2d/2 Ed − EN + k0 + p0 − (k+p)22MN + iε (D.6) because for simplicity we also neglect the tree-nucleon force. Note that the expression above corresponds to kinematics where the center of mass of the two particles is at rest. In the following discussion we will work in a general frame, where K = K(k, p;P ) with k = (k0,k), p = (p0,p), and P = (P0,P) = (E,P), but whenever we refer to the explicit form of the interaction we only need the kinematics as in Eq. (D.6). D.2 Bethe–Salpeter equation To start the discussion, we derive the Bethe–Salpeter equation for our model nucleon– deuteron system, being a little more careful than in [169] and, as already there, closely following the derivation in Lurie’s textbook [200]. The central object of interest is the full two-body nucleon–deuteron propagator (Green’s function) (Gij)βα(x1, x2;x3, x4) = 〈Ω|T ( dj(x1)N β(x2)d i†(x3)N †α(x4) ) |Ω〉 , (D.7) where |Ω〉 and T (· · · ) denote the interacting vacuum of our effective field theory and the time-ordering operator, respectively. Diagrammatically, we have (Gij)βα(x1, x2;x3, x4) = x4 x3 x2 x1 . (D.8) We furthermore define the Bethe–Salpeter wave functions (ψjEPa) β(x1, x2) = 〈Ω|T ( dj(x1)N β(x2) )|E,P, a〉 , (D.9a) (ψ†iEPa)α(x3, x4) = 〈E,P, a|T ( di†(x1)N †α(x2) )|Ω〉 , (D.9b) where |E,P, a〉 denotes a two-particle eigenstate with energy E and momentum P of the Hamiltonian corresponding to our model Lagrangian. The index a collectively denotes D.2. Bethe–Salpeter equation 165 the eigenvalues of any other operators (possibly) needed in order to form a complete set of states. For notational, we always work with the doublet-projected quantities like G(x1, x2;x3, x4) = (σ i)α ′ α (G ij)β ′ α′(x1, x2;x3, x4)(σ j)ββ′ ∣∣∣ α=β=1 (D.10) in the following, i.e., all equations are to be understood with the doublet projection already applied. By using Wick’s theorem and the general perturbation expansion we find the (inhomogeneous) Bethe–Salpeter equation G(x1, x2;x3, x4) = G0(x1, x2;x3, x4) + ∫ d4z1 d 4z2 d 4z3 d 4z4 G0(x1, x2; z1, z2) ×K(z1, z2; z3, z4)G(z3, z4;x3, x4) , (D.11) with G0(x1, x2;x3, x4) = ∆d(x1, x3)∆N(x2, x4) . (D.12) Diagrammatically, Eq. (D.11) can be represented as x4 x3 x2 x1 = x3 x1 x4 x2 + x3 z3 z2 x2 x4 z4 z1 x1 . (D.13) D.2.1 Momentum space By translation invariance, both G(x1, x2;x3, x4) and K(x1, x2;x3, x4) can only depend on the differences of coordinates, and the same of course holds for the propagators. We introduce center-of-mass and relative coordinates as x = x1 − x2 , X = ηdx1 + ηNx2 ; x′ = x3 − x4 , X ′ = ηdx3 + ηNx4 (D.14a) and X ′′ = X −X ′ , (D.14b) where ηd = Md Md +MN , ηN = MN Md +MN , i.e. ηd + ηN = 1 . (D.14c) Considering now G0(x1, x2;x3, x4) = ∫ d4q (2pi)4 ∫ d4q′ (2pi)4 e−iq·(x1−x3) e−iq ′·(x2−x4) ∆d(q)∆N(q′) (D.15) and making the change of variables q = ηdP + p , q ′ = ηNP − p , (D.16) 166 Appendix D. Bound states in nonrelativistic effective field theory we get G0(x1, x2;x3, x4) = ∫ d4P (2pi)4 ∫ d4p (2pi)4 e−iP ·X ′′ e−ip·(x−x ′) ∆d(ηdP + p) ∆N(ηNP − p) . (D.17) Inserting an identity operator expressed in momentum space, we arrive at G0(x1, x2;x3, x4) = ∫ d4P (2pi)4 ∫ d4p (2pi)4 ∫ d4k (2pi)4 e−iP ·X ′′ e−ip·x eik·x ′ (2pi)4δ(4)(k − p) ×∆d(ηdP + p) ∆N(ηNP − p) . (D.18) Moreover, we have G(x1, x2;x3, x4) ≡ G(X ′′, x, x′) = ∫ d4P (2pi)4 ∫ d4p (2pi)4 ∫ d4k (2pi)4 e−iP ·X ′′ e−ip·x eik·x ′ G(k, p;P ) , (D.19) and an analogous expression for K(x1, x2;x3, x4). Putting everything into Eq. (D.11) and making suitable changes of variables for the d4zi integrals yields the (inhomogeneous) Bethe–Salpeter equation in momentum space: G(k, p;P ) = G0(k, p;P ) + ∫ d4q (2pi)4 ∫ d4q′ (2pi)4 G(k, q;P )K(q, q′;P )G0(q′, p;P ) (D.20) with G0(k, p;P ) = (2pi) 4δ(4)(k − p) ·∆d(ηdP + p) ∆N(ηNP − p) . (D.21) Diagrammatically, it can be written as = + . (D.22) One of the integrals in Eq. (D.20) can be cancelled with the delta function in G0, yielding G(k, p;P ) = ∆d(ηdP + p) ∆N(ηNP − p) × [ (2pi)4δ(4)(k − p) + ∫ d4q (2pi)4 G(k, q;P ) ·K(q, p;P ) ] , (D.23) and finally∫ d4q (2pi)4 [ G−10 (q, p;P )−K(q, p;P ) ] G(k, q;P ) = (2pi)4δ(4)(k − p) (D.24) with G−10 (k, p;P ) = (2pi) 4δ(4)(k − p) · [∆d(ηdP + p)]−1 · [∆N(ηNP − p)]−1 (D.25) as an alternative formulation of Eq. (D.20). D.2. Bethe–Salpeter equation 167 D.2.2 Bound state contribution Going back to configuration space for the moment, we now assume the existence of a stable trinucleon bound state (which can be thought of as an unphysical triton if one identifies the uncharged single nucleon state in our model system with the neutron) with energy E = −EB < 0. For simplicity, we also assume that it is non-degenerate, i.e., that it is characterized by a unique set of quantum numbers a0, which we omit in the following. Selecting the time-ordering t1, t2 > t3, t4 and inserting a complete set of states, we get G(x1, x2;x3, x4) = ∑ a ∫ d3P (2pi)3 ∫ dE 2pi ψEPa(x1, x2)ψ † EPa(x3, x4) , (t1, t2 > t3, t4) (D.26) with dE = dP0. The bound-state contribution to this expression is GB(x1, x2;x3, x4) = ∫ d3P (2pi)3 ψBP(x1, x2)ψ † BP(x3, x4) , (t1, t2 > t3, t4) , (D.27) where we have introduced the short-hand notation ψBP ≡ ψ−EBPa0 for the bound-state wave function. By translation invariance, we can write ψBP(x1, x2) = 〈Ω|T ( d(x1 + a)N(x2 + a) )|−EB,P, a0〉 · eiP ·a (D.28) with P0 = E = −EB for any four-vector a. We use this with a = −X and an analogous expression for ψ†BP(x3, x4) to write ψBP(x1, x2)ψ † BP(x3, x4) = ψBP(x)ψ † BP(x ′) · e−iP ·X′′ , (D.29) implicitly defining the Bethe–Salpeter wave functions ψBP(x) and ψ † BP(x ′) for the relative motion. Using this, the bound-state contribution is GB(x1, x2;x3, x4) = ∫ d3P (2pi)3 e−iP ·X ′′ ψBP(x)ψ † BP(x ′) · θ ( X ′′0 − 1 2 |x0| − 1 2 |x′0| ) , (D.30) where we have inserted the theta function in order to impose the time-ordering t1, t2 > t3, t4. Using the formula θ(y0) = i ∫ dp0 2pi 1 p0 + iε e−i p0y0 (ε→ 0 implied) (D.31) we get GB(x1, x2;x3, x4) = i ∫ d3P (2pi)3 ∫ dE 2pi eiP·X ′′ e−iEX ′′ 0 e i 2 (E+EB)|x0| ψBP(x)e i 2 (E+EB)|x′0| ψ†BP(x ′) E + EB + iε , (D.32) where we have changed the integration variable p0 → E+EB. Defining new wave functions ψ˜BP(x) = e i 2 (E+EB)|x0| ψBP(x) , ψ˜ † BP(x ′) = e i 2 (E+EB)|x′0| ψ†BP(x ′) (D.33) and their Fourier transforms ψ˜BP(x) = ∫ d4p (2pi)4 e−ip·x ψ˜BP(p) , ψ˜ † BP(x ′) = ∫ d4k (2pi)4 eik·x ′ ψ˜†BP(k) , (D.34) 168 Appendix D. Bound states in nonrelativistic effective field theory we finally arrive at GB(x1, x2;x3, x4) = i ∫ d4P (2pi)4 ∫ d4p (2pi)4 ∫ d4k (2pi)4 e−iP ·X ′′ e−ip·xeik·x ′ ψ˜BP(p) ψ˜ † BP(k) E + EB + iε (D.35) with P0 ≡ E. Note that ψ˜BP → ψBP for E → −EB, so we have G(k, p;P ) = i ψBP(p)ψ † BP(k) E + EB + iε + terms regular at E = −EB (D.36) in momentum space. D.2.3 Homogeneous equation Inserting the factorization (D.36) into Eq. (D.24) and multiplying by (E + EB) we find the homogeneous Bethe–Salpeter equation∫ d4q (2pi)4 [ G−10 (q, p;P )−K(q, p;P ] ψBP(q) = 0 (D.37) after taking the limit E → −EB and cancelling non-zero factors. Equivalently, we can also obtain this in the form ψBP(p) = ∆d(ηdP + p) ∆N(ηNP − p) · ∫ d4q (2pi)4 K(q, p;P )ψBP(q) (D.38) from Eq. (D.20). D.2.4 Three-dimensional reduction We now consider a bound state at rest, P = (−EB,0), and define the amputated wave function B(p0,p) = ψB0(p0,p) · [∆d(−ηdEB + p0,p)]−1 · [∆N(−ηNEB − p0,p)]−1 , (D.39) which fulfills the equation B(p0,p) = ∫ d4q (2pi)4 K(q, p;−EB) ∆d(−ηdEB + q0,q) ∆N(−ηNEB − q0,q) B(q0,q) . (D.40) Carrying out the dq0 integration picks up the residue from the nucleon propagator pole at q0 = −ηNEB − q2/(2MN) + iε. From the resulting right-hand side of Eq. (D.40) we then find that the function B(p) ≡ B ( −ηNEB − p 2 2MN ,p ) (D.41) D.3. Operator formalism 169 fulfills the equation B(p) = ∫ d3q (2pi)3 K ( ηNE − q 2 2MN ,k, ηNE − p 2 2MN ,p;E ) ∆d ( −EB − q 2 2MN ,q ) B(q) . (D.42) For future reference we also define the wave function φ(p) = ∫ dp0 2pi ψ(p0,p) , (D.43) for which from Eq. (D.39) one immediately finds that φ(p) = ∆d ( −EB − p 2 2MN ,p ) B(p) . (D.44) D.3 Operator formalism In order to focus on the essential, we now reformulate the results derived above in an abstract operator notation. The Bethe–Salpeter equation (D.20) can be written as G = G0 +GKG0 = G0 +G0KG , (D.45) where the middle and the right-hand side are equivalent.1 Assuming the existence of a bound state with energy E = −EB, we have the factorization G ∼ i |ψ〉〈ψ| E + EB for E → −EB , (D.46) as given explicitly in Eq. (D.36). Inserting this into Eq. (D.45), multiplying by (E+EB), and acting on |ψ〉, we obtain the homogeneous equation |ψ〉 = G0K|ψ〉 (D.47) after taking the limit E → −EB and using that G0 is regular for E → −EB. This is, of course, just Eq. (D.38). Note that all the operators here are in general functions of the total energy, G = G(E), K = K(E), etc., but that for the sake of notational simplicity we have not written out this dependence explicitly. D.3.1 Lippmann–Schwinger equation Defining the T-matrix operator T via the relation KG = TG0 , (D.48) we can rewrite Eq. (D.45) in the form G = G0 +G0TG0 . (D.49) 1This can be seen, for example, by iterating both versions and noting that the results are the same. 170 Appendix D. Bound states in nonrelativistic effective field theory Inserting this into both sides of the original Eq. (D.45), we get G0 +G0TG0 = G0 +G0KG0 +G0KG0TG0 . (D.50) After cancelling the common term G0 and multiplying through by G −1 0 on both sides, we arrive at the familiar Lippmann–Schwinger equation T = K +KG0T . (D.51) The precise relation between this operator and the T -matrix elements used in Chapter 6 will be discussed in Section D.4.3 below. As a remark we add here that G, with all momenta put on-shell and the propagator term G0 removed (“amputated”), is just an S-matrix element. Going from Eq. (D.50) to (D.51) is essentially equivalent to using “S = 1 + T .” D.4 Normalization condition We are now finally equipped to derive the normalization for the Bethe-Salpeter wave functions and, subsequently, for the trinucleon wave functions used in Chapter 6. For the general derivation we stay in the abstract operator notation and only go back to the momentum-space formulation when we discuss the explicit form for our N–d model system in Section D.4.2 below. D.4.1 General derivation Multiplying Eq. (D.45) by G−1 from the right and by G−10 from the left, we get G−10 = G −1 +K , (D.52) and hence G(G−10 −K)G = G (D.53) from the trivial identity GG−1G = G. Inserting furthermore the factorization (D.46) at the bound-state pole, we find limE→−EB i 2 |ψ〉〈ψ|G−10 −K|ψ〉〈ψ| (E + EB)2 = limE→−EB i |ψ〉〈ψ| E + EB . (D.54) Multiplying this by (E + EB) and furthermore with |ψ〉 from both sides, we find limE→−EB i 〈ψ|G−10 −K|ψ〉 E + EB = 1 (D.55) after cancelling the common factor 〈ψ|ψ〉2. Using now l’Hoˆpital’s rule to evaluate the limit, we finally arrive at i〈ψ| d dE ( G−10 −K )|ψ〉∣∣∣ E=−EB = 1 . (D.56) D.4. Normalization condition 171 Note that for the standard case, where G0 ∝ (E − H0)−1 with the free Hamiltonian H0, and where the kernel (potential) does not depend on the energy, Eq. (D.56) reduces to the familiar condition 〈ψ|ψ〉 = 1. As already remarked in Chapter 6, this kind of normalization condition for energy-dependent interactions is not at all a new result but has been known for a long time [192]. D.4.2 Explicit form in three dimensions We now go back to the momentum-space basis and consider a bound state at rest, setting P = (−EB,0). We define a reduced two-body propagator G˜ that only depends on the relative three-momenta k and p by integration over the energies: G˜(k,p;−EB) = ∫ dk0 2pi ∫ dp0 2pi G(k, p;P ) . (D.57) By the definition (D.43), this implies G˜ ∼ i |φ〉〈φ| E + EB for E → −EB . (D.58) From Eq. (D.45) we get G˜ = G˜0 + G˜0KG = G˜0 ( 1 + G˜−10 G˜0KG ) (D.59) and hence G˜−1 = ( 1 + G˜−10 G˜0KG )−1 G˜−10 . (D.60) We now define G˜−1 − G˜−10 ≡ −V˜ (D.61) and find G˜ = G˜ ( G˜−10 − V˜ ) G˜ . (D.62) Repeating the procedure described in Section D.4.1 above then yields the normalization condition i〈φ| d dE ( G˜−10 − V˜ ) |φ〉 ∣∣∣ E=−EB = 1 . (D.63) So far it seems that we have hardly done anything but complicate the notation, but the advantage will become apparent soon. For the first term in Eq. (D.63), we find G˜0(k,p;E) = ∫ dk0 2pi ∫ dp0 2pi (2pi)4δ(4)(k − p) ·∆d(ηdE + p0,p) ∆N(ηNE − p0,−p) = (2pi)3δ(3)(k − p) · ∫ dp0 2pi ∆d (ηdE + p0,p) · i ηNE − p0 − p22MN + iε = (2pi)3δ(3)(k − p) ·∆d ( E − p 2 2MN ,p ) , (D.64) 172 Appendix D. Bound states in nonrelativistic effective field theory and thus G˜−10 (k,p;E) = (2pi) 3δ(3)(k − p) · [ ∆d ( E − p 2 2MN ,p )]−1 . (D.65) Now we consider V˜ . From Eqs. (D.60) and (D.61) we get V˜ = G˜−10 − ( 1 + G˜−10 G˜0KG )−1 G˜−10 = G˜ −1 0 − [ ∞∑ n=0 ( −G˜−10 G˜0KG )n] G˜−10 . (D.66) Furthermore, iterated application of the Bethe–Salpeter equation (D.45) yields G˜0KG = G˜0KG0 + ˜G0KG0KG0 + · · · , (D.67) such that we have a double expansion in Eq. (D.66). We write V˜ ≡ ∞∑ n=0 V˜n = G˜ −1 0 − G˜−10︸︷︷︸ V˜0=0 + G˜−10 G˜0KG0G˜ −1 0︸︷︷︸ V˜1 + · · · , (D.68) where the index indicates the number of insertions of K. In the following calculations we will frequently omit the arguments (k,p;E) on the left-hand side of equations. It will, however, always be clear what they should be from the corresponding right-hand sides. With G˜0KG0 = ∫ dk0 2pi ∫ dp0 2pi [ ∆d(ηdE + k0,k) ∆N(ηNE − k0,−k) ·K(k0,k, p0,p;E) ×∆d(ηdE + p0,p) ∆N(ηNE − p0,−p) ] = ∆d ( E − k 2 2MN ,k ) ·K ( ηNE − k 2 2MN ,k, ηNE − p 2 2MN ,p;E ) ·∆d ( E − p 2 2MN ,p ) (D.69) and the G˜−10 from Eq. (D.65) amputating the deuteron propagators, we find that V˜1(k,p;E) = K ( ηNE − k 2 2MN ,k, ηNE − p 2 2MN ,p;E ) . (D.70) The important point is now that all higher contributions vanish. To see this, we first look at V˜2 = G˜ −1 0 [ ˜G0KG0KG0 − G˜0KG0G˜−10 G˜0KG0 ] G˜−10 (D.71) and use the following Lemma. It holds that ·˜ · ·G0G˜−10 G˜0 · · · = ˜· · ·G0 · · · . (D.72) D.4. Normalization condition 173 Proof. We start by calculating the right hand side: ˜· · ·G0 · · · = ∫ d4q1 (2pi)4 ∫ d4q2 (2pi)4 [· · · ] (2pi)4δ(4)(q1 − q2) ×∆d ( ηdE + q 0 1,q1 ) · i ηNE − q01 − q 2 1 2MN + iε [· · · ] = ∫ d3q1 (2pi)3 [· · · ] ∆d ( E − q 2 1 2MN ,q1 ) [· · · ] . (D.73) The left hand side is ·˜ · ·G0G˜−10 G˜0 · · · = ∫ d4p1 (2pi)4 ∫ dq01 2pi ∫ d3q1 (2pi)3 ∫ d3q2 (2pi)3 ∫ dq02 2pi ∫ d4p2 (2pi)4 [· · · ] × (2pi)4δ(4)(p1 − q1) ·∆d ( ηdE + q 0 1,q1 ) · i ηNE − q01 − q 2 1 2MN + iε × (2pi)3δ(3)(q1 − q2) · [ ∆d ( E − q 2 1 2MN ,q1 )]−1 × (2pi)4δ(4)(q2 − p2) ·∆d ( ηdE + q 0 2,q2 ) · i ηNE − q02 − q 2 2 2MN + iε [· · · ] = ∫ d3q1 (2pi)3 [· · · ] ∆d ( E − q 2 1 2MN ,p ) [· · · ] = ˜· · ·G0 · · · . (D.74) The crucial point is that the residues of the nucleon propagators are always picked up in such a way that one deuteron propagator is cancelled by its inverse from the G˜−10 , cf. Eq. (D.65). This immediately shows that V˜2 vanishes, and for the higher terms it is just a matter of checking the relative signs that they also do. For example, we have V˜3 = G˜ −1 0 [ ˜G0KG0KG0KG0 − G˜0KG0G˜−10 ˜G0KG0KG0 − ˜G0KG0KG0G˜−10 G˜0KG0 + G˜0KG0G˜−10 G˜0KG0G˜−10 G˜0KG0 ] G˜−10 = 0 . (D.75) Altogether, we have found that V˜ (k,p;E) = K ( ηNE − k 2 2MN ,k, ηNE − p 2 2MN ,p;E ) = iMNy 2 d 2 · 1 k2 + k · p + p2 −MNE − iε . (D.76) D.4.3 Factorization of the T-matrix What remains is to establish the connection between the quantities defined here and the T -matrix elements and wave functions used in Chapter 6. Comparing Eqs. (D.42) 174 Appendix D. Bound states in nonrelativistic effective field theory and (D.76) with the interaction as given in Section 6.2.4 already suggests that there is a direct correspondence between the states B(p) introduced here and the trinucleon wave functions of Chapter 6. For the T operator introduced in Eq. (D.48) we find from Eq. (D.50) that G˜0TG0 = G˜0KG0 + ˜G0KG0TG0 . (D.77) Applying the Lemma (D.72) to the second term on the right-hand side gives ˜G0KG0TG0 = G˜0KG0G˜−10 G˜0TG0 . (D.78) Inserting into this the identity in the form 1 = G˜0G˜ −1 0 (D.79) and multiplying Eq. (D.77) with G˜−10 from both sides we find that G˜−10 G˜0TG0G˜ −1 0 = [ G˜−10 G˜0KG0G˜ −1 0 ] G˜0 [ G˜−10 G˜0TG0G˜ −1 0 ] , (D.80) where from the discussion in Section D.4.2 we see that the interaction is the same as in the normalization condition: G˜−10 G˜0KG0G˜ −1 0 = V˜ . (D.81) Comparing this now with the integral equations in Chapter 6, we can conclude that the T -matrix there is iT (E; k,p) = 〈k|G˜−10 G˜0TG0G˜−10 |p〉 , (D.82) where all operators are of course functions of the energy E. As E → −EB we now have, using Eq. (D.49) and noting that the bound state cannot be in G0 since it has to arise from the interaction, G˜−10 G˜0TG0G˜ −1 0 = G˜ −1 0 G˜G˜ −1 0 + regular terms = i G˜−10 |φ〉〈φ|G˜−10 E + EB + regular terms , (D.83) where the second identity follows from Eq. (D.58). Now, according to Eqs. (D.44) and (D.64) we have |φ〉 = G˜0|B〉 ⇐⇒ |B〉 = G˜−10 |φ〉 , (D.84) which implies 〈B| = 〈φ|(G˜−10 )† . (D.85) But, up to a delta function, G˜−10 is just the deuteron propagator ∆d which, from Eq. (D.5) we find to be a purely imaginary quantity.2 Hence,( G˜−10 )† = −G˜−10 , for E < 0 (D.86) 2Note the overall i in the prefactor in Eq. (D.5) and that the rest is real for p0 < 0 and ε→ 0. D.5. Some remarks 175 and G˜−10 G˜0TG0G˜ −1 0 = −i |B〉〈B| E + EB + regular terms as E → −EB . (D.87) For the T -matrix of Chapter 6 we then find from Eq. (D.82) that T (E; k,p) = −B †(k)B(p) E + EB + regular terms as E → −EB . (D.88) Finally, for the normalization condition (D.63) written in terms of the |B〉 we analogously find −i〈B|G˜0 [ d dE ( G˜−10 − V˜ )] G˜0|B〉 ∣∣∣ E=−EB = 1 . (D.89) D.5 Some remarks In order to fully establish the connection of the results derived here with the formalism of Chapter 6, a few remarks are in order. Starting from a formulation in terms of four-dimensional energy-momentum vectors we have, in the preceding sections, obtained functions that only depend on three-momenta by explicitly integrating over the zero components. Essentially, this corresponds to imposing an equal-time condition on the Bethe–Salpeter amplitudes in order to get back ordinary Schro¨dinger wave functions (cf. Section 9.1 in Ref. [200]). To see this, consider an arbitrary amplitude ψEP(x) = ψEP(x0,x) defined analogously to what was done for the bound-state contribution in Eqs. (D.28) and (D.29). Since its Fourier transform is ψEP(q) = ψEP(q0,q) = ∫ dx0 e iq0·x0 e−iq·x ψEP(x0,x) , (D.90) integrating over q0 gives a delta function δ(x0) = δ(t1 − t2). As used in Sections D.2.4 and D.4.2, in loops the integration over the zero components of the four-momenta always picks up the on-shell pole of a nucleon propagator. 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Black et al., Determination of proton-deuteron scattering lengths, Phys. Lett. B 471, 103–107 (1999). [200] D. Lurie´, Particles and Fields (Interscience Publishers, New York; London; Syd- ney, 1968). 188 Bibliography Acknowledgments I would like to express my sincere gratitude to a number of people who supported me in this work—both directly and indirectly and in various ways—over the course of the last three years. First of all I would like to thank Prof. Hans-Werner Hammer, my thesis advisor in Bonn, and Prof. Dean Lee at the North Carolina State University for suggesting many interesting topics, keeping me on the right track, and in general for providing advice, support, and encouragement, regarding both this thesis and my career in general. I consider myself very lucky to have been their student. To Dean Lee I am furthermore grateful for inviting my to stay at NC State for two months and for taking such good care of me there. My thanks also go to Prof. Ulf-G. Meißner for carefully reading the manuscript of this thesis, for providing many valuable comments and corrections, and beyond that for further support and encouragement. I would like to thank the “Studientiftung des deutschen Volkes” for providing me with a doctoral scholarship to cover my living expenses, and furthermore for offering summer schools and student meetings that I had a lot of fun at. I am also grateful to the Bonn– Cologne Graduate School of Physics and Astronomy (BCGS) for additional support. For the many opportunities to travel that I had over the course of this work, I would in particular like to thank the organizers of the APFB2011 conference for the chance to visit Seoul (and letting me stay at the university guest house for the weekend after the conference), the Institute of Nuclear Theory at the University of Washington for its hospitality for three weeks during the program “Light nuclei from first principles,” the organizers of which I would furthermore like to thank for inviting me. Thanks also to Profs. Roxanne Springer and Thomas Mehen for hosting me at Duke University for two weeks, and to Profs. Daniel Phillips and Richard Furnstahl for inviting me to give seminar talks at Ohio University and Ohio State University (a trip that I enjoyed very much) while I was at NC State, and for interesting discussions. Moreover, I would like to thank Prof. Harald Grießhammer for many useful discussions about pionless effective field theory (in particular, about the 3He problem at NLO), and for providing valuable comments on the corresponding part of this thesis. My thanks also go to all other people I had the chance to collaborate with, in particular to Serdar Elhatisari for putting so much work into the van der Waals causality project, and to Shahin Bour and Jared Vanasse. I would like to thank the group of Prof. Hammer and all my colleagues at the HISKP, 189 190 Bibliography both present and former, for providing a great working atmosphere. In particular, my thanks go to my office mate Philipp Hagen, and moreover to Sebastian Schneider, Martin Hoferichter, Bastian Kubis, and Simon To¨lle, for numerous interesting and enjoyable discussions—not at all limited to physics topics—throughout the years. Finally, I would like to thank my parents and grandparents for their continued support during the years, for steady encouragement, and most of all for providing a home that I could always come back to when I needed it. Last, but most certainly not least, thank you, Jan, for being there and everything.

Finite range effects in coulomb stripping

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