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Effective Quantum Theories with Short- and Long-Range Forces

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Effective Quantum Theories with Short- and Long-Range Forces Effective quantum theories with short- and long-range forces Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn vorgelegt von Sebastian K¨onig aus Northeim Bonn, August 2013 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Rheinischen Friedrich-Wilhelms-Universit¨atBonn 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. Konig¨ and H.-W. Hammer, Low-energy p-d scattering and He-3 in pionless • EFT, Phys. Rev. C 83, 064001 (2011) S. Konig¨ , D. Lee, and H.-W. Hammer, Volume Dependence of Bound States • with Angular Momentum, Phys. Rev. Lett. 107, 112001 (2011) S. Bour, S. Konig¨ , 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. Konig¨ , D. Lee, and H.-W. Hammer, Non-relativistic bound states in a finite • volume, Annals Phys. 327, 1450{1471 (2012) S. Konig¨ , D. Lee, and H.-W. Hammer, Causality constraints for charged parti- • cles, J. Phys. G: Nucl. Part. Phys. 40, 045106 (2013) S. Konig¨ and H.-W. Hammer, The Low-Energy p{d System in Pionless EFT, • Few-Body Syst. 54, 231{234 (2013) S. Elhatisari, S. Konig¨ , D. Lee, and H.-W. Hammer, Causality, universality, • and effective field theory for van der Waals interactions, Phys. Rev. A 87, 052705 (2013) f¨urmeinen 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 . ..
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