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Effective Field Theory for Halo Nuclei

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Effective Field Theory for Halo Nuclei Effective Field Theory for Halo Nuclei Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn vorgelegt von Philipp Robert Hagen aus Troisdorf Bonn 2013 Angefertigt mit der Genehmigung der Mathmatisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn 1. Gutachter: Prof. Dr. Hans-Werner Hammer 2. Gutachter: Prof. Dr. Bastian Kubis Tag der Promotion: 19.02.2014 Erscheinungsjahr: 2014 Abstract We investigate properties of two- and three-body halo systems using effective field theory. If the two-particle scattering length a in such a system is large compared to the typical range of the interaction R, low-energy observables in the strong and the electromagnetic sector can be calculated in halo EFT in a controlled expansion in R/ a . Here we will focus | | on universal properties and stay at leading order in the expansion. Motivated by the existence of the P-wave halo nucleus 6He, we first set up an EFT framework for a general three-body system with resonant two-particle P-wave interactions. Based on a Lagrangian description, we identify the area in the effective range parameter space where the two-particle sector of our model is renormalizable. However, we argue that for such parameters, there are two two-body bound states: a physical one and an addi- tional deeper-bound and non-normalizable state that limits the range of applicability of our theory. With regard to the three-body sector, we then classify all angular-momentum and parity channels that display asymptotic discrete scale invariance and thus require renor- malization via a cut-off dependent three-body force. In the unitary limit an Efimov effect occurs. However, this effect is purely mathematical, since, due to causality bounds, the unitary limit for P-wave interactions can not be realized in nature. Away from the unitary limit, the three-body binding energy spectrum displays an approximate Efimov effect but lies below the unphysical, deep two-body bound state and is thus unphysical. Finally, we discuss possible modifications in our halo EFT approach with P-wave interactions that might provide a suitable way to describe physical three-body bound states. We then set up a halo EFT formalism for two-neutron halo nuclei with resonant two- particle S-wave interactions. Introducing external currents via minimal coupling, we calcu- late observables and universal correlations for such systems. We apply our model to some known and suspected halo nuclei, namely the light isotopes 11Li, 14Be and 22C and the hypothetical heavy atomic nucleus 62Ca. In particular, we calculate charge form factors, relative electric charge radii and dipole strengths as well as general dependencies of these observables on masses and one- and two-neutron separation energies. Our analysis of the 62Ca system provides evidence of Efimov physics along the Calcium isotope chain. Exper- imental key observables that facilitate a test of our findings are discussed. Parts of this thesis have been published in: E. Braaten, P. Hagen, H.-W. Hammer and L. Platter. Renormalization in the Three- • body Problem with Resonant P-wave Interactions. Phys. Rev. A, 86:012711, (2012), arXiv:1110.6829v4 [cond-mat.quant-gas]. P. Hagen, H.-W. Hammer and L. Platter. Charge form factors of two-neutron halo • nuclei in halo EFT. Eur. Phys. J. A, 49:118, (2013), arXiv:1304.6516v2 [nucl-th]. G. Hagen, P. Hagen, H.-W. Hammer and L. Platter. Efimov Physics around the neu- • tron rich Calcium-60 isotope. Phys. Rev. Lett., 111:132501, (2013), arXiv:1306.3661 [nucl-th]. Contents 1 Introduction 1 1.1 From the standard model to halo effective field theory . .... 1 1.1.1 Overview ................................ 3 1.2 EFTwithlargescatteringlength . .. .. 4 1.2.1 Scatteringtheoryconcepts . 4 1.2.2 Universality, discrete scale invariance and the Efimov effect . .. 5 1.2.3 Halo EFT and halo nuclei . 7 1.3 Notationandconventions ........................... 9 2 Three-body halos with P-wave interactions 13 2.1 Fundamentals of non-relativistic EFTs . 14 2.1.1 Galilean invariance . 14 2.1.1.1 Galilean group . 14 2.1.1.2 Galilean invariants . 14 2.1.2 Auxiliary fields . 16 2.1.2.1 Equivalent Lagrangians . 16 2.1.2.2 Equivalence up to higher orders . 17 2.2 S-waveinteractions ............................... 19 2.2.1 EffectiveLagrangian .......................... 19 2.2.2 Discrete scale invariance and the Efimov effect . 20 2.3 P-waveinteractions............................... 22 2.3.1 EffectiveLagrangian .......................... 22 2.3.2 Two-bodyproblem ........................... 24 2.3.2.1 Effective range expansion . 25 2.3.2.2 Pole and residue structure . 26 2.3.3 Three-bodyproblem .......................... 27 2.3.3.1 Kinematics . 27 2.3.3.2 T-matrix integral equation . 29 2.3.3.3 Angular momentum eigenstates . 30 2.3.3.4 Renormalization . 33 2.3.3.5 Boundstateequation . .. 35 2.3.4 Discrete scale invariance and the Efimov effect . 35 2.3.4.1 Discrete scale invariance . 35 v 2.3.4.2 Bound-statespectrum . 43 3 Halo EFT with external currents 49 3.1 Two-neutronhaloEFTformalism . 49 3.1.1 EffectiveLagrangian .......................... 49 3.1.2 Two-bodyproblem ........................... 52 3.1.2.1 Effective range expansion . 53 3.1.3 Three-bodyproblem .......................... 54 3.1.3.1 Kinematics . 55 3.1.3.2 T-matrix integral equation . 56 3.1.3.3 Angular momentum eigenstates . 57 3.1.3.4 Renormalization . 60 3.1.3.5 Boundstateequation . .. 61 3.1.4 Trimer couplings . 63 3.1.4.1 Trimer residue . 64 3.1.4.2 Irreducible trimer-dimer-particle coupling . 65 3.1.4.3 Irreducible trimer-three-particle coupling . 66 3.2 Externalcurrents ................................ 68 3.2.1 Effective Lagrangian via minimal coupling . 68 3.2.2 Electric form factor and charge radius . 69 3.2.2.1 Formalism . 69 3.2.2.2 Results ............................ 72 3.2.3 Universalcorrelations. .. .. 76 3.2.3.1 Calcium halo nuclei . 78 3.2.4 Photodisintegration........................... 80 3.2.4.1 Formalism . 80 3.2.4.2 First results . 83 4 Summary and outlook 87 A Kernel analytics 91 A.1 Structureofthefulldimerpropagator. ... 91 A.1.1 Polegeometry.............................. 91 A.1.1.1 S-wave interactions . 93 A.1.1.2 P-wave interactions . 96 A.1.2 Cauchy principal value integrals . 102 A.2 Legendrefunctionsofsecondkind . 103 A.2.1 Recursionformula............................ 104 A.2.2 Analyticstructure ........................... 105 A.2.2.1 Geometry of singularities . 105 A.2.3 Hypergeometricseries . 106 A.2.3.1 Approximative expansion . 107 A.2.4 Mellintransform ............................ 108 B Kernel numerics 113 C Angular momentum coupling 117 C.1 Clebsch–Gordan-coefficients . 117 C.2 Sphericalharmonics............................... 118 C.3 Angular decomposition of the interaction kernel . 119 C.4 Eigenstatesoftotalangularmomentum . 122 C.5 Paritydecoupling ................................ 125 D Feynman diagrams 129 D.1 Feynmanrules.................................. 129 D.2 P-waveinteractions............................... 129 D.2.1 Dimerself-energy ............................ 129 D.2.2 Dimer-particle interaction . 131 D.3 Two-neutronhaloEFTwithexternalcurrents . 132 D.3.1 Dimerself-energy ............................ 132 D.3.2 Twoparticlescattering. 133 D.3.3 Dimer-particle interaction . 134 D.3.4 Formfactorcontributions . 136 D.3.4.1 Breitframe .......................... 137 D.3.4.2 Parallel term . 137 D.3.4.3 Exchangeterm ........................ 143 D.3.4.4 Loopterm........................... 147 D.3.5 Photodisintegration. 154 D.3.5.1 Dipole matrix element . 155 D.3.5.2 Dipole strength distribution . 157 Chapter 1 Introduction 1.1 From the standard model to halo effective field theory A vast amount of physical phenomena in nature can be described within the so-called stan- dard model (SM). Its fundamental degrees of freedom, the elementary particles, are ordered in three generations of quarks and leptons and a set of exchange particles, describing their interactions. The interactions are commonly divided into the electromagnetic, the weak, and the strong sector, where the first two were successfully unified to the electroweak force. Furthermore, within this picture, all elementary particles are point-like and their inertial masses are generated by the Higgs mechanism, which introduces at least one additional bosonic field. This mechanism was already proposed in 1964 by, among others, Higgs and Englert [1–3]. Recent experiments at CERN confirmed the existence of such a so-called Higgs field, awarding both authors the 2013 Nobel prize in physics. The theory of strong interactions is usually referred to as quantum chromodynamics (QCD). It describes how quark fields q interact with each other through gauge bosons G called gluons. Since gluons carry color charge, they can interact with each other. The fundamental object, the theory is mathematically based on, is the QCD-Lagrangian 0 µ λa a 1 a µν (q,G) = q† γ iγ ∂ ig G m q F F , LQCD f µ − s 2 µ − f f − 4 µν a (1.1) F a = ∂ Ga ∂ Ga + g f abcGbGc , µν µ ν − ν µ s µ ν where we implicitly sum over all double indices. Thereby, the ranges for flavor indices (f), color indices (a, b, c) and Lorentz indices (µ, ν) are 1,... 6 , 1,... 8 and 0,..., 3 , µ { } { } { } respectively. γ are the four Dirac matrices and λa are the eight Gell-Mann matrices with abc structure constants f . mf is the bare mass parameter for a quark of flavor f and gs is the strong coupling constant. In the course of the great progress in the understanding of nature, provided by the SM, various new questions and problems came up. On the one hand, there are in a way fundamental problems to the SM. For example, satisfying explanations for phenomena 1 2 CHAPTER 1. INTRODUCTION related to the gravitational sector, such as gravitation itself or dark matter and dark energy, are still missing.
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