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Self-Consistent Green's Functions with Three-Body Forces Self-consistent Green's functions with three-body forces Arianna Carbone Departament d'Estructura i Constituents de la Mat`eria Facultat de F´ısica,Universitat de Barcelona Mem`oriapresentada per optar al grau de Doctor per la Universitat de Barcelona arXiv:1407.6622v1 [nucl-th] 24 Jul 2014 11 d'abril de 2014 Directors: Dr. Artur Polls Mart´ıi Dr. Arnau Rios Huguet Tutor: Dr. Artur Polls Mart´ı Programa de Doctorat de F´ısica Table of contents Table of contentsi 1 Introduction1 1.1 The nuclear many-body problem . .3 1.2 The nuclear interaction . .9 1.3 Program of the thesis . 13 2 Many-body Green's functions 15 2.1 Green's functions formalism with three-body forces . 17 2.2 The Galistkii-Migdal-Koltun sum rule . 21 2.3 Dyson's equation and self-consistency . 24 3 Three-body forces in the SCGF approach 31 3.1 Interaction-irreducible diagrams . 32 3.2 Hierarchy of equations of motion . 40 3.3 Ladder and other approximations . 54 4 Effective two-body chiral interaction 61 4.1 Density-dependent potential at N2LO . 63 4.2 Partial-wave matrix elements . 72 5 Nuclear and Neutron matter with three-body forces 89 5.1 The self-energy . 93 5.2 Spectral function and momentum distribution . 99 5.3 Nuclear matter . 105 5.4 Neutron matter and the symmetry energy . 111 6 Summary and Conclusions 119 A Feynman rules 129 i Table of contents B Interaction-irreducible diagrams 137 C Complete expressions for the density-dependent force 141 C.1 Numerical implementation . 149 Bibliography 153 ii 1 Introduction The quantum many-body problem is an everlasting challenge for theoretical physics. The aim sought is to analyze, at the quantum level, the observables arising from a group of N interacting particles. If the N particles in the many-body system are non-interacting, the system is ideal and can be solved as the sum of N one-body (1B) problems. If the system is interacting, the only analytically solvable system is the two-body (2B) one. This can be recast in the form of two 1B problems, one for the center of mass and the other for the relative coordinate of the two particles. The three-body (3B) problem is already not solvable analytically, but an exact solution could be obtained by solving the Faddeev equations. As the number of particles increases, one is forced to resort to approximate solutions. The objective of many-body physics has constantly been that of dealing with systems formed from few to many interacting particles, from finite systems, such as nuclei, atoms or molecules, to the infinite ones, such as the electron gas or nuclear matter present in the interior of stars. The quest is to provide the best description of the system via the application of consistent approximations, with the aim of achieving results to be contrasted with the experimental ones. In the specific case of nuclear physics, the study and understanding of interacting nuclear systems has spanned from the finite nuclei to the infinite nuclear matter case. In order to characterize such systems, the interactions which govern the behavior of nucleons in their interior need to be sorted out. The nucleon-nucleon (NN) interactions devised to do so can be divided into two categories: microscopic and phenomenological. The microscopic poten- tials are models constructed to fit experimental data of NN scattering and properties of the deuteron. All microscopic potentials have a common one- pion exchange (OPE) long-range part and usually differ in the treatment of the intermediate and especially short-range part. Due to the strong repul- sive short-range part, and the presence of tensor components, these potentials cannot be used directly in perturbative calculations and need to be renormal- ized through the use of many-body methods. Phenomenological potentials are simpler, and are constructed fitting a number of coupling constants to 1 1. Introduction 2 empirical saturation properties of nuclear matter and observables of ground states of nuclei. For this reason their validity far from these conditions can be questionable. These potentials can be of a relativistic or nonrelativistic kind and are used directly in mean-field calculations, such as relativistic mean field theory or the Hartree-Fock (HF) approximation. In the present thesis we will focus our study on infinite nuclear matter, analyzing the properties of the two systems which stand at the extremes in terms of isospin asymmetry: symmetric nuclear matter (SNM), with an equal number of neutrons and protons, and pure neutron matter (PNM), with only neutrons. We will base our approach on the use of realistic microscopic poten- tials, hence we will need to choose a many-body technique in order to treat the potential. We choose to perform our calculations in the framework of the self-consistent Green's function's (SCGF) theory. This approach is based on a diagrammatic expansion of the single-particle (SP) Green's function, or propagator. This method was devised to treat the correlated, i.e. beyond mean-field, behavior of strongly interacting systems, such as nuclear matter. Unfortunately, it is a well established fact that whatever microscopic two- nucleon forces (2NFs) are used in the many-body calculation, empirical satu- ration properties of nuclear matter fail to be reproduced. Saturation densities appear at high values, with energies which are too attractive, overbinding nu- clear matter. This discrepancy with empirical values bears some similarities to the case of light nuclei where, on the contrary, the 2NF based theory un- derbinds experimental data. The inclusion of three-nucleon forces (3NFs) has been the indispensable strategy to cure this deficiency. Hence, microscopic 3NFs have been mostly devised to provide attraction in light finite systems, small densities, and repulsion in infinite systems, high densities. However, phenomenological ingredients have often been chosen to model these 3NFs. To bypass the need to adjust the potential with ad hoc contributions, an alternative has been provided by chiral effective effective field theory (χEFT). Featuring the consistent Hamiltonian at the 2B and 3B level provided by χEFT, in this thesis we will present calculations including both 2NFs and 3NFs. This will be dealt within an extended SCGF formalism, specifically formulated to include consistently 3B forces (3BFs). In the next section we present an overview of the different many-body ap- proaches which have been used to solve the infinite nuclear matter many-body problem. These approaches can treat the particles either relativistically or non-relativistically. The non-relativistic assumption seems reasonable, given that the energies involved in nuclear matter provide Fermi velocities which are less than one third the speed of light. All these approaches try to deal with the strong short-range force which acts in between nucleons. We will then focus, in the second section of this introduction, on the variety of microscopic potentials which, through the decades, have been devised to describe this kind of interactions. Finally, we will outline the general program of the thesis. 3 1.1. The nuclear many-body problem 1.1 The nuclear many-body problem In theoretical nuclear physics, the goal to describe from an ab initio point of view the basic properties of nuclear systems in terms of realistic interactions, has pushed the development of a variety of many-body methods. The quest to solve the nuclear many-body problem from first principles, i.e. starting from the Schr¨odinger equation, is appealing and very challenging. In the specific case of infinite nuclear matter, this aim is confronted with the lack of exper- imental results needed to constrain the construction of the theory from its building blocks, i.e. the Hamiltonian. However, the infinite system of nucle- ons interacting via the strong force, in spite of being ideal, is characterized by well-known empirical properties. These properties can be obtained extrapo- lating experimental results from finite nuclei. In this sense, the semi-empirical mass formula of Bethe and Weizs¨acker (Weizs¨acker, 1935; Bethe & Bacher, 1936), provides a way to quantify these empirical properties: 2 2 2=3 Z (N Z) B(N; Z) = aV A aSA aC aA − + δ(A) : (1.1) − − A1=3 − A The formula models the nucleus as an incompressible quantum drop (Baym et al., 1971) formed by a number N of neutrons and Z of protons, with mass number A = N + Z. Each term in Eq. (1.1) captures a specific physical prop- erty of the nucleus. The first two contributions, the volume and surface terms, define respectively the binding energy in the interior and the correction due to the existence of the surface of the nucleus. The third contribution accounts for the Coulomb repulsion in between protons. The next term is known as symmetry energy, and shows how the nuclei which are isospin symmetric, i.e. N = Z, are energetically favored. The last term in Eq. (1.1) takes into ac- count pairing effects and depends on wether the nuclei are even or odd in number of nucleons. All coefficients in Eq. (1.1), as well as δ(A), are fitted to reproduce the properties of nuclei along the nuclear chart. Via experiments of electron scattering on nuclei, it has been tested that the charge distribution in the interior of heavy nuclei reaches a constant value. This charge density is proportional to the density inside the nucleus. The ex- trapolated value for very large mass number A defines the empirical saturation −3 density of nuclear matter, ρ0 = 0:16 fm . In this limit, A , the binding energy per nucleon of a symmetric nucleus, N = Z, can be! obtained 1 from the semi-empirical mass formula, Eq.
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