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The University of Adelaide Thesis submitted for the degree of Doctor of Philosophy in Theoretical Physics Department of physics and Special research centre for the subatomic structure of matter Hadrons and Quarks in Dense Matter: From Nuclear Matter to Neutron Stars Supervisors: Author: Prof. A. W. Thomas Daniel L. Whittenbury Prof. A. G. Williams Dr R. Young November 10, 2015 Contents List of Tables vii List of Figures ix Signed Statement xv Acknowledgements xvii Dedication xix Abstract xxi List of Publications xxiii 1 Introduction 1 2 From Nuclear Physics to QCD and Back Again 9 2.1 NN Interaction and Early Nuclear Physics . .9 2.2 Basic Notions of Quantum Chromodynamics . 16 2.2.1 QCD Lagrangian and its Symmetries . 17 2.2.2 Asymptotic Freedom and Perturbation Theory . 21 2.2.3 Chiral Effective Field Theory . 24 2.3 Nuclear Matter . 25 2.4 Experimental and Theoretical Knowledge of the Nuclear EoS . 30 2.5 Quantum Hadrodynamics and the Relativistic Mean Field Approximation 35 2.6 The Quark-Meson Coupling Model . 42 3 General Relativity and the Astrophysics of Neutron Stars 47 3.1 Hydrostatic Equilibrium . 49 3.2 The Neutron Star EoS . 51 4 Hartree-Fock QMC Applied to Nuclear Matter 57 4.1 The Hartree-Fock QMC . 58 4.2 The Lagrangian Density . 59 4.3 Equations of Motion and the MFA . 61 4.4 The Hamiltonian Density . 62 4.5 Hartree-Fock Approximation . 64 4.6 The In-medium Dirac Equation . 72 iii 4.7 Detailed Evaluation of σ Meson Fock Term . 73 4.8 Final Expressions for the Fock Terms . 75 4.9 Model Parameters . 79 4.10 Nuclear Matter Properties . 82 4.11 Asymmetric Nuclear Matter . 85 4.12 Sensitivity to Parameter Variation . 89 4.13 Summary . 90 5 Hartree{Fock QMC Applied to Neutron Stars 93 5.1 Generalised Beta Equilibrium Matter and Neutron Stars . 93 5.2 Numerical Results and Discussions . 96 5.2.1 Comparison with Other Models . 102 5.3 Summary . 103 6 Quark Matter in the Nambu{Jona-Lasinio Model 107 6.1 A Brief Introduction to the NJL Model . 107 6.2 A Simple and Less Intuitive Explanation of the Mean Field Approximation110 6.3 A Fierz Invariant NJL Lagrangian Derived from the Colour Current Interaction . 111 6.4 A Better Explanation of the Mean Field Approximation Using the Path Integral Formalism . 113 6.5 Pion Phenomenology and the Fitting of the NJL Model Parameters . 123 6.6 At Finite Density . 127 6.7 Flavour Independent Vector Interaction . 129 6.8 Numerical Results and Discussion . 130 6.9 Summary . 148 7 Phase Transitions to Quark Matter 149 7.1 Hybrid Stars and the Phase Transition to Quark Matter . 149 7.2 Interpolation Construction . 153 7.3 Numerical Results and Discussion . 155 7.4 Summary . 166 8 Summary and Outlook 167 A Hadronic Matter Supplemental Material 173 A.1 Bag Model Mass Parametrisations . 173 A.2 Self-energy . 173 A.3 Subtraction of Contact Terms . 174 A.3.1 Sigma Meson . 174 A.3.2 Vector Meson (η = !/ρ) Vector-Vector Term . 174 A.3.3 Vector Meson (η = !/ρ) Vector-Tensor Term . 175 A.3.4 Vector Meson (η = !/ρ) Tensor-Tensor Term . 175 A.3.5 Pion Term . 176 B Quark Matter Supplemental Material 177 B.1 Supplementary Material on Path Integrals, Generating Functionals and the Stationary Phase Approximation . 177 B.1.1 The Generating Functional and Green's Functions in the Canon- ical Formalism . 177 B.1.2 The Path Integral Formalism . 179 B.1.3 Free Boson Theory . 180 B.1.4 Stationary and Saddle Point Approximation . 181 B.1.5 The Connected Generating Functional and the Effective Action 183 B.1.6 Connection to QFTs at Finite Temperature and Statistical Me- chanics . 185 B.1.7 Free Fermion Theory . 188 B.2 Symmetry Lie Group and Lie Algebra Conventions . 190 B.3 Fierz Transformations . 191 B.4 Useful Functional Formulas . 191 B.5 Useful Properties Involving Determinants . 192 B.6 Schwinger's Proper Time Regularisation . 193 B.7 Effective Potential Derivation . 193 B.7.1 Effective Potential in Vacuum . 193 B.7.2 Effective Potential at Finite Density . 194 B.7.3 Divergent Part of the Effective Potential . 198 B.8 The Gap Equation . 199 B.8.1 The Gap Equation in Vacuum . 199 B.8.2 The Gap Equation at Finite Density . 200 B.9 Pion Polarisation Function . 201 B.10 Pion-quark Coupling . 205 B.11 Pion Decay Constant . 205 B.12 Three Momentum Regularised NJL Model with t' Hooft Term . 206 Bibliography 209 List of Tables 2.2.1 Quark properties [91], masses are in units of GeV. 17 2.3.1 Typical parameter set for the Bethe-Weizacker formula [109] . 26 2.5.1 Two typical QHD parameter sets. Coupling constants are dimensionless except for κ which is given in MeV. The parameter sets are given using the convention that the scalar field is positive. See the above references for the details of the fitting procedure. 42 2.5.2 Masses and incompressibility for the two typical QHD parameter sets given in Table 2.5.1. The masses and incompressibility are given in MeV. 42 4.9.1 Relations between baryon magnetic moments and anomalous isoscalar B B and isovector magnetic moments κ(IS;IV ) =: κ(!,ρ) = fB(!,ρ)=gB(!,ρ) us- ing experimental magnetic moments [256]. 80 4.13.1 Couplings, nuclear matter properties and selected hyperon optical po- tentials determined for our standard case (for which Λ = 0:9 GeV, and free RN = 1:0 fm) and the effect of subsequent variations in which dif- ferences from the standard parameter set are indicated in column 1. The tabulated quantities at saturation are the slope and curvature of the symmetry energy, L0 and Ksym, the incompressibility K0, skewness coefficient Q0 and the volume component of isospin incompressibility Kτ;v, respectively. 92 5.3.1 Selected nuclear matter properties, hyperon optical potentials and neu- tron star properties determined for our standard case (for which Λ = free 0:9 GeV, and RN = 1:0 fm) and the effect of subsequent variations in which differences from the standard parameter set are indicated in column 1. The tabulated quantities at saturation are the incompress- ibility K0, the slope of the symmetry energy, L0, and hyperon optical potentials, respectively. Tabulated neutron star quantities are the stel- lar radius, maximum stellar mass and corresponding central density −3 (units ρ0 = 0:16 fm ). .......................... 106 vii 6.5.1 We fitted all parameter sets using low energy hadron phenomenology. The proper time parameter sets used fπ = 93 MeV and mπ = 140 MeV. The three momentum regularised parameter set HK [282, 308] used fπ = 93 MeV, mπ = 138 MeV, mK = 495:7 MeV, mη0 = 957:5 MeV and m` = 5:5 MeV. An (∗) marks variables used in the fitting procedure. A dagger (y) indicates parameter sets used in other plots. 128 7.3.1 Hybrid star properties in the percolation picture. The crossover region is chosen to be (¯ρ, Γ) = (3ρ0; ρ0). An astericks (∗) indicates that a con- sistent EoS could not be constructed between that variation of hadronic and quark models with the chosen interpolation method. 164 7.3.2 Hybrid star properties in the percolation picture under variation of ρ¯ 2 f3ρ0; 4ρ0; 5ρ0g. The \Standard" or baseline scenario is used for the hadronic model and the flavour dependent vector interaction is used in each of the quark models. An astericks (∗) indicates that a consistent EoS could not be constructed between that variation of hadronic and quark models with the chosen interpolation method. 165 2.1.1 Transformations upon Wick rotation and various definitions. 188 List of Figures 2.1.1 Division of configuration space into three regions. 12 1 1 2.1.2 Phase shifts versus energy for (a) S0 and (b) D2 partial waves. Data obtained through NN-Online [40] and INS DAC [41] on 7/2/2014. 13 2.2.1 Summary of symmetries and their breaking. 19 2.2.2 Running of couplings to one loop level in (a) QED and (b) QCD. 23 2.6.1 Schematic picture of the QMC Model. 44 3.1.1 Summary of typical estimates for the masses of main sequence star corpses and their progenitors [218]. 50 3.2.1 Schematic cross section of a neutron star, see for example Refs. [111, 189, 191]. Crust: nuclei, electrons and neutrons. Outer core: Nuclear liquid consisting of neutrons, protons, electrons and muons. Inner core: Content uncertain, commonly thought that hyperons, Bose condensates or quark matter could exist. 52 4.10.1 Pure neutron matter energy per particle as a function of density as obtained in the present work, in comparison with complete CEFT at N3LO order { for more details of the latter, see Ref. [260]. 83 4.10.2 Density dependence of pressure in PNM as predicted in BHF, DBHF, QuMoCa and CEFT with and without three-body forces. (a) With- out three-body forces. (b) With three-body forces. The QMC model prediction is shown in (b). For more details see the text and ref. [87]. 84 4.10.3 (a) Pressure in SNM as a function of density as predicted in QMC model. The shaded area is taken from Ref. [265]. (b) Pressure in PNM as a function of density as predicted in the QMC model. The upper and lower shaded areas correspond to two different estimates of the contribution of the symmetry pressure to the total pressure. For more detail see Ref. [265] . 86 4.11.1 (a) Symmetry energy, S, as a function of baryon number density, as calculated in this work.
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