QUANTUM FIELD THEORY FOR THE EARLY UNIVERSE A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (PHD) IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DANIELE TERESI SCHOOL OF PHYSICS AND ASTRONOMY 2015 2 CONTENTS List of Figures7 List of Abbreviations 11 Abstract 13 Lay Abstract 15 Declaration 17 Copyright 19 List of Publications 21 Acknowledgements 23 1 Introduction 29 2 Equilibrium Thermal Field Theory 35 2.1 Quantum Field Theory on functional contours............... 35 2.1.1 Closed-time path formalism.................... 36 2.1.2 Thermal equilibrium: imaginary-time contour........... 37 2.1.3 Thermal equilibrium: real-time contour.............. 38 2.2 Imaginary-time formalism......................... 38 2.2.1 Analytic continuations....................... 40 2.3 Real-time formalism............................ 43 3 CONTENTS 3 Interacting Quantum Fields at Finite Temperature 49 3.1 Feynman rules................................ 49 3.1.1 Imaginary time........................... 49 3.1.2 Real time.............................. 50 3.2 Dyson equation............................... 51 3.2.1 Resummation in the real-time formalism............. 53 3.2.2 KMS condition........................... 54 3.2.3 Narrow-width approximation.................... 55 3.3 Thermal mass and width.......................... 56 3.3.1 Thermal mass............................ 56 3.3.2 Cancellation of pinch singularities................. 57 3.3.3 Thermal width........................... 59 3.4 Boltzmann equations............................ 63 4 Leptogenesis 67 4.1 Thermal leptogenesis............................ 68 4.2 Resonant enhancement of the asymmetry................. 71 4.3 Flavour-diagonal rate equations for RL................... 75 4.3.1 Observed lepton asymmetry.................... 81 5 Flavour-covariant Formalism 85 5.1 Flavour transformations........................... 86 5.2 Flavour covariant discrete symmetries................... 92 5.3 Markovian master equation......................... 97 5.4 Transport equations............................. 102 6 Flavour-covariant Rate Equations 105 6.1 Rate equations for decay and inverse decay................ 106 6.2 Lepton asymmetry via heavy-neutrino oscillations............. 113 6.3 Decoherence in the charged-lepton sector................. 117 4 CONTENTS 6.4 Scattering terms............................... 120 6.5 Final rate equations............................. 125 7 Thermal Quantum Field Theory out of Equilibrium 129 7.1 Cornwall–Jackiw–Tomboulis effective action............... 129 7.2 Kadanoff–Baym equations......................... 133 7.3 Closed-time path formalism in the interaction picture........... 137 8 Kadanoff–Baym Approach to Resonant Leptogenesis 143 8.1 Flavour-covariant scalar model of Resonant Leptogenesis......... 143 8.2 Quantum transport equations........................ 146 8.2.1 Heavy-neutrino rate equations................... 148 8.2.2 Lepton asymmetry source term................... 152 8.3 Flavour mixing and Kadanoff–Baym ansaetze............... 153 8.4 Asymmetry via heavy-neutrino mixing and oscillations.......... 159 9 Radiative Model of Resonant `-genesis 163 9.1 No-go theorem for minimal radiative RL.................. 164 9.2 Next-to-minimal model of resonant τ-genesis............... 165 9.3 Benchmark points.............................. 167 9.4 Analytic strong-washout solutions..................... 169 9.4.1 Analytic solution for the asymmetry via heavy-neutrino mixing and oscillations........................... 169 9.4.2 Semi-analytic results for the charged lepton decoherence effect.. 172 9.5 Numerical results for the lepton asymmetry................ 174 10 Conclusions 183 A Structure of the Real-Time Propagator 187 B Absorptive Self-Energy in the Scalar φ3 Theory 191 5 CONTENTS C Resummed Yukawa Couplings in the KB Approach 195 D LFV and LNV Observables 199 Final word count: 29152 6 LIST OF FIGURES 2.1 Closed-time path contour.......................... 37 2.2 Schwinger-Keldysh real-time contour.................... 39 4 3.1 Contributions to ∆11(k) up to 1-loop order in real-time φ theory. The crosses denote the mass counterterms.................... 56 3.2 1-loop corrections to the imaginary-time propagator in the φ3 theory. Counterterm graphs are not shown...................... 59 3.3 Imaginary part of the self-energy in the scalar φ3 model.......... 61 3.4 Imaginary part of the scalar φ3 self-energy in the low-temperature and small-mass regime.............................. 61 3.5 Thermal decay processes obtained by cutting the 1-loop self-energy in the φ3 model. Lines attached to a grey blob denote particles in the thermal bath. 62 4.1 The two types of CP -violation involved in the heavy Majorana neutrino decay N LΦ. The notation for the fields used here will be explained ! in Section 5.1................................. 70 6.1 Feynman diagrams for 2 1 inverse heavy-neutrino decay, in the pres- ! ence of a statistical background. The flavour indices are shown explicitly, while other indices are suppressed..................... 110 6.2 Feynman diagrams for the self-energies of the heavy neutrinos. The cut, across which positive energy flows from unshaded to shaded regions, is associated with production rates in the thermal plasma, as described by the generalized optical theorem. See also Figure 6.1............ 111 7 LIST OF FIGURES 6.3 Feynman diagrams for ∆L = 0 scattering [(a), (b)] and ∆L = 2 scat- tering [(c), (d)], in the presence of a statistical background. The flavour indices are shown explicitly, while other indices are suppressed...... 127 6.4 Feynman diagrams for the self-energies of the lepton doublets. The cut, across which positive energy flows from unshaded to shaded regions, is associated with production rates in the thermal plasma, as described by the generalized optical theorem. See also Figure 6.3............ 128 7.1 Perturbation-theory diagrams resummed by the Hartree-Fock self-energy on the LHS in the 2PI formalism. The dressed propagator is denoted by a thick line, whereas thin lines represent the tree-level one.......... 133 7.2 Some of the perturbation-theory diagrams resummed by the 1-loop self- energies on the LHS in the 2PI formalism. Notation as in Figure 7.1. ....................................... 133 7.3 The CTP contour for the interaction-picture formulation of non-equilibrium QFT, indicating the relationship between macroscopic and microscopic times t = t~ t~ ............................... 138 f − i 8.1 Diagrammatic representation of the truncation procedure. The heavy- neutrino equation is truncated spectrally at zeroth loop order, whereas the charged-lepton one is not. Both equations are truncated statistically at 1-loop CJT-resummed order......................... 151 8.2 Iterative solution to the Schwinger-Dyson equation for the dressed heavy- neutrino matrix Wightman propagator. Here, the double lines are fully dressed propagators, whereas the single lines are the propagators dressed with dispersive corrections only. Unshaded circles denote the relevant self-energies, whereas the shaded ones are the amputated self-energy cor- rections to the vertices, which can be identified at leading order with the resummed Yukawa couplings (see (8.30) and AppendixC)......... 155 8 LIST OF FIGURES 8.3 Diagrammatic representation of the source term for the charged-lepton asymmetry in terms of the resummed Yukawa couplings and the spectrally- free heavy-neutrino propagator....................... 158 8.4 The evolution of the total asymmetry (black continuous line), starting N N L from the initial conditions η = 2 ηeq12 and δη = 0. The red dot- ted line is the contribution of flavour mixing and the blue dashed line is that of oscillations. For illustrative purposes, the parameters are cho- 12 1 sen as follows: m = 1 TeV, (m m )=m = 10− , h = N N; 2 − N; 1 N 6 iδ 2 6 iδ 0:5 10− (1 + 5 e ) m and h = 0:5 i 10− (1 5 e ) m , with × N b ×b − N b 5 δ = 2 10− . For simplicity, the effect of thermal masses and widths is × b neglected................................... 161 9.1 The deviation of the heavy-neutrino number-densities ηN = ηN /ηN αβ αβ eq − δ from their equilibrium values for the three benchmark points given αβ b b in Table 9.1. The different lines show the evolution of the diagonal (solid lines) and off-diagonal (dashed lines) number densities in the fully flavour- N N covariant formalism. The numerical values of η22 and η33 coincide with each other in all three cases......................... 175 b b 9.2 Lepton flavour asymmetries as predicted by the BP1 parameters given in Table 9.1. The top panel shows the comparison between the total asymmetry obtained using the fully flavour-covariant formalism (thick solid lines, with different initial conditions) with those obtained using the flavour-diagonal formalism (dashed lines). Also shown (thin solid line) is the semi-analytic result (9.31). The bottom panel shows the diagonal (solid lines) and off-diagonal (dashed lines) elements of the total lepton number asymmetry matrix in the fully flavour-covariant formalism. For details, see the text.............................. 177 9.3 Lepton flavour asymmetries as predicted by the BP2 minimal RLτ model parameters given in Table 9.1. The labels are the same as in Figure 9.2.. 178 9 LIST OF FIGURES 9.4 Lepton flavour asymmetries as predicted by the BP3 minimal RLτ model parameters given in Table 9.1. The labels are the same as in Figure 9.2.. 179 A.1 Cutting equation for the propagator. Type-2 vertices are denoted by cir- cles, whereas