Dissertation submitted to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences Put forward by Tibor Frossard born in Courgenay, Switzerland Oral examination: 19 June 2013 First principles approach to leptogenesis referees: Prof. Dr. Manfred Lindner Prof. Dr. J¨urgenBerges Von fundamentalen Prinzipien zu Leptogenese Baryogenese durch Leptogenese liefert eine elegante Erkl¨arungf¨urden Ursprung der Baryone- nasymmetrie des Universums durch den CP-verletzenden Zerfall von schweren rechtsh¨andigen Neutrinos (im fr¨uhenUniversum). Die nahe Verbindung zu Neutrinophysik hat in den letzten Jahrzehnten sehr zur Popularit¨atdieses Szenarios beigetragen. Eine pr¨aziseBerechnung der Baryonenasymmetrie ist schwierig, da sie es erfordert, der Entwicklung des warmen fr¨uhen Universums im thermischen Ungleichgewicht zu folgen. In dieser Arbeit diskutieren wir Lep- togenese mit Methoden der Nichtgleichgewichts-Quantenfeldtheorie. Diese Methode ist frei von zahlreichen Problemen der konventionellen Herangehensweise, welche auf der klassischen Boltzmann-Gleichung basiert. Wir leiten eine quanten-korrigierte Boltzmann-Gleichung f¨urdie Asymmetrie direkt aus den Grundprinzipien her. Die erhaltende Gleichung ist frei von dem Doppelz¨ahlungsproblem und beinhaltet konsistent die thermischen Korrekturen zu den Eigen- schaften der Quasiteilchen, insbesondere deren thermische Masse und Breite. Effekte durch die begrenzte Breite werden durch eine modifizierte Quasiteilchenn¨aherung ber¨ucksichtigt. Wir vergleichen numerisch die Ergebnisse dieses Nichtgleichgewichts-Quantenfeldtheorie-Ansatzes mit den konventionellen Methoden, und finden, dass die thermischen Effekte teilweise durch die thermischen Massen kompensiert werden. First principles approach to leptogenesis Baryogenesis via leptogenesis offers an elegant explanation of the origin of the baryon asym- metry of the universe by means of the CP-violating decay of heavy right-handed neutrinos in the early universe. This scenario has become very popular over the past decades due to its connection with neutrino physics. A precise computation of the baryon asymmetry pro- duced in the leptogenesis scenario is difficult since it requires to follow the out-of-equilibrium evolution of the hot early universe. We present here a nonequilibrium quantum field theory approach to leptogenesis. This method is free of many problems inherent to the conventional approach based on the classical Boltzmann equation. Starting from first principles we derive a quantum-corrected Boltzmann equation for the asymmetry. The obtained equation is free of the double counting problem and incorporates consistently thermal corrections to the quasi- particles properties, in particular thermal masses and thermal widths. Finite width effects are taken into account through the extended quasiparticle approximation. We compare nu- merically the reaction densities obtained from the conventional and nonequilibrium quantum field theory approaches. We find that the enhancement due to thermal effects is partially compensated by the suppression due to thermal masses. Contents 1 Introduction 7 1.1 Overview . .7 1.2 Baryon asymmetry of the universe . .9 1.2.1 Baryon asymmetry of the universe . .9 1.2.2 Type-I seesaw Lagrangian . 10 1.2.3 Production of baryon asymmetry in the leptogenesis scenario . 12 2 Boltzmann approach to leptogenesis 15 2.1 Boltzmann equation . 15 2.2 Boltzmann equation at O(h4)............................ 17 2.3 Higgs mediated processes . 23 3 Nonequilibrium quantum field theory 27 3.1 Kadanoff-Baym equations . 27 3.2 Quantum kinetic equation . 33 3.3 Boltzmann limit . 41 3.4 Extended quasiparticle ansatz . 44 4 Nonequilibrium approach to leptogenesis 49 4.1 Lepton asymmetry . 49 4.2 Heavy neutrino decay . 53 4.2.1 Tree-level contribution . 53 4.2.2 Equilibrium solution for the heavy neutrino propagator . 55 4.2.3 Self-energy contribution . 57 4.2.4 Vertex contribution . 59 4.3 j∆Lj = 2 scattering processes . 62 4.4 Higgs decay . 67 4.4.1 Tree-level amplitude . 68 4.4.2 Self-energy contribution . 69 4.4.3 Vertex contribution . 70 4.5 Higgs mediated scattering processes . 71 4.5.1 Tree-level amplitude . 73 4.5.2 Self-energy contribution . 75 4.5.3 Vertex contribution . 76 4.6 Heavy neutrino number density . 78 5 Comparison of the Boltzmann and nonequilibrium approaches 81 5.1 Rate equations . 81 5.2 Numerical comparison . 88 5.2.1 Heavy neutrino decay . 90 5 5.2.2 j∆Lj = 2 scattering processes . 91 5.2.3 Higgs decay . 95 5.2.4 Higgs mediated processes . 98 5.2.5 Heavy neutrino decay versus Higgs mediated processes . 104 6 Conclusion 109 Acknowledgements 113 A Notation 117 B Kinematics 119 B.1 Two-body decay . 119 B.2 Two-body Scattering . 120 B.3 Three-body decay . 121 C Properties of the propagators 123 C.1 Symmetry properties in coordinate representation . 123 C.2 Symmetry properties in Wigner representation . 124 C.3 CP-conjugated propagators in coordinate representations . 124 C.4 CP-conjugated propagators in Wigner representations . 125 D Wigner transform of a convolution product 127 D.1 Properties of the diamond operator . 128 E 2PI effective action and self-energies 129 E.1 2PI effective action . 129 E.2 Lepton self-energy . 129 E.3 Right-handed neutrino self-energy . 132 E.4 Higgs self-energy . 136 6 Chapter 1 Introduction 1.1 Overview At a fundamental level the vast majority of physical phenomena can be explained by the two basic theories of modern physics, the Standard Model (SM) of particle physics [1{3] and gen- eral relativity (GR). These theories have successfully passed numerous experimental tests over the past decades, the most recent one being the discovery at the Large Hadron Collider (LHC) of a Higgs boson [4{6] consistent with the SM predictions. Complementary to man-made experiments, the primordial universe constitutes a unique laboratory to test our current un- derstanding of the laws of nature at energies that cannot be reached by existing or forthcoming facilities. When extrapolating the SM with the concordance model of cosmology (ΛCDM) back in the hot early universe one arrives at the conclusion that matter and antimatter would com- pletely annihilate into radiation if they were present in exactly equal amount. An asymmetry between matter and antimatter in the primordial universe is therefore needed to explain the baryon asymmetry in the present universe. The generation of this asymmetry, baryogenesis, is one of the greatest challenges of modern physics. It is now widely accepted that the SM alone cannot be responsible for the matter-antimatter imbalance of the universe. The asymmetry as initial condition for the universe is very unsatisfactory and disfavoured when one includes a period of inflation in the very early universe to explain the flatness and horizon problems observed in the cosmic microwave background (CMB). Any scenario of a dynamical asymmetry generation must satisfy the three Sakharov conditions [7]: (i) Baryon number nonconservation, (ii) C- and CP-violation, (iii) Departure from thermal equilibrium. Interestingly enough, these conditions are fullfilled within the SM. Baryon number is violated by the triangle anomaly. Non-perturbative baryon number violating processes, which are called sphalerons, involve nine left-handed quarks and three-left-handed leptons. They conserve B−L (B is the baryon number and L the lepton number) but violate any other linear combinations. At zero temperature their amplitudes are highly suppressed [8] but become large at high temperature [9]. The chiral nature of the SM violates C (C denotes charge conjugation) and the phases in the Cabibbo-Kobayashi-Maskawa (CKM) matrix violate CP (P is the parity). Finally, the electroweak phase transition provides the out-of-equilibrium dynamics required by the condition (iii). However, in the SM, the CP-violation is too small and the departure from equilibrium not strong enough to generate the observed baryon asymmetry. 7 Chapter 1 Introduction Over the past decades many theories beyond the SM have been proposed to generate the baryon asymmetry dynamically. Amongst the most studied ones are GUT baryogenesis [10{ 19], electroweak baryogenesis [20{22], Affleck-Dine mechanism [23, 24] and leptogenesis [25]. Because of its connection with neutrino physics, the mechanism of baryogenesis via leptoge- nesis has become very popular over the past few years. In this scenario an asymmetry is first produced in the lepton sector by the decay of heavy right-handed neutrinos, and then transferred to the baryon sector through the sphaleron processes. A precise computation of the baryon asymmetry in any theory of baryogenesis is very chal- lenging because of the third Sakharov condition (iii): one needs to track down the out-of- equilibrium evolution of the baryon number in the early universe. In leptogenesis the conven- tional way to tackle this problem is a semiclassical approach: classical Boltzmann equations with vacuum transition amplitudes computed in the S-matrix formalism are used to follow the number densities of the different particle species. It is however clear that this approach is questionable: baryon number generation is a pure quantum phenomenon which takes place in the hot early universe. Moreover, the Boltzmann equations suffer from the so-called double counting problem, which can be solved only in some limiting cases. Even though a Boltzmann treatment of leptogenesis is not well justified, it has been used extensively to analyse