Electroweak Phase Transition in the Standard Model and its Inert Higgs Doublet Extension Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakult¨at der Universit¨at Bern vorgelegt von Manuel Meyer von Ohmstal LU Supervisor: Prof. Dr. Mikko Laine Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Universit¨at Bern Co-Supervisor: Dr. Germano Nardini Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Universit¨at Bern Electroweak Phase Transition in the Standard Model and its Inert Higgs Doublet Extension Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakult¨at der Universitat Bern vorgelegt von Manuel Meyer von Ohmstal LU Supervisor: Prof. Dr. Mikko Laine Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Universit¨at Bern Co-Supervisor: Dr. Germano Nardini Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Universit¨at Bern Von der Philosophisch-naturwissenschaftlichen Fakult¨at angenommen. Bern, 31.08.2017 Der Dekan: Prof. Dr. Gilberto Colangelo to Larissa May you live forever in our thoughts and hearts. Abstract The origin of the asymmetry between matter and anti-matter is an interest- ing but yet unresolved problem of modern physics. To realize a net production of more baryons than anti-baryons a system has to depart from thermal equi- librium. A first order electroweak phase transition in the evolution of the early universe could account for a stage of non-equilibrium. This thesis is meant to examine this transition within the Standard Model and its inert Higgs doublet extension. The main parts of this thesis are chapters 2 and 3 which are based on our papers [1] and [2] respectively. Even though there is no electroweak phase transition within the Standard Model, its thermodynamics across the crossover shows interesting features at temperatures around T 160 GeV. Although the system does not leave thermal equilibrium its dynamics∼ deviate from ideal gas thermodynamics. The Standard Model, lacking a first order phase transition, cannot account for the baryon asymmetry of the universe but its background can still have an impact on non- equilibrium physics taking place at temperatures around T 160 GeV. We study the relevant thermodynamical functions across the electroweak∼ crossover in a perturbative three-loop computation and by making use of results from lattice simulations based on a dimensionally reduced effective theory. Many extensions of the Standard Model include an extended Higgs sector where scalar couplings are faced with conflicting requirements. Small couplings are needed to predict the measured dark matter relic abundance, whereas large couplings strengthen a first order phase transition. Large couplings, however, can compromise perturbative studies and spoil the high-temperature expansion needed for dimensionally reduced lattice simulations. Within the Inert Doublet Model we compute the resummed two-loop effective potential and we compare physical parameters related to the electroweak phase transition, e.g. the latent heat, the critical temperature and the discontinuity of the transition, with the high-temperature expansion. We also provide master integral functions for a model independent computation of a two-loop potential. Acknowledgements First and foremost I thank my supervisor, Mikko Laine, for his support, advise, guidance and fruitful discussions in the past years. It was an exceptional experience to learn and work with him and to benefit from his expertise and knowledge. I also thank Germano Nardini for his co-supervision of my thesis. I cherished all the discussions and talks we had and I am grateful that he always showed me where to put my finger on by asking the right questions. I thank Kimmo Kainulainen for being the co-referee of my thesis and the external expert on the defense committee. For interesting conversations and discussions about various topics in physics I thank my fellow graduate students Stephan Caspar and Jason Aebischer. I also enjoyed our talks and activities beyond physics. I thank my parents, Sepp and Vreni and my brother Philipp for their love and support throughout all these years. I thank my roommates B¨a, Marc, Stylo and Br¨onni for putting up with me at all times and for providing distractions from continuous calculations. Never forget the simple truth that more gratinity equals more happiness. Of all my friends I especially thank Paikea for her support during difficult times and good ones. I also thank Gion for his relentless teasing to get my Ph.D. and helping me improve my cooking skills. For discussions about life, the universe and everything and for the diversion from sitting in the office all day long I thank my dancing partner Niki. This thesis is dedicated to my best friend Larissa. You will forever be kept in our hearts as if you never left. Contents 1 Introduction 1 1.1 Baryogenesis ............................... 2 1.2 PhaseTransitions ............................ 4 1.3 ThermalFieldTheory.......................... 6 1.3.1 ThermalIntegrals ........................ 7 1.3.2 ThermalMasses ......................... 11 1.3.3 Resummation .......................... 12 2 Phase Transition in the SM 15 2.1 MasterEquation ............................. 16 2.2 Scale violation by Quantum Corrections . 19 2.3 HiggsCondensate ............................ 21 2.4 VacuumSubtraction........................... 28 2.5 Results................................... 30 2.6 Implications................................ 32 3 Phase Transition in the IDM 35 3.1 TheInertDoubletModel ........................ 36 3.2 TheTwo-LoopPotential ........................ 39 3.2.1 One-Loop Finite Temperature Integrals . 44 3.2.2 Two-Loop Finite Temperature Integrals . 46 3.3 NumericalEvaluation .......................... 51 3.4 Constraints ................................ 55 3.4.1 TheoreticalConstraints. 55 3.4.2 ExperimentalConstraints . 61 4 Conclusions 63 4.1 Outlook .................................. 64 A Master Integrals 67 A.1 VacuumIntegrals............................. 68 A.2 Overview and High-Temperature Expansions . 70 B Two-Loop Diagrams 75 C Pole Masses 79 C.1 Counterterms ............................... 79 C.2 Polemasses ................................ 80 Notations In this thesis we use natural units where the speed of light c, the Boltzmann con- stant k as well as the reduced Planck constant h are set to unity. Furthermore, B ̵ there are a couple of definitions and notations we list here. For estimations of a certain scale we use a generic coupling constant 2 2 2 2 2 g g1,g2,g3,λ,ht , ∈ { } where the couplings g1,g2 and g3 are the gauge couplings of the gauge groups U 1 ,SU 2 and SU 3 respectively. The Higgs scalar self coupling is λ and ht is( the) top( Yukawa) coupling.( ) We consider all other fermions to be massless. We use the MS renormalization scheme at the scaleµ ¯ and dimensional reg- ularization in D d 1 4 2ǫ dimensions. Integrals over four-dimensional momentum space= are+ depicted= − as ǫ eγE µ¯2 d4 2ǫk − , 4π 2π 4 2ǫ ≡ − K ( ) where γE is the Euler-Mascheroni constant. At finite temperature, we introduce Matsubara frequencies for the temporal component k0 of the momentum 2 2 2 K k0, k K k0 k . = ( ⃗) ⇒ = + Therefore, an integration over 4 2ǫ -dimensional momentum space is expressed via the sum-integral ( − ) ǫ eγE µ¯2 d3 2ǫk T , − . k k 4π 2π 3 2ǫ = k0 ≡ − K ( ) Introducing the inverse temperature β 1 T , the integration over space-time in position space is = β γ 2 ǫ e µ¯ − 3 2ǫ dτ , d − x . x x 4π X = 0 ≡ The Matsubara frequencies are sometimes denoted ωn corresponding to their respective mode for bosons ωn 2πnT or fermions ωn 2n 1 πT , with n Z. = k =2 ( 2+ ) ∈ This should not be confused with the energy ωi k mi . Propagators can then equivalently be written as = + 1 1 1 . 2 2 2 2 2 2 k 2 K mi = k0 k mi = ωn ωi + + + + ( ) Chapter 1 Introduction The universe and our physical description thereof is fascinating on all length scales, ranging from cosmological large scale structures like galaxy clusters to subatomic particles and their elementary constituents. In the very early uni- verse, those scales were close to each other but through the expansion of the universe they have diverged significantly. Objects at a cosmological scale are described by the theory of general relativity, whereas the theory of subatomic particles and their interactions is described by the Standard Model. The Standard Model of particle physics (SM) is a gauge theory that incorpo- rates the theory of electromagnetism as well as the weak and strong interactions. The quantum field theory of electromagnetism is described by quantum electro- dynamics (QED) through the gauge group U 1 and the gauge group of the weak interactions is SU 2 . The mediators of electroweak( ) gauge group SU 2 U 1 ( ) ( ) × ( ) are the massless photon and the massive W ± and Z bosons. Quantum chro- modynamics (QCD) is the theory of strong interactions with symmetry group SU 3 and its gauge fields are the gluons. The gauge group of the Standard Model( ) is then written as GSM SU 3 SU 2 U 1 . = ( ) × ( ) × ( ) The three gauge groups all have different couplings which depend on the energy scale. This dependence is also called the running of the couplings and it is described by beta-functions in renormalization group equations. If we run the couplings to a higher temperature scale, it is like going back- ward in time to the early universe. And if the couplings were to meet in one point, it could give rise to a unified description in a so called Grand Unified Theory (GUT). Unfortunately, within the Standard Model this is not the case. The introduction of supersymmetry potentially solves this unification but there are also other models with different higher gauge groups claiming to explain grand unification. The idea in all these models is a theory with more symme- tries that are then broken through multiple steps to ultimately reproduce the Standard Model symmetry group. One such symmetry breaking pattern, or restoration if we consider going from low to high temperatures, has been described a long time ago.