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Aspects of False Vacuum Decay Technische Universität München Physik Department T70 Aspects of False Vacuum Decay Wenyuan Ai Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. Wilhelm Auwärter Prüfer der Dissertation: 1. Prof. Dr. Björn Garbrecht 2. Prof. Dr. Andreas Weiler Die Dissertation wurde am 22.03.2019 bei der Technischen Universität München ein- gereicht und durch die Fakultät für Physik am 02.04.2019 angenommen. Abstract False vacuum decay is the first-order phase transition of fundamental fields. Vacuum instability plays a very important role in particle physics and cosmology. Theoretically, any consistent theory beyond the Standard Model must have a lifetime of the electroweak vacuum longer than the age of the Universe. Phenomenologically, first-order cosmological phase transitions can be relevant for baryogenesis and gravitational wave production. In this thesis, we give a detailed study on several aspects of false vacuum decay, including correspondence between thermal and quantum transitions of vacuum in flat or curved spacetime, radiative corrections to false vacuum decay and, the real-time formalism of vacuum transitions. Zusammenfassung Falscher Vakuumzerfall ist ein Phasenübergang erster Ordnung fundamentaler Felder. Vakuuminstabilität spielt in der Teilchenphysik und Kosmologie eine sehr wichtige Rolle. Theoretisch muss für jede konsistente Theorie, die über das Standardmodell hinausgeht, die Lebensdauer des elektroschwachen Vakuums länger sein als das Alter des Univer- sums. Phänomenologisch können kosmologische Phasenübergänge erster Ordnung für die Baryogenese und die Produktion von Gravitationswellen relevant sein. In dieser Arbeit geben wir eine detaillierte Studie zu verschiedenen Aspekten des falschen Vakuumzer- falls, einschließlich der Korrespondenz zwischen thermischen und Quantenübergängen des Vakuums in flachen oder gekrümmten Raumzeiten, Strahlungskorrekturen zu falschem Vakuumzerfall und den Formalismus realer Zeiten für Vakuumübergänge. The bulk of this manuscript is based on the articles which were written during this Ph.D. thesis, a list of which is shown below: ? W. Y. Ai, B. Garbrecht and C. Tamarit, “Functional methods for false vacuum decay in real time,” arXiv:1905.04236 [hep-th]. ? W. Y. Ai, “Correspondence between Thermal and Quantum Vacuum Transitions around Horizons,” JHEP 1903 164 (2019) [arXiv:1812.06962 [hep-th]]. ? W. Y. Ai, B. Garbrecht and P. Millington, “Radiative effects on false vacuum decay in Higgs-Yukawa theory,” Phys. Rev. D 98, 076014 (2018) [arXiv:1807.03338 [hep-th]]. Notations • ZE[0] — Euclidean partition function without source • Z[β] — thermal partition function at temperature 1/β • µ, ν; ::: — taking values 0; :::; 3 or 1; :::; 4 in the Minkowski case or the Euclidean case, respectively. • i; j; ::: — taking values 1; 2; 3 denoting the spatial indices • ∆(4), @2 — four-dimensional Laplacian, both are interchangeably used in this thesis •r — three-dimensional derivative operator • B — mostly used as the bounce action, also as subscript for “bounce” • Γ — decay rate • Γ — effective action • , 0+ — infinitesimal positive number as used in the Feynman i-prescription • Except for appendicesA andD, we take ~ = c = kB = 1 throughout this thesis Contents 1 Introduction1 2 False Vacuum Decay in Flat Spacetime at Zero and Finite Temperature5 2.1 Quantum tunneling in quantum mechanics..................5 2.2 False vacuum decay at zero temperature................... 11 2.3 The bubble growth after the nucleation.................... 12 2.4 False vacuum decay at finite temperature.................. 14 2.4.1 Top-down................................ 15 2.4.2 Bottom-up............................... 17 3 Correspondence between Quantum and Thermal Vacuum Transitions 21 3.1 Unruh effect and Hawking radiation..................... 22 3.2 Vacuum transition in 1 + 1-dimensional spacetime............. 23 3.2.1 Quantum transitions in 1 + 1-dimensional spacetime for inertial ob- servers.................................. 24 3.2.2 Thermal transition in 1+1-dimensional spacetime for Rindler observers 27 3.3 False vacuum decay in Schwarzschild spacetime............... 29 3.4 A New Paradox from Black Holes?...................... 33 4 Radiative Effects on False Vacuum Decay I: Motivation and Formalism 37 4.1 Motivation.................................... 37 4.2 General formalism............................... 38 4.2.1 Prototypal Higgs-Yukawa model................... 38 4.2.2 Effective action............................. 39 4.2.3 One-loop corrections to the action.................. 40 4.2.4 Radiatively corrected decay rate................... 43 5 Radiative Effects on False Vacuum Decay II: Planar-Wall Limit 45 5.1 Green’s functions, functional determinants and bounce correction in the planar-wall approximation........................... 45 5.1.1 Green’s function and functional determinants in the planar-wall limit 45 5.1.2 One-loop correction to the bounce in the planar-wall limit..... 48 5.2 Renormalization................................ 49 5.2.1 Renormalization of the mass and the quartic coupling constant using the Coleman-Weinberg potential................... 50 5.2.2 Wave-function renormalization through adiabatic expansion of the Green’s functions............................ 51 5.2.3 Renormalized bounce, effective action and decay rate........ 54 5.3 Numerical studies................................ 54 5.3.1 Tadpoles and corrections to the bounce............... 56 5.3.2 Corrections to the action........................ 59 6 Real-time Picture of Quantum Tunneling I: Optical Theorem 65 6.1 Euclidean path integral revisited: Picard-Lefschetz theory......... 66 6.2 Optical theorem for unstable vacuum..................... 71 6.3 Minkowski path integral and complex bounce................ 74 7 Real-time Picture of Quantum Tunneling II: Flow Equations 77 7.1 Flow equations and flow eigenequations................... 77 7.2 Mapping flow eigenequations to ordinary eigenequations.......... 78 7.3 Analytic continuation of functional determinants.............. 81 7.3.1 Finite T and T ............................. 81 7.3.2 Taking the limit T ;T! 1 ....................... 83 7.4 The decay rate................................. 86 B 7.5 The physical meaning of the negative eigenvalue λ0 ............. 88 8 Conclusion and Perspectives 93 Acknowledgement 97 Appendix 99 A Functional Determinant 101 A.1 Gel’fand-Yaglom Method............................ 101 A.1.1 Gel’fand-Yaglom theorem....................... 101 A.1.2 Evaluating the ratio of the functional determinants......... 102 A.2 Green’s function method............................ 103 A.3 Gel’fand-Yaglom method vs. Green’s function method........... 104 B Fermionic Green’s Function 107 B.1 Angular-momentum recoupling........................ 107 B.2 Green’s function: hyperspherical problem.................. 111 B.3 Green’s function: planar problem....................... 112 C General Argument for the Analytic Continuation between Euclidean and Minkowski Functional Determinants 115 C.1 Analytic continuation of eigenfunctions and eigenvalues........... 115 C.2 Orthonormal property and the completeness of the analytically continued eigenfunctions.................................. 118 C.3 Analytic continuation of the functional determinants............ 120 C.4 Examples.................................... 121 D Decay Rate from the WKB Method 127 Bibliography 131 Chapter 1 Introduction It is remarkable that the abundant world that we live in can be described and explained at the very fundamental level by dozens of fundamental particles and four fundamental interactions between them: gravity, electromagnetic, weak and strong interactions. All the known particles that constitute the matter and interactions except for gravity can be very well described by the standard model (SM) of particle physics [1]. Gravity is described by general relativity (GR) [2]. It turns out that all the particles are excitations of some more fundamental objects that we call quantum fields. False vacuum decay is the first-order phase transition of such fundamental quantum fields. One surprising feature in the SM is that the electromagnetic interaction and the weak interaction are unified and they appear to be different aspects of this unified theory only after the electroweak symmetry breaking. Such an electroweak sector in the SM provides a crucial phenomenological motivation for the study of false vacuum decay. In the SM, the Higgs doublet is responsible for all the masses of other particles via the mechanism of spontaneous symmetry breaking (SSB) where the Higgs field obtains a non-vanishing vacuum expectation value (VEV). The non-vanishing VEV of the Higgs field generates masses for other particles through the Yukawa interactions. To illustrate SSB, let us write down the Lagrangian for the Higgs doublet µν LHiggs = η (@µφ)@νφ − V (φ); (1.1) where the Minkowski metric is ηµν = diag(1; −1; −1; −1) and V (φ) = −µ2φyφ + λ(φyφ)2: (1.2) Here φ is the Higgs doublet, in components φ φ = 1 : (1.3) φ2 For λ > 0, µ2 > 0, the potential has an infinite set of degenerate minima satisfying µ2 v2 φyφ = ≡ : 2λ 2 1 The degeneracy of the minima reflects the global SU(2)L×U(1)Y symmetry of the theory. The vacuum picks a particular point in the degenerate minima which we take as 1 0 h0jφj0i = p : (1.4) 2 v 1It must be gauged in order to introduce the electroweak gauge fields.
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