Precise calculation of the decay rate of false vacuum with multi-field bounce
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Springer
Received: August 12, 2020 Accepted: October 3, 2020 Published: November 4, 2020
So Chigusa,a Takeo Moroib and Yutaro Shojic a
KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan b Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan c Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya, Aichi 464-8602, Japan
E-mail: [email protected], [email protected], [email protected] Abstract: We study the decay rate of a false vacuum in gauge theory at the one-loop level. We pay particular attention to the case where the bounce consists of an arbitrary number of scalar fields. With a multi-field bounce, which has a curved trajectory in the field space, the mixing among the gauge fields and the scalar fields evolves along the path of the bounce in the field space and the one-loop calculation of the vacuum decay rate becomes complicated. We consider the one-loop contribution to the decay rate with an arbitrary choice of the gauge parameter, and obtain a gauge invariant expression of the vacuum decay rate. We also give proper treatments of gauge zero modes and renormalization. Keywords: Beyond Standard Model, Higgs Physics ArXiv ePrint: 2007.14124
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)006
JHEP11(2020)006
Precise calculation of the decay rate of false vacuum with multi-field bounce
Contents 1 Introduction
1 3 3 6 8 8 10
3 Decomposition of solutions 3.1 ` > 0 3.2 ` = 0
11 12 13
4 Functional determinants 4.1 ` > 0 4.2 ` = 0 4.3 Background gauge
13 13 18 20
5 Zero modes 5.1 General issues 5.2 Gauge zero modes 5.3 Translational zero modes
23 23 27 30
6 Semi-analytic expression of the decay rate 6.1 Contributions of FP ghosts and transverse modes 6.2 Contributions of (SLϕ) modes 6.3 Background gauge
32 32 33 34
7 Renormalization
35
8 Conclusions and discussion
37
A Evaluation of determinants A.1 Alternative fluctuation operators A.2 Evaluation of solutions for ` > 0 A.2.1 Behavior at infinity A.2.2 Translational zero modes A.2.3 Functional determinant (` > 1) A.2.4 Functional determinant (` = 1) A.3 Evaluation of solutions for ` = 0 A.3.1 Behavior at infinity
38 38 40 40 42 43 44 45 45
–i–
JHEP11(2020)006
2 Setup and formulation 2.1 Lagrangian and bounce 2.2 Fluctuation operators 2.2.1 FP ghosts 2.2.2 Gauge bosons and scalars 2.3 Prefactor and functional determinant
47 49
B Use of alternative fluctuation opeartors B.1 General discussion B.1.1 Setup B.1.2 Recursive formula B.1.3 Error evaluation formula B.2 Alternative fluctuation operators B.2.1 Extended fluctuation operators B.2.2 Linear approximation B.3 (SLϕ) modes B.4 (c¯ c) modes B.5 (ηλ) modes
50 50 50 51 51 54 54 56 58 59 59
C MS Counterterms
60
1
Introduction
The decay of a false vacuum has attracted theoretical and phenomenological interests in particle physics and cosmology. For example, in the standard model (SM) and models beyond the SM, there may exist a vacuum whose energy density
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