Precise calculation of the decay rate of false vacuum with multi-field bounce

Published for SISSA by 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: , , 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 Open Access, c The Authors. 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 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 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 1 Introduction 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 (cc̄) 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 is lower than that of the electroweak (EW) vacuum. If this is the case, the EW vacuum becomes a false vacuum and is not absolutely stable. Thus, the longevity of the EW vacuum often provides an important constraint on model parameters. In particular, assuming that the standard model is valid up to the Planck scale, the EW vacuum decays within a timescale shorter than the present cosmic age if the top-quark mass is too large or the Higgs mass is too small [1–12].1 In addition, the decay of the false vacuum is also important for the studies of phase transitions in cosmological history, which may be related to inflation or the baryon asymmetry of the Universe. Thus, the precise calculation of the decay rate of the false vacuum is of great importance. The calculation of a vacuum decay rate has been formulated in [27, 28], where the field configuration called the bounce plays a central role. The bounce is a saddle-point solution of the Euclidean equation of motion, which dominates the path integral for the decay process of the false vacuum. With the bounce configuration being given, the decay rate of a vacuum in unit volume is expressed as γ = Ae−B , (1.1) where B is the action of the bounce and A contains the effects of the quantum corrections to the action. At the one-loop level, A is obtained by evaluating the functional determinants 1 For discussion of the absolute stability of the EW vacuum in the SM, see [13–26]. –1– JHEP11(2020)006 A.3.2 Gauge zero modes A.3.3 Functional determinant 2 The numerical impact of the prefactor is demonstrated in [12, 29, 30]. When the true vacuum appears as a result of the renormalization group improvement of the potential, a careful treatment is required to avoid the double-counting of quantum corrections. If the properties of the bounce are well described by the (renormalizable) Lagrangian with choosing relevant renormalization scale, however, the functional-determinant method can give the one-loop contribution to the prefactor without the double counting; examples include the standard model (see [10–12]). 4 For other studies about the stability of the electroweak vacuum in models beyond the SM, see, for example, [35–53]. 3 –2– JHEP11(2020)006 of the fluctuation operators around the false vacuum and those around the bounce. For the precise determination of a vacuum decay rate, the calculation of A is necessary not only because it fixes the overall factor but also because it cancels out the renormalization scale dependence of B at the one-loop level [29].2 If scalar fields responsible for the bounce couple to the gauge fields, the fluctuation operator generally depends on the gauge-fixing parameter (which we call ξ) and ξ appears everywhere in the calculation of the prefactor A. On the other hand, the decay rate of the false vacuum should be independent of ξ because the effective action is gauge independent at its extrema [31, 32]. An explicit check of the gauge invariance at the one-loop level is quite formidable and the first calculation appeared only recently in [33, 34]. In these papers, a manifestly gauge-invariant expression of the decay rate has been obtained for the case where the bounce consists of a single field (single-field bounce). They also address another issue that arises when a gauge symmetry preserved in the false vacuum is broken by the bounce configuration. In such a case, there appears a flat direction of the action corresponding to the global part of the gauge symmetry; it can be seen as a gauge zero mode in the calculation of the functional determinant. Since the fluctuation toward such a flat direction cannot be treated with the saddle point method in the path integral, we need special treatment; a correct prescription for the gauge zero mode has been developed for the single-field bounce [34]. The prescriptions to calculate the decay rate of false vacuum given in [33, 34] are essential to perform a complete one-loop calculation of the (...truncated)