On the gauge invariance of the decay rate of false vacuum

Abstract

We study the gauge invariance of the decay rate of the false vacuum for the model in which the scalar field responsible for the false vacuum decay has gauge quantum number. In order to calculate the decay rate, one should integrate out the field fluctuations around the classical path connecting the false and true vacua (i.e., so-called bounce). Concentrating on the case where the gauge symmetry is broken in the false vacuum, we show a systematic way to perform such an integration and present a manifestly gauge-invariant formula of the decay rate of the false vacuum.where the boundary condition of fWe also introduce the function fThe explicit form of fis given bywhere I 2J+1 is the modified Bessel function. Then,For the contributions of the S-, L-, and ϕ-modes with J > 0, we need the functions Ψ and Ψ, which are regular at the origin, satisfyingHereafter, the boundary conditions for Ψ and Ψ at the origin are taken to be the same.With three independent solutions of the above equations (which we denote Ψ I and Ψ I , with I = 1, 2, and 3), the functional determinants of our interests are given bywhere D (S,L,ϕ) J (r) ≡ det(Ψ 1 (r) Ψ 2 (r) Ψ 3 (r)),D (S,L,ϕ) J (r) ≡ det( Ψ 1 (r) Ψ 2 (r) Ψ 3 (r)). (40) (FP) J and f (η) J (T ) J and f (T ) J satisfy (∆ J − g 2φ2 )f (T ) J = 0,

False vacuum decay in gauge theory

“…Recently, a gauge-invariant expression of the decay rate has been derived for a case where gauge symmetry is spontaneously broken in the false vacuum [6]. In [6], a gauge fixing function which reduces to the R ξ gauge around the false vacuum has been adopted.…”

“…In [6], a gauge fixing function which reduces to the R ξ gauge around the false vacuum has been adopted. (We call such a gauge as an R ξ -like gauge.)…”

“…(We call such a gauge as an R ξ -like gauge.) However the procedure proposed in [6] cannot be applied if gauge symmetry is preserved in the false vacuum. This is because, in such a case, there shows up a class of bounce configurations related by the internal symmetry, all of which contribute to the false vacuum decay.…”

“…1 We improve the analysis of [6] and present a prescription giving rise to a gauge-invariant expression of the decay rate, which is applicable to the symmetry-preserving false vacuum. We use the…”

“…It also satisfies 6) where the center of the bounce is set to be r = 0. We assume thatφ is a real function of r. The bounce configuration is given by (A µ , Φ) = (0,φ/ √ 2) when v = 0, while that for the case of v = 0 is not unique as we will explain later.…”