Precise Calculation of the Decay Rate of False Vacuum with Multi-Field Bounce

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.


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][2][3][4][5][6][7][8][9][10][11][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 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 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]. If scalar fields responsible for the bounce couple to the gauge fields, the gauge invariance of the prefactor A is non-trivial because, in such a case, the fluctuation operator generally depends on the gauge-fixing parameter (which we call ξ). 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 [30,31]. In [32,33], a manifestly gauge-invariant expression of the decay rate has been obtained for the case where the bounce consists of a single field (singlefield bounce). In addition, in gauge theory, there exists another difficulty especially 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 [33]. The prescriptions to calculate the decay rate of false vacuum given in [32,33] are essential to perform a complete one-loop calculation of the decay rate of the EW vacuum in the SM [10][11][12], which update the previous result [1]. In addition, they are also applied to models beyond the SM [12,34]. #2 In this paper, we extend the results of [32,33] to the case where the bounce consists of more than one field (multi-field bounce). We give a prescription to obtain a gauge-invariant expression of the vacuum decay rate, adopting two different gauge-fixing conditions; one is the Fermi gauge and the other is the background gauge. In the Fermi gauge, the treatment of the gauge zero mode can be understood easily, but the numerical calculation becomes difficult due to a severe cancellation. On the other hand, in the background gauge, the treatment of the gauge zero mode is complicated, but the numerical calculation becomes easier because of better behavior of fluctuation operators. Thus, we give a prescription to convert the result in the background gauge to that in the Fermi gauge which is guaranteed to be gauge invariant.
This paper is organized as follows. In Section 2, we explain our basic setup for the calculation of the vacuum decay rate. We show that the vacuum decay rate (in particular, the prefactor A) can be expressed by using solutions of a set of differential equations. In Section 3, we provide a decomposition of the solutions. In Section 4, we construct a set of solutions and calculate their asymptotic behavior, which is needed for the evaluation of A. In Section 5, we provide a general method to treat the zero modes and apply it to the zero modes in association with the gauge and the translational symmetries. In Section 6, we summarize the analytic results. In Section 7, issues related to the renormalization are discussed. Section 8 is devoted to conclusions and discussion.

Lagrangian and bounce
We consider a Euclidean four-dimensional gauge theory with a direct-product gauge group, G. We concentrate on the contributions from scalar fields and gauge fields; the effects of fermions, if they exist, can be taken into account separately. The Lagrangian is given by where V is a scalar potential, while L (GF) and L (ghost) include the gauge fixing terms and the terms containing the ghosts, respectively, which will be defined later. In addition, F a µν is the field strength of the gauge field A a µ (with a being the adjoint index of G), while φ i (with i = 1 − n φ being the index distinguishing scalars) are real scalar fields. #3 Here and hereafter, the summation over the repeated scalar indices is implicit. The covariant derivative of φ is #2 For other studies about the stability of the electroweak vacuum in models beyond the SM, see, for example, [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. #3 In our convention, complex scalar fields are understood to be decomposed into pairs of real scalar fields.
defined as In the real basis we are working with, the generators satisfy 3) where the superscript "T " denotes transpose and f abc is the structure constant. The gauge coupling constants, denoted as g a , can be different for different subgroups of G. We assume that the scalar potential, V (φ), has two minima, i.e., the false vacuum and the true vacuum (the false vacuum has higher potential energy than the true vacuum).
For the process of the false vacuum decay, the path integral is dominated by the field configuration called "bounce," an O(4) symmetric saddle-point solution of the Euclidean equations of motion [52,53]. Due to the gauge symmetry, there exist an infinite number of solutions. We adopt one solution with A µ = 0. Then, the bounce configuration, which we denote asφ(r), satisfies with the following boundary conditions: where v i denotes the scalar amplitude at the false vacuum. Here, r ≡ √ x µ x µ is the radius from the center of the bounce, and ∂ r denotes the derivative with respect to r.
The bounce solution has the following properties. First, since the potential is symmetric under the infinitesimal gauge transformation φ → φ + δθ a T a φ with an arbitrary choice of transformation parameters δθ a , ∂V ∂φ i T a ij φ j = 0. (2.8) Differentiating it with respect to φ k , we obtain Then, from Eq. (2.5), where Ω ij = ∂ 2 V ∂φ i ∂φ j φ→φ . (2.11) 3 From Eq. (2.10), we obtain , Ω]φ = 0, (2.12) because Ω is gauge invariant. It implies that (∂ rφ ) T T aφ should be proportional to r −3 or zero. Since it should vanish at the origin, For the later convenience, we define M ia (r) = −g a T a ikφ k (r), (2.14) which satisfies Using Eq. (2.13), the following relation holds: 16) where M ′ = ∂ r M . Notice that the gauge boson mass matrix in the false vacuum is given by M T M (r → ∞).
In our analysis, we concentrate on the case where the following conditions hold: • The rank of the matrix M T M (r) is unchanged for r < ∞. (At the false vacuum, some of the broken gauge symmetries may be recovered so that the ranks of M T M (r < ∞) and M T M (∞) may be different.) • There are no zero modes except for the gauge zero modes and the translational zero modes.
• At a large r, M T M (r) approaches to M T M (∞) exponentially.
The second condition is just for simplicity and our results can be extended to the cases with additional zero modes. The third condition is violated when the theory has the (approximate) scale invariance, which has already been discussed for the single-field case [10][11][12] and the multi-field case [34]. We define the subset of gauge fields that acquire masses from the bounce at r < ∞. Then, we define the following numbers: • n G : the number of the gauge bosons which acquire masses from the bounce at r < ∞.
• n B : the number of the gauge bosons which remain massive at the false vacuum.
• n U : the number of the gauge bosons which become massless at the false vacuum (but are massive at r < ∞).
• n ϕ : the total number of scalar fields.
• n H : the number of the scalar fields that do not mix with the gauge bosons at r < ∞.
Notice that n ϕ = n G + n H and n G = n B + n U . Now we introduce the gauge fixing terms. We consider the following two gauge fixing conditions: 1. Fermi gauge: These gauge fixing terms are consistent with the bounce of our choice since their contributions to the equation of motion vanish in the limit of A µ → 0 and φ →φ. They are also consistent with the spacial translational invariance; it corresponds to the fluctuation of δφ = δx µ ∂ µφ . As for the global gauge transformation, δφ = δθ a T aφ , the gauge fixing term of the Fermi gauge is invariant [33,54]. Then, in the Fermi gauge, a prescription to take care of the gauge zero modes is available. On the contrary, in the background gauge, the proper treatment of the gauge zero modes becomes more complicated, as we will discuss. The background gauge is, however, useful for numerical calculation especially with ξ = 1. Thus, we also discuss how the prefactors based on these two gauge fixing conditions are related in this paper.

Fluctuation operators
To evaluate the decay rate of the false vacuum, we integrate out the fluctuations around the bounce configuration. As is explicitly given in the next subsection, the prefactor is expressed with the functional determinant of the fluctuation operators, which are the second derivative of the action with respect to the fluctuations.
In discussing the contributions to the prefactor in our setup, we should consider the fluctuation operator for the bosons and that for the Faddeev-Popov (FP) ghosts. In the Fermi gauge, they are given by #4 and respectively. Note that M (Aµϕ) is an (n G + n ϕ ) × (n G + n ϕ ) object and M (cc) is n G × n G . In addition, in the background gauge, In order for the following discussion, let us define For later convenience, we choose the following basis of the gauge bosons and the scalars: where W andm 2 are full rank diagonal matrices. #5 Such a choice always exists because of Eq. (2.27). In this basis, we also define submatrices of M as (2.30) #4 We do not put the subscript "Fermi" on the fluctuation operators in the Fermi gauge for notational simplicity while we put the subscript "BG" on those in the background gauge. #5 We consider the case where there is no massless physical scalar in the false vacuum, and hencem 2 does not have zero eigenvalues.

6
In addition, we define The fluctuation around the bounce can be expanded using the hyperspherical functions on S 3 , which are functions ofx µ ≡ x µ r. The rotational Lie algebra of the four dimensional Euclidean space is equivalent to SU (2) L ×SU (2) R and the hyperspherical functions, Y P (x µ ), are labeled by indices P = ( , m A , m B ). Here, = 0, 1, 2, ⋯ is the azimuthal quantum number and m A and m B are the magnetic quantum numbers that take values of − 2, 2+1, ⋯, 2. #6 Using mode functions that depend only on r and the hyperspherical functions, we expand fields around the bounce as where the summation over P is implicit. Here, V (1) ν and V (2) ν are arbitrary two independent vectors and Note that (L) and (T ) modes do not have = 0 mode. For the following discussion, it is convenient to define the derivative operator that corresponds to the Laplacian acting on the modes with the azimuthal quantum number : Similarly, we define the fluctuation operators at the false vacuum, which can be obtained by replacement M → M and Ω →Ω. We denote them as M (Aµϕ) and M (cc) .
In the following, we will show the hyperspherical expansion of the fluctuation operators around the bounce and around the false vacuum.

FP ghosts
The fluctuations of the FP ghosts can be expanded by using the hyperspherical functions. Correspondingly, the fluctuation operator for the FP ghosts can be block-diagonalized as (2.39) #6 The variable J used in [32,33] and are related as = 2J.

7
where the power comes from ( + 1) 2 different choices of (m A , m B ) and the complexity of the FP ghosts. In the Fermi gauge, Notice that the fluctuation operators for the ghosts are n G × n G objects. At the false vacuum, we have a similar block-diagonalization: where, in the Fermi gauge,

Gauge bosons and scalars
Due to the mixing between the gauge bosons and the scalars, we cannot discuss their effects separately. Since only the mode functions with the same P = ( , m A , m B ) mix, the fluctuation operator M (Aµϕ) can be block-diagonalized as Here, independently of the choice of the gauge fixing, the fluctuation operator for the (T ) modes is given by Meanwhile, the fluctuation operators of (S), (L), and (ϕ) modes depend on the gauge fixing.
In the Fermi gauge, At the false vacuum, similar block-diagonalizations hold. For the (T ) mode, The fluctuation operator for (SLϕ) modes at the false vacuum can be further blockdiagonalized thanks to the choice of the basis of Eqs. (2.28) and (2.29). For > 0, the fluctuation operator can be expressed as corresponding to the contributions from massive gauge bosons and Nambu-Goldstone (NG) bosons (due to broken gauge symmetry), massless gauge bosons (in association with unbroken gauge symmetry), and physical scalars. Firstly, contributions to M (B) are from (S) and (L) modes of gauge bosons corresponding to broken symmetries (a = 1, . . . , n B ) and the corresponding NG modes (i = 1, . . . , n B ). Then, M (B) is a 3n B × 3n B object and is given by Secondly, contributions to M (U ) are from massless gauge bosons in (S) and (L) modes. We obtain a 2n U × 2n U fluctuation operator as For the other massive scalars, which is an (n U + n H ) × (n U + n H ) object. Similarly, for = 0, we obtain

Prefactor and functional determinant
The prefactor A is expressed as where A (cc) denotes the contributions from the FP ghosts and A (Aµϕ) denotes those from the gauge bosons and the scalars. For the evaluation of the prefactor at the one-loop level, the following quantities are necessary [28]: (2.62) Here and hereafter, the "prime" is used for quantities after the proper subtraction of the zero modes if necessary. In particular, because of the translational invariance, the = 1 contribution inevitably contains the effects of translational zero modes [28]. In addition, if n U > 0, both M More details about the zero mode subtraction and the relation between A (Aµϕ) and A ′(Aµϕ) will be explained in Section 5.
To evaluate the ratio of two functional determinants, we adopt the method developed in [55][56][57][58][59]. Let M (X) and M (X) be n × n fluctuation operators. We first prepare n linearly independent n-dimensional functions ψ (X)(I) (r) andψ (X)(I) (r) (with I = 1 − n) that satisfy and are regular at r → 0. Then, we define n × n objects Ψ (X) (r) andΨ (X) (r) as with which the ratio of functional determinants can be evaluated as where r 0 and r ∞ are abbreviations of r → 0 and r → ∞, respectively. Thus, for the evaluation of the prefactor A, we need to understand the asymptotic behavior of Ψ (X) andΨ (X) (and hence those of det Ψ (X) (r) and detΨ (X) (r)) at r → 0 and r → ∞. In particular, the behavior of det Ψ (SLϕ) and det Ψ (Sϕ) 0 is non-trivial because of the mixing among (S), (L), and (ϕ) modes. In the following sections, we discuss how we can evaluate those quantities.

Decomposition of Solutions
In this section, we provide a decomposition of the set of solutions Ψ (SLϕ) and Ψ (Sϕ) 0 , generalizing the results of [32,33].

3.1
> 0 Let us consider the solutions of the following equation in the Fermi gauge: , there exist (2n G + n ϕ ) linearly independent solutions of Eq. (3.1).
Solutions of Eq. (3.1) can be decomposed by using four functions, χ, η, ζ, and λ: Here, the functions χ, η, ζ, and λ are dependent on the azimuthal quantum number ; the subscript " " is omitted from these functions for notational simplicity. The shape of χ, η, and ζ is n G × (2n G + n ϕ ), while that of λ is n ϕ × (2n G + n ϕ ); here, the (2n G + n ϕ ) columns are linearly independent and are distinguished by the boundary conditions at r = 0. The evolution of functions χ and ζ is governed by the following differential equations: while η and λ satisfy In addition, λ satisfies the following constraint: The above constraint is consistent with the evolution equations; one can derive ∆ (M T λ) = 0, so that Eq. (3.7) holds if it is satisfied at r = 0. So far, we have discussed in the Fermi gauge. We note here that the decomposition given in Eq.  The solutions can be decomposed as where χ, ζ, and λ for = 0 satisfy Eqs. Notice that now χ and ζ are n G × (n G + n ϕ ) objects and λ is n ϕ × (n G + n ϕ ); the columns correspond to (n G + n ϕ ) independent choices of the boundary conditions at r = 0. Thus, Ψ (Sϕ) 0 is an (n G + n ϕ ) × (n G + n ϕ ) object.

Functional Determinants
In this section, we study the behavior of the functional determinants of the fluctuation operators in the Fermi gauge. Equivalence to the results in the background gauge is also discussed.

> 0
Let us consider the case with > 0. For the study of the functional determinants, we should first understand the behavior of the solutions of Eq. (3.1). For this purpose, we define an r-dependent n ϕ × n H matrix V H , which is given by with u p (p = 1, ⋯, n H ) being zero eigenvectors of M T , i.e., The columns of the solution of Eq. (3.1) can be classified into the following three types: • Type 2: ζ (2) ≠ 0.

13
where the superscripts "(1)," "(2)," and "(3)" indicate solutions for the Type 1, 2, and 3, respectively. Note that the Type 1, 2, and 3 solutions include n G , n G , and n ϕ independent solutions, respectively. The full matrix of solutions is constructed as and similar for χ, λ, and ζ. For each type of solutions, the boundary conditions at r → 0 are imposed as follows.
The asymptotic behavior at r → ∞ can be understood by carefully observing the differential equation. We leave the precise discussion to Appendix A and, in this section, only show the results. Neglecting terms irrelevant for the calculation of the functional determinant of our interest, the columns of Ψ (SLϕ) (r → ∞) can be obtained by linear combinations of those of the following objects: 14) Here, I B and I U are the n B × n B and n U × n U unit matrices, respectively, whileη (B) andη (U ) are n G × n B and n G × n U objects both of which satisfy Among the solutions,η (B) corresponds to n B solutions that exponentially grow at r → ∞, whileη (U ) corresponds to n U solutions that behave as r −2 at r → ∞. In addition,λ is n ϕ ×n H , satisfying and Then, Ψ (SLϕ) (r → ∞) can be expressed as with τ (ηλ) being an n ϕ × n ϕ orthogonal matrix. In the above expression, elements irrelevant for the following discussion are neglected. In order to calculate the prefactor A, we should also consider the fluctuation operator around the false vacuum. In particular, we need to derive solutions of M (SLϕ) Ψ (SLϕ) = 0. For this purpose, we can use the fact that the fluctuation operator around the false vacuum can be block-diagonalized as Eq. (2.52); the blocks are for the fluctuations of massive gauge bosons and NG bosons, for massless gauge bosons, and for physical scalars. Thus, we can discuss their contributions separately. Similarly to the discussion above, we defineΨ (X) (X = B, U, σ) obeying M (X) Ψ (X) = 0, which describe independent solutions of the differential equation.

Note thatΨ
Effects of the fluctuations of massive gauge boson and NG bosons around the false vacuum are embedded inΨ (B) , which behaves aŝ Next, we consider the fluctuations of massless gauge bosons. For those, we can obtain the solutions in the following form: Furthermore, the solutions related to the physical scalars are given by an (n U +n H )×(n U +n H ) object,Ψ (σ) , whose evolution is governed by where I H is the n H × n H unit matrix. Then, detΨ (SLϕ) (r) can be calculated as By using det Ψ (SLϕ) and detΨ (SLϕ) , the functional determinant of our interest can be expressed as In order to evaluate the above quantity, we define which can be obtained by solving Eqs. (3.5) and (3.6) with taking the boundary conditions given in Eq. (4.8) or (4.9). Then, by using the following relation: where the subscript "∞" implies that the quantity should be evaluated at r = r ∞ ; one can check that r 0 dependence and r ∞ dependence cancel out after taking r 0 → 0 and r ∞ → ∞. It is important to notice that Eq. = 0, corresponding to the Type 1, 2, and 3 solutions, are given in the following form: • Type 1: (4.33) • Type 2: where we neglect terms that are irrelevant for the calculation of the functional determinant of our interest.
Then, we define We also defineΨ Using quantities defined above, the determinant is calculated as We obtain the following expression: Importantly, however, extra treatment is needed when there exists an unbroken gauge symmetry at the false vacuum. If it exists, there shows up a gauge zero mode, which makes Eq. (4.42) vanish. Indeed, it is easy to see the existence of zero modes explicitly. For = 0, the Type 1 solution given in Eq. vanishes if there exists an unbroken gauge symmetry at the false vacuum. In our setup, the number of gauge zero modes is n U .
If there exist zero modes, a naive calculation of the prefactor A makes it divergent. Such a divergence is an artifact arising from the flat directions of the Euclidean action, implying the break down of the saddle point method in the path integral. The proper treatments of the zero modes will be discussed in Section 5. 19

Background gauge
Although it is convenient to use the Fermi gauge to discuss the gauge invariance of the decay rate, the background gauge is useful in performing the numerical calculation of the decay rate. Here, we show that the calculations based on the Fermi and the background gauges give the same result for > 0. (For = 0, the zero mode subtraction is non-trivial in the background gauge. The treatments of the zero modes will be discussed in the next section.) In order to discuss the functional determinants in the background gauge, we first define the function Ψ (cc) ,BG which is n G × n G . It obeys the following differential equation: with the boundary condition In our convention, because M T M (r → ∞) reduces to the block-diagonal form (see Eq. (2.28)), the relevant part of Ψ (cc) ,BG for our analysis behaves as where f (cc) ,B and f (cc) ,U are n B ×n B and n U ×n U , respectively, while τ (cc) is an n G ×n G orthogonal matrix. Notice that f Similarly, we also define the functionΨ (cc) ,BG , obeying Because of the block-diagonal nature of M T M in our convention,Ψ ,BG can be expressed aŝ ,B is an n B × n B object, whilef ,BG , we should derive the solutions of the following differential equation: As in the case of the Fermi gauge, we can find a set of functions that are relevant for the determinant:ψ where terms irrelevant for our discussion are neglected. With these functions, Ψ ,BG (r → ∞) can be expressed as BG being an n ϕ × n ϕ orthogonal matrix. Meanwhile, the behavior of Ψ (SLϕ) ,BG around r = 0 is given by can be again obtained using the block-diagonalization given in Eq. (2.52). We definê Ψ ,BG = 0, which describe independent solutions of the differential equation. ForΨ For the others, we can findΨ ,BG (r) is given by and consequently, In the Fermi and the background gauges, the functional determinants of the fluctuation operators of the (SLϕ) modes differ from each other. In the calculation of the prefactor A, the difference is compensated by the contributions from the FP ghosts. Indeed, we can find The above relation guarantees the equivalence of the total functional determinants (on which the prefactor A depends) in the Fermi and the background gauges for ≥ 1.
In performing the calculation in the background gauge, we should also study the = 0 modes. The boundary condition for = 0 (for the study of M (Sϕ) 0,BG ) are also obtained as

Zero Modes
When the action has flat directions around the bounce, mode functions corresponding to those directions become zero modes of fluctuation operators. Here, we consider the zero modes in association with continuous symmetries of the theory. In particular, in the case of our interest, the translational and the gauge symmetries result in zero modes and their proper treatments are essential to calculate the prefactor, A. In this section, we discuss how we can deal with these zero modes.

General issues
Let us first discuss the treatment of zero modes in general. As we have mentioned in the previous section, the ratio of the functional determinants of n × n fluctuation operators, M and M, #7 is evaluated as where Ψ(r) andΨ(r) satisfy MΨ = 0 and MΨ = 0, respectively. Notice that Ψ andΨ are n × n matrices, whose columns are linearly independent solutions, i.e., with Mψ (I) = 0 and Mψ (I) = 0. When M has zero eigenvalues, det Ψ(r ∞ ) = 0 and some of ψ (I) (r ∞ )'s are not independent. Here, the number of the dependent columns of Ψ(r ∞ ) #7 Here, we omit the superscripts and the subscript for notational simplicity.
matches that of the zero modes, which we denote as n zero . In this subsection, we take a basis in which (In the following subsections, we may use a different convention.) The functional determinant is due to the use of the saddle point method in the path integral. To evaluate the path integral, we first expand the fields around the bounce configuration. Let us denote the fields as Φ; in the case of our interest, Φ contains the gauge and scalar fields, i.e., Φ ∋ (A µ , φ). Then, we expand Φ around the bounce configuration (which is denoted asΦ) as where c a 's are expansion coefficients and G a (x)'s denote the eigenfunctions of the fluctuation operator: with ω a 's being the corresponding eigenvalues. The eigenfunctions should satisfy G a (r ∞ ) = 0, and are normalized as #8 Then, the path integral is evaluated as By performing the Gaussian integrals, we obtain If there exist zero modes, some of ω a 's vanish and the above result diverges. In such a case, we cannot use the naive saddle point method to evaluate the path integral. When the zero mode is related to the symmetry of the theory, we may properly eliminate the zero modes and avoid the divergence of the transition amplitude mentioned above, as #8 The inner product is defined as discussed in [28] for the case of translational zero modes. Let us denote a generic symmetry transformation of the bounce configuration as where z A denotes the transformation parameter. With the symmetry transformation, the bounce action is invariant so that F A satisfies Thus, the following relation holds where I 0 denotes the set of indices for the zero modes, resulting in (5.14) Using the Jacobian to convert the variable c a to z A is found to be where det AB denotes the determinant of the matrix with the indices A and B. Based on the above argument, we reinterpret the path integral containing zero modes related to the symmetry as where Det ′ M denotes the minor determinant of M with its zero eigenvalues eliminated: With properly interpreting the integrals over z A 's, the divergence originating from the zero modes may be avoided.

25
The minor determinant, Det ′ M, can be calculated by regulating the fluctuation operator as where ν is a constant. With ν being small enough, non-zero eigenvalues of M are (almost) unchanged while the zero eigenvalues are lifted by ν. Consequently, (5.20) We can calculate Det M reg by using the procedure mentioned above. We can obtain a solution of M reg Ψ reg = 0, where Ψ reg is n × n and has n linearly independent columns: Then, Det M reg can be obtained by using Ψ reg . Because our purpose is to evaluate det Ψ reg (r ∞ ) up to O(ν nzero ), we only need to calculate the columns in association with the zero eigenvalues up to O(ν); for the other columns, we can take ψ (I) reg ≃ ψ (I) (I > n zero ). We can calculate ψ (I) reg related to the zero modes by treating ν as a perturbation. We introduce the functionψ (I) as where the superscript "(I ≤ n zero )" indicates the columns in association with the zero modes. Then,ψ (I≤nzero) should satisfy with ψ (I≤nzero) (r 0 ) = 0. Withψ (I≤nzero) being obtained by solving the above equation, we can take care of the zero modes as where withΨ being n × n function containingψ (I≤nzero) : Then, using the fact that we obtain The actual calculations of Det ′ M for the gauge and translational zero modes will be discussed in the following subsections. 26

Gauge zero modes
We first consider the gauge zero modes. As we have mentioned in the previous section, if a gauge symmetry, which is broken by the bounce, is restored at the false vacuum, there show up zero modes in = 0 fluctuation operators. In this subsection, we apply the discussion in the previous subsection to the gauge zero modes. The gauge zero modes can be given in the following form (see Eq. (4.33)): where U (A) 's are defined as Notice that ψ (A) is a column of Ψ The functionψ (A) can be decomposed aš As r becomes large,ζ (A) (r) becomes constant because M T M U (A) approaches to zero exponentially. Then, the relevant part ofψ (A) for the calculation of the prefactor A is given byψ where I U and X (A,B) U are n U × n U objects whose (A, B) elements are given by and withλ being the function introduced in Eq. (4.35), we can find The gauge zero modes are associated with the gauge symmetries that are restored at the false vacuum; the NG bosons in association with the global gauge transformations are not eaten by the gauge bosons and appear as the zero modes. Thus, the path integration over the gauge zero modes can be replaced by the integration over the gauge volume of the unbroken gauge symmetry at the false vacuum.
We parameterize such a global gauge transformation of the bounce configuration as whereT A 's are generators of unbroken gauge symmetry at the false vacuum withT Aφ being required to be orthogonal, i.e., Here, the generatorsT A are introduced to set a diagonal basis for the integration over the gauge volume, and are given by linear combinations of the generators of our original choice, i.e.,T Note that the ambiguity in κ is absorbed into the normalization of the gauge volume V U (see the following discussion).
Since the fluctuation operator in the Fermi gauge is invariant under the transformation given in Eq. (5.45), the path integral over the gauge zero modes can be interpreted as where c (gauge) a 's are the expansion coefficients in association with the gauge zero modes, V U is the volume of the moduli space arising from the spontaneous symmetry breaking, and (5.49) Then, using detΨ and hence the = 0 contribution, containing the path integral over the gauge zero modes, can be written as  .43)). The proper treatment of such zero modes is, however, complicated because the relation between the path integral over such zero modes and the integration over the gauge parameter is non-trivial in the background gauge. Here, we consider a way to reconstruct the result in the Fermi gauge from that in the background gauge. Even though the study in the Fermi gauge is enough for the gauge invariant formulation of the decay rate, the background gauge is advantageous for numerically calculating the decay rate. This is because the numerical calculation in the background gauge shows a better convergence at r → ∞. Thus, it is desirable to understand the procedure to transform the results in the background gauge to those in the Fermi gauge. We will explicitly show such a transformation in the following.
In discussing the treatment of the gauge zero modes in the background gauge, we first define Here, the regulator is chosen so that the final result becomes simple. Then, we can find (see Appendix A) In numerical calculations, Eq. (5.55) can be used to obtain the Fermi gauge result from the calculation with the fluctuation operators in the background gauge. Here, one should note that, even though our regulator for the background gauge becomes equal to that of Eq. (5.19) when ξ = 1, The above inequality is not surprising because the gauge transformations relating different bounce configurations differ in two choices of the gauge fixing. In particular, in the background gauge, different bounces are related by a local, not global, gauge transformation [33]. Thus, if we worked only in the background gauge, it would become very non-trivial to relate the path integral over the zero modes to the integration over the gauge volume. 30
We need to eliminate the zero eigenvalues from Det M can be calculated. The bounce is localized in space-time, and the shift of the position of the bounce does not change the action. Thus, in four space-time dimensions, there are four translational zero modes which can be parameterized as or as with N (tr) being the normalization constant: The path integral over the translational zero modes can be understood as the integration over the position of the center of the bounce as [28]: Consequently, the functional determinant from the path integral is interpreted as Integration over the space-time volume will disappear from the expression of the decay rate per unit volume (i.e., the transition probability per unit time and unit volume). We note that, in the background gauge, we can obtain Eq. (5.68) with the fluctuation operators in the Fermi gauge being replaced by those in the background gauge.

Semi-Analytic Expression of the Decay Rate
Now, we summarize the semi-analytic expression of the decay rate of the false vacuum. We are particularly interested in the expression of the prefactor A and its gauge invariance. The prefactor A can be given as a product of the contributions of various fluctuations as .) For A (cc) and A (SLϕ) , 0 ≤ < ∞ and for A (T ) , 1 ≤ < ∞. The final result A does not depend on the choice of the gauge fixing (as far as the gauge zero modes are irrelevant), although A (cc) and A (SLϕ) in the Fermi and the background gauges differ from each other. In the following, we first summarize the results in the Fermi gauge and then discuss those in the background gauge.

Contributions of FP ghosts and transverse modes
First, we consider the contributions of the FP ghosts. In the Fermi gauge, the FP ghosts do not couple to the bounce, and hence the fluctuation operator of FP ghosts does not contain φ. Consequently, With these quantities, Then, (6.9)

Contributions of (SLϕ) modes
Contributions of the (SLϕ) modes are complicated especially because of the zero modes. The prefactor originating from the (SLϕ) modes can be written as (6.10) For > 1, we do not expect zero modes. Based on Eq. (4.32), we obtain For = 1, the translational zero modes exist. The zero eigenvalues from the translational zero modes are removed by replacing η (tr) and λ (tr) byη (tr) andλ (tr) , respectively, in det Ψ (SLϕ) 1 (r → ∞), with taking account of the Jacobian (see Eq. (5.68)). As a result, A (SLϕ) 1 is obtained as Here, the definitions of ψ (tr) andψ (tr) are given in Eqs. ( where V U is the gauge volume and κ is the matrix orthogonalizing T Aφ (see Eq. (5.47)). In addition, Ψ (λ) 0 is given in Eq. (5.44).

Background gauge
Here, we summarize the relation between the results in the Fermi gauge and in the background gauge. In the background gauge, the FP contribution becomes The functions Ψ As we have seen in Eq. (4.61), the following relation holds for > 0: Thus, we can explicitly see that the Fermi and the background gauges give the same decay rate as far as the modes with > 0 are concerned. For = 0, the proper treatment of the gauge zero modes in the background gauge is nontrivial, as we have explained in Section 5. Thus, in the background gauge, we just introduce a convenient way of regulating the fluctuation operator with gauge zero modes and relate its functional determinant with that in the Fermi gauge. With the regulated fluctuation operators given in Eqs. (5.53) and (5.54), we can obtain with K AB being given in Eq. (5.56). The above equality is useful for numerical calculations: the right-hand side of Eq. (6.18) is written only with the quantities in the background gauge and hence we can calculate them numerically.

Renormalization
The functional determinants diverge once the contributions of all are taken into account, and the renormalization is necessary. Here, we discuss how the divergences are removed, adopting the MS scheme.
In calculating the one-loop contribution to the decay rate of the false vacuum, we are interested in the ratio of the functional determinants in the following form: where M (X) and M (X) are fluctuation operators around the bounce and that around the false vacuum, respectively. Our procedure of the renormalization is as follows [1]: (i) We first identify terms which contain the divergence in ln A (X) (which we call δ ln A (X) div ).
(ii) Next, we perform the MS subtraction of δ ln A (X) div . The result is denoted as δ ln A (X) div,MS . Then, the divergence is removed as [1] ln A (X) → ln A (X) Notice that, in the above expression, ln A (X) and δ ln A Before proceeding further, we comment on the choice of the gauge fixing which affects the calculation of Det M (SLϕ) . The calculation can be performed in any gauge; as we have discussed in Section 4, the Fermi and the background gauge calculations give the same result of Det M (SLϕ) for ≥ 1 and hence the calculation of the divergent part is possible in either gauge. In practice, however, it is convenient to work in the background gauge because the number of diagrams necessary to calculate the divergent part is reduced. This is because, in the background gauge, δM does not contain derivative operators; thus, only s ,I , (7.5) where the following equations are satisfied: with δΨ (X) ,I (r → 0) ≃ 0. Notice that δΨ (X) ,I can be understood as the term which is of the order of (δM) I . Then, ln A (X) becomes from which we obtain In order to perform the MS subtraction from the divergent part, we can use  in the square bracket should be understood as those given in Eqs. (7.9) and (7.10), respectively. Notice that the quantity in the square bracket can be evaluated order by order in . At a large , the contribution behaves as −2 and the sum over converges. One may truncate higher , or fit the terms with large enough and sum them up for better precision.

Conclusions and Discussion
In this paper, we have studied the decay rate of the false vacuum in gauge theory for the case with a multi-field bounce. If more than one scalar fields contribute to the bounce, the mixing among the gauge fields and the scalars becomes r-dependent, making it complicated to calculate the one-loop contributions to the decay rate. We have extended the results of [32,33] to perform a gauge invariant calculation of the decay rate at the one-loop level. The one-loop contributions to the decay rate, which are denoted as A in this article, are given by the ratio of the functional determinants of the fluctuation operators around the bounce and those around the false vacuum (see Eqs. (2.61) and (2.62)). Using the fact that the functional determinant of the fluctuation operator, M (X) , can be related to the asymptotic behavior of the function, ψ (X) , obeying M (X) ψ (X) = 0, we have derived an expression of the decay rate at the one-loop level, which is manifestly independent of the gauge parameter, ξ. In particular, we have discussed how the functional determinant of M (Aµϕ) , i.e. the fluctuation operator for the gauge and scalar fields, can be evaluated in a gauge invariant way. Our main results are summarized in Section 6.
In our study, we have worked in both the Fermi gauge and the background gauge. We have shown that the Fermi gauge has an advantage to show the gauge invariance of the result especially when there exist gauge zero modes. The path integral over the gauge zero modes should be replaced by the integration over the gauge volume; the rule of the replacement can be straightforwardly obtained in the Fermi gauge. The background gauge has, however, an advantage in numerically calculating the decay rate because of the better convergence. In order to utilize the calculation in the background gauge, we derived a prescription to translate the result in the background gauge to that in the Fermi gauge.
We have also discussed how we can remove ultraviolet divergences from the decay rate. We have shown a procedure to subtract the divergences from the one-loop results and to perform the renormalization in the MS scheme (see Section 7).
Our results apply to various types of models with many scalar fields having gauge charges; in such models, it is often the case that there shows up a true vacuum whose energy density is lower than that of the electroweak vacuum. Our results can be used to calculate the decay rate of the electroweak vacuum in such models. Phenomenological application of our results may be discussed elsewhere.

Acknowledgements
The work of SC is supported by JSPS KAKENHI grant. The work of TM is supported by JSPS KAKENHI grant Nos. 16H06490 and 18K03608. The work of YS is supported by JSPS KAKENHI grant No. 16H06492.

A.1 Alternative fluctuation operators
One of the important steps in our calculation is to rewrite det Ψ (SLϕ) using η and λ at r ∼ ∞. Since the fluctuation matrices approach those at the false vacuum as r → ∞, one may expect that the behavior of Ψ (SLϕ) at r ∼ ∞ is the same as that of the solution at the false vacuum, Ψ (SLϕ) . However, it is not always the case because they are related as where G is a square matrix. Since the elements ofΨ (SLϕ) generally become hierarchical as r → ∞, larger elements easily contaminate smaller elements due to the mixing induced by G. The crucial problem here is that the contamination cannot be avoided just by taking a linear combination of the solutions since G is r-dependent. Such contamination leads to a severe cancellation when we calculate the determinant of Ψ (SLϕ) . In the following, we discuss a way to avoid this problem.
We first define the projection operators as which satisfy P H + P G = I ϕ and P H P G = P G P H = 0. Then, we define r-dependent orthogonal matrices O ϕ (r) and O G (r) having the following properties: where m 2 , W , W B and W U are diagonal matrices. Since we work in the field basis given in Eqs. The basic idea is to use the following matrices instead of M and Ω.
Here, r * is taken to be much larger than the typical scale of the bounce. Then, the behavior of the solution at r → ∞ can be understood easily since the fluctuation operators become block-diagonalized. The behavior at r ∼ r * will be defined in Appendix B. The decomposed solution in Section 3 cannot be used as it is since M alt does not satisfy Eqs. (2.15) and (2.16) for r ≳ r * . However, we find that the solution becomes exact if we add extra scalar mass terms to the fluctuation operators as ,alt , which satisfy the same differential equations as Ψ (cc) and Ψ (ηλ) but with M → M alt and Ω → Ω alt . For later convenience, we also define The deformation at r ∼ r * should be smooth enough and the decomposition of the solutions should be well-defined for the entire region of r. In addition, we need to ensure the use of these alternative fluctuation operators does not affect the final results. We will discuss these issues in Appendix B.

A.2.1 Behavior at infinity
Here, we evaluate Ψ where T (cc) and T (ηλ) are constant square matrices. We examine the behavior of each type of the solutions at r → ∞ below.

A.2.2 Translational zero modes
Here, we discuss the translational zero modes. The following discussion applies to both the Fermi gauge and the background gauge. With the alternative fluctuation operator, M 1,alt , the existence of the translational zero modes is not guaranteed; det Ψ (SLϕ) 1,alt (r → ∞) can be non-vanishing. However, it does not cause a problem since it approaches zero as r * → ∞ and we always take ν → 0 after r * → ∞.
Let us start with the original fluctuation matrix (the one without "alt"). To construct the solution explicitly, we regularize the translational zero modes using we can use the same discussion for the zero modes as in Section 5. Then, we define its alternative as M ,alt (r → ∞) = det Ψ Keeping only the leading terms in r, we can interchange r → ∞ and r * → ∞ and det Ψ (SLϕ) (r → ∞) In the background gauge, and detΨ (SLϕ) Here, we have used W B (r → ∞) = W . Thus, and hence det Ψ From Eqs. (4.5), (4.7) and (4.10), independently of the choice of the gauge fixing. Taking the ratio between the determinants with r → 0 and r → ∞, we get Eq. (4.32).

45
Notice that E can be non-vanishing when the translational zero modes disappear in the alternative fluctuation operator. As r * → ∞, E should approach zero. Keeping only the leading terms in r, we can interchange r → ∞ and r * → ∞. Then, (A. 53) In the background gauge, det Ψ (A. 56) and hence Taking the ratio between the determinants with r → 0 and r → ∞, we obtain Eq. (5.68).

A.3.1 Behavior at infinity
The behavior of Ψ (A.60) In the background gauge, (A.61) • Type 2 There are some useful relations: where Y is given in Eq. (A.30) and Γ is a function obeying In the background gauge, ζ (2) = Ψ (cc) 0,alt , and The behavior of W U is given by ,BG,alt (r → ∞) ≃ in the background gauge. Here, K alt is the n U × n U matrix defined by Notice that its elements are finite and its determinant is non-vanishing.
• Type 3 We can take ζ

A.3.2 Gauge zero modes
Here, we discuss the gauge zero modes. Even with the alternative fluctuation operator, there appear gauge zero modes, which are given by with which we defineΨ The following relation also holds. (

B Use of Alternative Fluctuation Opeartors
In this appendix, we justify the use of the alternative fluctuation operators in the evaluation of the determinants at r → ∞.

B.1.1 Setup
We compare the determinants of the two functions, F and F alt , which satisfy −∂ 2 r − Λ(r)∂ r − δΛ(r)∂ r + Ξ(r) + δΞ(r) F(r) = 0, (B.1) where Λ, δΛ, Ξ, δΞ, F and F alt are n × n matrices. The support of δΛ(r) and δΞ(r) is r > r * and we take We assume that there exist constants, C F , C δΛ and C δΞ , such that where A is the induced (spectral) norm of the real matrix A, which is defined as where k > 2 is a non-integer constant and µ(A) is the logarithmic norm of matrix A, which is defined as Notice that µ(A) gives the largest eigenvalue of 1 2 (A + A T ). The goal of this subsection is to show that the following quantity has an upper bound: lim r→∞ ln det F(r) det F alt (r) .

B.1.2 Recursive formula
Let us construct a formal solution of Eq. (B.1) using F alt . We express F as where Θ is a function satisfying Here, the path-ordered exponential of matrix A(s) is defined as #10 P exp Treating δΛ and δΞ as perturbations, we expand Θ as where Θ (p) is the p-th order term with respect to fluctuations. Formally, we obtain This recursive formula can be solved order by order and we can obtain Θ if the sum over p converges.

B.1.3 Error evaluation formula
Let us first evaluate the right path-ordered exponential in Eq. (B.12). Using Eq. (B.5) as well as P exp s r dtF ′ alt (t)F −1 alt (t) y R (r) = y R (s), (B.14) the following relation holds: One may also write the path-ordered exponential as Then, where D − is the left-hand derivative. After integration, we obtain Similarly, we evaluate the left path-ordered exponential in Eq. (B.12). From Eq. (B.5), Then, we obtain where D + is the right-hand derivative, based on which we find Thus, the following inequality holds: where C (p) 's are positive constants. Indeed, it is the case for p = 1 as Next, let us assume Θ (q) (r) < C (q) r * r 2 for all q < p, and show that the inequality (B.26) holds. Indeed, where Thus, the inequality (B.26) is valid for all p. In Fig. 1, we show the contours of constant C tot , taking k = 2.5. As we can see from the figure, C tot is actually convergent for small enough C (1) and C δΛ r 2 * , and C tot approaches zero as C (1) and C δΛ r 2 * go to zero. Thus, if C tot converges, we obtain

B.2 Alternative fluctuation operators
Let us consider a one-parameter family of differential equations with different transition point, r * , and we denote the solution as F alt (r * ; r). Then, C δΞ , C δΛ and C F are dependent on r * . From the above discussion, sufficient conditions for In this subsection, we give several general remarks. Then, in the following subsections, for (SLϕ) modes, (cc) modes and (ηλ) modes, we construct F alt (r * ; r) explicitly and show that the above conditions as well as Eq. (B.6) are satisfied.

B.2.1 Extended fluctuation operators
Since the conditions (B.4) are sensitive to the behavior of the fluctuation operators around r ∼ r * , we need to specify their deformation explicitly.
In particular, the differential equations for Ψ where Ω sp is an n G × n G matrix, which will be determined later. We take the same Ω sp for the false vacuum. Thus, the spectator scalars do not affect the ratio of the determinants. We also introduce P ext G,alt and P ext H,alt , which are defined as where I ext ϕ is the (n ϕ + n G ) × (n ϕ + n G ) identity matrix. We assume that the deformation begins at r = r * = r 1 and ends at r = r 5 with r 5 − r 1 ≪ r * . We construct the alternative matrices, M ext alt and Ω ext alt , which satisfy M ext alt (r) = M ext (r) and Ω ext alt (r) = Ω ext (r) for r < r 1 . For r 1 < r < r 5 , we deform them in the following way. where ω expresses the elements for the spectator scalars and ω(r 1 ) = 0. Keeping the following relations: we deform M ext alt in the following steps. (a) r 1 < r < r 2 : Turn on ω keeping ω ∝ ω(r 2 ).

B.2.2 Linear approximation
To evaluate the effect of the deformation on the functional determinants, we construct each step of the deformation more concretely. Since the difference between the original matrices and the alternative matrices are expected to be very small, we can work in the linear approximation. As in the previous sections, we choose the basis of the fields so that Notice that the first condition is somewhat stronger than Eq. (A.6) since some of the elements of W −1 (r) diverge as r → ∞. The existence of such a field basis is guaranteed by the one-toone correspondence between the massive gauge bosons and the NG bosons.
Since O ϕ and O G are very close to the identity matrices for a large enough r * , we approximate them as where δ ϕ and δ G are constant anti-symmetric matrices and satisfy δ ϕ max < δ 2 max , δ G max < δ 2 max , (B.52) with ⋯ max being the max norm and δ max ≪ 1.
We also approximate that M (r) and Ω(r) are almost constant during the deformation and express them as Then, we construct the deformation as follows.
• r 1 < r < r 2 : We turn on ω as ω(r) = ς 1 (r)δ Notice that the elements of ε ϕ and ε G are much smaller than one due to Eq. (B.52). In this step, Ω ext alt is changed according to Eq. (B.45).
• r 3 < r < r 4 : We turn off ω as ω(r) = (1 − ς 3 (r))ω(r 3 Here, we have ignored ω, which is non-vanishing only around r ∼ r * . One can show that its effect disappears as r * → ∞. Then, we can show the following. • All of the eigenvalues of (Λ + Λ T ) 2 are 3 r and hence µ(−Λ) = −3 r. ) becomes positive. However, it is harmless since we use the fluctuation operator with regulator, with which all the solutions are increasing.
• Since the growth of F (SLϕ) alt is either exponential or power at r → ∞, C F is finite.
• Since we can take a smaller δ max for a larger r * , it is easy to find C δΛ that satisfies Eq. (B.35). Notice that the max norm and the spectral norm of an n × n matrix, A, are related through A ≤ n A max . and Ω ext − Ω ext alt and δΩ ext are obtained by appropriate projections. Since its elements decrease as δ max becomes smaller, we can find C δΞ that satisfies Eq. (B.36).
The same discussion applies to = 0.