On the Gauge Invariance of the Decay Rate of False Vacuum

TL;DR

This paper develops a systematic, gauge-invariant method to calculate the decay rate of false vacuum in models where the scalar field carries gauge charge, addressing gauge dependence issues in such decay processes.

Contribution

It introduces a new, systematic approach to perform gauge-invariant calculations of false vacuum decay rates in gauge theories.

Findings

Derived a manifestly gauge-invariant formula for decay rate

Provided a systematic method for integrating out gauge field fluctuations

Ensured gauge invariance in false vacuum decay calculations

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.

Equations

$$\gamma =\mathcal{A}{e}^{-\mathcal{B}},$$

$$\mathcal{L}=\frac{1}{4}{F}_{\mu \nu }{F}_{\mu \nu }+[({\partial }_{\mu }+ig{A}_{\mu }){\Phi }^{†}][({\partial }_{\mu }-ig{A}_{\mu })\Phi ]+V+{\mathcal{L}}_{G.F.}+{\mathcal{L}}_{\mathrm{ghost}},$$

$$\mathcal{F}={\partial }_{\mu }{A}_{\mu }-2\xi g(\mathrm{Re}\Phi )(\mathrm{Im}\Phi )={\partial }_{\mu }{A}_{\mu }+\frac{i}{2}\xi g({\Phi }^2-{{\Phi }^{†}}^2),$$

$${\mathcal{L}}_{G.F.}=\frac{1}{2\xi }{\mathcal{F}}^2,$$

$${\mathcal{L}}_{\mathrm{ghost}}=\bar{c}\left[-{\partial }_{\mu }{\partial }_{\mu }+\xi g^2({\Phi }^2+{{\Phi }^{†}}^2)\right]c.$$

$$(A_{\mu },\Phi {)}_{\mathrm{false}\mathrm{vacuum}}=(0,v/\sqrt{2}),$$

$$(A_{\mu },\Phi {)}_{\mathrm{bounce}}=(0,\bar{\varphi }(r)/\sqrt{2}),$$

$${\left[{\partial }_r^2\Phi +\frac{3}{r}{\partial }_r\Phi -V_{\Phi }\right]}_{\Phi \to \bar{\varphi }/\sqrt{2}}=0,$$

$${\partial }_r\bar{\varphi }(r=0)=0,$$

$$\bar{\varphi }(r=\infty )=v.$$

$${\partial }_r^2\bar{\varphi }+\frac{3}{r}{\partial }_r\bar{\varphi }-m_h^2(\bar{\varphi }-v)\simeq 0,$$

$$\bar{\varphi }(r\to \infty )\simeq v+\kappa \frac{e^{-m_{h}r}}{r^{3/2}},$$

$$A_{\mu }=a_{\mu },\Phi =\frac{1}{\sqrt{2}}\left(\bar{\varphi }+h+i\phi \right),$$

$$a_{\mu }(x)\ni$$

$$+{\alpha }_{T1}(r)i{ϵ}_{\mu \nu \rho \sigma }{V}_{\nu }^{(1)}{L}_{\rho \sigma }{\mathcal{Y}}_{J,m_A,m_B}+{\alpha }_{T2}(r)i{ϵ}_{\mu \nu \rho \sigma }{V}_{\nu }^{(2)}{L}_{\rho \sigma }{\mathcal{Y}}_{J,m_A,m_B},$$

$$h(x)\ni$$

$$\phi (x)\ni$$

$$L\equiv \sqrt{4J(J+1)}.$$

$${\mathcal{M}}_J^{(S,L,\phi )}\equiv$$

$$\displaystyle+\left(1-\frac{1}{\xi}\right)\left(\begin{array}[]{ccc}\displaystyle{\partial_{r}^{2}+\frac{3}{r}\partial_{r}-\frac{3}{r^{2}}}&\displaystyle{-L\left(\frac{1}{r}\partial_{r}-\frac{1}{r^{2}}\right)}&0\\[8.53581pt] \displaystyle{L\left(\frac{1}{r}\partial_{r}+\frac{3}{r^{2}}\right)}&\displaystyle{-\frac{L^{2}}{r^{2}}}&0\\[8.53581pt] 0&0&0\end{array}\right),$$

$${\Delta }_J\equiv {\partial }_r^2+\frac{3}{r}{\partial }_r-\frac{L^2}{r^2}.$$

$${\mathcal{M}}_{J=0}^{(S,\phi )}\equiv$$

$${\mathcal{M}}_J^{(T)}=-{\Delta }_J+g^2{\bar{\varphi }}^2$$

$${\mathcal{M}}_J^{(h)}={\left[-{\Delta }_J+V_{\Phi {\Phi }^{†}}\right]}_{\Phi \to \bar{\varphi }/\sqrt{2}},$$

$${\mathcal{M}}_J^{(\bar{c},c)}=-{\Delta }_J+\xi g^2{\bar{\varphi }}^2,$$

$$\mathcal{A}=\frac{{\mathcal{B}}^2}{4{\pi }^2}{\mathcal{A}}^{\prime (h)}{\mathcal{A}}^{(S,L,\phi )}{\mathcal{A}}^{(T)}{\mathcal{A}}^{(\bar{c},c)}{\mathcal{A}}^{(\mathrm{extra})},$$

$${\mathcal{A}}^{\prime (h)}=$$

$${\mathcal{A}}^{(S,L,\phi )}=$$

$${\mathcal{A}}^{(T)}=$$

$${\mathcal{A}}^{(\bar{c},c)}=$$

$$\frac{\text{mbox}Det{\mathcal{M}}^{(X)}}{\text{mbox}Det{\hat{\mathcal{M}}}^{(X)}}=\underset{r\to \infty }{\lim }\frac{\text{mbox}det({\psi }_1(r)\cdots {\psi }_N(r))}{\text{mbox}det({\hat{\psi }}_1(r)\cdots {\hat{\psi }}_N(r))}.$$

$$({\Delta }_J-\xi g^2{\bar{\varphi }}^2)f_J^{(\mathrm{FP})}=0,$$

$$f_J^{(\mathrm{FP})}(r\to 0)\simeq r^{2J}.$$

$$({\Delta }_J-\xi g^2{v}^2){\hat{f}}_J^{(\mathrm{FP})}=0,$$

$${\hat{f}}_J^{(\mathrm{FP})}(r\to 0)\simeq r^{2J}.$$

Full text

KEK-TH-1964

UT-17-09

March, 2017

**On the Gauge Invariance

of the Decay Rate of False Vacuum

**

Motoi Endo(a,b,c), Takeo Moroi(d,c), Mihoko M. Nojiri(a,b,c), Yutaro Shoji(e)

(a)KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan

(b)*The Graduate University of Advanced Studies (Sokendai),

Tsukuba, Ibaraki 305-0801, Japan*

(c)Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan

(d)Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

(e)Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan

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.

There have been continuous interest in the theoretically correct calculation of the decay rate of the false vacuum. One of the recent motivations has been provided by the discovery of the Higgs boson at the LHC [1] and the precision measurement of the top quark mass at the LHC and Tevatron [2]; in the standard model, we are facing the possibility to live in a metastable electroweak vacuum with lifetime much longer than the age of the universe [3, 4, 5, 6, 7, 8, 9]. Furthermore, the false and true vacua may show up in various models of physics beyond the standard model. One important example is supersymmetric standard model in which the electroweak symmetry breaking vacuum may become unstable with the existence of the color or charge breaking vacuum at which colored or charged sfermion fields acquire vacuum expectation values; the condition that the electroweak vacuum has sufficiently large lifetime constrains the parameters in supersymmetric models [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Thus, detailed understanding of the decay of the false vacuum is important in particle physics and cosmology.

In [20, 21, 22], the calculation of the decay rate of the false vacuum was formulated with the so-called bounce configuration which is a solution of the 4-dimensional (4D) Euclidean equation of motion connecting false vacuum and true vacuum (more rigorously, the other side of the potential wall). The decay rate of the false vacuum per unit volume is given in the following form:

[TABLE]

where ${\cal B}$ is the bounce action, while the prefactor ${\cal A}$ is obtained by integrating out field fluctuations around the bounce configuration as well as those around the false vacuum.

In gauge theories, if a scalar field with gauge quantum number acquires non-vanishing amplitude at the true or false vacuum, the gauge, Higgs and the ghost sectors contribute to ${\cal A}$. The decay rate should be calculated with the gauge-fixed Lagrangian which contains the gauge parameter $\xi$. In the present study, we concentrate on the gauge dependence (i.e., the $\xi$-dependence) of the decay rate of the false vacuum. Formally, the $\xi$-dependence of ${\cal A}$ should cancel out exactly. This is due to the fact that the decay rate is derived from the effective action of the bounce configuration, and also that the effective action for any solution of the equation of motion is assured to be gauge invariant [23, 24]. In the actual calculation, however, the gauge independence is not manifest because the $\xi$-dependence should cancel out among the contributions of gauge field, Nambu-Goldstone (NG) boson, and Faddeev-Popov (FP) ghosts.#1#1#1The gauge invariance of the effective potential of the model we consider was discussed in [25]; however, the scalar configuration was assumed to be space-time independent, and hence the result is not applicable to the present case. The gauge independence of the sphaleron transition rate was studied in [26] using functional determinant method which is also adopted in our analysis. In particular, the gauge boson and the NG mode, whose fluctuation operator is $\xi$-dependent, mix with each other around the bounce configuration. This makes the study of the decay rate complicated. Furthermore, it is difficult to check the gauge independence even numerically because a stable numerical implementation proposed so far requires $\xi=1$.

In this letter, we show a procedure to integrate out the field fluctuations, which gives rise to a manifestly gauge invariant expression of the decay rate overcoming the difficulties mentioned above. In the current study, we concentrate on the case where

the gauge symmetry is $U(1)$,#2#2#2The application of our prescription to the case of non-abelian gauge symmetry is straightforward. 2. 2.

there is only one charged scalar field $\Phi$ which affects the decay of the false vacuum, 3. 3.

the $U(1)$ symmetry is spontaneously broken in the false vacuum.

More general cases, in particular, the case where the $U(1)$ symmetry is preserved at the false vacuum, is discussed in [27].

First, let us explain the set up of our analysis. The Euclidean Lagrangian is given by

[TABLE]

where $A_{\mu}$ is the gauge field, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and $V$ is the scalar potential. In addition, ${\cal L}_{\rm G.F.}$ and ${\cal L}_{\rm ghost}$ are the gauge-fixing term and the terms containing FP ghosts (denoted as $c$ and $\bar{c}$), respectively. We use the following gauge-fixing function:#3#3#3Previous studies used different type of the gauge-fixing functions: $\partial_{\mu}A_{\mu}-\sqrt{2}\xi g\bar{\phi}{\rm Re}\Phi$, around the bounce (i.e., $\Phi=\bar{\phi}/\sqrt{2}$), and $\partial_{\mu}A_{\mu}-\sqrt{2}\xi gv{\rm Im}\Phi$, around the false vacuum (i.e., $\Phi=v/\sqrt{2}$). Expanding the fields around the solution of the classical equation of motion, we obtain the same gauge-fixing functions as the previous studies at least at the one-loop level, although our gauge-fixing function can be used both around the bounce and around the false vacuum.

[TABLE]

with which

[TABLE]

and

[TABLE]

The scalar potential $V$ has true and false vacua. We assume that the true and false vacua exist at the tree-level; we do not consider the case where the second vacuum is radiatively generated. The field configuration of the false vacuum is expressed as#4#4#4The field amplitude at the false vacuum (as well as the bounce configuration) may be shifted due to loop effects; the shifts are $\xi$-dependent in general. However, at the one-loop level, the shifts do not affect the extremum values of the effective action to which the decay rate of the false vacuum is related.

[TABLE]

with $v$ being a constant which is non-vanishing in this letter.

The false vacuum decay is dominated by the classical path, so-called the bounce [20]. When $v\neq 0$, the bounce solution, which is $O(4)$ symmetric [28, 29], is given in the following form:

[TABLE]

where $r\equiv\sqrt{x_{\mu}x_{\mu}}$ is the radius of the 4D Euclidean space. Here, the function $\bar{\phi}$ is a solution of the classical equation of motion:

[TABLE]

where $V_{\Phi}$ denotes the derivative of the scalar potential with respect to $\Phi$. It also satisfies the following boundary conditions:

[TABLE]

We assume that $\bar{\phi}$ is a real function of $r$. At $r\rightarrow\infty$, $\bar{\phi}$ settles on the false-vacuum; in such a limit, $\bar{\phi}$ (approximately) obeys the following equation:

[TABLE]

where $m_{h}$ is the mass of the (massive) scalar boson around the false vacuum. Then, the asymptotic behavior of $\bar{\phi}$ can be expressed as

[TABLE]

with $\kappa$ being a constant.

For the calculation of the decay rate of the false vacuum, it is necessary to integrate out the fluctuations around the bounce. The gauge and scalar fields are decomposed around the bounce as

[TABLE]

where the “Higgs” mode $h$ and the “NG” mode $\varphi$ are real fields. We expand the field fluctuations as#5#5#5For notational simplicity, we omit the subscripts $J$, $m_{A}$, and $m_{B}$ from the radial function $\alpha$’s, and the summations over $J$, $m_{A}$, and $m_{B}$ are implicit.

[TABLE]

where ${\cal Y}_{J,m_{A},m_{B}}$ denotes the 4D hyperspherical harmonics; the eigenvalues of $S_{A}^{2}$, $S_{B}^{2}$, $S_{A,3}$ $S_{B,3}$ (with $S_{A}$ and $S_{B}$ being generators of the rotational group of the 4D Euclidean space, i.e., $SU(2)_{A}\times SU(2)_{B}$) are $J(J+1)$, $J(J+1)$, $m_{A}$, and $m_{B}$, respectively. Notice that $J=0$, $\frac{1}{2}$, $1$, $\cdots$. In addition, $V^{(1)}_{\nu}$ and $V^{(2)}_{\nu}$ are (arbitrary) two independent vectors, $L_{\rho\sigma}\equiv\frac{i}{\sqrt{2}}(x_{\rho}\partial_{\sigma}-x_{\sigma}\partial_{\rho})$, and

[TABLE]

For $J>0$, the fluctuation operator for $(\alpha_{S},\alpha_{L},\alpha_{\varphi})$ is obtained as

[TABLE]

where $\bar{\phi}^{\prime}\equiv\partial_{r}\bar{\phi}$, and

[TABLE]

For $J=0$, $\alpha_{L}$-mode does not exist, and the fluctuation operator is in $2\times 2$ form as

[TABLE]

In addition, the fluctuation operator for the transverse modes, the Higgs mode, and the FP ghost mode are given by

[TABLE]

with $V_{\Phi\Phi^{\dagger}}\equiv\partial^{2}V/\partial\Phi\partial\Phi^{\dagger}$.

We also need the fluctuation operators around the false vacuum, denoted as $\widehat{\cal M}_{J}^{(S,L,\varphi)}$, $\widehat{\cal M}_{J}^{(T)}$, and so on. (Here and hereafter, the “hat” is used for objects related to the false vacuum.) They can be obtained from the fluctuation operators Eqs. (25), (29), (30), (31), and (32) by replacing $\bar{\phi}\rightarrow v$, and $\bar{\phi}^{\prime}\rightarrow 0$.

The prefactor ${\cal A}$ in Eq. (1) is related to the functional determinants of the fluctuation operators introduced above. It can be expressed as [21]

[TABLE]

where ${\cal A}^{\prime(h)}$, ${\cal A}^{(S,L,\varphi)}$, ${\cal A}^{(T)}$, and ${\cal A}^{(\bar{c},c)}$ are contributions of the Higgs mode, $(\alpha_{S},\alpha_{L},\varphi)$, $(\alpha_{T_{1}},\alpha_{T2})$, and FP ghosts, respectively, which are given by

[TABLE]

Here, “prime” in Eq. (34) indicates that the effect of the zero modes in association with the translational invariance is omitted in calculating the functional determinant [21]. The contributions of extra fields other than those introduced above are expressed by ${\cal A}^{(\rm extra)}$; we do not consider them in this letter. We are interested in the gauge dependence of the decay rate, therefore we focus on the $S$, $L$, and NG modes as well as FP ghosts whose fluctuation operators are dependent on $\xi$.

Our main task is to calculate the functional determinants mentioned above. For this purpose, we use the method discussed in [22, 30, 31, 32]. With $N\times N$ fluctuation operators ${\cal M}^{(X)}$ and $\widehat{\cal M}^{(X)}$ being given, we introduce two sets of $N$ linearly independent functions $\psi^{(X)}_{I}$ and $\widehat{\psi}^{(X)}_{I}$ ($I=1-N$), obeying ${\cal M}^{(X)}\psi_{I}=0$ and $\widehat{\cal M}^{(X)}\widehat{\psi}_{I}=0$. Here, $\psi_{I}$ and $\widehat{\psi}_{I}$ satisfy the same boundary condition at $r=0$. Then, the ratio of the functional determinants is related to their asymptotic behaviors at $r\rightarrow\infty$ as

[TABLE]

In the following, we use the above relation to evaluate the functional determinants of the fluctuation operators given in Eqs. (34) $-$ (37). For our study, $\psi_{I}$ and $\widehat{\psi}_{I}$ are required to be regular at $r=0$ for the finiteness of the effective action.

The fluctuation operator for the ghost is given in Eq. (32). For the calculation of its functional determinant, we define the function $f^{\rm(FP)}_{J}$ which obeys

[TABLE]

where the boundary condition of $f^{\rm(FP)}_{J}$ is taken to be

[TABLE]

We also introduce the function $\widehat{f}^{\rm(FP)}_{J}$ which obeys

[TABLE]

with

[TABLE]

The explicit form of $\widehat{f}^{\rm(FP)}_{J}$ is given by

[TABLE]

where $I_{2J+1}$ is the modified Bessel function. Then,

[TABLE]

For the contributions of the $S$-, $L$-, and $\varphi$-modes with $J>0$, we need the functions $\Psi$ and $\widehat{\Psi}$, which are regular at the origin, satisfying

[TABLE]

Hereafter, the boundary conditions for $\Psi$ and $\widehat{\Psi}$ at the origin are taken to be the same. With three independent solutions of the above equations (which we denote $\Psi_{I}$ and $\widehat{\Psi}_{I}$, with $I=1$, $2$, and $3$), the functional determinants of our interests are given by

[TABLE]

where

[TABLE]

Hereafter, we use the fact that the solution of Eq. (45) can be decomposed as

[TABLE]

where the functions $\chi$, $\eta$, and $\zeta$ obey the following equations:

[TABLE]

Then, the following identities hold:#6#6#6At the leading order in fluctuations, Eq. (66) is equivalent to $\alpha_{\cal F}+\xi\zeta=0$, where $\alpha_{\cal F}$ is the radial mode function of the gauge fixing function, i.e., ${\cal F}(x)\ni\alpha_{\cal F}(r){\cal Y}_{J,m_{A},m_{B}}$.

[TABLE]

Hereafter, we give three independent solutions $\Psi_{I}$ ($I=1-3$) of Eq. (45), and show their boundary conditions at $r=0$. The solutions can be constructed with the following three sets of the functions $(\chi_{I},\eta_{I},\zeta_{I})$:

For $\Psi_{1}$, we take $\eta_{1}=\zeta_{1}=0$, and

[TABLE]

with which Eqs. (63), (64) and (65) are satisfied. Then,

[TABLE]

where

[TABLE] 2. 2.

For $\Psi_{2}$, we can take $\zeta_{2}=0$, and

[TABLE]

Then,

[TABLE] 3. 3.

For $\Psi_{3}$, we take

[TABLE]

while $\chi_{3}(r\rightarrow 0)$ and $\eta_{3}(r\rightarrow 0)$ are both $O(r^{2J+2})$. The contributions to the top and middle components of $O(r^{2J+1})$ vanish, and

[TABLE]

The solutions around the false vacuum, denoted as $\widehat{\Psi}_{I}$, satisfy the same boundary conditions at $r\rightarrow 0$ as those of $\Psi_{I}$, and obey the following differential equation:

[TABLE]

Notice that the evolution equation of the bottom component of $\widehat{\Psi}_{I}$ does not contain the top and middle components and vice versa.

For the following discussion, it is convenient to define the function $f^{(\eta)}_{J}$, which obeys

[TABLE]

and

[TABLE]

(For $J=0$, $f^{(\eta)}_{0}(r\rightarrow 0)\simeq 1$.) We emphasize here that the function $f^{(\eta)}_{J}$ is independent of $\xi$. The homogeneous solutions of Eqs. (63) and (65) (that of Eq. (64)) are given by $f^{\rm(FP)}_{J}$ ($f^{(\eta)}_{J}$); thus, in particular, $\eta_{2}=f^{(\eta)}_{J}$. We also define the function $\widehat{f}^{(\eta)}_{J}$ which obeys

[TABLE]

with

[TABLE]

Next, we consider the mode with $J=0$. The fluctuation operators ${\cal M}_{J=0}^{(S,\varphi)}$ and $\widehat{\cal M}_{J=0}^{(S,\varphi)}$ are in $2\times 2$ form. For the calculation of their functional determinants, we need the solutions of the following equations:

[TABLE]

with which

[TABLE]

where

[TABLE]

Solutions of (90) are given in the following form:

[TABLE]

where the functions $\chi$ and $\zeta$ obey Eq. (63) with $\eta=0$ and Eq. (65), respectively. Two independent solutions of Eq. (90) can be chosen as follows:

For $\Psi_{1}$, we take $\zeta_{1}=0$, and

[TABLE]

Then,

[TABLE] 2. 2.

For $\Psi_{2}$, we take

[TABLE]

while $\chi_{2}(r\rightarrow 0)$ is $O(r^{2})$, with which

[TABLE]

Notice that $\chi_{1}=\zeta_{2}=f^{\rm(FP)}_{0}$.

With the solutions introduced above, we now discuss the decay rate of the false vacuum. For this purpose, we study the asymptotic behaviors of the solutions at $r\rightarrow\infty$. First, we consider the modes with $J\neq 0$. Each of Eq. (39) and (86) has only one growing solution at $r\rightarrow\infty$. The other solutions are exponentially suppressed at $r\rightarrow\infty$; those dumping modes are irrelevant for the following discussion. At $r\rightarrow\infty$, $f^{\rm(FP)}_{J}$ and $f^{(\eta)}_{J}$ behave as

[TABLE]

where $c_{\rm FP}$ and $c_{\eta}$ are constants.

The behaviors of $\chi_{I}$, $\eta_{I}$, and $\zeta_{I}$ can be understood by using $f^{\rm(FP)}_{J}$ and $f^{(\eta)}_{J}$, using the fact that $\bar{\phi}^{\prime}$ is exponentially suppressed at $r\rightarrow\infty$ (see Eq. (12)). Obviously,

[TABLE]

Because $\chi_{2}$ is given by the sum of a homogeneous solution and a particular solution (which we denote $\delta\chi^{(\eta)}$), $\Psi_{2}$ can be expressed by

[TABLE]

where $a_{1}$ is a constant. The function $\delta\chi^{(\eta)}$ satisfies the following equation:

[TABLE]

At $r\rightarrow\infty$, we can see that $\delta\chi^{(\eta)}$ behaves as

[TABLE]

Furthermore, the functions for $\Psi_{3}$ behave as

[TABLE]

with $b_{1}$ and $b_{2}$ constants. The functions $\delta\chi^{(\zeta)}$ and $\delta\eta^{(\zeta)}$ obey the following equations:

[TABLE]

and their asymptotic behaviors are#7#7#7In Eq. (122), we do not explicitly show the effect of $\delta\eta^{(\zeta)}$ on $\delta\chi^{(\zeta)}$, because it is subdominant.

[TABLE]

Using the asymptotic behaviors given above, the determinant defined in Eq. (48) has the following structure:

[TABLE]

The determinant is dominated by the product of the diagonal elements, and is given by

[TABLE]

Because $\chi_{1}$ and $\zeta_{3}$ obey the same equation as that of the FP mode while $\eta_{2}$ is $\xi$-independent, ${\cal D}^{(S,L,\varphi)}_{J}(r\rightarrow\infty)$ has a $\xi$-dependence which can be cancelled out by the contribution from the FP ghosts.

In order to evaluate $\widehat{\Psi}_{I}$, we can use the fact that the upper two components of $\widehat{\Psi}_{I}$ and the bottom component are decoupled in the evolution equation given in Eq. (85). We can find

[TABLE]

The functional determinant around the false vacuum is given by

[TABLE]

The discussion for $J=0$ is similar to that for $J\neq 0$. The asymptotic behaviors of $\Psi_{I}$ are given by

[TABLE]

For the false vacuum solutions, $\widehat{\Psi}_{I}$ ($I=1$ and $2$), we can use the fact that $\widehat{\Psi}^{\rm(top)}$ and $\widehat{\Psi}^{\rm(bot)}$ evolve independently. Requiring that $\widehat{\Psi}_{I}$ satisfies the same boundary condition as ${\Psi}_{I}$ at $r\rightarrow 0$, we obtain

[TABLE]

The functional determinant around the bounce and that around the false vacuum are given by

[TABLE]

and

[TABLE]

respectively.

Combining the effects of the gauge field, the NB boson, and the FP ghosts, we obtain

[TABLE]

We emphasize that the above result is manifestly gauge invariant. For completeness, we also summarize the contributions of the transverse and Higgs modes. The contribution of transverse mode is given by

[TABLE]

Here, the functions $f^{(T)}_{J}$ and $\widehat{f}^{(T)}_{J}$ satisfy

[TABLE]

with

[TABLE]

For the Higgs mode contribution, we first define the functions $f^{(h)}_{J}$ and $\widehat{f}^{(h)}_{J}$, satisfying

[TABLE]

with

[TABLE]

As we have mentioned, we need to omit the zero eigenvalues in association with the translational invariance. Such zero eigenvalues show up in the $J=\frac{1}{2}$ mode [21, 19], and can be eliminated by using the function $\check{f}^{(h)}_{1/2}$ which obeys

[TABLE]

With $\check{f}^{(h)}_{1/2}$, the Higgs mode contribution is given by

[TABLE]

By substituting Eqs. (156), (157), and (165) into Eq. (33), we obtain the prefactor ${\cal A}$ for the calculation of the decay rate of the false vacuum. With the present prescription, it is related to the asymptotic behaviors of the solutions of second-order differential equations which are $\xi$-independent. Notice that the formula of the decay rate we obtained is manifestly gauge invariant.

In summary, in this letter, we have studied the false vacuum decay in theory with $U(1)$ gauge symmetry, paying particular attention to the gauge invariance of the decay rate. Concentrating on the case where the gauge symmetry is broken in the false vacuum, we derived a manifestly gauge-invariant expression of the decay rate. Although we have studied the case with $U(1)$ gauge symmetry, application of our result to models with non-abelian gauge groups is straightforward.

We emphasize that our result not only guarantees the gauge invariance of the decay rate but also simplifies the numerical calculation. In order to evaluate the prefactor ${\cal A}$ numerically, one should calculate the functions $\Psi$, which are three- or two-component objects, by solving Eq. (45) or (90). With a general value of $\xi$, each mode grows differently at $r\rightarrow\infty$, and the numerical calculation of the functional determinant is difficult. This problem can be avoided if we take the so-called ’t Hooft-Feynman gauge with $\xi=1$ because, in such a gauge, the fluctuation operators of the gauge and NG fields become simple. Even taking the ’t Hooft-Feynman gauge, however, one has to solve the coupled equations, which is numerically demanding. In addition, if one adopts a particular choice of gauge, like the ’t Hooft-Feynman gauge, the gauge independence of the result is not explicit. On the contrary, with our results, only the asymptotic behaviors of functions which are manifestly $\xi$-independent are necessary to obtain the decay rate of the false vacuum. Our simple formula, which is manifestly gauge independent, would greatly reduce the numerical costs compared to the previous procedures.

Finally, several comments are in order.

• •

When the gauge symmetry is preserved at the false vacuum, i.e., $v=0$, we can find a class of solution of the classical equation of motion. With the function $\bar{\phi}$ obeying Eq. (8), the following field configuration satisfies the condition for the bounce:

[TABLE]

Here, the function $\Theta$ obeys

[TABLE]

and $\Theta^{\prime}(0)=0$ (with the “prime” being the derivative with respect to $r$). With such a boundary condition, the function $\Theta$ is determined by its value at $r=0$. In calculating the decay rate, we need to take account of the effect of all the possible bounce configuration parametrized by $\Theta(0)$. Importantly, the fluctuation operator depends on $\Theta(0)$ with the present choice of gauge-fixing function, which makes the calculation of the decay rate complicated. As hinted in [33], the calculation is simplified with a different gauge-fixing function which is independent of the scalar field.

• •

Related to the previous comment, when the gauge symmetry is preserved at the false vacuum, there shows up a zero mode related to internal symmetry. The path integral over such a zero mode should be reinterpreted as the integration over the possible bounce configuration parametrized by $\Theta(0)$.

• •

What we are calculating is the one-loop effective action of the bounce, therefore renormalization is necessary. In other words, in the calculation of the prefactor ${\cal A}$, contribution from each angular momentum $J$ is finite, but the infinite sum of those contributions diverges. The divergences should be subtracted by including the counter terms [21].

We discuss these issues in a separate publication [27].

Acknowledgements: This work was supported by the Grant-in-Aid for Scientific Research on Scientific Research B (No.16H03991 [ME and MMN] and No.26287039 [MMN]), Scientific Research C (No.26400239 [TM]), Young Scientists B (No.16K17681 [ME]) and Innovative Areas (No.16H06490 [TM] and 16H06492 [MMN]), and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.