Simple Gradient Flow Equation for the Bounce Solution

TL;DR

This paper introduces a new simple gradient flow equation to efficiently derive bounce solutions for false vacuum decay, building on classical methods and demonstrating robustness in convergence.

Contribution

The paper presents a novel gradient flow approach for bounce solutions, simplifying the derivation process and ensuring stable convergence regardless of initial conditions.

Findings

The method reliably finds bounce solutions in false vacuum decay.

The approach is robust against different initial configurations.

It simplifies the computational process for decay rate analysis.

Abstract

Motivated by the recent work of Chigusa, Moroi, and Shoji, we propose a new simple gradient flow equation to derive the bounce solution which contributes to the decay of the false vacuum. Our discussion utilizes the discussion of Coleman, Glaser, and Martin and we solve a minimization problem of the kinetic energy while fixing the potential energy. The bounce solution is derived as a scale-transformed of the solution of this problem. We also show that the convergence of our method is robust against a choice of the initial configuration.

Equations

$$\mathcal{S}[\varphi ]$$

$$\mathcal{T}[\varphi ]$$

$$\mathcal{V}[\varphi ]$$

$$-{\nabla }^2{\varphi }_i+\frac{\partial V}{\partial {\varphi }_i}$$

$$\underset{|x|\to \infty }{\lim }{\varphi }_i(x)$$

$$\mathcal{T}[\varphi ]>0,\mathcal{V}[\varphi ]<0.$$

$$-\frac{d^2{\varphi }_i}{d{r}^2}-\frac{d-1}{r}\frac{d{\varphi }_i}{dr}+\frac{\partial V}{\partial {\varphi }_i}=0.$$

$$\frac{\partial }{\partial \tau }{\phi }_i(r,\tau )$$

$$\lambda [\phi ]$$

$$\mathcal{V}[\phi ]{|}_{\tau =0}<0.$$

$$\frac{d}{d\tau }\mathcal{V}[\phi ]$$

$$\frac{d}{d\tau }\mathcal{T}[\phi ]$$

$$\left(\sum _i{\int }_0^{\infty }dr{r}^{d-1}({\nabla }^2{\phi }_i{)}^2\right)\left(\sum _i{\int }_0^{\infty }dr{r}^{d-1}{\left(\frac{\partial V(\phi )}{\partial {\phi }_i}\right)}^2\right)\ge {\left(\sum _i{\int }_0^{\infty }dr{r}^{d-1}\frac{\partial V(\phi )}{\partial {\phi }_i}{\nabla }^2{\phi }_i\right)}^2.$$

$${\nabla }^2{\phi }_i=\lambda \frac{\partial V(\phi )}{\partial {\phi }_i}$$

$${\varphi }_B(r)=\varphi ({\lambda }^{1/2}r).$$

$$V(\varphi )=\frac{1}{2}{\varphi }^2-\frac{1}{3}{\varphi }^3.$$

$$\phi (r,0)=\{\begin{array}{ll}5(1-r)& (0\le r\le 1)\\ 0& (r>1)\end{array}.$$

$$V=$$

$$+c\left(\frac{1}{4}{\varphi }_2^4-\frac{1}{3}{\varphi }_2^3\right).$$

$$-{\nabla }^2{\varphi }_{({\mathcal{V}}_0)i}+\lambda \frac{\partial V}{\partial {\varphi }_i}=0.$$

$${\varphi }_B(x)={\varphi }_{({\mathcal{V}}_0)}({\lambda }^{1/2}x).$$

$$(d-2)\mathcal{T}[\tilde{\varphi }]+d\mathcal{V}[\tilde{\varphi }]=0.$$

$$\mathcal{T}[{\varphi }_{(\mathcal{V}[\tilde{\varphi }])}]\le \mathcal{T}[\tilde{\varphi }].$$

$$\mathcal{T}[{\varphi }_B]$$

$$\mathcal{V}[{\varphi }_B]$$

$$\mathcal{T}[{\varphi }_B]\le \mathcal{T}[\tilde{\varphi }].$$

$$\mathcal{S}[{\varphi }_B]\le \mathcal{S}[\tilde{\varphi }].$$

Full text

I Introduction

The decay of the false vacua is an important topic in particle physics and cosmology. The decay rate of the false vacua can be calculated from “the imaginary part” of the Euclidean path integral Coleman:1977py 111 For earlier discussions, see, e.g., Refs. Lee:1974ma ; Frampton:1976kf . . In the path integral formalism, we can see that the main contribution comes from the bounce solution $\phi_{B}$ which is a non-trivial solution of the equation of motion with the least action. Thus, the bounce solution plays a crucial role in the decay of the false vacua. To calculate the bounce solution, we have to solve the equation of motion with the boundary condition at infinity. In general, it is not easy to calculate the bounce solution, and this is particularly the case for models with multi scalar fields.

Several algorithms to calculate the bounce action have been discussed so far, e.g., gradient flow with modifications Claudson:1983et ; Cline:1998rc ; Cline:1999wi , modified actions which have the bounce solution as a local minimum Kusenko:1995jv ; Kusenko:1996jn ; Moreno:1998bq ; John:1998ip , changing gradually a coefficient of the friction term (the second term on the LHS of Eq. (7)) Konstandin:2006nd ; Park:2010rh , machine learning Jinno:2018dek ; Piscopo:2019txs and so on. Also, public codes to calculate the bounce solution are available, such as CosmoTransitions Profumo:2010kp ; Wainwright:2011kj , AnyBubble Masoumi:2016wot , and BubbleProfiler Akula:2016gpl ; Athron:2019nbd . Some works discuss the bounce solution/action avoiding the direct calculation, e.g., some approximations Johnson:2008kc ; Masoumi:2017trx ; Guada:2018jek , upperbounds Dasgupta:1996qu ; Sarid:1998sn ; Brown:2017cca , lowerbounds, Aravind:2014aza ; Sato:2017iga ; Brown:2017cca , and an alternative formulation Espinosa:2018hue ; Espinosa:2018voj ; Espinosa:2018szu .

One of the reasons for the technical difficulty is that the bounce solution is a saddle point of the action, i.e., the bounce is not a stable solution of a simple minimization problem. Recently, Chigusa, Shoji, and Moroi Chigusa:2019wxb proposed a new method to obtain the bounce solution. They proposed a gradient flow equation whose fixed point is the bounce solution. Their flow equation has the gradient of the action and an additional term to lift up unstable direction around the bounce solution. Motivated by Ref. Chigusa:2019wxb , in this paper, we propose a new simple flow equation. Coleman, Glaser, and Martin (CGM) Coleman:1977th showed that the calculation of the bounce solution is equivalent to the minimization of the kinetic energy ${\cal T}$ while fixing the potential energy ${\cal V}<0$. This minimization problem can be naturally formulated in a flow equation. In the end, the bounce solution is obtained as a scale-transformed of the solution of this problem. In Sec. II, we describe our formulation to calculate the bounce solution. In Sec. III, we discuss numerical analysis on several examples by using our flow equation, and show that our flow equation works well.

II Formulation

In this paper, we focus on the Euclidean action with $n$ scalar fields with the canonical kinetic term.

[TABLE]

Here $d$ is the dimension of the space, and we assume $d$ is larger than 2. The scalar potential $V$ satisfies $V(0)=0$, $\partial V/\partial\phi_{i}=0$, all of the eigenvalues of the Hessian of $V$ at $\phi_{i}=0$ are non-negative, and $V$ is somewhere negative.

The bounce solution which contributes to the decay of the false vacuum satisfies the equation of the motion and the boundary condition at infinity:

[TABLE]

Also, the bounce solution should be a non-trivial solution, i.e., $\exists i,x,~{}\phi_{i}(x)\neq 0$. Thus,

[TABLE]

Note that ${\cal V}[\phi]<0$ is required in order for the bounce solution to be an extremum under the scale transformation: $\phi_{i}(x)\to\phi_{i}(\lambda x)$. See, e.g., Ref. Coleman:1977th . The bounce solution has the least action among configurations which satisfy the above conditions Eqs. (4, 5, 6). It is known that the bounce solution has spherical symmetry Coleman:1977th ; lopes1996radial ; byeon2009symmetry ; Blum:2016ipp . Therefore, Eq. (4) can be simplified as

[TABLE]

In order to discuss the bounce solution, CGM Coleman:1977th introduced the reduced problem, which is defined as the problem of finding a configuration vanishing at infinity which minimizes $\cal T$ for some fixed negative $\cal V$. The existence of the solution of this problem is ensured by CGM’s theorem B in Ref. Coleman:1977th and Ref. brezis1984minimum . Also, CGM’s theorem A ensures that the bounce solution can be obtained as a scale-transformed of a solution of the reduced problem. See the Appendix. Here we solve the CGM’s reduced problem by using a gradient flow equation. We introduce functions $\varphi_{i}(r,\tau)$ and propose the following gradient flow equations:

[TABLE]

Here $\tau$ is “the time” for the flow of $\varphi$ and $\nabla^{2}\varphi_{i}=\partial_{r}^{2}\varphi_{i}+(d-1)(\partial_{r}\varphi)/r$. We take the initial $\varphi(r,0)$ such that

[TABLE]

Note that $\lim_{r\to\infty}\varphi_{i}(r,\tau)=0$ should hold in order for ${\cal V}[\phi]$ to be finite. By using Eq. (8) and Eq. (9), we can show

[TABLE]

To show Eq. (12), we used the following Cauchy-Schwarz inequality:

[TABLE]

Also, we can see that the equalities of Eq. (12) and Eq. (13) hold if and only if

[TABLE]

is satisfied. Eqs. (11, 12) tell us that ${\cal T}[\varphi]$ monotonously decreases while ${\cal V}[\varphi]$ is constant during the flow of $\varphi$. In the limit of $\tau\to\infty$, $\varphi$ converges to a configuration which satisfies $\nabla^{2}\varphi_{i}-\lambda(\partial V(\varphi)/\partial\varphi_{i})=0$. The convergence of $\varphi$ is guaranteed by the existence of the minimizer Coleman:1977th ; brezis1984minimum . Note that this fixed point cannot be the false vacuum $\varphi_{i}=0$ because ${\cal V}[\varphi]$ in the neighborhood of the false vacuum is positive and ${\cal V}[\varphi]$ is always negative during the flow. As long as the initial condition is not fine-tuned, $\varphi$ at $\tau\to\infty$ should be stable solution under the small perturbation, i.e., ${\cal T}[\varphi]$ should be a local minimum under the small perturbation such that ${\cal V}[\phi]$ is not changed. In principle, the reduced problem could have several local minima. Physically, this case happens if there exist several directions of tunneling. In this case, $\varphi$ at $\tau\to\infty$ depends on the initial condition, and we can find the global minimum among those local minima. The configuration which gives the smallest value of ${\cal T}$ is the solution of the CGM’s reduced problem.

Let $\phi_{i}(r)(\equiv\lim_{\tau\to\infty}\varphi_{i}(r,\tau))$ be the solution of the reduced problem, and derive the bounce solution. The bounce solution $\phi_{B}(r)$ can be obtained by a scale transformation of $\phi$ as

[TABLE]

The above $\lambda$ is calculated as $\lim_{\tau\to\infty}\lambda[\varphi]$. Although the CGM’s theorem A ensures that this $\phi_{B}$ is the bounce solution, let us see this more explicitly. We can immediately see that (i) $\phi_{B}$ satisfies the EOM (Eq. (4)) and (ii) $\lim_{r\to\infty}\phi_{B}(r)=0$ because ${\cal V}[\phi_{B}]$ is finite. Also, we can see that (iii) ${\cal S}$ has only one unstable direction around $\phi_{B}$. Since $\phi_{B}$ is a scale-transformed of $\phi$, $\phi_{B}$ is the global minimum of the action ${\cal S}$ if the potential energy ${\cal V}$ is fixed. The direction in which ${\cal S}$ decreases is the direction which changes ${\cal V}[\phi]$, i.e., the scale transformation. Therefore, $\phi_{B}$ which is defined in Eq. (15) is the bounce solution.

An essential point of our method is that the negative eigenmode around the bounce solution can be related to the scale transformation. By fixing the potential energy ${\cal V}$, we freeze fluctuation in this direction. Note that a method which is proposed in Ref. Claudson:1983et also utilizes this property.

III Example

In the previous section, we have seen that the CGM’s reduced problem can be solved by the flow equation Eq. (8) and Eq. (9), and the bounce solution can be obtained from Eq. (15). In this section, we discuss numerical results for several examples, and show that our method works well.

First, let us take the following single scalar potential in $d=4$ Euclidean space.

[TABLE]

We take the initial configuration at $\tau=0$ as

[TABLE]

The flow of this field configuration is shown in Fig. 1. We can see the convergence of the configuration. In Fig. 3 and Fig. 3, we show the flow of the configuration from different initial conditions. We can see that the convergence of the configuration is robust for different initial conditions. The final configuration depends only on the value of ${\cal V}[\varphi]$, and those different final results are connected with each other by an appropriate scale transformation. By using this result, we can obtain the bounce solution from Eq. (15). We compare our bounce solution with the result by CosmoTransitions Wainwright:2011kj in Fig. 4. We can see that the two results agree well and our method works.

Next, let us discuss a case with two scalar fields. We take the following potential:

[TABLE]

Again, we compare our bounce solutions with the results by CosmoTransitions. The case with $c=2$ is shown in Figs. 5 and 6, and the case with $c=80$ in Figs. 7 and 8. We can see that our result agrees with that of CosmoTransitions.

IV Conclusion

In this paper, motivated by a recent work of Chigusa, Shoji, and Moroi Chigusa:2019wxb , we proposed a new simple gradient flow equation which is defined in Eq. (8) and Eq. (9). Our flow equation solves the CGM’s reduced problem Coleman:1977th , i.e., the minimization problem of kinetic energy ${\cal T}$ while fixing potential energy ${\cal V}$. This minimization problem can be naturally formulated in a flow equation, and the bounce solution can be obtained as a scale-transformed of this solution as Eq. (15). Since our flow equation solves the minimization problem and the existence of the minimizer is guaranteed by Refs. Coleman:1977th ; brezis1984minimum , the convergence of this method is robust against the choice of initial configuration as long as ${\cal V}[\varphi]<0$ is satisfied. A numerical package using this method is presented in Ref. Sato:2019wpo 222 See also https://github.com/rsato64/SimpleBounce. . This package calculates the Euclidean bounce action in ${\cal O}(0.1)$ s with ${\cal O}(0.1)~{}\%$ accuracy for models with 1–8 scalar field(s), which is faster than CosmoTransitions Profumo:2010kp ; Wainwright:2011kj , AnyBubble Masoumi:2016wot , and BubbleProfiler Akula:2016gpl ; Athron:2019nbd .

Acknowledgements

The author thanks Takeo Moroi for useful discussions.

Appendix A CGM’s theorem A

In this Appendix, we briefly summarize the theorem A in Ref. Coleman:1977th . We denote the solution of the reduced problem for given $\cal V$ as $\phi_{(\cal V)}$. This theorem ensures that the bounce solution is given by a scale transformation of $\phi_{(\cal V)}$.

$\phi_{({\cal V}_{0})}$ is a stationary point of ${\cal T}[\phi]+\lambda({\cal V}[\phi]-{\cal V}_{0})$, where $\lambda$ is the Lagrange multiplier. Thus, $\phi_{({\cal V}_{0})}$ satisfies

[TABLE]

Here $\lambda$ should be appropriately chosen for the value of ${\cal V}_{0}$. We define the following configuration $\phi_{B}$:

[TABLE]

We can see that this is the bounce solution. First, by using Eqs. (19, 20), we can check that $\phi_{B}$ satisfies the EOM Eq. (4). Next, let us show that the action of any non-trivial solution of Eq. (4) is equal to or larger than ${\cal S}[\phi_{B}]$. Let $\tilde{\phi}$ be a non-trivial solution of Eq. (4). The action of $\tilde{\phi}$ is extremized under the scale transformation of $\tilde{\phi}$. Therefore,

[TABLE]

There exists a solution of the reduced problem for ${\cal V}={\cal V}[\tilde{\phi}]$, and the kinetic energy is not larger than ${\cal T}[\tilde{\phi}]$:

[TABLE]

${\cal T}[\phi_{B}]$ and ${\cal V}[\phi_{B}]$ are given as

[TABLE]

Here $\lambda\geq 1$ because of $(d-2){\cal T}[\phi_{B}]+d{\cal V}[\phi_{B}]=0$ and Eqs. (21, 22). Thus, by using Eqs. (22, 23), we can show that

[TABLE]

${\cal S}=(2/d){\cal T}$ is satisfied for solutions of Eq. (4). Then,

[TABLE]

Thus, $\phi_{B}$ has the least action among the non-trivial solutions of the EOM.