Absolute Lower Bound on the Bounce Action

The decay rate of a false vacuum is determined by the minimal action solution of the tunnelling field: bounce. In this Letter, we focus on models with scalar fields which have a canonical kinetic term in $N(>2)$ dimensional Euclidean space, and derive an absolute lower bound on the bounce action. In the case of four-dimensional space, we show the bounce action is generically larger than $24/\lambda_{\rm cr}$, where $\lambda_{\rm cr} \equiv {\rm max} [ -4V(\phi) /|\phi|^4] $ with the false vacuum being at $\phi=0$ and $V(0)=0$. We derive this bound on the bounce action \textit{without solving the equation of motion explicitly}. Our bound is derived by a quite simple discussion, and it provides useful information even if it is difficult to obtain the explicit form of the bounce solution. Our bound offers a sufficient condition for the stability of a false vacuum, and it is useful as a quick check on the vacuum stability for given models. Our bound can be applied to a broad class of scalar potential with any number of scalar fields. We also discuss a necessary condition for the bounce action taking a value close to this lower bound.


I. INTRODUCTION
The stability condition of a vacuum is one of the important constraints on viable models of particle physics. Even in the standard model, it gives a nontrivial constraint on the Higgs boson mass and the top quark mass [1]. Furthermore, physics beyond the standard model often introduces additional scalar fields, and they could destabilize the standard model vacuum by giving a deeper vacuum. In these situations, the standard model vacuum is a false vacuum and its lifetime should be longer than the age of the Universe.
The lifetime of a false vacuum in quantum field theory can be calculated by using Coleman's semiclassical method [2]. In this method, the decay rate of a false vacuum per volume is evaluated as Γ/V ∼ Ae −S where A is a prefactor and S is the action for a nontrivial solution of the equation of motion which gives the minimal action. Such a solution is called a bounce solution. To obtain the bounce solution, we have to solve the equation of motion of scalar fields with an appropriate boundary condition. However, it is not always easy to obtain the explicit solution of the equation of motion. In particular, we have to solve a large number of coupled equations of motion if we consider some model with a large number of scalar fields such as the landscape scenario [3,4].
It is convenient if we can discuss a possible range of the minimal bounce action value without solving the equation of motion explicitly. In this context, for example, a generic upper bound on the minimal bounce action is discussed in Refs. [5,6]. A lower bound is discussed in Ref. [7] which focuses on quartic scalar potential, and Ref. [8] which reduces the problem to the effective single scalar problem.
In this Letter, we derive a generic lower bound on the minimal bounce action, which can be applied to a broad class of scalar potential with any number of scalar fields. Our bound can be derived by using a quite simple discussion which is based on the Lagrange multiplier method. The bound has a simple form, and it provides a sufficient condition for the stability of a false vacuum. Therefore, even if it is difficult to obtain the explicit form of the bounce solution, our bound is useful as a quick check on the stability of the false vacuum. In section II, we discuss the lower bound on the minimal bounce action. In section III, we compare our lower bound with the actual value or the upper bound for some representative examples.

II. AN LOWER BOUND ON THE BOUNCE ACTION
Here, we derive an absolute lower bound on the bounce action. We consider m scalar fields with the canonical kinetic term in N -dimensional Euclidean space. The action is given by where φ ≡ (φ 1 , φ 2 , .., φ m ) and U is the inverted potential: U (Φ) ≡ −V (Φ) with V (Φ) being the actual one. Throughout this Letter, we set the false vacuum at φ = 0 (and V ( 0) = 0) without loss of generality. Let us consider N = 4 dimensional Euclidean space for a while. If φ is a solution of equation of motion, it stationalizes the action. Considering the rescaling of the Euclidean space coordinates φ(x) → φ(ξx), we have the following relation which leads to Thus, the problem of finding the minimal action solution can be reduced to that of finding the minimal kinetic energy solution. Since the minimal action bounce solution is known to be O(N ) symmetric for N > 2 and even for multiscalar cases [9][10][11][12], we consider an O(4) symmetric bounce whose radial coordinate is r. With O(4) symmetry, the kinetic energy T is given by The equation of motion is where we denote the "dot" as a derivative with respect to r.
To discuss the minimum kinetic energy T , we define a class of bounce solutions. We characterize them by two parameters: field difference ∆φ a ≡ φ a (0) − φ a (∞) and potential difference ∆U ≡ U [φ(0)] − U [φ(∞)]. Our first goal is to derive a lower bound for such a class of solution. We can easily see ∆φ a and ∆U are functional ofφ a 's. By multiplyingφ a to Eq. (7) and integrating from zero to infinity, we obtain On the other hand, holds. To consider the minimization problem on T with fixed ∆φ a and ∆U , we introduce the Lagrange multiplier α a and β, and defineT as An extremum condition δT /δφ a = 0 giveṡ φ a = − α a r π 2 r 4 + 3β (a = 1, · · · , m).
In the above solution, the Lagrange multiplier α a and β are determined from the constraints ∞ 0 drφ a = −∆φ a and m a=1 ∞ 0 dr(3/r)φ 2 a = ∆U as where |∆φ| = m a=1 ∆φ 2 a . At this point, the solution Eqs. (11,12) is just an extremum, and it is not clear whether this point is the global minimum or not. To check this point, let us seeT again with Eq. (12).
The above equations tells us that the solution Eqs. (11,12) does give the global minimum on T for fixed ∆φ a and ∆U .
Then, we can write the following inequality on the bounce action S (if it exists) by using |∆φ| and ∆U as The above inequality is saturated if and only ifφ a = −α a r/(π 2 r 4 + 3β) holds 1 . To calculate this bound, we need ∆φ a , i.e., φ a (r = 0). Although we do not know about ∆φ a unless we explicitly solve the equation of motion, we can set a bound on the minimal action even without solving the equation of motion. Suppose there exists λ(> 0) such that We can find λ for the potential such that V (φ)/|φ| 4 is bounded below. Then, we can define λ cr as We can see this λ cr is the minimum of a set of λ which satisfy Eq. (15). Then, the bounce action has an absolute lower bound because λ φ ≤ λ cr is satisfied for any value of φ a . As a reference, the Fubini instanton [13], which is a bounce solution with negative quartic potential V = (λ 4 /4)φ 4 , has S = 8π 2 /3λ 4 26.3/λ 4 > 24/λ cr because λ cr = λ 4 holds in this case. To derive the above bound Eq. (17), we do not need the explicit form of the bounce solution. Although the bound Eq. (17) may be weaker than Eq. (14), we can derive Eq. (17) only from the information of the potential. In N (> 2) dimensional case, the same procedure gives the lower bound: N −2 . One may be interested in the condition in which the lower bound Eq. (17) becomes close to the actual value. As long as the true vacuum and the false vacuum are not degenerated, our method can give a good estimation on the lower bound of the decay rate. For detailed discussion, see the Appendix.
So far, we have derived a lower bound on the bounce action. Here let us comment on an upper bound on the bounce action. As is discussed above, by finding a point φ cr which maximizes [−4V (φ)/|φ| 4 ], we can obtain a lower bound on the action. By using this φ cr , we can easily obtain an upper bound as discussed in Ref. [5]. First, we restrict the field space into φ cr direction, which is a straight line passing through the false vacuum φ = 0 and φ cr . We obtain a reduced single field theory on this straight line, and we can easily estimate the bounce action of this reduced action. Then, the resulting bounce action becomes an upper bound on the actual minimal bounce action. Thus, by finding a point φ cr which maximizes [−4V (φ)/|φ| 4 ] , we can obtain both a lower and an upper bound on the actual minimal bounce action at the same time.
Now, let us briefly discuss the applicability of our results. As long as the kinetic term is canonical and once the potential of the scalar fields is determined, our method gives a lower bound on the classical bounce action in a simple way. In some models, quantum corrections or thermal loop corrections are essential to generate a barrier between the false and the true vacuum. In such cases, the effective potential can be used for our method, and our method gives a good estimation on the lower bound of the bounce action as long as the perturbative calculation around the bounce can be used. In general, to obtain the vacuum decay rate precisely, we need to estimate the prefactor by integrating out fluctuations around the actual bounce solution 2 : 2 The gauge dependence and renormalization scale dependence are canceled by considering loop corrections [14,15].
where Γ/V denotes decay rate per unit four-dimensional volume, µ denotes a typical energy scale of the bounce dynamics, S cl denotes classical bounce action, and A is a (normalized) prefactor. As long as the theory is perturbative, we may expect ln A ∼ O(1) although there is a little ambiguity on the definition of µ. In the case of S cl O(1), the vacuum decay rate is mainly determined by the classical bounce action and our bound becomes useful to determine the order of the decay rate. Actually, when we consider the cosmological history of the vacuum, the relevant range of the action is S cl ∼ O(100). Also, the condition for the thermal transition in the expanding Universe is H 4 ∼ T 4 e −S3/T , where H is the Hubble expansion rate and T is temperature. The typical size of dimensionless action S 3 /T is O(100). In these cases, our bound can provide a lower bound on the vacuum decay rate.
Finally, we derive a sufficient condition of vacuum stability in our present Universe. In order to have a stable Universe, the vacuum decay should not happen within a Hubble volume in a Hubble time: where H 0 ∼ 10 −42 GeV is the Hubble constant today.
On the other hand, by using Eq. (14), the vacuum decay rate per volume to the point φ a (if it exists) is bounded as where we assume that the size of prefactor is roughly given by |φ| 4 . By using Eqs. (20) and (21), we can show a sufficient condition of vacuum stability on the shape of the potential: This is a sufficient condition for the stability of a false vacuum. Any false vacuum with any potential which satisfies Eq. (22) has a lifetime which is longer than the age of the Universe.

III. COMPARISONS WITH THE ACTUAL VALUE
In this section, we discuss several explicit examples in four-dimensional space. We will see the consistency of Eq. (17), and furthermore, see that the lower bound Eq. (17) becomes close to the actual value of the bounce action in many cases. In this sense, the lower bound Eq. (17) is a quite useful tool to estimate the value of the bounce action when an explicit calculation is difficult. The first example is a single scalar field theory with a polynomial potential: This potential gives us good insight into a relationship between our bound Eq. (17) and the minimal bounce action S. As discussed in Ref. [6], we can parametrize the minimal bounce action as  HereŜ is a function which only depends on κ. According to the definition given in Eq. (16), λ cr is calculated as By using this λ cr , we obtain the bound onŜ(κ) aŝ Ref. [6] gives the numerical result ofŜ(κ) by calculating the bounce configuration, and we show a comparison between the result of Ref. [6] and the bound Eq. (27) in Fig. 1 and Fig. 2. We can see that our bound becomes close for large negative κ. In this regime, the bounce solution is well described by the Fubini instanton [13]. On the other hand, our bound departs from the numerical value of the minimal bounce action if κ is close to 1/4, in this regime, the false and true vacua are almost degenerate and the bounce solution is well described by thin-wall approximation. There exists a potential barrier between the true vacuum and false vacuum.

B. Multi scalar fields
The second example is a polynomial potential with multiscalar field φ 1 , ..., φ m . We consider a term up to the quartic interaction, and parametrize it as follows where M is a mass scale which does not affect the value of classical action and µ i , γ ijk , λ ijkl denote some dimensionless coupling. Here we do not calculate the bounce configuration explicitly. Instead of the explicit calculation, we estimate a lower and upper bound on the bounce action. The upper bound is estimated by the straight line method described in the later part of Sec. II. We define the ratio between the upper and lower bound as R ≡ S upper /S lower . If this R is close to 1, our lower bound is close to the actual value of the bounce action. Here we take µ = 700 GeV and mτ R = mL + 200 GeV. The blue line is written by using the RHS of Eq. (17). The red dashed line is written by using a fitting formula given in Ref. [16].
We calculate the ratio R by taking µ's, γ's, and λ's as random variables as in Ref. [4]. The ranges of the parameters are taken as We take the range of γ and λ so that the theory remains stable against loop corrections. We generate 1000 parameter points, and show the distribution of R in Tab. I. This result shows the lower bound Eq. (17) becomes close to the actual value of the bounce action in the case of a large number of scalar fields. This feature can be understood as follows. As we have seen in the previous single scalar example, λ cr depends on quartic coupling and the cubic coupling square (see Eq. (26)). In the present case, the typical value of quartic coupling is 1/m and that of cubic coupling square is 1/m 2 . With larger m, quartic coupling becomes more and more relevant and the bounce action becomes close to our lower bound.

C. MSSM
The last example is the MSSM. Supersymmetric models introduce a lot of scalar partners of the standard model fermions, and sometimes they destabilize the standard model-like vacuum. For example, Ref. [16] discussed a vacuum stability in a direction of the third generation slepton with large tan β. The scalar potential for the up-type Higgs H u , the left-handed stauL, and the right-handed stauτ R is given as Here we do not consider the down-type Higgs H d because its VEV is suppressed by 1/ tan β. δ H expresses a radiative correction from the top quark and the stop, and its typical value is δ H 1. A cubic term H * uLτR in the last line destabilizes the standard model-like vacuum. Its coupling constant is proportional to µ tan β. In Fig. 3, we show a comparison between the lower bound on the bounce action which is given in Eq. (17) and Ref. [16]. The lower bound on the bounce action S is 400 at the blue line, and the standard model-like vacuum is sufficiently stable in the lower right region of the blue line. By using the result in Ref. [16], in Fig. 3, we show the red line on which S = 400 is satisfied. We can see our bound Eq. (17) is consistent with the result of Ref. [16].
To discuss the stability in the upper left region, Eq. (17) is not enough in general. However, Fig. 3 shows that the sufficient stability condition by the blue line only differs by 5 % from the upper bound on tan β by the red line. This means that Eq. (17) gives a good estimation on the upper bound of tan β. Actually, Figs. 1, 2 show the lower bound on the bounce action gives a good estimation on the actual value unless the true and false vacua are degenerated. Such a degenerated situation is a special situation in the sense that it requires a tuning of the parameters or an approximate symmetry between two vacua. Thus, we can expect that our discussion is useful to discuss more complicated models.

IV. CONCLUSION
In this Letter, we derived a generic lower bound Eq. (17) on the bounce action by using a quite simple discussion with the Lagrange multiplier. Our bound can be applied to a broad class of scalar potential with any number of scalar field. Necessary information to derive this bound is only λ cr which is defined by Eq. (16). In particular, our bound provides useful information for a model with a large number of scalar fields such as the landscape scenario because we do not need the explicit form of the bounce solution. By using this result, in Eq. (22), we derived a sufficient condition of the stable vacuum of the Universe for a general scalar potential. The bound Eq. (22) can be used as a quick check on the stability of a false vacuum in a broad class of models. As we discussed in section III, the lower bound Eq. (17) gives a good estimation on the actual value in many cases. We also investigated a condition for when the bounce action becomes close to the lower bound. As long as two vacua are not almost degenerated the minimal bounce action can be close to the lower bound. We have seen this feature in some representative examples.
We can see as long as m typ r 2 * r c 1, (A14) the contribution from S r>rc is suppressed. The left hand side of Eq. (A14) is small if we consider small and mild shape of V + . We conclude a necessary condition for the minimal action which is close to the lower bound is If this condition is violated, the minimal action S is significantly deviated from S 0 . Now, let us generalize the previous discussion. For given potential V (φ), we can define λ cr as a minimal λ with We also define ∆U and ∆φ at the point where the equality holds: We can also define φ c as the maximal value of φ with Then, V max is given by a maximal value of V (φ) in φ < φ c . As before, we define r 2 * ≡ ∆φ 2 /∆U , r c ≡ r * (V (φ c )/∆U ) −1/6 and m 2 typ ≡ V max /φ 2 c . Then, if the condition Eq. (A14) does not hold, bounce action will deviate from the lower bound. Thus, this condition can be regarded as a necessary condition for the bounce action to have a value close to the lower bound.
The condition Eq. (A14) characterizes a smallness of the potential barrier. This is because if V max is small, m typ also becomes small. In addition, if V + is small, r c becomes large. And if the barrier is relatively large, the bounce action will deviate from the lower bound 24/λ.