False vacuum decay in kink scattering

In this work we consider kink-antikink and antikink-kink collisions in a modified $\phi^4$ model with a false vacuum characterized by a dimensionless parameter $\epsilon$. The usual $\phi^4$ model is recovered for $\epsilon=0$. We investigate the $\epsilon<<1$ regime where the kink in the presence of false vacuum can be understood as a small deformation of the standard kink for the $\phi^4$ model. We show that the attractive interaction between the kink-antikink pair leads to a rich scattering pattern, in some cases delaying considerably the false vacuum decay.

Embedded in higher dimensional spaces, kinks give rise to domain walls and branes [69].
In some cyclic universe scenarios, domain walls were considered as a possibility to generate the Big Bang conditions [70][71][72][73], where planar symmetry is assumed and the dynamics is effectively (1, 1) dimensional. Solitons are also studied in the context of bubble collisions in the early universe. In the limit of a high nucleation rate per unit four-volume in comparison to H 4 (where H is the Hubble expansion rate), one can neglect the spacial curvature of the universe and consider the bubbles as in flat spacetime [74]. In this regime, collisions of two bubbles have SO(2, 1) symmetry and are described by a (1, 1) dimensional wave equation.
The mechanism of false vacuum decay in scalar field theories is well understood [75,76].
In particular, for an asymmetric double well potential, domain wall collisions with planar and SO(2, 1) symmetry were considered in Ref. [77]. Domain wall collision with such high degree of symmetry can be described as collisions between a kink and an antikink in (1, 1) dimensions. In this background the effects of small initial planar [77] and nonplanar [78] small fluctuations were investigated. However, Ref. [77] only provided some examples for the output of background collisions, with no description of their dependence either with initial velocity or with the parameter controlling the difference in potential energy between the two wells.
The present work intends to fill this gap, with the aim of investigating the kink scattering and the corresponding false vacuum evolution, considering the situation where the false vacuum initially occupies the restricted region between the kink-antikink pair and that where the false vacuum occupies the unbounded region outside the area defined by the antikink-kink system. We consider symmetric kink-antikink and antikink-kink collisions, with the pair initially at equal distances from the center of mass and initial velocities with same modulus. We show how the attractive interaction between the kink and antikink is the main factor affecting the structure of the scattering and, in some cases, delaying considerably the complete decay of the false vacuum even in a restricted region around the center of mass of the kink-antikink pair.
In the next section we review some of the main properties of the φ 4 model. The numerical results are presented in the Sect. III, and we conclude in the Sect. IV.

II. THE MODEL
We consider the action in (1, 1)-dimensions in Minkowski spacetime with a Lagrangian with standard dynamics and a modified φ 4 potential where ǫ is a dimensionless self-coupling parameter. From here on we will consider ǫ ≥ 0 without loosing generality. For ǫ > 0 the potential has a local minimum at φ = −1 (the false vacuum), a local maximum at φ = 0 and a global minimum at φ = 1 (the true vacuum).
Here we are interested in the |ǫ| ≪ 1 limit, where the presence of the linear term in Eq. (2) leads to a small force on the φ 4 kink that favors the expansion of the true vacuum region [81]. For sake of definiteness, we take λ = M 2 = 2. In this case equation of motion is given byφ Linear stability analysis around the static kink solution with φ(x, t) = φ K (x)+cos(ωt)η(x, t) leads to a Schrödinger-like equation: Static solutions can be found for ǫ = 0, given by the usual φ 4 kink φ K (x) = tanh(x) and antikink φK(x) = − tanh(x). These solutions, of minimal energy, can also be obtained by first-order BPS equations [79,80]. Stability analysis of the φ 4 kink have eigenvalues (and corresponding eigenfunctions) [31]: ω 2 0 = 0 (the zero-mode or translational mode) and ω 2 1 = 3 (the vibrational mode), followed by a continuum of states with ω 2 k = 4 + k 2 . The existence of vibrational modes is important for the comprehension of the complex behavior of kinkantikink scattering. One such effect is known as two-bounces, and it is characterized by a collision process where the translational energy is stored in the kink-antikink pair during an amount of time. As a result, the pair, after being scattered, oscillates around the contact point and retrocede for a second collision. This mechanism was described in the Ref. [33] as an exchange of energy between the translational and vibrational modes. Despite applicable in the present case, we remark there are some known exceptions to this mechanism: in the φ 6 model, two-bounces occur even in the absence of a vibrational mode [41]; the presence of more than one vibrational mode can result in the suppression of two-bounce windows [51].
Also, quasinormal modes can store energy during a collision in the φ 4 model [40]. Now back to our problem for ǫ = 0, as far as we know, there are no explicit solutions of the equation of motion. In particular, corrections φ n to the kink field in nth order in ǫ were presented in Ref. [81], and are described in terms of the basis of eigenfunctions {η n }.

III. NUMERICAL RESULTS
The kink scattering processes have a strong dependence on whether the initial region with false vacuum is finite or not. Then in the following we will consider separately kink-antikink and antikink-kink collisions. to ǫ = 0.

A. False vacuum in an unbounded interval: kink-antikink collisions
First of all let us consider a symmetric kink-antinkink collision. We take as initial conditions a φ 4 kink with velocity v and a φ 4 antikink with velocity −v: We fixed x 0 = 15 as the initial kink position, i.e., the kink solution centered at −x 0 and the antikink at x 0 . Note that the initial condition is only a solution for ǫ = 0. We apply this even for ǫ ≪ 1 considering that the system will relax by emitting scalar radiation during their free propagation toward the collision region. To solve Eq. The structure of such two-bounce windows is better visualized in Fig. 2a, which shows the number of bounces as a function of the initial velocity. As shown in Fig. 1a  inelastic scattering behavior with 1-bounce for v > v crit , but with v crit growing with ǫ. This is a direct effect of the tendency of the false vacuum to decay. One example of 1-bounce collision is presented in Fig. 3a. We observed an effect that appears only for ǫ = 0, that is the lack of bounces or even bion states for v < v * : the kink-antikink pair has not enought energy to encounter at the center of mass before the false vacuum decay.
An example of this effect is shown in Fig. 3b. Also, 2-bounce windows are still present, but are thinner in velocity for larger values of ǫ. See an example of this in Fig. 3c.
Another interesting aspect of kink-antikink collisions with the growing of ǫ is the appearing of a two-bounce window of zero order for ǫ 0.008. An example of such effect is presented in Fig. 4. There one can see that there is no oscillations of φ(x = 0, t) between the bounces, meaning that m = 0. The appearing of zero order two-bounce windows was reported before in models with degenerate vacua in the context of boundary scattering [66] and in the study of transition .

B. False vacuum in a bounded interval: antikink-kink collisions
We take as initial condition a φ 4 antikink with velocity v and a φ 4 kink with velocity −v: In the context of false vacuum, the output for antikink-kink collisions is much simpler than the one described above for kink-antikink. This is due to the fact that both the kink-antikink interaction and the tendency of the false vacuum to decay favor a kink-antikink pair to form.
The output of mechanism is a two-step process: firstly the field bounces around the false vacuum with low frequency. Then the antikink-kink pair radiates and the field oscillates in a high-frequency, approaching in the long run to the true vacuum state. This is showed in Figs. 5a-5d for ǫ = 0.001. Surprisingly larger initial velocities are not more effective in inducing the process of vacuum decay. On the contrary, with the growing of v the lowfrequency oscillation lasts even more before the decaying process to the false vacuum. It seems that, with false vacuum, larger initial velocities excite the vibrational state of the kink and antikink in a way that does not occurs for the degenerate vacuum, and the antikinkkink pair can bounce several times and be even more departed during such low-frequency oscillations (compare Figs. 5a-5b).
The relation between high-and low-frequency oscillations is more evident in a plot of the scalar field at the center of mass, φ(0, t), as shown in Figs. 6a-b. There one can see that for a too low initial velocity the field is not even able to bounce and the high-frequency process already takes place, as can be see in Fig. 6a for v = 0.1. As an example for larger velocities we take v = 0.25 as in Fig. 6b. There one can see three low-frequency oscillations around the false vacuum φ = −1 before the field jumps and acquires the high-frequency We found that it is relatively easier for a finite region in a true vacuum to expand through the false vacuum region, as described here in kink-antikink collisions. This is evident for low initial velocities, where the kink-antikink pair scatters at non-null distance. In comparison, it demands considerably more time for a finite region in false vacuum to decay completely to the true vacuum. We showed this for antikink-kink collisions. That is, the attractive kinkantikink interaction favors the formation of a long-lived bion state that inhibits the vacuum decay. The antikink-kink collisions were described in a two-step process with a low-frequency oscillation around the false vacuum followed by a high-frequency oscillation tending to the true vacuum in the long run. This two-step process resembles the behavior of what is known in the literature as false n-bounce-windows. In the end such bounce windows must be false since the scalar field will not remain in the initial state. We also investigated the effect of increasing the gap parameter ǫ. For kink-antikink collisions we have the increasing of both velocities v * and v crit .
For antikink-kink collisions the center of mass belongs to the false vacuum region and we have a two-step process for decay of the false vacuum: a low-frequency fast decaying bouncing state, followed by another long-lived high frequency bion state that radiates continuously.
For antikink-kink collisions, the more notable effect of the increasing of ǫ is the reduction of the maximum separation of the antikink-kink pair during the low-frequency oscillations.
This shows the stronger tendency of the false vacuum to decay. The final state is the same, with the field oscillating fastly around the true vacuum and radiating.

V. ACKNOWLEDGEMENTS
The authors thank FAPEMA and CNPq (brazilian agencies) for financial support. Gomes thanks CAUP from University of Porto for the hospitality. P.P.A. acknowledges the support provided by the FCT grant UID/FIS/04434/2013.