Degenerate vacua to vacuumless model and $K\bar K$ collisions

In this work we investigate a $Z_2$ symmetric model of one scalar field $\phi$ in $(1,1)$ dimension. The model is characterized by a continuous transition from a potential $V(\phi)$ with two vacua to the vacuumless case. The model has kink and antikink solutions that minimize energy. Stability analysis are described by a Schr\"odinger-like equation with a potential that transits from a volcano-shape with no vibrational states (in the case of vacuumless limit) to a smooth valley with one vibrational state. We are interested on the structure of 2-bounce windows present in kink-antikink scattering processes. The standard mechanism of Campbell-Schonfeld-Wingate (CSW) requires the presence of one vibrational state for the occurrence of 2-bounce windows. We report that the effect of increasing the separation of vacua from the potential $V(\phi)$ has the consequence of trading some of the first 2-bounce windows predicted by the CSW mechanism by false 2-bounce windows. Another consequence is the appearance of false 2-bounce windows of zero-order.


INTRODUCTION
Solitary waves are solutions from nonlinear physics characterized by the very special property of localized density energy that can freely propagate without loosing form. The realization of solitary waves in nature is now amply recognizable, with interesting effects in solid state and atomic physics [1]. In cosmology solitary waves are also studied in the context of bubble collisions in the primordial universe, where in some limit the effect of gravitational interaction between the bubbles can be negligible and the process of collisions can be described by an effective (1, 1) dimensional model [2]. The simplest solitary wave is described by the (1, 1) dimensional kink. Usually kink models are constructed starting from a Lagrangian density where the φ is a real scalar field and a potential V (φ) with minima at non-zero values of φ that suffers a spontaneous symmetry breaking. The equation of motion for the field φ(x, t) is given by where V φ = dV /dφ. If we can write the potential in terms of the superpotential V (φ) = 1 2 W 2 φ , we note that the energy will obeys the first-order differential equation The defects formed with this prescription minimize energy and are known as BPS defects [3,4].
Stability analysis considers small fluctuations resulting in a Schödinger-like equation As one knows [5], this equation is factorizable, since it can be written as which forbids the existence of tachionic modes. Zero-mode exist, corresponding to a translational degree of freedom of the kink/antinink, and is given by with C a normalization constant.
Nonintegrable models like those considered here have the scattering process with a very rich character. The archetype model is the φ 4 , where the main aspects where studied in deep (see for instance refs. [6][7][8]). There one knows that large initial velocities v > v crit lead to a simple scattering process where the KK pair encounter and after a single contact recede from themselves. This is called a 1-bounce process. Small initial velocities lead to the formation of a bound KK state called bion that radiate continuously until be completely annihilated. In the bion region and near to the frontier region in velocities v ∼ v crit it can occur the formation of 2-bounce windows that accumulates toward v = v crit with lower and lower thickness. Between a border of a 2-bounce windows and the bion region another system of 3-bounce windows can be found. This substructure is verified for even high levels of bounce windows in a fractal structure.
Stability analysis of the kink lead to a Schrödinger-like equation where the presence of zero-mode in related to the translational invariance of the kink (invariance under Lorentz boosts). Usually one interprets the presence of non-null bound states as vibrational states.
In the standard Campbell-Schonfeld-Wingate (CSW) mechanism, a resonant exchange of energy between the translational and vibrational modes is responsible for the structure of 2-bounce windows [9]. As far as we know there are two exceptions to this scenario: i) despite the absence of vibrational mode in the perturbation of a kink, considering the effect of collectiveKK structure, it was explained the occurrence of 2-bounce windows in the φ 6 model [10]; ii) the presence of more that one vibrational state can in some circumstances destroy the 2-bounce structure [11].
In this work we are interested in study the effect of the separation of the vacua of the potential V (φ) in the process of KK collision, focusing mainly on the appearance and structure of 2-bounce windows. In the Sect. II we will investigate a model with unusual scattering properties that can also contribute for the understanding of the mechanism of formation of 2-bounce windows. In Sect. III we present the numerical analysis of the KK scattering process. Our main conclusions are reported in Sect. IV.

II. THE MODEL
An exception of the known mechanism for constructing kinks are the so called vacuumless defects which are constructed in models with a potential that has a local maximum but no minima. One example is the model proposed by Cho and Vilenkin [12]: Potentials of this type appear also in non-perturbative effects in supersymmetric gauge theories [13]. The equation of motion for the scalar field has the solution which has finite energy.
Static solutions of this model where further studied both in their gravitational aspects in (3, 1) dimensions [14] and concerning to their topological structure and trapping of fields in (1, 1) dimensions [15]. More recently, Dutra and Faria Jr [16] considered an extension described by the potential  Static solutions for the scalar field are where plus and minus signs are for kink (K) and antikink (K), respectively. The vacua of the model are described by [16] φ(

III. NUMERICAL RESULTS
In this section we present our main results of the KK scattering process. We are interested in the effect of the separation of vacua of V (φ) given by Eq. We considered a symmetric KK collision. Initial conditions are given by sufficiently distant defects described by boosted free solutions: Here φ K means free kink solution. We fixed bounce windows at all, in accord with what expected from the standard CSW mechanism [9].
Note that for this case the critical velocity v crit is close to 1, showing that the scattering process leads almost always to bion states with just a small interval in initial velocities leading to collisions with 1-bounce. We also noted that v crit grows with the reduction of A also the small critical velocity v crit ∼ 0.28 (when compared to v crit ∼ 0.97 for A = 0.1), indicating that potentials with degenerated vacua with smaller |φ| favor the enlargement of the 1-bounce region. We note also the presence of a first smooth peak unexpected from CSW mechanism. This is called zero-order false 2-bounce windows, and where first observed in the context of φ 6 model [10]. Now if we slowly reduce the value of A back to zero, the structure of zero-order false 2-bounce windows is enlarged, reducing their height until finally disappear for A 0.2.
Also for A ∼ 0.4 there is the formation of two more false 2-bounce windows beside the zeroth-order. This is possibly connected to the presence of a maximum in the energy of the vibrational state, as seen in Table I. This behavior of false windows grows until a complete disappearance of 2-bounce windows in Fig. 4a for

IV. CONCLUSIONS
In KK scattering process where the 2-bounces occur, but instead of being separated, the pair is turned either to an oscillon or to a bion state, finally resulting in total anihilation. In particular, the oscillon state seems to have more general relation to the nonlinearity of the model, having been observed for some velocities even far from the 2-bounce windows (true of false).
This process continues when one reduces the parameter to the region where the vacua are more departed. In particular case A = 0.3 is characterized by the total suppression of 2-bounce windows, substituted by a sequence of false 2-bounce windows, despite having one vibrational state. Here the CSW mechanism works in the sence of predicting the existence and characteristics of the scattering processes from each 2-bounce windows. However, despite the initial scattering mechanism occurs accordingly, in the end it is frustrated, the pair being not allowed to separate anymore. In some sense the model introduces an asymmetry in the influence of A on the behavior of the scattering process: region of A < A c = 0.4 (where A c marks the maximum of energy of vibrational state) favors the frustration of the scattering mechanism followed by a changing of vacuum of the solutions. On the other hand, region A > A c there is no such process. This is a clear indication of the influence of a vacuumless character on the frustration of 2-bounce windows.

V. ACKNOWLEDGEMENTS
The authors thank FAPEMA and CNPq for financial support.