Scattering of metastable lumps in a model with a false vacuum

In this work we consider the scalar field model with a false vacuum proposed by A. T. Avelar, D. Bazeia, L. Losano and R. Menezes, Eur. Phys. J. C 55, 133-143 (2008). The model depends on a parameter $s>0$. The model has unstable nontopological lump solutions with a bell shape for small $s$, acquiring a flat plateau around the maximum for large $s$. For $s\to\infty$ the $\phi^4$ model is recovered. We show that for $s\gtrsim 2$ the lump is metastable with the only negative mode very close to zero. Metastable lumps can propagate and survive long enough to produce dynamical effects. Due to their simplicity, they can be an alternative to the procedure of stabilization which requires, for instance, a complex scalar field to construct nontopological solitons. We study lump-lump collisions in this model, describing the main characteristics of the scattering products at their dependence on $s$ and the initial velocity modulus of each lump.


INTRODUCTION
In classical field theory, some solutions can be classified with respect to the topological mapping between the space of coordinates and the internal space of configurations, and are of interest in physics [1][2][3]. In a scalar field theory in (1,1) dimensions, the topological defect is the kink (and antikink). In this case, the scalar field φ(x) has different asymptotic limits for x → ±∞ and the solution is linearly stable. The static kink solution φ K (x) is related to the static antikink solution φK(x) by φK(x) = −φ K (x). The nontopological defect is named lump [4,5]. In this case, the scalar field has equal asymptotic limits for x → ±∞ and is unstable under linear perturbations.
In some of these applications, the lump can be stabilized if described by a complex scalar field, as done in q-ball models [22], or coupled with other charged matter fields [23]. In this case the stable solution has a conserved Noether charge and is named by some authors as nontopological solitons.
Models with analytic static lump solutions were studied in the Refs. [23][24][25][26]. In the Ref. [27], the authors investigated the fermion transfer due to the scattering between an antikink and a lump in the φ 4 model with a Yukawa coupling. The collisions between lump and solitary waves in (3, 1) dimensions were studied in the context of extended Kadomtsev-Petviashvili equation, which describes the multi-component plasma model [28]. In the Ref. [29], the generation and the evolution of lump Benney-Luke solitary waves was analyzed.
These are solutions of the Kadomtsev-Petviashvili I equation, which model small amplitude shallow-water waves. The existence of such waves for the Benney-Luke equation with surface tension it has been discussed in the Ref. [30], and in appropriate limits, this equation reduces to the integrable Korteweg-de Vries (KdV) equation. In the Ref. [31] it was discussed the scattering between lumps with periodic waves, with other lumps and with kink solutions.
There, it was observed interesting effects such as the fission of lump waves. Motivated by the interest in non-diffractive and non-dispersive wave packets propagating in optical media, in the Ref. [32] the authors showed the existence and interactions of dark-lump solitary wave solutions of the nonlinear Schrödinger equation in (2, 1) dimensions. The results showed connections between nonlinear wave propagation in optics and hydrodynamics.
In this work, we considered the lump scattering in (1, 1) dimensions in a model with false vacuum that is an extension of the φ 4 model. In the next section, we introduce the model and the nontopological solutions. Furthermore, we also discuss the linear stability analysis of the solutions. In the Sect. III, we present the structure of lump-lump scattering. We report the formation of localized oscillations and kink-antikink pairs. We present our main conclusion in the Sect. IV.

II. THE MODEL
We consider the following action with standard dynamics (we work in a flat metric with signature (+−)) The equation of motion is given by Static solutions that minimize the energy satisfy the following first-order equations: In this work, we consider the potential given by [26] V where s > 0 is a real parameter. In the limit s → ∞ the φ 4 model is achieved. The Fig. 1a shows this potential for some value of s. The Fig. 1b shows the main characteristics of the potential for s = 1. For general s, the potential has a local minimum at a global minimum at and a local maximum at and crosses the φ axis at φ back = tanh(s).
We note that the value of V (φ) at φ + is always positive. On the other hand, the value of the potential at φ − is negative, but getting to zero for s → ∞. Then, the model is characterized The lump solution of the Eq. (4) centered at x = 0 is given by [26] The The linear stability analysis considers perturbations around the static solution of the form . This leads to a Schrödinger-like equation, A plot for this potential for several values of s is depicted in the Fig. 3. This potential can  be separated in two regions from the values = (1/2) ln( √ 3 + 2) ≃ 0.66 [33]. From s ≤s the potential has a single-well, whereas for s >s the potential takes the shape of a double-well potential. Moreover, for s 3 the potential acquires a plateau around x = 0 whose tickness increase with s [26]. In this region the potential is similar to that of a kink-antikink pair.

III. LUMP-LUMP COLLISIONS
In this Section we will analyze lump scattering for 0.5 < s < 9, covering all qualitatively (2), we use the following initial conditions: where Some examples for a large value of s is presented in the Fig. 7. There one sees that the lump solutions propagate without radiating significatively before scattering, as expected from a metastable solution. After the scattering one can see the production of a central bion (Fig. 7a), two (Fig. 7b) and three (Fig. 7c) escaping oscillations along with continuously emitted radiation. Note that the bion has a more irregular aspect than the propagating oscillations. In all cases one can see that after the scattering the output states are acompanied by a larger rate of radiation emission in comparison to the travelling lumps.
For some large values of s one can also see the production of one or two kink-antikink-like pairs. For instance, in the Fig. 8a one sees that after the interaction, the lump-lump pair produces a kink-antikink-like pair that escape to infinity, accompanied by radiation. In the Fig. 8b we have the production of two kink-antikink-like pairs. The Fig. 9 is a phase diagram v × s that shows the final state of the scalar field at x = 0.
The unstable behavior of the lump is evident for s 0. identified in blue.
The diagram of the Fig. 9 shows that metastable lumps, where s 3, has an intrincate pattern of scattering for small velocities, where the lump-lump pair has more time to interact.
On the other hand, for ultralarge velocities, the diagram assumes a blue pattern, showing that it is very difficult to produce oscillations at x = 0 or just one kink-antikink pair. In these situations, metastable lumps will produce or a pair of oscillations, or two kink-antikink pairs. For even larger values of s, the results tend to agree with those of the scattering of in the φ 4 model, corresponding to s → ∞. The Fig. 10 shows that, for low initial velocities, it is more probable the production of one kink-antikink pair intercalated for some specific velocities with the production of two oscillations for small values of s. This agrees with the random sector of the diagram of the Fig. 9, a green region with some pixels in blue.
Also, we note the presence of windows in velocity for the production of two oscillations, corresponding to the fringes in blue in the Fig. 9. For large velocities, the Fig. 10a shows that, for s = 3.5, it is more frequent the occurrence of two propagating oscillations than the production of two kink-antikink pairs. Both situations correspond to the blue region of the upper part of the Fig. 9. Comparing the Figs. 10a-d we see that the increasing of s reduces the occurrence of two propagating oscillations, favoring the production of two kinkantikink pairs. The production of one or three oscillations are events with lower frequency.
Also events with four oscillations was registered, but they are rare, occurring only for very specific velocities.

IV. CONCLUSIONS
In