Kink scattering in a Lorentz-violating $\phi^{6}$ model

In $(1+1)$-dimensional nonlinear scalar field models with more than one degenerated vacua, there may exist a type of classical solution called kink, which smoothly connects different vacua and has localized energy density. In the past decades, kink and its higher-dimensional extension, domain wall, have been extensively studied in both condensed matter physics and high energy physics [1].

A related topic that has been repeatedly discussed recently, is the scattering of non-integrable kinks. Unlike the collision of integrable kinks where the kinks simply pass through each other, a pair of incoming non-integrable kinks (for instance, $\phi^{4}$ kinks) can be bounced back after a finite time of the collision or form a bound state called bion/oscillon, depending on their initial velocities, say $v_{0}$ and $-v_{0}$ for symmetric collisions [2, 3, 4, 5]. The velocity intervals within which kinks are bounced back after $n$ collisions are called $n$-bounce windows ($n$BWs). For the $\phi^{4}$ model, the $n$-bounce windows form an interesting fractal structure [6].

Bounce windows have also been found in many other non-integrable kink models [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. A well-accepted explanation for the appearance of bounce windows is the resonant energy exchange between the translational and vibrational modes of the kink [5, 9]. This idea is shown recently using a collective coordinate model [26]. In addition to double kink collisions, the simultaneous collisions of multi-kink at the same point have also been investigated in many works [27, 28, 29, 30, 31, 32]. Such collisions can produce a very large energy density around the collision point, and the properties of the high energy region are important for predicting the maximal number of kink-antikink pairs formed in wave train collisions [33]. In some models [27, 34], one can also pinpoint the location of bounce windows by observing the behaviors of maximal energy densities, which vary smoothly within the bounce windows but become chaos elsewhere. Other interesting topics include deformed $\phi^{4}$ theory [17], deformed sine-Gordon model[35], etc.

Despite many interesting discoveries, non-integrable kink collisions are rarely considered in Lorentz-violating models. Considering the fact that Lorentz symmetry breaking naturally arises in many fundamental theories, such as canonical and loop quantum gravity [36, 37, 38], noncommutative quantum field theory [39], and might have observational implications [40, 41] (see [42] for more references), it would be interesting to consider kink collisions in Lorentz-violating non-integrable scalar field models. However, the breaking of Lorentz symmetry hinders one from deriving moving kink solutions by simply boosting static ones, which are necessary for the construction of the initial configuration. Therefore, the study of kink collisions was limited to the models that preserve the Lorentz invariance for a long time, until Bazeia and others proposed a model in which there is a simple transformation relationship between the static and dynamic kink solutions [43].

In this paper, we study the kink interactions in a Lorentz-violating $\phi^{6}$ model using the traveling kink solution reported in Ref. [43]. In the following section, the Lorentz-violating $\phi^{6}$ model is considered and the kink solutions in the Lorentz-violating $\phi^{6}$ model are derived. Then we analyze the linear perturbation modes of both the single kink and double kink solutions. The numerical simulation of the kink collisions, as well as the dependencies of several physical quantities on the Lorentz-violating parameter, is conducted subsequently.

The $(1+1)$-dimensional Lorentz-violating $\phi^{6}$ theory is defined by the Lagrangian density:

where we introduced the conventional Minkowski metric $\eta^{\mu\nu}\equiv\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ and a tensor $\kappa^{\mu\nu}\equiv\begin{pmatrix}0&\alpha\\ \alpha&0\end{pmatrix}$, representing the Lorentz violation.

Since the $\phi^{6}$ model contains three degenerate vacua ${\{-1,0,1\}}$, kinks and antikinks take two types, each of which is connected to neighboring vacua and has to be studied separately.

The Lagrangian (Eq. 1) yields the EoM for the field

The static kink solution interpolating $(0,1)$ is found to be the same as the ordinary $\phi^{6}$ model solution

The corresponding antikink is

and kinks belong to the other topological sector are

To obtain a moving kink solution, one has to resort to a more sophisticated method, for the ordinary Lorentz boost simply does not hold in this Lorentz-violating model. However, one can still carry out a boost-like approach by deforming the Lorentz factor $\gamma\equiv 1/\sqrt{1-v^{2}+2\alpha v}$ [43]. The traveling kink solution can then be expressed as

where $x_{0}$ and $v_{0}$ are the initial position and velocity of the kink.

As is known, the breaking of the Lorentz invariance breaks the symmetry (the width, the energy density, etc) between moving kink and antikink, and kink collisions would be asymmetrical. The spatial configurations of kink-antikink ($K\bar{K}$) and antikink-kink ($\bar{K}K$) pairs at $\alpha=0,1,2$ are shown in Fig. 1. As $\alpha$ increases, the asymmetry between the kink and the antikink increases. Note that the energy density distribution of the same kink is also different if it is moving oppositely, as indicated by the dashed lines.

To prove the stability of the kink solutions and get a better understanding of the bounce windows, the linear perturbative states of the static kink solutions should be examined. Consider a small perturbation $\delta\phi(x,t)$ around a background (a static kink or a two-kink configuration), by taking the variation of the action with respect to $\delta\phi(x,t)$ and Fourier expand the field $\phi(x,t)$, we immediately arrive at a Schrödinger-like equation:

where a factorizable Hamiltonian $\mathscr{H}\equiv-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}x^{2}}+\frac{1}{\theta}\frac{{\mathrm{d}}^{2}\theta}{{\mathrm{d}}x^{2}}=\left(\frac{{\mathrm{d}}}{{\mathrm{d}}x}+\frac{1}{\theta}\frac{{\mathrm{d}}\theta}{{\mathrm{d}}x}\right)\left(-\frac{{\mathrm{d}}}{{\mathrm{d}}x}+\frac{1}{\theta}\frac{{\mathrm{d}}\theta}{{\mathrm{d}}x}\right)$ (with $\theta=\partial_{x}\phi_{s}$) is introduced, and $\tilde{\omega}_{n}^{2}\equiv\left(1+\alpha^{2}\right)\omega_{n}^{2}$ is then always non-negative according to the supersymmetric quantum mechanics, thus proving our claim that the static solutions are stable against small perturbations. Note that $f_{n}(x)$ is only the expansion coefficient of perturbation, and $g_{n}(x)\equiv f_{n}(x)\mathrm{e}^{{i}\alpha\omega_{n}x}$ is the complete eigenfunction.

For the single kink solution $\phi_{s(0,1)}(x)=\sqrt{\frac{1+\tanh(x)}{2}}$, the effective potential in the Schrödinger-like equation is $V_{\mathrm{eff}}=\frac{\partial_{x}^{3}\phi_{s}}{\partial_{x}\phi_{s}}=\frac{15}{4}(\tanh(x)+1)^{2}-6(\tanh(x)+1)+1$ and only translational mode survives.

For the two-kink static configurations,

the effective potential becomes $V_{\mathrm{eff}}^{2K}=15\phi_{2K}^{4}-12\phi_{2K}^{2}+1$, where $\phi_{2K}$ denotes either $K\bar{K}$ or $\bar{K}K$ pair. It turns out that the $K\bar{K}$ pair still supports only translational mode, but $\bar{K}K$ configuration supports many of the vibrational modes, just as the case in the ordinary $\phi^{6}$ model [9]. The eigenvalues $\tilde{\omega}_{n}^{2}$ depend both on the half-separation $x_{0}$ and the Lorentz-violating parameter $\alpha$. The eigenfunctions $g_{n}(x)$ for the first five modes of $\bar{K}K$ pair with half-separation $x_{0}=10$ ($14$ modes in total) are shown in Fig. 2. The $\alpha-$dependence of the vibrational modes indicates that the kink collision results will be affected by the Lorentz violation. Various novel kink-collision phenomena will be discussed in the following sections.

In this section, we study the asymmetrical kink collisions. Due to the absence of exact kink scattering solutions of the non-integrable model, one has to resort to numerical simulation. The initial conditions of kink scattering are set by superpositions of individual kinks and antikinks, which is Eq. 8 and

Since a single kink has an exponentially decaying tail, the overlap in the initial conditions can be safely ignored at a large distance. Initial conditions for more kinks can be obtained in the same way.

The Fourier spectrum method [44] with periodic boundary conditions is used to solve the EoM numerically, and one may control the validity of the simulation by insisting on energy conservation. The total energy of the configuration is calculated by

and can be approximated by

where $E\left(v_{0}\right)=\gamma\left(1+\alpha v_{0}\right)M_{K}$ is the energy of a moving kink and the mass of the kink is $M_{K}=\int[(\partial_{x}\phi_{(0,1)})^{2}/2+V]dx=\frac{1}{4}$. The relative error between $E_{\mathrm{th}}$ and the calculated energy $E_{\mathrm{num}}$ will be checked to guarantee the validity of simulation results. The EoM is discretized, with the time step chosen automatically by MATLAB ode45 solver, typically around $0.002$, to ensure the relative error of energy to be less than $10^{-9}$. The space step is chosen to be $0.1$ and $0.05$ to guarantee the results are convergent and no significant change happens with further precision improvement.

In this study, we were concerned mainly with the kink scattering of a Lorentz-violating $\phi^{6}$ model. The static and boosted kink solutions were obtained in the spirit of Ref. [43]. It was found that with Lorentz invariance broken, traveling kinks and antikinks became asymmetrical. The following analysis proved the stability of kink solutions. The eigenfunctions of the $\bar{K}K$ linear perturbation for $\alpha=0,1,2$, $0\leq n\leq 4$ and separation $x_{0}=10$ were calculated, as shown in 2.

After the general remarks, the asymmetrical kink collisions are studied for both the $K\bar{K}$ and $(\bar{K}K)$ configurations. The Lorentz-violating $\phi^{6}$ model has no BW for $K\bar{K}$ collisions and no 3BW or higher bounce windows for $\bar{K}K$ collisions. The structure in Fig. 3 shows how $v_{c}$ decreases due to the presence of $\alpha$. We observed rising windows in the structure of $K\bar{K}$ scatterings, and the damping of the field value at the origin, or equivalently, the deviation of bions from the origin under Lorentz violation. To the author’s knowledge, these two effects have not been reported yet.

We also discovered an empirical formula Eq. 12 of $v_{c}$ dependencies on $\alpha$ for the two topologies in the $\phi^{6}$ model and that of the $\phi^{4}$ model. One can study higher-order Lorentz-violating models to generalize the formula. We then studied the impacts of $\alpha$ on maximal energy densities. Due to the lacking of different modes of single kinks, the dependencies have no intricate structure as that of the $\phi^{4}$ model.

The impact of Lorentz violation on the multi-kink collisions was also studied. We found that the presence of small $\alpha$ would change the collision process. In those examples, the kinetic energy is transferred to forming bions or enlarging their amplitude.

The author is grateful to Yuan Zhong for helpful discussions.