Kink properties in Lorentz-violating scalar field theory

Nonlinear field models are very interesting in various areas of physics including particle physics, condensed matter physics, cosmology and optics [1, 2]. Solitons and solitary waves that have localized energy and propagate without losing the shape are important solutions of some nonlinear field equations. We can cite the following examples of solitons; vertices in superconductors [3] and fluids [4], cosmic strings [5], monopoles [6] and instantons [7]. There are also many works on supersymetric solitons, see for example [8, 9, 10, 11]. The kinks which appear in (1+1)-dimensional models with a potential possessing two or more degenerated minima, are the simplest examples of solitons [12]. Some realistic models in (3+1) or (2+1) dimensions can be reduced to the effective (1+1)-dimensional dynamics. In some theories such as $\lambda\phi^{4}$, the existence of kink in higher dimensions is banned [13]. Actually, to construct a modified kink in such a case we should interpolate from field theory in the Minkowski space-time to field theory in curved space-time background [14].

One of the most important symmetries which should be considered in theories for high energy physics is the Lorentz symmetry. So far, it has not been observed any deviation of this symmetry by experimental inspections. However, Lorentz symmetry may be violated due to the quantum gravity in the Plank scale [15]. Noncommutative theories also constitute an essential motivation for study of Lorentz-breaking theories and involve a Lorentz-breaking tensor $\Theta^{\mu\nu}$ [16, 17]. Therefore, it is interesting to restudy a Lorentz covariant theory when a small violation in Lorentz symmetry is allowed. The standard model and QED, for instance, were extended by the LV terms [18]. The Casimir effect in an LV theory is also considered in some works, see for example [19, 20, 21].

So far, various issues and problems have been studied within the LV extended standard model framework (for instance see [22] and references therein). This investigations have been done with basically a twofold purpose: the determination of new physical effects induced by the Lorentz-violating coefficients and the imposition of stringent upper bounds on the LV coefficients that constrain the magnitude of Lorentz breaking. Our goal in this paper is to study the former. Such studies have been done in several papers such as Refs. [23, 24, 25]. In Ref. [23], authors have investigated the static limit of Lorentz-violating electrodynamics in vacuum and in media and obtained explicit solutions for some simple cases. More details of this theory have been investigated within the Refs. [23, 24] and classical solutions are also properly examined [26, 27].

In addition, LV effects on topological defects are considered e.g. in Refs. [28, 29, 30, 31] and, in particular, for scalar field theories in Refs. [32, 33, 34]. In Ref. [34], authors considered two models in (1+1) dimensions (with one and two scalar fields) in which the Lorentz and CPT symmetries are violated. Actually, in the case of a single scalar field, the LV term, modeled by $-\frac{1}{2}k^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, where $k^{\mu\nu}$ is LV parameter, can be removed through a linear transformation of coordinates, such a transformation will generate a Jacobian multiplier in the new action. In fact, one can imagine $k^{\mu\nu}=cu^{\mu}u^{\nu}$ characterized by the LV vector $u^{\mu}$ which is a spacelike two-vector denoting the aether [35, 36]. Therefore, in (1+1) dimensions, we can say that there does not exist any observable effect due to the leading order of this CPT even Lorentz violating interaction of scalar field to the aether. Therefore, the nontrivial contributions due to this term appear when we couple the model with more sophisticated fields. However, the effects of higher order LV terms have not been yet considered. One may naturally expect that there must exist some effects due to more complicated LV interacting terms. In this paper, we investigate the unremovable LV effects that occurred in the next to leading order. Explicitly, we study the LV interacting scalar field theory modeled by $-\frac{1}{2}c\phi^{2}(u\cdot\partial\phi)^{2}$ in (1+1) dimensions. This is the lowest order term which leads to concrete effects on the kink properties. We discuss two different forms of LV terms $cu^{\mu}$, time-like and space-like, i.e. Lorentz symmetry breaking in the direction of time and space, respectively. For the case of the time-like vector, we see there is no concrete effect. Therefore, we study only the space-like LV vector.

After a very brief review of the standard kink in Sec. 2, we first, find the modification to the shape of the kink in subsection 2.1 and then calculate the mass correction due to this LV term. The stability equation is obtained in Sec. 3a and based on this new corrected stability equation we find the modification of bound and continuum states. The last section is devoted to summarizing the results and conclusions.

Here we first, very briefly, explain the standard kink and derive its mass. The appropriate Lagrangian for spontaneously broken symmetry in 1+1 dimensions within the real scalar $\phi^{4}$ theory is:

By making the following scaling:

the dimensionless Lagrangian is achieved as follows:

Equation of motion for Lagrangian (3) reads as

This equation has four static solutions: two trivial non-topological solutions $\phi_{\rm{tri.}}(x)=\pm 1$, and two topological ones (kink and anti-kink):

which approach two different values as $x\to\pm\infty$ corresponding to the two possible vacuum states. (For more details see please [37]).
Putting the solutions (5) in the related Hamiltonian

one can find the familiar classical mass of the kink as

As we know, the trivial vacuum and kink are classical static solutions of a nonlinear relativistic field equation. In principle, they can be associated with quantum extended particle states. Therefore, we are now going to study the relevance of them to the corresponding quantum field theory. To do this, it may be thought that one needs either the Feynman path-integral or the canonical quantization methods. However, another simple method which is used in the literature is an approach through which one expands the energy corresponding to the static states via a semiclassical method. Then, using the classical solutions one can obtain a little information about the quantum behavior of the system. This semiclassical quantization technique has been firstly introduced for relativistic field theories, using an appropriate generalization of the WKB approximation in quantum mechanics by Dashen, Hasslacher and Neveu [38, 39]. For future use, here, we very briefly review this approach and some of its conclusions.

The energy of static states involves only the potential energy which can be expanded about a minimum as follows:

where $\zeta(x)=\phi(x)-\phi_{0}(x)$ and $\cal U$ is the interacting potential density along with the mass term. In fact, this is functional Taylor expansion of $V[\phi(x)]$ about the static solution $\phi_{0}(x)$. In Eq. (14), integration by part has been used and dots represent cubic and higher terms which can be treated by perturbation. This method can be used when the weak coupling approximation is established. The first term in Eq. (14) is related to classical contribution in energy. The second term is the leading order quantum correction which is similar to the harmonic oscillator with well-known quantum physics. We can find the normal modes by solving the following stability equation:

Now, one can construct the energy spectrum of a static state corrected by quantum fluctuations as follows:

where $n_{i}$ is the excitation number of the $i$th normal mode. Using the Lagrangian Eq. (3), the Eq. (15) reduces to

For the trivial vacuum $\phi_{0}=\phi_{\rm{tri.}}=\pm 1$, this equation has plane wave solutions $e^{\pm ikx}$ on which we can impose the Periodic Boundary Condition (PBC) in a box with length $L$. Therefore, $\omega_{n}^{2}=k_{n}^{2}+2$ with $k_{n}=2n\pi/L$, and the energy becomes

For the $\phi_{0}=\phi_{\rm{k}}=\pm\tanh\left(\frac{x-x_{0}}{\sqrt{2}}\right)$, the Eq. (17) becomes

where we choose $x_{0}=0$. The Eq. (19) has two bound state solutions (regardless of normalization factors)

followed by a continuum

The zero mode is corresponding to the translational invariance of the potential. The asymptotic behavior of $\zeta_{q}(x)$ at $\pm\infty$ is $\exp[iqx/\sqrt{2}\pm\delta(q)/2]$ where $\delta(q)=-2\tan^{-1}\left(\frac{3q}{2-q^{2}}\right)$ is the phase shift of scattering states. Again, the PBC gives the allowed mumenta

and the energy spectrum is

For more details please see [37], for instance.

The Lorentz symmetry may be broken in high energies where the gravity must be quantized. Therefore, we need to check what causes the Lorentz violation in low energy theories. In this paper, we have studied the effects of an LV potential on the energy spectrum predicted for the $\phi^{4}$ scalar field theory in two space-time dimensions. We have considered an additional fundamental interaction $\frac{-c}{2}\phi^{2}(u.\partial\phi)^{2}$ that couples the derivative of a field to a constant vector describing LV. This is the lowest term which leads to nontrivial effects in the theory. The modifications of wave functions and the corresponding energy spectrum have been investigated for both trivial vacuum and kink. The trivial vacuum is not affected by the LV potential. However, up to the first order of the LV coefficient $c$, we have found the kink and its excitation modes (all bound and continuum states) are modified. One may expect such modifications because kink is a result of potential and this new LV term is also in the same order in $\phi$’s.