Explicit kinks in higher-order field theories

We study an example of higher-order field-theoretic model with an eighth-degree polynomial potential -- the $\varphi^8$ model. We show that for some certain ratios of constants of the potential, the problem of finding kink-type solutions in $(1+1)$-dimensional space-time reduces to solving algebraic equations. For two different ratios of the constants, which determine positions of the vacua, we obtained explicit formulae for kinks in all topological sectors. The properties of the obtained kinks are also studied -- their masses are calculated, and the excitation spectra which could be responsible for the appearance of resonance phenomena in kink-antikink scattering are found.


Introduction
Topological solitons is an important class of solutions of field-theoretic models which are of great importance to high energy physics, cosmology and condensed matter [1][2][3][4][5]. In this context, models with polynomial potentials, in turn, are widely used. Apart from applications in high energy physics theory, such models are used to simulate spontaneous symmetry breaking in the Ginzburg-Landau model of superconductivity [6,7], (consecutive) phase transitions in materials [8,9], field evolution in the early Universe [10] etc., see also [5,11,12] for review.
Recently, models with potentials in the form of polynomials of eighth degree and higher are of growing interest [8,[30][31][32][33][34]49]. In particular, the excitation spectra of the ϕ 8 kinks with exponential asymptotics, as well as resonance phenomena in the scattering of such kinks at low energies, were studied [31]. As a separate branch of study, one can emphasize investigation of properties of kinks with power-law asymptotics. Such kinks are topological solutions of, e.g., the ϕ 8 , ϕ 10 , or ϕ 12 model with particular form of potential. More specifically, the potential must have a minimum, which is a zero of the fourth or higher order. Then the corresponding kink has a power-law asymptotic behavior at that spatial infinity at which the field approaches the aforementioned minimum, see, e.g., [32,Sec. II.A].
Due to the presence of power-law tails, kinks acquire new properties. In particular, a kink and an antikink (or a kink and a kink) placed at a certain distance from each other interact much more strongly than in the case of exponential asymptotics. This phenomenon is called long-range interaction of kinks with power-law tails [32][33][34]49]. In Ref. [33] scattering of the ϕ 8 kinks with power-law asymptotic behavior has been studied numerically. Besides, it was shown that resonance phenomena in the kink-antikink collisions could be a consequence of the presence of the vibrational modes of the "kink+antikink" system as a whole.
Recent works [32] and [34] continued the study of interactions of kinks with power-law asymptotics. It was demonstrated that the presence of the power-law tails entails longrange interaction which, in turn, requires special approach to constructing of the initial conditions for the numerical simulations of the kink-antikink and kink-kink collisions. The problem is that the conventional initial conditions which were used in the case of exponential asymptotics, being applied to the kinks with power-law tails lead to appearance of significant disturbances due to radiation. This, in turn, creates the illusion of repulsion between kink and antikink [33]. Several methods for "distilling" the initial configurations into suitable ansätze were proposed, it was shown how these approaches capture the attractive nature of interactions between the kink and antikink in the presence of long-range interaction [32]. The general results on the interactions of kinks with power-law asymptotics in ϕ 2n+4 models for n ≥ 2 have been obtained in [34]. It was found that the interaction between kink and antikink is generically attractive, while the interaction between two kinks is generically repulsive. The force of interaction falls off with distance as its 2n/(n − 1)-th power. The obtained analytic estimation is in good agreement with the results of numerical simulations for n = 2 (the ϕ 8 model), n = 3 (the ϕ 10 model) and n = 4 (the ϕ 12 model). It is worth to mention also Refs. [49,50], where various properties of field-theoretic models with high-degree polynomial potentials are also considered.
Despite some certainly interesting attempts to obtain explicit expressions for kinks of models with polynomial potentials [15,51] with very specific set of vacua, kinks of the ϕ 8 model and of higher degree models so far could mainly be obtained in the implicit form. The purpose of this paper is to show that in some more general cases (for some relations between model parameters) it is possible to obtain explicit formulae for kinks. We will consider the example of the ϕ 8 model with a potential of a certain type, which will be described below.
This our paper is organized as follows. In Section 2 we briefly describe the ϕ 8 model. In Section 3 we show how the explicit formulae for kinks can be obtained for some particular model parameters. Section 4 presents some properties of the obtained kink solutions. Finally, we conclude in Section 5.

The ϕ 8 model
Consider a field-theoretic model in (1 + 1)-dimensional space-time with a real scalar field ϕ(x, t). Assume that the dynamics of the system is determined by the Lagrangian For the topological kinks to exist, it is necessary that the potential V (ϕ) be a (usually non-negative) function of the field ϕ that has two or more degenerate minima. The model considered by us is described by the potential in the form of eighth degree polynomial: where a and b are constants, 0 < a < b, see Fig. 1. The energy functional for the Lagrangian (2.1) is which in the static case becomes From the Lagrangian (2.1) one can obtain the equation of motion for the field ϕ(x, t): which in the static case ϕ = ϕ(x) takes the form This second order ordinary differential equation can be easily transformed into the first order differential equation The kinks and antikinks of the model are solutions of Eq. (2.7) that interpolate between neighboring vacua of the model (i.e., connect adjacent minima of the potential (2.2)). This means that are two neighboring minima of the potential (2.2). A static solution having these asymptotics is called a configuration belonging to the topological sector (ϕ ). The potential (2.2) has four degenarate minima, ϕ (vac) = ±a and ϕ (vac) = ±b, hence there are three topological sectors: (−b, −a), (−a, a), and (a, b). As always, the terms "kink" and "antikink" stand for configurations described by increasing and decreasing functions of coordinate, respectively.
It is important to notice that for non-negative potential (2.2) we can introduce the superpotential W (ϕ) -a smooth function of ϕ such as Then the energy (2.4) of a time-independent configuration can be rewritten as the following: From Eq. (2.10) one can see that a static configuration belonging to a given topological sector has the minimal energy if the integrand vanishes, i.e., This equation obviously coincides with Eq. (2.7). Solutions of Eq. (2.7), i.e., kinks and antikinks which are called BPS configurations (or BPS-saturated solutions) [52,53], have the minimal energy among all possible field configurations in a given topological sector. The energy (2.11) is also called kink's (antikink's) mass.
For the model under consideration with the potential (2.2) we can take the superpotential in the form Then the masses of all kinks and antikinks are and At a = 0 the mass of the kink in the sector (−a, a) vanishes, while in the sector (a, b) from Eq. (2.14) we obtain 2b 5 15 , which coincides with [33,Eq. (11)] taking into account the difference in the definition of the potential (2.2) and the potential [33,Eq. (8)]. Besides that, the masses (2.14) and (2.15) coincide with the results of [31,Sec. 3.1] at λ = 1/ √ 2. Moreover, the mass of the kink in the sector (a, b) quite naturally vanishes at a = b.
Until now, it was believed that kinks of the ϕ 8 model can be obtained only in an implicit form, i.e., in the form of the dependence x = x(ϕ) (apparently with the exception of a particular case [51]). However, as we will demonstrate below, a detailed analysis of the solutions of the static equation of motion shows that, at least for particular values of the ratio b/a, kinks can be obtained in the explicit form ϕ = ϕ(x).

Explicit kinks
First, consider topological sectors (a, b) and (−b, −a). Substituting the potential (2.2) into the equation of motion (2.7) and integrating with taking into account that 0 < a < |ϕ| < b, we obtain an implicit kink solution: For future convenience we transform this equation to the following form: Then, denoting b/a = n and setting b = 1, as well as introducing we obtain n ϕ − 1 n ϕ + 1 (n − 1) 3 (n 2 + 3n + 1) n 5 . (3.5) In the limit n → ∞ Eq.  In the topological sector (−a, a) we have |ϕ| < a and therefore we obtain the following algebraic equation 1 + n ϕ 1 − n ϕ and the kink/antikink mass In the limit n → ∞ Eq. (3.7) yields obvious result M (− 1 n , 1 n ) → 0. In Fig. 2 we show the dependences (3.5) and (3.7). It is curious to notice that the two lines intersect at , which is the square of the golden ratio. Since we are considering the model with three topological sectors, it is obvious that a = b. Therefore, the minimum value of n is 2, with which we begin our consideration.

The case n = 2
At n = 2 equation (3.4) looks like where which yields for ϕ K (x) = ϕ It is easy to see, that this is exactly the same that could be obtained from Eqs. (3.5), (3.7) for n = 2.

Kink's excitation spectra
Having at hand explicit formulae for kinks, now we can study the kinks' excitation spectra. The problem is formulated as follows (see, e.g., [31,Sec. 2], [33,Sec. 4]). We add a small perturbation δϕ(x, t) to the static kink ϕ K (x), the excitation spectrum of which we are looking for, i.e., ϕ(x, t) = ϕ K (x) + δϕ(x, t), ||δϕ|| ||ϕ K ||. Substituting this ϕ(x, t) into the equation of motion (2.5), we obtain in a linear approximation in δϕ: We can separate the variables x and t or, in other words, we can look for a solution of this equation in the form δϕ(x, t) = ψ(x) cos ωt. Then Eq. (4.2) yields the following eigenvalue problem: which is similar to the one-dimensional stationary Schrödinger equation with the Hamilto-nianĤ The function U (x) is the stability potential which can be viewed as "quantum-mechanical" potential. The "energy levels" of the discrete spectrum in the potential well U (x) are nothing else than eigenvalues ω 2 i , which are our ultimate goal. It is easy to find that . (4.6) Notice that it can also be easily shown that there is always a zero level in the kink's excitation spectrum, see, e.g., [33,Eqs. (25), (26) [33,Sec. 4]. To be brief, the essence of the method is as follows. The ordinary differential equation (4.4) at a particular value of ω is solved numerically separately at x < 0 and x > 0 starting from the left and the right infinity, respectively. Then the two obtained solutions ψ L (x) and ψ R (x) are matched in some point x = x match near the origin. If the selected value of ω is an eigenvalue of the Hamiltonian (4.5), the "left" and the "right" solutions would be parts of the same eigenfunction ofĤ. This entails zeroing out the Wronskian of the functions ψ L (x) and ψ R (x) at x = x match . To find out the functions ψ L (x) and ψ R (x), we solved the ordinary differential equation (4.4) numerically using the classic fourth-order Runge-Kutta method with the step h = 10 −5 .
Substituting explicit formulae for kinks at n = 2 and n = 3 into Eq. (4.6), we can get the "quantum-mechanical" potentials for each kink. The final formulae are too bulky, and we do not give them here, however obtaining them either manually or using computer algebra system does not present fundamental difficulties. For convenience, consider the two cases n = 2 and n = 3 separately and in more detail.
The case n = 2. The potentials U (x) for all three kinks are shown in Fig. 5. For the kinks in the topological sectors ( 1 2 , 1) and (−1, − 1 2 ) the potentials are obviously mirror symmetric, i.e., . Therefore, we focus on the kink ( 1 2 , 1) only. The corresponding potential has asymptotics and The discrete spectrum of the operatorĤ is thus localized in the range 0 ≤ ω 2 ≤ 0.5625. We performed the numerical search for discrete levels in the potential well U ( 1 2 ,1) (x) and found only the zero mode ω 2 0 ≈ 2 · 10 −13 . For the kink in the topological sector (− 1 2 , 1 2 ) the "quantum-mechanical" potential x 256/81 256/729 U Figure 6. The "quantum-mechanical" (stability) potential U (x) for kinks in the case n = 3.
The minimal value U min = U ( 1 3 ,1) (x min ) ≈ −0.969014 at x min ≈ 0.300959, and the discrete spectrum of the operatorĤ is localized in the range 0 ≤ ω 2 ≤ 0.351166. As a result of the numerical search for discrete levels in the potential well U ( 1 3 ,1) (x) we obtained only the zero mode with frequency ω 2 0 ≈ 5 · 10 −11 . The kink in the topological sector (− 1 3 , 1 3 ) has the symmetric "quantum-mechanical" potential,

Conclusion
We have considered a field-theoretic model with a real scalar field in the (1+1)-dimensional space-time with field self-interaction (model potential) in the form of the eighth degree polynomial (2.2) with four degenerate minima. We were interested in the possibility of obtaining kink-type solutions in explicit form ϕ = ϕ(x). Despite being in demand, kinks of the ϕ 8 model were still known only in an implicit form x = x(ϕ) until now. This situation significantly limited the study of their properties, especially analytically. We have shown that in the case of a ratio of constants b/a = n equal to positive integers, in order to obtain explicit formulae for kinks, it is necessary to solve an algebraic equation of degree n + 1. As an example, we have considered cases of n = 2 and 3 and obtained analytical formulae for kinks in all topological sectors of the model. For n = 3, the expressions for kinks look rather cumbersome; nevertheless, this is a significant step forward in the study of topological solitons of the ϕ 8 model.
Further, using the obtained formulae for kinks, we have calculated the kinks' masses, which surely coincide with the values obtained using the superpotential. The point is that the topological solitons under consideration are BPS-saturated static configurations that have the smallest possible energy in their topological sectors. This energy can be easily found using superpotential.
Besides that, using explicit formulae for kinks, we investigated the excitation spectra of all kinks at n = 2 and n = 3. A thorough search of levels in the discrete part of the spectrum of the eigenvalue problem (4.4) has shown the presence of only zero levels with