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Kink solutions in a generalized scalar φ 4 G field model

We study a scalar field model in a two dimensional space-time with a generalized ϕG4 potential which has four minima, obtaining novel kink solutions with well defined properties although the potential is non-analytical at the origin. The model contains a control parameter δ that breaks the degeneracy of the potential minima, giving rise to two different phases for the system. The δ < 0 phases do not possess solitary wave solutions. At the transition point δ = 0 all the potential minima are degenerate and three different kink solutions result. As the transition to the δ > 0 phase takes place, the minima of the potential are no longer degenerate and a unique kink ϕ δ solution is produced. Remarkably, this kink is a coherent structure that results from the merge of three kinks that can be identified with those observed at the transition point. To support the interpretation of ϕ δ as a bound state of three kinks, we calculate the force between the kink-kink pair components of ϕ δ , obtaining an expression that has both exponentially repulsive and constant attractive contributions that yields an equilibrium configuration, explaining the formation of the ϕ δ multi-kink state. We further investigate kink properties including their stability guaranteed by the positive defined spectrum of small fluctuations around the kink configurations. The findings of our work together with a semiclassical WKB quantization, including the one loop mass renormalization, enable computing quantum corrections to the kink masses. The general results could be relevant to the development of effective theories for non-equilibrium steady states and for the understanding of the formation of coherent structures.


Introduction
Solitons and solitary waves are remarkable properties of nonlinear field theories. Solitary waves are finite energy non-dispersive localised solutions of classical field equations of motion ( [1][2][3][4]). When two solitary waves collide and each preserves its form after scattering, we refer to them as solitons [5]. Solitons and solitary waves frequently display particle-like properties and are relevant to the understanding of a plethora of non linear phenomena in many areas of physics, for example: hydrodynamics [6], nonlinear optics [7], condensed matter [8,9], nuclear physics [10], quantum field theory [11], and cosmology [12].
Solitary waves satisfy the Euler-Lagrange equations of motion, yet it is also essential to identify the stability condition in many cases related to the existence of conserved charges of topological origin [13]. In the case of two dimensions (space and time), the scalar field theory with a f 4 potential gives rise to the kink solution [14][15][16], that owes its stability to the existence of two degenerate minima in the potential; the solution approaches different minima as the field comes near to spatial infinity in different directions. Also to be highlighted is the importance of the f 4 model for its connection to the study of phase transitions [17], the Ginzburg Landau theory of superconductivity [18,19], and the spontaneous symmetry breaking and Higgs mechanism [20][21][22].
The study of kinks in f 2 n models with n 3 has recently attracted considerable attention [23][24][25] given that the number and properties of kink solutions are extended by including polynomial potentials that have a greater number of minima. These models can be used to analyse systems with multiple phase transitions, in which it is possible to describe successive first order alternating with second order phase transitions [26]. Another finding is that in some cases the kink-kink and kink-anti-kink forces present a polynomial behavior with respect to the separation [27,28], as compared to the exponential short range expression that is observed in the f 4 model.
In this work we consider a generalized f G 4 model that possesses four inequivalent minima, resulting from the addition of non-analytical odd powers of f n , n < 4 to the potential. In another context similar models have been used to study first order phase transitions [29,30], and it was recently found that a Landau theory for nonequilibrium steady-states can be constructed if one exempts the assumption of analyticity in the effective potential [31]. However in this work we are interested in the kink solutions of the relativistic f G 4 model. We prove that although the f G 4 potential is non analytical at the origin f = 0, a careful treatment of the potential discontinuities enables kink solutions with well defined properties. The kinks obtained in the different phases of the model are studied in detail, including the existence of a static multi-kink that results from the bound state of three kinks. This result is possible because the kink-kink interaction between the components of the multi-kink is given by a confining potential. Additionally using the scheme based on the semiclassical functional quantization including the one loop mass renormalization [32][33][34][35] we calculate the quantum mass corcections for the kinks of the f G 4 model. The paper is organized as follows. Section 2 contains a general review of the formalism required to study kink solutions in scalar field theories in one space dimension. Section 3 deal with the description of the generalized f G

( )
The energy functional corresponding to the Lagrangian (1) is given by the following expression As far as the field potential is concerned we propose the following This potential is a generalization of the usual f 4 potential The case to be studied includes odd and even powers of f n , n 4, and the symmetry breaking associated with inequivalent potential minima will appear when v 1 > 0 and v 2 > 0. Additionally, the incorporation of the δ parameter breaks the degeneracy of the potential minima. Usually the odd powers of f are not incorporated in a scalar field potential because they lead to non-analytical terms in the theory [16,17,19]. However it will be demonstrated that a correct treatment of the discontinuities leads to a model with well defined properties.
The potential U(f) in equation (4) has minima at f = ±V 1 , ±V 2 ; where V 1 , V 2 and the value of the maxima V m are given by the following expressions , 5 where V m (V 1 ) correspond to the +(−) selection. Throughout this work we consider v 2 > v 1 , in such a way that the extrema points of the potential satisfy V 1 < V m < V 2 . We assume that U(f) has four different potential minima (V m and V 1 are real), hence the δ parameter must satisfy the conditions The potential in equation (4) gives rise to the spontaneous symmetry breaking of the discrete symmetry f(x) → − f(x), which can be verified considering excitations around any of the minima defining f(x) = V i + σ i (x). Plugging this expression into equation (1) yields two independent Lagrangians  i , the details of which are quoted in the appendix (A). As expected  i do not contain linear terms in σ i (x), but they include a quadratic mass term and also cubic and quartic interaction terms The masses of the two normal modes are determined from the second derivative of the potential evaluated at f = ± V i : where m 1 , m 2 correspond to the masses of the particle excitations around ±V 1 and ±V 2 respectively. When the cubic and quartic terms in  i are neglected, the dynamics of the system is described by a set of uncoupled harmonic oscillations, with eigenvalues w = and plane wave solutions . A perturbative incorporation of the cubic and quartic terms in the formalism enables the calculation of higher order effects.

2-dimensional topological solitary waves
We now focus on time independent field configurations. The existence of stable solitary wave solutions requires that the potential have two or more degenerate minima. Furthermore, in order that equation (3) be finite, its integrand should vanish as x goes to either plus or minus infinity. This implies that as x → ±∞ the field must approach one of the minima V i of the potential and also that U(V i ) = 0, hence a stable solitary wave is obtained when the field f(x) interpolates between two different contiguous absolute minima V i ≠ V j . By multiplying the time-independent version of equation (2) by df/dx and integrating, a first order Bogomolny equation [36] is obtained where, according to the previous discussion, the integration constant is zero. Equation (8) describes a system that is mathematically identical to the problem of a unit mass particle with null total energy, that moves in a −U(f) potential; the equivalent 'position' and 'time' correspond to f and x respectively [1]. Equation (8) leads to where the +(−) signs correspond to the kink (anti-kink), and the kink position x 0 is arbitrary because of the translational invariance symmetry of the Lagrangian. In order to get a consistent solution f(x 0 ) = f m is selected as f m = V m , where V m is the field value between V i and V j at which the potential acquires its maxima value. The stability of the kink results from the existence of a non-trivial topological charge that can be assigned to each configuration. In two dimensions the topological current is defined as J μ = C ò μν ∂ ν f(x), where C is a constant and ò μν is the two dimensional Levi-Civita symbol. The current is automatically conserved because ò μν is antisymmetric ∂ ν J μ = 0. The corresponding conserved charge is where we selected C = 1/2v 2 .

Kinks in the generalized f 4 G model
As mentioned, kink solutions are obtained when f(x) interpolates between two contiguous absolute minima The previous condition cannot be satisfied when δ < 0. Bubble solutions may be obtained [37] in this region, but the asymptotic conditions f(+∞) = f(−∞) = 0 yield a vanishing topological charge, implying that the bubbles are unstable. The cases of interest appear with the phase transition to positive δ values. The δ = 0 and δ > 0 regions show markedly different properties and will be considered separately.

Kink solutions: degenerate minima case (δ = 0)
We first look at the potential U 0 (f) in equation (4), obtained when δ = 0. In this case the minima of the potential are located at ±v 1 and ±v 2 and they are all degenerate since U 0 ( ±v 1 ) = U 0 ( ±v 2 ) = 0. Therefore, we expect kink solutions that interpolate between the following pairs of potential minima: (−v 2 , −v 1 ), (−v 1 , v 1 ) and (v 1 , v 2 ), in addition to the corresponding anti-kinks that invert the direction in which the potential minima are connected.
Consider first the kink f A localised in the topological sector (−v 2 , −v 1 ). Substituting the potential U 0 (f) into equation (9) and taking into account that f m = − (v 1 + v 2 )/2, it is straightforward to integrate equation (9); inverting the result to obtain Here the central position of the kink was selected at x = 0, and the masses of the scalar excitations in equation (7) reduce to ). According to equation (9) the corresponding anti-kink configuration is obtained as ) is directly calculated using equation (11) to obtain The kink mass M A is obtained substituting equation (11) into equation (3), whereas the topological charge is computed from equation (10), resulting in We point out that f A and f Ā have the same mass value, whereas their charges have opposite signs.
For the kink f C in the topological sector (v 1 , v 2 ) the calculations are completely analogous. The kink profile is given by f C (x) = f A (x) + v 1 + v 2 and the corresponding mass and topological charge coincide with those of f A : thus changes sign, hence we must perform separate calculations for positive and negative values of f B (x). We evaluate equation (9) separately in the intervals (− v 1 , 0) and (0, v 1 ), considering that f m = 0 and selecting x 0 = 0. After evaluating the integrals, the results can be inverted and written in a single equation using a y variable that is piecewise defined as follows is then given by

Taking into account that
, both the function f B (0) = 0 and its first ) are continuos at the kink position; however the second derivative is discontinuous. We The mass and topological charge are calculated as Figure 1 displays the plots for the three kink solutions and the corresponding energy densities. We observe that the profiles of the three field configurations display a characteristic kink behavior. The energy densities for f A (x) and f C (x) are smooth energy packets localised around the kink position, whereas the energy density for f B (x) presents a spike maxima because the second order derivative of f B (x) is discontinuous at the kink position. We point out that we can define two parity symmetry operations P x and P f that invert the coordinate or field sign respectively: e call P x and P f the charge and mirror symmetries.

Kink solution for non degenerate minima (δ ≠ 0)
As the transition to the δ > 0 phase takes place the potential minima are no longer degenerate, There are now only two absolute minima, hence a unique kink interpolates between f(−∞) = −V 2 to f(∞) = V 2 , as well as its corresponding anti-kink. To determine the kink configuration we substitute the potential equation (4) into equation (9) and perform the integrations in the intervals (0, v 2 ) and ( − v 2 , 0) separately. In the entire interval (0, v 2 ) the field f is positive, considering that f m = 0, and selecting the kink position at An analogous expression is obtained for x 0. Both relations can be explicitly inverted and the results are summarized in the following expression We can directly verify that f δ (x) and its first derivative are continuous everywhere, but its second derivative is discontinuous at the origin (f(0) = 0) in agreement with equation (2) and the discontinuity that ∂U/∂f presents at that point. The topological charge for f δ is given as Q δ = ± 1. The kink mass is evaluated using equations (3), central energy packet presents a spike configuration. Based on these results we conclude that f δ is a bound state resulting from the merge of the f A , f B and f C kinks. To support this claim we notice that the following charge equality is trivially fulfilled Additionally, it follows that for small values of δ the kink mass in equation (22) can be approximated as where we used the values for M A = M C equation (13) and M B equation (17). The mass of the f δ kink results from the addition of the masses of the constituent kinks and a term that, as shown in section (4), represents the potential energy of the system evaluated at the equilibrium configuration equation (24).
Using equation (19) evaluated at x = x m and f = V m we obtain the value of x m that determines the distance of f A and f C relative to f B . For small δ it is approximated as where according to equation (25) there is a large but finite separation. In section 4 we shall prove that ± x m represent the equilibrium positions of the forces that act on f A and f C , considering that f B is fixed at x = 0.
A fundamental property of solitary waves is that they are non-dispersive, namely they represent localised energy packages that move with constant speed, maintaining their initial structure. To verify that the previously obtained solutions can be considered as true solitary waves, we apply a boost by defining f X (γ(x − vt)), with g = -v 1 1 2 ( )where f X is any of the kink solutions equations (11), (15), (21). We confim that f X (ξ) is a solution of the time dependent equation (2), hence the four kinks (f A , f B , f C , f δ ) and their corresponding antikinks are indeed solitary waves.

Kink interactions
In this section we analyse the effect of combining two kinks or a kink with an anti-kink, and compute the forces that act between them. In general these configurations will be time dependent, but if we consider a KK or KK pair that at an initial time is separated by a large distance R ? 1/m, for a short period it will experiment a rigid displacement and the force can be determined. In order to have a continuous finite energy configuration it is required that a kink K is followed by an antinkink K defined in the same topological sector or a kink K' defined in a contiguous topological sector. Thus we consider three independent configurations: CC , BB and BC that will give rise to different expressions for the corresponding forces. These results will be extended to analyse the forces that act within the multi-kink f δ .
The momentum density for a scalar field obtained from Noether's theorem is given as - Integrating this expression in the interval (X, ∞ ) and utilising equation (8) the force acting on the field to the right of X is obtained as [3,27] ò Here we took into account that the last two terms in the previous equation cancel out as x → ∞.

δ = 0 case
First we analyse the CC configuration in which the kink C localised at x = − q occupies the x < 0 region, whereas C defined for x > 0 is situated at x = q; the ansatz for the CC field is represented as where R = 2q is the separation between the kink and the anti-kink. In order to compute F BB we define given in equation (15), the following asymptotic expression (x = 1, q ? 1/m) follows up f

( )
We now turn the attention to the BC system. The field ansatz configuration is written as figure (3) shows that f BC (x) represents a field that interpolates between the −v 1 and the +v 2 potential minima. When x = 1, q ? 1/m the asymptotic expression for f BC (x) reduces to ( ) The previous results show that generically the kink-anti-kink interaction is attractive, while the kink-kink interaction is repulsive, and that in both cases the interaction decays exponentially as the kink separation increases. However the interaction strengths are not equal in absolute value, but rather the ratio of the forces are The parameters are selected as v 1 = 1, v 2 = 4, λ = 1, δ = 0, and q = 5. The kink separation is R = 2q = 10 in both cases.
given as follows 1 .

δ > 0 case
When δ > 0, f A , f B and f C are no longer exact solutions of equation (8), instead we have a new kink f δ that interpolates between the two absolute potential minima ±v 2 . However, as mentioned in section (3), the f δ (x) profile resembles a multi-kink formed from the merge of f A , f B and f C , which leads us to analyse the configuration defined by the following ansatz We compute the pair interactions between the components of f ABC (x; q) and determine if it is possible to find a value of q for which the configuration is stable. The force exerted on the field f C (x − q) in equation (32) is dominated from the f B − f C interaction, that was already calculated in equation (30). However that calculation was carried out for δ = 0, which only includes the potential U 0 (f) in equation (4). Hence we must add the contribution from ) evaluated at f BC (0) using equation (29). A direct calculation yields ΔU(f BC (0)) ≈ m 2 δ/2. Adding this result to F BC given in equation (30) the total force acting on f C results in Remarkably, in addition to the repulsive contribution, there is an attractive constant long range force. Thus, the dynamics of the position of the kink f C takes place in an effective potential V AB;C (q) that is obtained from the space integral of equation (33) leading to where K is a constant. This potential has a stable equilibrium point q = x m , that coincides with the separation between the contiguous kinks components of f δ given in equation (25). As the force acting on f A is identical to equation (33) and the the force on f B cancels, it follows that the effective potential for the f ABC multikink is V ABC (q) = 2V AB;C (q). When this potential is evaluated at the equilibrium point q = x m and selecting the constant in equation (34) as it follows that V ABC (x m ) exactly coincides with the correction of order δ to the f δ mass in equation (24), confirming that it represents the potential energy of the multi-kink at the equilibrium configuration.
Finally, to complete the interpretation of f δ as a f A + f B + f C multi-kink state, we notice that for δ = 1, both f δ (x) equation (21) and f ABC (x; x m ) equation (32) reduce to the same expression Figure 4 compares the plots obtained from the previous expression with the exact solution given in equation (21). It is clearly shown that the difference between the two expressions becomes imperceptible as the δ parameter decreases.

Quantum mass corrections, and renormalization
We now turn our attention to the study of field excitations around the kink configurations. The kinks are expected to be stable, this is verified by showing that the spectrum of the fluctuations is positive defined. Furthermore the use of a WKB approximation enables computing the quantum corrections to the kink mass. In what follows we mainly focus on the case of degenerate potential minima δ = 0, hence the field potential is given by U 0 (f) in equation (4).
The field f(x, t) is written as the sum of the classical kink solution f K (x) and a small fluctuation η(x, t) as follows f(x, t) = f K (x) + η(x, t). If we substitute the previous decomposition into the energy functional equation (3), integrating by parts and taking into account that f K (x) satisfy equation (8), we find that Compared with the usual results [2,32,33,38], this expression includes an extra term that takes into account that ∂η/∂x is discontinuous at the point (36) is diagonalised if the functions η m (x) are selected as eigenfunctions of the following one dimensional Schrödinger equation (4) is a quadratic form permits making contact with the supersymmetric quantum mechanics formalism [38,39]. Writing

the energy in equation
) results in the superpotential W(f) and the SUSY-QM potentials  V x S ( ) being expressed as follows The potential in the stability equation k ik x ( ) and (ii) when the background is the kink configuration, the explicit solutions of which will be presented in following paragraphs, but generically include a discrete ω n and a continuous ω k spectrum whose eigenfunctions have an asymptotic behavior η k (x) = e ikx± δ( k)/2 as x → ± ∞, where δ(k) is the phase shift. Notice that the asymptotic behavior of η k (x) implies a reflectionless potential.
The contribution of the continuous modes is computed incorporating a regularization scheme in which the system is enclosed in a finite box of length L. Hence the kink excitation spectrum w = + m k j j 2 2 2 , and that of the vacuum fluctuations ω′ 2 j = m 2 + k′ 2 j becomes discrete. Taking into account the periodic boundary conditions, , we calculate the zero point energy contribution to the kink mass, subtracting the vacuum energy, to obtain where in the last equality we set å  p L j dk 1 2 as L → ∞. This quantity is still logarithmically divergent and it requires the addition of a mass counter-term obtained by a one loop perturbative renormalization scheme [32,35]. The mass counter-term takes the form d -á ñ  m where x m is given in equation (25). The corresponding energy densities are also compared. As observed the two configurations coincide for small δ. The parameter values are The asymptotic behavior of the continuous wave functions takes the following form (40), integrating by parts, and separating the finite contribution from the one that diverge with Λ, leads to It is noteworthy that the stability equation (43), its solutions, and the quantum mass (47) for the f C kink, coincide with the results previously obtained for the f 4 model [32], notwithstanding the f 4 and f G 4 models are notoriously different. This can be explained by the reconstruction method [41,42], in which the structure of the scalar field theory is obtained from the stability equations and the knowledge of the bound spectrum. In particular it has been show that when the spectrum has two bound states, the reconstruction is not unique [43]. Hence we conclude that considering the stability equation and its corresponding spectrum, the application of the reconstruction method should produce both the f 4 and f G 4 models.

f B quantum mass corrections
Consider now the quantum fluctuations around the f B kink. In equation (15) we recall that f B (x) is separately defined in the positive and negative x-axis. Using the expression for f B (x) to evaluate the effective potential in equation (38)  ( ) ( ( )) ( ). Note that according to equations (37), (38) the delta term potential has to be included because f B (x) cancels at x = 0, so the derivative of η k is expected to be discontinuous at the origin. Equation (48) and the function H(y) is defined in equation (44). The spectrum coincides with that calculated in the previous subsection, but the eigenfunctions are now given in terms of y coth instead of x tanh functions. All the eigenfunctions in equation (49) are continuous at the origin, but their first derivatives are discontinuous. In particular, as expected, the zero energy mode is given by For the two discrete modes the values of the coefficients D k in equation (48) are negative, thus the delta terms represent attractive potentials that explain the existence of bound states.
For the continuous states, we observe that the potential in equation (48) is again transparent, hence ( ) ( ) as x → ± ∞, and the phase shift computed from equation (49) becomes The first part of δ B (k) exactly coincides with the phase shift δ C (k) obtained in the previous subsection. Hence we can separate the contribution of the continuos modes to the kink energy in equation (40) as where M cont−C is given in equation (45) and ΔM cont−B is obtained substituting the second part of equation (50) into equation (40) and integrating by parts resulting in ò p p The denominator in the preceding integral can be factorized according to 4(k 2 − m 2 r 1 )(k 2 − m 2 r 2 ), where The integral in equation (51)  This proves that the current formalism allows us to adequately analyse the discontinuities that appear in the case of kink B, giving rise to a finite value for the quantum mass of the kink. Unlike the results obtained in the previous subsection, the stability equation and the quantum mass corrections of the f B kink equations (48), (49), (56) have not been previously obtained. As mentioned, the use of the reconstruction method applied to the f C kink leads to either the f 4 or the f G 4 models. However, we propose . From these expressions, the standard procedure can be followed to extract the Feynman rules that allow performing perturbative calculations. Clearly the results will depend on the minimum around which the perturbation is considered. However, in the degenerate potential minima limit (δ = 0) all the parameters of the Lagrangians  1 and  2 coincide, so a perturbative calculation result is indistinct of the selected minima. For example the evaluation of the tadpole diagram leads to the result δm 2 in equations (41), already known in literature [32].

(
) required to satisfy the condition imposed to the solutions by the δ(x) potential term in equations (48), (B1).
In the scattering regime ò k = 2k/m, and the solution to equation (B1) is given by ( ) is expressed in terms of the hypergeometric function [40] 2 F 1 that reduces to the last term in equation (B2) with H(ξ) defined in equation (44). Finally taking into account that y defined in equation (44) is piecewise, the normalization constants for η k (x) have to be separately selected for x < 0 and x > 0 in order to enforce the continuity condition at the origin, giving rise to the following result for the continuous eigenfunctions