# One-dimensional soliton system of gauged kink and Q-ball

## Abstract

In the present paper, we consider a \((1 + 1)\)-dimensional gauge model consisting of two complex scalar fields interacting with each other through an Abelian gauge field. When the model’s gauge coupling constants are set to zero, the model possesses non-gauged Q-ball and kink solutions that do not interact with each other. It is shown here that for nonzero gauge coupling constants, the model has a soliton solution describing the system that consists of interacting Q-ball and kink components. These two components of the kink-Q-ball system have opposite electric charges, meaning that the total electric charge of the system vanishes. The properties of the kink-Q-ball system are studied both analytically and numerically. In particular, it was found that the system possesses a nonzero electric field and is unstable with respect to small perturbations in the fields.

## 1 Introduction

It is known that in the case of Maxwell electrodynamics, any one-dimensional or two-dimensional field configuration with a nonzero electric charge possesses infinite energy. The reason for this is simple: at large distances, the electric field of this configuration does not depend on the coordinate in the one-dimensional case, and behaves as \(r^{-1}\) in the two-dimensional case, meaning that the energy of the electric field diverges linearly in the one-dimensional case and logarithmically in the two-dimensional case. Hence, there are no electrically charged solitons in one and two dimensions; such solitons appear only in three dimensions (e.g. the three-dimensional electrically charged dyon [1] or Q-ball [2, 3, 4, 5]). It should be noted, however, that electrically charged two-dimensional vortices exist in both the Chern–Simons [6, 7, 8, 9, 10] and the Maxwell–Chern–Simons [11, 12, 13, 14] gauge models. Furthermore, it was shown in [15, 16] that Chern–Simons gauge models also possess one-dimensional domain walls. The domain walls have finite linear densities of magnetic flux and electric charge, and thus there is a linear momentum flow along the domain walls.

However, even in the case of Maxwell gauge field models, there are electrically neutral low-dimensional soliton systems with a nonzero electric field in their interior areas. In particular, a one-dimensional soliton system consisting of electrically charged Q-ball and anti-Q-ball components was considered in [17], and a two-dimensional soliton system consisting of vortex and Q-ball components interacting through an Abelian gauge field was described in [18].

In the present paper, we examine a one-dimensional soliton system consisting of Q-ball and kink components with opposite electric charges, meaning that the system, which has a nonzero electric field, is electrically neutral as a whole. The properties of this kink-Q-ball system are investigated using both analytical and numerical methods. In particular, we find that unlike the non-gauged one-dimensional Q-ball, the kink-Q-ball system does not enter the thin-wall regime.

An interesting problem arises concerning the stability of the kink-Q-ball system with respect to small perturbations in the fields. Recall that the Abelian Higgs model possesses an electrically neutral kink solution [19, 20]. Formally, this gauged kink solution is the usual kink of a self-interacting real scalar field up to gauge transformations. The properties of these two kink solutions differ considerably, since the classical vacua of the corresponding field models have a different topology. While the real kink is topologically stable, the gauged kink has a single unstable mode. From a topological point of view, the gauged kink lies between two topologically distinct vacua of the Abelian Higgs model, and thus is a sphaleron [19, 20]. Note, however, that the gauged kink is a static field configuration modulo gauge transformation, whereas the kink-Q-ball system will depend on time in any gauge. Due to this fact, the kink-Q-ball system cannot be a sphaleron, and its classical stability therefore requires separate consideration.

This paper has the following structure. In Sect. 2, we describe briefly the Lagrangian, the symmetries, the field equations, and the energy-momentum tensor of the Abelian gauge model under consideration. In Sect. 3, we investigate the properties of the kink-Q-ball system. Using the Hamiltonian formalism and the Lagrange multipliers method, we establish the time dependence of the soliton system’s fields. An important differential relation for the kink-Q-ball solution is derived and a system of nonlinear differential equations for ansatz functions is obtained. We then establish some general properties of the kink-Q-ball system. In particular, we examine the asymptotic behaviour of the system’s fields at small and large distances, establish some important properties of the electromagnetic potential, and derive the virial relation for the soliton system. In Sect. 4, we study the properties of the kink-Q-ball system in three extreme regimes, i.e. in the thick-wall regime and the regimes of small and large gauge coupling constants. We also establish the basic properties of the plane-wave field configuration of the model. In Sect. 5, we present and discuss the numerical results obtained. They include the dependences of the energy of the kink-Q-ball system on its phase frequency and Noether charge, along with numerical results for the ansatz functions, the energy density, the electric charge density, and the electric field strength. We also present results for the classical stability of the kink-Q-ball system.

Throughout the paper, we use the natural units \(\hbar = c = 1\).

## 2 The gauge model

*C*,

*P*, and

*T*transformations.

## 3 The kink-Q-ball system and its properties

*E*and the Noether charge \(Q_{\chi }\) of the soliton system are gauge-invariant, relation (28) is also gauge-invariant. In a similar way to the case of non-gauged nontopological solitons [21, 22, 23], relation (28) determines the basic properties of the gauged kink-Q-ball system.

*f*(

*x*) and

*s*(

*x*) are assumed to be complex functions of the real argument

*x*. Substituting Eq. (27) into field equations (9)–(11), we can easily check that the real and imaginary parts of

*f*(

*x*) satisfy the same differential equation with real coefficients. Similarly, the real and imaginary parts of

*s*(

*x*) also satisfy the same differential equation with real coefficients. It follows that the functions

*f*(

*x*) and

*s*(

*x*) have the forms \(f(x) = \exp ( i \alpha ){\tilde{f}}(x)\), \(s( x ) = \exp ( i\beta ) {\tilde{s}}(x)\), where \({\tilde{f}}( x )\) and \({\tilde{s}}( x )\) are real functions, whereas \(\alpha \) and \(\beta \) are constant phases. However, these phases can be cancelled by global gauge transformations (7), and the functions

*f*(

*x*) and

*s*(

*x*) can therefore be assumed to be real without loss of generality. The functions \(a_{0}(x)\),

*f*(

*x*), and

*s*(

*x*) satisfy the system of ordinary nonlinear differential equations:

*f*(

*x*), and

*s*(

*x*) as

*C*-invariance of the Lagrangian (1). Using Eqs. (32), (33), and (37), we can find the behaviour of the energy

*E*and the Noether charges \(Q_{\phi }\) and \(Q_{\chi }\) under the transformation \(\omega \rightarrow -\omega \):

The *P*-invariance of the Lagrangian (1) leads to the invariance of system (29)–(31) under the space inversion \(x\rightarrow -x\). Due to the space homogeneity, system (29)–(31) is also invariant under the coordinate shift \(x \rightarrow x + x_{0}\). Furthermore, due to Eqs. (7), system (29)–(31) is invariant under the two independent discrete transformations: \(f \rightarrow -f\) and \(s \rightarrow -s\). These facts and the symmetry properties of boundary conditions (34) lead to the conclusion that \(a_{0}\left( x\right) \) and \(s\left( x \right) \) are even functions of *x*, while \(f\left( x \right) \) is an odd function of *x*. This is consistent with the fact that the non-gauged kink solution is an odd function of *x*, whereas the non-gauged Q-ball solution is an even function of *x*.

*x*is obtained by substitution of the power expansions for \(a_{0}(x)\),

*f*(

*x*), and

*s*(

*x*) into Eqs. (29)–(31) and equating the resulting Taylor coefficients to zero. In this way, we obtain:

*x*-axis. Due to the symmetry \(j^{0}\left( -x \right) = j^{0}\left( x \right) \), these points (nodes of \(j^{0}\left( x \right) \)) are symmetric with respect to the origin \(x = 0\). Next, according to Gauss’s law \(a_{0}^{\prime \prime }\left( x\right) = -j^{0} \left( x\right) \), the second derivative \(a_{0}^{\prime \prime }\left( x \right) \) vanishes at the nodes of \(j^{0}\left( x \right) \). Thus, the nodes of \(j^{0}\left( x\right) \) are the inflection points of the electromagnetic potential \(a_{0}\left( x\right) \). From Eq. (29) it follows that at an inflection point \(x_{\mathrm {i}}\), the electromagnetic potential \(a_{0}\left( x_{\mathrm {i}} \right) \) can be expressed in terms of \(f\left( x_{\mathrm {i}}\right) \) and \(s\left( x_{\mathrm {i}} \right) \):

*x*. If we let \(x_{\mathrm {n}}\) be a conjectural point in which \(a_{0}\left( x \right) \) vanishes, then from Eq. (29) we have the relation

*x*. The case of negative \(\omega \) is treated similarly. Thus, we come to the important conclusion that the electromagnetic potential \(a_{0}\left( x \right) \) cannot vanish at any finite

*x*, and so the sign of the electromagnetic potential coincides with that of the phase frequency over the whole range of

*x*:

*x*. Of course, this conclusion is valid only for adopted gauge (27c).

*E*of the soliton system is the sum of terms (54)–(57). Using this fact and virial relation (54), we obtain two representations for the energy of the soliton system:

*g*and \(\epsilon \) as follows:

*g*to rescale the fields of the model and the remaining coupling constants as follows: \(\phi =g^{-1}{\bar{\phi }}\), \(\chi =g^{-1}{\bar{\chi }}\), \(\eta = g^{-1}{\bar{\eta }}\), \(A^{\mu } = g^{-1}{\bar{A}}^{\mu }\), \(\lambda = g^{2} {\bar{\lambda }}\), \(e =g{\bar{e}}\), and \(q = g {\bar{q}}\). Note that the mass \(m_{\phi }=\sqrt{2\lambda }\eta \) of the Higgs field \(\phi _{H}\), the mass \(m_{A} = \sqrt{2} e \eta \) of the gauge field \(A^{\mu }\), and the parameter \(\epsilon \) are invariant under rescaling, whereas the mass \(m_{\chi }\) of the complex scalar field \(\chi \) is not subjected to rescaling.

*g*is factorized. Using Eqs. (65) and (66), it can be shown that the Lagrangian (1) has the following behaviour under rescaling:

*g*. Next, Eq. (67) can be written in the form

*e*, and

*q*, then \(\left( \kappa ^{-1}\phi ,\kappa ^{-1}\chi , \kappa ^{-1} A^{\mu }\right) \) is also a solution corresponding to the parameters \(m_{\chi }\), \(\kappa ^{2 }g_{\chi }\), \(\kappa ^{4}h_{\chi }\), \(\kappa ^{-1} \eta \), \(\kappa ^{2} \lambda \), \(\kappa e\), and \(\kappa q\).

## 4 Extreme regimes of the kink-Q-ball system

*x*-coordinate:

*e*and

*q*tend to zero, whereas in the second they tend to infinity. In both regimes, the ratio \(\varrho = e q^{-1}\) of the gauge coupling constants is a constant value. Since system of differential equations (29)–(31) is invariant under the transformation \(e \rightarrow -e\), we may suppose without loss of generality that \(\varrho \) is positive. Moreover, system (29)–(31) is also invariant under the transformation \(e \rightarrow - e\), \(q \rightarrow - q\), \(a_{0} \rightarrow -a_{0}\), as are the soliton energy and the Noether charges. It follows that both gauge coupling constants may be considered as positive without loss of generality.

*e*and

*q*vanish, the gauge field \(A_{\mu } = \left( a_{0}\left( x \right) ,\, 0 \right) \) is decoupled from the kink-Q-ball system, which thus becomes a set of non-gauged kink and Q-ball solutions that do not interact with each other. In this connection, we want to ascertain the behaviour of the gauge potential \(a_{0}\left( x \right) \) as \(e = \varrho q \rightarrow 0\). To do this, we use expressions (40a) and (42c) for \(a_{0} \left( x \right) \), which are valid for small and large values of \(\left| x \right| \), respectively. We also assume that Eqs. (40a) and (42c) qualitatively describe the behaviour of \(a_{0}\) at intermediate \(\left| x \right| \). Equations (40a) and (42c) depend on the free parameters \(a_{0}\) and \(a_{\infty }\), respectively, which can be determined by the condition of continuity of \(a_{0}\left( x\right) \) and \(a_{0}^{\prime }\left( x\right) \) at some intermediate

*x*. As a result, \(a_{0}\) and \(a_{\infty }\) become functions of the parameters of the model, including the gauge coupling constants

*e*and

*q*. It can be shown that \(a_{0}\) and \(a_{\infty }\) tend to the same nonzero limit as \(e = \varrho q \rightarrow 0\). It follows that at any finite

*x*, the gauge potential \(a_{0}\left( x \right) \) tends asymptotically to a constant as \(e = \varrho q \rightarrow 0\):

Let us consider the behaviour of the gauge potential \(a_{0}\left( x \right) \) at small values of *e* and *q* in more detail. If the gauge coupling constants *e* and *q* are arbitrarily small but different from zero, the gauge potential \(a_{0}\left( x \right) \) satisfies boundary conditions (34a) and (35). The first boundary condition provides the finite contribution of the term \(e^{2}a_{0}\left( x\right) ^{2}f\left( x\right) ^{2}\) to the soliton energy, while the second one does the same for the contribution of the electric field energy density \(a_{0}^{\prime \,2}/2\). When the gauge coupling constants *e* and *q* tend to zero (i.e., *e* and *q* are arbitrarily small but nonzero), boundary conditions (34a) and (35) hold. In this case, according to Eq. (78), the gauge potential \(a_{0}(x)\) tends asymptotically to the constant \(\alpha \) at any finite *x*. That is, for any finite *x* and arbitrarily small positive \(\epsilon \), there exists the arbitrarily small positive \(\delta \left( \epsilon ,x\right) \) such that \(\left| a_{0}(x)-\alpha \right| < \epsilon \) if \(e = \varrho q < \delta \). Nevertheless, at arbitrarily small but nonzero *e* and *q*, the gauge potential \(a_{0}(x)\) tends to zero as \(x \rightarrow \pm \infty \). However, if \(e = \varrho q = 0\), the term \(e^{2}a_{0}\left( x\right) ^{2}f\left( x\right) ^{2}\) in Eq. (33) vanishes, meaning that boundary condition (34a) becomes unnecessary and only the less stringent boundary condition (35) holds. In this case, we have the nonzero constant solution \(a_{0}(x) = \alpha \), where \(\alpha \) is equal to limiting value (78) to which the electromagnetic potential \(a_{0}(x)\) tends asymptotically as \(e = \varrho q \rightarrow 0\). We conclude that the behaviour of the electromagnetic potential \(a_{0}(x,e,q)\) is nonregular in neighbourhoods of the infinitely remote points with \(\left( e, q, x \right) = \left( 0, 0, \pm \infty \right) \). Indeed, it follows from the foregoing that in a close neighbourhood of \(e = \varrho q = 0\), \(\underset{ x \rightarrow \pm \infty }{\lim }a_{0} \left( x, e, \varrho ^{-1} e \right) = \alpha \delta _{0 e}\), where \(\delta _{0 e}\) is the Kronecker delta.

*e*and

*q*) and \(\varrho \). Note that due to the relation \(e = \varrho q\), we use only one expansion parameter

*e*. Substituting Eqs. (78) and (79) into Eq. (32), we find the asymptotic behaviour of the Noether charges \(Q_{\phi }\) and \(Q_{\chi }\) as \(e = \varrho q \rightarrow 0\):

*e*and

*q*) and \(\varrho \). Similarly to Eq. (80), we obtain the asymptotic behaviour of the components of the soliton energy (54)–(57) and the total soliton energy:

*e*and

*q*) and \(\varrho \). Note that the non-gauged solutions \(f_{\mathrm {k}}\) and \(s_{\mathrm {q}}\) can be expressed in analytical form, as can the corresponding energies and the Noether charges. The corresponding expressions for the one-dimensional non-gauged Q-ball are given in [17]. Thus the coefficients \(Q_{0}\), \(E_{0}^{\left( G \right) }\), \(E_{0}^{\left( T \right) }\), \(E_{0}^{\left( P \right) }\), and \(E_{0}\) can also be expressed in analytical form.

*q*: \(a_{0}\left( x, -q\right) = -a_{0}\left( x, q\right) \). Thus, we conclude that \(a_{2} \propto q^{-1}\) in the leading order, and so from Eq. (41a) it follows that \(a_{0} = \omega q^{-1}+ a_{-3}q^{-3} + O \left( q^{-5}\right) \), where \(a_{-3}\) is a constant. This suggests that the electromagnetic potential \(a_{0}\left( x\right) \) has a similar asymptotic expansion

*e*and

*q*) and \(\varrho \). Using Eqs. (30), (31), and (82), we obtain the asymptotic expansions for \(f\left( x \right) \) and \(s\left( x\right) \):

*e*and

*q*) and \(\varrho \). In a similar way to Eqs. (78) and (79), we can use Eqs. (82) and (83) to obtain asymptotic expansions for the components of the soliton energy, the total energy, and the Noether charges:

*E*, and the Noether charges \(Q_{\chi }\) and \(Q_{\phi }\) also tend to finite values as \(e=\varrho q \rightarrow \infty \). Hence, the electric charges of the kink and Q-ball components increase indefinitely in this regime, despite the fact that the electric field energy \(E^{\left( E\right) }\) tends to zero. This is only possible if electric charge density (32) tends to zero as \(e = \varrho q \rightarrow \infty \). Based on this, we can obtain the limiting relation between the coefficient functions of asymptotic expansions (82) and (83):

Let us discuss the behaviour of the asymptotic expansions in the two extreme regimes. In the regime \(e = \varrho q \rightarrow 0\), the leading terms of asymptotic expansions (80)–(81) are linear in *e*. This is due to the linear dependence of \(s\left( x \right) \) on *e* in Eq. (79b). In turn, this linear dependence results from the fact that differential equation (31) contains the term \(-2 q\omega a_{0}(x)s(x) \rightarrow -2 \varrho ^{-1} e \omega \alpha s(x) \), which is linear in *e*. From Eq. (31) it follows that *s*(*x*) is invariant under the transformation \(e \rightarrow - e\), \(q \rightarrow - q\), \(a_{0} \rightarrow -a_{0}\), and thus coefficient function \(s_{1}(x)\) changes sign under this transformation. Due to this, the coefficients \(E_{1}^{(E)}\), \(E_{1}^{(G)}\), \(E_{1}^{(P)}\), \(E_{1}^{(T)}\), and \(Q_{1}\) also change sign under this transformation. Thus, the Noether charges, the components of the soliton energy, and the total soliton energy are invariant under the transformation \(e \rightarrow - e\), \(q \rightarrow - q\), \(a_{0} \rightarrow -a_{0}\) as expected. We can compare this situation with that of the opposite regime, \(e = \varrho q \rightarrow \infty \). In this case, the asymptotic expansion of the electromagnetic potential \(a_{0}(x)\) has the form (82), and so the term \( -2 q\omega a_{0}(x)s(x)\) in Eq. (31) transforms into the term \(-2\varrho ^{-1}\omega \left( a_{-1}(x) + e^{-2}a_{-3}(x)\right) s(x)\), which has no linear dependence on *e*. As a result, differential equations (30) and (31) contain only even inverse powers of *e*, and asymptotic expansions (83a) and (83b) therefore also contain only even inverse powers of *e*. Due to this, asymptotic expansions (84a)–(84f) also include only even inverse powers of *e*. The behaviour of the kink-Q-ball system in the extreme regimes \(e = \varrho q \rightarrow 0\) and \(e = \varrho q \rightarrow \infty \) was investigated using numerical methods, and was found to be in accordance with Eqs. (80), (81), (84), and (85).

Finally, we consider the plane-wave solution of gauge model (1). In this case, the gauge field \(A^{\mu }\) and the scalar fields \(\phi \) and \(\chi \) are spread over the one-dimensional space and fluctuate around their vacuum values. Since the scalar field \(\phi \) has the nonzero vacuum value \(\left| \phi _{ \text {vac}}\right| = \eta \), the classical vacuum of model (1) is not invariant under local gauge transformations (6), meaning that the local gauge symmetry is spontaneously broken. For this reason, it is convenient to perform an investigation of the plane-wave solution in the unitary gauge \(\text {Im}\left( \phi \left( x,t\right) \right) = 0\). In this gauge, the Higgs mechanism is explicitly realized, and hence we can read off the particle composition of model (1). In the neighbourhood of the gauge vacuum \(\phi _{\text {vac}} = \eta \), \(\chi _{\text {vac}} = 0\), we have the complex scalar field \(\chi \) with mass \(m_{\chi }\), the real scalar Higgs field \(\phi _{H}\) with mass \(m_{\phi } = \sqrt{2\lambda }\eta \), and the massive gauge field \(A^{\mu }\) with mass \(m_{A} = \sqrt{2} e \eta \).

*L*of the spatial dimension, and is presented as a series in inverse powers of

*L*:

## 5 Numerical results

To analyse the kink-Q-ball system, we must solve system of differential equations (29)–(31) subject to boundary conditions (34). This first boundary value problem can be solved only numerically. For this purpose, we use the boundary value problem solver provided in the Maple package [27]. To check the correctness of the numerical solutions, we use Eqs. (28), (36), and (53).

*e*,

*q*, \(m_{\phi } = \sqrt{2 \lambda } \eta \), \(\lambda \), \(m_{\chi }\), \(g_{\chi }\), and \(h_{\chi }\). Without loss of generality, the mass \(m_{\chi }\) can be chosen as the energy unit, and the dimensionless functions \(a_{0}\left( x\right) \), \(f\left( x\right) \), and \(s\left( x\right) \) therefore depend only on the seven dimensionless parameters \({\tilde{\omega }} = \omega /m_{\chi }\), \({\tilde{e}} = e/m_{\chi }\), \({\tilde{q}} = q/m_{\chi }\), \({\tilde{m}}_{\phi } = m_{ \phi }/m_{\chi }\), \({\tilde{\lambda }} = \lambda /m_{\chi }^{2}\), \({\tilde{g}}_{\chi } = g_{\chi }/m_{\chi }^{2}\), and \({\tilde{h}}_{\chi } = h_{\chi }/m_{\chi }^{2}\). In the present paper, we consider the kink-Q-ball system for which the dimensionless non-gauged parameters \({\tilde{m}}_{\phi }=\sqrt{2}\), \({\tilde{\lambda }} = 1\), \({\tilde{g}}_{\chi } = 2.3\), and \({\tilde{h}}_{\chi } = 1\) have the same order of magnitude. The dimensionless gauge coupling constants \({\tilde{e}}\) and \({\tilde{q}}\) are assumed to be equal to each other, and can take the values 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, and 0.5.

*A*,

*B*, and

*C*are positive constants. We were unable to find any solutions to the boundary value problem for \(\tilde{ \omega }< {\tilde{\omega }}_{\min }\), and we therefore conclude that the kink-Q-ball system does not enter the thin-wall regime, in which both \({\tilde{E}}\) and \(Q_{\chi }\) must tend to infinity. Note in this connection that a one-dimensional non-gauged Q-ball enters the thin-wall regime as the phase frequency \({\tilde{\omega }}\) tends to its minimal value [23]. In particular, in the case of self-interaction potential (4), the energy, the Noether charge, and the linear size of the one-dimensional Q-ball are proportional to \(\ln \left( {\tilde{\omega }}-{\tilde{\omega }}_{\min }\right) ^{-1}\), meaning that they diverge logarithmically as \({\tilde{\omega }} \rightarrow {\tilde{\omega }}_{\min }\). However, this divergence is rather weak; for example, the energy and the Noether charge of a three-dimensional Q-ball with the same self-interaction potential are proportional to \(\left( {\tilde{\omega }} - {\tilde{\omega }}_{\min } \right) ^{-3}\), whereas the size of the Q-ball is proportional to \(\left( {\tilde{\omega }}-{\tilde{\omega }}_{\min }\right) ^{-1}\). At the same time, the electromagnetic interaction is strongest in one spatial dimension, as the Coulomb force does not depend on the distance between charges as long as the gauge symmetry is not broken. Thus we may suppose that the Coulomb interaction prevents the kink-Q-ball system from entering the thin-wall regime.

*D*,

*F*, and

*G*are positive constants. The right branch starts from the point with the phase frequency \(\tilde{\omega }_{\text {l}} < {\tilde{\omega }}_{\text {r}}\) and continues up to the maximum possible value of \({\tilde{\omega }}_{\text {tk}} = 1\). The energy \({\tilde{E}}\) and the Noether charge \(Q_{\chi }\) of the kink-Q-ball system reach maximum values at the starting point, as well as the electric charge \({\tilde{q}} Q_{\chi }\) of the Q-ball component. At the same time, the derivatives \(d{\tilde{E}}/d{\tilde{\omega }}\) and \(dQ_{\chi }/d{\tilde{\omega }}\) are finite at the starting point. Note that a similar situation also occurs for some three-dimensional gauged Q-balls [2, 28]. The dependences \(E\left( \omega \right) \) and \(Q\left( \omega \right) \) of these Q-balls also reach maximum values at the limiting point of the phase frequency, while the derivatives \(dE/d\omega \) and \(dQ/d\omega \) are finite in this point. As \({\tilde{\omega }} \rightarrow 1\), the kink-Q-ball system enters the thick-wall regime. It was found numerically that in the neighbourhood of \({\tilde{\omega }}_{\text {tk }} = 1\), \(\varDelta {\tilde{E}} \sim Q_{\chi } \sim H \left( {\tilde{\omega }}_{\text {tk }} - {\tilde{\omega }}\right) ^{1/2}\), in accordance with Eqs. (75) and (76).

Based on the dependences \({\tilde{E}}\left( {\tilde{\omega }} \right) \) and \(Q_{\chi }\left( {\tilde{\omega }}\right) \), we can obtain the dependence \({\tilde{E}}\left( Q_{ \chi }\right) \). Figures 4 and 5 show the dependence \({\tilde{E}}\left( Q_{ \chi }\right) \) for the gauge coupling constants \({\tilde{e}} = {\tilde{q}} = 0.05\) and \({\tilde{e}} = {\tilde{q}} = 0.4\), respectively. The straight lines \({\tilde{E}} = Q_{\chi }\) in these figures correspond to the plane-wave solution. We see that for \({\tilde{e}} = {\tilde{q}} = 0.05\), the dependence \({\tilde{E}}\left( Q_{\chi }\right) \) is a single connected curve, whereas for \({\tilde{e}} ={\tilde{q}} = 0.4\), it consists of two separate curves. Of course, the number of curves in Figs. 4 and 5 is determined by the number of corresponding curves in Fig. 2. The curves in Figs. 4 and 5 possess cusps, the number of which is determined by the number of extremes of the corresponding curves in Figs. 1 and 2. The second derivative \(d^{2}{\tilde{E}}/dQ_{\chi }^{2}\) changes sign when passing through the cusps or discontinuities, meaning that the convex and concave sections of the curves change each other. Note that in Figs. 4 and 5, the energy of the kink-Q-ball system turns out to be greater than the energy of the plane-wave solution with the same value of \(Q_{\chi }\). This also turns out to be true for all other cases considered in the present paper. It follows that the kink-Q-ball system may transit into the plane-wave field configuration through quantum tunnelling.

*x*, whereas \(f\left( x\right) \) is an odd function of

*x*. From Fig. 7, it follows that the kink-Q-ball system possesses symmetrical energy and electric charge densities, and a nonzero electric field strength that is an odd function of the space coordinate. The distribution of the electric charge density is a centrally symmetric peak with a positive \(j_{0}\) surrounded by two areas with a negative \(j_{0}\), and thus the total electric charge of the kink-Q-ball system vanishes. The central positive peak is due to the contribution of the field \(\chi \), whereas the two negative areas to the sides are due to the contribution of the field \(\phi \). Indeed, from Eq. (32), it follows that the electric charge density of the field \(\phi \) is \(-2a_{0}e^{2}\!f^{2}\). We see that the electromagnetic potential \(a_{0}\) can induce a nonzero electric charge density even if the scalar field \(\phi \) approaches the vacuum value \(\left| \eta \right| \). As a result, a substantial part of the electric charge of the complex scalar field \(\phi \) comes from the two side regions where \(\left| \phi \right| \approx \left| \eta \right| \).

In Fig. 8, we can see the kink-Q-ball solution that corresponds to the same phase frequency \({\tilde{\omega }} = 0.985\) as the previous one but belongs to the right branch in Fig. 2. The densities of the energy and the electric charge are presented in Fig. 9 along with the electric field strength. The solution presented in Fig. 8 differs drastically from the solution presented in Fig. 6. Unlike Fig. 6, the ansatz functions \(s\left( {\tilde{x}} \right) \) and \(a_{0}\left( {\tilde{x}} \right) \) have two widely separated symmetrical maxima. At the same time, the form of the kink component changed insignificantly in comparison with that in Fig. 6. Note that in accordance with Eqs. (48) and (50), the electromagnetic potential \(a_{0}\left( {\tilde{x}} \right) \) also has a weak local maximum at \({\tilde{x}} = 0\), which is not discernible in Fig. 8. The densities \({\mathcal {E}}\) and \(j^{0}\) in Fig. 9 also differ considerably from those in Fig. 7. Firstly, the energy density \({\mathcal {E}}\) has the sharp central peak and the two symmetrical side peaks. The central peak is due to the kink component, whereas the two side peaks result from the Q-ball component of the soliton solution. Secondly, the distribution of the electric charge density \(j_{0}\) consists of two symmetrical side peaks with a positive \(j_{0}\), each of which surrounded by two regions with a negative \(j_{0}\). Note that in accordance with Eq. (50), there is also the local positive maximum of \(j_{0}\) at \({\tilde{x}} = 0\).

We now discuss the issue of the stability of the kink-Q-ball system. As has already been pointed out, the energy of the kink-Q-ball system turns out to be greater than the energy of the plane-wave solution with the same \(Q_{\chi }\), for all cases considered in the present paper. It follows that the kink-Q-ball system is unstable against transit into a plane-wave configuration via quantum tunnelling. We still need to consider the classical stability of the kink-Q-ball system with respect to fluctuations in the fields \(\phi \), \(\chi \), and \(A_{\mu }\) in the functional neighbourhood of the kink-Q-ball solution.

It is known that the gauge model described by the first line of the Lagrangian (1) has a kink solution [19, 20]. In the adopted gauge \(A_{x} = 0\), this kink solution is given by Eq. (5). The gauged kink has zero electric charge, and therefore has finite energy. However, unlike the kink of a self-interacting real scalar field [29, 30], the gauged kink is not a topologically stable field configuration. Due to the topological structure of the vacuum of the Abelian Higgs model, the gauged kink is a sphaleron [19, 20], the existence of which is due to the paths in the functional space that connect the topologically distinct vacua of the Abelian Higgs model [31]. The sphaleron lies between two topologically distinct neighbouring vacua and has only one unstable mode.

The gauged kink is a static solution modulo gauge transformations. However, in the case of the kink-Q-ball solution, we have a different situation. It can easily be shown that the kink-Q-ball solution will depend on time in any gauge, and thus is not a static solution. It follows that the point in the functional space corresponding to the kink-Q-ball solution will vary with time in any gauge. This fact does not allow the kink-Q-ball solution to be a sphaleron, so the issue of the unstable modes of the kink-Q-ball solution should be investigated separately.

To investigate the classical stability of the kink-Q-ball system, it is necessary to consider the second-order variation \(\delta ^{2}E\) in the energy of system. Moreover, the fields of the model must fluctuate so that the Noether charges \(Q_{\phi }\) and \(Q_{\chi }\) remain fixed [21] and the perturbed electromagnetic potential satisfies Gauss’s law. As a result, the second-order variation for fixed \(Q_{\phi }\) and \(Q_{\chi }\) is written as \(\delta ^{2}E = \iint \nolimits _{-\infty }^{+\infty }\psi ^{T}\left( x \right) {\mathcal {K}} \left( x, x^{\prime } \right) \psi \left( x^{\prime }\right) dx dx^{\prime }\), where \({\mathcal {K}}\) is a linear symmetric integro-differential operator and \(\psi = \left( \delta \phi _{1}, \delta \phi _{2}, \delta \chi _{1}, \delta \chi _{2} \right) \) is a fluctuation in the model’s scalar fields \(\phi = \phi _{1} + i\phi _{2}\) and \(\chi = \chi _{1} + i\chi _{2}\). Note that the perturbation in the electromagnetic potential is not included in \(\psi \), since it must be expressed in terms of the fluctuations in the scalar fields using Gauss’s law. Thus to study the stability of the kink-Q-ball system, we must find the eigenvalues and eigenmodes of the complicated integro-differential operator: \(\int \nolimits _{ -\infty }^{ +\infty }{\mathcal {K}}\left( x, x^{\prime } \right) \psi _{i}\left( x^{\prime }\right) dx = \lambda _{i}\psi _{i}\left( x\right) \), where the eigenmodes \(\psi _{i} \) form a complete orthonormal set of real functions: \(\int _{-\infty }^{+\infty }\psi _{i}^{T}\psi _{j} dx = \delta _{ij}\).

All of these factors make it difficult to study the spectrum of \({\mathcal {K}}\), even through the use of numerical methods. However, these difficulties can be avoided if we solve field equations (9)–(11) numerically, with a perturbed initial field configuration in a close neighbourhood of the kink-Q-ball solution. To examine the stability of the kink-Q-ball system, we work in the temporal gauge \(A_{0} = 0\). In this gauge, the time evolution of the fields \(\phi _{1}\), \(\phi _{2}\), \(\chi _{1}\), \(\chi _{2}\), and \(A_{x}\) is determined by a system of five differential equations, each of which includes the second-order time derivative of one of the fields. This circumstance make it easier to study the time evolution of the perturbed kink-Q-ball system by numerical methods. At \(t = 0\), the perturbed kink-Q-ball system must satisfy Gauss’s law and must have the same \(Q_{\phi }\) and \(Q_{\chi }\) as the unperturbed kink-Q-ball system. The field equations then guarantee that for \(t > 0\), Gauss’s law is satisfied and the perturbed kink-Q-ball system possesses the same \(Q_{\phi }\) and \(Q_{\chi } \) as the unperturbed one. Having obtained the perturbed and unperturbed kink-Q-ball solutions, we can observe the behaviour of the field perturbations as time passes. If any field perturbation oscillates in a neighbourhood of the unperturbed kink-Q-ball solution, then the solution is classically stable. If at least one field perturbation exists that increases exponentially with time, then the kink-Q-ball solution is classically unstable.

The unstable perturbation \(\varPsi _{1}\) corresponds to the unstable eigenmode of the non-gauged complex kink [31], whereas the unstable perturbation \(\varPsi _{2}\) has no analogue in the non-gauged case since the one-dimensional non-gauged Q-ball corresponding to self-interaction potential (4) is stable [23]. The unstable perturbations \(\varPsi _{1}\) and \(\varPsi _{2}\) are orthogonal \(( \int _{ -\infty }^{ +\infty } \varPsi _{1}^{T} \varPsi _{2} dx = 0 )\) and have opposite parities. It follows that the perturbations \(\varPsi _{1} \) and \(\varPsi _{2} \) are expanded in terms of different eigenmodes of the operator \({\mathcal {K}}\), and thus the instabilities in \(\varPsi _{1}\) and \(\varPsi _{2}\) result from different unstable eigenmodes of \({\mathcal {K}}\). Thus the linear integro-differential operator \({\mathcal {K}}\) has at least two unstable eigenmodes. Note that the perturbations \(\varPsi _{1}\) and \(\varPsi _{2}\) are representatives of broader classes of unstable perturbations; that is, the initial field perturbations \(\varXi _{1} = \left( 0, \delta \phi _{2}(x),0,0\right) \) and \(\varXi _{2} = \left( 0, 0, 0, \delta \chi _{2}(x)\right) \), where \(\delta \phi _{2}(x)\) is an arbitrary even function and \(\delta \chi _{2}(x)\) is an arbitrary odd function, are also unstable because their scalar products with the unstable perturbations \(\varPsi _{1}\) and \(\varPsi _{2} \) are different from zero: \(\int _{-\infty }^{+\infty } \varXi _{1}^{T} \varPsi _{1} dx \ne 0\) and \(\int _{-\infty }^{+\infty }\varXi _{2}^{T} \varPsi _{2} dx \ne 0\).

Using the graphical tools of the Maple package, we found that the unstable perturbations \(\varPsi _{1}\) and \(\varPsi _{2}\) increase exponentially with time and eventually destroy the kink-Q-ball system. We were able to estimate the exponential growth rate \(\nu \) for the perturbations \(\varPsi _{1}\) and \(\varPsi _{2}\). For \(\varPsi _{1}\), the exponential growth rate \(\nu \approx 0.6\div 0.8\), whereas for \(\varPsi _{2}\), the exponential growth rate \(\nu \approx 0.3\div 0.5\), where the intervals in \(\nu \) are due to the use of different gauge coupling constants and phase frequencies. We see that the exponential growth rates of \(\varPsi _{1}\) and \(\varPsi _{2}\) are different, so the time evolution of the perturbations \(\varPsi _{1}\) and \(\varPsi _{2}\) is determined by different unstable modes of the operator \({\mathcal {K}}\). Note that for the unstable eigenmode of the non-gauged kink, the exponential growth rate \(\nu = m_{\phi } /2\approx 0.707\). Thus, we can conclude that the time evolution of the perturbation \(\varPsi _{1}\) is determined by the unstable eigenmode of \({\mathcal {K}}\), which turns into the unstable eigenmode of the non-gauged complex kink as the gauge coupling constants vanish. The results obtained here show that the kink-Q-ball system is not a sphaleron since it has at least two unstable modes, whereas a sphaleron must have only one unstable mode.

## 6 Conclusion

In the present paper, we consider one-dimensional model (1), which consists of two self-interacting complex scalar fields interacting through an Abelian gauge field. It is shown that the model possesses a soliton solution consisting of a gauged kink and a gauged Q-ball. Since the finiteness of the energy of the one-dimensional soliton system leads to its electric neutrality, the gauged kink and the gauged Q-ball have opposite electric charges. Due to the neutrality of the Abelian gauge field, the opposite electric charges of the kink and Q-ball components are conserved separately, and despite the neutrality of the kink-Q-ball system, it possesses a nonzero electric field.

In the kink-Q-ball system, the energy and the Noether charge have rather unusual dependences on the phase frequency. It is shown here that the energy and the Noether charge of the kink-Q-ball system do not tend to infinity as the phase frequency tends to its minimum value. It follows that there is no thin-wall regime for the kink-Q-ball system. We also find that when the magnitude of the phase frequency is in the neighbourhood of \(m_{\chi }\), the dependences of the energy and the Noether charge on the phase frequency consist of two separate branches, provided that the model’s gauge coupling constants are large enough. The solutions from the left branches have a one-peak form, whereas those from the right branches have a two-peak form. In all cases, however, the kink-Q-ball system enters the thick-wall regime as the magnitude of the phase frequency tends to \(m_{\chi }\).

In addition to the kink-Q-ball solution, the model also possesses a plane-wave solution. For all sets of model parameters considered in the present paper, the energy of the kink-Q-ball solution turns out to be greater than the energy of the plane-wave solution for the same value of the Noether charge. Due to the topological structure of the vacuum of the model, the kink-Q-ball solution is not topologically stable, meaning that it can transit into the plane-wave configuration through quantum tunnelling.

It is well-known that the Abelian Higgs model possesses a gauge kink solution. The gauge kink is electrically neutral and has one unstable mode. From the viewpoint of topology, the gauge kink is a static (modulo gauge transformations) field configuration lying between two topologically distinct adjacent vacua. In contrast, the kink-Q-ball solution depends on time in any gauge, meaning that it is not a static field configuration. Hence, the kink-Q-ball solution cannot be a sphaleron, and its classical stability requires separate consideration. We investigate the classical stability of the kink-Q-ball system by means of a numerical solution of the field equations, with initial field configurations perturbed in a close neighbourhood of the kink-Q-ball solution. In all cases considered, it was found that the kink-Q-ball solution has at least two unstable modes, and thus it is even more unstable than the gauged kink.

## Notes

### Acknowledgements

This work was supported by the Russian Science Foundation, Grant No. 19-11-00005.

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