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

TL;DR

This paper investigates a (1+1)-dimensional gauge model with two interacting scalar fields, revealing a novel soliton system combining Q-ball and kink solutions with unique charge and stability properties.

Contribution

It introduces a new interacting kink-Q-ball soliton system in a gauge model, analyzing its properties both analytically and numerically, which was not previously known.

Findings

The system exists only at nonzero gauge coupling constants.

The kink-Q-ball system has a nonzero electric field.

The system is unstable under small perturbations.

Abstract

In the present paper, we consider a $\left(1 + 1\right)$-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 equal to zero, the model possesses non-gauged Q-ball and kink solutions that do not interact with each other. It is shown that at nonzero gauge coupling constants, the model possesses the soliton solution describing the system consisting of interacting Q-ball and kink components. The kink and Q-ball components of the kink-Q-ball system have opposite electric charges, so the total electric charge of the kink-Q-ball system vanishes. Properties of the kink-Q-ball system are researched analytically and numerically. In particular, it was found that the kink-Q-ball system possesses a nonzero electric field and is unstable with respect to small perturbations of fields.

Equations

$$\mathcal{L}$$

$$D_{\mu }\varphi ={\partial }_{\mu }\varphi +ie{A}_{\mu }\varphi ,D_{\mu }\chi ={\partial }_{\mu }\chi +iq{A}_{\mu }\chi .$$

$$V\left(\left|\varphi \right|\right)$$

$$U\left(\left|\chi \right|\right)$$

$${\varphi }_{\mathrm{k}}\left(x\right)=\eta \tanh \left(\frac{m_{\varphi }x}{2}\right)\exp \left(-i\delta \right),$$

$$\varphi \left(x\right)$$

$$\chi \left(x\right)$$

$$A_{\mu }\left(x\right)$$

$$\varphi \left(x\right)$$

$$\chi \left(x\right)$$

$$j_{\varphi }^{\mu }$$

$$j_{\chi }^{\mu }$$

$${\partial }_{\mu }{F}^{\mu \nu }$$

$$D_{\mu }{D}^{\mu }\varphi +\frac{\partial V}{\partial \left|\varphi \right|}\frac{\varphi }{2\left|\varphi \right|}$$

$$D_{\mu }{D}^{\mu }\chi +\frac{\partial U}{\partial \left|\chi \right|}\frac{\chi }{2\left|\chi \right|}$$

$$j^{\nu }=e{j}_{\varphi }^{\nu }+q{j}_{\chi }^{\nu }.$$

$$T_{\mu \nu }$$

$$T_{00}=\mathcal{E}$$

$$F=\underset{-\infty }{\overset{\infty }{\int }}\mathcal{E}dx-\omega {\int }_{-\infty }^{\infty }{j}_{\chi }^{0}dx=E-\omega Q_{\chi },$$

$$\mathcal{H}$$

$${\partial }_x^2{A}_0+ie\left\{{\varphi }^{\ast }{\pi }_{{\varphi }^{\ast }}-\varphi {\pi }_{\varphi }\right\}+iq\left\{{\chi }^{\ast }{\pi }_{{\chi }^{\ast }}-\chi {\pi }_{\chi }\right\}=0,$$

$$\mathcal{H}-\mathcal{E}$$

$${\partial }_t\varphi$$

$${\partial }_t\chi$$

$$\delta F=\delta E-\omega \delta Q_{\chi }=0,$$

$$\delta Q_{\chi }=-i\int \left({\pi }_{\chi }\delta \chi +\chi \delta {\pi }_{\chi }-\text{c.c.}\right)dx.$$

$${\partial }_t\chi$$

$${\partial }_t{\pi }_{\chi }$$

$${\partial }_t{\chi }^{\ast }$$

$${\partial }_t{\pi }_{{\chi }^{\ast }}$$

$$\varphi \left(x,t\right)$$

$$\chi \left(x,t\right)$$

$$A_{\mu }\left(x,t\right)$$

$$\frac{dE}{d{Q}_{\chi }}=\omega ,$$

$$a_0^{\prime \prime }(x)-2a_0\left(x\right)\left(e^{2}f{\left(x\right)}^2+q^{2}s{\left(x\right)}^2\right)+2q\omega s{\left(x\right)}^2=0,$$

$$f^{\prime \prime }\left(x\right)+\left(\lambda {\eta }^2+e^2{a}_0{\left(x\right)}^2\right)f\left(x\right)-\lambda f{\left(x\right)}^3=0,$$

$$s^{\prime \prime }\left(x\right)-\left(m_{\chi }^2-{\left(\omega -q{a}_0\left(x\right)\right)}^2\right)s\left(x\right)+g_{\chi }s{\left(x\right)}^3$$

Full text

††journal: Eur. Phys. J. C

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 is simple: at large distances, the electric field of such a field configuration does not depend on coordinate in the one-dimensional case and behaves as $r^{-1}$ in the two-dimensional case, so the energy of the electric field diverges linearly in the one-dimensional case and logarithmically in the two-dimensional case. That is why there are no electrically charged solitons in one and two dimensions; such solitons appear only in three dimensions (e.g. three-dimensional electrically charged dyon julia_zee or Q-ball klee ; anag ; ardoz_2009 ; gulamov_2014 ). It should be noted, however, that electrically charged two-dimensional vortices exist in both the Chern-Simons hong ; jw1 ; jw2 ; bazeia_1991 ; ghosh and the Maxwell-Chern-Simons paul ; khare_rao_227 ; khare_255 ; loginov_plb_784 gauge models. Furthermore, it was shown in dsantos ; losano that Chern-Simons gauge models also possess one-dimensional domain walls. The domain walls have finite linear densities of magnetic flux and electric charge, so there is a linear momentum flow along the domain walls.

Nevertheless, 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, the one-dimensional soliton system consisting of electrically charged Q-ball and anti-Q-ball components has been considered in loginov_prd_99 and the two-dimensional soliton systems consisting of vortex and Q-ball components interacting through an Abelian gauge field has been described in loginov_plb_777 .

In the present paper, we research the one-dimensional soliton system consisting of Q-ball and kink components, possessing opposite electric charges, so the system with a nonzero electric field is electrically neutral as a whole. Properties of this kink-Q-ball system are investigated by analytical and numerical methods. In particular, we found that in contrast to the non-gauged one-dimensional Q-ball, the kink-Q-ball system does not go into the thin-wall regime.

There is an interesting problem concerning the stability of the kink-Q-ball system with respect to small perturbations of fields. Recall that the Abelian Higgs model possesses an electrically neutral kink solution bochkarev_mpl_2 ; grigoriev . Formally, this gauged kink solution is the usual kink of a self-interacting real scalar field up to gauge transformations. However, properties of these two kink solutions differ considerably, because the classical vacua of the corresponding field models have a different topology. While the real kink is topologically stable, the gauged kink has exactly one unstable mode. From the topological point of view, the gauged kink lies between two topologically distinct vacua of the Abelian Higgs model, so it is a sphaleron bochkarev_mpl_2 ; grigoriev . 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, so its classic stability requires separate consideration.

The paper has the following structure. In Sec. 2, we describe briefly the Lagrangian, the symmetries, the field equations, and the energy-momentum tensor of the Abelian gauge model under consideration. In Sec. 3, we research 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. The important differential relation for the kink-Q-ball solution is derived and the system of nonlinear differential equations for ansatz functions is obtained. Then we establish some general properties of the kink-Q-ball system. In particular, we research asymptotic behavior 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 Sec. 4, we study properties of the kink-Q-ball system in the three extreme regimes, namely, in the thick-wall regime and in the regimes of small and large gauge coupling constants. We also establish basic properties of the plane-wave field configuration of the model. In Sec. 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.

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

2 The gauge model

The Lagrangian density of the $\left(1+1\right)$-dimensional gauge model under consideration has the form

[TABLE]

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the strength of the Abelian gauge field and $\phi$, $\chi$ are complex scalar fields minimally interacting with the Abelian gauge field through the covariant derivatives:

[TABLE]

The self-interaction potentials of the scalar fields have the form

[TABLE]

Let us suppose that the self-interaction potential $U\left(\left|\chi\right|\right)$ has a global zero minimum at $\chi=0$ and admits the existence of usual non-gauged Q-balls. Then the parameters $g_{\chi}$ and $h_{\chi}$ are positive and satisfy the condition $3g_{\chi}^{2}<16h_{\chi}m_{\chi}^{2}$. Unlike the sixth-order potential $U\left(\left|\chi\right|\right)$, the fourth-order potential $V\left(\left|\phi\right|\right)$ reaches a zero minimum on the circle $\left|\phi\right|=\eta$. The potential $V\left(\left|\phi\right|\right)$ allows the existence of the complex non-gauged kink solution

[TABLE]

where $m_{\phi}=\sqrt{2\lambda}\eta$ is the mass of the scalar $\phi$-particle and $\delta$ is an arbitrary phase.

Besides the local gauge transformations:

[TABLE]

the Lagrangian (1) is also invariant under the two independent global gauge transformations:

[TABLE]

As a consequence, we have the two Noether currents:

[TABLE]

and the two separately conserved Noether charges: $Q_{\phi}=\int j_{\phi}^{0}dx$ and $Q_{\chi}=\int j_{\chi}^{0}dx$. Note also that in addition to the local and global gauge transformations, the Lagrangian (1) is invariant under the discrete $C$, $P$, and $T$ transformations.

The field equations of the model are written as

[TABLE]

where the electromagnetic current $j^{\nu}$ is written in terms of two Noether currents (8)

[TABLE]

From Eq. (12) it follows that the electric charges $eQ_{\phi}$ and $qQ_{\chi}$ of the complex scalar fields $\phi$ and $\chi$ are conserved separately. This fact is a consequence of the neutrality of the Abelian gauge field $A_{\mu}$.

The symmetric energy-momentum tensor of the model can be obtained by using the well-known formula $T_{\mu\nu}=2\partial\mathcal{L}/\partial g^{\mu\nu}-g_{\mu\nu}\mathcal{L}$:

[TABLE]

Thus, we have the following expression for the energy density of the model

[TABLE]

where $E_{x}=F_{01}=\partial_{t}A_{1}-\partial_{x}A_{0}$ is the electric field strength.

3 The kink-Q-ball system and its properties

It is known lee ; fried that any nontopological soliton, in particular, a Q-ball, is an extremum of an energy functional at a fixed value of the corresponding Noether charge. Using this basic property of a Q-ball and taking into account that the self-interaction potential $U\left(\left|\chi\right|\right)$ admits the existence of Q-balls formed from the complex scalar field $\chi$, we shall search for a soliton solution of model (1) that is an extremum of the energy functional $E=\int\nolimits_{-\infty}^{\infty}\mathcal{E}dx$ at a fixed value of the Noether charge $Q_{\chi}=\int\nolimits_{-\infty}^{\infty}j_{\chi}^{0}dx$. According to the method of Lagrange multipliers, such a solution is an unconditional extremum of the functional

[TABLE]

where $\omega$ is the Lagrange multiplier. Let us determine the time dependence of the soliton solution. To do this, we shall use Eq. (15) and the Hamiltonian formalism. In the axial gauge $A_{x}=A^{1}=0$, the Hamiltonian density of model (1) is written as

[TABLE]

We see that in the adopted gauge, the model is described in terms of the eight canonically conjugated fields: $\phi$, $\pi_{\phi}=\left(D_{0}\phi\right)^{\ast}$, $\phi^{\ast}$, $\pi_{\phi^{\ast}}=D_{0}\phi$, $\chi$, $\pi_{\chi}=\left(D_{0}\chi\right)^{\ast}$, $\chi^{\ast}$, and $\pi_{\chi^{\ast}}=D_{0}\chi$, while the time component $A_{0}$ of the gauge field is determined in terms of the canonically conjugated fields by Gauss’s law

[TABLE]

and so it is not an independent dynamic field. Although energy density (14) is not equal to Hamiltonian density (16):

[TABLE]

the integral of Eq. (18) over the one-dimensional space vanishes provided that field configurations of the model possess finite energy and satisfy Gauss’s law (17).

In the adopted axial gauge $A_{x}=0$, field equations (10) and (11) can be recast in the Hamiltonian form:

[TABLE]

where we use the relation $E=\int\nolimits_{-\infty}^{\infty}\mathcal{E}dx=H=\int\nolimits_{-\infty}^{\infty}\mathcal{H}dx$. On the other hand, the first variation of functional (15) vanishes on the soliton solution:

[TABLE]

where the first variation of the Noether charge $Q_{\chi}$ is expressed in terms of the canonically conjugated fields as follows:

[TABLE]

Combining Eqs. (19) – (22), we find that in the adopted gauge, only the time derivatives of the canonically conjugated fields $\chi$, $\pi_{\chi}$, $\chi^{\ast}$, and $\pi_{\chi^{\ast}}$ are different from zero:

[TABLE]

while the time derivatives of $\phi$, $\pi_{\phi}$, $\phi^{\ast}$, and $\pi_{\phi^{\ast}}$ are equal to zero. Recalling that $\pi_{\chi}=\left(D_{0}\chi\right)^{\ast}=\partial_{t}\chi^{\ast}-iqA_{0}\chi^{\ast}$ and taking into account Eqs. (24) and (25), we conclude that the time derivative of $A_{0}$ also vanishes. It follows that only the scalar field $\chi$ of the soliton system has nontrivial time dependence:

[TABLE]

Let us return to Eq. (21). This equation holds for arbitrary variations of fields on the soliton solution, including those that transfer the soliton solution to an infinitesimally close one. It follows that the energy of the soliton system satisfies the important relation

[TABLE]

where the Lagrange multiplier $\omega$ is some function of the Noether charge $Q_{\chi}$. Since the energy $E$ and the Noether charge $Q_{\chi}$ of the soliton system are gauge-invariant, relation (28) is also gauge-invariant. Like the case of non-gauged nontopological solitons fried ; coleman ; lee , relation (28) determines basic properties of the gauged kink-Q-ball system.

In Eqs. (27), functions $f\left(x\right)$ and $s\left(x\right)$ are assumed to be some complex functions of the real argument $x$. Substituting Eqs. (27) into field equations (9) – (11), we can easily check that the real and imaginary parts of $f\left(x\right)$ satisfy the same differential equation with real coefficients. Similarly, the real and imaginary parts of $s\left(x\right)$ also satisfy the same differential equation with real coefficients. It follows that the functions $f\left(x\right)$ and $s\left(x\right)$ have the form: $f\left(x\right)=\exp\left(i\alpha\right)\tilde{f}\left(x\right)$, $s\left(x\right)=\exp\left(i\beta\right)\tilde{s}\left(x\right)$, where $\tilde{f}\left(x\right)$ and $\tilde{s}\left(x\right)$ are real functions, whereas $\alpha$ and $\beta$ are constant phases. However, these phases can be cancelled by global gauge transformations (7), so the functions $f\left(x\right)$ and $s\left(x\right)$ can be supposed to be real without loss of generality. The functions $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$ satisfy the system of ordinary nonlinear differential equations:

[TABLE]

[TABLE]

[TABLE]

which is obtained by substituting Eqs. (27) into field equations (9) – (11).

The most important among the local quantities of the kink-Q-ball system are the electromagnetic current density and the energy density. Their expressions in terms of $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$ are written as

[TABLE]

[TABLE]

The energy $E=\int\nolimits_{-\infty}^{\infty}\mathcal{E}dx$ of the kink-Q-ball system must be finite. Using this fact and Eq. (3), we obtain the boundary condition for $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$:

[TABLE]

Note that the finiteness of the electric field’s energy $E^{\left(E\right)}=\int\nolimits_{-\infty}^{\infty}a_{0}^{\prime 2}/2dx$ leads to one more boundary condition for $a_{0}\left(x\right)$:

[TABLE]

This condition, however, is equivalent to Eq. (34a) provided that $a_{0}\left(x\right)$ is regular as $x\rightarrow\pm\infty$.

Gauss’s law (29) can be written as $a_{0}^{\prime\prime}=-j^{0}$, where $j^{0}$ is electric charge density (32). Integrating this equation over $x\in\left(-\infty,\infty\right)$ and taking into account boundary conditions (35), we conclude that the total electric charge $Q=\int\nolimits_{-\infty}^{\infty}j^{0}dx$ of a field configuration with finite energy vanishes:

[TABLE]

where $Q_{\phi}$ and $Q_{\phi}$ are the Noether charges defined by Eqs. (8).

It can easily be checked that system (29) – (31) is invariant under the discrete transformation

[TABLE]

This invariance is a consequence of the $C$-invariance of the Lagrangian (1). Using Eqs. (32), (3), and (37), we find the behavior of the energy $E$ and the Noether charges $Q_{\phi}$ and $Q_{\chi}$ under the transformation $\omega\rightarrow-\omega$:

[TABLE]

We see that the energy of the kink-Q-ball system is an even function of $\omega$, whereas the Noether charges $Q_{\phi}$ and $Q_{\chi}$ are odd functions of $\omega$.

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, the system (29) – (31) is also invariant under the coordinate shift $x\rightarrow x+x_{0}$. Furthermore, due to Eqs. (7), the system (29) – (31) is invariant under the two independent discrete transformations: $f\rightarrow-f$ and $s\rightarrow-s$. These facts and 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 $x$.

The asymptotic form of the soliton solution for small $x$ is obtained by substitution of the power expansions for $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$ into Eqs. (29) – (31) and equating the resulting Taylor coefficients to zero. By acting in this way, we obtain:

[TABLE]

where the next-to-leading coefficients

[TABLE]

are expressed in terms of the three leading coefficients $a_{0}$, $f_{1}$, $s_{0}$, and the model’s parameters.

For large $\left|x\right|$, system (29) – (31) is linearized and we obtain the asymptotic form of the soliton solution satisfying boundary conditions (34):

[TABLE]

where $m_{\phi}=\sqrt{2\lambda}\eta$, $\Delta=\left(m_{\chi}^{2}-\omega^{2}\right)^{1/2}$​, and $m_{A}=\sqrt{2}e\eta$.

Let us discuss the global behavior of the electromagnetic potential $a_{0}\left(x\right)$. Since the total electric charge $Q=\int\nolimits_{-\infty}^{+\infty}j^{0}\left(x\right)dx$ of the kink-Q-ball system vanishes, the electric charge density $j^{0}\left(x\right)$ must vanish at some points of the $x$-axis. Because of 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 in 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 $s\left(x_{\mathrm{i}}\right)$:

[TABLE]

Two conclusions follow from Eq. (43). Firstly, at an inflection point $x_{\mathrm{i}}$, the sign of $a_{0}\left(x_{\mathrm{i}}\right)$ coincides with the sign of $\omega$ (we suppose that the gauge coupling constants are positive by definition):

[TABLE]

Secondly, at an inflection point $x_{\mathrm{i}}$, the following inequality holds:

[TABLE]

Next, from Eq. (32), we obtain the expression for the electric charge density at the origin:

[TABLE]

from which it follows that the sign of the curvature of $a_{0}\left(x\right)$ at $x=0$ is opposite in sign to $\omega-qa_{0}\left(0\right)$:

[TABLE]

An elementary graphical analysis made using Eqs. (43) – (47) leads us to the following conclusions about the behavior of $a_{0}\left(x\right)$:

[TABLE]

and

[TABLE]

where $\pm x_{\mathrm{i1}}$ is the two symmetric inflection points closest to the origin $x=0$. From Eqs. (46), (48), and (49) it follows that the sign of the electric charge density at the origin coincides with that of the phase frequency

[TABLE]

We can also make conclusions about the behavior of $a_{0}\left(x\right)$ for $\left|x\right|>x_{\mathrm{i1}}$. In particular, $a_{0}\left(x\right)$ cannot vanish at any finite $x$. Indeed, let $x_{\mathrm{n}}$ be a conjectural point in which $a_{0}\left(x\right)$ vanishes. Then from Eq. (29) we have the relation

[TABLE]

We see that the sign of the curvature of $a_{0}\left(x\right)$ at the point $x_{\mathrm{n}}$ is opposite to the sign of $\omega$. Let $\omega$ be positive. Then from Eq. (51) it follows that in some neighborhood of $x_{\mathrm{n}}$, the functions $a_{0}\left(x\right)$ and $a_{0}^{\prime\prime}\left(x\right)$ are negative. But according to Eq. (44), there are no inflection points for negative $a_{0}\left(x\right)$, so $a_{0}^{\prime\prime}\left(x\right)$ can never change the sign, $a_{0}\left(x\right)$ decreases indefinitely, and boundary condition (34a) cannot be satisfied. It follows that $a_{0}\left(x\right)$ cannot vanish at any finite $x$. The case of negative $\omega$ is treated similarly. Thus, we come to an 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 $x$:

[TABLE]

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

Let $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$ be a solution of system (29) – (31) that satisfy boundary conditions (34). When we perform the scale transformation $x\rightarrow\lambda x$ of the argument of the solution, the Lagrangian $L=\int\nolimits_{-\infty}^{\infty}\mathcal{L}dx$ becomes a simple function of the scale parameter $\lambda$. The function $L\left(\lambda\right)$ must have an extremum at $\lambda=1$, so the derivative $dL/d\lambda$ vanishes at this point. Using this fact, we obtain the virial relation for the soliton system:

[TABLE]

where

[TABLE]

is the electric field’s energy,

[TABLE]

is the gradient part of the soliton’s energy,

[TABLE]

is the kinetic part of the soliton’s energy, and

[TABLE]

is the potential part of the soliton’s energy.

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

[TABLE]

Another representation for the soliton system’s energy can be obtained by integrating the term $a_{0}^{\prime}{}^{2}/2$ in Eq. (3) by parts and using Eqs. (29), (32), and (34):

[TABLE]

Finally, using Eq. (59), we obtain the relation between the Noether charge $Q_{\chi}$, the electric field’s energy $E^{\left(E\right)}$, and the kinetic energy $E^{\left(T\right)}$:

[TABLE]

4 Extreme regimes of the kink-Q-ball system

In this section, we will first study the properties of the kink-Q-ball system in the thick-wall regime kusenko_1997 ; multam ; paccetti . In this regime, the parameter $\Delta=\left(m_{\chi}^{2}-\omega^{2}\right)^{1/2}$ tends to zero, so the absolute value of phase frequency tends to $m_{\chi}$. From Eqs. (42) it follows that in the thick-wall regime, the functions $s\left(x\right)$ and $a_{0}\left(x\right)$ are spread over the one-dimensional space, whereas the asymptotic behavior of $f\left(x\right)$ remains unchanged. In the thick wall regime, functions $s\left(x\right)$ and $a_{0}\left(x\right)$ uniformly decrease as $\Delta$ and $\Delta^{2}$, respectively, while the function $f\left(x\right)$ tends to non-gauged kink solution (5). For this reason, we perform the following scale transformation of the fields and $x$-coordinate:

[TABLE]

while the field $f\left(x\right)$ is taken equal to that of kink solution (5). To research the properties of the kink-Q-ball system in the thick-wall regime, we shall use functional (15) that is related to the energy functional through the Legendre transformation: $F\left(\omega\right)=E\left(Q_{\chi}\right)-\omega Q_{\chi}$. Using scale transformation (61), we can determine the leading term of the dependence of the functional $F\left(\omega\right)$ on $\omega$ in the thick-wall regime:

[TABLE]

where $E_{k}=4\eta^{3}\sqrt{2\lambda}/3=4\eta^{2}m_{\phi}/3$ is the rest energy of the non-gauged kink and the dimensionless functional $\bar{F}$ does not depend on $\omega$:

[TABLE]

In Eq. (62), higher-order terms in $\Delta$ may be neglected in the thick-wall regime, so we obtain sequentially:

[TABLE]

where known properties of Legendre transformation are used. Using Eqs. (64) and (65), we obtain the energy of the kink-Q-ball system as a function of its Noether charge in the thick-wall regime:

[TABLE]

It has been found numerically that the kink-Q-ball system does not turn into the thin-wall regime as the magnitude of the phase frequency tends to its minimum value. Such behavior can be qualitatively explained as follows. Gauss’s law $a_{0}^{\prime\prime}\left(x\right)=-j_{0}\left(x\right)$ has an obvious mechanical analogy. It describes a one-dimensional motion of a unit mass particle along the coordinate $a_{0}$ in time $x$. The particle is subjected to the time-dependent force $-j_{0}\left(x\right)$. It starts to move at the time $x=0$ from the point $a_{0}\left(0\right)>0$ (we suppose that $\omega>0$) with zero initial velocity $a_{0}^{\prime}\left(0\right)$. Since $j_{0}\left(x\right)>0$ as $x<x_{\mathrm{i}}$ ($x_{\mathrm{i}}$ is the inflection point of $a_{0}\left(x\right)$), the particle’s coordinate $a_{0}$ is decreased with increasing of $x$. At the time $x_{\mathrm{i}}$, the particle is at the point $a_{0}\left(x_{\mathrm{i}}\right)$ given by Eq. (43) and possesses the velocity $a_{0}^{\prime}\left(x_{\mathrm{i}}\right)=-Q_{+}/2$, where $Q_{+}=\int\nolimits_{-x_{\text{i}}}^{x_{\text{i}}}j_{0}\left(x\right)dx$ is the positive electric charge of the central area of the soliton system. The negative electric charge $\int\nolimits_{x_{\text{i}}}^{\infty}j_{0}\left(x\right)dx$ of the side area $x>x_{\mathrm{i}}$ corresponds to the impulse of force acting in the positive direction. According to boundary conditions (34a) and (35), this impulse of force must bring the particle to rest at the point $a_{0}=0$ as time $x$ tends to infinity.

As the magnitude of the phase frequency $\omega$ approaches to the minimum value, the spatial size of the kink-Q-ball system increases, whereas the positive electric charge of the central area weakly depends on $\omega$. As a result, the velocity of the particle at the inflection point $x_{\text{i}}$ remains approximately constant, whereas the decelerating force $-j_{0}\left(x\right)$ decreases because of the spreading of the impulse of force $-\int\nolimits_{x_{\text{i}}}^{\infty}j_{0}\left(x\right)dx$ (which is equal to half the positive electric charge $Q_{+}$ of the central area) over the side area $\left[x_{\text{i}},\infty\right)$. Thus the decelerating force $j_{0}\left(x\right)$ decreases and so it cannot stop the particle. Because of that, the particle reaches the point $a_{0}=0$ for a finite period of time, so boundary conditions (34a) and (35) are not satisfied and the kink-Q-ball system cannot exist.

Next, we consider the regime of small gauge coupling constants. When the gauge coupling constants $e$ 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 the set of non-gauged kink and Q-ball that do not interact with each other. In this connection, we want to ascertain the behavior of the gauge potential $a_{0}\left(x\right)$ as $e=\varrho q\rightarrow 0$, where $\varrho$ is a positive constant. To do this, we use asymptotic expressions (40a) and (42) for $a_{0}\left(x\right)$ that are valid for small and large values of $\left|x\right|$, respectively. We also suppose that Eqs. (40a) and (42) describe qualitatively the behavior of $a_{0}$ at intermediate $\left|x\right|$. Eqs. (40a) and (42) depend on the free parameters $a_{0}$ and $a_{\infty}$, respectively. These parameters 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 model’s parameters, 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 $e=\varrho q\rightarrow 0$:

[TABLE]

where the limit value $\alpha$ depends on the model’s parameters and $\varrho$. This fact and linearization of Eqs. (30) and (31) lead us to the asymptotic forms of $f\left(x\right)$ and $s\left(x\right)$ at small gauge coupling constants:

[TABLE]

where $f_{\text{k}}\left(x\right)$ and $s_{\text{q}}\left(x\right)$ are the non-gauged kink and Q-ball solutions, respectively, whereas $f_{2}\left(x\right)$ and $s_{1}\left(x\right)$ are some regular functions that depend on the model’s parameters (except for $e$ and $q$) and $\varrho$. Note that due to the relation $e=\varrho q$, we use only one expansion parameter $e$. Substituting Eqs. (67) and (68) into Eq. (32), we find the asymptotic behavior of the Noether charges $Q_{\phi}$ and $Q_{\chi}$ as $e=\varrho q\rightarrow 0$:

[TABLE]

where $Q_{0}=2\omega\int\nolimits_{-\infty}^{+\infty}s_{\mathrm{q}}^{2}dx$ is the Noehter charge of the non-gauged Q-ball and the coefficient $Q_{1}$ depends on the model’s parameters (except for $e$ and $q$) and $\varrho$. Similarly to Eq. (69), we obtain the asymptotic behavior of the soliton energy’s components (54) – (57) and the total soliton energy:

[TABLE]

where $E_{0}^{\left(G\right)}=\int\nolimits_{-\infty}^{+\infty}\left(f_{\mathrm{k}}^{\prime 2}+s_{\mathrm{q}}^{\prime 2}\right)dx$, $E_{0}^{\left(T\right)}=\omega^{2}\int\nolimits_{-\infty}^{+\infty}s_{\mathrm{q}}^{2}dx$, and $E_{0}^{\left(P\right)}=\int\nolimits_{-\infty}^{+\infty}\left(V\left(f_{\mathrm{k}}\right)+U\left(s_{\mathrm{q}}\right)\right)dx$ are the gradient, kinetic, and potential parts of the non-gauged soliton system’s energy, respectively, $E_{0}=E_{0}^{\left(G\right)}+E_{0}^{\left(T\right)}+E_{0}^{\left(P\right)}$ is the total energy of the non-gauged soliton system, and the coefficients $E_{1}^{\left(E\right)}$, $E_{1}^{\left(G\right)}$, $E_{1}^{\left(T\right)}$, and $E_{1}^{\left(P\right)}$ depend on the model’s parameters (except for $e$ and $q$) and $\varrho$. Note that the non-gauged solutions $f_{\mathrm{k}}$ and $s_{\mathrm{q}}$ can be expressed in analytical form, as well as the corresponding energies and the Noether charges. The corresponding expressions for the one-dimensional non-gauged Q-ball are given in loginov_prd_99 . 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.

Next let us consider the opposite regime in which both gauge coupling constants tend to infinity: $e=\varrho q\rightarrow\infty$. We suppose that the behavior of the electromagnetic potential $a_{0}\left(x\right)$ in a neighborhood of $x=0$ is regular, so the coefficient $a_{2}$ given by Eq. (41a) is either finite or tending to zero as $e=\varrho q\rightarrow\infty$. But from Eqs. (29) and (31) it follows that the electromagnetic potential is an odd function of $q$: $a_{0}\left(x,-q\right)=-a_{0}\left(x,q\right)$. Thus, we conclude that in the leading order, $a_{2}\propto q^{-1}$, so from Eq. (41a) it follows that $a_{0}\sim\omega q^{-1}-a_{-3}q^{-3}$, where $a_{-3}$ is a positive constant. This fact suggests that the electromagnetic potential $a_{0}\left(x\right)$ has a similar asymptotic expansion in the inverse powers of $e$:

[TABLE]

where $a_{-1}\left(x\right)$ and $a_{-3}\left(x\right)$ are some regular functions depending on the model’s parameters (except for $e$ and $q$) and $\varrho$. Using Eqs. (30), (31), and (71), we obtain the general form of asymptotic expansions for $f\left(x\right)$ and $s\left(x\right)$:

[TABLE]

where $f_{0}\left(x\right)$, $f_{-2}\left(x\right)$, $s_{0}\left(x\right)$, and $s_{-2}\left(x\right)$ are regular functions depending on the model’s parameters (except for $e$ and $q$) and $\varrho$. Similar to Eqs. (67) and (68), we can use Eqs. (71) and (72) to obtain the asymptotic expansions for the soliton energy’s components, the total energy, and the Noether charges:

[TABLE]

where we use the tilde to distinguish corresponding coefficients from those of Eqs. (70a) – (70d). We see that as $e=\varrho q\rightarrow\infty$, the gauge field $a_{0}\left(x\right)$ tends to zero, so the electric field’s energy $E^{\left(E\right)}$ also vanishes in this regime. At the same time, the products $ea_{0}\left(x\right)$ and $qa_{0}\left(x\right)$ tend to nonzero limits $a_{-1}\left(x\right)$ and $\varrho^{-1}a_{-1}\left(x\right)$, respectively, so the gauge field $a_{0}\left(x\right)$ does not decouple from the kink-Q-ball system. Due to this, the limit solutions $f_{0}\left(x\right)$ and $s_{0}\left(x\right)$ are different from the corresponding non-gauged solutions $f_{\mathrm{k}}\left(x\right)$ and $s_{\mathrm{q}}\left(x\right)$, respectively. From Eqs. (73b) – (73f) it follows that the soliton energy’s components $E^{\left(G\right)}$, $E^{\left(T\right)}$, $E^{\left(P\right)}$, the total soliton energy $E$, and the Noether charges $Q_{\chi}$ and $Q_{\phi}$ also tend to some finite values as $e=\varrho q\rightarrow\infty$. It follows that the electric charges of the kink and the Q-ball increase indefinitely in this regime, despite the fact that the electric field’s energy $E^{\left(E\right)}$ tends to zero. This is because the electric charges of the kink and the Q-ball tend to cancel each other at any spatial point as $e=\varrho q\rightarrow\infty$. Note that the behavior of the kink-Q-ball system in the extreme regimes $e=\varrho q\rightarrow 0$ and $e=\varrho q\rightarrow\infty$ was investigated by numerical methods. It was found that it is in accordance with Eqs. (69), (70), and (73).

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

We want to find the spatially uniform solution of field equations (9) – (11) possessing the Noether charges $Q_{\phi}$ and $Q_{\chi}$ (recall that $eQ_{\phi}+qQ_{\chi}=0$ for any finite energy field configuration) and to determine its energy. For this, we use field equations (9) – (11) in the unitary gauge. We suppose that with unlimited spreading, the amplitudes of the complex scalar field $\chi$ and the real scalar Higgs field $\phi_{H}$ tend to zero, so we can neglect higher-order terms in the Lagrangian (1). On spatially uniform fields, the field equations for $A^{\mu}$ and $\phi_{H}$ become algebraic ones, whereas the field equation for $\chi$ determines the time dependence of $\chi$. The results obtained are presented as series in inverse powers of the plane-wave solution’s spatial size $L$:

[TABLE]

[TABLE]

[TABLE]

We see that as $L\rightarrow\infty$, the amplitudes of the fields $A^{0}$, $\phi_{H}$, and $\chi$ tend to zero, whereas the phase frequency of $\chi$ tends to $m_{\chi}$. Note that the fields $A^{0}$, $\phi_{H}$, and $\chi$ of the plane-wave solution tend to zero as $L^{-1}$, $L^{-2}$, and $L^{-1/2}$, respectively, so the Higgs field $\phi_{H}$ tends to zero much more quickly than the complex scalar field $\chi$. Substituting Eqs. (74) – (4) into Eqs. (8), we obtain the Noether charge densities $j_{\phi}^{0}$ and $j_{\chi}^{0}$ of the plane-wave solution:

[TABLE]

From Eqs. (77) and (78) it follows that the electric charge of the plane-wave solution vanishes:

[TABLE]

as it should be. Next, we calculate the energies of the $\phi$ and $\chi$ components of the plane-wave solution:

[TABLE]

and

[TABLE]

so the total energy of the plane-wave solution turns out to be equal to

[TABLE]

Let us discuss the results obtained. First of all, from Eqs. (80) – (82) it follows that the $\phi$ component does not contribute to the energy of the plane-wave solution as $L\rightarrow\infty$. At the same time, from Eqs. (77) and (79) it follows that the electric charge of the $\phi$ component does not vanish and is opposite to that of the $\chi$ component, so the total electric charge of the plane-wave solution vanishes. Thus, the $\phi$ component contributes to the electric charge of the plane-wave solution, but does not contribute to its energy. This can be explained as follows. In the unitary gauge $\text{Im}\left(\phi\right)=0$, the real Higgs field $\phi_{H}$ fluctuates around the real vacuum average $\eta$, so the initial scalar field $\phi$ is written as $\phi=\eta+\phi_{H}$. Further, from Eqs. (8a) and (14) we obtain the electric charge and energy densities of the $\phi$ component of the plane-wave solution:

[TABLE]

and

[TABLE]

where Eqs. (74) and (75) have been used. We see that in the leading order in $L^{-1}$, the Higgs field $\phi_{H}$ contributes neither to $j_{\phi}^{0}$ nor to $\mathcal{E}_{\phi}$, whereas the vacuum average of the scalar field $\phi$ contributes in both cases. We also see that as $L\rightarrow\infty$, the behavior of $j_{\phi}^{0}$ and $\mathcal{E}_{\phi}$ is determined only by the electromagnetic potential $A^{0}$. At the same time, from Eq. (74) it follows that for the plane-wave solution, $A^{0}\sim L^{-1}$, so $j_{\phi}^{0}\sim L^{-1}$ and $\mathcal{E}_{\phi}\sim L^{-2}$ in the leading order in $L^{-1}$. It follows that $Q_{\phi}=\int\nolimits_{-\infty}^{+\infty}j_{\phi}^{0}dx=O\left(1\right)$ and $E_{\phi}=\int\nolimits_{-\infty}^{+\infty}\mathcal{E}_{\phi}dx=O\left(L^{-1}\right)$, so the $\phi$ component does not contribute to the plane-wave solution’s energy as $L\rightarrow\infty$.

5 Numerical results

To study the kink-Q-ball system, we must solve system of differential equations (29) – (31) satisfying boundary conditions (34). This first boundary value problem can be solved only numerically. To solve the boundary value problem, we use the method of finite differences and subsequent Newtonian iterations realized in the Maple package maple . To check the correctness of a numerical solution, we use Eqs. (28), (36), and (53).

To solve the boundary value problem, we need to know the eight dimensional parameters: $\omega$, $e$, $q$, $m_{\phi}=\sqrt{2\lambda}\eta$, $\lambda$, $m_{\chi}$, $g_{\chi}$, and $h_{\chi}$. Without loss of generality the mass $m_{\chi}$ can be chosen as the energy unit, so the dimensionless functions $a_{0}\left(x\right)$, $f\left(x\right)$, and $s\left(x\right)$ 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 $\tilde{h}_{\chi}=1$ are of the same order of magnitude. The dimensionless gauge coupling constants $\tilde{e}$ and $\tilde{q}$ were taken to be equal to each other and could take the values $0.05$, $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$.

Let us denote by $\Delta E$ the difference between the energies of the kink-Q-boll system and the non-gauged kink: $\Delta E=E-E_{\text{k}}$, where $E_{\mathrm{k}}=4\eta^{3}\sqrt{2\lambda}/3$. Figures 1 and 2 present the dependence of the dimensionless energy difference $\Delta\tilde{E}=\Delta E/m_{\chi}$ on the dimensionless phase frequency $\tilde{\omega}$. The curves in these figures correspond to gauge coupling constants $\tilde{e}=\tilde{q}$ taking the values from the set $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$. Figure 1 presents the curves in the range from the minimum values of $\tilde{\omega}$, which we managed to reach by numerical methods, to the value $\tilde{\omega}=0.88$. Figure 2 presents the same curves in the range from $\tilde{\omega}=0.88$ to the maximum possible value $\tilde{\omega}=1$. We use the two figures for a better representation of the dependences $\Delta\tilde{E}\left(\tilde{\omega}\right)$. For the same values of gauge coupling constants, the dependences $\Delta\tilde{E}\left(\tilde{\omega}\right)$ and $Q_{\chi}\left(\tilde{\omega}\right)$ are qualitatively similar, so the dependences $Q_{\chi}\left(\tilde{\omega}\right)$ are not given in the present paper.

Let us discuss the main features of the curves in Figs. 1 and 2. First of all, we note that the energy of the kink-Q-ball system does not tend to infinity as $\tilde{\omega}$ tends to its minimum values (that depend on the gauge coupling constants). Indeed, it was found numerically that the dependences $\tilde{E}\left(\tilde{\omega}\right)$ and $Q_{\chi}\left(\tilde{\omega}\right)$ have a branching point at $\tilde{\omega}_{\text{min}}$:

[TABLE]

where $A$, $B$, and $C$ are positive constants. We were unable to find any solutions of the boundary value problem for $\tilde{\omega}<\tilde{\omega}_{\min}$, so we conclude that the kink-Q-ball system does not turn into the thin-wall regime in which both $\tilde{E}$ and $Q_{\chi}$ must tend to infinity.

The behavior of the curves in the neighborhood of $\tilde{\omega}=1$ is also rather unusual. We see that for the gauge coupling constants $\tilde{e}=\tilde{q}$ from the set $0.2$, $0.3$, $0.4$, and $0.5$, the dependence $\tilde{E}\left(\tilde{\omega}\right)$ consists of two separate curves. The left curve starts from the minimal phase frequency $\tilde{\omega}_{\min}$ (where it has branching point (85)) and continues until the maximal phase frequency $\tilde{\omega}_{\text{r}}$. The behavior of the left curve in the neighborhood of $\tilde{\omega}_{\text{r}}$ is similar to that in the neighborhood of $\tilde{\omega}_{\min}$:

[TABLE]

where $D$, $F$, and $G$ are positive constants. The right curve starts from some phase frequency $\tilde{\omega}_{\text{l}}<\tilde{\omega}_{\text{r}}$ and continues up to the maximum possible value $\tilde{\omega}_{\text{tk}}=1$. The curve has no singularity in the neighborhood of $\tilde{\omega}_{\text{l}}$. According to Seq. 5, the kink-Q-ball system goes into the thick-wall regime as $\tilde{\omega}\rightarrow 1$. Indeed, it was found numerically that in the neighborhood of $\tilde{\omega}_{\text{tk}}=1$, $\Delta\tilde{E}\sim Q_{\chi}\sim H\left(\tilde{\omega}_{\text{tk}}-\tilde{\omega}\right)^{1/2}$ in accordance with Eqs. (64) and (65).

Figure 3 shows the dependences $\Delta\tilde{E}\left(\tilde{\omega}\right)$ for $\tilde{e}=\tilde{q}=0.05$ and $\tilde{e}=\tilde{q}=0.1$ in the neighborhood of $\tilde{\omega}_{\text{tk}}=1$. We see that with decreasing gauge coupling constants, the left and right curves are merged into one, so the dependence $\Delta\tilde{E}\left(\tilde{\omega}\right)$ becomes single-valued. In accordance with Seq. 4, both curves go into the thick-wall regime as $\tilde{\omega}\rightarrow 1$.

Knowing 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 the corresponding curves in Figs. 2. The curves in Figs. 4 and 5 possess cusps, whose number 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 the sign when passing through the cusps or discontinuities, so 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 more 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 tunneling.

Figure 6 shows the kink-Q-ball solution with the dimensionless phase frequency $\tilde{\omega}=0.3$. The energy density, the electric charge density, and the electric field strength corresponding to this kink-Q-ball solution are presented in Fig. 7. We see that in accordance with Seq. 3, $a_{0}\left(x\right)$ and $s\left(x\right)$ are even functions of $x$, whereas $f\left(x\right)$ is an odd function of $x$. We also see that $f\left(x\right)$ and $s\left(x\right)$ reach neighborhoods of their boundary values (34) faster than $a_{0}\left(x\right)$. From Fig. 7 it follows that the kink-Q-ball system possesses the symmetric energy and electric charge densities. It also possesses the nonzero electric field strength that is an odd function of the space coordinate. The distribution of the electric charge density is a central symmetric peak with a positive $j_{0}$ surrounded by two areas with a negative $j_{0}$, so 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 side negative areas are due to the contribution of the field $\phi$. Note that the energy density $\mathcal{E}$ reaches a close neighborhood of zero faster than the electric charge density $j_{0}$ and the electric field strength $E_{x}$. The reason is the similar behavior of the electromagnetic potential $a_{0}$ in Fig. 6. 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 $\phi$ reaches a close neighborhood of 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 parts of the two side areas where $\left|\phi\right|\approx\left|\eta\right|$, $\chi\approx 0$, and $\mathcal{E}\approx 0$.

Let us now discuss the issue of stability of the kink-Q-ball system. As has already been pointed out, the energy of the kink-Q-ball system turned out to be more 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 through quantum tunneling. It remains to consider the stability of the kink-Q-ball system with respect to classic fluctuations of the fields $\phi$, $\chi$, and $A_{\mu}$ in a functional neighborhood of the kink-Q-ball solution.

It is known that the gauge model described by the first line of the Lagrangian (1) possesses the kink solution bochkarev_mpl_2 ; grigoriev . In the adopted gauge $A_{x}=0$, this kink solution is given by Eq. (5). The gauged kink has zero electric charge, so it possesses finite energy. However, unlike the kink of a self-interacting real scalar field dashen ; polyakov , 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 bochkarev_mpl_2 ; grigoriev . The existence of the sphaleron is due to the paths in the functional space that connect topologically distinct vacua of the Abelian Higgs model Manton . The sphaleron lies between two topologically distinct neighboring vacua and has exactly 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, so it 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 question about the unstable modes of the kink-Q-ball solution should be investigated separately.

To investigate the classic stability of the kink-Q-ball system, it is necessary to study the spectrum of the operator of second variational derivatives in the functional neighborhood of the kink-Q-ball solution. Wherein, the model’s fields must fluctuate so that the Noether charges $Q_{\phi}$ and $Q_{\chi}$ remain fixed and the perturbed electromagnetic potential $A_{0}+\delta A_{0}$ continues to satisfy Gauss’s law. All these factors make it difficult to study the spectrum even using numerical methods. However, these difficulties can be avoided if we numerically solve field equations (9) – (11) in the temporal gauge $A_{0}=0$ and with a perturbed initial field configuration in the close neighborhood of the kink-Q-ball solution. Indeed, Gauss’s law can be easily implemented in the temporal gauge at $t=0$, whereupon Gauss’s law will be automatically satisfied for $t>0$. The perturbed initial field configuration must have the same $Q_{\phi}$ and $Q_{\chi}$ as the kink-Q-ball system; then the field equations guarantee that the field configuration will also have the same $Q_{\phi}$ and $Q_{\chi}$ at later times. Having perturbed and unperturbed kink-Q-ball solutions of the field equations, we can observe how field fluctuations behave as time increases. If any fluctuation of fields oscillates in a close neighborhood of the kink-Q-ball solution then the solution is classically stable. If there exists at least one fluctuation of fields that increases exponentially with time then the kink-Q-ball solution is classically unstable.

In the present paper, we research the stability of the kink-Q-ball system for $\tilde{e}=\tilde{q}=0.05$, $0.1$, $0.2$, $0.3$, $0.4$, and $0.5$. For each value of the gauge coupling constants, we took the step of changing of $\tilde{\omega}$ equal to $0.1$. To solve field equations (9) – (11), we use the solver of partial differential equations realized in the Maple package maple . We have found that there are at least two unstable modes for the all considered gauge coupling constants and phase frequencies. The first unstable mode corresponds to an initial symmetric perturbation of $\text{Im}\phi$, whereas the second one corresponds to an initial antisymmetric perturbation of $\text{Im}\chi$. It follows that the kink-Q-ball system is not a sphaleron since it has at least two unstable modes, whereas a sphaleron must have exactly one unstable mode.

6 Conclusion

In the present paper, we consider one-dimensional model (1) consisting of two self-interacting complex scalar fields interacting through an Abelian gauge field. It was shown that the model possesses the 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. Despite the neutrality of the kink-Q-ball system, it possesses a nonzero electric field.

The kink-Q-ball system has rather unusual dependences of the energy and the Noether charge on the phase frequency. Indeed, it was found 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, but instead have the branch point. It follows that there is no thin-wall regime for the kink-Q-ball system. We also found that when the magnitude of the phase frequency is in the neighborhood 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. In all cases, however, the kink-Q-ball system goes into 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 the model parameters considered in the present paper, the energy of the kink-Q-ball solution turns out to be more than the energy of the plane-wave solution with the same value of the Noether charge. Due to the topological structure of the model’s vacuum, the kink-Q-ball solution is not topologically stable, so it can transit into the plane-wave configuration through quantum tunneling.

It is known that the Abelian Higgs model possesses the 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 the two topologically distinct adjacent vacua. Unlike this, the kink-Q-ball solution depends on time in any gauge, so it is not a static field configuration. Hence, the kink-Q-ball solution cannot be a sphaleron, and so the question of its classic stability requires separate consideration. We research the classic stability of the kink-Q-ball system by means of numerical solution of the field equations with initial field configurations perturbed in the close neighborhood of the kink-Q-ball solution. It was found that in all considered cases, the kink-Q-ball solution has at least two unstable modes, so it is even more unstable than the gauged kink.

Acknowledgements.

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

Appendix A Quasi-classical limit of the soliton system

In this Appendix, we consider the quasi-classical limit of the kink-Q-ball system. In this limit, quantum field fluctuations are localized in a close neighborhood of the classical solution, so the soliton system behaves like a classical object. Following lee , we form from the coupling constants $g_{\chi}$ and $h_{\chi}$ the two dimensionless combinations: $\epsilon=\left[16h_{\chi}m_{\chi}^{2}/\left(3g_{\chi}^{2}\right)-1\right]^{1/2}$ and $g=2\left[h_{\chi}/\left(3g_{\chi}\right)\right]^{1/2}$. Let us remember that the condition $3g_{\chi}^{2}<16h_{\chi}m_{\chi}^{2}$ is assumed to be fulfilled, so the parameter $\epsilon$ is real. The coupling constants $g_{\chi}$ and $h_{\chi}$ are expressed in terms of $g$ and $\epsilon$ as follows:

[TABLE]

where the rescaled coupling constants $\bar{g}_{\chi}$ and $\bar{h}_{\chi}$ are

[TABLE]

so $\bar{h}_{\chi}=3\bar{g}_{\chi}/4$. Next, we use the dimensionless parameter $g$ to rescale the model’s fields and 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 the rescaling, whereas the mass $m_{\chi}$ of the complex scalar field $\chi$ is not subjected to the rescaling.

In terms of the rescaled fields and coupling constants, self-interaction potentials (3) and (4) are written as:

[TABLE]

[TABLE]

Using Eqs. (92) and (93), it can be shown that the Lagrangian (1) has the following behavior under the rescaling:

[TABLE]

where rescaled Lagrangian $\bar{\mathcal{L}}\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu},m_{\phi},\bar{\eta},m_{\chi},\epsilon,\bar{e},\bar{q}\right)$ is $\mathcal{L}\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu},m_{\phi},\bar{\eta},m_{\chi},1,\epsilon,\bar{e},\bar{q}\right)$ and does not depend on the scale factor $g$. From Eq. (94) it follows that if $\left(\phi,\chi,A^{\mu}\right)$ is a solution corresponding to the parameters $m_{\chi}$, $g_{\chi}$, $h_{\chi}$, $\eta$, $\lambda$, $e$, 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 $\kappa q$, where $\kappa$ is an arbitrary positive constant. Thus, if we know a particular soliton solution, we, in fact, know the one-parameter family of rescaled soliton solutions.

We want to find the area of the model’s parameters that correspond to the quasi-classical limit of the soliton system. To do this, we suppose that the dimensionless combinations of parameters of the rescaled Lagrangian $\bar{\mathcal{L}}$ are of the order of unity:

[TABLE]

In this case, the rescaled soliton solution $\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu}\right)$ is also of the order of unity, whereas the action of the soliton system over the period $T=2\pi/\omega$ is of the order of $g^{-2}$: $S_{T}=g^{-2}\int\nolimits_{0}^{2\pi/\omega}\!\!\!\int dtdx\bar{\mathcal{L}}\sim g^{-2}$. Next, we suppose that the dimensionless scale factor $g$ tends to zero:

[TABLE]

Conditions (95) and (96) can be rewritten in terms of the initial model’s parameters as follows:

[TABLE]

where the tilde means the same order of magnitude.

Under conditions (97), the action $S_{T}\left(\phi,\chi,A^{\mu}\right)=g^{-2}\bar{S}_{T}\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu}\right)$ is proportional to the large factor $g^{-2}$, so the exponential integrand $\exp\!\left[ig^{-2}S_{T}\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu}\right)\right]$ of functional integrals of quantum field theory quickly oscillates as the rescaled fields $\bar{\phi}$, $\bar{\chi}$, and $\bar{A}^{\mu}$ fluctuate in a neighborhood of the classical soliton solution. Due to this, the main contribution to functional integrals comes from the close functional neighborhood of the classical soliton solution, so we may expand the action $S_{T}\left(\bar{\phi},\bar{\chi},\bar{A}^{\mu}\right)$ into the functional series in field variations and retain only quadratic terms (linear terms of the functional series vanish due to field equations). According to Rajaraman , such a behavior corresponds to the quasi-classical limit of the soliton system. Thus, the kink-Q-ball system transits into the quasi-classical limit if conditions (97) are satisfied. From Eqs. (97) it follows that the quasi-classical limit corresponds to small coupling constants $h_{\chi}$, $g_{\chi}$, $\lambda$, $e$, and $q$, although the coupling constants differently tend to zero as $g\rightarrow 0$. At the same time, from the field rescaling it follows that the quasi-classical limit corresponds to large amplitudes of the fields $\phi$, $\chi$, and $A^{\mu}$.

Finally, under the rescaling, the energy density and the Noether charge densities behave as follows:

[TABLE]

where in Eq. (98), we omit the lists of parameters (which are the same as in Eq. (94)) for brevity. From Eqs. (98) – (100) it follows that the energy and the Noether charges of the kink-Q-ball system become large in the quasi-classical limit.