One-dimensional soliton system of gauged $Q$-ball and anti-$Q$-ball

The ($1+1$)-dimensional gauge model of two complex self-interacting scalar fields that interact with each other through an Abelian gauge field and a quartic scalar interaction is considered. It is shown that the model has nontopological soliton solutions describing soliton systems consisting of two $Q$-ball components possessing opposite electric charges. The two $Q$-ball components interact with each other through the Abelian gauge field and the quartic scalar interaction. The interplay between the attractive electromagnetic interaction and the repulsive quartic interaction leads to the existence of symmetric and nonsymmetric soliton systems. Properties of these systems are investigated by analytical and numerical methods. The symmetric soliton system exists in the whole allowable interval of the phase frequency, whereas the nonsymmetric soliton system exists only in some interior subinterval. Despite the fact that these soliton systems are electrically neutral, they nevertheless possess nonzero electric fields in their interiors. It is found that the nonsymmetric soliton system is more preferable from the viewpoint of energy than the symmetric one. Both symmetric and nonsymmetric soliton systems are stable against decay into massive scalar bosons.

Figure 1

The dependence of the dimensionless soliton energy $\tilde{E}=m_{\varphi }^{-1}E$ on the dimensionless phase frequency $\tilde{\omega }=m_{\varphi }^{-1}\omega$. The solid curve corresponds to the symmetric soliton system, and the dashed curve corresponds to the nonsymmetric one.

Figure 2

The dependence of the soliton Noether charge $Q_{\chi }$ on the dimensionless phase frequency $\tilde{\omega }=m_{\varphi }^{-1}\omega$. The solid curve corresponds to the symmetric soliton system, and the dashed curve corresponds to the nonsymmetric one.

Figure 3

The dependence of the dimensionless energy $\tilde{E}=m_{\varphi }^{-1}E$ of the symmetric soliton system on the Noether charge $Q_{\chi }$ (solid curve). The dash-dotted line is the straight line $\tilde{E}={\tilde{\omega }}_{\mathrm{tk}}{Q}_{\chi }=(1+m_{\chi }/m_{\varphi })Q_{\chi }$.

Figure 4

The dependence of the energy difference $\mathrm{\Delta }\tilde{E}={\tilde{E}}_{\mathrm{s}}-{\tilde{E}}_{\mathrm{a}}$ between the symmetric and nonsymmetric soliton solutions on the Noether charge $Q_{\chi }$.

Figure 5

The nonsymmetric numerical solution for $f(\tilde{x})$ (solid curve), $s(\tilde{x})$ (dashed curve), and $\tilde{e}{a}_0(\tilde{x})$ (dotted curve). The dimensionless phase frequency $\tilde{\omega }=1.5$.

Figure 6

The dimensionless versions of the energy density $\tilde{\mathcal{E}}=m_{\varphi }^{-2}\mathcal{E}$ (solid curve), the scaled electric charge density ${\tilde{e}}^{-1}\widetilde{j_0}={\tilde{e}}^{-1}{m}_{\varphi }^{-2}{j}_0$ (dashed curve), and the scaled electric field strength ${\tilde{e}}^{-1}{\tilde{E}}_x={\tilde{e}}^{-1}{m}_{\varphi }^{-1}{E}_x$ (dotted curve), corresponding to the nonsymmetric solution in Fig. 5.

Figure 7

The symmetric numerical solution for $f(\tilde{x})$ (solid curve), $s(\tilde{x})$ (dashed curve), and $\tilde{e}{a}_0(\tilde{x})$ (dotted curve). The dimensionless phase frequency $\tilde{\omega }=1.5$.

Figure 8

The dimensionless versions of the energy density $\tilde{\mathcal{E}}=m_{\varphi }^{-2}\mathcal{E}$ (solid curve), the scaled electric charge density ${\tilde{e}}^{-1}\widetilde{j_0}={\tilde{e}}^{-1}{m}_{\varphi }^{-2}{j}_0$ (dashed curve), and the scaled electric field strength ${\tilde{e}}^{-1}{\tilde{E}}_x={\tilde{e}}^{-1}{m}_{\varphi }^{-1}{E}_x$ (dotted curve), corresponding to the symmetric solution in Fig. 7.