Vacuum charge and current densities in the supercritical two-dimensional Dirac–Coulomb system in a magnetic field with an axial-vector potential

We consider nonperturbative vacuum polarization effects in the supercritical region for a planar Dirac–Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge $Z > Z_{{\rm cr},1}$ and radius $R_0$ in the magnetic field with an axial-vector potential. We study the behavior of the vacuum charge and vacuum current densities, $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec r\,)$ and $\vec{j}_{\scriptscriptstyle\mathrm{VP}}(\vec r\,)$. We focus on the divergence in the theory corresponding to the renormalization and convergence of partial series for $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec r\,)$ and $\vec{j}_{\scriptscriptstyle\mathrm{VP}}(\vec r\,)$. We stress that in contrast to the vacuum charge density, the partial channels with large values of the third projection of the total angular momentum $|m_j|$ must be taken into account in calculating the vacuum current density in the presence of an external magnetic field localized in the range $R_1>R_0$. We show that in the presence of a supercritical Coulomb source, the induced magnetic field can enhance the original magnetic field for certain values of parameters of the external vector potential.