Essentially nonperturbative vacuum polarization effects in a two-dimensional Dirac–Coulomb system with $Z>Z_\mathrm{cr}$: Vacuum charge density

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge $Z>Z_\mathrm{cr,1}$ and radius $R_0$, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom”; we study the behavior of both the vacuum density $\rho_{{\textrm{VP}}}(\vec r\,)$ and the total induced charge and also the vacuum-polarization energy $\mathcal{E}_{_\textrm{VP}}$. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for $\rho_{\textrm{VP}}(\vec r\,)$ and $\mathcal{E}_{\textrm{VP}}$, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for $\rho_{\textrm{VP}}$ and $\mathcal{E}_{\textrm{VP}}$. The most important result is that for $Z \gg Z_\mathrm{cr,1}$ in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge $Z$, which reaches large negative values and whose behavior is estimated from below (in absolute value) as $\sim-|\eta_\mathrm{eff}Z^3| /R_0$. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with $m_j=\pm 1/2,\pm 3/2$. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge $Q_{\textrm{VP}}$ in the overcritical region in detail.