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

For a planar Dirac–Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge $Z>Z_{\mathrm{cr},1}$ and radius $R_0$, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$, we thoroughly consider the calculation of the vacuum energy $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$ based on the renormalization, the convergence of the partial expansion for $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$, and the behavior of the integral induced charge $Q_{\scriptscriptstyle\mathrm{VP}}$ in the overcritical region. In particular, we show that the renormalization using the fermionic loop with two external lines turns out to be a universal technique, which eliminates the divergence of the theory in the purely perturbative and essentially nonperturbative modes for $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$ and $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$. The most significant 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 $Z$ reaching large negative values; its behavior is estimated from below (in absolute value) as $\sim-|\eta_{\mathrm{eff}}Z^3|/R_0$. We also study the dependence of the effect of the decrease in $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$ on the cutoff of the Coulomb asymptotics of the external field at different scales $R_1>R_0$ and $R_1\gg R_0$.