Non-perturbative vacuum polarization effects in two-dimensional supercritical Dirac-Coulomb system. II. Vacuum energy

Non-perturbative vacuum polarization effects are explored for a supercritical Dirac-Coulomb system with $Z > Z_{cr,1}$ in 2+1 D, based on the original combination of analytical methods, computer algebra and numerical calculations, proposed recently in Refs. [1]-[3]. Both the vacuum charge density $\rho_{VP}(\vec{r})$ and vacuum energy $\mathcal{E}_{VP}$ are considered. Due to a lot of details of calculation the whole work is divided into two parts I and II. Taking account of results, obtained in the part I [4] for $\rho_{VP}$, in the present part II the evaluation of the vacuum energy $\mathcal{E}_{VP}$ is investigated with emphasis on the renormalization and convergence of the partial expansion for $\mathcal{E}_{VP}$. It is shown that the renormalization via fermionic loop turns out to be the universal tool, which removes the divergence of the theory both in the purely perturbative and essentially non-perturbative regimes of the vacuum polarization. The main result of calculation is that for a wide range of the system parameters in the overcritical region $\mathcal{E}_{VP}$ turns out to be a rapidly decreasing function $\sim - \eta_{eff}\, Z^3/R\,$ with $\eta_{eff}>0$ and $R$ being the size of the external Coulomb source. To the end the similarity in calculations of $\mathcal{E}_{VP}$ in 2+1 and 3+1 D is discussed, and qualitative arguments are presented in favor of the possibility for complete screening of the classical electrostatic energy of the Coulomb source by the vacuum polarization effects for $Z \gg Z_{cr,1}$ in 3+1 D.