Nonperturbative vacuum polarization effects are explored for a supercritical Coulomb source with Z > Zcr in 1+1 D. Both the vacuum charge density ρvac(x) and vacuum energy Evac are considered. It is shown that in the overcritical region the behavior of vacuum energy could be significantly different from perturbative quadratic growth up to decrease reaching large negative values.

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Non-perturbative vacuum polarization effects are explored for a supercritical Dirac-Coulomb system with Z > Zcr,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 ρV P ( r) and vacuum energy EV P 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 ρV P , in the present part II the evaluation of the vacuum energy EV P is investigated with emphasis on the renormalization and convergence of the partial expansion for EV P . 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 EV P turns out to be a rapidly decreasing function ∼ −η ef f Z 3 /R with η ef f > 0 and R being the size of the external Coulomb source. To the end the similarity in calculations of EV P 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 Zcr,1 in 3+1 D.

Based on the original combination of analytical methods, computer algebra tools and numerical calculations, proposed recently in Refs.[1]-[3], the non-perturbative vacuum polarization effects in the 2+1 D supercritical Dirac-Coulomb system with Z > Zcr,1 are explored. Both the vacuum charge density ρV P ( r) and vacuum energy EV P are considered. The main result of the work is that in the overcritical region EV P turns out to be a rapidly decreasing function ∼ −η ef f Z 3 /R with η ef f > 0 and R being the size of the external Coulomb source. Due to a lot of details of calculation the whole work is divided into two parts I and II. In the present part I we consider the evaluation and behavior of the vacuum density ρV P , which further is used in the part II for evaluation of the vacuum energy, with emphasis on the renormalization, convergence of the partial expansion for ρV P and behavior of the integral induced charge QV P in the overcritical region.

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