Non-perturbative vacuum polarization effects in two-dimensional supercritical Dirac-Coulomb system. I. Vacuum charge density

TL;DR

This paper investigates non-perturbative vacuum polarization effects in a 2+1 dimensional supercritical Dirac-Coulomb system, focusing on vacuum charge density and energy, revealing a rapid decrease of vacuum energy with increasing charge.

Contribution

It introduces a combined analytical, algebraic, and numerical approach to study vacuum polarization in supercritical regimes, providing detailed analysis of vacuum charge density and energy behavior.

Findings

Vacuum energy decreases rapidly as Z increases in the overcritical region.

Vacuum charge density behavior is characterized and analyzed.

The work emphasizes renormalization and convergence in the calculations.

Abstract

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 $Z > Z_{cr,1}$ are explored. Both the vacuum charge density $\rho_{VP}(\vec{r})$ and vacuum energy $\mathcal{E}_{VP}$ are considered. The main result of the work is that 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. 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 $\rho_{VP}$, 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 $\rho_{VP}$ and behavior of the integral induced charge $Q_{VP}$ in the overcritical region.