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

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.