Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac-Coulomb System for Z > Zcr: Vacuum Polarization Effects

Abstract

For a planar Dirac-Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge Z > Z_cr,1and radius R₀, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density \({\rho _{{\rm{VP}}}}(\vec r)\), we thoroughly consider the calculation of the vacuum energy ε_VPbased on the renormalization, the convergence of the partial expansion for \({\rho _{{\rm{VP}}}}(\vec r)\), and the behavior of the integral induced charge Q_VPin 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 \({\rho _{{\rm{VP}}}}(\vec r)\) and ε_VP. The most significant result is that for Z ≫ Z_cr,1in 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 ∼ −|η_effZ³|/R₀. We also study the dependence of the effect of the decrease in ε_VPon the cutoff of the Coulomb asymptotics of the external field at different scales R₁ gt; R₀and R₁ ≫ R₀.