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Vacuum polarization of planar Dirac fermions by a superstrong Coulomb potential

We study the vacuum polarization of planar charged Dirac fermions by a strong Coulomb potential. Induced vacuum charge density is calculated and analyzed at the subcritical and supercritical Coulomb potentials for massless and massive fermions. For the massless case the induced vacuum charge density is localized at the origin when the Coulomb center charge is subcritical while it has a power-law tail when the Coulomb center charge is supercritical. The finite mass contribution into the induced charge due to the vacuum polarization is small and insignificantly distorts the Coulomb potential only at distances of order of the Compton length. The induced vacuum charge has a screening sign. As is known the quantum electrodynamics vacuum becomes unstable when the Coulomb center charge is increased from subcritical to supercritical values. In the supercritical Coulomb potential the quantum electrodynamics vacuum acquires the charge due to the so-called real vacuum polarization. We calculate the real vacuum polarization charge density. Screening of the Coulomb center charge are briefly discussed. We expect that our results will be helpful for more deep understanding of the fundamental problem of quantum electrodynamics and can as a matter of principle be tested in graphene with a supercritical Coulomb impurity.