Planar density of vacuum charge induced by a supercritical Coulomb potential

Analytical expressions for the planar density of an induced vacuum charge are obtained in a strong Coulomb potential in coordinate space. Treatment is based on a self-adjoint extension approach for constructing of the Green's function of a charged fermion in an external electromagnetic field. Induced vacuum charge density is calculated and analyzed in subcritical and supercritical Coulomb potentials for massless and massive fermions. We argue that the virtual and so-called real vacuum polarizations contribute in an induced vacuum charge in a supercritical Coulomb potential. The behavior of the polarization vacuum charge density is investigated at long and short distances from the Coulomb center. The induced vacuum charge has a screening sign. Screening of a Coulomb impurity in graphene is briefly discussed. The real vacuum polarization charge density that acquires the quantum electrodynamics vacuum in a supercritical Coulomb potential due to the real vacuum polarization is calculated. It is shown that the vacuum charge densities essentially differ in massive and massless cases. We expect that our results can, as a matter of principle, be tested in graphene with a supercritical Coulomb impurity.