Vacuum polarization of charged massless fermions in Coulomb and Aharonov--Bohm fields

Vacuum polarization of charged massless fermions is investigated in the superposition of Coulomb and Aharonov--Bohm (AB) potentials in 2+1 dimensions. For this purpose we construct the Green function of the two-dimensional Dirac equation with Coulomb and AB potentials (via the regular and irregular solutions of the radial Dirac equation) and calculate the vacuum polarization charge density in these fields in the so-called subcritical and supercritical regimes. The role of the self-adjoint extension parameter is discussed in terms of the physics of problem. We hope that our results will be helpful in the more deep understanding the fundamental problem of quantum electrodynamics and can be applied to the problems of charged impurity screening in graphene with taking into consideration the electron spin.