Effect of vacuum polarization of charged massive fermions in an Aharonov-Bohm field

The effect of vacuum polarization of charged massive fermions in an Aharonov-Bohm (AB) potential in 2 + 1 dimensions is investigated. The causal Green function of the Dirac equation with the AB potential is represented via the regular and irregular solutions of the two-dimensional radial Dirac equation. It is shown that the vacuum current density contains the contribution from free filled states of the negative energy continuum as well as that from a bound unfilled state, which can emerge in the above background due to the interaction of the fermion's spinmagnetic moment with the AB magnetic field, while the induced charge density contains only the contribution from the bound state. The expressions for the vacuum charge and induced current densities are obtained (recovered for massless fermions) for the graphene in the field of infinitesimally thin solenoid perpendicular to the plane of a sample. We also find the bound state energy as a function of magnetic flux, fermion spin, and the radius of solenoid, and we discuss the role of the so-called self-adjoint extension parameter and determine it in terms of the physics of the problem.