Polarization of a momentum-cutoff vacuum in a Coulomb field

This paper discusses the vacuum polarization in a Coulomb field -Zα/r when the electron momentum has been cut off at a given value Λ. Radiative corrections are neglected. We carry out initially a first-order perturbative calculation when vertical bar Zα vertical bar is small and study the prerenormalization short-distance (i.e., r≅(ℎ/2π)/Λ) behavior of the induced charge-density distribution as Λ goes to infinity. Then the perturbative results are compared with nonperturbative ones that are obtained by calculating numerically the set of eigenstates of a discretized Dirac equation in the Coulomb field. Our numerical investigations show that (i) when vertical bar Zα vertical bar <1 the nonperturbative results agree well with the first-order perturbative approximations; (ii) the inverse cubic law exhibited by the Uehling term of the induced charge density at distances (ℎ/2π)/Λ>1; (iii) the contribution to the induced charge density coming from extremely high energy states (i.e., the states just below the upper edge of the positive energy band and the states just above the lower edge of the negative energy band) is everywhere negative around a positively charged point source, while the contribution coming from all other states is everywhere positive. These two contributions cancel each other almost completely and result in a small negative term in the vicinity of the source