BCS approximation to the effective vector vertex of superfluid fermions

Abstract

We examine the effective interaction of nonrelativistic fermions with an external vector field in superfluid systems. In contrast to the complicated vertex equation, usually used in this case, we apply the approach that does not employ an explicit form of the pairing interaction. This allows us to obtain a simple analytic expression for the vertex function only in terms of the order parameter and other macroscopic parameters of the system. We use this effective vertex to analyze the linear response function of the superfluid medium at finite temperatures. At the time-like momentum transfer, the imaginary part of the response function is found to be proportional to ${V}_{F}^{4}$, i.e., the energy losses through vector currents are strongly suppressed. As an application, we calculate the neutrino energy losses through neutral weak currents caused by the pair recombination in the superfluid neutron matter at temperatures lower than the critical one for the ${}^{1}{S}_{0}$ pairing. This approach confirms a strong suppression of the neutrino energy losses as predicted in L. B. Leinson and A. P\'erez [Phys. Lett. B638, 114 (2006)].