BCS approximation to the effective vector vertex of superfluid fermions

TL;DR

This paper presents a simplified analytical approach to the effective vector vertex in superfluid fermions, enabling analysis of the medium's response and neutrino energy losses with suppressed energy dissipation at low temperatures.

Contribution

It introduces a novel method that avoids explicit pairing interaction forms to derive the effective vertex function in superfluid fermions.

Findings

Imaginary part of response function is proportional to the fourth power of Fermi velocity.

Neutrino energy losses are strongly suppressed at low temperatures.

The approach confirms previous predictions of suppressed neutrino emission in superfluid neutron matter.

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 which does not employ an explicit form of the pairing interaction. This allows 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 the fourth power of small Fermi velocity, 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 S-wave pairing. This approach confirms a strong suppression of the neutrino energy losses as predicted in Ref.[4].