Effect of BCS pairing on entrainment in neutron superfluid current in neutron star crust

The relative current density $n^i$ of conduction neutrons in a neutron star crust beyond the neutron drip threshold can be expected to be related to the corresponding particle momentum covector $p_i$ by a linear relation of the form $n^i=K^{ij}{p}_j$ in terms of a physically well defined mobility tensor $K^{ij}$. This result is describable as an ``entrainment’’ whose effect - wherever the crust lattice is isotropic - will simply be to change the ordinary neutron mass m to an effective mass $m_{\star }$ such that in terms of the relevant number density n of unconfined neutrons we shall have $K^{ij}=n\mathrm{/}{m}_{\star }{\gamma }^{ij}$. In a preceding analysis based on an independent particle treatment using Bloch boundary conditions to obtain the distribution of energy $E_k$ and associated group velocity $v_k^i=\mathrm{\partial }{E}_k\mathrm{/}\mathrm{\partial }\mathrm{\hslash }{k}_i$ as a function of wavenumber $k_i$, it was shown that the mobility tensor would be given by $K^{ij}\propto \int d^{3}k{v}_k^i{v}_k^j\delta \{E_k-\mu \}$, where $\mu$ is the Fermi energy. Using the approach due to Bogoliubov, it is shown here that the effect of BCS pairing with an energy gap \Delta_F and corresponding quasiparticle energy function \EE_k=\sqrt{(E_k-\mu)^2+\Delta_F^2}$ will just be to replace the Dirac distributional integrand by a smoother distribution : $K^{ij}\propto \int d^{3}k{v}_k^i{v}_k^j{\mathrm{\Delta }}_F^2\mathrm{/}{\text{EE}}_k^3$. It is also shown how the pairing condensation gives rise to superfluidity in the technical sense of providing (meta) stability against resistive perturbations for a current that is not too strong (its momentum $p_i$ must be small enough to give $2|p_i v_k^i|<\EE_k^2/|E_k-\mu|$ for all modes). It is concluded that the prediction of a very large effective mass enhancement in the middle layers of the crust will not be significantly affected by the pairing mechanism.