$^1S_0$ pairing for neutrons in dense neutron matter induced by a soft pion

The possibility of neutron pairing in the $^1S_0$ channel is studied for dense neutron matter in a vicinity of the $\pi^0$ condensation point. The $^1S_0$ pairing gap $\Delta$ is shown to occur in a model with a pairing force induced by the exchange of a soft neutral pionic mode. The soft pion induced potential $V_{\pi}(r)$ is characterized by an attenuating oscillatory behavior in coordinate space, while in momentum space all $S$-wave matrix elements $V_{\pi}(p,p')$ are positive. The solution of the gap equation reveals strong momentum dependence.