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

The possibility of neutron pairing in the ${}^{1}S_0$ channel is studied for dense neutron matter in a vicinity of the ${\pi }^0$ condensation point. The ${}^{1}S_0$ 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 $V_{\pi }\left(r\right)$ is characterized by an attenuating oscillatory behavior in coordinate space, while in momentum space all $S$-wave matrix elements $V_{\pi }(p,p^{\prime })$ are positive. The solution of the gap equation reveals strong momentum dependence.

Figure 1

The soft ${\pi }^0$ potential $V_{\pi }\left(r\right)$ in coordinate space in the spin-singlet state and the radial wave function $r\psi \left(r\right)$ of the lowest $S$ state at parameter $\eta =0.3$.

Figure 2

The $S$-wave matrix elements $V_{\pi }(p,p^{\prime })$ as a function of momentum: diagonal, $p^{\prime }=p$; some off-diagonal, $p^{\prime }=p_{\mathrm{F}}$ and $p^{\prime }=0$.

Figure 3

(a) The gap function normalized by the value ${\Delta }_{\mathrm{F}}=\Delta \left(p_{\mathrm{F}}\right)$ and (b) the anomalous density for several values of the parameter $\eta$.

Figure 4

Quasiparticle momentum distribution for several values of the parameter $\eta$.

Figure 5

The correlation length $\xi$ and its approximation ${\xi }_{\mathrm{F}}$ (both in units of $p_{\mathrm{F}}^{-1}$) as functions of the parameter $\eta$. The horizontal dotted line represents the average interparticle distance $r_s$ (same units).

Figure 6

Diagonal matrix elements of the effective pairing interaction found for the model space with $\delta p/p_{\mathrm{F}}=0.2$ for several values of the parameter $\eta$.