Generalized Nambu-Goldstone pion in dense matter: a schematic NJL model

Chiral symmetry is always broken in cold, dense matter, by chiral condensation at low densities and by diquark condensation at high density. We construct here, within a schematic Nambu-Jona-Lasinio (NJL) model, the corresponding generalized Nambu-Goldstone pion, ${\pi }_G$. As we show, the ${\pi }_G$ mode naturally emerges as a linear combination of the $\langle \bar{q}q\rangle$vacuumpion$\pi$andthe$\langle qq \rangle$ diquark condensate pion $\tilde{\pi }$, with $q$ the quark field, and continuously evolves with increasing density from being $\pi$-like in the vacuum to $\tilde{\pi }$-like in the high density diquark pairing phase. We calculate the density-dependent mass, decay constant, and coupling to quarks of the ${\pi }_G$, and derive a generalized Gell-Mann–Oakes–Renner (GMOR) relation in the presence of a finite bare quark mass $m_q$. We briefly discuss the implications of the results to possible Bose condensation of ${\pi }_G$ in more realistic models.