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

## TL;DR

This paper constructs a generalized Nambu-Goldstone pion in dense matter using a schematic NJL model, showing its evolution from vacuum to high-density phases and calculating its properties.

## Contribution

It introduces a generalized pion mode in dense matter as a mixture of vacuum and diquark condensate pions within a schematic NJL model, with detailed property calculations.

## Key findings

- The generalized pion evolves from vacuum to high-density phases.
- Density-dependent mass, decay constant, and coupling are calculated.
- A generalized GMOR relation is derived for finite quark mass.

## Abstract

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 $\langle \bar{q}q\rangle$ vacuum pion $\pi$ and the $\langle qq \rangle$ 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.