# Chiral imbalanced hot and dense quark matter: NJL analysis at the physical point and comparison with lattice QCD

## TL;DR

This paper uses the NJL model to explore the phase structure of hot, dense quark matter with isospin and chiral imbalances, revealing dualities and predicting phase transitions consistent with lattice QCD results.

## Contribution

It demonstrates an approximate duality between chiral symmetry breaking and pion condensation at the physical point and compares NJL predictions with lattice QCD, supporting the duality hypothesis.

## Key findings

- Chiral isospin chemical potential induces pion condensation at high temperatures.
- Pseudo-critical temperature increases with chiral chemical potential.
- Lattice QCD results support the NJL model's duality predictions.

## Abstract

Hot and dense quark matter with isospin and chiral imbalances is investigated in the framework of the (3+1)-dimensional Nambu--Jona-Lasinio model (NJL) in the large-$N_c$ limit ($N_c$ is the number of quark colors). Its phase structure is considered in terms of baryon -- $\mu_B$, isospin -- $\mu_I$ and chiral isospin -- $\mu_{I5}$ chemical potentials. It is shown in the paper that (i) in the chiral limit there is a duality between chiral symmetry breaking (CSB) and charged pion condensation (PC) phenomena. (ii) At the physical point, i.e. at nonzero bare quark mass $m_0$, and temperature this duality relation is only approximate, although rather accurate. (iii) We have shown that the chiral isospin chemical potential $\mu_{I5}$ in dense quark matter generates charged pion condensation both at zero and nonzero $m_0$, and at $\mu_{I5}\ne 0$ this phase might be observed up to temperatures as high as 100 MeV. (iv) Pseudo-critical temperature of the chiral crossover transition rises in the NJL model with increasing $\mu_{I5}$. (v) It has been found an agreement between particular sections of the phase diagram in the framework of NJL model and corresponding ones in lattice QCD simulations. Two different plots from different lattice simulations that are completely independent and are not connected at the first sight are in reality dual to each other, it means that lattice QCD simulations support the hypothesis that in real quark matter there exists the (approximate) duality between CSB and charged PC. Moreover, we can reverse the logic and we can predict the increase of pseudo-critical temperature with chiral chemical potential, the much debated effect recently, just by the duality notion, hence bolster confidence in this result (lattice QCD showed this feature for unphysically large pion mass) and put it on the considerably more solid ground.