# Spin polarized phases in strongly interacting matter: interplay between axial-vector and tensor mean fields

## TL;DR

This paper investigates how axial-vector and tensor interactions induce spin polarization in strongly interacting matter at high densities, revealing a sequential emergence of these phases and associated Nambu-Goldstone modes.

## Contribution

It demonstrates the order of spin polarization phases driven by axial-vector and tensor mean fields and analyzes their symmetry-breaking patterns and Nambu-Goldstone modes.

## Key findings

- Tensor mean field induces spin polarization first at high density.
- Axial-vector mean field emerges after tensor mean field, requiring broken chiral symmetry.
- Different Nambu-Goldstone modes appear in each spin polarized phase.

## Abstract

The spontaneous spin polarization of strongly interacting matter due to axial-vector and tensor type interactions is studied at zero temperature and high baryon-number densities. We start with the mean-field Lagrangian for the axial-vector and tensor interaction channels, and find in the chiral limit that the spin polarization due to the tensor mean field ($U$) takes place first as the density increases for sufficiently strong coupling constants, and then that due to the axial-vector mean field ($A$) emerges in the region of finite tensor mean field. This can be understood that making the axial-vector mean field finite requires a broken chiral symmetry somehow, which is achieved by the finite tensor mean field in the present case. It is also found from symmetry argument that there appear the type I (II) Nambu-Goldstone modes with a linear (quadratic) dispersion in the spin polarized phase with $U\neq0$ and $A=0$ ($U\neq0$ and $A\neq0$), although these two phases exhibit the same symmetry breaking pattern.