# Finite Density Two Color Chiral Perturbation Theory Revisited

## TL;DR

This paper revisits two-color, two-flavor chiral perturbation theory at finite isospin and baryon densities, analyzing phase transitions and symmetry properties using both Ginzburg-Landau and full chiral Lagrangian approaches.

## Contribution

It provides a detailed analysis of the phase diagram and the nature of phase transitions in two-color QCD with two flavors at finite densities, highlighting a novel change in the order parameter.

## Key findings

- Identification of a non-standard phase transition when chemical potentials are equal and exceed the pion mass.
- The order parameter becomes an $SU(2)$ doublet across the transition.
- Construction of the effective Lagrangian for Goldstone modes at equal chemical potentials.

## Abstract

We revisit two-color, two-flavor chiral perturbation theory at finite isospin and baryon density. We investigate the phase diagram obtained varying the isospin and the baryon chemical potentials, focusing on the phase transition occurring when the two chemical potentials are equal and exceed the pion mass (which is degenerate with the diquark mass). In this case, there is a change in the order parameter of the theory that does not lend itself to the standard picture of first order transitions. We explore this phase transition both within a Ginzburg-Landau framework valid in a limited parameter space and then by inspecting the full chiral Lagrangian in all the accessible parameter space. Across the phase transition between the two broken phases the order parameter becomes an $SU(2)$ doublet, with the ground state fixing the expectation value of the sum of the magnitude squared of the pion and the diquark fields. Furthermore, we find that the Lagrangian at equal chemical potentials is invariant under global $SU(2)$ transformations and construct the effective Lagrangian of the three Goldstone degrees of freedom by integrating out the radial fluctuations.