# Two-flavor chiral perturbation theory at nonzero isospin: Pion condensation at zero temperature

## TL;DR

This paper computes the equation of state for two-flavor chiral perturbation theory at finite isospin, including quantum corrections, and compares the results with lattice QCD simulations, confirming the second-order phase transition to pion condensation.

## Contribution

It provides the first next-to-leading order quantum corrections to the equation of state in the pion-condensed phase at zero temperature.

## Key findings

- Quantum corrections increase with isospin chemical potential.
- Good agreement with lattice QCD results, improving at higher order.
- Transition from vacuum to Bose-condensed phase is second order.

## Abstract

In this paper, we calculate the equation of state of two-flavor finite isospin chiral perturbation theory at next-to-leading order in the pion-condensed phase at zero temperature. We show that the transition from the vacuum phase to a Bose-condensed phase is of second order. While the tree-level result has been known for some time, surprisingly quantum effects have not yet been incorporated into the equation of state. We find that the corrections to the quantities we compute, namely the isospin density, pressure, and equation of state, increase with increasing isospin chemical potential. We compare our results to recent lattice simulations of 2+1 flavor QCD with physical quark masses. The agreement with the lattice results is generally good and improves somewhat as we go from leading order to next-to-leading order in $\chi$PT.