Pion condensation in dense QCD, the dilute Bose gas, and speedy Goldstone bosons

TL;DR

This paper uses chiral perturbation theory to analyze pion condensation in dense QCD, deriving higher-order corrections to the pressure and comparing results with lattice simulations, revealing good agreement up to certain isospin densities.

Contribution

It provides a detailed next-to-next-to-leading order calculation of the pressure in pion-condensed QCD and explores the properties of the Goldstone mode, extending previous nonrelativistic results.

Findings

Agreement with lattice data up to 180-200 MeV isospin chemical potential.

Derived higher-order corrections to the dilute Bose gas energy density.

Characterized the Goldstone boson properties in the condensed phase.

Abstract

We consider pion condensation in QCD at finite isospin density $\mu_I$ and zero temperature using two-flavor chiral perturbation theory ($\chi$PT). The pressure is calculated to next-to-leading order (NLO) in the low-energy expansion. In the nonrelativistic limit, we recover the classic result by Lee, Huang, and Yang for the energy density of a dilute Bose gas with an $s$-wave scattering length that includes loop corrections from $\chi$PT. In the chiral limit, higher-order calculations are tractable. We calculate the pressure to next-to-next-to-leading order (NNLO) in the low-energy expansion, which is an expansion in powers of $\mu_I^2/(4\pi)^2f^2$, where $f$ is the (bare) pion decay constant. The spontaneous breakdown of the global internal symmetry $U(1)_{I_3}$ gives rise to a massless Goldstone boson or phonon. We discuss the properties of the low-energy effective theory describing this mode. Finally, we compare our results for the pressure and the speed of sound with recent lattice simulations with 2+1 flavors. The agreement is very good for isospin chemical potentials up to 180-200 MeV, depending on the physical quantity.