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

Abstract

We consider pion condensation in QCD at finite isospin density $\hat{I}{¼}_I$ and zero temperature using two-flavor chiral perturbation theory ($\ddot{I}‡$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 $\ddot{I}‡$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 $\hat{I}{¼}_I^2/(4\ddot{I}€{)}^2{f}^2$, 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.

11 pages, one figure. V2: few typos fixed and added references. V3: Added two figures of pressure and speed of sound. Comparison with lattice. New abstract. Matches printed version in PRD