Pion condensation in QCD at finite isospin density, the dilute Bose gas, and speedy Goldstone bosons

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 2I=(4 pi)2f2, where f is the (bare) pion decay constant. The spontaneous breakdown of the global internal symmetry U(1)I3 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.