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

Abstract

We consider pion condensation in QCD at finite isospin density μI and zero temperature using two-flavor chiral perturbation theory (χPT). 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 χPT. 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 μ2I/(4π)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.