Chiral perturbation theory and Bose-Einstein condensation in QCD

Abstract

We present recent results in three-flavor chiral perturbation theory at finite isospin μI and strangeness μS chemical potentials at zero temperature. The tree-level spectrum for the mesons and gauge bosons in the pion-condensed phase is derived. The phase diagram to O(p2) in the μI–μS plane is mapped out with and without electromagnetic effects. The phase diagram consists of a vacuum phase and three Bose-condensed phases with condensates of π±, K±, and K0/K¯0, respectively. Including electromagnetic interactions, the charged Bose-condensed phases become Higgs phases via the Higgs mechanism. We calculate the pressure, energy density, isospin density, and speed of sound in the pion-condensed phase to O(p4). The results are compared with recent lattice simulations and the agreement is very good for isospin chemical potentials up to approximately 180 MeV. Moreover, by integrating out the s-quark, we show that the thermodynamic quantities can be mapped onto their two-flavor counterparts with renormalized parameters. The breaking of the U(1) symmetry in the Bose-condensed phases gives rise to a Goldstone boson, whose dispersion is linear for small momenta. We use Son’s prescription to construct an effective theory for the Goldstone mode in the pion-condensed phase, which is valid for momenta p≪μI. It is shown that its damping rate is of order p5 in the nonrelativistic limit, which is Beliaev’s result for a dilute Bose gas. It is also shown that in the nonrelativistic limit the energy density can be matched onto the classic result by Lee, Huang and Yang (LHY) for a dilute Bose, with an s-wave scattering length that includes radiative corrections. Published by the American Physical Society 2025