Chiral perturbation theory and Bose-Einstein condensation in QCD

Abstract: We present recent results in three-flavor chiral perturbation theory at finite isospin $\mu_I$ and strangeness $\mu_s$ chemical potentials at zero temperature. The phase diagram to ${\cal O}(p^2)$ in the $\mu_I$--$\mu_S$ 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 $\pi^{\pm}$, $K^{\pm}$, and $K^{0}/\bar{K}^0$, respectively. Including electromagnetic interactions, the Bose-condensed phases become Higgs phases via the Higgs mechanism. The tree-level spectrum for the mesons and gauge bosons is also derived. We calculate the pressure, energy density, isospin density, and speed of sound in the pion-condensed phase to ${\cal O}(p^4)$ for three-flavor $\chi$PT. The results are compared with recent lattice simulations and the agreement is very good for isospin chemical potentials up to approximately 200 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. %to ${\cal O}(p^6)$ for two-flavor $\chi$PT in the chiral limit. We also consider the nonrelativistic limit. It is shown that 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. The breaking of the $U(1)$ symmetry in the Bose-condensed phases gives rise to a Goldstone bosons, whose dispersion is linear for momenta $p\ll\mu_I$. In this regime, we use Son's prescription to construct an effective theory for the Goldstone field which is valid in this regime. It is shown that its damping rate is of order $p^5$. This result is in agreement with Beliav's for a dilute Bose gas.