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QCD at finite isospin density: chiral perturbation theory confronts lattice data

We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential ($μ_I$) and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at $μ_I=m_π$. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for $μ_I<200$ MeV. For larger values of $μ_I$, the agreement between lattice data and the two-flavor predictions is surprisingly good and better than with the three-flavor predictions. Finally, in the limit $m_{s}\gg m_{u}=m_{d}$, we show that the three-flavor observables reduce to the two-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor $χ$PT can largely be explained by the differences in the measured low-energy constants.