On the effective lagrangian at nonzero isospin chemical potential
We revisit the most general effective lagrangian within Chiral Perturbation Theory at nonzero isospin chemical potential. In addition to the contributions already considered in the literature, we discuss the effects of new terms allowed by the symmetries, derived within the external source method including spurion fields, as well as of linear-field corrections. We study the influence of those new contributions on the energy density at zero temperature and observables derived from it, such as the pion and quark condensates and the isospin density. Corrections are shown to be compatible with lattice results, which favor a nonzero value for the only undetermined low-energy constant (LEC) to leading order ${\cal O}(p^2)$, rendering in particular a shift of the critical value for Bose-Einstein condensation. To ${\cal O}(p^4)$ we study the physical constraints on the new LEC, which renormalize the energy density and whose numerical effect is estimated within natural values. The new ${\cal O}(p^4)$ corrections give rise to more significant deviations than those previously considered and remain compatible with lattice results.