Effective Lagrangian at nonzero isospin chemical potential

Abstract

We revisit the most general effective Lagrangian within chiral perturbation theory at nonzero isospin chemical potential ${\ensuremath{\mu}}_{I}$ up to $\mathcal{O}({p}^{4})$. 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 linear-field corrections relevant to $\mathcal{O}({p}^{4})$. We study the influence of those new contributions to the free 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 nonzero values for the low-energy constants (LECs) multiplying the new $\mathcal{O}({p}^{2})$ and $\mathcal{O}({p}^{4})$ field operators in the Lagrangian. In particular, the $\mathcal{O}({p}^{4})$ LECs are renormalized to render the free energy density finite. Constraints on the LECs arise from preserving the physical condition ${n}_{I}({\ensuremath{\mu}}_{I}<{\ensuremath{\mu}}_{c})=0$, while ${\ensuremath{\mu}}_{c}={M}_{\ensuremath{\pi}}$ still holds to leading order and can be maintained to next-to-leading order through an additional constraint requiring the new LECs.