In-medium chiral condensate beyond linear density approximation

Abstract

In-medium chiral perturbation theory is used to calculate the density dependence of the quark condensate $\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}$. The corrections beyond the linear density approximation are obtained by differentiating the interaction contributions to the energy per particle of isospin-symmetric nuclear matter with respect to the pion mass. Our calculation treats systematically the effects from one-pion exchange (with ${m}_{\ensuremath{\pi}}$-dependent vertex corrections), iterated $1\ensuremath{\pi}$-exchange, and irreducible $2\ensuremath{\pi}$-exchange including intermediate $\ensuremath{\Delta}(1232)$-isobar excitations, with Pauli-blocking corrections up to three-loop order. We find a strong and nonlinear dependence of the ``dropping'' in-medium condensate on the actual value of the pion (or light quark) mass. In the chiral limit, ${m}_{\ensuremath{\pi}}=0$, chiral restoration appears to be reached already at about $1.5$ times normal nuclear matter density. By contrast, for the physical pion mass, ${m}_{\ensuremath{\pi}}=135$ MeV, the in-medium condensate stabilizes at about $60%$ of its vacuum value above that same density. Effects from $2\ensuremath{\pi}$-exchange with virtual $\ensuremath{\Delta}(1232)$-isobar excitations turn out to be crucial in generating such pronounced deviations from the linear density approximation above ${\ensuremath{\rho}}_{0}$. The hindered tendency toward chiral symmetry restoration provides a justification for using pions and nucleons as effective low-energy degrees of freedom at least up to twice nuclear matter density.