In order to study partial restoration of the chiral symmetry at finite density, we investigate the density corrections of the chiral condensate up to the next-to-leading order of density expansion using the chiral Ward identity and an in-medium chiral perturbation theory. In our study, we assume that all the in-vacuum quantities for the pion, the nucleon and the πN interaction are determined in vacuum and focus on density expansion of the in-medium physical quantities. We perform diagrammatic analysis of the correlation functions which provide the in-medium chiral condensate. This density expansion scheme shows that the medium effect to the chiral condensate beyond the linear density comes from density corrections to the πN sigma term as a result of the interactions between pion and nucleon in nuclear matter. We also discuss that higher density contributions beyond order of ρ 2 cannot be fixed only by the in-vacuum πN dynamics and we encounter divergence in the calculation of the ρ 2 order corrections of the chiral condensate. To remove the divergence, we need NN two-body contact interaction, which can be fixed in vacuum.

The in-medium pion properties, i.e. the temporal pion decay constant f t , the pion mass m * π and the wave function renormalization, in symmetric nuclear matter are calculated in an in-medium chiral perturbation theory up to the next-to-leading order of the density expansion O(k 4 F ). The chiral Lagrangian for the pion-nucleon interaction is determined in vacuum, and the low energy constants are fixed by the experimental observables. We carefully define the in-medium state of pion and find that the pion wave function renormalization plays an essential role for the in-medium pion properties. We show that the linear density correction is dominant and the next-to-leading corrections are not so large at the saturation density, while their contributions can be significant in higher densities. The main contribution of the next-to-leading order comes from the double scattering term. We also discuss whether the low energy theorems, the Gell-Mann-Oakes-Renner relation and the Glashow-Weinberg relation, are satisfied in nuclear medium beyond the linear density approximation. We also find that the wave function renormalization is enhanced as largely as 50% at the saturation density including the next-to-leading contribution and the wave function renormalization could be measured in the in-medium π 0 → γγ decay.

Abstract. We discuss density corrections of the chiral condensate up to a NLO order using the chiral Ward identity and an in-medium chiral perturbation theory. The in-medium chiral condensate is calculated by a correlation function of the axial current and pseudoscalar density in the nuclear matter as a consequence of the chiral Ward identity. The correlation function is evaluated using the chiral perturbation theory with the hadronic quantities of pion-nucleon dynamics. We assume that the in-vacuum interaction vertices are known, which means that the in-vacuum loop corrections are renormalized to the tree chiral couplings by taking the values of the couplings in chiral Lagrangian as the physical values. We focus on density order in the physical quantities in our perturbative calculation. This study shows that the medium effects to the chiral condensate beyond the linear density come from density corrections to the πN sigma term. It implies that calculating the density dependence of the chiral condensate in nuclear matter is essentially equivalent to describe nuclear matter in chiral effective theory.

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