In-medium chiral condensate beyond linear density approximation

In-medium chiral perturbation theory is used to calculate the density dependence of the quark condensate $\langle \overline{q}q\rangle$. 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 $m_{\pi }$-dependent vertex corrections), iterated $1\pi$-exchange, and irreducible $2\pi$-exchange including intermediate $\Delta (1232)$-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, $m_{\pi }=0$, chiral restoration appears to be reached already at about $1.5$ times normal nuclear matter density. By contrast, for the physical pion mass, $m_{\pi }=135$ MeV, the in-medium condensate stabilizes at about $60\%$ of its vacuum value above that same density. Effects from $2\pi$-exchange with virtual $\Delta (1232)$-isobar excitations turn out to be crucial in generating such pronounced deviations from the linear density approximation above ${\rho }_0$. 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.

Figure 1

One-pion exchange Fock diagram with pion self-energy. Its combinatoric factor is 1/4.

Figure 2

Hartree and Fock diagrams related to the chiral $\pi \pi \mathit{NN}$ contact vertex proportional to $c_1$. The combinatoric factors of these diagrams are 1/2 and 1, in the order shown.

Figure 3

The nuclear matter saturation curve underlying our calculation of the in-medium quark condensate. Apart from the interaction contributions described in Refs. [9, 10] and those proportional to $c_1$, it includes one single adjusted term linear in density, $\overline{E}(k_f){}^{(\mathrm{adj})}=-7.64{\mathrm{GeV}}^{-2}{k}_f^3$.

Figure 4

The nucleon-nucleon potential in the ${}^1{S}_0$ channel from lattice QCD [16] for three different pion masses, $m_{\pi }=(380,529,731)$ MeV.

Figure 5

The ratio of the in-medium chiral condensate to its vacuum value as a function of the nucleon density ρ for three different values of the pion mass, $m_{\pi }=(0,70,135)$ MeV. The dashed line corresponds to the linear density approximation using the empirical central value ${\sigma }_N=45$ MeV [4].

Figure 6

Interaction contributions to the ratio between in-medium and vacuum chiral condensate. The five classes described in subsections 2a, 2b, 2c, 2d, 2e are consecutively added in the sequence: $\mathrm{linear}\to 1\pi \to \mathrm{iterated}\to 2\pi \to \Delta \to c_1$.

Figure 7

The effective in-medium nucleon sigma-term ${\sigma }_{N,\mathrm{eff}}(\rho )$ versus the nucleon density ρ with its error band $\pm 8$ MeV.