Nuclear energy density functional from chiral pion-nucleon dynamics revisited

We use a recently improved density-matrix expansion to calculate the nuclear energy density functional in the framework of in-medium chiral perturbation theory. Our calculation treats systematically the effects from $1\pi$-exchange, iterated $1\pi$-exchange, and irreducible $2\pi$-exchange with intermediate $\Delta$-isobar excitations, including Pauli-blocking corrections up to three-loop order. We find that the effective nucleon mass $M^*(\rho)$ entering the energy density functional is identical to the one of Fermi-liquid theory when employing the improved density-matrix expansion. The strength $F_\nabla(\rho)$ of the $(\vec\nabla \rho)^2$ surface-term as provided by the pion-exchange dynamics is in good agreement with that of phenomenological Skyrme forces in the density region $\rho_0/2 <\rho <\rho_0$. The spin-orbit coupling strength $F_{so}(\rho)$ receives contributions from iterated $1\pi$-exchange (of the ``wrong sign'') and from three-nucleon interactions mediated by $2\pi$-exchange with virtual $\Delta$-excitation (of the ``correct sign''). In the region around $\rho_0/2 \simeq 0.08 $fm$^{-3}$ where the spin-orbit interaction in nuclei gains most of its weight these two components tend to cancel, thus leaving all room for the short-range spin-orbit interaction. The strength function $F_J(\rho)$ multiplying the square of the spin-orbit density comes out much larger than in phenomenological Skyrme forces and it has a pronounced density dependence.