Chiral approach to nuclear matter: Role of explicit short-range NN-terms
Abstract.We extend a recent chiral approach to nuclear matter by including the most general (momentum-independent) NN-contact interaction. Iterating this two-parameter contact vertex with itself and with one-pion exchange the emerging energy per particle exhausts all terms possible up to and including fourth order in the small momentum expansion. Two (isospin-dependent) cut-offs $\Lambda_{0,1}$ are introduced to regularize the (linear) divergences of some three-loop in-medium diagrams. The equation of state of pure neutron matter, $\bar E_n(k_n)$, can be reproduced very well up to quite high neutron densities of $\rho_n = 0.5 \mathrm{fm}^{-3}$ by adjusting the strength of a repulsive nn-contact interaction. Binding and saturation of isospin-symmetric nuclear matter is a generic feature of our perturbative calculation. Fixing the maximum binding energy per particle to $-\bar E(k_{f0}) = 15.3 $MeV we find that any possible equilibrium density $\rho_0$ lies below $\rho_0^{\rm max} = 0.191 \mathrm{fm}^{-3}$. The additional constraint from the neutron matter equation of state leads however to a somewhat too low saturation density of $\rho_0 = 0.134 \mathrm{fm}^{-3}$. We also investigate the effects of the NN-contact interaction on the complex single-particle potential U(p,kf) + i W(p,kf). We find that the effective nucleon mass at the Fermi surface is bounded from below by $M^*(k_{f0}) \geq 1.4 M$. This property keeps the critical temperature of the liquid-gas phase transition at somewhat too high values $T_{\rm c} \geq 21 $MeV. The downward bending of the asymmetry energy A(kf) above nuclear-matter saturation density is a generic feature of theapproximation to fourth order. We furthermore investigate the effects of the NN-contact interaction on the $(\mathop{\nabla}\limits^{\to} \rho)^2$-term in the nuclear energy density functional ${\cal E}[\rho,\tau]$. Altogether, there is within this complete fourth-order calculation no “magic” set of adjustable short-range parameters with which one could reproduce simultaneously and accurately all semi-empirical properties of nuclear matter. In particular, the conditions for a good neutron matter equation of state and for good single-particle properties are mutually exclusive.