Quartic isospin asymmetry energy of nuclear matter from chiral pion-nucleon dynamics

Abstract

Based on a chiral approach to nuclear matter, the quartic term in the expansion of the equation of state of isospin-asymmetric nuclear matter is calculated. The contributions to the quartic isospin asymmetry energy ${A}_{4}({k}_{f})$ arising from $1\ensuremath{\pi}$ exchange and chiral $2\ensuremath{\pi}$ exchange in nuclear matter are calculated analytically together with three-body terms involving virtual $\mathrm{\ensuremath{\Delta}}(1232)$ isobars. From these interaction terms one obtains at saturation density ${\ensuremath{\rho}}_{0}=0.16\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ the value ${A}_{4}({k}_{f0})=1.5\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$, more than three times as large as the kinetic energy part. Moreover, iterated $1\ensuremath{\pi}$ exchange exhibits components for which the fourth derivative with the respect to the isospin asymmetry parameter $\ensuremath{\delta}$ becomes singular at $\ensuremath{\delta}=0$. The genuine presence of a nonanalytical term ${\ensuremath{\delta}}^{4}ln|\ensuremath{\delta}|$ in the expansion of the energy per particle of isospin-asymmetric nuclear matter is demonstrated by evaluating an $s$-wave contact interaction at second order.