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

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 $A_4\left(k_f\right)$ arising from $1\pi$ exchange and chiral $2\pi$ exchange in nuclear matter are calculated analytically together with three-body terms involving virtual $\mathrm{\Delta }\left(1232\right)$ isobars. From these interaction terms one obtains at saturation density ${\rho }_0=0.16{\mathrm{fm}}^{-3}$ the value $A_4\left(k_{f0}\right)=1.5\mathrm{MeV}$, more than three times as large as the kinetic energy part. Moreover, iterated $1\pi$ exchange exhibits components for which the fourth derivative with the respect to the isospin asymmetry parameter $\delta$ becomes singular at $\delta =0$. The genuine presence of a nonanalytical term ${\delta }^{4}ln\left|\delta \right|$ 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.

Figure 1

Quartic isospin asymmetry $A_4\left(k_f\right)$ as a function of the nucleon density $\rho =2k_f^3/3{\pi }^2$.

Figure 2

Dependence of the second-order energy per particle ${\overline{E}}_{\mathrm{as}}{(k_p,k_n)}^{\left(2\mathrm{nd}\right)}$ on the isospin asymmetry $\delta$. Three different choices for the scattering lengths, $a_s=a_t,a_t=0$, and $a_s=0$, are considered.