Correlated two-pion exchange and large-Ncbehavior of nuclear forces

Abstract

The effect of correlated scalar-isoscalar two-pion exchange (CrTPE) modes is considered in connection with central and spin-orbit parts of the nucleon-nucleon $(NN)$ force. The two-pion correlation function is coupled directly to the scalar form factor of the nucleon which we calculate in the large-${N}_{c}$ limit where the nucleon can be described as a soliton of an effective chiral theory. The results for the central $NN$ force show a strong repulsive core at short internucleon distances supplemented by a moderate attraction beyond $1\phantom{\rule{0.3em}{0ex}}\mathrm{fm}$. The long-range tail of the central $NN$ potential is driven by the pion-nucleon sigma term and consistent with the effective $\ensuremath{\sigma}$ meson exchange. The spin-orbit part of the $NN$ potential is repulsive. The large-${N}_{c}$ scaling behavior of the scalar-isoscalar $NN$ interaction is addressed. We show that the spin-orbit part is $\mathcal{O}(1∕{N}_{c}^{2})$ in strength relative to the central force resulting in the ratio $\ensuremath{\simeq}1∕9$ suggested by the $1∕{N}_{c}$ expansion for ${N}_{c}=3$. The latter is in agreement with our numerical analysis and with the Kaplan-Manohar large-${N}_{c}$ power counting. Unitarization of the $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude plays here an important role and improves the tree level results. Analytical representations of the CrTPE $NN$ potential in terms of elementary functions are derived and their chiral content is discussed.