Chiral 3$π$-exchange NN-potentials: Results for dominant next-to-leading order contributions

Abstract

We calculate in (two-loop) chiral perturbation theory the local NN-potentials generated by the three-pion exchange diagrams with one insertion from the second order chiral effective pion-nucleon Lagrangian proportional to the low-energy constants $c_{1,2,3,4}$. The resulting isoscalar central potential vanishes identically. In most cases these $3π$-exchange potentials are larger than the ones generated by the diagrams involving only leading order vertices due to the large values of $c_{3,4}$ (which mainly represent virtual $Δ$-excitation). A similar feature has been observed for the chiral $2π$-exchange. We also give suitable (double-integral) representations for the spin-spin and tensor potentials generated by the leading-order diagrams proportional to $g_A^6$ involving four nucleon propagators. In these cases the Cutkosky rule cannot be used to calculate the spectral-functions in the infinite nucleon mass limit since the corresponding mass-spectra start with a non-vanishing value at the $3π$-threshold. Altogether, one finds that chiral $3π$-exchange leads to small corrections in the region $r\geq 1.4$ fm where $1π$- and chiral $2π$-exchange alone provide a very good strong NN-force as shown in a recent analysis of the low-energy pp-scattering data-base.