${\mathrm{\Delta }}_{33}$-isobar contribution to the soft nucleon-nucleon potentials. I. 2π-exchange potentials

Two-pion-exchange (TPE) nucleon-nucleon potentials are derived for one or two Δ isobars in the intermediate states. Strong dynamical pair suppression is assumed. At the NNπ and the NΔπ vertices Gaussian form factors are incorporated into the relativistic two-body framework by using a dispersion representation for the one-pion-exchange amplitudes. The Fourier transformations are performed using factorization techniques for the energy denominators, taking into account the mass difference between the nucleon and the Δ isobar. Analytic expressions for the TPE potentials are obtained, which contain at most one-dimensional integrals. The TPE potentials are first calculated up to orders (${\mathit{f}}_{\mathit{N}\mathit{N}\mathrm{\pi }}$ ${\mathrm{f}}_{\mathit{N}\mathrm{\Delta }\mathrm{\pi }}$${)}^2$ and ${\mathit{f}}_{\mathit{N}\mathrm{\Delta }\mathrm{\pi }}^4$. These come from the adiabatic contributions of all planar and crossed three-dimensional momentum-space TPE diagrams. We also give the contributions of the OPE iteration, which can be subtracted or not, depending on whether one performs a coupled-channel calculation for, e.g., the NN, NΔ system, or a single NN-channel calculation. Next, we calculate the (${\mathit{m}}_{\mathrm{\pi }}$/M) corrections. These are due to the 1/M terms in the pion-nucleon vertices, and the 1/M terms in the nonadiabatic expansion of the nucleon energies in the intermediate states.