Three-nucleon force in chiral EFT with explicit $\Delta(1232)$ degrees of freedom: Longest-range contributions at fourth order

We analyze the longest-range two-pion exchange contributions to the three-nucleon force at leading-loop order in the framework of heavy-baryon chiral effective field theory with explicit $\Delta(1232)$ degrees of freedom. All relevant low-energy constants which appear in the calculation are determined from pion-nucleon scattering. Comparing our results with the ones obtained in the $\Delta$-less theory at N4LO, we find effects of the $\Delta$ isobar for this particular topology to be rather well represented in terms of resonance saturation of various low-energy constants in the $\Delta$-less approach.