Chiral three-nucleon force at N${}^4$LO. II. Intermediate-range contributions

We derive the subleading contributions to the two-pion–one-pion exchange and ring three-nucleon force topologies emerging at next-to-next-to-next-to-next-to-leading order in chiral effective field theory. The resulting expressions do not involve any unknown parameters. To study convergence of the chiral expansion we work out the most general operator structure of a local isospin-invariant three-nucleon force. Using the resulting operator basis with 22 independent structures, we compare the strength of the corresponding potentials in configuration space for individual topologies at various orders in the chiral expansion. As expected, the subleading contributions from the two-pion–one-pion exchange and ring diagrams are large, which can be understood in terms of intermediate excitation of the $\Delta$(1232) isobar.