Chiral2π-exchangeNNpotentials: Two-loop contributions

We calculate in heavy baryon chiral perturbation theory the local $\mathrm{NN}$ potentials generated by the two-pion exchange diagrams at two-loop order. We give explicit expressions for the mass spectra (or imaginary parts) of the corresponding isoscalar and isovector central, spin-spin and tensor $\mathrm{NN}$ amplitudes. We find from two-loop two-pion exchange, a sizable isoscalar central repulsion that amounts to 62.3 MeV at $r=1.0\mathrm{}\mathrm{fm}.$ There is a similarly strong isovector central attraction that, however, originates mainly from the third-order low-energy constants $d_j$ entering the chiral $\pi N$-scattering amplitude. We also evaluate the one-loop $2\pi$-exchange diagram with two second-order chiral $2\pi \mathrm{NN}$ vertices proportional to the low-energy constants $c_{1,2,3,4}$ as well as the first relativistic $1/M$ correction to the $2\pi$-exchange diagrams with one such vertex. The diagrammatic results presented here are relevant components of the chiral $\mathrm{NN}$ potential at next-to-next-to-next-to-leading order.