Nucleon-Nucleon scattering from dispersion relations: next-to-next-to-leading order study

We study nucleon-nucleon ($NN$) scattering by applying the $N/D$ method in chiral perturbation theory up to next-to-next-to-leading (NNLO) order in the calculation of the imaginary part of the $NN$ partial-wave amplitudes along the left-hand-cut, which is the dynamical input for this approach. A quite good reproduction of the Nijmegen partial-wave analysis phase shifts and mixing angles is obtained, which implies a steady improvement in the accurateness achieved by increasing the chiral order in the calculation of the dynamical input. A power counting for the subtraction constants is established, which is appropriate for those subtractions attached to both the left- and right-hand cuts. We discuss that it is not necessary to modify the $NN$ chiral potential at NNLO to agree with data, but instead one should perform the iteration of two-nucleon intermediate states to finally achieve analytic and unitarity $NN$ partial-wave amplitudes in a well-defined way. We also confirm at NNLO the long-range correlations between the $NN$ $S$-wave effective ranges and scattering lengths, when employing only once-subtracted dispersion relations, that holds up to around 10% when compared with experimental values.