Nucleon-nucleon interactions from dispersion relations: Coupled partial waves

Abstract

We consider nucleon-nucleon interactions from chiral effective field theory applying the $N/D$ method. The case of coupled partial waves is treated, extending Albaladejo and Oller [Phys. Rev. C 84, 054009 (2011)], where the uncoupled case was studied. As a result, three $N/D$ elastic-like equations have to be solved for every set of three independent coupled partial waves. As in the previous reference the input for this method is the discontinuity along the left-hand cut of the nucleon-nucleon partial wave amplitudes. It can be calculated perturbatively in chiral perturbation theory because it involves only irreducible two-nucleon intermediate states. We apply our method to the leading-order result consisting of one-pion exchange as the source for the discontinuity along the left-hand cut. The linear integral equations for the $N/D$ method must be solved in the presence of $\ensuremath{\ell}\ensuremath{-}1$ constraints, with $\ensuremath{\ell}$ the orbital angular momentum, in order to satisfy the proper threshold behavior for $\ensuremath{\ell}\ensuremath{\geqslant}2$. We dedicate special attention to satisfy the requirements of unitarity in coupled channels. We also focus on the specific issue of the deuteron pole position in ${}^{3}\phantom{\rule{-0.16em}{0ex}}{S}_{1}{\ensuremath{-}}^{3}\phantom{\rule{-0.16em}{0ex}}{D}_{1}$ scattering. Our final amplitudes are based on dispersion relations and chiral effective field theory, involving only convergent integrals. They are amenable to a systematic improvement order by order in the chiral expansion.