The N/D method with non-perturbative left-hand-cut discontinuity and the S01 NN partial wave

Abstract

In this letter we introduce an integral equation that allows to calculate the exact left-hand-cut discontinuity for an uncoupled S-wave partial-wave amplitude in potential scattering for a given finite-range potential. In particular this is applied here to the S01 nucleon–nucleon (NN) partial wave. The calculation of Δ(A) is completely fixed by the potential because short-range physics (corresponding to integrated out degrees of freedom within the low-energy Effective Field Theory) does not contribute to Δ(A). The results obtained from the N/D method for a partial-wave amplitude are rigorous, since now the discontinuities along the left-hand cut and right-hand cut are exactly known. This solves in this case the open question with respect to the N/D method and the effect on the final result of the non-perturbative iterative diagrams in the evaluation of Δ(A). The solution of this problem also implies the equivalence of the N/D method and the Lippmann–Schwinger (LS) equation for the nonsingular one-pion exchange S01 NN potential (Yukawa potential). The equivalence between the N/D method with one extra subtraction and the LS equation renormalized with one counterterm or with subtractive renormalization also holds for the singular attractive S01 NN potentials calculated by including higher orders in Chiral Perturbation Theory (ChPT). However, the N/D method is more flexible and, rather straightforwardly, it allows to evaluate partial-wave amplitudes with a higher number of extra subtractions, that we fix in terms of shape parameters within the effective range expansion. We give results up to three extra subtractions in the N/D method, which provide a rather accurate reproduction of the S01 NN phase shifts when the NNLO ChPT potential is employed. Our new method then provides a general theory to renormalize non-perturbatively singular and regular potentials in scattering that can be extended to higher partial waves as well as to coupled channel scattering.