The N/D method with non-perturbative left-hand-cut discontinuity and the $^1S_0$ $NN$ partial wave

TL;DR

This paper introduces a rigorous integral equation for calculating the exact left-hand-cut discontinuity in potential scattering, improving the N/D method's accuracy and applicability to various nuclear potentials.

Contribution

It derives an exact integral equation for the left-hand-cut discontinuity, demonstrating the N/D method's equivalence to the Lippmann-Schwinger equation and extending its use to higher partial waves.

Findings

Exact calculation of left-hand-cut discontinuity for S-wave amplitudes.

Demonstrated equivalence of N/D and Lippmann-Schwinger methods.

Accurate description of phase shifts at NNLO with additional physical effects.

Abstract

In this letter we deduce 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. The results obtained from the $N/D$ method for the partial-wave amplitude are rigorous, since now the discontinuities along the left-hand cut and right-hand cut are exactly known. This solves 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 $\Delta(A)$. A big advantage of the method is that short-range physics (corresponding to integrated out degrees of freedom within low-energy Effective Field Theory) does not contribute to $\Delta(A)$ and it manifests through the extra subtractions that are implemented within the method. We show the equivalence of the $N/D$ method and the Lippmann-Schwinger (LS) equation for a nonsingular $^1S_0$ $NN$ potential (Yukawa potential). The equivalence between the $N/D$ method with one extra subtraction and the LS equation renormalized with one counter term or with subtractive renormalization also holds for the singular attractive higher-order ChPT potentials. The $N/D$ method also 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. The result at NNLO shows that the $^1S_0$ phase shifts might be accurately described once the electromagnetic and charge symmetry breaking terms are included. Our present results can be extended to higher partial waves as well as to coupled channel scattering.