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In this letter we deduce an integral equation that allows to calculate theexact left-hand-cut discontinuity for an uncoupled $S$-wave partial-waveamplitude in potential scattering for a given finite-range potential. Theresults obtained from the $N/D$ method for the partial-wave amplitude arerigorous, since now the discontinuities along the left-hand cut and right-handcut 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 iterativediagrams in the evaluation of $\Delta(A)$. A big advantage of the method isthat short-range physics (corresponding to integrated out degrees of freedomwithin low-energy Effective Field Theory) does not contribute to $\Delta(A)$and it manifests through the extra subtractions that are implemented within themethod. We show the equivalence of the $N/D$ method and the Lippmann-Schwinger(LS) equation for a nonsingular $^1S_0$ $NN$ potential (Yukawa potential). Theequivalence between the $N/D$ method with one extra subtraction and the LSequation renormalized with one counter term or with subtractive renormalizationalso holds for the singular attractive higher-order ChPT potentials. The $N/D$method also allows to evaluate partial-wave amplitudes with a higher number ofextra subtractions, that we fix in terms of shape parameters within theeffective range expansion. The result at NNLO shows that the $^1S_0$ phaseshifts might be accurately described once the electromagnetic and chargesymmetry breaking terms are included. Our present results can be extended tohigher partial waves as well as to coupled channel scattering.

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