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

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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.

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Is Open Access

Renormalization of the 1S0 One-Pion-Exchange NN Interaction in Presence of Derivative Contact Interactions

(2003)

We use standard distorted wave theory techniques and dimensional regularization to find out solutions of the nucleon-nucleon Lippman--Schwinger equation with a kernel determined by the Weinberg's next-to-leading potential, which consists of one--pion exchange and additional contact terms with derivatives. Though for simplicity, we restrict the discussion to the $^1S_0$ channel and to contact terms containing up to two derivatives, the generalization to higher waves and/or number of derivatives is straightforward. The undetermined low energy constants emerging out of the renormalization procedure are fitted to data.