Contact potentials in presence of a regular finite-range interaction using dimensional regularization and the $N/D$ method

TL;DR

This paper develops a method to solve the Lippman-Schwinger equation with finite-range and contact interactions using dimensional regularization and the N/D method, ensuring unitarity and proper analytic structure in nucleon-nucleon scattering.

Contribution

It introduces a renormalization approach for the LSE with finite-range and derivative contact terms, demonstrating equivalence with the N/D method and detailed analysis of the spin singlet nucleon-nucleon S-wave.

Findings

The LSE solution satisfies elastic unitarity.

The amplitude inherits the left-hand cut from OPE.

The method generalizes to higher derivatives easily.

Abstract

We solve the Lippman-Schwinger equation (LSE) with a kernel that includes a regular finite-range potential and additional contact terms with derivatives. We employ distorted wave theory and dimensional regularization, as proposed in Physics Letters B 568 (2003) 109. We analyze the spin singlet nucleon-nucleon $S-$wave as case of study, with the regular one-pion exchange (OPE) potential in this partial wave and up to ${\cal O}(Q^6)$ (six derivatives) contact interactions. We discuss in detail the renormalization of the LSE, and show that the scattering amplitude solution of the LSE fulfills exact elastic unitarity and inherits the left-hand cut of the long-distance OPE amplitude. Furthermore, we proof that the LSE amplitude coincides with that obtained from the exact $N/D$ calculation, with the appropriate number and typology of subtractions to reproduce the effective range parameters taken as input to renormalize the LSE amplitude. The generalization to higher number of derivatives is straightforward.