On inclusion of the long-range proton-proton Coulomb force in the three-nucleon scattering Faddeev calculations

TL;DR

This paper introduces a simplified method to include the long-range proton-proton Coulomb force in three-nucleon scattering calculations, enabling more straightforward computation of elastic and breakup observables without complex partial wave decomposition.

Contribution

The authors develop a new approach using a 3D screened Coulomb t-matrix that simplifies inclusion of Coulomb effects in three-nucleon scattering calculations, avoiding partial wave decomposition.

Findings

Elastic scattering observables can be obtained directly without renormalization.

Breakup observables require renormalization of on-shell amplitudes.

The method is effective across a wide energy range for proton-deuteron scattering.

Abstract

We propose a simplified approach to incorporate the long-range proton-proton (pp) Coulomb force in the three-nucleon (3N) scattering calculations, based on exact formulation presented in Eur. Phys. Journal A {\bf{41}}, 369 (2009) and {\bf{41}}, 385 (2009). It permits us to get elastic proton-deuteron (pd) scattering and breakup observables relatively simply by performing standard Faddeev calculations as known for the neutron-deuteron (nd) system. The basic ingredient in that approach is a 3-dimensional screened pp Coulomb t-matrix obtained by numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. Based on this t-matrix pure Coulomb transition terms contributing to elastic scattering and breakup are calculated without any need for partial wave decomposition. For elastic scattering such a term removes the Rutherford amplitude for point deuteron proton-deuteron (pd) scattering. For breakup it has never been applied in spite of the fact that its contributions could become important in some regions of the breakup phase space. We demonstrate numerically that the pd elastic observables can be determined directly from the resulting 3N amplitudes without any renormalization, simply by increasing the screening radius in order to reach the existing screening limit. However,for pd breakup the renormalization of the contributing on-shell amplitudes is required. We apply our approach in a wide energy range of the incoming proton for pd elastic scattering as well as for pd breakup reaction.