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

We propose a simplified approach to incorporate the long-range proton-proton ($pp$) Coulomb force in three-nucleon (3N) scattering calculations, based on the exact formulation presented by Wita\l{}a, Skibi\ifmmode \acute{n}\else \'{n}\fi{}ski, Golak, and Gl"ockle [Eur. Phys. J. A 41, 369 (2009) and Eur. Phys. J. A 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 three-dimensional screened $pp$ Coulomb $t$-matrix obtained by numerical solution of the three-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 provides contributions which are 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 the $pd$ breakup reaction.