Incorporating the Coulomb potential into a finite, unitary perturbation theory

TL;DR

This paper develops a perturbation theory that incorporates the Coulomb interaction exactly, allowing direct calculation of phase shifts in nuclear scattering problems, and validates it with model comparisons showing accurate results.

Contribution

It introduces a novel perturbation approach that treats Coulomb interactions exactly, extending previous methods to directly compute phase shifts in nuclear physics.

Findings

Second-order phase shift formulas match exact solutions within third-order error.

The theory provides physically acceptable scattering results.

Cross sections are of correct order of magnitude.

Abstract

We have constructed a perturbation theory to treat interactions that can include the Coulomb interaction, describing a physical problem that is often encountered in nuclear physics. The Coulomb part is not treated perturbatively; the exact solutions are employed. The method is an extension of the results presented in Hoffmann (2021 J. Math. Phys. 62 032105). It is designed to calculate phase shifts directly rather than the full form of the wavefunctions in position space. We present formulas that allow calculation of the phase shifts to second order in the perturbation. The phase shift results to second order, for a short-range potential, were compared with the exact solution, where we found an error of third order in the coupling strength. A different model, meant as a simple approximation of nuclear scattering of a proton on Helium-4 and including a Coulomb potential and a spherical well, was constructed to test the theory. The wavepacket scattering formalism of Hoffmann (2017 J. Phy. B: At. Mol. Opt. Phys 50 215302), known to give everywhere finite results, was employed. We found physically acceptable results and a cross section of the correct order of magnitude.