Effect of Coulomb Forces on the Position of the Pole in the Scattering Amplitude and on Its Residue

TL;DR

This paper derives explicit formulas for the vertex constant in charged particle decay, analyzing how Coulomb forces influence the pole position and residue in scattering amplitudes, with applications to nuclear bound and resonant states.

Contribution

It provides new explicit expressions for the vertex constant considering Coulomb effects, including cases with poles in the effective-range function, and studies pole trajectories as Coulomb forces vary.

Findings

Derived explicit formulas for vertex constants with Coulomb interactions.

Analyzed pole trajectories transitioning from resonant to virtual states.

Applied results to specific nuclear systems like ${}^3 m{He}$, ${^2}$He, ${}^5$He, and ${}^5$Li.

Abstract

Explicit expressions of the vertex constant for the decay of a nucleus into two charged particles for an arbitrary orbital momentum $l$ are derived for the standard expansion of the effective-range function $K_l(k^2)$, as well as when the function $K_0(k^2)$ has a pole. As physical examples, we consider the bound state of the nucleus ${}^3\rm{He}$ and the resonant states of the nuclei ${^2}$He and ${^3}$He in the s-wave, and those of ${}^5\rm{He}$ and ${}^5\rm{Li}$ in the p-wave. For the systems $Np$ and $Nd$ the pole trajectories are constructed in the complex planes of the momentum and of the renormalized vertex constant. They correspond to a transition from the resonance state to the virtual state while the Coulomb forces gradually decrease to zero.