Numerical Calculation of Coulomb Corrections in Forward Elastic $\boldsymbol{p}^\uparrow\!\!\;\boldsymbol{p}$ and $\boldsymbol{p}^\uparrow\!\boldsymbol{A}$ Scattering

The analysis of RHIC hydrogen gas jet target polarimeter measurements of transverse analyzing powers $A_\text{N}(t)$ in proton-nucleus scattering requires accurate Coulomb corrections to both spin-flip and non-flip amplitudes. These corrections must cover a wide range of nuclear charges $Z$ and form factor slopes, with flexibility to vary form factors during data fitting. To avoid technically challenging calculations involving a small but finite fictitious photon mass, the Coulomb correction to the non-flip electromagnetic amplitude with an exponential form factor was related to the corresponding correction for the spin-flip amplitude. This approach allows soft photon contributions to all amplitudes, including those with non-exponential form factors, to be calculated in the massless photon limit using only analytical expressions and numerically stable integrals with nonsingular integrands and finite integration limits. In addition, an absorptive correction to the spin-flip electromagnetic amplitude, which plays a critical role in spin effects in forward polarized proton-nucleus scattering, was accurately evaluated.