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

TL;DR

This work develops a numerically stable eikonal framework to compute Coulomb corrections to forward elastic scattering of transversely polarized protons on protons and nuclei, essential for RHIC/HJET analyses in the Coulomb-nuclear interference region. By relating the nonflip electromagnetic correction to the spin-flip correction, it enables a massless-photon treatment of soft-photon contributions across both exponential and nonexponential form factors, and it incorporates absorptive corrections from soft magnetic photon exchange and higher-order magnetic-photon exchanges. The results show near equality of nonflip and spin-flip Coulomb corrections in the $pp$ CNI region (to within about $10^{-8}$), while for heavy nuclei the corrections become sizable and significantly influence $A_N^{pA}(t)$, underscoring the need for Glauber-based form-factor inputs for precise analyses. The methodology provides a practical tool for HJET data interpretation and lays groundwork for detailed future$pA$ Coulomb-correction studies.

Abstract

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.

Figures (12)

• Figure 1: Preliminary results for the proton-nucleus elastic analyzing powers $A_\text{N}^{pA}(t)$ measured at HJET Poblaguev:2023stj. The displayed analyzing powers are normalized to the proton-proton value, calculated for $E_\text{beam} = 100\,\text{GeV}$, assuming no hadronic single spin-flip ($r_5 = 0$).
• Figure 2: Three types of elastic proton-proton scattering: (C) electromagnetic, including multiphoton exchange; (N) bare hadronic; and (NC) combined hadronic and electromagnetic interactions.
• Figure 3: Dependence of the eikonal function on the impact parameter. The Coulomb eikonal, $\chi_C(b)$, is normalized according to Eq. \ref{['eq:fbC']}. The relative normalization of the other functions corresponds to $pp$ scattering with $B_X = 11.3\,\text{GeV}^{-2}$ and $q_c^2 = 1.9 \times 10^{-3}\,\text{GeV}^2$.
• Figure 4: Comparison of the numerically computed Coulomb phase (dependent on photon mass $\lambda^2$) with the analytical result Kopeliovich:2000ez obtained in the leading-order approximation.
• Figure 5: Comparison of the nonflip $|\mathcal{F}(q^2)|e^{i\Phi_C(q^2)}$ and spin-flip $|\mathcal{F}(q^2)|e^{i\Phi_C(q^2)}$ Coulomb corrections to the forward elastic $p^{\uparrow}p$ electromagnetic amplitudes.