Phenomenological reexamination of two-photon-exchange corrections to the proton recoil-polarization ratio ${\mu }_p{G}_E/G_M$ utilizing electron-proton elastic scattering experimental data

In this work, we have phenomenologically reexamined the $\varepsilon$ dependence of the proton's recoil-polarization ratio $R={\mu }_p{G}_E/G_M$ for possible two-photon-exchange (TPE) corrections beyond the one-photon-exchange (OPE) or Born approximation. High-precision Rosenbluth measurements of ${\sigma }_R(\varepsilon ,Q^2)$ taken at $Q^2=2.64$, 3.20, and $4.10{\mathrm{GeV}}^2$ were used to extract the TPE coefficients needed to provide an estimate of the size of the TPE corrections to the ratio $R$, and construct the two TPE $Y_{(M,E)}(\varepsilon ,Q^2)$ amplitudes. Our results suggest that the ratio $R$ shows very small (negligible) enhancement with $\varepsilon$, less than 0.40% in magnitude, with values consistent with the OPE prediction for the $Q^2$ range studied. We also compare our results to previous phenomenological extractions and several theoretical calculations.