Phenomenological extraction of two-photon exchange amplitudes from elastic electron-proton scattering cross section data

In this work, I improve on and extend to low and high $Q^2$ values the extractions of the $\varepsilon$ dependence of the real parts of the two-photon exchange (TPE) amplitudes relative to the magnetic form factor, as well as the ratio $P_l/P_l^{Born} (\varepsilon,Q^2)$ by using world data on $\sigma_R(\varepsilon,Q^2)$ with an emphasis on precise new data covering the low-momentum region which is sensitive to the large-scale structure of the nucleon. I provide simple parametrizations of the TPE amplitudes, along with an estimate of the fit uncertainties. The extracted TPE amplitudes are compared with previous phenomenological extractions and TPE calculations. The $P_l/P_l^{Born}$ ratio is extracted by using the new parametrizations of the TPE amplitudes and compared to previous extractions, TPE calculations, and direct measurements at $Q^2$ = 2.50 (GeV/c)$^2$. The extracted TPE amplitudes are on the few-percentage-points level and behave roughly linearly with increasing $Q^2$ where they become nonlinear at high $Q^2$. Contrary to $Y_M$, which is influenced mainly by elastic contributions, I find $Y_E$ to be influenced by inelastic contributions at large $Q^2$ values. While $Y_E$ and $Y_3$ differ in magnitude, they have opposite sign and tend to partially cancel each other. This suggests that the TPE correction to $\sigma_R(\varepsilon,Q^2)$ is driven mainly by $Y_M$ and to a lesser extent by $Y_3$ in agreement with previous phenomenological extractions and hadronic TPE calculations.