Empirical parametrization of the two-photon-exchange effect contributions to the electron-proton elastic scattering cross section
The most recent electron-proton elastic scattering data were re-analyzed using an empirical parametrization of the two-photon-exchange (TPE) effect contributions to ${\ensuremath{\sigma}}_{R}$. The TPE effect contribution $F({Q}^{2},\ensuremath{\varepsilon})$ was double Taylor series expanded as a polynomial of order $n$ keeping only terms linear in $\ensuremath{\varepsilon}$ to account for the experimentally observed and verified linearity of the Rosenbluth plots. We fix the ratio $R={G}_{\mathit{Ep}}/{G}_{\mathit{Mp}}$ to be that obtained from a fit to the recoil-polarization data and parametrize ${\ensuremath{\sigma}}_{R}$ first by a three-parameter formula (fit I) and then by a two-parameter formula (fit III). In contrast to previous analyses, the fit parameter ${G}_{\mathit{Mp}}^{2}$ as obtained from these fits is either smaller or equal to the values obtained from our conventional Rosenbluth fit (fit II) but never larger. The ratio $g({Q}^{2})/{G}_{\mathit{Mp}}^{2}$ which represents the ratio of the TPE and one-photon-exchange (OPE) effect contributions to the intercept of ${\ensuremath{\sigma}}_{R}$ is large and it ranges 3%$\ensuremath{-}$88%. The ratio ${R}_{1\ensuremath{\gamma}\ensuremath{\bigotimes}2\ensuremath{\gamma}}=\ensuremath{\tau}f({Q}^{2})/{G}_{\mathit{Ep}}^{2}$ which represents the ratio of the TPE and OPE effect contributions to the slope of ${\ensuremath{\sigma}}_{R}$ is also large, reaching a value of 12.0$\ensuremath{-}$14.4 at ${Q}^{2}=$ 5.25 (GeV/c)${}^{2}$. The ratio ${R}_{1\ensuremath{\gamma}\ensuremath{\bigotimes}2\ensuremath{\gamma}}$ as obtained from fits I and III is consistent, within error, with those obtained from previous analyses. Our formulas seem to explain the linearity of ${\ensuremath{\sigma}}_{R}$. Moreover, our analysis shows that the extracted ${G}_{\mathit{Ep}}^{2}$ and ${G}_{\mathit{Mp}}^{2}$ using the conventional Rosenbluth separation method can in fact be broken into the usual OPE and TPE contributions. Therefore, ${\ensuremath{\sigma}}_{R}$ can in fact be derived under weaker conditions than those imposed by the Born approximation. Our results show that the TPE amplitudes, $g({Q}^{2})/{G}_{\mathit{Mp}}^{2}$ and $f({Q}^{2})/{G}_{\mathit{Mp}}^{2}$, are sizable and grow with ${Q}^{2}$ value up to ${Q}^{2}~$ 6 (GeV/c)${}^{2}$ in agreement with previous studies. A revision of and comparison to previous analyses are also presented.