Background: The inconsistency in the results obtained from the Rosenbluth separation method and the high-Q 2 recoil polarization results on the ratio µpG p E /G p M implies a systematic difference between the two techniques. Several studies suggested that missing higher order radiative corrections to elastic electron-proton scattering cross section σR(ε, Q 2 ) and in particular hard two-photon-exchange (TPE) effect contributions could account for the discrepancy.Purpose: In this work, I improve on and extend to low-and high-Q 2 values the extractions of the ε dependence of the real parts of the TPE amplitudes, relative to the magnetic form factor, as well as the ratio P l /P Born l (ε, Q 2 ) using world data on σR(ε, 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.Method: I combine cross section and polarization measurements of elastic electron-proton scattering to extract the TPE amplitudes. Because the recoil polarization data were confirmed "experimentally" to be essentially independent of ε, I constrain the ratio Pt/P l (ε, Q 2 ) to its ε-independent term (Born value) by setting the TPE contributions to zero. That allows for the amplitude YM (ε, Q 2 ) and σR(ε, Q 2 ) to be expressed in terms of the remaining two amplitudes YE(ε, Q 2 ) and Y3(ε, Q 2 ) which in turn were parametrized as second-order polynomials in ε and Q 2 to reserve as possible the linearity of σR(ε, Q 2 ) as well as to account for possible nonlinearities in the TPE amplitudes. Further, I impose the Regge limit which ensures the vanishing of the TPE contributions to σR(ε, Q 2 ) and the TPE amplitudes in the limit ε → 1.Results: I provide simple parametrizations of the TPE amplitudes, along with an estimate of the fit uncertainties. The extracted TPE amplitudes are compared to previous phenomenological extractions and TPE calculations. The P l /P Born l ratio is extracted 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 .
Conclusions: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 . On the contrary to YM which is influenced mainly by elastic contributions, I find YE to be influenced by inelastic contributions at large Q 2 values. While YE and Y3 differ in magnitude, they have opposite sign and tend to partially cancel each other. This suggests that the TPE correction to σR(ε, Q 2 ) is driven mainly by YM and to a lesser extent by Y3 in agreement with previous phenomenological extractions and hadronic TPE calculations.