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Two-photon exchange correction in elastic unpolarized electron-proton scattering at small momentum transfer

We evaluate the two-photon exchange (TPE) correction to the unpolarized elastic electron-proton scattering at small momentum transfer $ Q^2 $. We account for the inelastic intermediate states approximating the double virtual Compton scattering by the unpolarized forward virtual Compton scattering. The unpolarized proton structure functions are used as input for the numerical evaluation of the inelastic contribution. Our calculation reproduces the leading terms in the $ Q^2 $ expansion of the TPE correction and goes beyond this approximation by keeping the full $ Q^2 $ dependence of the proton structure functions. In the range of small momentum transfer, our result is in good agreement with the empirical TPE fit to existing data.