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Two-photon exchange (TPE) contributions to elastic electron-proton scatteringin the forward regime and in leading log $\sim t\ln|t|$ approximation in themomentum transfer $|t|$ are considered. The imaginary part of TPE amplitude inthese kinematics is related to the total photo absorption cross section. Thereal part of the TPE amplitude is obtained from the imaginary part by means ofa fixed-$t$ dispersion relation. The dispersion integrals for the relevantelastic $ep$-scattering amplitude converge and do not need subtraction. Thisallows us to make clean a prediction for the real part of the TPE amplitude atforward angles with the leading term $\sim t\ln|t|$. Numerical estimates arecomparable to the experimental precision in extracting the charge radius fromexperimental data.

Forward sum rule for the $2\gamma$-exchange correction to the charge radius extraction from elastic electron scattering