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