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We relate the forward two-photon exchange (TPE) amplitudes to integrals ofthe inclusive lepton-proton scattering cross sections. These relations yield analternative way for the evaluation of the TPE correction to hyperfine-splitting(HFS) in the hydrogen-like atoms with an equivalent to the standard approach(Iddings, Drell and Sullivan) result implying the Burkhardt-Cottingham sumrule. For evaluation of the individual effects (e.g., elastic contribution) ourapproach yields a distinct result. We compare both methods numerically onexamples of the elastic contribution and the full TPE correction to HFS inelectronic and muonic hydrogen.

Forward two-photon exchange in elastic lepton-proton scattering and hyperfine-splitting correction