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

## TL;DR

This paper develops a new method to evaluate the two-photon exchange correction to hyperfine splitting in hydrogen-like atoms by relating TPE amplitudes to inclusive scattering cross sections, offering an alternative to traditional approaches.

## Contribution

It introduces a novel approach linking TPE amplitudes to scattering cross sections, providing a different evaluation method for hyperfine splitting corrections.

## Key findings

- The new method reproduces the standard sum rule results.
- Numerical comparisons show differences in elastic contributions.
- The approach offers a distinct evaluation of TPE effects in hydrogen.

## Abstract

We relate the forward two-photon exchange (TPE) amplitudes to integrals of the inclusive lepton-proton scattering cross sections. These relations yield an alternative 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 sum rule. For evaluation of the individual effects (e.g., elastic contribution) our approach yields a distinct result. We compare both methods numerically on examples of the elastic contribution and the full TPE correction to HFS in electronic and muonic hydrogen.