$\mathrm{\Delta }(1232)$-Resonance in the Hydrogen Spectrum

The electromagnetic excitation of the $\mathrm{\Delta }(1232)$-resonance plays an appreciable role in the Lamb shift and hyperfine structure of muonic and electronic hydrogen. Its effect appears at the subleading order $\mathcal{O}({\alpha }^5)$, together with other proton-polarizability contributions from forward two-photon exchange. We use the large-$N_c$ relations for the nucleon-to-delta transition form factors to compute the effect of the $\mathrm{\Delta }(1232)$ in the hydrogen spectrum. We pay particular attention to a subtile difference between predictions based on a direct calculation of the two-photon exchange (or Compton scattering amplitudes) and predictions based on the $\mathrm{\Delta }(1232)$-production photoabsorption cross sections. The mismatch is explained by studying the dispersion relations for tree-level Compton scattering off the proton in more details.