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

TL;DR

This paper investigates the impact of the $ ext{Δ}(1232)$ resonance on hydrogen spectral lines, using large-$N_c$ relations to compute its effects and analyzing discrepancies in theoretical predictions.

Contribution

It introduces a novel calculation of the $ ext{Δ}(1232)$ resonance contribution to hydrogen spectra using large-$N_c$ relations and examines differences in prediction methods.

Findings

The $ ext{Δ}(1232)$ resonance significantly affects hydrogen Lamb shift and hyperfine structure.

Discrepancies between direct two-photon exchange calculations and photoabsorption-based predictions are explained.

Dispersion relation analysis clarifies the source of prediction mismatches.

Abstract

The electromagnetic excitation of the $\Delta(1232)$-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 $\Delta(1232)$ 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 $\Delta(1232)$-production photoabsorption cross sections. The mismatch is explained by studying the dispersion relations for tree-level Compton scattering off the proton in more details.