Few-Body Syst (2018) 59:93
Δ(1232)-Resonance in the Hydrogen Spectrum
Received: 29 January 2018 / Accepted: 11 May 2018
© Springer-Verlag GmbH Austria, part of Springer Nature 2018
Abstract The electromagnetic excitation of the Δ(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
), together with other proton-polarizability contributions from forward two-photon exchange. We use the
relations for the nucleon-to-delta transition form factors to compute the effect of the Δ(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) (Faustov et al. in Phys At Nucl
62:2099, 1999) and predictions based on the Δ(1232)-production photoabsorption cross sections (Buchmann
in Can J Phys 87:773–783, 2009). The mismatch is explained by studying the dispersion relations for tree-level
Compton scattering off the proton in more details.
Spectroscopy of muonic hydrogen (μH) has great potential for precise extractions of proton structure infor-
mations, such as the proton charge radius. The μH Lamb shift experiment, performed by the CREMA col-
laboration [1,2], provided the currently most precise determination of the proton charge radius. Their value
is about 10 times more accurate than the CODATA average of experiments with electronic probes , but
5.6 σ smaller—hence, posing the proton radius puzzle. Evidently, the extraction of the charge radius from the
experimental Lamb shift or the Zemach radius from the measured hyperﬁne splitting (HFS), strongly depends
on the quality of the theoretical input (summarized f.i. in Ref. ). The biggest theoretical uncertainty comes
from the forward two-photon exchange (TPE) between muon and proton, or rather, the proton-polarizability
effect given by the non-Born contributions to the TPE, see Fig. 1. These effects are of the order
therefore subleading with respect to the proton charge radius contribution which is of order
At present, the experimental information on the HFS in μH, used to extract the Zemach radius of the
proton [2,4], only comes from the 2S level. In the future, the planned measurements of the ground-state 1S
HFS in μH (CREMA , FAMU  and J-PARC/Riken-RAL ) will improve the experimental HFS accuracy
considerably, and thereby call for at least a factor of 10 improvement in precision of the theory predictions of
proton-polarizability effects .
In this conference proceedings, we discuss the polarizability effect on the hydrogen spectrum generated
by the Δ(1232)-resonance through the diagram in Fig. 2 (Lamb shift in Sect. 4, HFS in Sect. 5).
See Ref.  for a recent review on polarizabilities in Compton scattering and hydrogen.
The results have been previously presented in Ref. .
This article belongs to the Topical Collection “NSTAR 2017—The International Workshop on the Physics of Excited Nucleons”.
F. Hagelstein (
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012 Bern,