Proton Polarization Shifts in Electronic and Muonic Hydrogen

The contribution of virtual excitations to the energy levels of electronic and muonic hydrogen is investigated combining a model-independent approach for the main part with quark model predictions for the remaining corrections. Precise values for the polarization shifts are obtained in the long-wavelength dipole approximation by numerically integrating over measured total photoabsorption cross sections. These unretarded results are considerably reduced by including retardation effects in an approximate way since the average momentum transfer (together with the mean excitation energy) turns out to be larger than usually assumed. Transverse and seagull contributions are estimated in a simple harmonic oscillator quark model and found to be non-negligible. Possible uncertainties and improvements of the final results are discussed. PACS numbers: 12.20.Ds, 12.39.Jh, 14.20.Dh, 36.10.Dr ∗E-mail address: [email protected] 1. There has been tremendous progress in the laser spectroscopy of hydrogen and deuterium atoms [1] which now are even sensitive to small nuclear and proton structure effects. One of these traditionally the least understood is the virtual excitation of the nucleus which acts back on the bound lepton. While the effect in deuterium is comparatively large and has been evaluated theoretically with increasing sophistication and reliability [2], the proton polarization shifts have not received much attention up to now. Khriplovich and Sen’kov [3] have estimated the shift in the electronic 1S-state as −71 ± 11 ± 7 Hz using values for the static proton polarizabilities and assuming a mean excitation energy of ∼ 300 MeV. They attribute the quoted errors to the use of a relativistic approximation for the electron and to the experimental values of the polarizabilities. However, experience gained in many decades of nuclear polarization calculations has told us that the use of an average excitation energy (the “closure approximation”) can be a considerable source of uncertainty unless one calculates it precisely. In addition, one has to make sure that other ingredients to the polarization shift (higher multipoles, transverse excitations) are well under control before a definite answer can be given. It is the purpose of this note to re-evaluate the polarization shift without the questionable use of a mean excitation energy and other simplifying assumptions. Since an experiment is in progress at PSI to measure the Lamb shift in muonic hydrogen [4] we will also evaluate the polarization shift for this case. Actually, with the experimental accuracies achievable in the near future, it turns out that the proton polarization shifts are of much greater importance for muonic than for electronic hydrogen. 2. For light electronic and muonic atoms the energy shift due to virtual excitations can be written as an integral over the forward virtual Compton amplitude. This quantity in turn may be expressed by its imaginary part, i.e. the structure functions W1/2(ν,Q ) which are measurable in the inclusive reaction e+ p −→ e +X. In the absence of detailed experimental information for all relevant values of momentum transferQ and energy transfer ν it is customary to apply the long wavelength (or unretarded dipole) approximation which should be valid for Q̄ 〈r2〉 ≪ 1 where 〈r2〉 is the root-mean-square radius of the proton and Q̄ an average momentum transfer. In this limit it is possible to express the structure functions by the experimentally measured photoabsorption cross section σγ(ν). Bernabéu and Ericson have derived in this way the following expression for the energy shift [5] ∆Enl = − α 2π2 |ψnl(0)| m ∫ ∞ 0 dν σγ(ν) f ( ν 2m )