Nuclear-size correction to the Lamb shift of one-electron atoms

Abstract

The nuclear size effect on the one-loop self energy and vacuum polarization is evaluated for the 1s, 2s, 3s, 2p 1/2 , and 2p 3/2 states of hydrogen-like ions. The calculation is performed to all orders in the binding nuclear strength parameter Zα. Detailed comparison is made with previous all-order calculations and calculations based on the expansion in the parameter Zα. Extrapolation of the all-order numerical results obtained towards Z = 1 provides results for the radiative nuclear size effect on the hydrogen Lamb shift.PACS numbers: 31.30.jf, 12.20.Ds, 31.15.A- IntroductionThe distribution of charge of the nucleus influences the Dirac energies of atomic systems as well as the quantum electrodynamical corrections to the energy levels. This effect, termed as the nuclear size (NS) correction, is important for comparison of theoretical predictions with experimental data for the whole range of the nuclear charges Z, from hydrogen (Z = 1) up to uranium (Z = 92). The NS corrections to the one-loop self energy and vacuum polarization have been previously investigated both within the approach based on the expansion in the nuclear binding strength parameter Zα [1][2][3][4][5][6] (where α is the fine structure constant) and within the numerical approach that accounts the parameter Zα to all orders [7,8]. In the high-Z region, the numerical allorder approach provides accurate predictions for the NS effect on the radiative corrections. For lower Z, however, the NS effect diminishes and becomes increasingly more difficult to identify in a numerical calculation. On the contrary, the Zα-expansion approach provides accurate predictions for the low-Z ions and only the qualitative estimates in the high-Z region. A quantitative cross-check of the two complementary approaches is not simple and has not been accomplished up to now.The NS corrections became of particular interest recently, after the results of the muonic hydrogen Lambshift experiment were announced [9]. It turned out that the value for the proton charge radius r p deduced from the muonic hydrogen differs by five standard deviations from the spectroscopic value of r p derived from the hydrogen atom. This unexplained disagreement stimulates the scientific community to double-check all contributions originating from the nuclear charge distribution, both for the muonic and normal atoms.The aim of the present investigation is to perform an accurate numerical all-order calculation of the NS correction to the one-loop self energy and vacuum polarization and to make a detailed comparison with the Zα-expansion results available.The relativistic units (m = = c = 1) and the charge units α = e 2 /(4π) are used in this paper. I. NS CORRECTION TO DIRAC ENERGYThe leading-order NS correction to the energy levels of a hydrogen-like atom is defined as the difference of the corresponding eigenvalues of the Dirac equation with the point-Coulomb and the extended-nucleus potentials. The two most commonly used models of the nuclear charge distribution are the uniformly charged sphere ("sp...

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