Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms

Abstract

We report high-precision calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter Zα. The results are in excellent agreement with the known terms of the Zα expansion and allow an accurate identification of the nonperturbative higher-order remainder. For hydrogen, the higher-order remainder was found to be much larger than anticipated. This result resolves the long-standing disagreement between the numerical all-order and the analytical Zα-expansion approaches to the recoil effect and completely removes the second-largest theoretical uncertainty in the hydrogen Lamb shift of the 1S and 2S states. PACS numbers: 06.20.Jr, 31.30.jf, 21.10.Ft The hydrogen atom is one of the simplest and the most accurately measured atomic systems in physics. The highest precision has been achieved for the 1S-2S transition, whose frequency was measured [1] with the relative uncertainty of 4.2 × 10 −15 or 10 Hz. In future, it should be possible to increase the experimental accuracy even further, closely approaching the 1.3 Hz natural linewidth of the 2S level. A combination of measured transition frequencies in hydrogen with sophisticated theoretical calculations provides the method for determination of the Rydberg constant [2], which is currently the most accurately known fundamental constant. A coproduct of the determination procedure of the Rydberg constant is the proton charge radius.The proton charge radius has recently received much attention, after the experiments on the muonic hydrogen [3,4] reported a large (7σ) unexplained difference of the proton charge radius values extracted from the muonic hydrogen and the usual (electronic) hydrogen. One of the possible explanation of this puzzle might be a yet undiscovered problem in the theory of the electronic hydrogen. For this reason, investigations of possible inconsistencies in the hydrogen theory are of particular importance today. In the present Letter we report calculations that remove the second-largest theoretical uncertainty in the hydrogen theory and resolve a long-standing discrepancy between the analytical and the numerical approach to the nuclear recoil effect.General expressions for the nuclear recoil effect to first order in the electron-nucleus mass ratio m/M and to all orders in the nuclear binding strength parameter Zα (Z is the nuclear charge number and α is the fine-structure constant) were obtained by one of us [5,6] (see also [7]) and later rederived by other authors [8][9][10]. Numerical calculations to all orders in Zα were reported in Refs. [11][12][13]. The results of these calculations agreed well with the first terms of the Zα expansion [9,14,15]. However, a significant disagreement was later found for the higher-order Zα expansion terms. Specifically, the Zα-expansion result for the (Zα) 7 m/M contribution for the 1s state obtained within the leading logarithmic approximation [16, 17] yielded −(18/π) (Zα) 7 m 2 /M ...

Nuclear recoil corrections to the Lamb shift of hydrogen and light hydrogenlike ions

“…Our first results for n = 1 and n = 2 states were published in Ref. [19]. In the present paper we extend our calculations to a larger range of the nuclear charge numbers and to higher excited states and describe details of the calculation.…”

Three-photon-exchange nuclear structure correction in hydrogenic systems

“…where G rec (1S, Z = 1) = 9.720 (3), G rec (2S, Z = 1) = 14.899 (3), δ fns P (nS, H) = −0.000 184 (1) in the case of hydrogen and δ fns P (nS, D) = −0.000 786 (6) for deuteron [28,30]. In the result, the updated contribution is…”

“…(33) of Ref. [21]) is replaced by the complete all-order (in Zα) result by Yerokhin and Shabaev [28,30]. The corresponding correction to the energy is δE(nS) = m 2 M (Zα) 5 πn 3 Zα 4 ln 2 − 7 2 π + (Zα) 2 G rec + δ fns P ,…”

Tabulation of the bound-state energies of atomic hydrogen

“…(5). A very small contribution to the difference comes from the recent improvement [13] in the determination of E (fs) from the determination of D Tables III, IV, and V, respectively, with higher-ℓ energies given in Table VI. The uncertainties listed are dominated by the uncertainties in E (fs) (both due to the fine structure of the state and due to the effect of E (fs) on the determination of the Rydberg constant).…”