Lamb shift corrections

Prom the theoretical point of view the accuracy of calculations is limited by the magnitude of the yet uncalculated contributions to the Lamb shift. Corrections to the P levels are known now with a higher accuracy than the corrections to the S levels, and do not limit the results of the comparison between theory and experiment. 

Hi) In circular atoms, the Rydberg electron remains always very far from the nucleus. Hence, all the contact terms, which become significant corrections at the 10-AO level in the optical experiments and which depend upon the not-so-well known proton form factor, are in circular states completely negligible. Lamb-shift corrections are also very small for these states. From the point of view of Q. E. D. corrections, circular atoms are, by far, the best candidate for R metrology. 

The quantity L(0) = In I, where I is the mean excitation potential introduced by Bethe, which controls the stopping of fast particles (see Sect. 2.3.4) L(2) = In K, where K is the average excitation energy, which also enters into the expression for Lamb shift (Bethe, 1947). Various oscillator sum rules have been verified for He and other rare gases to a high degree of accuracy. Their validity is now believed to such an extent that doubtful measurements of photoabsorption and electron-impact cross sections are sometimes altered or corrected so as to satisfy these. 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

Another obvious contribution to the Lamb shift of the same leading order is connected with the polarization insertion in the photon propagator (see Fig. 2.2). This correction also induces a correction to the Coulomb potential... 

It is clear now that the scale setting factor which should be used for qualitative estimates of the high order corrections to the Lamb shift is equal to Am Za) /n. Note the characteristic dependence on the principal quantum number 1/n which originates from the square of the wave function at the origin V (0) 1/n . All corrections induced at small distances (or at high... 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

The diagrams in Fig. 3.11 (e) with the light by light scattering insertions in the external photons do not generate corrections of the previous order in Za. They are both ultraviolet and infrared finite and respective calculations are in principle quite straightforward though technically involved. Only numerical results were obtained for the contributions to the Lamb shift [47, 52]... 

The Darwin potential generates the logarithmic correction to the nonrela-tivistic Schrodinger-Coulomb wave function in (3.65), and the result in (3.97) could be obtained by taking into account this correction to the wave function in calculation of the contribution to the Lamb shift of order a Za.ym. This logarithmic correction is numerically equal 14.43 kHz for the IS -level in hydrogen, and 1.80 kHz for the 2S level. 

This value is two times larger than the result cited above in Table 3.6, and taken at face value shifts the value of the IB Lamb shift in hydrogen by 7 kHz. However, the authors of [123] show that extrapolation of their numerical data to Za) —> 0 leads to the value of Beo twice as large as the anal3dic result for Bgo in [109, 110]. Clearly any phenomenological conclusions in this situation are premature, and the problem of corrections of order a Za) "m requires... 

Concluding our discussion of the purely radiative corrections to the Lamb shift let us mention once more that the main sources of the theoretical uncertainty in these contributions is connected with the nonlogarithmic corrections of order a (Za)" and uncalculated contributions of orders a Za), which may be as large as a few kHz for IS-state and a few tenths of kHz for the 2S-state in hydrogen. All other unknown purely radiative contributions to the Lamb shift are much smaller. 

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