Radiative Corrections of Order a Za Ep

Corrections of order a Za)Ep are similar to the corrections of order a (Za) Ep, and can be calculated in the same way. These corrections are generated by three-loop radiative insertions in the skeleton diagram in Fig. 9.2. Their natural scale is determined by the factor a Za)/Tr Ep, that is about 1 kHz. 

The validity of the scattering approximationj or calculation of all radiative and radiative-recoil corrections of order a Za)Ep greatly facilitates the calculations. One may obtain a compact general expression for all such corrections (both logarithmic and nonlogarithmic) induced by the radiative insertions in the electron line in Fig. 10.5 (see, e.g., [30])... 

Nontrivial interplay between radiative corrections and binding effects first arises in calculation of the combined expansion over a and Za. The simplest contribution of this type is of order a Za)Ep and was calculated a long time ago in classical papers [12, 13, 14],... 

See one more comment on this discrepancy below in Subsect. 10.2.10 where the radiative-recoil correction of order a Za) m/M) Ep is discussed. 

Higher order in mass ratio radiative-recoil corrections of order a Za) m/M) Ep, n > 2, are generated by the same set of diagrams in Fig. 10.5, in Fig. 10.6 and in Fig. 10.7 with the radiative insertions in the electron and muon lines, and with the polarization insertions in the photon lines, as the respective corrections of the previous order in the mass ratio. Analytic calculation of the correction of order a Za) m/M) Ep in [52] proceeds as in that case, the only difference is that now one has to preserve all contributions which are of second order in the small mass ratio. It turns out that all such corrections are generated at the scale of the electron mass, and one obtains for the sum of all corrections [52]... 

Radiative-recoil corrections of order a Za)" m/M)Ep were never calculated completely. As we have mentioned in Subsect. 9.4.1.1 the leading logarithm squared contribution of order a Za) EF may easily be calculated if one takes as one of the perturbation potentials the potential corresponding to the electron electric form factor and as the other the potential responsible for the main Fermi contribution to HFS (see Fig. 10.16). Then one obtains the leading logarithm squared contribution in the form [18]... 

The relatively large magnitude of the correction in (10.33) demonstrates that a calculation of all radiative-recoil corrections of order a Za) m/M)Ep is warranted. The error of the total radiative-recoil correction in the last line in Table 10.2 includes, besides the errors of individual contributions in the upper lines of this Table, also an educated guess on the magnitude of yet uncalculated contributions. 

Radiative-recoil corrections of order a Za) m/M)Ep are similar to the radiative corrections to the Zemach contribution, and in principle admit a straightforward calculation in the framework of the skeleton integral approach. Leading logarithmic contributions of this order were considered in [6, 7]. The logarithmic estimate in [7] gives... 

The single-logarithmic correction of order Z a Za) m/M)Ep originating from radiative insertion in the muon line was calculated in [23, 24]... 

---