Corrections of Order a Za Ep

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], 

The contribution to HFS induced by the skeleton diagram with two external photons in Fig. 9.2 is given by the infrared divergent integral 

1 Correction Induced by the Radiative Insertions in the Electron Line 

For calculation of the contribution to HFS of order a Za)EF induced by the one-loop radiative insertions in the electron line in Fig. 9.3 we have to substitute in the integrand in (9.9) the gauge invariant electron factor F k). This electron factor is equal to the one loop correction to the amplitude of the forward Compton scattering in Fig. 9.4. Due to absence of bremsstrahlung in the forward scattering the electron factor is infrared finite. 

The subtracted electron factor generates a finite radiative correction after substitution in the integral in (9.9). The contribution to HFS is equal to 

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We used in (9.17) the subtracted electron factor. However, it is easy to see that the one-loop anomalous magnetic moment term in the electron factor generates a correction of order a Za)Ep in the diagrams in Fig, and also should be taken into account. An easy direct calculation of the anomalous magnetic moment contribution leads to the correction... 

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. 

All corrections of order a Za)Ep connected with the diagrams containing at least one one-loop or two-loop polarization insertion were obtained in [28]... 

Corrections of Order a Za) Ep 9.4.2.1 Leading Double Logarithm Corrections... 

Corrections of order a Za) Ep are suppressed by an extra factor ck/tt in comparison with the leading contributions of order a Za) Ep and are too small to be of any phenomenological interest now. All corrections of order a Za) Ep are collected in Table 9.4, and their total uncertainty is determined by the error of the nonlogarithmic contribution of order a Za) Ep. 

Calculation of the leading logarithmic corrections of order a Za) Ep to HFS parallels the calculation of the leading logarithmic corrections of order a(Za) to the Lamb shift, described above in Subsect. 3.5.1. Again all leading logarithmic contributions may be calculated with the help of second order perturbation theory (see (3.71)). 

The results in (9.45) and (9.46) include not only nonlogarithmic corrections of order a Za) Ep but also contributions of all terms of the form a Za) EF with n > 4. Comparing contributions in (9.44) and in (9.45) with the contribution of order a Za) Ep in (9.44) we could come to the conclusion that the contribution of the corrections of order a Za) Ep with n > 4 is relatively large, about 16 Hz. On the other hand, even allowing for the presence of higher order terms in (9.45) and (9.46), all three results [49, 51, 54] are compatible within the error bars. Clearly further work on nonlogarithmic corrections of order a Za) Ep is warranted. We use the result [54] for numerical calculation of the magnitude of HFS in muonium. 

Calculation of the nonlogarithmic polarization operator contributions of order a Za) Ep goes exactly like calculation of the respective corrections of order a Za) Ep and is connected with the same diagrams. In fact both logarithmic and nonlogarithmic polarization operator corrections of orders a Za) Ep and a Za) Ep were obtained in one and the same calculation in [55]. Nonlogarithmic corrections of order a Za) Ep have the form... 

Thus all corrections of order a Za) Ep collected in Table 9.5 are now known with an uncertainty of about 0.008 kHz. Scattering of the results in [49, 51, 53, 54] for the nonlogarithmic contribution of order a Za) Ep shows that due to complexity of the numerical calculations a new consideration of this correction would be helpful. 

One should expect that corrections of order a Za) Ep are suppressed relative to the contributions of order a Za) Ep by the factor a/n. This means that at the present level of experimental accuracy one may safely neglect these corrections, as well as corrections of even higher orders in a. 

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])... 

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