Nonrecoil Corrections to HFS

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

Recoil contributions in (10.6), and (10.7) are symmetric with respect to masses of the light and heavy particles. As in the case of the leading recoil correction, they were obtained without expansion in the mass ratio, and hence an exact dependence on the mass ratio is known (not just the first term in the expansion over m/M). Let us mention that while for the nonrecoil nonlogarithmic contributions of order Za), both to HFS and the Lamb shift, only numerical results were obtained, the respective recoil contributions are known anal3dically in both cases (compare discussion of the Lamb shift contributions in Subsect. 4.2.3). 

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