Recoil Corrections to HFS

The very presence of the recoil factor m/M emphasizes that the external field approach is inadequate for calculation of recoil corrections and, in principle, one needs the complete machinery of the two-particle equation in this case. However, many results may be understood without a cumbersome formalism. 

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Radiative-Recoil Corrections to HFS 199 Table 10.1. Recoil Corrections... 

Nonlogarithmic radiative-recoil corrections to HFS were first calculated numerically in the Yennie gauge [35, 25] and then analytically in the Feynman gauge [31]... 

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). 

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 hyperfine structure interval in hydrogen is known experimentally on a level of accuracy of one part in 1012, while the theory is of only the 10 ppm level [9]. In contrast to this, the muonium hfs interval [12] is measured and calculated for the ground state with about the same precision and the crucial comparison between theory and experiment is on a level of accuracy of few parts in 107. Recoil effects are more important in muonium (the electron to nucleus mass ratio m/M is about 1/200 in muonium, while it is 1/2000 in hydrogen) and they are clearly seen experimentally. A crucial experimental problem is an accurate determination of the muon mass (magnetic moment) [12], while the theoretical problem is a calculation of fourth order corrections (a(Za)2m/M and (Za)3m/M) [11]. 

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