Heavy Particle Polarization Contributions

Muon and heavy particle polarization contributions to h3rperfine splitting in muonium were considered in Subsubsect. 9.3.1.2 and Subsect. 10.2.7. 

In the external field approximation the skeleton integral with the muon polarization insertion coincides with the respective integral for muonium (compare (9.12) and the discussion after this equation) and one easily obtains [33] 

This result gives a good idea of the magnitude of the muon polarization contribution since the muon is relatively light in comparison to the scale of the proton form factor which was ignored in this calculation. 

The total muon polarization contribution may be calculated without great efforts but due to its small magnitude such a calculation is of minor phenomenological significance and was never done. Only an estimate of the total muon polarization contribution exists in the literature [7] 

Hadronic vacuum polarization in the external field approximation for the pointlike proton also was calculated in [33]. Such a calculation may serve only as an order of magnitude estimate since both the external field approximation and the neglect of the proton form factor are not justified in this case, because the scale of the hadron polarization contribution is determined by the same yo-meson mass which determines the scale of the proton form factor. Again a more accurate calculation is feasible but does not seem to be warranted, and only an estimate of the hadronic polarization contribution appears in the literature [7] 

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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... 

The contribution of the muon polarization operator was already considered above. One might expect that contributions of the diagrams in Fig. 10.8 with the heavy particle polarization loops are of the same order of magnitude as the contribution of the muon loop, so it is natural to consider this contribution here. Respective corrections could easily be calculated by substituting the expressions for the heavy particle polarizations in the unsubtracted skeleton integral in (10.3). The contribution of the heavy lepton t polarization operator was obtained in [37, 38] both numerically and analytically... 

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