Recoil correction radiative

Radiative-Recoil corrections are the expansion terms in the expressions for the energy levels which depend simultaneously on the parameters a, m/M and Za. Their calculation requires application of all the heavy artillery of QED, since we have to account both for the purely radiative loops and for the relativistic two-body nature of the bound states. 

Let us emphasize once more that hyperfine structure, radiative, recoil, radiative-recoil, and nonelectromagnetic corrections are all missing in the Dirac energy spectrum. Discussion of their calculations is our main topic below. 

In the standard nomenclature the name radiative-recoil is reserved for the recoil corrections to pure radiative effects, i.e., for corrections of the form... 

The hrst nontrivial radiative-recoil correction is of order a Za). We have already discussed the nonrecoil contribution of this order in Subsect. 3.3.2. Due to the wave function squared factor this correction naturally contained an explicit factor rrir/m). Below we will discuss radiative-recoil corrections of order a Za) with mass ratio dependence beyond this factor (nirfm). ... 

Over the years different methods were applied for calculation of the radiative-recoil correction of order a Za). It was first considered in the diagrammatic approach [1, 3, 2]. Later it was reconsidered on the basis of the Braun formula [4]. The Braun formula depends on the total electron Green function... 

Technically calculations of the diagrams in these two papers were organized in completely different ways, but both groups obtained one and the same analytic expression for the radiative-recoil correction of order a Zaf" [5, 6, 7]... 

The method of direct analysis of the integration regions applied to the bound state problem in [5, 6] allowed these authors also to obtain quadratic in mass ratio radiative-recoil corrections of order a Za) ... 

Calculation of the radiative-recoil correction generated by the one-loop polarization insertions in the exchanged photon lines in Fig. 5.2 follows the same path as calculation of the correction induced by the insertions in the electron line. The respective correction was independently calculated analytically both in the skeleton integral approach [8] and with the help of the Braun formula... 

The radiative-recoil correction to the Lamb shift induced by the polarization insertions in the exchanged photons was also calculated in [9]. The result of that work contradicts the results in [8, 4]. The calculations in [9] are made in the same way as the calculation of the recoil correction of order (Za) (m/M)m in [10], and lead to a wrong result for the same reason. 

We have already considered these corrections together with other radiative-recoil corrections above, in Subsect. 5.1.3. This discussion will be partially reproduced here in order to make the present section self-contained. 

Standard Radiative, Recoil and Radiative-Recoil Corrections... 

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 corrections induced by the polarization loops containing other heavy particles will be considered below in Sect. 10.2 together with other radiative-recoil corrections. 

This calculation of the leading logarithm squared term [30] (see Fig. 9.11) also produces a recoil correction to the nonrecoil logarithm squared contribution. We will discuss this radiative-recoil correction below in the Subsect. 10.2.11 dealing with other radiative-recoil corrections, and we will consider in this section only the nonrecoil part of the logarithm squared term. 

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])... 

The crucial property of the integrand in Eq. (10.16), which facilitates calculation, is that the denominator admits expansion in the small parameter /i prior to momentum integration. This is true due to the inequality j 2 2 2 which is valid according to the definitions of the functions a and b. In this way, we may easily reproduce the nonrecoil skeleton integral in (9.9), and obtain once again the nonrecoil corrections induced by the radiative insertions in the electron line [32, 33, 34]. This approach admits also an analytic calculation of the radiative-recoil corrections of the first order in the mass ratio. 

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

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. 

Analytic calculations of the muon-line radiative-recoil correction are carried out in the same way as in the electron-line case and the purely numerical [25, 35] and analytic [36, 1] results for this contribution are in excellent agreement... 

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