Skeleton integral

Skeleton Integral Approach to Calculations of Radiative Corrections... 

The correction of order a Za) induced by the polarization operator insertions in the external photon lines in Fig. 3.10 was obtained in [40, 41, 42] and may again be calculated in the skeleton integral approach. We will use the simplicity of the one-loop polarization operator, and perform this calculation in more detail in order to illustrate the general considerations above. For calculation of the respective contribution one has to insert the polarization operator in the skeleton integrand in (3.33)... 

Of course, the skeleton integral still diverges in the infrared after this substitution since... 

We have restored in (3.39) the characteristic factor 1/(1 — rn /MX which was omitted in (3.33), but which naturally arises in the skeleton integral. However, it is easy to see that an error generated by the omission of this factor is only about 0.02 kHz even for the electron-line contribution to the IS level shift, and, hence, this correction may be safely omitted at the present level of experimental accuracy. 

The contribution to the Lamb shift is given by the insertion of the one-loop polarization operator squared Ii k) in the skeleton integral in (3.33), and taking into account the multiplicity factor 3 one easily obtains [46, 47, 48]... 

Direct substitution of the radiatively corrected electron factor C k) in the skeleton integral in (3.33) would lead to an infrared divergence. This divergence reflects existence in this case of the correction of the previous order in Za generated by the two-loop insertions in the electron line. The magnitude of this previous order correction is determined by the nonvanishing value of the electron factor C k) at zero... 

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... 

As usual we start with the skeleton integral contribution in (3.33) corresponding to the two-photon skeleton diagram in Fig. 3.8. Insertion of the factor GE —k ) — 1 in the proton vertex corresponds to the presence of a nontrivial proton form factor . 

Dimensionless integration momentum in (3.33) was measured in electron mass. We return here to dimensionful integration momenta, which results in an extra factor in the numerators in (6.10), (6.11) and (6.12) in comparison with the factor in the skeleton integral (3.33). Notice also the minus sign before the momentum in the arguments of form factors it arises because in the equations below k = fc. ... 

The main feature of the polarizability contribution to the energy shift is its logarithmic enhancement [26, 30]. The appearance of the large logarithm may easily be understood with the help of the skeleton integral. The heavy particle factor in the two-photon exchange diagrams is now described by the photon-nucleus inelastic forward Compton amplitude [31]... 

Inserting the electron line factor [47, 48], the proton slope contribution (6.1), and the combinatorial factor 2 in the skeleton integral in (3.33), one obtains an integral for the electron-line contribution which does not depend on any parameters, and can be easily calculated numerically with an arbitrary precision [49]. Like a more complicated integral for the corrections of order a Za) ui [48] this integral also admits an analytic evaluation, and the analytic result was obtained in [50, 51]... 

Explicit expression for the electron loop polarization contribution to HFS in Fig. 9.5 is obtained from the skeleton integral in (9.9) by the standard substitution in (3.35). One also has to take into account an additional factor 2 which corresponds to two possible insertions of the polarization operator in... 

Some of the diagrams in Fig. 9.8 also generate corrections of the previous order in Za, which would naively induce infrared divergent contributions after substitution in the skeleton integral in (9.9). 

Technically the lower order contributions to HFS are produced by the constant terms in the low-frequency asymptotic expansion of the electron factor. These lower order contributions are connected with integration over external photon momenta of the characteristic atomic scale mZa and the approximation based on the skeleton integrals in (9.9) is inadequate for their calculation. In the skeleton integral approach these previous order contributions arise as the infrared divergences induced by the low-frequency terms in the electron factors. We subtract leading low-frequency terms in the low-frequency as Tnp-totic expansions of the electron factors, when necessary, and thus get rid of the previous order contributions. 

The next correction of order a Za.)EF is generated by the gauge invariant set of diagrams in Fig. 9.8(c). The respective analytic expression is obtained from the skeleton integral by simultaneous insertion in the integrand of the one-loop polarization function Ii k) and of the electron factor F k). 

The total contribution of all nineteen diagrams to HFS was first calculated purely numerically in the Feynman gauge in the NRQED framework in [22, 23]. The semianalytic skeleton integral calculation in the Yennie gauge was completed a bit later in [26, 27]... 

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. 

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... 

Parametrically the result in (11.4) is of order Za) m/A)Ep, where A is the form factor scale. This means that this correction should be considered together with other recoil corrections, even though it was obtained from a nonrecoil skeleton integral. 

Calculation of the respective radiative-recoil correction of order a Za) m/M)EF in the skeleton integral approach is quite straightforward and may readily be done. However, numerically the correction in (11.24) is smaller than the uncertainty of the Zemach correction, and calculation of corrections to this result does not seem to be an urgent task. 

Radiative-recoil corrections of order a Za) m/M)Ep are similar to the radiative corrections to the Zemach contribution, and in principle admit a straightforward calculation in the framework of the skeleton integral approach. Leading logarithmic contributions of this order were considered in [6, 7]. The logarithmic estimate in [7] gives... 

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]... 

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