Corrections of Order Za Ep

We will first consider the contributions generated only by the elastic intermediate nuclear states. This means that calculating this correction we will treat the nucleus as a particle which interacts with the photons via its nontrivial Sachs electric and magnetic form factors in (6.8). 

As usual we start consideration of the contributions of order Za)Ep with the infrared divergent integral (9.9) corresponding to the two-photon skeleton diagram in Fig. 9.2. Insertion of factors GE —k ) — 1 or Gm(— )/(l + k) — 1 in one of the external proton legs corresponds to the presence of a nontrivial proton form factor.  

We need to consider diagrams in Fig. 11.1 with insertion of one factor GE M) —k ) — 1 in the proton vertex  

Subtraction is necessary in order to avoid double counting since the diagrams with the subtracted term correspond to the pointlike proton contribution, already taken into account in the expression for the Fermi energy in (8.2). 

Dimensionless integration momentum in (9.9) is measured in electron mass. We return here to dimensionful integration momenta, which results in an extra factor m in the numerators in (11.1) and (11.2). Notice also the minus sign before the momentum in the arguments of form factors, it arises because in the equations below k = fc.  

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Fig. 11.2. Elastic nuclear size correction of order Za)Ep with two form factor insertions. Empty dot corresponds either to Gsi—k ) — 1 or Gm(—A )/(l + /t) — 1...

Fig. 11.4. Diagrams with two form factor insertions for total elastic nnclear size corrections of order Za)Ep...

The recoil part of the proton size correction of order Za)Ep was first considered in [9, 10]. In these works existence of the nontrivial nuclear form factors was ignored and the proton was considered as a heavy particle without nontrivial momentum dependent form factors but with an anomalous magnetic moment. The result of such a calculation is most conveniently written in terms of the elementary proton Fermi energy Ep which does not include the contribution of the proton anomalous magnetic moment (compare (10.2) in the muonium case). Calculation of this correction coincides almost exactly... 

Since the Zemach and recoil corrections are parametrically of the same order of magnitude only their sum was often considered in the literature. The first calculation of the total proton size correction of order Za)Ep with form factors was done in [12], followed by the calculations in [13, 11]. Separately the Zemach and recoil corrections were calculated in [5, 6]. Results of all these works essentially coincide, but some minor differences are due to the differences in the parameters of the dipole nucleon form factors used for numerical calculations. 

The leading recoil correction of order Za m/M)Ep is generated by the graphs with two exchanged photons in Fig. 10.1, similar to the case of the recoil... 

The total proton size dependent contribution of order (Za)Ep, which is often called the Zemach correction, has the form... 

A very important point is the overall uncertainty of our calculations and we consider here particularly the contribution of the QED part to the uncertainty. The second order vacuum polarization effects (relativistic corrections are of order (Za) Ep)). Another source of uncertainty arises from higher order recoil effects (<5 ) which can be estimated as =... 

We used in (9.17) the subtracted electron factor. However, it is easy to see that the one-loop anomalous magnetic moment term in the electron factor generates a correction of order a Za)Ep in the diagrams in Fig, and also should be taken into account. An easy direct calculation of the anomalous magnetic moment contribution leads to the correction... 

Corrections of order a Za)Ep are similar to the corrections of order a (Za) Ep, and can be calculated in the same way. These corrections are generated by three-loop radiative insertions in the skeleton diagram in Fig. 9.2. Their natural scale is determined by the factor a Za)/Tr Ep, that is about 1 kHz. 

All corrections of order a Za)Ep connected with the diagrams containing at least one one-loop or two-loop polarization insertion were obtained in [28]... 

Corrections of Order a Za) Ep 9.4.2.1 Leading Double Logarithm Corrections... 

Corrections of order a Za) Ep are suppressed by an extra factor ck/tt in comparison with the leading contributions of order a Za) Ep and are too small to be of any phenomenological interest now. All corrections of order a Za) Ep are collected in Table 9.4, and their total uncertainty is determined by the error of the nonlogarithmic contribution of order a Za) Ep. 

Calculation of the leading logarithmic corrections of order a Za) Ep to HFS parallels the calculation of the leading logarithmic corrections of order a(Za) to the Lamb shift, described above in Subsect. 3.5.1. Again all leading logarithmic contributions may be calculated with the help of second order perturbation theory (see (3.71)). 

The results in (9.45) and (9.46) include not only nonlogarithmic corrections of order a Za) Ep but also contributions of all terms of the form a Za) EF with n > 4. Comparing contributions in (9.44) and in (9.45) with the contribution of order a Za) Ep in (9.44) we could come to the conclusion that the contribution of the corrections of order a Za) Ep with n > 4 is relatively large, about 16 Hz. On the other hand, even allowing for the presence of higher order terms in (9.45) and (9.46), all three results [49, 51, 54] are compatible within the error bars. Clearly further work on nonlogarithmic corrections of order a Za) Ep is warranted. We use the result [54] for numerical calculation of the magnitude of HFS in muonium. 

Calculation of the nonlogarithmic polarization operator contributions of order a Za) Ep goes exactly like calculation of the respective corrections of order a Za) Ep and is connected with the same diagrams. In fact both logarithmic and nonlogarithmic polarization operator corrections of orders a Za) Ep and a Za) Ep were obtained in one and the same calculation in [55]. Nonlogarithmic corrections of order a Za) Ep have the form... 

Thus all corrections of order a Za) Ep collected in Table 9.5 are now known with an uncertainty of about 0.008 kHz. Scattering of the results in [49, 51, 53, 54] for the nonlogarithmic contribution of order a Za) Ep shows that due to complexity of the numerical calculations a new consideration of this correction would be helpful. 

One should expect that corrections of order a Za) Ep are suppressed relative to the contributions of order a Za) Ep by the factor a/n. This means that at the present level of experimental accuracy one may safely neglect these corrections, as well as corrections of even higher orders in a. 

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