Proton form factor

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 . 

The actual calculation essentially coincides with the calculation of the corrections of order a Za) to the Lamb shift in Subsect. 3.3.3 but is technically simpler due to the triviality of the proton form factor slope contribution in (6.1). 

Fig. 6.5. Electron-line radiative correction to the nuclear size effect. Bold dot corresponds to proton form factor slope...

Due to the analogy between contributions of the diagrams with muon and hadron vacuum polarizations, it is easy to see that insertion of hadron vacuum polarization in one of the exchanged photons in the skeleton diagrams with two-photon exchanges generates a correction of order (x Zotf (see Fig. 7.12). Calculation of this correction is straightforward. One may even take into account the composite nature of the proton and include the proton form factors in photon-proton vertices. Such a calculation was performed in [51, 52] and produced a very small contribution... 

However, the experimental data on the proton form factors used in [54[ contained some misprints. Corrections to this experimental data were taken into account in [21]. 

The last term in the braces is ultraviolet divergent, but it exactly cancels in the sum with the point proton contribution in (11.12). The sum of contributions in (11.12) and (11.13) is the total proton size correction, including the Zemach correction. According to the numerical calculation in [6] this is equal to AE = —33.50 (55) x lO Ep. As was discussed above, the Zemach correction included in this result strongly depends on the precise value of the proton radius, while numerically the much smaller recoil correction is less sensitive to the small momenta behavior of the proton form factor and has smaller uncertainty. For further numerical estimates we will use the estimate AE = 5.22 (1) X 10 Ep of the recoil correction obtained in [6]. 

Calculation of the nonlogarithmic part of the polarization operator insertion requires more detailed information on the proton form factors, and using the dipole parametrization one obtains [7]... 

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. 

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]... 

Hi) In circular atoms, the Rydberg electron remains always very far from the nucleus. Hence, all the contact terms, which become significant corrections at the 10-AO level in the optical experiments and which depend upon the not-so-well known proton form factor, are in circular states completely negligible. Lamb-shift corrections are also very small for these states. From the point of view of Q. E. D. corrections, circular atoms are, by far, the best candidate for R metrology. 

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