Vacuum polarization contribution

The energy shift ( order a2 ) due to the vacuum polarization contribution is given by the well-known expression (Lyubovitskji and Rusetsky, 2000). The calculation of the electromagnetic energy-level shift is now complete. 

Detailed analysis of the VP and SE energy eontributions shows that for ions with small Z the QED eontribution is not signifieant, but with growth of Z (Z > 40) the QED contribution becomes very important. Moreover, for heavy and superheavy ions its role is of main importance. Now let us consider the role of the nuelear finite-size effeet. As calculations show, for multicharged ions with Z < 20 its contribution is very small, but for ions with Z > 70 it can equal the vacuum polarization contribution. In Table 3 there are displayed the results of calculations for the nuelear eorrection to the energy of low transitions for Li-like ions. Our calculations also show that a variation of the nuelear radius by a... 

It is not obvious that the hadronic vacuum polarization contribution should be included in the phenomenological analysis of the Lamb shift measurements, since experimentally it is indistinguishable from an additional contribution to the proton charge radius. We will return to this problem below in Sect. 6.1.3. 

This potential and its effect on the energy levels were first considered in [87]. Since each external Coulomb line brings an extra factor Za the energy shift generated by the Wichmann-Kroll potential increases for large Z. For practical reasons the effects of the Uehling and Wichmann-Kroll potentials were investigated mainly numerically and without expansion in Za, since only such results could be compared with the experiments. Now there exist many numerical results for vacuum polarization contributions. In accordance... 

The effects connected with the electron vacuum polarization contributions in muonic atoms were first quantitatively discussed in [4]. In electronic hydrogen polarization loops of other leptons and hadrons considered in Subsect. 3.2.5 played a relatively minor role, because they were additionally suppressed by the typical factors (mg/m). In the case of muonic hydrogen we have to deal with the polarization loops of the light electron, which are not suppressed at all. Moreover, characteristic exchange momenta mZa in muonic atoms are not small in comparison with the electron mass rUg, which determines the momentum scale of the polarization insertions m Za)jme 1.5). We see that even in the simplest case the polarization loops cannot be expanded in the exchange momenta, and the radiative corrections in muonic atoms induced by the electron loops should be calculated exactly in the parameter m Za)/me-... 

Expression for the two-loop vacuum polarization contribution to HFS in Fig. 9.8(b) is obtained from the skeleton integral in (9.9) by the substitution... 

This is in good agreement with the measurements (19), (20), and (21). The uncertainty in (34) comes mainly from the hadronic vacuum-polarization contribution (30). It must be improved by at least a factor of two before we can extract useful physical information from the new high precision measurement of j. Fortunately, this contribution can be calculated from the measured value of R (= Future measurements of R at VEPP-2M, VEPP-4M, DA< NE and BEPS as well as analysis of the hadronic tau decay data [61,62,63], is expected to reduce the uncertainty of this contribution to a satisfactory level. 

We have reproduced the vacuum polarization contribution and found that for arbitrary ns... 

Again, the symmetry of two equally contributing diagrams was taken into account by an additional factor 2 in (18). A reducible part of the vacuum polarization contributions can be shown to vanish [32], The vacuum polarization potential introduced in (18) is given by... 

Table 3. Comparison of the vacuum polarization contribution Eq. (12) against [9]. The uncertainty of the analytic result was discussed in Ref. [4]...

Secondly, one can assume for the form factor the existing experimental value 1 2 = (1.676 + 0.008) fm [3] (as seen by an electron probe), assume muon-electron universality, and then give a limit within which the QED vacuum polarization contribution is tested by these measurements. In doing this, one can see that such a QED correction is tested (at the momentum transfer implied by the experiment) to the level of 0.17% the result of this experiment represents to my knowledge one of the best direct tests, so far performed, of a vacuum polarization correction. 

Fig. 10. Contributions of self energy, vacuum polarization, and finite size to the Lamb shift of the lsi/2-state. For the energy values a dimensionless quantity similar to F Za) in eq. (10) is used. The vacuum polarization contribution is separated into Uehling and Wichmann-Kroll parts. The negative of the Uehling contribution is shown. For high Z, the finite nuclear size becomes a major contribution to the total Lamb shift. Also the Wichmann-Kroll part becomes more important for large Z.

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