Electron vacuum polarization

Here u(+), w(z) are related to U(z), W(z) through the definitions (31) and Evac stands for the contribution due to the electron vacuum polarization which is added by hand . The results of calculations for the matrix elements determining the energy-label shift are the same as in Ref. (Lyubovitskji and Rusetsky, 2000) ... 

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-... 

Then the analogue of the Breit potential induced by the electron vacuum polarization insertion is given by the integral... 

In the case of electronic hydrogen this hadronic insertion in the radiative photon is additionally suppressed in comparison with the contribution of the electron vacuum polarization roughly speaking as (mg/rnTr). ... 

Fig. 7.16. Electron vacuum polarization correction to nuclear polarizability contribution...

Three types of processes are divergent as a result of this coupling to virtual quanta the self-energy of the electron, vacuum polarizations, and vertex functions. 

Fig. 1. Electronic vacuum polarization diagrams one-potential (a) and two-potential (b) contributions. The bold line is for the non-relativistic reduced Green function of a muon in the Coulomb field of nucleus...

Table 1. Contributions of electronic vacuum polarization to the difference hfsO) -8 f hfa(2.s)- Units axe 10 4EF...

It is not enough to consider the free vacuum polarization. The relativistic corrections to the free vacuum polarization in Eqs. (11-14) are of the same order as the so-called Wichmann-Kroll term due to Coulomb effects inside the electronic vacuum-polarization loop. To estimate this term we fitted its numerical values from Ref. [17], which are more accurate for some higher Z 30, by expression... 

Here, / is the boundary of the closed shells n>f indicating the unoccupied bound and the upper continuum electron states m negative continuum (accounting for the electron vacuum polarization). All the vacuum polarization and the self-energy corrections to the sought-for values are omitted. Their numerical smallness compared... 

Here, C is the gauge constant, / is the boundary of the closed shells n > f indicating the vacant band and the upper continuum electron states matomic core and the states of a negative continuum (accounting for the electron vacuum polarization). The minimization of the functional ImSEninv leads to the Dirac-Kohn-Sham-like equations for the electron density that are numerically solved. Finally an optimal set of the IQP functions results. In concrete calculation it is sufficient to use the simplified procedure, which is reduced to the functional minimization using the variation of the correlation potential parameter b in Eq. 3.11 [20, 32]. The Dirac equations for the radial functions F and G (the large and small Dirac components respectively) are ... 

Solving this iterative process gives rise to a set of orbitals to construct the ground state four-current, J (x), including vacuum polarization corrections due to the external field as well as the field mediating the interaction between the electrons. As the charge density in the nonrelativistic case, the four-current has the form of a reference noninteracting A-electron system,... 

When innermost core shells must be treated explicitly, the four-component versions of the GREGP operator can be used, in principle, together with the all-electron relativistic Hamiltonians. The GRECP can describe here some quantum electrodynamics effects (self-energy, vacuum polarization etc.) thus avoiding their direct treatment. One more remark is that the... 

The a4 term requires evaluation of 891 four-loop Feynman diagrams. They fall naturally into five (gauge-invariant) groups according to the way closed electron loops (of vacuum-polarization (v-p) type and light-by-light (l-l) scattering... 

QED contributions to the Lamb shift consist of electron self-energy and vacuum polarization terms. In one-electron atoms the former is both the larger and the more difficult to calculate and has been the focus of much recent theoretical work. Up to Feynman diagrams including two-loops the self-energy contribution to a hydrogenic energy level can be written as [32]... 

The contribution due to the three-loop slope of the Dirac form factor was the last unknown contribution to the hydrogen energy levels at order a3(Za)4. The two other contributions come from the three-loop electron anomalous magnetic moment and the three-loop vacuum polarization correction to the Coulomb propagator. These contributions can be extracted from the literature [10,13]. 

We have displayed the contributions due to the three-loop slope of the Dirac form factor, the three-loop anomalous magnetic moment of the electron and the three-loop photon vacuum polarization separately. Thanks to the cancellation between these contributions, the correction turns out to be quite small numerically. 

Fig. 1. The QED contributions of order a/it) to the bound-electron gj factor depicted as Feynman diagrams. Double lines indicate bound fermions, wavy bnes indicate photons. The interaction with the magnetic field is denoted by a triangle. Diagram (a) is also termed SE, ve (self-energy vertex correction), diagrams (c) and (e) SE, wf (self-energy wave-function correction), diagram (b) VP, pot (vacuum-polarization potential correction), and diagrams (d) and (f) VP, wf (vacuum-polarization wave-function correction)...

Table 5. Lamb shift contribution for the ground state of 208Pb81+ i0n (in eV). The notations are the same as in Table 4. The finite nuclear size correction is calculated for a Fermi distribution with (r2 1,/2 = 5.505 0.001 fm. The SESE (a) (irred) correction is obtained by an interpolation from the known values for Z = 70, 80,92. The inaccuracy of the Uehling approximation for VPVP (f) and S(VP)E corrections is neglected. The zero value presented for the nuclear polarization is due to the cancellation of the usual nuclear polarization [35] with the mixed nuclear polarization (NP)-vacuum polarization correction [36]. The latter effect arises when the nucleus interacts with a virtual electron-positron pair. For lead, due to the collective monopole vibrations, specific for this nucleus, mixed NP-VP effect becomes rather large. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for 208Pb, making this ion especially suitable for the most precise theoretical predictions...

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