Polarization vacuum

According to Equation (1.15) the exact first-order vacuum polarization potential induced by the static external Coulomb field of a nucleus reads (in the Feynman 

The zero-potential term (Za)°, as well as all terms of even power (Za)2n, vanish by virtue of Furry s theorem (Furry 1951). 

The Uehling potential represents the dominant vacuum-polarization correction of order a(Za) to the one-photon exchange potential between the nuclear-chaige distribution and the bound electron. Finally, we derive the analytically known renormalized polarization function in terms of an integral representation. For spherically symmetric external charge distributions we obtain the renormalized Uehling potential (Klarsfeld 1977)  

The representation (1.18) implies a subtraction scheme for calculating the finite part of the Wichmann-Kroll potential and the vacuum polarization charge density It was first considered by Wichmann and Kroll (1956). A detailed discussion of the evaluation of this contribution for high-Z nuclei of finite extent is presented in Soff and Mohr (1988) and Soff (1989). A special application of the computed vacuum polarization potential to muonic atoms has been presented in Schmidt et al. (1989). 

The numerical approach developed in Soff and Mohr (1988) utilizes the decomposition of the exact Green function into partial waves and derives the radial vacuum polarization charge density after the integration contour has been Wick-rotated 

In Eq(231) G. f2 f2) is the Green function of the Dirac equation (4) and the symbol Sp concerns the spinor indeces. 

Expression (231) and, consequently, Eq(229) are divergent. This divergency follows from the singularity of the Green function 

The asymptotic behaviour of the Uehling potential is defined by the expressions  

Remembering that the Bohr radius in relativistic units is ro = 1 /ma, we see that at characteristic atomic distances in the neutral hydrogen atom (Z—l) the deviation from the Coulomb potential is exponentially small  

However, this is not the case for a hydrogenlike ion with a high Z value (puiro) puO-/maZ) w 1. 

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For other modem discussions of the problem of vacuum polarization by an external field, see ... 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

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 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. 

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,... 

The summation starts in -me2, and the energy in the current units is just m. To simplify the iterative process, a standard approximation neglects the vacuum polarization effects giving a simpler structure of the four-current [5,7] ... 

Ho, W.C.G., Lai, D. (2003), Transfer of polarized radiation in strongly magnetized plasmas and thermal emission from magnetars effect of vacuum polarization , MNRAS 338, 233. 

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... 

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

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... 

Only the value of the leading coefficient in the low energy expansion of the hadronic vacuum polarization is needed for calculation of the hadronic contribution to the Lamb shift (see the LHS of (3.32)). A model independent value of this coefficient may be obtained for the analysis of the experimental data on the low energy e+e annihilation. Respective contribution to the 15 Lamb shift [39] is —3.40(7) kHz. This value is compatible but more accurate than the result in (3.32). ... 

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. 

Two-loop Vacuum Polarization Baranger, Dyson, Salpeter (1952) [29] -1 910.67 -238.83... 

The simplest correction is induced by the diagrams in Fig. 3.11 (a) with two insertions of the one-loop vacuum polarization in the external photon lines. 

The naive insertion 1/k l2 k) of the irreducible two-loop vacuum polar-... 

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-... 

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

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... 

The muon mass is only slightly lower than the pion mass, and we should expect that insertion of hadronic vacuum polarization in the radiative photon in Fig. 7.13 will give a contribution to the anomalous magnetic moment comparable with the contribution induced by insertion of the muon vacuum polarization. 

Respective corrections are written via the slope of the Dirac form factor and the anomalous magnetic moment exactly as in Subsect. 7.3.4. The only difference is that the contributions to the form factors are produced by the hadronic vacuum polarization. 

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...

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