Wichmann-Kroll correction

There are two contributions of order a Zay m to the energy shift induced by the Uehling and the Wichmann-Kroll potentials (see Fig. 3.10 and Fig. 3.16, respectively). Respective calculations go along the same lines as in the case of the Coulomb-line corrections of order a Zay considered above. 

Contribution of the Wichmann-Kroll diagram in Fig. 3.16 with three external fields attached to the electron loop [26] may be considered in the same way as the polarization insertions in the Coulomb potential, and as we will see below it generates a correction to the Lamb shift of order a Za) m. 

For the potential correction, no Uehling-like contribution exists for a homogeneous external magnetic field [31,32], and the remaining Wichmann-Kroll part can be written as [40]... 

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... 

Given the Wichmann-Kroll density we can calculate first the contribution to the vacuum polarization potential and then the corresponding energy shift. The energy correction associated with the Wichmann-Kroll potential caused by the density (1.20) is usually expressed in terms of a function Hwk- Again for bound ns states we may write similarly to Equation (1.17)... 

Vacuum polarization Uehling-like loop correction Uehling corr. of wave function Wichmann-Kroll corr. of wave f. 0.0093 0.0260 -0.0007... 

Corrections to Fyp nlj, aZ) of order (aZ) with n > 1 have been considered by Wichmann and KroU [18] and lead to an expansion similar to that given above for the self-energy. Coefficients of higher-order terms in the Uehling and Wichmann and Kroll potentials are given, for example, in Ref. [16]. 

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