content_image/387225.jpg content_image/387228.jpg content_image/387224.jpg content_image/387226.jpg content_image/387223.jpg content_image/387238.jpg content_image/387227.jpg
and δE = 〈 ⃗ p$_{1}$ · ⃗ p$_{2}$ 〉 . The perturbed wave function δφ is defined by 

where δ$_{M}$ stands for the first-order perturbation induced by the operator ⃗ p$_{1}$ · ⃗ p$_{2}$ . The asymptotic expansion of J$_{M}$ ( k ) is much simpler than that of J ( k ) and w$_{M}$ ( k ) is just 

<img src='content_image/387225.jpg'>

Correspondingly, the function 

The numerical evaluation of Eq. (A7) was performed in a way similar to that for the plain Bethe logarithm. In particular, the same sets of optimized nonlinear param- eters were used. Since a high accuracy is not needed for this correction, a somewhat simplified calculational scheme was used in this case. The high-energy part of the photon-momentum integral, k ∈ [100 , ∞ ), was eval- uated by integrating the fitted asymptotic expansion for w$_{M}$ ( k ), which was taken to be of the form (A6) with 6 − 9 fitting parameters. 

## Appendix B: Expectation value of 1 / r 3 

The definition of the expectation value of the regular- ized operator 1 /r 3 is given by Eq. (12). With the basis- set representation of the wave function employed in this work, a typical singular integral to be calculated is 

<img src='content_image/387228.jpg'>

The straightforward way is to evaluate this integral an- alytically for a finite value of the regulator ϵ and then 

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<img src='content_image/387224.jpg'>

<img src='content_image/387226.jpg'>

expand it in small ϵ . This way is possible, but we prefer to use a simpler procedure, which is also the closest to the method of evaluation of the regular integrals. 

We recall that all regular integrals are immediately ob- tained from the master integral 

<img src='content_image/387223.jpg'>

by formal differentiation or integration with respect to the corresponding parameters. The differentiation over α and β and an integration over γ delivers a result for the integral of the type 1 /r $^{2}$. This integral is conver- gent, so the result is exact. The second integration over γ (which would yield an integral of the type 1 /r $^{3}$) is di- vergent. The simplest way to proceed is as follows. We introduce a cutoff parameter for large values of γ , eval- uate the integral over γ , and drop all cutoff-dependent terms. The expression obtained in this way differs from the correct one by a γ -independent constant only, which can be proved by differentiating with respect to γ . 

The missing constant is most easily recovered by con- sidering the behavior of the integral I when γ → ∞ . For very large γ , only the region of very small r contributes and we have 

<img src='content_image/387238.jpg'>

Therefore, 

<img src='content_image/387227.jpg'>

The above equation yields the necessary condition for determining the missing constant term in the general ex- pression for the regularized integral I$_{reg}$ . 

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