9 1 16 π 2 ∫ d $^{3}$r$_{1}$ d $^{3}$r$_{2}$ exp( − αr$_{1}$ − βr$_{2}$ − γr ) r$_{1}$r$_{2}$r = 1 ( α + β )( β + γ )( α + γ ) (B2) | δφ 〉 = | δ$_{M}$ φ 〉 − | φ 〉 δ$_{M}$ D D , (A9) w$_{M}$ ( k ) = k 3 D J$_{M}$ ( k ) + k 2 D 2 〈 φ | ⃗ ∇ $^{2}$| δφ 〉 . (A10) R$_{M}$ ( κ ) = 2 κ D 〈 φ | ⃗ ∇ $^{2}$| δφ 〉 . (A11) I$_{reg}$ ≡ lim ϵ → 0 [ I$_{ϵ}$ + γ + ln ϵ ] γ →∞ = − 2 ( α + β ) 3 ln γ . (B4) I$_{ϵ}$ = 1 16 π 2 ∫ d $^{3}$r$_{1}$ d $^{3}$r$_{2}$ exp( − αr$_{1}$ − βr$_{2}$ − γr ) r 3 Θ( r − ϵ ) . (B1) 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 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 Correspondingly, the function R$_{M}$ ( κ ) is 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 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. 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 Therefore, I$_{ϵ}$ = 2 ∫ ∞ ϵ dr r ∫ ∞ 0 dr$_{1}$ r$_{1}$ ∫ r 1 + r | r 1 − r | dr$_{2}$ r$_{2}$ e − αr 1 − βr 2 − γr r 3 ≈ 2 ( α + β ) 3 ∞ ϵ dr e − γr r . (B3) 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 The above equation yields the necessary condition for determining the missing constant term in the general ex- pression for the regularized integral I$_{reg}$ . The straightforward way is to evaluate this integral an- alytically for a finite value of the regulator ϵ and then [1] P. C. Pastor, G. Giusfredi, P. DeNatale, G. Hagel, C. de Mauro, and M. Inguscio, Phys. Rev. Lett. 92 , 023001 (2004). [2] H. Araki, Prog. Theor. Phys. 17 , 619 (1957). [3] J. Sucher, Phys. Rev. 109 , 1010 (1958). [4] V. M. Shabaev, Phys. Rep. 356 , 119 (2002). [5] G. W. F. Drake, Can. J. Phys. 66 , 586 (1988). [6] K. Pachucki, Phys. Rev. A 74 , 022512 (2006). [7] K. Pachucki, Phys. Rev. A 74 , 062510 (2006). [8] A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plu- nien, and G. Soff, Phys. Rev. A 71 , 062104 (2005). [9] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Berlin, Springer, 1957). [10] K. Pachucki and J. Sapirstein, J. Phys. B 33 , 455 (2000). [11] K. Pachucki, Phys. Rev. Lett. 84 , 4561 (2000). [12] K. Pachucki, J. Phys. B 35 , 3087 (2002). [13] G. W. F. Drake, I. B. Khriplovich, A. I. Milstein, and A. S. Yelkhovsky, Phys. Rev. A 48 , R15 (1993). [14] M. Douglas and N. Kroll, Ann. Phys. (NY) 82 , 89 (1974). [15] K. Pachucki and V. A. Yerokhin, Phys. Rev. A 79 , 062516 (2009), [ ibid. 80 , 019902 (E) (2009)]. [16] V. Korobov and A. Yelkhovsky, Phys. Rev. Lett. 87 , 193003 (2001). [17] A. Yelkhovsky, Phys. Rev. A 64 , 062104 (2001). [18] K. Pachucki, Phys. Rev. Lett. 97 , 013002 (2006). [19] P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod.