n Fig.1. A simple calculation gives the following correction $\delta Q_W/Q_W=\delta_{cd}$ related to this renormalization $$\label{cd}
\delta_{cd}={{4\alpha Z}\over{3\pi Q_W}}
(1-4\sin^2\theta_W)\ln(\lambda_C/r_0)\approx -0.1\%.$$ Where $\theta_W\approx $ is the Weinberg angle, $\sin^2\theta_W\approx
0.2230$, see Ref. [@RPP]. Note that this correction is practically independent of $Z$ because $Z/Q_W \approx -Z/N\approx -0.7$, where $N$ is the number of neutrons. One can also obtain the correction (\[cd\]) using Eqs. (2a,b), and (3b) from Ref.[@Mar1].

Next we consider the contribution of the electron self-energy operator $\Sigma$. This operator being substituted to the Dirac equation, $m \to m+\Sigma$, leads to the Lamb shift of the energy level and to the modification of the electron wave function, see, e.g., Ref. [@BLP]. As shown in Fig.1e, this modification influences the matrix element of the weak interaction. The diagram Fig.1e is not invariant with respect to the gauge transformations of the electromagnetic field. However, the sum of the diagrams Fig.1e and Fig.1f (the vertex correction) is gauge invariant. It is convenient to represent the self energy operator as a series in powers of the Coulomb field of the nucleus, $\Sigma=\Sigma_0+\Sigma_1+\Sigma_2+...$, see Fig.2.

=8.cm

We need $\Sigma({\bf r},{\bf r'}|\epsilon)$ at $r \sim r' \ll \lambda_C$, $\epsilon\approx m$. In this limit a calculation in the Feynman gauge with logarithmic accuracy gives $$\label{ss}
\Sigma_0={\hat p}{{\alpha}\over{4\pi}}\ln(p^2/m^2), \ \ \
\Sigma_1={{Z\alpha^2}\over{4\pi r}}\gamma_0\ln(p^2/m^2),$$ where ${\hat p}=p^{\mu}\gamma_{\mu}$, $\gamma_{\mu}$ is the Dirac matrix, and $p^{\mu}$ is the momentum operator. All the higher terms are not logarithmically enhanced. Further calculation in Feynman gauge is rather involved because the diagram Fig.1f is also logarithmically enhanced and there is a delicate cancellation between a part of the $\Sigma$-contribution and the logarithmic part of Fig.1f. To avoid all these complications it