as a function of $M$.

The corresponding continuum renormalization factors in the ${{\overline {\rm MS}}}$ scheme are calculated employing the DRED scheme as the regularization in the Feynman gauge with the fictitious gluon mass $\lambda$. The vertex correction for ${\cal O}_{PD}$ is $$\begin{aligned}
V_{PD}^{{{\overline {\rm MS}}}} &=& 1+ \frac{g^2}{16\pi^2}\delta_{PD}
\left[-\log (\lambda/\mu)^2 +1 \right],\end{aligned}$$ giving $v_{PD}^{{{\overline {\rm MS}}}}=\delta_{PD}$. We remark that in this case the evanescent operator does not appear at the one-loop level.

Combining this result with the previous lattice one we finally obtain the relation between the operators ${\cal O}_{PD}^{{{\overline {\rm MS}}}}$ and ${\cal O}_{PD}^{lat}$: $$\begin{aligned}
{\cal O}_{PD}^{{{\overline {\rm MS}}}}(\mu) & = &\frac{1}{(1-w_0^2)^{3/2} Z_w^{3/2}}
Z_{PD} (\mu a ) {\cal O}_{PD}^{lat} (1/a),\end{aligned}$$ with $$\begin{aligned}
Z_{PD} (\mu a) &=& \frac{ (Z_2^{{{\overline {\rm MS}}}})^{3/2} V_{PD}^{{{\overline {\rm MS}}}}}
{(Z_2)^{3/2} V_{PD}} \nonumber \\
&=& 1 + \frac{g^2}{16\pi^2}\left[
(\delta_{PD} - 3 C_F/2)\log (\mu a)^2 + z_{PD} \right],
 \\
z_{PD} &=& v_{PD}^{{{\overline {\rm MS}}}} - v_{PD} + \frac{3}{2} C_F\{\Sigma_1^{{{\overline {\rm MS}}}}
-\Sigma_1\}.\end{aligned}$$ We present numerical values for $z_{PD}$ in Table \[tab:total\] and those for the mean-field improved one, $z_{PD}^{MF}$, in Table \[tab:totalMF\].

Renormalization factor for $B_K$ {#sec:bk}
================================

As an application of results in the previous sections, we estimate a renormalization factor for the kaon $B$ 