parameter $B_K$, defined by $$B_K =\frac{\langle \overline{K}^0 \vert {\cal O}_+ \vert K^0 \rangle}
{\frac{8}{3} \langle \overline{K}^0 \vert A_4 \vert 0 \rangle
\langle 0 \vert A_4 \vert K^0 \rangle}$$ with $q_1=q_3 = s$ and $q_2=q_4= d$ in ${\cal O}_+$.

Denoting the renormalization factor between the continuum $B_K$ at scale $\mu$ and the lattice one at scale $1/a$ as $Z_{B_K}( \mu a )$, we obtain $$Z_{B_K}( \mu a ) =\frac{(1-w_0^2)^{-2} Z_w^{-2} Z_+ (\mu a )}
{(1-w_0)^{-2} Z_w^{-2} Z_A(\mu a)^2}
=\frac{Z_+ (\mu a )}{Z_A(\mu a)^2},$$ where $$Z_+(\mu a) = 1 + \frac{g^2}{16\pi^2}\left[ -4\log (\mu a) + z_+ \right]$$ from eq. (\[eq:4fermi\]) in this paper, and $$Z_A(\mu a) = 1 + \frac{g^2 C_F}{16\pi^2} z_A$$ from Ref. [@AIKT98], so that $$Z_{B_K}(\mu a) = 1 +\frac{g^2}{16\pi^2}\left[ -4\log (\mu a) + z_+ 
-2 C_F z_A \right] .$$ Note that $z_A$ in Ref. [@AIKT98] is evaluated in the NDR scheme while the DRED scheme is used for $z_+$ in this paper. From the result in Appendix B we have $$z_A({\rm DRED}) = z_A({\rm NDR}) +1/2, \qquad
z_+({\rm DRED}) = z_+({\rm NDR}) +3 .$$

In Ref.[@Blum-Soni] $B_K$ has been evaluated at $\beta = 5.85$, 6.0 with $ M= 1.7$ and $\beta = 6.3$ with $M=1.5$, using domain-wall QCD with 