the quenched approximation. Here we explicitly calculate $Z_{B_K}( \mu a )$ for these parameters. From Table \[tab:total\] and the previous result[@AIKT98], $z_+ = -41.854 (-42.399)$, $z_A = -17.039 (-16.827)$ and $z_+ -2C_F z_A = 3.583 (2.473)$ for $M=1.7 (1.5)$ in the DRED scheme, and $z_+ = -44.854 (-45.399)$, $z_A = -17.539 (-17.327)$ and $z_+ -2C_F z_A = 1.917 (0.8063)$ for $M=1.7 (1.5)$ in the NDR scheme. Taking $\mu =1/a$ and $g^2 = g^2_{\overline{MS}}(1/a)$, estimated by the formula $$\frac{1}{g^2_{\overline{MS}}}(1/a) = P\frac{\beta}{6} -0.13486$$ for the quenched QCD with $P$ being the average value of the plaquette, we have $Z_{B_K} = $ 1.053 (1.029), 1.049 (1.026) and 1.030 (1.010) at $\beta =$ 5.85, 6.0 and 6.3, respectively, in the DRED (NDR) scheme. Sizes of one-loop corrections for $B_K$ are not so large, $1-5\%$, at these $\beta$ values even without mean-field improvement, since the large contribution, which comes from a $(1-w_0) Z_w$ factor, cancels out in the ratio of ${\cal O}_+$ and $A_4^2$.

If we employ the mean-field improvement by replacing $M\rightarrow
{\widetilde{M}}=M+4(u-1)$ with $u = P^{1/4}$, we obtain $Z_{B_K} = $ 1.018 (0.994), 1.017 (0.994) and 1.009 (0.988) at $\beta =$ 5.85, 6.0 and 6.3, respectively, in the DRED (NDR) scheme. See Appendix 