A for some remarks.

Note that there is no mean-field improvement factor for $B_K$ in actual simulations since, as mentioned before, it is defined by the ratio. Therefore the difference between values of $Z_{B_K}$ with and without mean-field improvement comes from higher order ambiguity in perturbation theory.

Necessary informations for the analysis in this section are given in Table \[tab:bk\], together with values of $Z_{B_K}$.

Conclusion {#sec:concl}
==========

In this paper we have calculated the one-loop contributions for the renormalization factors of the three- and four-quark operators in DWQCD. We have demonstrated that the three- and four-quark operators in DWQCD can be renormalized without any operator mixing between different chiralities as opposed to the Wilson case. This desirable property in DWQCD would practically surpass the cost of the introduction of an unphysical fifth dimension. The numerical values for the finite parts $z_X$ with $X=\pm,1,2,PD$ settle in reasonable magnitude with the mean-field improvement, while unimproved values are rather large in general.

In this work we do not treat the operators which yield the so-called “penguin” diagram. It seems feasible to carry out the calculation of their renormalization factors, which we leave to future investigation.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work is 