make a perturbative calculation of the renormalization factors for the three- and four-quark operators consisting of physical quark fields in the domain-wall QCD(DWQCD). This work is an extension of the previous paper[@AIKT98], in which we developed a perturbative renormalization procedure for DWQCD demonstrating the calculation of the renormalization factors for quark wave function, mass and bilinear operators. We focus on whether or not the renormalization of the three- and four-quark operators in DWQCD is free from the notorious operator mixing problem. In the Wilson case it is well known that the mixing problem is not adequately manipulated by the perturbation theory, leading to an “incorrect” value for the $B_K$ matrix element.

This paper is organized as follows. In Sec. \[sec:model\] we briefly introduce the DWQCD action and the Feynman rules relevant for the present calculation to make this paper self-contained. In Sec. \[sec:4fermi\] our calculational procedure of the renormalization factors for the four-quark operators is described in detail. We also evaluate the renormalization factors for three-quark operators in Sec. \[sec:3fermi\]. In Secs. \[sec:4fermi\] and \[sec:3fermi\] numerical results for one-loop coefficients of the renormalization factors are given with and without the mean field improvement. In Sec. \[sec:bk\], using our results, we 