Perturbative renormalization factors of $ΔS = 1$ four-quark operators for domain-wall QCD

Renormalization factors for $\Delta S =1$ four-quark operators in the effective weak Hamiltonian are perturbatively evaluated in domain-wall QCD. The one-loop corrections of $\Delta S=1$ four-quark operators consist of two types of diagrams: one is gluon exchange between quark lines and the other is penguin diagrams containing quark loops. Combining both contributions, our results allow a lattice calculation of the amplitude for $K\to \pi\pi$ decays with $O(g^2)$ corrections included.