---
abstract: 'Renormalization factors for three- and four-quark operators, which appear in the low energy effective Lagrangian of the proton decay and the weak interactions, are perturbatively calculated in domain-wall QCD. We find that the operators are multiplicatively renormalizable up to one-loop level without mixing with any other operators that have different chiral structures. As an application, we evaluate a renormalization factor for $B_K$ at the parameters where previous simulations have been performed, and find one-loop corrections to $B_K$ are 1-5% in these cases.'
address: |
    $^1$Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan\
    $^2$Department of Physics, Washington University, St. Louis, Missouri 63130, USA\
author:
- '$^1$Sinya Aoki, $^1$Taku Izubuchi, $^{2}$Yoshinobu Kuramashi [^1] and $^1$Yusuke Taniguchi'
title: |
    [hep-th/9902008]{}\
    [UTHEP-398]{}\

    Perturbative renormalization factors of three- and four-quark operators for domain-wall QCD
---

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Introduction
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Calculation of hadron matrix elements of phenomenological interest represents an inevitable application of lattice QCD. In the past decade much efforts have been devoted for the calculation of three- and four-quark hadron matrix elements relevant to the proton decay amplitude and the weak interaction ones using the Wilson and the Kogut-Susskind(KS) quark actions. However, the satisfactorily precise measurement of the matrix elements has 