analyze a renormalization factor for $B_K$. Our conclusions are summarized in Sec. \[sec:concl\].

The physical quantities are expressed in lattice units and the lattice spacing $a$ is suppressed unless necessary. We take SU($N$) gauge group with the gauge coupling $g$ and the second Casimir $C_F = \displaystyle \frac{N^2-1}{2N}$, while $N=3$ is specified in the numerical calculations.

Action and Feynman rules {#sec:model}
========================

We take the Shamir’s domain-wall fermion action[@Shamir93], $$\begin{aligned}
S_{\rm DW} &=&
\sum_{n} \sum_{s=1}^{N_s} \Biggl[ \frac{1}{2} \sum_\mu
\left( {\overline{\psi}}(n)_s (-r+\gamma_\mu) U_\mu(n) \psi(n+\mu)_s
+ {\overline{\psi}}(n)_s (-r-\gamma_\mu) U_\mu^\dagger(n-\mu) \psi(n-\mu)_s \right)
{\nonumber}\\&&
+ \frac{1}{2}
\left( {\overline{\psi}}(n)_s (1+\gamma_5) \psi(n)_{s+1}
+ {\overline{\psi}}(n)_s (1-\gamma_5) \psi(n)_{s-1} \right)
+ (M-1+4r) {\overline{\psi}}(n)_s \psi(n)_s \Biggr]
{\nonumber}\\&+&
 m \sum_n \left( {\overline{\psi}}(n)_{N_s} P_{+} \psi(n)_{1}
+ {\overline{\psi}}(n)_{1} P_{-} \psi(n)_{N_s} \right),
\label{eqn:action}\end{aligned}$$ where $n$ is a four dimensional space-time coordinate and $s$ is an extra fifth dimensional or “flavor” index, the Dirac “mass” $M$ is a parameter of the theory which we set $0 < M < 2$ to realize the massless fermion at tree level, $m$ is a physical quark mass, and the Wilson parameter is set to $r=-1$. It is important to notice that we have boundaries for the flavor space; $1 \le s \le N_s$. In our one-loop calculation we 