factor and vertex corrections in the NDR scheme.

Renormalization factors for three-quark operators {#sec:3fermi}
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The three-quark operators relevant to the proton decay amplitude are given by $$\left({\cal O}_{PD}\right)_\delta =\varepsilon^{abc} 
\left((\bar q^C_1)^a \Gamma_X (q_2)^b\right) (\Gamma_Y (q_3)^c)_\delta,
\label{eq:O_PD}$$ where $\bar q^C = -q^T C^{-1}$ with $C=\gamma_0\gamma_2$ is a charge conjugated field of $q$ and $\Gamma_X\otimes\Gamma_Y = P_R\otimes P_R, P_R\otimes P_L, 
P_L\otimes P_R, P_L\otimes P_L$. The summation over repeated color indices $a,b,c$ is assumed. We should note that the domain-wall fermion action [(\[eqn:action\])]{} is transformed identically into that with the conjugated field by using transpose and matrix $C$. The resultant action and Feynman rules for the conjugated field is obtained by the replacement that $$\begin{aligned}
ig T^A &\rightarrow & -ig (T^A)^T ,
$$ where the superscript $T$ means the transposed matrix.

In order to evaluate the vertex corrections we consider the following Green function: $$\langle \left({\cal O}_{PD}\right)_\delta 
\rangle_{\alpha\beta\gamma}^{ijk}
\equiv {\langle  \left({\cal O}_{PD}\right)_\delta 
(q^C_1)_\alpha^i(\bar q_2)_\beta^j
(\bar q_3)_\gamma^k  \rangle},$$ where $\alpha,\beta,\gamma$ and $i,j,k$ are spinor and color indices respectively. Truncating the external quark propagators of $\langle {\cal O}_{PD}\rangle$ we obtain the vertex function $$(1-w_0^2)^3\left(\Lambda_{PD}\right)_{\alpha\beta;\delta\gamma}^{ijk}
=(1-w_0^2)^3\left(\Lambda_\Gamma^{(0)}
+\Lambda_\Gamma^{(1)}\right)_{\alpha\beta;\delta\gamma}^{ijk},$$ where the trivial factor $(1-w_0^2)^3$ is factored out for the convenience. We suppress the external momenta $p_i$ since 