will take $N_s\to\infty$ limit to avoid complications arising from the finite $N_s$. $P_{R/L}$ is a projection matrix $P_{R/L}=(1\pm\gamma_5)/2$. For the gauge part we employ a standard four dimensional Wilson plaquette action and assume no gauge interaction along the fifth dimension.

In the DWQCD the zero mode of domain-wall fermion is extracted by the “physical” quark field defined by the boundary fermions $$\begin{aligned}
q(n) = P_R \psi(n)_1 + P_L \psi(n)_{N_s},
{\nonumber}\\
{\overline{q}}(n) = {\overline{\psi}}(n)_{N_s} P_R + {\overline{\psi}}(n)_1 P_L.
\label{eq:quark}\end{aligned}$$ We will consider the QCD operators constructed from this quark fields, since this field has been actually used in the previous simulations. Moreover our renormalization procedure is based on the Green functions consisting of only the “physical” quark fields, in which we have found that the renormalization becomes simple[@AIKT98].

Weak coupling perturbation theory is developed by expanding the action in terms of gauge coupling. The gluon propagator can be written as $$\begin{aligned}
G_{\mu\nu}^{AB}(k) =\delta_{\mu\nu}\delta_{AB} \frac{1}{4\sin^2(k/2)+\lambda^2}\end{aligned}$$ in the Feynman gauge with the infrared cut-off $\lambda^2$, where $\sin^2(k/2) =\sum_\mu \sin^2(k_\mu/2)$. Quark-gluon vertices are also identical to those in the $N_s$ flavor Wilson fermion. We need only one gluon vertex for our present calculation: $$\begin{aligned}
V_{1\mu}^A (k,p)_{st}
&=& -i g T^A \{ \gamma_\mu \cos(-k_\mu/2 + 