\Bigr]\end{aligned}$$ with $$\begin{aligned}
W &=& 1-M -r \sum_\mu (1-\cos p_\mu),
\\
\cosh (\alpha) &=& \frac{1+W^2+\sum_\mu \sin^2 p_\mu}{2|W|},
\label{eq:alpha}
\\
F &=& 1-e^{\alpha} W-m^2 \left(1-W e^{-\alpha}\right),
\label{eq:F}\end{aligned}$$ where the argument $p$ in the factors $\alpha$ and $W$ is suppressed.

In the perturbative calculation of Green functions the external quark momenta and masses are assumed to be much smaller than the lattice cut-off, so that we can expand the external quark propagators in terms of them. We have the following expressions as leading term of the expansion: $$\begin{aligned}
\langle q\bar q \rangle (p) & = & \frac{1-w_0^2}{i{p\kern-1ex /}+ (1-w_0^2)m},
\\
\langle q \bar \psi_s \rangle (p) &=&
\langle q\bar q \rangle (p)
\left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right),
\label{eqn:qpsi}
\\
\langle \psi_s \bar q\rangle (p) &=&
\left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right) 
\langle q\bar q \rangle (p),
\label{eqn:psiq}\end{aligned}$$ where $w_0 = 1-M$.

Renormalization factors for four-quark operators {#sec:4fermi}
================================================

We consider the following four-quark operators: $$\begin{aligned}
{\cal O}_\pm & = & \frac{1}{2} \left[
(\bar q_1 \gamma_\mu^L q_2)(\bar q_3 \gamma_\mu^L q_4)
\pm
(\bar q_1 \gamma_\mu^L q_4)(\bar q_3 \gamma_\mu^L q_2) \right], \\
{\cal O}_1 & = & 
-C_F (\bar q_1 \gamma_\mu^L q_2)(\bar q_3 \gamma_\mu^R q_4)
+ (\bar q_1 T^A \gamma_\mu^L q_2)(\bar q_3 T^A \gamma_\mu^R q_4), \\
