\ .
\label{W-c}\end{aligned}$$ In Eq. (\[W-c\]), $T_{qF}^A(x_B,\mu^2)$ is defined by Eq. (\[T4qA\]) with the factorization scale $\mu^2$. $\hat{H}^{(D)}_{\mu\nu}(p_c)$ represents the spinor trace of the partonic part of the double scattering and is given by $$\hat{H}^{(D)}_{\mu\nu}(p_c) =\frac{4}{3}
\pi^2 \alpha_s \alpha_{em} e_q^2 \,
{\rm Tr}\left[\gamma \cdot p \gamma_{\mu} \gamma \cdot (xp+q) 
\gamma^{\sigma} \gamma \cdot p_c \gamma^{\rho} \gamma \cdot (xp+q)
\gamma_{\nu}\right] \, p_{\rho}\,p_{\sigma}\ ,
\label{hatH}$$ where a color factor $1/2N_c$ with $N_c=3$ was included. Using the definition $z=\ell/p_c$, see Eq. (\[z\]) below, we can reexpress the momentum $p_c$ in $\delta'(p^2_{c_T})$ in terms of the momentum $\ell$ of the observed pion, $$-\delta '(p_{c_T}^2)
=- \frac{d}{d(\frac{\ell_{T}^2}{z^2})} 
\left( \delta (\frac{\ell_{T}^2}{z^2}) \right) 
= - z^4 \delta '(\ell_{T}^2) \ .
\label{dpi}$$ Combining Eqs. (\[sigma-pi\]), (\[sigma-c\]), (\[W-c\]), and (\[dpi\]), we obtain $$\frac{d\sigma^{(D)}_{eA\rightarrow e\pi X}}{dx_B  dQ^2 d\ell_T^2}
=\frac{e^2}{8\pi x_B^2 s^2 Q^2} \, 
\frac{\pi}{(2p\cdot q)^3}
\sum_q\int dz\, z^2 \, D_{q\rightarrow \pi}(z)\, 
T_{qF}^A(x_B,Q^2)\, 
L^{\mu\nu} \hat{H}^{(D)}_{\mu\nu}(\ell/z) \,
\left[-\delta ' (\ell_{T}^2) \right]\, ,
\label{sigma-pi2}$$ where the $\sum_q$ runs over all quark and antiquark flavors, and $z=\ell/p_q$ is the momentum fraction of the quark flavor $q$ carried by the observed pion. At the lowest order in $\alpha_s$, hard gluon initiated double scattering, which is proportional to the four-gluon correlation function $T_{gF}$, does not contribute [@LQS1; 