quark and gluons. These two poles (which are not pinched) can be used to carry out the momentum fraction $dx_1$ and $dx_2$ integrals [@LQS1; @LQS2]. After taking the poles, the hard-scattering factor (the interaction between the virtual photon and the initial quark) and the final-state interaction between the “on-shell” outgoing quark and the gluon are separately gauge invariant.

To derive the leading contribution from the double scattering, we follow the derivation in Ref. [@Guo2]. By taking the poles to fix the integrations of momentum fractions $x_1$ and $x_2$, we derive contributions of all three diagrams in Fig. \[fig2\] to the hadronic tensor $W^{(D)}_{\mu\nu}$ in Eq. (\[sigma-c\]) as $$\begin{aligned}
W^{(D)}_{\mu\nu}(x_B,Q^2,p_c) &=& \frac{1}{4\pi}\,
\int \frac{dy^-}{4\pi}\, \frac{dy_1^-dy_2^-}{(2\pi)^2}\,
\frac{d^2y_T}{(2\pi)^2} \,
\int d^2 k_T\,  e^{ik_T \cdot y_{1T}} \, 
e^{i \frac{k_T^2-2k_T \cdot q}{2p \cdot q} p^+ (y_1^--y_2^-)} \,
\nonumber \\ 
&\ & \times\,
\langle p_A | \bar{\psi}_q(0)\,\gamma^+ \, 
A^+(y_{1}^{-},y_{1T})\,  A^+(y_{2}^{-})
            \psi_q(y^{-})\, | p_{A}\rangle 
\nonumber \\
&\ & \times (2\pi) \theta (y_1^--y^-) \, 
(2\pi) \theta (y_2^-) \, e^{i x_B p^+ y^- } 
\nonumber \\
&\ & \times \frac{2\pi}{(2p\cdot q)^3} 
      \cdot \hat{H}^{(D)}_{\mu\nu}(p_c) 
\cdot  \left[ \delta (p_{c_T}^2-k_T^2) - \delta (p_{c_T}^2) \right] 
dp_{c_T}^2 \\
& \approx & \left[
\frac{\pi}{(2p\cdot q)^3}\, \hat{H}^{(D)}_{\mu\nu}(p_c) \,
\left[-\delta ' (p_{c_T}^2) \right] T_{qF}^A(x_B,\mu^2) \right]
\, dp_{c_T}^2 