by the diagram in Fig. \[fig1\]a, $$L^{\mu\nu}(k_1,k_2)
=\frac{1}{2}\, {\rm Tr}(\gamma \cdot k_1 \gamma^{\mu}
\gamma \cdot k_2 \gamma^{\nu}) \ ,
\label{L}$$ and $W^{(D)}_{\mu\nu}$ is the hadronic tensor due to double scattering, which is given by the diagram shown in Fig. \[fig1\]b. For comparison, the lowest order differential cross section for producing a parton of momentum $p_c$ due to single scattering is given by $$\left[
\frac{d{\sigma}^{(0)}_{eA\rightarrow ecX}}{dx_B dQ^2 dp_{c_T}^2}
\right]\,
dp_{c_T}^2
= \left[
\frac{d{\sigma}^{(0)}_{eA\rightarrow ecX}}{dx_B dQ^2}\, 
  \delta(p_{c_T}^2) \right]\,
dp_{c_T}^2
\label{single}$$ where $d{\sigma}^{(0)}/dx_B dQ^2$ is the leading order inclusive DIS cross section, which appears in the definition of $\Delta\langle\ell^2_T\rangle_{1/3}$ in Eq. (\[dlT20\]).

The complete double scattering contributions to the production of a leading quark of momentum $p_c$ in DIS at the leading order of $\alpha_s$ come from the diagrams shown in Fig. \[fig2\], which represent the final-state interactions, plus the same order diagrams involving initial-state interactions shown in Fig. \[fig2p\]. In the following, we first discuss the role of initial-state interactions, and argue that although required by gauge invariance, they contribute to the $a$ term of Eq. (\[BA13\]) only.

After the collinear expansion of the parton momenta, the gluon interactions on the initial quark lines, as shown in Fig. \[fig2p\], can be reduced 