\label{dlT20}\end{aligned}$$ where superscript “(D)” represents the double scattering contribution, and “(0)” stands for the leading power contribution. Our formalism in Eqs. (\[DLTa\]) and (\[dlT20\]) can be verified from the $A$-dependence of the measured transverse momentum broadening.

From the factorized formula in Eq. (\[Hexp\]), the double scattering contribution $H_A^{(2)}$ depends only on the twist-2 parton-to-pion fragmentation function $D(z)$. Therefore, the differential cross section in Eq. (\[dlT20\]) for the pion production in DIS can be expressed as $$\frac{d\sigma^{(D)}_{eA\rightarrow e\pi X}}{dx_B  dQ^2 d\ell_T^2}
=\sum_{c}\, \int 
\frac{d{\sigma}^{(D)}_{eA\rightarrow ecX}}{dx_B dQ^2 dp_{c_T}^2}
\cdot D_{c \rightarrow \pi} (z) \cdot \frac{dz}{z^2} \ ,
\label{sigma-pi}$$ where $\sum_c$ runs over all parton flavors $c$, $z=\ell/p_c$ is the momentum fraction carried by the produced pion, and $D_{c\rightarrow\pi}(z)$ is the fragmentation function for the parton of momentum $p_c$ to become a pion of momentum $\ell$. In Eq. (\[sigma-pi\]), $d{\sigma}^{(D)}_{eA\rightarrow ecX}/
dx_B dQ^2 dp_{c_T}^2$ is the double scattering contributions to the differential cross section to produce a parton of momentum $p_c$ in DIS, and it can be written as $$\left[
\frac{d{\sigma}^{(D)}_{eA\rightarrow ecX}}{dx_B dQ^2 dp_{c_T}^2}
\right]\, dp_{c_T}^2
=\frac{1}{8\pi}\, \frac{e^2}{x_B^2s^2Q^2} \,
L^{\mu\nu}(k_1,k_2)\, W^{(D)}_{\mu\nu}(x_B,Q^2,p_c)\ ,
\label{sigma-c}$$ where $s=(p+k_1)^2$ is the total invariant mass of the lepton-nucleon system. In Eq. (\[sigma-c\]), the leptonic tensor $L^{\mu\nu}$ is given 