soft-gluons, and has the following operator definition [@LQS1; @Guo2], $$\begin{aligned}
T_{qF}^{A}(x,\mu^2)&=& \int \frac{dy^-}{2\pi} 
e^{ix p^+ y^-} \frac{dy_1^- dy_2^-}{2\pi}
\theta(y_1^--y^-)\theta(y_2^-)
\nonumber \\
&\times &
\langle P_A| \bar{\psi}_q(0)\, \frac{\gamma^+}{2}\, 
F^+_{\ \alpha}(y_2^-) \, F^{+\alpha}(y_1^-)\, 
\psi_q(y^-)\, |P_A\rangle \, ,
\label{T4qA}\end{aligned}$$ with $A$ the atomic weight, and $q$ is the quark flavor and $F$ is the gluon field strength. In Eq. (\[T4qA\]), $x$ is the effective momentum fraction and $\mu$ represents the scale-dependence of the correlation functions.

According to QCD factorization theorem [@CSS_fac; @QS_fac], all information on the identity of the target is factorized into the matrix elements. The $A^{1/3}$-type enhancement must be a property of the relevant nuclear matrix elements, like the one in Eq. (\[T4qA\]). Experimentally, the $A^{1/3}$-type nuclear enhancement have been observed in the transverse momentum broadening of Drell-Yan pairs in hadron-nucleus collisions [@DY_PT]. Theoretically, Eq. (\[T4qA\]) shows how such enhancement can occur, through integrals over the relative positions of the fields, $y_i$, that appear in the expectation values. For example, enhancement can occur when the two quark fields are close together and the two gluon fields are close together, and they pair off into color singlets that are separated by no more than a nucleon diameter in the $y_i$ integrals; but 