@LQS2].

In the photon-nucleus frame, we choose the target momentum $p$ along the $-\vec{z}$-axis, such that $p^{\mu}=(p^0,p^x,p^y,p^z)=(P,0,0,-P)$, and only $p^-=(p^0-p^z)/\sqrt{2}$ is nonvanishing. In this frame, we have the following expression for the momentum fraction $z=\ell/p_c$, $$\begin{aligned}
z &\equiv& \frac{\ell}{p_{c}}=\frac{\ell^{+}}{p_{c}^{+}}
=\frac{p\cdot \ell }{p \cdot p_{c}} \nonumber \\ 
&\approx & \frac{p\cdot \ell }{p\cdot q} \ .
\label{z}\end{aligned}$$ In deriving the last equation, we used $p_c=x_Bp+q$ and $p^2\approx
0$.

After working out the algebra for $L^{\mu\nu} \hat{H}^{(D)}_{\mu\nu}$, and substituting Eq. (\[sigma-pi2\]) into Eq. (\[dlT20\]), we obtain $$\Delta\langle \ell_{T}^2\rangle_{1/3} \left(\ell_{\rm min}\right) = 
\frac{4\pi^2\alpha_s(\mu^2)}{3}\,
\frac{\sum_q e_q^2\,\int_{z_{\rm min}}^1 
             dz\, z^2 D_{q\rightarrow\pi}(z) T_{qF}^{A}(x_B,\mu^2)}
     {\sum_q e_q^2\,q^{A}(x_B,\mu^2)}\, ,
\label{pilt}$$ where $D_{q\rightarrow\pi}(z)$ are the quark-to-pion fragmentation functions, and $q^A(x_B,\mu^2)=\phi_{q/A}(x_B,\mu^2)$ are the effective quark distributions in the nucleus normalized by the atomic weight $A$. In Eq. (\[pilt\]), $z_{\rm min}=p\cdot \ell_{\rm 
min}/(p\cdot q)$ is defined in terms of $\ell_{\rm min}$, which is the momentum cut for the pions measured in the experiment. One can choose $\ell_{\rm min}$ to be large enough to ensure that the observed pions are from the fragmentation of energetic quarks.

In Eq. (\[pilt\]), the $\mu^2$ in $\alpha_s(\mu^2)$ is the renormalization scale, and the $\mu^2$ in $T_{qF}^{A}(x_B,\mu^2)$ and $q^{A}(x_B,\mu^2)$ are the factorization scale. Since the short-distance part of the 