$A^{1/3}$-type of nuclear enhancement [@QS_fac]. Since the short distance contributions of Feynman diagrams in Fig. \[fig2p\] have at least one propagator given by the contact term, these diagrams do not contribute to the $A^{1/3}$-type of nuclear enhancement. At the same time, the contact term of the initial-state interaction is important for the gauge invariance of the complete double scattering (twist-4) process [@Qiu], but only at order $A^0$.

Although the Feynman diagrams with the final-state interactions in Fig. \[fig2\] are not gauge invariant in general, their contributions to the $A^{1/3}$-behavior of nuclear enhancement are observable and hence gauge invariant [@LQS2]. The three lowest order diagrams in Fig. \[fig2\] are all convoluted with the same two-quark-two-gluon matrix element through three independent parton momenta. The leading power contributions of these diagrams come from the region phase space when all partons are collinear to the direction of the target, as shown in Fig. \[fig2\]. In order to convert the gluon field $A^+$ to the corresponding field strength $F^{+\perp}$ in covariant gauge, we expand the gluon momenta in an extra transverse component $k_T$ in Fig. \[fig2\] [@LQS2]. All three Feynman diagrams in Fig. \[fig2\] have two potential poles due to the interactions between the final-state 