from the gauge invariant $g^{*}g^{*}\rightarrow Q\overline{Q}$ amplitude and $\mathcal{F}_{g^{*}/A}$, $\mathcal{F}_{g^{*}/B}$ are UGDs for hadrons $A$ and $B$. The contributions with off-shell quarks are suppressed. The UGDs are assumed to undergo the BFKL evolution equations. In Ref. [@Catani:1994sq] it was argued that similar factorization holds to all orders for DIS heavy quark structure function, although the argumentation misses the details comparing to collinear factorization proofs [@Collins:2011zzd], especially the definitions of UGDs and complications arising at higher orders in the axial gauge [@Avsar:2012hj].

            

In the works [@Deak2010],[@Kutak:2012rf],[@VanHameren2013],[@vanHameren2013a; @vanHameren:2014lna; @vanHameren:2014ala] as well as in this paper the CCH factorization was extended to model the cross section for jet production in hadron-hadron collisions. The first difficulty arises because now one has to consider also gluons in the final state, e.g. $g^{*}g^{*}\rightarrow gg$ sub-process for dijet production. The corresponding amplitude is however not gauge invariant when calculated from ordinary Feynman diagrams. A few approaches have been proposed to calculate a gauge invariant extension of such amplitudes [@vanHameren2012; @vanHameren2013a; @vanHameren:2013csa; @vanHameren:2014iua; @Kotko2014a]. These [*gauge invariant off-shell amplitudes*]{} in fact correspond to a vertex that can be calculated from the well-known Lipatov’s effective action [@Lipatov:1995pn; @Antonov:2004hh] (see Fig. \[fig:HEF\]B). The approaches [@vanHameren2012; @vanHameren2013a; @vanHameren:2013csa; @vanHameren:2014iua; @Kotko2014a] were 