Quark jets scattering from a gluon field: from saturation to high $p_t$

We continue our studies of possible generalization of the Color Glass Condensate (CGC) effective theory of high energy QCD to include the high $p_t$ (or equivalently large $x$) QCD dynamics as proposed in [JJM-elastic]. Here we consider scattering of a quark from both the small and large $x$ gluon degrees of freedom in a proton or nucleus target and derive the full scattering amplitude by including the interactions between the small and large $x$ gluons of the target. We thus generalize the standard eikonal approximation for parton scattering which can now be deflected by a large angle (and therefore have large $p_t$) and also lose a significant fraction of its longitudinal momentum (unlike the eikonal approximation). The corresponding production cross section can thus serve as the starting point toward derivation of a general evolution equation that would contain DGLAP evolution equation at large $Q^2$ and the JIMWLK evolution equation at small $x$. This amplitude can also be used to construct the quark Feynman propagator which is the first ingredient needed to generalize the Color Glass Condensate (CGC) effective theory of high energy QCD to include the high $p_t$ dynamics. We outline how it can be used to compute observables in the large $x$ (high $p_t$) kinematic region where the standard Color Glass Condensate formalism breaks down.