  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.
