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We revisit the high energy semi-classical description of the exclusive
processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman
$x$ dependence in both the hard and the hadronic matrix elements. This is
achieved by a modification of the standard shock wave approximation to derive
the effective Feynman rules, which leads to a generic expression on which we
then perform a partial twist expansion to get rid of quantities suppressed by
the proper physical scales. We obtain a compact factorized master formula that
can be used to investigate the Bjorken limit at leading twist. In particular,
we recover the full one-loop result in the collinear limit for pure gluon
exchange with the target. Finally, we discuss the subtleties in taking the
simultaneous collinear and small $x$ limit.
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