  The dynamical cascade of momentum, spin, charge, and other quantum numbers
from an ultra-violet process into the infra-red is a fundamental concern for
asymptotically free or conformal gauge field theories. It is also a practical
concern for any high energy scattering experiment with energies above tens of
GeV. We present a formulation of the evolution equation that governs this
cascade, the Banfi-Marchesini-Smye equation, from both an effective field
theory point of view and a direct diagrammatic argument. The equation uses
exact momentum conservation, and is applicable to both scattering with initial
and final state hard partons. The direct diagrammatic formulation is organized
by constructing a generating functional. This functional is also automatically
realized with soft wilson lines and collinear field operators coupled to
external currents. The two approaches are directly connected by reverse
engineering the Lehman-Symanzik-Zimmermann reduction procedure to insert states
within the soft and collinear matrix elements. At leading order, the cascade is
completely controlled by the soft anomalous dimension. By decomposing the
anomalous dimension into on-shell and off-shell regions as would be realized in
the effective field theory approach with a Glauber mediating potential, we are
forced to choose a transverse momentum ordering in order to trivialize the
overlap between Glauber potential contributions and the pure soft region. The
evolution equation then naturally incorporates factorization violating effects
driven by off-shell exchanges for active partons. Finally, we examine the
consequences of abandoning exact momentum conservation as well as terminating
the evolution at the largest inclusive scale, procedures often used to simplify
the analysis of the cascade.
