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.
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