Probabilistic derivation of parton splitting functions

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Abstract
We show that the parton splitting functions defined by Altarelli and Parisi can be derived in a completely probabilistic manner. In this approach, the apparent infrared divergences in the Altarelli-Parisi calculation cancel between gain and loss terms in the splitting functions, and momentum and quark quantum number conservation are satisfied automatically without the imposition of extra constraints. We discuss the relation of the probabilistic approach to the usual leading-logarithm diagrammatic analysis.