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Splitting functions govern the scale evolution of parton distributionfunctions. Through a Mellin transformation, they are related to anomalousdimensions of twist-two operators in the operator product expansion. We studyoff-shell operator matrix element, where the physical operators mix underrenormalization with other gauge-variant operators of the same quantum numbers.We devise a new method to systematically extract the Feynman rules resultingfrom those operators without knowing the operators themselves. As a firstapplication of the new approach, we independently reproduce the well-knownthree-loop singlet splitting functions obtained from computations of on-shellquantities.

Renormalization of twist-two operators in QCD and its application to singlet splitting functions