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Abstract: The evolution of parton densities, which are crucial for making quantitative predictions in all high-energy hadron collider processes, is governed by the splitting functions. To consistently determine N3LO parton densities, which are becoming increasingly important with the inclusion of several matching coefficients at N3LO, it is necessary to know the 4-loop splitting functions. To make the computation possible and save computational resources, finding an efficient method to compute 4-loop splitting functions is of great importance. The method of computing the off-shell matrix elements with a twist-two operator insertion is among the most efficient techniques. Nonetheless, the off-shell nature of external gluons poses renormalization issues related to gauge invariance. Addressing these concerns entails identifying all gauge-variant operators that mix with the physical twist-two operators.
We recently introduced a new framework in which we systematically extract all gauge-variant operator Feynman rules. We applied this framework to rederive the unpolarized singlet splitting functions up to the three-loop order, and we demonstrated that our approach is valid to all loop orders. Furthermore, our framework enables the derivation of all-n operator Feynman rules, which makes the calculation of splitting functions with all-n dependence easier.
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