The OPE Approach to Renormalization: Operator Mixing
Operator mixing is a generic and unavoidable feature in realistic quantum field theories, including QCD and effective field theories. Yet existing methods for computing anomalous dimensions in the presence of mixing remain limited by intricate sub-divergence subtractions and escalating computational complexity. In this work, we extend the recently developed OPE-based renormalization algorithm to systematically handle operator mixing. The key innovation is a recursive framework built on a strategically chosen basis of lower-dimensional symmetric traceless tensor operators as ``soft'' operators, which uniquely and fully determines the $Z$-factors of higher-dimensional ``hard'' operators from their lower-dimensional counterparts. This approach entirely eliminates the need for sub-divergence subtraction that burdens traditional methods such as the $R^{*}$ operation, thereby achieving a substantial reduction in computational complexity. As the first application of this framework, we obtain the five-loop anomalous dimensions for operators with $\Delta\leq 5$ in the $\phi^{4}$ model and the two-loop anomalous dimensions for operators with $\Delta\leq 10$ in the $\phi^{3}$ model.