The OPE Approach to Renormalization: Operator Mixing

Abstract

We extend the OPE-based renormalization algorithm to composite operators with operator mixing, focusing on scalar operators in $\phi^4$ and $\phi^3$ models. Using the OPE of operators with a fundamental field, we show that the $Z$-factors of these composite operators are determined by OPE coefficients of lower-dimensional traceless symmetric tensor operators, and establish a recursive renormalization framework. We report the five-loop anomalous dimensions for operators with $\Delta\le5$ in the $\phi^4$ model and the two-loop anomalous dimensions for operators with $\Delta\le10$ in the $\phi^3$ model. These results further demonstrate the versatility and efficiency of the OPE-based algorithm.