The OPE Approach to Renormalization: Operator Mixing

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 $\mathrm{\Delta }\le 5$ in the ${\phi }^4$ model and the two-loop anomalous dimensions for operators with $\mathrm{\Delta }\le 10$ in the ${\phi }^3$ model. These results further demonstrate the versatility and efficiency of the OPE-based algorithm.