The OPE Approach to Renormalization: Operator Mixing

TL;DR

This paper extends the operator product expansion (OPE) renormalization method to composite operators with mixing, providing recursive formulas and high-loop anomalous dimensions in scalar field theories.

Contribution

It introduces a recursive framework for renormalization of mixed operators using OPE coefficients and reports high-order anomalous dimensions in scalar models.

Findings

Derived five-loop anomalous dimensions for operators with Δ≤5 in φ^4 model.

Computed two-loop anomalous dimensions for operators with Δ≤10 in φ^3 model.

Demonstrated the efficiency of the OPE-based renormalization algorithm.

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.