Four-loop Anomalous Dimensions of Scalar-QED Theory from Operator Product Expansion

TL;DR

This paper computes the four-loop anomalous dimensions of the fixed-charge operator in scalar-QED using the Operator Product Expansion, extending previous three-loop results and validating the OPE algorithm's effectiveness.

Contribution

It introduces a four-loop calculation of operator anomalous dimensions in scalar-QED and proposes a new graph decomposition method for loop integrand construction.

Findings

Four-loop anomalous dimension of $\,\phi^Q$ operator computed.

Validation of the OPE algorithm for higher-loop renormalization.

New graph decomposition technique for integrand derivation.

Abstract

We apply the Operator Product Expansion (OPE) algorithm to the renormalization of scalar-QED theory, with a specific focus on the fixed-charge operator $\phi^Q$. Within the OPE framework, the anomalous dimension of the $\phi^Q$ operator is perturbatively computed to four-loop order in the modified minimal subtraction scheme, extending beyond the previously available three-loop result. The beta functions, as well as the mass and field anomalous dimensions, are also computed at this order. An alternative loop-integrand construction method is proposed, based on graph decomposition and skeleton expansion techniques, for deriving the integrands of one-Particle-Irreducible correlation functions. This work represents the first non-trivial validation of the OPE algorithm for higher-loop renormalization beyond pure scalar theories. The present successful computations further confirm the efficiency and versatility of the OPE algorithm in renormalization analysis.