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

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.