Five-loop Anomalous Dimensions of Cubic Scalar Theory from Operator Product Expansion

TL;DR

This paper calculates the five-loop anomalous dimensions of the $\,\phi^Q$ operator in six-dimensional cubic scalar theory using advanced operator product expansion and graphical function methods, extending previous loop order records.

Contribution

It introduces a systematic algorithm and computational tools for high-loop anomalous dimension calculations in quantum field theories, demonstrated by the five-loop result.

Findings

Computed five-loop anomalous dimensions of $\,\phi^Q$ operator.

Extended large N expansion to order 1/N^5.

Validated the efficiency of the new computational algorithm.

Abstract

In this work, we compute the anomalous dimensions of the $\phi^Q$ operator in six-dimensional cubic scalar theory. The renormalization analysis is carried out within the framework of the Operator Product Expansion method, while the ultraviolet divergences of Feynman integrals are evaluated using the graphical function method. Inspired by the intrinsic connection between Wilson coefficients and anomalous dimensions, an algorithm was proposed recently, which provides a practical and systematic framework for calculating the anomalous dimensions of masses, fields, and composite operators, with broad potential applicability to generic quantum field theories. Notably, the HyperlogProcedures package, developed based on the graphical function method, enables the computation of two-point propagator-type integrals, derived herein for capturing ultraviolet divergences, to very high loop orders. With these advanced techniques, we have successfully computed the anomalous dimensions of the $\phi^Q$ operator up to five loops. Furthermore, we present a large $N$ expansion of the scaling dimensions at the Wilson-Fisher fixed point, extended to the $1/N^5$ order. This computation sets a new loop-order record for the anomalous dimension of the $\phi^Q$ operator in cubic scalar theory, while further validating the efficiency and versatility of the proposed algorithm in renormalization analyses.