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

TL;DR

This work computes the $5$-loop anomalous dimensions of the $φ^Q$ operator in the six-dimensional cubic scalar theory by combining an OPE-based renormalization algorithm with the graphical function method. The authors transform complex multi-leg operator insertions into two-point propagator-type integrals using a $Φ$-field representation and derive $Z$-factors from UV finiteness of Wilson coefficients, enabling high-loop renormalization without sub-divergence subtractions. They achieve explicit $5$-loop results for the $φ^Q$ operator and provide the large-$N$ expansion of the Wilson-Fisher fixed-point scaling dimensions up to $1/N^5$, with fixed-point data consistent with prior literature and semi-classical predictions. The results establish a new loop-order benchmark and demonstrate the method’s versatility for high-precision renormalization in quantum field theories, with potential extensions to higher loops and to non-scalar theories.

Abstract

In this work, we compute the anomalous dimensions of the $φ^Q$ 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 $φ^Q$ 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 $φ^Q$ operator in cubic scalar theory, while further validating the efficiency and versatility of the proposed algorithm in renormalization analyses.

Figures (5)

• Figure 1: Examples of coupling factors in the $\Phi$-field representations and their corresponding component field representations. The internal $\Phi$ fields represent all possible internal state configurations in component fields.
• Figure 2: The 1-loop and 2-loop 1PI topologies of two-point functions in $\Phi$-field representation.
• Figure 3: The 1-loop and 2-loop 1PI topologies of three-point functions in $\Phi$-field representation. A gray arrow points to the cut graphs of each topology. They are generated by the non-isomorphic computation of all possible removals of one external leg.
• Figure 4: The tree-level, 1-loop and 2-loop 1PI topologies of $\varphi^2$-$2\varphi$ correlation function, where the $_\otimes$ in subscript denotes a $\phi^Q$ operator. A gray arrow points to the cut graphs of each topology. They are generated by the non-isomorphic computation of all possible removals of one external leg. The UV divergences are computed from these cut graphs.
• Figure 5: The tree-level, 1-loop and 2-loop irreducible 1PI topologies of $\varphi^3$-$3\varphi$ correlation function. The topologies in the first row are equivalent to those corresponding to the $\varphi^2$-$2\varphi$ correlation function. There are only 2 new irreducible topologies shown in the second row. A gray arrow points to the cut graphs of each new irreducible topology, which are generated by the non-isomorphic computation of all possible removals of two external legs.