A graphical representation of gluonic operators

TL;DR

This paper introduces a graphical operator-diagram formalism to represent gluonic operators in $D$ dimensions and develops a systematic method to construct primitive (evanescent) operator bases up to length $\le 7$. By separating kinematic and color factors and encoding Lorentz indices as diagrammatic edges and $D$-indices as arrows, the authors identify primitive configurations via skeleton diagrams and a sink-based counting framework, enabling explicit enumeration of primitive operators. The work provides detailed length-by-length constructions (lengths 2–7) and demonstrates symmetry-based refinement of the primitive basis, with implications for SMEFT operator bases and multi-loop renormalization including evanescent contributions. The approach paves the way for scalable, structure-aware generation of gauge-invariant operator bases and suggests extensions to broader gauge theories and non-local operators.

Abstract

Composite local operators are central to effective field theories (EFTs), as they define interaction vertices in effective Lagrangians and play a fundamental role in investigating the structure of quantum field theories. The contribution of high-dimensional operators in the Standard Model Effective Field Theory (SMEFT) grows increasingly important as experimental precision improves at the Large Hadron Collider (LHC) and in future colliders. However, the number of operators increases very rapidly with dimension, making it extremely challenging to identify their complete set. In our previous work \cite{Jin:2020pwh}, we proposed a systematic method for generating gluonic operators using primitive operators. In this paper, we introduce a graphical representation of gluonic operators and, based on this representation, present a method to systematically construct primitive operators. Using this method, we derive primitive operators corresponding to gluonic operators of length 2 to length 7 in $D$-dimensions.

Figures (37)

• Figure 1: A vertex representating the DF-block $D_{12\cdots(n-2)}F_{n-1,n}$.
• Figure 2: The operator diagram of $F_{12}D_1F_{34}D_2F_{34}$.
• Figure 3: ($a$): DF rule 2 and ($b$): Bianchi identity in operator diagram language.
• Figure 4: A disconnected operator diagram.
• Figure 5: Different arrow configurations of a diagram.