A graphical representation of gluonic operators

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.