An application of diagrammatic renormalization to 2++ tensor di-gluonium

In this paper, we apply the diagrammatic renormalization method to the NLO analysis of the [Formula: see text] tensor di-gluonium channel within the Quantum chromodynamics (QCD) sum-rules approach. Diagrammatic renormalization eliminates nonlocal divergences directly, avoiding the construction of renormalization factors and complications arising from operator mixing in the conventional renormalization method. The local divergences in QCD correlation functions contribute only to subtraction terms in dispersion relations in QCD sum-rules, making it particularly well-suited for diagrammatic renormalization as the local divergences do not enter sum-rules analysis. We provide a detailed example of renormalizing a representative NLO diagram and perform a comprehensive comparison of all nonzero NLO diagrams for [Formula: see text] tensor di-gluonium treated with both diagrammatic and conventional operator-mixing methods. The results from both approaches are in agreement, confirming the validity of diagrammatic renormalization. By simplifying the renormalization process, the diagrammatic renormalization method offers a practical alternative for higher-loop analysis of gluonium states and extensions to multi-quark systems.