An Application of Diagrammatic Renormalization to $2^{++}$ Tensor Di-Gluonium

We apply the diagrammatic renormalization method to the NLO analysis of the $2^{++}$ tensor di-gluonium channel within the QCD sum-rules approach. Diagrammatic renormalization eliminates non-local 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 non-zero NLO diagrams for $2^{++}$ 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.