Independent Operators at Different Dimension

TL;DR

This paper systematically identifies independent operators for lattice QCD calculations of glueball spectra across different dimensions, and analyzes their group representations to clarify the nature of certain glueball candidates.

Contribution

It introduces a method to find all independent operators and their group representations for lattice QCD, aiding in accurate glueball spectrum calculations.

Findings

Identified all independent operators for $SO(3)^{PC}$ at various dimensions.

Decomposed these operators into irreducible representations of $O^{PC}$.

Argued that $f_J(2220)$ and $g_T$ cannot be tensor glueballs simultaneously.

Abstract

To apply lattice QCD in the calculation of glueball spectrum it is needed firstly to know associated operators acting on vacuum. We show how to find all the independent representations and operators, of group $SO(3)^{PC}$ at different dimension, since the work is not trivial. Then, we decompose these representation into irreducible representation of $O^{PC}$ group, which are listed in the note. At last we argue that $f_J(2220)$ and $g_T$ states can not be tensor glueball simultaneously.