Fierz analyses on the decay properties of two- and three-gluon glueballs

The Fierz rearrangement, based on the various internal symmetries of hadrons, can be used to study their decay properties in a largely model-independent way. In this Letter we apply this method to calculate the relative branching ratios of two-gluon glueballs with $J^{PC}=0^{++}\mathrm{/}{0}^{-+}$ and three-gluon glueballs with $J^{PC}=0^{++}\mathrm{/}{1}^{+-}$. In total, we derive nearly one hundred ratios for these glueballs. Our results suggest that the $f_0(1710)$ and $\eta (2370)$ likely contain a significant gluon component, whereas the gluon component in $f_0(1500)$ appears to be small. Furthermore, we propose observing the three-gluon glueball with $J^{PC}=0^{++}$ in the $\pi \pi \omega$ and $K\bar{K}\phi$ channels, and the three-gluon glueball with $J^{PC}=1^{+-}$ in the $\pi \pi \omega$, $\pi \pi \phi$, and $K\bar{K}\phi$ channels. This study enhances our understanding of the gluonic structure of exotic hadrons and will assist future experimental searches in high-energy physics.