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 $J^{PC} = 0^{++}/0^{-+}$ and three-gluon glueballs with $J^{PC} = 0^{++}/1^{+-}$. In total, we derive nearly one hundred ratios for these glueballs. Our results suggest that the $f_0(1710)$ and $η(2370)$ 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 $ππω$ and $K\bar{K}ϕ$ channels, and the three-gluon glueball with $J^{PC} = 1^{+-}$ in the $ππω$, $ππϕ$, and $K\bar{K}ϕ$ channels. This study enhances our understanding of the gluonic structure of exotic hadrons and will assist future experimental searches in high-energy physics.

💡 Research Summary

In this paper the authors employ the Fierz rearrangement—a technique rooted in the internal color and Lorentz symmetries of QCD—to study the strong decay patterns of pure gluonic bound states (glueballs) in a largely model‑independent fashion. They focus on two‑gluon glueballs with quantum numbers J^{PC}=0^{++} and 0^{-+}, and three‑gluon glueballs with J^{PC}=0^{++} and 1^{+-}. Starting from the interpolating currents J_{0}=g_{s}^{2}G^{i}{\mu\nu}G^{i\mu\nu} (scalar), η{0}=g_{s}^{2}G^{i}{\mu\nu}\tilde G^{i\mu\nu} (pseudoscalar), and the corresponding three‑gluon currents constructed with the antisymmetric f^{ijk} and symmetric d^{ijk} color tensors, the authors model the decay as the excitation of the gluon fields into vector quark–antiquark currents. By applying a single Fierz rearrangement in both color and Lorentz spaces for the two‑gluon case, and two successive rearrangements for the three‑gluon case, they rewrite the four‑quark operators as products of two mesonic currents (or three currents for the three‑gluon case). The color algebra uses identities such as λ^{i}{ab}λ^{i}{cd}=2δ{ad}δ_{cb}−(2/3)δ_{ab}δ_{cd}, while the Lorentz algebra employs the standard set of Dirac matrices (γ, σ, γ_{5}) to obtain the coefficients C_{1…4} and Dirac structures Γ_{1,2}.

The resulting decay amplitudes are then expressed in terms of well‑known meson decay constants (λ_{π}, f_{π}, f_{ρ}, f_{Tρ}, etc.). For the scalar two‑gluon glueball they compute relative branching ratios for eight channels (ππ, K\bar K, ηη, ρρ, σσ, ηη′, ωω, etc.) and find, for example, B(ρρ)/B(ππ)=6.49×10^{-2}. Comparing these ratios with experimental data, they conclude that the f_{0}(1710) matches the predicted pattern (K\bar K/ππ≈1.93) and is therefore a strong candidate for the 0^{++} two‑gluon glueball, whereas the f_{0}(1500) shows significant discrepancies, indicating a small gluonic component. For the pseudoscalar two‑gluon glueball they derive twenty‑four ratios and identify η(2370) as a promising candidate, especially because the channels ϕϕ, ωω, and ϕω are predicted to have large branching fractions.

The three‑gluon analysis is more involved: two successive Fierz rearrangements introduce additional color structures involving f^{ijk} and d^{ijk}. The authors calculate 27 relative branching ratios for the 0^{++} three‑gluon glueball (mass ≈4.2 GeV) and 29 ratios for the 1^{+-} state (mass ≈3.2 GeV). They propose that the 0^{++} three‑gluon glueball should be searched for in the ππ ω and K\bar K ϕ final states, while the 1^{+-} state is expected to appear prominently in ππ ω, ππ ϕ, and K\bar K ϕ channels.

Overall, the paper presents nearly one hundred relative branching ratios, demonstrating that the Fierz rearrangement provides a powerful, largely model‑independent framework for predicting glueball decay patterns. The results not only reinforce the gluonic nature of f_{0}(1710) and η(2370) but also give concrete experimental targets for ongoing and future high‑luminosity facilities such as BESIII, GlueX, LHCb, and PANDA. This work thus advances both theoretical understanding of exotic hadrons and practical strategies for their discovery.