Three-gluon decays of radially excited quarkonia $ψ(2S)$ and $Υ(2S)$ with both relativistic and QCD radiative corrections

Abstract

For radially excited heavy quarkonia $ V=ψ(2S) $ and $ Υ(2S) $, the nodal structure of their wave function renders the three-gluon decay $ V\to ggg $ acutely sensitive to relativistic corrections, presenting a longstanding challenge for reliable theoretical predictions. We perform a comprehensive analysis of these decays within the Bethe-Salpeter formalism, constructing analytic harmonic oscillator wave functions that explicitly incorporate the $2S$ nodal structure. Model-independent relations among polarized decay widths are derived from helicity-flip and phase-space symmetries. To obtain physically consistent results beyond $ q^2 $-order relativistic corrections, we introduce a concise phenomenological treatment that effectively incorporates partial higher-order contributions while preserving the correct low-momentum limit. Taking into account both relativistic and QCD radiative corrections, we compute $ Γ(V\to ggg) $, $ Γ(V\to e^+e^-) $, and their ratio $ R_V = Γ(V\to ggg)/Γ(V\to e^+e^-) $, which allows a clean extraction of the harmonic oscillator parameter $ β_V $, with the extracted values lying at the lower end of the ranges from typical phenomenological models. Our predictions for the branching ratios $ \mathcal{B}(V\to ggg) $ and $ \mathcal{B}(V\to e^+e^-) $ are in excellent agreement with experimental data. A striking feature is the distinct convergence behavior of the relativistic expansion: the leptonic width converges rapidly already at $ q^2 $ order, whereas the gluonic width converges far more slowly due to destructive interference induced by the nodal structure in the multi-gluon convolution. These results offer new insights into the dynamics of excited quarkonia and provide valuable constraints on non-perturbative descriptions of heavy-quark bound states.

Figures (1)

• Figure 1: Dependence of the ratio $R_{V}$ on the harmonic oscillator parameter $\beta_V$. The green curve shows our theoretical prediction. The yellow band represents the experimental value with its $1\sigma$ uncertainty. The intersection determines the extracted $\beta_V$ range.