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

TL;DR

This paper analyzes three-gluon decays of radially excited heavy quarkonia $(2S)$ and $(2S)$, incorporating relativistic and QCD radiative corrections, and compares predictions with experimental data to deepen understanding of quarkonium dynamics.

Contribution

It introduces a comprehensive Bethe-Salpeter approach with harmonic oscillator wave functions and a phenomenological method to include higher-order effects, improving decay width predictions.

Findings

Predicted decay ratios agree with experimental data.

Leptonic widths converge rapidly with relativistic expansion, gluonic widths converge slowly.

Extracted harmonic oscillator parameters are at the lower end of typical ranges.

Abstract

For radially excited heavy quarkonia $ V=\psi(2S) $ and $ \Upsilon(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 $ \Gamma(V\to ggg) $, $ \Gamma(V\to e^+e^-) $, and their ratio $ R_V = \Gamma(V\to ggg)/\Gamma(V\to e^+e^-) $, which allows a clean extraction of the harmonic oscillator parameter $ \beta_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.