Radiative leptonic decay of heavy quarkonia -

\begin{document}$ \bar{Q} $\end{document} at the leading-order level within the non-relativistic quantum chromodynamics (NRQCD) framework, where Q represents either a charm or bottom quark. The branching ratios for the radiative leptonic decays

\begin{document}$ X\rightarrow \gamma l^{+} l^{-} $\end{document}

are revisited, and the angular and energy/momentum distributions of the final state particles are analyzed in the rest frame of X. Furthermore, we apply Lorentz transformations from the rest frame of X to the center-of-mass frame of

\begin{document}$ l^+ l^- $\end{document}

to establish the connection between the widths

\begin{document}$ {\Gamma_{X \rightarrow \gamma l^{+} l^{-}}} $\end{document}

and

\begin{document}$ {\Gamma_{X \rightarrow l^{+} l^{-}}} $\end{document}

. The comparison of the connection to those documented in literature (divided by

\begin{document}$ 2\pi $\end{document}

) for various X states, such as

\begin{document}$ J/\Psi $\end{document}

,

\begin{document}$ \Psi(2S) $\end{document}

,

\begin{document}$ \Upsilon(1S) $\end{document}

, and

\begin{document}$ \Upsilon(2S) $\end{document}

, shows relative differences typically around or below 10%, comparable to the next-to-leading order corrections of

\begin{document}$ O(\alpha) $\end{document}

and

\begin{document}$ O(v^4) $\end{document}

. However, we observe a significant disparity in the ratio between

\begin{document}$ {\Gamma_{\Psi(2S) \to \gamma \tau^+ \tau^-}} $\end{document}

and

\begin{document}$ {\Gamma_{\Psi(2S) \to \tau^+ \tau^-}} $\end{document}

, with our prediction being four times larger than those in literature. The outcomes derived from this study have practical implications in describing the quantum electrodynamics radiative processes and contribute to the investigation of QCD processes associated with the decays of heavy quarkonia and searches for new physics."> Radiative leptonic decay of heavy quarkonia -

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