Revisiting the Heavy Vector Quarkonium Leptonic Widths

We revisit the heavy quarkonium leptonic decays $\psi (nS)\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and $\mathrm{{\rm Y}}(nS)\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ using the Bethe-Salpeter method. The emphasis is on the relativistic correction. For the $\psi (1S-5S)$ decays, the relativistic effects are $22_{-2}^{+3}\mathrm{\%}$, $34_{-5}^{+5}\mathrm{\%}$, $41_{-6}^{+6}\mathrm{\%}$, $52_{-13}^{+11}\mathrm{\%}$ and $62_{-12}^{+14}\mathrm{\%}$, respectively. For the $\mathrm{{\rm Y}}(1S-5S)$ decays, the relativistic effects are $14_{-2}^{+1}\mathrm{\%}$, $23_{-3}^{+0}\mathrm{\%}$, $20_{-2}^{+8}\mathrm{\%}$, $21_{-7}^{+6}\mathrm{\%}$ and $28_{-7}^{+2}\mathrm{\%}$, respectively. Thus, the relativistic corrections are large and important in heavy quarkonium leptonic decays, especially for the highly excited charmonium. Our results for $\mathrm{{\rm Y}}(nS)\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ are consistent with the experimental data.