Revisiting the Heavy Vector Quarkonium Leptonic Widths

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