# Revisiting the Heavy Vector Quarkonium Leptonic Widths

## TL;DR

This paper reevaluates heavy quarkonium leptonic decay widths using the Bethe-Salpeter method, emphasizing relativistic corrections, which are significant especially for excited states, and finds results consistent with experimental data.

## Contribution

It provides a detailed analysis of relativistic effects on heavy quarkonium leptonic decays using the Bethe-Salpeter approach, highlighting their importance for excited states.

## Key findings

- Relativistic effects are substantial, reaching up to 62% for certain states.
- Results for Υ(nS) decays align with experimental measurements.
- Relativistic corrections increase with excitation level in heavy quarkonium.

## Abstract

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.