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

    J. H. Kuhn, A. A. Penin, and A. A. Pivovarov, Nucl. Phys. B,534: 356 (1998)

  • [2]

    M. Beneke, Y. Kiyo, and K. Schuller, Phys. Lett. B,658: 222 (2008)

  • [3]

    A. A. Penin and N. Zerf, JHEP,1404: 120 (2014)

  • [4]

    M. Beneke, Y. Kiyo, and K. Schuller, Nucl. Phys. B,714: 67 (2005)

  • [5]

    M. Beneke and A. Signer, Nucl. Phys. B,471: 233 (1999)

  • [6]

    A. M. Badalian, B. L. G. Bakker, and I. V. Danilkin, Phys. Atom. Nucl.,73: 138 (2010)

  • [7]

    P. Gonzalez, A. Valcarce, H. Garcilazoet al., Phys. Rev. D,68: 034007 (2003)

  • [8]

    B.-Q. Li and K.-D. Chao, Phys. Rev. D,79: 094004 (2009)

  • [9]

    D. Cinabroet al. (CLEO Collaboration), Phys. Lett.,340: 129 (1994)

  • [10]

    M. A. Sanchis-Lozano, Int. J. Mod. Phys. A,19: 2183 (2004)

  • [11]

    G. Adamset al. (CLEO Collaboration), Phys. Rev. Lett.,94: 012001 (2005)

  • [12]

    D. Bessonet al. (CLEO Collaboration), Phys. Rev. Lett.,98: 052002 (2007)

  • [13]

    D. Aloni, A. Efrati, Y. Grossmanet al., JHEP,1706: 019 (2017)

  • [14]

    P. Sanchezet al. (BABAR Collaboration), Phys. Rev. Lett.,104: 191801 (2010)

  • [15]

    R. Barbieri, R. Gatto, R. Kogerleret al., Phys. Lett. B,57: 455 (1975)

  • [16]

    L. Bergstrom, H. Snellman, and G. Tengstrand, Z. Phys.,4: 215 (1980)

  • [17]

    B. Niczyporuket al. (LENA Collaboration), Phys. Rev. Lett.,46: 92 (1981)

  • [18]

    E. J. Eichten and C. Quigg, Phys. Rev. D,49: 5845 (1994)

  • [19]

    E. Eichten, K. Gottfried, T. Kinoshitaet al., Phys. Rev. D,17: 3090 (1978)

  • [20]

    W. Buchmuller and S.-H. H. Tye, Phys. Rev. D,24: 132 (1981)

  • [21]

    C. Quigg and J. L. Rosner, Phys. Lett. B,71: 153 (1977)

  • [22]

    B. Colquhoun, R. J. Dowdall, C. T. H. Davieset al., Phys. Rev. D,91: 074514 (2015)

  • [23]

    A. Pineda, Nucl. Phys. B,494: 213 (1997)

  • [24]

    A. Pineda, Phys. Rev. D,66: 054022 (2002)

  • [25]

    M. Beneke, A. Signer, and V. A. Smirnov, Phys. Rev. Lett.,80: 2535 (1998)

  • [26]

    A. Pineda and A. Signer, Nucl. Phys. B,762: 67 (2007)

  • [27]

    M. Beneke, Y. Kiyo, and A. A. Penin, Phys. Lett. B,653: 53 (2007)

  • [28]

    B. A. Kniehl, A. A. Penin, V. A. Smirnovet al., Phys. Rev. Lett.,90: 212001 (2003), Erratum Phys. Rev. Lett.,91: 139903 (2003)

  • [29]

    P. Marquard, J. H. Piclum, D. Seidelet al., Nucl. Phys.,758: 144 (2006)

  • [30]

    M. Beneke, Y. Kiyo, P. Marquardet al., Phys. Rev. Lett.,112: 151801 (2014)

  • [31]

    P. Marquard, J. H. Piclum, D. Seidelet al., Phys. Rev. D,89: 034027 (2014)

  • [32]

    S. J. Brodsky and X. G. Wu, Phys. Rev. D,85: 034038 (2012)

  • [33]

    S. J. Brodsky and X. G. Wu, Phys. Rev. Lett.,109: 042002 (2012)

  • [34]

    M. Mojaza, S. J. Brodsky, and X. G. Wu, Phys. Rev. Lett.,110: 192001 (2013)

  • [35]

    S. J. Brodsky, M. Mojaza, and X. G. Wu, Phys. Rev. D,89: 014027 (2014)

  • [36]

    J. M. Shen, X. G. Wu, H. H. Maet al., JHEP,1506: 169 (2015)

  • [37]

    X. D. Huang, X. G. Wu, J. Zenget al., Eur. Phys. J. C,79: 650 (2019)

  • [38]

    G. T. Bodwin, Eric Braaten, and G. P. Lepage, Phys. Rev. D,51: 1125 (1995)

  • [39]

    G. T. Bodwin and A. Petrelli, Phys. Rev. D,66:094011 (2002), Erratum Phys. Rev. D,87: 039902 (2013)

  • [40]

    Z.-K. Geng, T. Wang, Y. Jianget al., Phys. Rev. D,99: 013006 (2019)

  • [41]

    X. H. Mo, C. Z. Yuan, and P. Wang, Phys. Rev. D,82: 077501 (2010)

  • [42]

    H.-F. Fu, X.-J. Chen, and G.-L. Wang, Phys. Lett. B,692: 312 (2010)

  • [43]

    E. E. Salpeter and H. A. Bethe, Phys. Rev.,84: 1232 (1951)

  • [44]

    E. E. Salpeter, Phys. Rev.,87: 328 (1952)

  • [45]

    C.-H. Chang, J.-K. Chen, X.-Q. Liet al., Commun. Theor. Phys.,43: 113 (2005)

  • [46]

    G.-L. Wang, Phys. Lett. B,633: 492 (2006)

  • [47]

    S. Godfrey and N. Isgur, Phys. Rev. D,32: 189 (1985)

  • [48]

    E. Laermann, F. Langhammer, I. Schmittet al., Phys. Lett. B,173: 437 (1986)

  • [49]

    D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D,79: 114029 (2009)

  • [50]

    M. Tanabashiet al. (Particle Data Group), Phys. Rev. D,98: 030001 (2018)

  • [51]

    N. Brambilla, E. Mereghetti, and A. Vairo, JHEP,0608: 039 (2006), Erratum JHEP,1104: 058 (2011)

  • [52]

    G. C. Donald, C. T. H. Davies, R. J. Dowdallet al., Phys. Rev. D,86: 094501 (2012)

  • [53]

    N. R. Soni, B. R. Joshi, R. P. Shahet al., Eur. Phys. J. C,78: 592 (2018)

  • [54]

    A. M. Badalian and B. L. G. Bakker, Phys. Rev. D,96: 014030 (2017)

  • [55]

    S. F. Radford and W. W. Repko, Phys. Rev. D,75: 074031 (2007)

  • [56]

    M. Shah, A. Parmar, and P. C. Vinodkumar, Phys. Rev. D,86: 034015 (2012)

  • [57]

    A. K. Rai, B. Patel, and P. C. Vinodkumar, Phys. Rev. C,78: 055202 (2008)

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