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Within the nonrelativistic QCD (NRQCD) factorization framework, we computethe order-$\alpha_s^3$ perturbative corrections to the leptonic decay of the$\Upsilon$ and $J/\psi$ with high numerical accuracy, at the lowest order invelocity expansion. We confirm the existing three-loop results in literature.Furthermore, we explicitly consider the complex-valued light-by-light (singlet)contributions. We for the first time also consider the finite charm quark masseffect for $\Upsilon$ leptonic decay, and obtain a new piece of 3-loopcontribution to the anomalous dimension related to the composite NRQCD bilinearof vector current that arises from the nonzero charm quark mass. Based on thecomplete three-loop NRQCD short-distance coefficients, we also present acomprehensive phenomenological analysis for $\Upsilon(J/\psi)$ leptonic width.

Complete three-loop QCD corrections to leptonic width of vector quarkonium