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

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the order-$α_s^3$ perturbative corrections to the leptonic decay of the $Υ$ and $J/ψ$ with high numerical accuracy, at the lowest order in velocity 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 mass effect for $Υ$ leptonic decay, and obtain a new piece of 3-loop contribution to the anomalous dimension related to the composite NRQCD bilinear of vector current that arises from the nonzero charm quark mass. Based on the complete three-loop NRQCD short-distance coefficients, we also present a comprehensive phenomenological analysis for $Υ(J/ψ)$ leptonic width.