${\cal O}(\alpha_sv^2)$ corrections to the hadronic decay of vector quarkonia

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the O(αsv2) corrections to the hadronic decay rate of vector quarkonia, exemplified by J/ψ and ϒ. Setting both the renormalization and NRQCD factorization scales to be mQ, we obtain Γ(J/ψ→LH)=0.0716αs3mc2⟨O1(S31)⟩J/ψ[1-1.19αs+(-5.32+3.03αs)⟨v2⟩J/ψ] and Γ(ϒ→LH)=0.0716αs3mb2⟨O1(S31)⟩ϒ[1-1.56αs+(-5.32+4.61αs)⟨v2⟩ϒ]. We confirm the previous calculation of O(αs) corrections on a diagram-by-diagram basis, with the accuracy significantly improved. For J/ψ hadronic decay, we find that the O(αsv2) corrections are moderate and positive, nevertheless unable to counterbalance the huge negative corrections. On the other hand, the effect of O(αsv2) corrections for ϒ(nS) is sensitive to the O(v2) NRQCD matrix elements. With the appropriate choice of the NRQCD matrix elements, our theoretical predictions for the decay rates may be consistent with the experimental data for ϒ(1S,2S)→LH. As a by-product, we also present the theoretical predictions for the branching ratio of J/ψ(ϒ)→3γ accurate up to O(αsv2).