$\mathcal{O}({\alpha }_s{v}^2)$ corrections to hadronic decay of vector quarkonia

Abstract

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the $\mathcal{O}({\alpha }_s{v}^2)$ corrections to the hadronic decay rate of vector quarkonia, exemplified by $J/\psi$ and ${\rm Y}$. Setting both the renormalization and NRQCD factorization scales to be $m_Q$, we obtain $\Gamma (J/\psi \to \mathrm{LH})=0.0716\frac{{\alpha }_s^3}{m_c^2}⟨{\mathcal{O}}_1({}^3{S}_1){⟩}_{J/\psi }[1-1.19{\alpha }_s+(-5.32+3.03{\alpha }_s)⟨v^2{⟩}_{J/\psi }]$ and $\Gamma ({\rm Y}\to \mathrm{LH})=0.0716\frac{{\alpha }_s^3}{m_b^2}⟨{\mathcal{O}}_1({}^3{S}_1){⟩}_{{\rm Y}}[1-1.56{\alpha }_s+(-5.32+4.61{\alpha }_s)⟨v^2{⟩}_{{\rm Y}}]$. We confirm the previous calculation of $\mathcal{O}({\alpha }_s)$ corrections on a diagram-by-diagram basis, with the accuracy significantly improved. For $J/\psi$ hadronic decay, we find that the $\mathcal{O}({\alpha }_s{v}^2)$ corrections are moderate and positive, nevertheless unable to counterbalance the huge negative corrections. On the other hand, the effect of $\mathcal{O}({\alpha }_s{v}^2)$ corrections for ${\rm Y}(nS)$ is sensitive to the $\mathcal{O}(v^2)$ 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 ${\rm Y}(1S,2S)\to \mathrm{LH}$. As a byproduct, we also present the theoretical predictions for the branching ratio of $J/\psi ({\rm Y})\to 3\gamma$ accurate up to $\mathcal{O}({\alpha }_s{v}^2)$.

arXiv comment21 pages, 5 figures, 3 tables

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