Next-to-next-to-leading-order QCD corrections to hadronic width of pseudoscalar quarkonium

Abstract

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the hadronic decay rates of the pseudoscalar quarkonia, at the lowest order in velocity expansion. The validity of NRQCD factorization for inclusive quarkonium decay process, for the first time, is verified to relative order $α_s^2$. As a byproduct, the renormalization group equation (RGE) of the leading NRQCD 4-fermion operator ${\cal O}_1({}^1S_0)$ is also deduced to this perturbative order. By incorporating this new piece of correction together with available relativistic corrections, we find that there exists severe tension between the state-of-the-art NRQCD predictions and the measured $η_c$ hadronic width, and in particular the branching fraction of $η_c\to γγ$. NRQCD appears to be capable of accounting for $η_b$ hadronic decay to a satisfactory degree, and our most refined prediction is ${\rm Br}(η_b\to γγ) = (4.8\pm 0.7)\times 10^{-5}$.