arXiv2023
In the framework of nonrelativistic QCD (NRQCD) factorization, we compute both the polarized and the unpolarized decay widths for the processes $η_b(χ_{bJ})\to J/ψJ/ψ$, accurate up to next-to-next-to-leading-order (NNLO) in $α_s$. For the first time, we confirm that the NRQCD factorization does hold at NNLO for the process involving triple quarkonia. We find the radiative corrections are considerable. In particular for $χ_{b2}$, both $\mathcal{O}(α_s)$ and $\mathcal{O}(α_s^2)$ corrections are sizable and negative, and can significantly reduce the leading order prediction. At NNLO, the branching fractions are $8.2\times 10^{-7}$, $6.2\times 10^{-6}$, $7.2\times 10^{-7}$ and $2.7\times 10^{-6}$ for $η_b$, $χ_{b0}$, $χ_{b1}$ and $χ_{b2}$ decay, respectively. Our theoretical predictions are consistent with the upper limits measured by the {\tt Belle} Collaboration. Moreover, we investigate the dependence of the theoretical predictions on the ratio of the charm quark mass and the bottom quark mass. By fixing $m_b$ and varying $m_c$ from $1.25$ to $1.9$ GeV, we find the branching fraction can change a factor of $2$, $3$, and $6$ for $η_b$, $χ_{b0}$, and $χ_{b1}$, respectively. In the phenomenological analysis, with the integrate luminosity $\mathcal{L}=100\,{\rm fb}^{-1}$, we expect about $(5-10)\times 10^3$ $η_b(χ_{bJ})\to J/ψJ/ψ\to \ell \bar{\ell}\ell \bar{\ell}$ events produced at the {\tt LHC}, thus it might be hopeful to search for these processes. On the other hand, there are less than $100$ $η_b(χ_{bJ})\to J/ψJ/ψ$ signal events at the B factory, so it seems the experimental measurements on these channels are quite challenging based on current dataset.