Two-Loop QCD Corrections to C even Bottomonium Exclusive Decays to Double $J/\psi$

Abstract

In the framework of nonrelativistic QCD (NRQCD) factorization, we compute both the polarized and the unpolarized decay widths for the processes ${\eta }_b({\chi }_{bJ})\to J/\psi J/\psi$, accurate up to next-to-next-to-leading-order (NNLO) in ${\alpha }_s$. 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 ${\chi }_{b2}$, both $\mathcal{O}({\alpha }_s)$ and $\mathcal{O}({\alpha }_s^2)$ 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 ${\eta }_b$, ${\chi }_{b0}$, ${\chi }_{b1}$ and ${\chi }_{b2}$ 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 ${\eta }_b$, ${\chi }_{b0}$, and ${\chi }_{b1}$, respectively. In the phenomenological analysis, with the integrate luminosity $\mathcal{L}=100{\mathrm{fb}}^{-1}$, we expect about $(5-10)\times 10^3$ ${\eta }_b({\chi }_{bJ})\to J/\psi J/\psi \to \ell \bar{\ell }\ell \bar{\ell }$ events produced at the {\tt LHC}, thus it might be hopeful to search for these processes. On the other hand, there are less than $100$ ${\eta }_b({\chi }_{bJ})\to J/\psi J/\psi$ signal events at the B factory, so it seems the experimental measurements on these channels are quite challenging based on current dataset.

Comments: 17 pages, 3 figures, 4 tables; The affiliation and acknowledgement of the first author are updated