Exploring the $Υ(6S)\to χ_{bJ}ϕ$ and $Υ(6S)\to χ_{bJ}ω$ hidden-bottom hadronic transitions

Abstract

In this work, we investigate the hadronic loop contributions to the $Υ(6S) \to χ_{bJ} ϕ~(J=0,1,2)$ along with $Υ(6S) \to χ_{bJ} ω~(J=0,1,2)$ transitions. We predict that the branching ratios of $Υ(6S) \to χ_{b0} ϕ$, $Υ(6S) \to χ_{b1} ϕ$ and $Υ(6S) \to χ_{b2} ϕ$ are $(0.68 \sim 4.62) \times 10^{-6}$, $(0.50 \sim 3.43) \times 10^{-6}$ and $(2.22 \sim 15.18) \times 10^{-6}$, respectively and those of $Υ(6S) \to χ_{b0} ω$, $Υ(6S) \to χ_{b1} ω$ and $Υ(6S) \to χ_{b2} ω$ are $(0.15 \sim 2.81) \times 10^{-3}$, $(0.63 \sim 11.68) \times 10^{-3}$ and $(1.08 \sim 20.02) \times 10^{-3}$, respectively. Especially, some typical ratios, which reflect the relative magnitudes of the predicted branching ratios, are given, i.e., for $Υ(6S)\to χ_{bJ}ϕ$ transitions, $\mathcal{R}^ϕ_{10}={\mathcal{B}[Υ(6S) \to χ_{b1} ϕ]}/{\mathcal{B}[Υ(6S) \to χ_{b0} ϕ]} \approx 0.74$, $\mathcal{R}^ϕ_{20}= {\mathcal{B}[Υ(6S) \to χ_{b2} ϕ]}/{\mathcal{B}[Υ(6S) \to χ_{b0} ϕ]} \approx 3.28$, and $\mathcal{R}^ϕ_{21} = {\mathcal{B}[Υ(6S) \to χ_{b2} ϕ]}/{\mathcal{B}[Υ(6S) \to χ_{b1} ϕ]} \approx 4.43$, and for $Υ(6S)\to χ_{bJ}ω$ transitions, $\mathcal{R}^ω_{10}={\mathcal{B}[Υ(6S) \to χ_{b1} ω]}/{\mathcal{B}[Υ(6S) \to χ_{b0} ω]} \approx 4.11$, $\mathcal{R}^ω_{20}= {\mathcal{B}[Υ(6S) \to χ_{b2} ω]}/{\mathcal{B}[Υ(6S) \to χ_{b0} ω]} \approx 7.06$, and $\mathcal{R}^ω_{21} = {\mathcal{B}[Υ(6S) \to χ_{b2} ω]}/{\mathcal{B}[Υ(6S) \to χ_{b1} ω]} \approx 1.72$. With the running of BelleII in the near future, experimental measurement of these two kinds of transitions will be a potential research issue.