Improved measurement of Born cross sections for $χ_{bJ}\,ω$ and $χ_{bJ}\,(π^+π^-π^0)_{\rm non-ω}$ ($J$ = 0, 1, 2) at Belle and Belle II

Abstract: We study the processes $\chi_{bJ}\,\omega$ and $$\chi_{bJ}\,(\pi^{+\pi^{-\pi^0)_{\rm}} non-\omega}$$ ($J$ = 0, 1, 2) at center-of-mass energies $\sqrt{s}$ from 10.73--11.02 GeV using a $142.5\,\mathrm{fb}<sup>{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e⁺ e<sup>-$ collider; and at $\sqrt{s}\sim10.75$ GeV using a $19.8\,\mathrm{fb}<sup>{-1}$ sample collected with Belle II at SuperKEKB. We find that the $\Upsilon(10753)$ state decays into $\chi_{bJ}\,\omega$ but not into $$\chi_{bJ}\,(\pi^{+\pi^{-\pi^0)_{\rm}} non-\omega}$$, while the $\Upsilon(10860)$ state, in contrast, decays into $$\chi_{bJ}\,(\pi^{+\pi^{-\pi^0)_{\rm}} non-\omega}$$ but not into $\chi_{bJ}\,\omega$. The mass and width of the $\Upsilon(10753)$ state are measured to be $(10756.1\pm3.4({\rm stat.})\pm2.7({\rm syst.}))$ MeV/$c<sup>2$ and $(32.2\pm11.3({\rm stat.})\pm14.9({\rm syst.}))$ MeV. The products of the partial width to $e<sup>+e<sup>-$ and branching fractions for $\Upsilon(10753)\to\chi_{b1}\,\omega$ and $\Upsilon(10753)\to\chi_{b2}\,\omega$ are ($1.46\pm0.25({\rm stat.})\pm 0.20({\rm syst.})$) eV and ($1.29\pm0.38({\rm stat.})\pm 0.31({\rm syst.})$) eV.