Improved measurement of Born cross sections for ${\chi }_{bJ}\omega$ and ${\chi }_{bJ}({\pi }^{+}{\pi }^{-}{\pi }^0{)}_{\mathrm{n}\mathrm{o}\mathrm{n}-\omega }$ ($J$ = 0, 1, 2) at Belle and Belle II

We study the processes ${\chi }_{bJ}\omega$ and ${\chi }_{bJ}({\pi }^{+}{\pi }^{-}{\pi }^0{)}_{\mathrm{n}\mathrm{o}\mathrm{n}-\omega }$ ($J$ = 0, 1, 2) at center-of-mass energies $\sqrt{s}$ from 10.73 to 11.02 GeV using a $142.5{\mathrm{f}\mathrm{b}}^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider; and at $\sqrt{s}\sim 10.75$ GeV using a $19.8{\mathrm{f}\mathrm{b}}^{-1}$ sample collected with Belle II at SuperKEKB. We find that the $\mathrm{{\rm Y}}(10753)$ state decays into ${\chi }_{bJ}\omega$ but not into ${\chi }_{bJ}({\pi }^{+}{\pi }^{-}{\pi }^0{)}_{\mathrm{n}\mathrm{o}\mathrm{n}-\omega }$, while the $\mathrm{{\rm Y}}(10860)$ state, in contrast, decays into ${\chi }_{bJ}({\pi }^{+}{\pi }^{-}{\pi }^0{)}_{\mathrm{n}\mathrm{o}\mathrm{n}-\omega }$ but not into ${\chi }_{bJ}\omega$. The mass and width of the $\mathrm{{\rm Y}}(10753)$ state are measured to be $(10756.1\pm 3.4(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 2.7(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))$ MeV/$c^2$ and $(32.2\pm 11.3(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 14.9(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))$ MeV. The products of the partial width to $e^{+}{e}^{-}$ and branching fractions for $\mathrm{{\rm Y}}(10753)\to {\chi }_{b1}\omega$ and $\mathrm{{\rm Y}}(10753)\to {\chi }_{b2}\omega$ are ($1.57\pm 0.27(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.22(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.})$) eV and ($1.39\pm 0.41(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.33(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.})$) eV.