Observation of $e^+e^- \to \pi^+ \pi^- \pi^0 \chi_{bJ}$ and search for $X_b \to \omega \Upsilon(1S)$ at $\sqrt{s}\sim 10.867$ GeV

The $e^+e^- \to \pi^+ \pi^- \pi^0 \chi_{bJ}$ ($J=0,~1,~2$) processes are studied using a 118~fb$^{-1}$ data sample collected at a center-of-mass energy of 10.867 GeV, in the $\Upsilon(10860)$ energy range, with the Belle detector. The $\pi^+ \pi^- \pi^0 \chi_{b1}$, $\pi^+\pi^-\pi^0\chi_{b2}$, $\omega\chi_{b1}$ signals and the evidence of $\omega\chi_{b2}$ are observed for the first time and the cross sections are measured. No significant $\pi^+\pi^-\pi^0\chi_{b0}$ or $\omega\chi_{b0}$ signal is observed and 90\% confidence level upper limits on the cross sections for these two processes are obtained. In the $\pi^+\pi^-\pi^0$ invariant mass spectrum, significant non-$\omega$ signals are also observed. We search for the $X(3872)$-like state with a hidden $b\bar{b}$ component (named $X_b$) decaying into $\omega \Upsilon(1S)$; no significant signal is observed with a mass between $10.55$ and $10.65$ GeV/$c^2$.