Hunting for $X_b$ via hidden bottomonium decays $X_b\to \pi\pi\chi_{bJ}$

In this work, we investigate the isospin breaking decay $X_b \to \pi^0\chi_{bJ}$ and the isospin conserved decay $X_b \to \pi\pi\chi_{bJ}$, where $X_b$ is taken to be the heavy quark flavor symmetry counterpart of $X(3872)$ in the bottomonium sector as a $B^*\bar B$ molecule candidate. Since the mass of this state may be far below the $B {\bar B}^*$ threshold and the mass difference between the neutral and charged bottom meson is small compared to the binding energy of the $X_b$, the isospin-violating decay channel $X_b \to \pi^0 \chi_{bJ}$ would be highly suppressed. The calculated partial width of $X_b \to \pi\pi\chi_{b1}$ is found to be about tens of $\rm{keVs}$, $1\sim 2$ order(s) of magnitude larger than those of $X_b \to \pi\pi\chi_{b2}$ and $X_b \to \pi\pi\chi_{b0}$. Taking into account the fact that the total width of $X_b$ may be smaller than a few MeV like $X(3872)$, the calculated branching ratios $X_b\to \pi\pi \chi_{b1}$ may reach to orders of $10^{-2}$, which makes it a possible channel for the experimental searching of the $X_b$.