Sum rules for interaction of $Υ$ resonances with $Z_b π$

The strength of the amplitudes for the coupling between the bottomonium $Υ(nS)$ states, the bottomonium-like isovector resonances $Z_b$ and a pion, $Υ(nS) Z_b π$, is considered. These amplitudes describe the decays $Z_b \to Υ(nS) π$ for $n=1,\,2,\,3$, and the processes $Υ(nS) \to Z_b π$ for $n=5,\,6, \ldots$ with either $Z_b(10610)$ or $Z_b(10650)$. It is pointed out that analyticity and unitarity impose a sum rule for these couplings to each of the $Z_b$ resonances. With the currently available data it appears to be difficult or impossible to simultaneously satisfy the sum rules for the $Z_b(10610)$ and $Z_b(10650)$ resonances. This difficulty can be resolved if there is a considerable dissimilarity in the yield of the states $Z_b(10610) π$ and $Z_b(10650) π$ in the $e^+e^-$ annihilation at energies above the $Υ(5S)$ resonance.

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