Sum rules for interaction of $\Upsilon$ resonances with $Z_b \pi$

The strength of the amplitudes for the coupling between the bottomonium $\Upsilon(nS)$ states, the bottomonium-like isovector resonances $Z_b$ and a pion, $\Upsilon(nS) Z_b \pi$, is considered. These amplitudes describe the decays $Z_b \to \Upsilon(nS) \pi$ for $n=1,\,2,\,3$, and the processes $\Upsilon(nS) \to Z_b \pi$ 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 $Z_b(10610) \pi$ and $Z_b(10650) \pi$ in the $e^+e^-$ annihilation at energies above the $\Upsilon(5S)$ resonance.