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

Understand

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.