Sum rules for interaction of ${\rm Y}$ resonances with $Z_b\pi$

Abstract

The strength of the amplitudes for the coupling between the bottomonium ${\rm Y}(nS)$ states, the bottomonium-like isovector resonances $Z_b$ and a pion, ${\rm Y}(nS)Z_b\pi$, is considered. These amplitudes describe the decays $Z_b\to {\rm Y}(nS)\pi$ for $n=1,2,3$, and the processes ${\rm Y}(nS)\to Z_b\pi$ for $n=5,6,\dots$ 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 ${\rm Y}(5S)$ resonance.

Comments: 8 pages. Corrected an omission in the original version