Sum rules for interaction of resonances with Zbπ

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