Charged charmonium-like Z+(4430) from rescattering in conventional B decays

In a previous paper we suggested an explanation for the peak designated as $Z(4430)^+$ in the $\psi^\prime\pi^+$ mass spectrum, observed by Belle in ${\bar{\!B}} \to {\psi^\prime} \pi^+ K$ decays, as an effect of $\bar{\!D}{}^{\,*0}D^{\,+} \to \psi^\prime\pi^+$ rescattering in the decays ${\bar{\!B}} \to {D_{s}^{\,\prime -}} {D}$, where the $D_{s}^{\,\prime -}$ is an as-yet unobserved radial excitation of the pseudoscalar ground state $D_{s}^{\,-}$-meson. In this paper, we demonstrate that this hypothesis provides an explanation of the double $Z^+$-like peaking structures, which were studied by LHCb with much higher statistics. While according to our hypothesis, the origin of the peaking structures is purely kinematical, reflecting the presence of a conventional resonance in the hidden intermediate state, the amplitude of the $Z(4430)^+$ peak carries a Breit-Wigner-like complex phase, arising from the intermediate $D_{s}^{\,\prime -}$ resonance. Thus, our hypothesis is entirely consistent with the recent LHCb measurement of the resonant-like amplitude behaviour of the $Z(4430)^+$. We perform a toy fit to the LHCb data, which illustrates that our approach is also consistent with all the observed structure in the LHCb $M({{\psi^\prime}\pi^+})$ spectrum. We suggest a critical test of our hypothesis that can be performed experimentally.