Resonant structures in $B^0\to\psi'\pi^K^+$ decays are analyzed by performing a fourdimensional fit of the decay amplitude, using $pp$ collision data corresponding to $\rm 3 fb^{1}$ collected with the LHCb detector. The data cannot be described with $K^+\pi^$ resonances alone, which is confirmed with a modelindependent approach. A highly significant $Z(4430)^\to\psi'\pi^$ component is required, thus confirming the existence of this state. The observed evolution of the $Z(4430)^$ amplitude with the $\psi'\pi^$ mass establishes the resonant nature of this particle. The mass and width measurements are substantially improved. The spinparity is determined unambiguously to be $1^+$.
Backgroundsubtracted and efficiencycorrected $m_{\psi'\pi^}$ distribution (black data points), superimposed with the reflections of $\cos\theta_{K^*}$ moments up to order four allowing for $J(K^*)\le 2$ (blue line) and their correlated statistical uncertainty (yellow band bounded by blue dashed lines). The distributions have been normalized to unity. 
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Distributions of the fit variables (black data points) together with the projections of the 4D fit. The red solid (brown dashed) histogram represents the total amplitude with (without) the $ Z_1^$ . The other points illustrate various subcomponents of the fit that includes the $ Z_1^$ : the upper (lower) blue points represent the $ Z_1^$ component removed (taken alone). The orange, magenta, cyan, yellow, green, and red points represent the $K^*(892)$, total $S$wave, $K^*(1410)$, $K^*(1680)$, $K^*_2(1430)$ and background terms, respectively. 
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Fitted values of the $ Z_1^$ amplitude in six $m_{\psi'\pi^}^2$ bins, shown in an Argand diagram (connected points with the error bars, $m_{\psi'\pi^}^2$ increases counterclockwise). The red curve is the prediction from the BreitWigner formula with a resonance mass (width) of 4475 (172) $\mathrm{ Me V}$ and magnitude scaled to intersect the bin with the largest magnitude centered at (4477 MeV)$^2$. Units are arbitrary. The phase convention assumes the helicityzero $K^*(892)$ amplitude to be real. 
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Distribution of $m_{\psi'\pi^}^2$ in the data (black points) for $1.0<m_{K^+\pi^}^2<1.8$ GeV$^2$ ($K^*(892)$, $K^*_2(1430)$ veto region) compared with the fit with two, $0^$ and $1^+$ (solidline red histogram) and only one $1^+$ (dashedline green histogram) $Z^$ resonances. Individual $Z^$ terms (blue points) are shown for the fit with two $Z^$ resonances. 
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Animated gif made out of all figures. 
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Created on 07 August 2020.