The first full amplitude analysis of $B^+\to J/\psi \phi K^+$ with $J/\psi\to\mu^+\mu^$, $\phi\to K^+K^$ decays is performed with a data sample of 3 fb$^{1}$ of $pp$ collision data collected at $\sqrt{s}=7$ and $8$ TeV with the LHCb detector. The data cannot be described by a model that contains only excited kaon states decaying into $\phi K^+$, and four $J/\psi\phi$ structures are observed, each with significance over $5$ standard deviations. The quantum numbers of these structures are determined with significance of at least $4$ standard deviations. The lightest has mass consistent with, but width much larger than, previous measurements of the claimed $X(4140)$ state.
Distribution of $m_{ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi}$ for the data and the fit results with a model containing only $K^{*+}\rightarrow \phi K^+$ contributions. 
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Masses of kaon excitations obtained in the default amplitude fit to the LHCb data, shown as red points with statistical (thicker bars) and total (thinner bars) errors, compared with the predictions by GodfreyIsgur [41] (horizontal black lines) for the most likely spectroscopic interpretations labeled with $n{}^{2S+1}L_J$ (see the text). Experimentally established states are also shown with narrower solid blue boxes extending to $\pm1\sigma$ in mass and labeled with their PDG names [31]. Unconfirmed states are shown with dashed green boxes. The long horizontal red lines indicate the $\phi K$ mass range probed in $B^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi K^+$ decays. Decays of the $2{}^{3}{\rm P}_{0}$ state ($J^P=0^+$) to $\phi K^+$ are forbidden. 
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Distributions of (top left) $\phi K^+$, (top right) $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K^+$ and (bottom) $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi$ invariant masses for the $B^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi K^+$ candidates (black data points) compared with the results of the default amplitude fit containing eight $K^{*+}\rightarrow \phi K^+$ and five $X\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi$ contributions. The total fit is given by the red points with error bars. Individual fit components are also shown. 
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Animated gif made out of all figures. 
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Previous results related to the $X(4140)\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi$ mass peak. The number of reconstructed $B^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi K^+$ decays ($N_B$) is given if applicable. Significances ($\sigma$) correspond to numbers of standard deviations. Upper limits on the $X(4140)$ fraction of the total $B^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi K^+$ rate are at 90% confidence level. The statistical and systematic errors are added in quadrature and then used in the weights to calculate the averages, excluding unpublished results (shown in italics). 
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Results for significances, masses, widths and fit fractions (\rm FF) of the components included in the default amplitude model. The first (second) errors are statistical (systematic). Possible interpretations in terms of kaon excitation levels are given for the resonant $\phi K^+$ fit components. Comparisons with the previously experimentally observed kaon excitations [31] and $X\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \phi$ structures are also given. 
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Created on 19 April 2019.Citation count from INSPIRE on 24 April 2019.