Angular correlations in $B^+\to X(3872) K^+$ decays, with $X(3872)\to \rho^0 J/\psi$, $\rho^0\to\pi^+\pi^$ and $J/\psi \to\mu^+\mu^$, are used to measure orbital angular momentum contributions and to determine the $J^{PC}$ value of the $X(3872)$ meson. The data correspond to an integrated luminosity of 3.0 fb$^{1}$ of protonproton collisions collected with the LHCb detector. This determination, for the first time performed without assuming a value for the orbital angular momentum, confirms the quantum numbers to be $J^{PC}=1^{++}$. The $X(3872)$ is found to decay predominantly through S wave and an upper limit of $4\%$ at $95\%$ C.L. is set on the fraction of D wave.
Distribution of $\Delta M$ for $B^+\rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K^+\pi^+\pi^$ candidates. The fit of the $X(3872)$ signal is displayed. The solid (blue), dashed (red), and dotted (green) lines represent the total fit, signal component, and background component, respectively. 
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Likelihoodweighted distribution of the Dwave fraction. The distribution is normalized to unity. 
dfracL.pdf [17 KiB] HiDef png [136 KiB] Thumbnail [47 KiB] *.C file 

Distributions of the test statistic $t \equiv 2\ln[{\cal L}(J_X^{\rm alt}))/{\cal L}(1^{++})]$, for the simulated experiments under the $J^{PC}=J_X^{\rm alt}$ hypothesis (blue solid histograms) and under the $J^{PC}=1^{++}$ hypothesis (red dashed histograms). The values of the test statistics for the data, $t_{\rm data}$, are shown by the solid vertical lines. 
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Backgroundsubtracted distributions of all angles for the data (points with error bars) and for the $1^{++}$ fit projections (solid histograms). 
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Backgroundsubtracted distribution of $\cos\theta_X$ for candidates with $\cos\theta_{\rho}>0.6$ for the data (points with error bars) compared to the expected distributions for various $X(3872)$ $J^{PC}$ assignments (solid histograms) with the $B_{LS}$ amplitudes obtained by the fit to the data in the fivedimensional angular space. The fit displays are normalized to the observed number of the signal events in the full angular phase space. 
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Animated gif made out of all figures. 
PAPER2015015.gif Thumbnail 
Parityallowed $LS$ couplings in the $X(3872)\rightarrow \rho^0 { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} $ decay. For comparison, we also list a subset of these couplings corresponding to the lowest $L$, used in the previous determinations [14,16,17] of the $X(3872)$ quantum numbers. 
Table_1.pdf [44 KiB] HiDef png [109 KiB] Thumbnail [17 KiB] tex code 
Created on 09 December 2018.Citation count from INSPIRE on 18 December 2018.