Analysis of the Quantum Numbers<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>J</mml:mi><mml:mrow><mml:mi>P</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msup></mml:math>of the<i>X</i>(3872) Particle

We present an analysis of angular distributions and correlations of the $X(3872)$ particle in the exclusive decay mode $X(3872)\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ with $J/\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. We use $780\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of data from $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the CDF II detector at the Fermilab Tevatron. We derive constraints on spin, parity, and charge conjugation parity of the $X(3872)$ particle by comparing measured angular distributions of the decay products with predictions for different ${J}^{PC}$ hypotheses. The assignments ${J}^{PC}={1}^{++}$ and ${2}^{\ensuremath{-}+}$ are the only ones consistent with the data.

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