A Study of the Angular Properties of the $X(3872) \rightarrow J/\psi \pi^{+} \pi^{-}$ Decay

A study of the angular properties of the decay $B^\pm \rightarrow K^\pm X(3872)(\rightarrow J/\psi \pi^+ \pi^-)$ is presented with the aim of measuring the $X(3872)$ $J^{PC}$ quantum numbers. The expected angular distributions in the hypotheses of the $X(3872)$ as a $J^{PC}$ $1^{++}$ or $2^{-+}$ particle are calculated. A simulation has been performed and is in agreement with the calculation. The cosine of the angle between the $X(3872)$ and the $J/\psi$ momenta in their respective mother's rest frames shows strong differences between the two hypotheses. This is peculiar to the decay chain $B^\pm \rightarrow K^\pm X(3872)(\rightarrow J/\psi \pi^+ \pi^-)$, and can be exploited by the LHCb experiment to measure the $X(3872)$ $J^{PC}$ quantum numbers. A possible experimental approach and some preliminary expectations are shown for this measurement.