Dispersive analysis of the isospin breaking in the $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}$ and $X(3872)\to J/\psi {\pi }^{+}{\pi }^0{\pi }^{-}$ decays

Abstract

We analyze the latest LHCb data on the ${\pi }^{+}{\pi }^{-}$ spectrum in the isospin-violating $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}$ decay, based on dispersion theory to deal with the $\pi \pi$ final state interactions. Additionally, the isospin breaking effects are properly introduced, allowing for a reliable and accurate extraction of the ratio, $R_X$, between the $X(3872)$ couplings to the $J/\psi \rho$ and $J/\psi \omega$ channels from the data. We find very good agreement with the LHCb data for the whole range of the ${\pi }^{+}{\pi }^{-}$ invariant mass, and $R_X$ is determined to be {$0.26\pm 0.03$}. Using this value, we make predictions for the ${\pi }^{+}{\pi }^0{\pi }^{-}$ mass distribution in the $X(3872)\to J/\psi {\pi }^{+}{\pi }^0{\pi }^{-}$ process, which is currently accessible by the BESIII Collaboration, and update a prediction for the pole positions of the isovector partner states of the $X(3872)$, $W_{c1}$, with $I(J^{PC})=1(1^{++})$.

Comments: 9 pages, 1 table and 2 figures. Version to be published in Phys. Rev. D; results updated with a mistake on the rho-omega mixing corrected and a new appendix explaining how the mixing was obtained