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

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 $I(J^{PC})=1(1^{++})$.