Shedding light on the $ X(3930) $ and $ X(3960) $ states with the $B^- \to K^- J/\psi \omega $ reaction

We have studied the contribution of the state $X(3930)$, coming from the interaction of the $D \overline{D}$ and $D^{+}_s D^{-}_s$ channels, to the $B^- \to K^- J/\psi \omega $ decay. The purpose of this work is to offer a complementary tool to see if the $X(3930)$ state observed in the $D^+ D^-$ channel is the same or not as the $X(3960)$ resonance claimed by the LHCb collaboration from a peak in the $D^{+}_s D^{-}_s$ mass distribution around threshold. We present results for what we expect in the $J/\psi \omega $ mass distribution in the $B^- \to K^- J/\psi \omega $ decay and conclude that a clear signal should be seen around $3930\,\rm MeV$. At the same time, finding no extra resonance signal at $3960\,\rm MeV$ would be a clear indication that there is not a new state at $3960\,\rm MeV$, supporting the hypothesis that the near-threshold peaking structure peak in the $D^{+}_s D^{-}_s$ mass distribution is only a manifestation of a resonance below threshold.
Comment: 15 pages; 10 figures; new version with improved discussion related to available experimental results