Triangle singularity in $B^{-}\to K^{-}X(3872);X\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and the X(3872) mass

Abstract

We evaluate the contribution to the $X(3872)$ width from a triangle mechanism in which the $X$ decays into $D^{\ast 0}{\bar{D}}^0-cc$, then the $D^{\ast 0}({\bar{D}}^{\ast 0})$ decays into $D^0{\pi }^0$ (${\bar{D}}^0{\pi }^0$) and the $D^0{\bar{D}}^0$ fuse to produce ${\pi }^{+}{\pi }^{-}$. This mechanism produces an asymmetric peak from a triangle singularity in the ${\pi }^{+}{\pi }^{-}$ invariant mass with a shape very sensitive to the $X$ mass. We evaluate the branching ratios for a reaction where this effect can be seen in the $B^{-}\to K^{-}{\pi }^0{\pi }^{+}{\pi }^{-}$ reaction and show that the determination of the peak in the invariant mass distribution of ${\pi }^{+}{\pi }^{-}$ is all that is needed to determine the $X$ mass. Given the present uncertainties in the $X$ mass, which do not allow to know whether the $D^{\ast 0}{\bar{D}}^0$ state is bound or not, measurements like the one suggested here should be most welcome to clarify this issue.

Comments: 10 pages,10 figures