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

We evaluate the contribution to the X(3872) width from a triangle mechanism in which the X decays into D${}^{\text{â}\text{ˆ}\text{—}0}$${\overline{D}}^0$ - cc, then the D${}^{\text{â}\text{ˆ}\text{—}0}$(${\overline{D}}^{\text{â}\text{ˆ}\text{—}0}$) decays into D${}^0$π${}^0$ (${\overline{D}}^0$π${}^0$) and the D${}^0$${\overline{D}}^0$ fuse to produce π${}^{+}$π${}^{-}$. This mechanism produces an asymmetric peak from a triangle singularity in the π${}^{+}$π${}^{-}$ 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${}^{-}$ → K${}^{-}$π${}^0$π${}^{+}$π${}^{-}$ reaction and show that the determination of the peak in the invariant mass distribution of π${}^{+}$π${}^{-}$ 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${}^{\text{â}\text{ˆ}\text{—}0}$${\overline{D}}^0$ state is bound or not, measurements like the one suggested here should be most welcome to clarify this issue.