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

We evaluate the contribution to the $X(3872)$ width from a triangle mechanism in which the $X$ decays into $D^{*0}\bar{D}^0 -cc$, then the $D^{*0} (\bar{D}^{*0})$ decays into $D^0 π^0$ ($\bar{D}^0 π^0$) and the $D^0 \bar{D}^0$ 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 $B^- \to K^- π^0 π^+ π^-$ 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 $D^{*0} \bar{D}^0$ state is bound or not, measurements like the one suggested here should be most welcome to clarify this issue.