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

Abstract

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