Triangle singularity appearing as an $X(3872)$-like peak in $B\to (J/\psi {\pi }^{+}{\pi }^{-})K\pi$

Abstract

We consider a triangle diagram for $B^0\to (J/\psi {\pi }^{+}{\pi }^{-})K^{+}{\pi }^{-}$ where an $X(3872)$ peak has been observed experimentally. We demonstrate that a triangle singularity inherent in the triangle diagram creates a sharp peak in the $J/\psi {\pi }^{+}{\pi }^{-}$ invariant mass distribution when the final $(J/\psi {\pi }^{+}{\pi }^{-})\pi$ invariant mass is at and around the $D^{\ast }{\bar{D}}^{\ast }$ threshold. The position and width of the peak is 3871.68 MeV (a few keV above the $D^{\ast 0}{\bar{D}}^0$ threshold) and $\sim$0.4 MeV, respectively, in perfect agreement with the precisely measured $X(3872)$ mass and width: $3871.69\pm 0.17$ MeV and $<1.2$ MeV. This remarkable agreement is virtually parameter-free. The result indicates that the considered mechanism has to be understood in advance when separating an $X(3872)$-pole contribution from $B^0\to (J/\psi {\pi }^{+}{\pi }^{-})K^{+}{\pi }^{-}$ data; the separation yields an $X(3872)\pi$ lineshape that could be used to determine the $X(3872)$ mass. We suggest a method to set a constraint on the triangle mechanism by analyzing a charge analogous process $B^0\to (J/\psi {\pi }^0{\pi }^{-})K^{+}{\pi }^0$ where a similar triangle singularity generates an $X^{-}(3876)$-like peak.

Comments: 9 pages, 7 figures; In v2, conclusion changed, new contents and figures added