Triangle singularities in ${\bar{B}}^0\to {\chi }_{c1}{K}^{-}{\pi }^{+}$ relevant to $Z_1(4050)$ and $Z_2(4250)$

Abstract

$Z_1(4050)$ and $Z_2(4250)$ observed in ${\bar{B}}^0\to {\chi }_{c1}{K}^{-}{\pi }^{+}$ by the Belle Collaboration are candidates of charged charmonium-like states that minimally includes two quarks and two antiquarks. While $Z_1(4050)$ and $Z_2(4250)$ have been interpreted as tetraquark states previously, we propose a completely different scenario based on a kinematical effect called the triangle singularity. We demonstrate that the triangle singularities cause in the ${\chi }_{c1}{\pi }^{+}$ invariant mass distribution resonance-like bumps that fit very well the Belle data. If these bumps are simulated by the $Z_1(4050)$ and $Z_2(4250)$ resonance excitations, the spin-parity of them are predicted to be $1^{-}$ for $Z_1(4050)$ and $1^{+}$ or $1^{-}$ for $Z_2(4250)$. The bump corresponding to $Z_1(4050)$ has a highly asymmetric shape, which the Belle data exactly indicate. We show that the asymmetric shape originates from an interplay between the triangle singularity and the opening of the $X(3872){\pi }^{+}$ channel near the triangle-singularity energy. This characteristic lineshape could be used to discriminate different interpretations of $Z_1(4050)$. An interesting prediction from interpretting $Z_1(4050)$ and $Z_2(4250)$ as the triangle singularities is that similar bumps caused by the same mechanisms possibly appear also in ${\bar{B}}^0\to J/\psi K^{-}{\pi }^{+}$ data; the already observed $Z_c(4200)$ corresponds to $Z_2(4250)$ of $J^P=1^{+}$.

Comments: 6 pages, 5 figures, (v2) numerical results corrected; changes are small, discussion on B0 -> J/psi K pi added; (v3) published version: relativistic energy-momentum relation is used for all particles; study on cutoff-dependence added