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 $J^P=1^+$.
Highlights
Z1ð4050Þ and Z2ð4250Þ (Xð4050Þ and Xð4250Þ in the Particle Data Group (PDG) notation [1]) were observed in the Belle experiment as resonancelike structures in the χc1πþ invariant mass distribution of B 0 → χc1K−πþ [2].1 It was not possible to determine the spin (J)-parity (P) of these states
While Z1ð4050Þ and Z2ð4250Þ have been interpreted as tetraquark states previously, we propose a completely different scenario based on a kinematical effect called the triangle singularity
We show that the asymmetric shape originates from an interplay between the triangle singularity and the opening of the Xð3872Þπþ channel near the triangle-singularity energy
Summary
Z1ð4050Þ and Z2ð4250Þ (Xð4050Þ and Xð4250Þ in the Particle Data Group (PDG) notation [1]) were observed in the Belle experiment as resonancelike structures in the χc1πþ invariant mass distribution of B 0 → χc1K−πþ [2].1 It was not possible to determine the spin (J)-parity (P) of these states. We propose a completely different interpretation of Z1ð4050Þ and Z2ð4250Þ Zcð4430Þ [32,33,34] and Zcð4200Þ [35], charged charmoniumlike state candidates, observed in B 0 → ψð2SÞK−πþ and J=ψK−πþ We successfully explained their properties (JP, mass, width, Argand plot) extracted in the experiments. We demonstrate that the triangle diagram A [B] creates a Z1ð4050Þ [Z2ð4250Þ]-like bump in the χc1πþ invariant mass (mχc1π) distribution of B 0 → χc1K−πþ. We clarify that the opening of the Xð3872Þπþ channel near the TS energy of mχc1π ∼ 4.02 GeV is responsible for it This characteristic bump shape could discriminate different interpretations of Z1ð4050Þ. A general formula for the decay amplitude is given by
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