Triangle singularities in ${\overline{B}}^{0}\ensuremath{\rightarrow}{\ensuremath{\chi}}_{c1}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ relevant to ${Z}_{1}(4050)$ and ${Z}_{2}(4250)$

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Triangle singularities in ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ relevant to $Z_{1} (4050)$ and $Z_{2} (4250)$

Satoshi X. Nakamura

Phys. Rev. D 100, 011504(R) – Published 25 July 2019

Abstract

$Z_{1} (4050)$ and $Z_{2} (4250)$ observed in ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ by the Belle Collaboration are candidates of charged charmoniumlike states that minimally include 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 $χ_{c 1} π^{+}$ invariant mass distribution resonancelike 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 $X (3872) π^{+}$ channel near the triangle-singularity energy. This characteristic lineshape could be used to discriminate different interpretations of $Z_{1} (4050)$ . An interesting prediction from interpreting $Z_{1} (4050)$ and $Z_{2} (4250)$ as the triangle singularities is that similar bumps caused by the same mechanisms possibly appear also in ${\bar{B}}^{0} \to J / ψ K^{-} π^{+}$ data; the already observed $Z_{c} (4200)$ corresponds to $Z_{2} (4250)$ of $J^{P} = 1^{+}$ .

Received 14 April 2019

DOI:https://doi.org/10.1103/PhysRevD.100.011504

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Hadron-hadron interactions Quantum chromodynamics Quark model

Exotic mesons

Particles & Fields

Authors & Affiliations

Satoshi X. Nakamura ^1,2,*

¹University of Science and Technology of China, Hefei 230026, People’s Republic of China
²State Key Laboratory of Particle Detection and Electronics (IHEP-USTC), Hefei 230036, People’s Republic of China

^*satoshi@ustc.edu.cn

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Vol. 100, Iss. 1 — 1 July 2019

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Figure 1
Triangle diagrams contributing to ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ (a,b). The triangle singularity from the diagram (a) [(b)] generates a $Z_{1} (4050)$ [ $Z_{2} (4250)$ ]-like bump in the $χ_{c 1} π^{+}$ invariant mass distribution. A generic triangle diagram (c) defines particle labels and, in the parentheses, their momenta.
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Figure 2
$χ_{c 1} π^{+}$ invariant mass distributions for ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ . The red (blue and magenta) solid curve in the panel (a) [(b) and (c)] is obtained from the triangle diagram A [B]. The interactions $v_{a b; 23}$ of Eqs. (2), (3), and (4) are used for (a), (b), and (c), respectively. The green dash-dotted curves are from Breit-Wigner amplitudes fitted to the solid curves. The dotted curves are the phase-space distributions. The solid and dotted curves are normalized to give unity when integrated with respect to $m_{χ_{c 1} π}$ . The scale for the panels (b) and (c) has been doubled.
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Figure 3
Cutoff ( $Λ$ ) dependence of the spectrum shapes generated by triangle diagrams. The panels (a) and (b) correspond to the triangle diagrams A and B, respectively. The red and blue solid curves are the same as those in Fig. 2 and 2, respectively, where $Λ = 1 GeV$ . The green dashed, and black dash-dotted curves are obtained with $Λ = 1.5$ , and 2 GeV, respectively. All the curves are normalized as in Fig. 2.
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Figure 4
$χ_{c 1} π^{+}$ invariant mass distributions for ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ . The red, blue, and magenta solid curves in Figs. 2 are modified to include only the contributions in $1.0 {GeV}^{2} < m_{K^{-} π^{+}}^{2} < 1.75 {GeV}^{2}$ , and superimposed on the Belle data (Fig. 14 of Ref. [2]) from the same kinematical constraint. Each of the curves is multiplied by a constant factor and an incoherent constant background is added to fit the data.
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Figure 5
$χ_{c 1} π^{+}$ invariant mass distributions for ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ calculated with the triangle diagram A. The red solid, blue dashed, green dotted, and magenta dash-dotted curves are calculated using the $X (3872) π^{+}$ threshold energy smaller than the PDG value by 0, 50, 100, and 150 MeV, respectively; see the text for details. All these curves, being scaled, have the same peak height. The black dash-two-dotted curve is obtained from the red solid one by turning off the on-shell $X (3872) π^{+}$ contribution.
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Figure 6
Argand plots from the triangle diagrams A (left) and B (right), corresponding to the spectra of Fig. 2 and 2, respectively. Six curved segments belong to six bins equally-separating the range of $M_{B W} - Γ_{B W} \leq m_{χ_{c 1} π} \leq M_{B W} + Γ_{B W}$ ; $M_{B W} = 4042$ (4194) MeV and $Γ_{B W} = 97$ (278) MeV for triangle diagram A (B). $m_{χ_{c 1} π}$ increases counterclockwise.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Triangle singularities in ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ relevant to $Z_{1} (4050)$ and $Z_{2} (4250)$

Satoshi X. Nakamura

Phys. Rev. D 100, 011504(R) – Published 25 July 2019

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Physical Review D

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Triangle singularities in B¯0→χc1K−π+ relevant to Z1(4050) and Z2(4250)

Satoshi X. Nakamura

Phys. Rev. D 100, 011504(R) – Published 25 July 2019

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Triangle singularities in ${\bar{B}}^{0} \to χ_{c 1} K^{-} π^{+}$ relevant to $Z_{1} (4050)$ and $Z_{2} (4250)$