# Coupling Hadron-Hadron Thresholds within a Chiral Quark Model Approach

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Naive Chiral Quark Model

## 3. The ${}^{\mathbf{3}}{\mathbf{P}}_{\mathbf{0}}$ Model

## 4. The Unquenched Quark Model

## 5. Coupled Channel Effects

#### 5.1. Isospin Breaking Effects

#### 5.2. HQSS and HFS Breaking

#### 5.3. Threshold Cusps

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HQQS | Heavy Quark Spin Symmetry |

HFS | Heavy Flavor Symmetry |

$\chi $PT | Chiral Perturbation Theory |

QCD | Quantum Chromodynamics |

## References

- Aubert, J.J.; Becker, U.; Biggs, P.J.; Burger, J.; Chen, M.; Everhart, G.; Goldhagen, P.; Leong, J.; McCorriston, T.; Rhoades, T.G.; et al. Experimental Observation of a Heavy Particle J. Phys. Rev. Lett.
**1974**, 33, 1404–1406. [Google Scholar] [CrossRef] [Green Version] - Augustin, J.E.; Boyarski, A.M.; Breidenbach, M.; Bulos, F.; Dakin, J.T.; Feldman, G.J.; Fischer, G.E.; Fryberger, D.; Hanson, G.; Jean-Marie, B.; et al. Discovery of a Narrow Resonance in e
^{+}e^{−}Annihilation. Phys. Rev. Lett.**1974**, 33, 1406–1408. [Google Scholar] [CrossRef] [Green Version] - Glashow, S.L.; Iliopoulos, J.; Maiani, L. Weak Interactions with Lepton-Hadron Symmetry. Phys. Rev. D
**1970**, 2, 1285–1292. [Google Scholar] [CrossRef] - Herb, S.W.; Hom, D.C.; Lederman, L.M.; Sens, J.C.; Snyder, H.D.; Yoh, J.K.; Appel, J.A.; Brown, B.C.; Brown, C.N.; Innes, W.R.; et al. Observation of a Dimuon Resonance at 9.5 GeV in 400-GeV Proton-Nucleus Collisions. Phys. Rev. Lett.
**1977**, 39, 252–255. [Google Scholar] [CrossRef] - Kelly, R.L.; Horne, C.P.; Losty, M.J.; Rittenberg, A.; Shimada, T.; Trippe, T.G.; Wohl, C.G.; Yost, G.P.; Barash-Schmidt, N.; Bricman, C.; et al. Review of particle properties. Rev. Mod. Phys.
**1980**, 52, S1–S286. [Google Scholar] [CrossRef] [Green Version] - Hagiwara, K.; Montanet, L.; Barnett, R.M.; Groom, D.E.; Trippe, T.G.; Wohl, C.G.; Armstrong, B.; Wagman, G.S.; Murayama, H.; Stone, J.; et al. Review of Particle Properties. Phys. Rev. D
**2002**, 66, 010001. [Google Scholar] [CrossRef] [Green Version] - Particle Data Group; Zyla, P.A.; Barnett, R.M.; Beringer, J.; Dahl, O.; Dwyer, D.A.; Groom, D.E.; Lin, C.J.; Lugovsky, K.S.; Pianori, E.; et al. Review of Particle Physics. Prog. Theor. Exp. Phys.
**2020**, 2020, 083C01. [Google Scholar] [CrossRef] - Cazzoli, E.G.; Cnops, A.M.; Connolly, P.L.; Louttit, R.I.; Murtagh, M.J.; Palmer, R.B.; Samios, N.P.; Tso, T.T.; Williams, H.H. Evidence for ΔS=−ΔQ Currents or Charmed-Baryon Production by Neutrinos. Phys. Rev. Lett.
**1975**, 34, 1125–1128. [Google Scholar] [CrossRef] - Eichten, E.; Gottfried, K.; Kinoshita, T.; Lane, K.D.; Yan, T.M. Charmonium: The Model. Phys. Rev. D
**1978**, 17, 3090–3117, Erratum in**1980**, 21, 313. [Google Scholar] [CrossRef] - Eichten, E.; Gottfried, K.; Kinoshita, T.; Lane, K.D.; Yan, T.M. Charmonium: Comparison with Experiment. Phys. Rev. D
**1980**, 21, 203–233. [Google Scholar] [CrossRef] - Gupta, S.N.; Radford, S.F.; Repko, W.W. Semirelativistic Potential Model for Charmonium. Phys. Rev. D
**1985**, 31, 160. [Google Scholar] [CrossRef] [PubMed] - Barnes, T.; Close, F.E.; Page, P.R.; Swanson, E.S. Higher quarkonia. Phys. Rev. D
**1997**, 55, 4157–4188. [Google Scholar] [CrossRef] [Green Version] - Ebert, D.; Faustov, R.N.; Galkin, V.O. Properties of heavy quarkonia and B
_{c}mesons in the relativistic quark model. Phys. Rev. D**2003**, 67, 014027. [Google Scholar] [CrossRef] [Green Version] - Eichten, E.; Godfrey, S.; Mahlke, H.; Rosner, J.L. Quarkonia and their transitions. Rev. Mod. Phys.
**2008**, 80, 1161–1193. [Google Scholar] [CrossRef] [Green Version] - Danilkin, I.V.; Simonov, Y.A. Channel coupling in heavy quarkonia: Energy levels, mixing, widths and new states. Phys. Rev. D
**2010**, 81, 074027. [Google Scholar] [CrossRef] [Green Version] - Ferretti, J.; Santopinto, E. Higher mass bottomonia. Phys. Rev. D
**2014**, 90, 094022. [Google Scholar] [CrossRef] [Green Version] - Godfrey, S.; Moats, K. Bottomonium Mesons and Strategies for their Observation. Phys. Rev. D
**2015**, 92, 054034. [Google Scholar] [CrossRef] [Green Version] - Caswell, W.E.; Lepage, G.P. Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories. Phys. Lett. B
**1986**, 167, 437–442. [Google Scholar] [CrossRef] - Pineda, A.; Soto, J. Effective field theory for ultrasoft momenta in NRQCD and NRQED. Nucl. Phys. B Proc. Suppl.
**1998**, 64, 428–432. [Google Scholar] [CrossRef] [Green Version] - Brambilla, N.; Pineda, A.; Soto, J.; Vairo, A. Effective Field Theories for Heavy Quarkonium. Rev. Mod. Phys.
**2005**, 77, 1423. [Google Scholar] [CrossRef] [Green Version] - Dudek, J.J.; Edwards, R.G.; Mathur, N.; Richards, D.G. Charmonium excited state spectrum in lattice QCD. Phys. Rev. D
**2008**, 77, 034501. [Google Scholar] [CrossRef] [Green Version] - Gray, A.; Allison, I.; Davies, C.T.H.; Dalgic, E.; Lepage, G.P.; Shigemitsu, J.; Wingate, M. The Upsilon spectrum and m(b) from full lattice QCD. Phys. Rev. D
**2005**, 72, 094507. [Google Scholar] [CrossRef] [Green Version] - Meinel, S. The Bottomonium spectrum from lattice QCD with 2+1 flavors of domain wall fermions. Phys. Rev. D
**2009**, 79, 094501. [Google Scholar] [CrossRef] [Green Version] - Gottfried, K. The spectroscopy of massive quark-antiquark systems, Progress in Particle and Nuclear Physics. Prog. Part. Nucl. Phys.
**1982**, 8, 49–71. [Google Scholar] [CrossRef] - Gostev, V.B.; Mineev, V.S.; Frenkin, A.R. The inverse problem of quantum mechanics for a linear potentia. Theor. Math. Phys.
**1983**, 56, 682–686. [Google Scholar] [CrossRef] - Sumino, Y. QCD potential as a “Coulomb-plus-linear” potential. Phys. Lett. B
**2003**, 571, 173–183. [Google Scholar] [CrossRef] [Green Version] - Mateu, V.; Ortega, P.G.; Entem, D.R.; Fernández, F. Calibrating the Naïve Cornell Model with NRQCD. Eur. Phys. J. C
**2019**, 79, 323. [Google Scholar] [CrossRef] - Copley, L.A.; Isgur, N.; Karl, G. Charmed baryons in a quark model with hyperfine interactions. Phys. Rev. D
**1979**, 20, 768–775. [Google Scholar] [CrossRef] - Stanley, D.P.; Robsen, D. Do Quarks Interact Pairwise and Satisfy the Color Hypothesis? Phys. Rev. Lett.
**1980**, 45, 235–238. [Google Scholar] [CrossRef] - Choi, S.K.; Olsen, S.L.; Abe, K.; Abe, T.; Adachi, I.; Ahn, B.S.; Aihara, H.; Akai, K.; Akatsu, M.; Akemoto, M.; et al. Observation of a Narrow Charmoniumlike State in Exclusive B
^{±}→K^{±}π^{+}π^{−}J/ψ Decays. Phys. Rev. Lett.**2003**, 91, 262001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Acosta, D.; Affolder, T.; Ahn, M.H.; Akimoto, T.; Albrow, M.G.; Ambrose, D.; Amerio, S.; Amidei, D.; Anastassov, A.; Anikeev, K.; et al. Observation of the Narrow State X(3872)→J/ψπ
^{+}π^{−}in $\overline{p}p$ Collisions at $\sqrt{s}$ = 1.96 TeV. Phys. Rev. Lett.**2004**, 93, 072001. [Google Scholar] [CrossRef] [Green Version] - Abazov, V.M.; Abbott, B.; Abolins, M.; Acharya, B.S.; Adams, D.L.; Adams, M.; Adams, T.; Agelou, M.; Agram, J.L.; Ahmed, S.N.; et al. Observation and Properties of the X(3872) Decaying to J/ψπ
^{+}π^{−}in $p\overline{p}$ Collisions at $\sqrt{s}$ = 1.96 TeV. Phys. Rev. Lett.**2004**, 93, 162002. [Google Scholar] [CrossRef] [Green Version] - Aubert, B.; Barate, R.; Boutigny, D.; Couderc, F.; Gaillard, J.M.; Hicheur, A.; Karyotakis, Y.; Lees, J.P.; Tisserand, V.; Zghiche, A.; et al. Study of the B
^{−}→J/ψK^{−}π^{+}π^{−}decay and measurement of the B^{−}→X(3872)K^{−}branching fraction. Phys. Rev. D**2005**, 71, 071103. [Google Scholar] [CrossRef] [Green Version] - Aaij, R.; Adeva, B.; Adinolfi, M.; Affolder, A.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; et al. Observation of J/ψp Resonances Consistent with Pentaquark States in ${\Lambda}_{b}^{0}$→J/ψK
^{−}p Decays. Phys. Rev. Lett.**2015**, 115, 072001. [Google Scholar] [CrossRef] [Green Version] - Aaij, R.; Beteta, C.A.; Adeva, B.; Adinolfi, M.; Aidala, C.A.; Ajaltouni, Z.; Akar, S.; Albicocco, P.; Albrecht, J.; Alessio, F.; et al. Observation of a Narrow Pentaquark State, P
_{c}(4312)^{+}, and of the Two-Peak Structure of the P_{c}(4450)^{+}. Phys. Rev. Lett.**2019**, 122, 222001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Manohar, A.; Georgi, H. Chiral quarks and the non-relativistic quark model. Nucl. Phys. B
**1984**, 234, 189–212. [Google Scholar] [CrossRef] - Fernandez, F.; Valcarce, A.; Straub, U.; Faessler, A. The nucleon-nucleon interaction in terms of quark degrees of freedom. J. Phys. G Nucl. Part. Phys.
**1993**, 19, 2013–2026. [Google Scholar] [CrossRef] - Vijande, J.; Fernández, F.; Valcarce, A. Constituent quark model study of the meson spectra. J. Phys. G Nucl. Part. Phys.
**2005**, 31, 481–506. [Google Scholar] [CrossRef] [Green Version] - Burgio, G.; Schröck, M.; Reinhardt, H.; Quandt, M. Running mass, effective energy, and confinement: The lattice quark propagator in Coulomb gauge. Phys. Rev. D
**2012**, 86, 014506. [Google Scholar] [CrossRef] [Green Version] - Bali, G.S. QCD forces and heavy quark bound states. Phys. Rep.
**2001**, 343, 1–136. [Google Scholar] [CrossRef] [Green Version] - Bali, G.S.; Neff, H.; Düssel, T.; Lippert, T.; Schilling, K. Observation of string breaking in QCD. Phys. Rev. D
**2005**, 71, 114513. [Google Scholar] [CrossRef] [Green Version] - De Rújula, A.; Georgi, H.; Glashow, S.L. Hadron masses in a gauge theory. Phys. Rev. D
**1975**, 12, 147–162. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Yasser, A.M.; Entem, D.R.; Fernández, F. J
^{PC}=1^{−−}hidden charm resonances. Phys. Rev. D**2008**, 78, 114033. [Google Scholar] [CrossRef] - Segovia, J.; Entem, D.R.; Fernandez, F.; Hernandez, E. Constituent quark model description of charmonium phenomenology. Int. J. Mod. Phys. E
**2013**, 22, 1330026. [Google Scholar] [CrossRef] [Green Version] - Kamimura, M. Nonadiabatic coupled-rearrangement-channel approach to muonic molecules. Phys. Rev. A
**1988**, 38, 621–624. [Google Scholar] [CrossRef] [PubMed] - Hiyama, E.; Kino, Y.; Kamimura, M. Gaussian expansion method for few-body systems. Prog. Part. Nucl. Phys.
**2003**, 51, 223–307. [Google Scholar] [CrossRef] - Hiyama, E. Gaussian expansion method for few-body systems and its applications to atomic and nuclear physics. Prog. Theor. Exp. Phys.
**2012**, 2012, 01A204. [Google Scholar] [CrossRef] [Green Version] - Morton, D.; Wu, Q.; Drake, G.W. Energy Levels for the Stable Isotopes of Atomic Helium (4He I and 3He I). Can. J. Phys.
**2006**, 84, 83–105. [Google Scholar] [CrossRef] - Bhaduri, R.K.; Cohler, L.E.; Nogami, Y. A unified potential for mesons and baryons. Il Nuovo Cimento A
**1981**, 65. [Google Scholar] [CrossRef] - Silvestre-Brac, B. Spectrum and Static Properties of Heavy Baryons. Few-Body Syst.
**1996**, 20. [Google Scholar] [CrossRef] - Kandula, D.Z.; Gohle, C.; Pinkert, T.J.; Ubachs, W.; Eikema, K.S.E. Extreme Ultraviolet Frequency Comb Metrology. Phys. Rev. Lett.
**2010**, 105, 063001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, W.C. Energy Levels of Neutral Helium (4He I). J. Phys. Chem. Ref. Data
**1973**, 2, 257–266. [Google Scholar] [CrossRef] [Green Version] - Tech, J.L.; Ward, J.F. Accurate Wavelength Measurement of the 1s2p
^{3}P^{0}− 2p^{2}^{3}P Transition in^{4}He I. Phys. Rev. Lett.**1971**, 27, 367–370. [Google Scholar] [CrossRef] - Micu, L. Decay rates of meson resonances in a quark model. Nucl. Phys.
**1969**, B10, 521–526. [Google Scholar] [CrossRef] - Le Yaouanc, A.; Oliver, L.; Pène, O.; Raynal, J.C. “Naive” Quark-Pair-Creation Model of Strong-Interaction Vertices. Phys. Rev. D
**1973**, 8, 2223–2234. [Google Scholar] [CrossRef] - Le Yaouanc, A.; Oliver, L.; Pène, O.; Raynal, J.C. Naive quark-pair—Creation model and baryon decays. Phys. Rev. D
**1974**, 9, 1415–1419. [Google Scholar] [CrossRef] - Yaouanc, A.L.; Oliver, L.; Pene, O.; Raynal, J.C. Strong decays of ψ(4028) as a radial excitation of charmonium. Phys. Lett. B
**1977**, 71, 397–399. [Google Scholar] [CrossRef] - Yaouanc, A.L.; Oliver, L.; Pène, O.; Raynal, J. Why is ψ(4414) so narrow? Phys. Lett. B
**1977**, 72, 57–61. [Google Scholar] [CrossRef] - Ackleh, E.S.; Barnes, T.; Swanson, E.S. On the mechanism of open-flavor strong decays. Phys. Rev. D
**1996**, 54, 6811–6829. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Entem, D.; Fernández, F. Scaling of the P03 strength in heavy meson strong decays. Phys. Lett. B
**2012**, 715, 322–327. [Google Scholar] [CrossRef] [Green Version] - Bijker, R.; Santopinto, E. Unquenched quark model for baryons: Magnetic moments, spins, and orbital angular momenta. Phys. Rev. C
**2009**, 80, 065210. [Google Scholar] [CrossRef] [Green Version] - Heikkilä, K.; Törnqvist, N.A.; Ono, S. Heavy $c\overline{c}$ and $b\overline{b}$ quarkonium states and unitarity effects. Phys. Rev. D
**1984**, 29, 110–120. [Google Scholar] [CrossRef] - Baru, V.; Hanhart, C.; Kalashnikova, Y.S.; Kudryavtsev, A.E.; Nefediev, A.V. Interplay of quark and meson degrees of freedom in a near-threshold resonance. Eur. Phys. J A
**2010**, 44, 93. [Google Scholar] [CrossRef] [Green Version] - Ortega, P.G.; Entem, D.R.; Fernández, F. Unquenching the Quark Model in a Nonperturbative Scheme. Adv. High Energy Phys.
**2019**, 2019, 3465159. [Google Scholar] [CrossRef] - Abulencia, A.; Acosta, D.; Adelman, J.; Affolder, T.; Akimoto, T.; Albrow, M.G.; Ambrose, D.; Amerio, S.; Amidei, D.; Anastassov, A.; et al. Measurement of the Dipion Mass Spectrum in X(3872)→J/ψπ
^{+}π^{−}Decays. Phys. Rev. Lett.**2006**, 96, 102002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abe, K. Evidence for X(3872) —> γ J / ψ and the sub-threshold decay X(3872) —> ω J / ψLepton and photon interactions at high energies. In Proceedings of the 22nd International Symposium, LP 2005, Uppsala, Sweden, 30 June–5 July 2005. [Google Scholar]
- Aushev, T.; Eidelman, S.; Gabyshev, N.; Shwartz, B.; Usov, Y.; Zhulanov, V.; Zyukova, O.; Drutskoy, A.; Goldenzweig, P.; Lange, J.S.; et al. Study of the B→X(3872)(→D
^{*0}${\overline{D}}^{0}$)K decay. Phys. Rev. D**2010**, 81, 031103. [Google Scholar] [CrossRef] [Green Version] - Guo, F.K. Novel Method for Precisely Measuring the X(3872) Mass. Phys. Rev. Lett.
**2019**, 122, 202002. [Google Scholar] [CrossRef] [Green Version] - Aaij, R.; Abellan Beteta, C.; Ackernley, T.; Adeva, B.; Adinolfi, M.; Afsharnia, H.; Aidala, C.A.; Aiola, S.; Ajaltouni, Z.; Akar, S.; et al. Study of the ψ
_{2}(3823) and χ_{c1}(3872) states in B^{+}→Jψπ^{+}π^{−}K^{+}decays. JHEP**2020**, 08, 123. [Google Scholar] [CrossRef] - Swanson, E.S. Diagnostic decays of the X(3872). Phys. Lett. B
**2004**, 598, 197–202. [Google Scholar] [CrossRef] [Green Version] - Gamermann, D.; Oset, E. Isospin breaking effects in the X(3872) resonance. Phys. Rev. D
**2009**, 80, 014003. [Google Scholar] [CrossRef] [Green Version] - Gamermann, D.; Nieves, J.; Oset, E.; Arriola, E.R. Couplings in coupled channels versus wave functions: Application to the X(3872) resonance. Phys. Rev. D
**2010**, 81, 014029. [Google Scholar] [CrossRef] [Green Version] - Ortega, P.G.; Segovia, J.; Entem, D.R.; Fernández, F. Coupled channel approach to the structure of the X(3872). Phys. Rev. D
**2010**, 81, 054023. [Google Scholar] [CrossRef] [Green Version] - Ferretti, J.; Galatà, G.; Santopinto, E. Interpretation of the X(3872) as a charmonium state plus an extra component due to the coupling to the meson-meson continuum. Phys. Rev. C
**2013**, 88, 015207. [Google Scholar] [CrossRef] [Green Version] - Ortega, P.G.; Entem, D.R.; Fernández, F. Molecular structures in the charmonium spectrum: TheXYZpuzzle. J. Phys. G Nucl. Part. Phys.
**2013**, 40, 065107. [Google Scholar] [CrossRef] [Green Version] - Burns, T.J. Phenomenology of Pc(4380)+, Pc(4450)+ and related states. Eur. Phys. J. A
**2015**, 51, 152. [Google Scholar] [CrossRef] [Green Version] - Guo, F.K.; Jing, H.J.; Meißner, U.G.; Sakai, S. Isospin breaking decays as a diagnosis of the hadronic molecular structure of the P
_{c}(4457). Phys. Rev. D**2019**, 99, 091501. [Google Scholar] [CrossRef] [Green Version] - Nieves, J.; Pavón Valderrama, M. Heavy quark spin symmetry partners of the X(3872). Phys. Rev. D
**2012**, 86, 056004. [Google Scholar] [CrossRef] [Green Version] - Hidalgo-Duque, C.; Nieves, J.; Valderrama, M.P. Light flavor and heavy quark spin symmetry in heavy meson molecules. Phys. Rev. D
**2013**, 87, 076006. [Google Scholar] [CrossRef] [Green Version] - Baru, V.; Epelbaum, E.; Filin, A.A.; Hanhart, C.; Nefediev, A.V. Molecular partners of the X(3872) from heavy-quark spin symmetry: A fresh look. EPJ Web Conf.
**2017**, 137, 06002. [Google Scholar] [CrossRef] [Green Version] - Guo, F.K.; Hidalgo-Duque, C.; Nieves, J.; Pavón Valderrama, M. Consequences of heavy-quark symmetries for hadronic molecules. Phys. Rev. D
**2013**, 88, 054007. [Google Scholar] [CrossRef] [Green Version] - Entem, D.R.; Ortega, P.G.; Fernández, F. Partners of the X(3872) and heavy quark spin symmetry breaking. AIP Conf. Proc.
**2016**, 1735, 060006. [Google Scholar] [CrossRef] [Green Version] - Cincioglu, E.; Nieves, J.; Ozpineci, A.; Yilmazer, A.U. Quarkonium Contribution to Meson Molecules. Eur. Phys. J. C
**2016**, 76, 576. [Google Scholar] [CrossRef] [Green Version] - Ortega, P.G.; Segovia, J.; Entem, D.R.; Fernández, F. Charmonium resonances in the 3.9 GeV/c2 energy region and the X(3915)/X(3930) puzzle. Phys. Lett. B
**2018**, 778, 1–5. [Google Scholar] [CrossRef] - Batley, J.; Lazzeroni, C.; Munday, D.J.; Slater, M.W.; Wotton, S.A.; Arcidiacono, R.; Bocquet, G.; Cabibbo, N.; Ceccucci, A.; Cundy, D.; et al. Observation of a cusp-like structure in the π
^{0}π^{0}invariant mass distribution from K^{±}→π^{±}π^{0}π^{0}decay and determination of the ππ scattering lengths. Phys. Lett. B**2006**, 633, 173–182. [Google Scholar] [CrossRef] - Budini, P.; Fonda, L. Pion-Pion Interaction from Threshold Anomalies in K
^{+}Decay. Phys. Rev. Lett.**1961**, 6, 419–421. [Google Scholar] [CrossRef] - Cabibbo, N. Determination of the a
_{0}− a_{2}Pion Scattering Length from K^{+}→π^{+}π^{0}π^{0}Decay. Phys. Rev. Lett.**2004**, 93, 121801. [Google Scholar] [CrossRef] [Green Version] - Cabibbo, N.; Isidori, G. Pion-pion scattering and the K→3π decay amplitudes. J. High Energy Phys.
**2005**, 2005, 021. [Google Scholar] [CrossRef] [Green Version] - Bugg, D.V. An explanation of Belle states Z b (10610) and Z b (10650). EPL (Europhys. Lett.)
**2011**, 96, 11002. [Google Scholar] [CrossRef] [Green Version] - Swanson, E.S. Z
_{b}and Z_{c}exotic states as coupled channel cusps. Phys. Rev. D**2015**, 91, 034009. [Google Scholar] [CrossRef] [Green Version] - Dong, X.K.; Guo, F.K.; Zou, B.S. Why there are many threshold structures in hadron spectrum with heavy quarks. arXiv
**2020**, arXiv:2011.14517. [Google Scholar] - Guo, F.K.; Hanhart, C.; Wang, Q.; Zhao, Q. Could the near-threshold XYZ states be simply kinematic effects? Phys. Rev. D
**2015**, 91, 051504. [Google Scholar] [CrossRef] [Green Version] - Aaltonen, T. Observation of the Y(4140) structure in the J/ψϕ Mass Spectrum in B
^{±}→J/ψϕK decays. arXiv**2011**, arXiv:1101.6058. [Google Scholar] [CrossRef] - Abazov, V.M. Inclusive Production of the X(4140) State in $p\overline{p}$ Collisions at D0. Phys. Rev. Lett.
**2015**, 115, 232001. [Google Scholar] [CrossRef] [PubMed] - Chatrchyan, S. Observation of a peaking structure in the J/ψϕ mass spectrum from B
^{±}→J/ψϕK^{±}decays. Phys. Lett.**2014**, B734, 261–281. [Google Scholar] [CrossRef] - Shen, C.P.; Yuan, C.Z.; Aihara, H.; Arinstein, K.; Aushev, T.; Bakich, A.M.; Balagura, V.; Barberio, E.; Bay, A.; Belous, K.; et al. Evidence for a new resonance and search for the Y(4140) in the gamma gamma —> phi J/psi process. Phys. Rev. Lett.
**2010**, 104, 112004. [Google Scholar] [CrossRef] [PubMed] - Lees, J.P. Study of B
^{±,0}→J/ψK^{+}K^{−}K^{±,0}and search for B^{0}→J/ψϕ at BABAR. Phys. Rev.**2015**, D91, 012003. [Google Scholar] [CrossRef] [Green Version] - Aaij, R. Observation of J/ψϕ structures consistent with exotic states from amplitude analysis of B
^{+}→J/ψϕK^{+}decays. arXiv**2016**, arXiv:1606.07895. [Google Scholar] - Ortega, P.G.; Segovia, J.; Entem, D.R.; Fernández, F. Canonical description of the new LHCb resonances. Phys. Rev. D
**2016**, 94, 114018. [Google Scholar] [CrossRef] [Green Version] - Albaladejo, M.; Guo, F.K.; Hidalgo-Duque, C.; Nieves, J. Zc(3900): What has been really seen? Phys. Lett. B
**2016**, 755, 337–342. [Google Scholar] [CrossRef] [Green Version] - Pilloni, A.; Fernández-Ramírez, C.; Jackura, A.; Mathieu, V.; Mikhasenko, M.; Nys, J.; Szczepaniak, A. Amplitude analysis and the nature of the Zc(3900). Phys. Lett. B
**2017**, 772, 200–209. [Google Scholar] [CrossRef] [Green Version] - Ikeda, Y.; Aoki, S.; Doi, T.; Gongyo, S.; Hatsuda, T.; Inoue, T.; Iritani, T.; Ishii, N.; Murano, K.; Sasaki, K. Fate of the Tetraquark Candidate Z
_{c}(3900) from Lattice QCD. Phys. Rev. Lett.**2016**, 117, 242001. [Google Scholar] [CrossRef] [PubMed] - Ortega, P.G.; Segovia, J.; Entem, D.R.; Fernández, F. The Z
_{c}structures in a coupled-channels model. Eur. Phys. J. C**2019**, 79, 78. [Google Scholar] [CrossRef] - Ablikim, M.; Achasov, M.N.; Albayrak, O.; Ambrose, D.J.; An, F.F.; An, Q.; Bai, J.Z.; Ferroli, R.B.; Ban, Y.; Becker, J.; et al. Observation of a Charged Charmoniumlike Structure in e
^{+}e^{−}→π^{+}π^{−}J/ψ at $\sqrt{s}$ = 4.26 GeV. Phys. Rev. Lett.**2013**, 110, 252001. [Google Scholar] [CrossRef] [Green Version] - Liu, Z.Q.; Shen, C.P.; Yuan, C.Z.; Adachi, I.; Aihara, H.; Asner, D.M.; Aulchenko, V.; Aushev, T.; Aziz, T.; Bakich, A.M.; et al. Study of e
^{+}e^{−}→π^{+}π^{−}J/ψ and Observation of a Charged Charmoniumlike State at Belle. Phys. Rev. Lett.**2013**, 110, 252002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablikim, M.; Achasov, M.N.; Albayrak, O.; Ambrose, D.J.; An, F.F.; An, Q.; Bai, J.Z.; Baldini Ferroli, R.; Ban, Y.; Becker, J.; et al. Observation of a Charged ($D{\overline{D}}^{*}$)
^{±}Mass Peak in e^{+}e^{−}→π$D{\overline{D}}^{*}$ at $\sqrt{s}$ = 4.26 GeV. Phys. Rev. Lett.**2014**, 112, 022001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablikim, M.; Achasov, M.N.; Albayrak, O.; Ambrose, D.J.; An, F.F.; An, Q.; Bai, J.Z.; Baldini Ferroli, R.; Ban, Y.; Becker, J.; et al. Observation of a Charged Charmoniumlike Structure Z
_{c}(4020) and Search for the Z_{c}(3900) in e^{+}e^{−}→π^{+}π^{−}h_{c}. Phys. Rev. Lett.**2013**, 111, 242001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablikim, M.; Achasov, M.N.; Ai, X.C.; Albayrak, O.; Albrecht, M.; Ambrose, D.J.; Amoroso, A.; Lou, X.; BESIII Collaboration. Observation of e
^{+}e^{−}→π^{0}π^{0}h_{c}and a Neutral Charmoniumlike Structure Z_{c}(4020)^{0}. Phys. Rev. Lett.**2014**, 113, 212002. [Google Scholar] [CrossRef] [Green Version] - Ablikim, M.; Achasov, M.N.; Ai, X.C.; Albayrak, O.; Albrecht, M.G.; Ambrose, D.J.; BESIII Collaboration. Confirmation of a charged charmoniumlike state Z
_{c}(3885)^{∓}in e^{+}e^{−}→π^{±}($D{\overline{D}}^{*}$)^{∓}with double D tag. Phys. Rev.**2015**, D92, 092006. [Google Scholar] [CrossRef] [Green Version] - Ablikim, M.; Achasov, M.N.; Ai, X.C.; Albayrak, O.; Albrecht, M.; Ambrose, D.J.; Amoroso, A.; An, F.F.; An, Q.; Bai, J.Z.; et al. Determination of the Spin and Parity of the Z
_{c}(3900). Phys. Rev. Lett.**2017**, 119, 072001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablikim, M.; Achasov, M.N.; Albayrak, O.; Ambrose, D.J.; An, F.F.; An, Q.; Bai, J.Z.; Ferroli, R.B.; Ban, Y.; Becker, J.; et al. Observation of a charged charmoniumlike structure in e
^{+}e^{−}→(${D}^{*}{\overline{D}}^{*}$)^{±}π^{∓}at $\sqrt{s}$ = 4.26 GeV. Phys. Rev. Lett.**2014**, 112, 132001. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Diagrams of the ${}^{3}{P}_{0}$ model that contributes to the decay of a meson into two mesons (

**a**) and a baryon into a baryon and a meson (

**b**).

**Figure 2.**Probabilities of different components in the physical $X\left(3872\right)$ state obtained in the unquenched chiral quark model as a function of the mass. The mass is varied using the ${}^{3}{P}_{0}$ strength parameter. Lines are $D{\overline{D}}^{*}$, charged neutral in black and charged channel in red, ${\chi}_{c1}\left(2P\right)$ in dashed magenta and ${\chi}_{c1}\left(1P\right)$ in thin black.

**Figure 3.**Diagonal matrix elements of the two meson interaction in momentum space for the ${D}^{(*)}{\overline{D}}^{(*)}$ sector (left panel) and ${B}^{(*)}{\overline{B}}^{(*)}$ sector (right panel). For the left panel, the dashed blue line gives the ${D}^{*}{\overline{D}}^{*}\left({0}^{++}\right)$ matrix element, the dashed red line the $D\overline{D}\left({0}^{++}\right)$, the solid blue line the right-hand side of Equation (61), and the solid red line left-hand side of the same equation. For the right panel, it is the same but for ${B}^{(*)}{\overline{B}}^{(*)}$ channels.

**Figure 4.**Diagonal matrix elements of the two meson interaction in momentum space for the ${D}^{(*)}{\overline{D}}^{(*)}$ sector (left panel) and ${B}^{(*)}{\overline{B}}^{(*)}$ sector (right panel). For the left panel, the dashed blue line gives the ${D}^{*}{\overline{D}}^{*}\left({0}^{++}\right)$ matrix element, the dashed red line the $D\overline{D}\left({0}^{++}\right)$, the solid blue line the right-hand side of Equation (62), the solid red line the left-hand side of the same equation, and the solid green line the right-hand side of Equation (63). At exact HQSS, the three solid lines should be equal. For the right panel, it is the same but for ${B}^{(*)}{\overline{B}}^{(*)}$ channels.

**Figure 5.**Charmonium spectrum in the energy region of $1P$ and $2P$ states. Blue boxes show the states in the Particle Data Group [7]. The $X\left(3915\right)$ has been included in the ${0}^{++}$ and in the ${2}^{++}$ channels since the J quantum number is not known. The quantum numbers of the $X\left(3940\right)$ are also not known, but it has been included in the ${1}^{++}$ channel since it has been seen in $D{\overline{D}}^{*}$ but not in $DD$. The states in red are naive $Q\overline{Q}$ states predicted by the model.

**Figure 6.**Bottomonium spectrum in the energy region of $2P$ and $3P$ states. Blue boxes show the states in the Particle Data Group [7]. The states in red are the pure $Q\overline{Q}$ states predicted by the model.

**Figure 7.**Line shapes for $D{\overline{D}}^{*}$ (left panel) and $\pi J/\psi $ (right panel) at $\sqrt{s}=4.26$ GeV [103]. Experimental data are from Refs. [109,110], respectively. The theoretical line shapes have been convoluted with the experimental resolution. The line-shape’s $68\%$ uncertainty is shown as a shadowed area.

**Figure 8.**Line shapes for ${D}^{*}{\overline{D}}^{*}$ (left panel) and ${\eta}_{c}\rho $ (right panel) at $\sqrt{s}=4.26$ GeV [103]. Experimental data for ${D}^{*}{\overline{D}}^{*}$ are from Ref. [111]. The theoretical line shapes have been convoluted with the experimental resolution. The line-shape’s $68\%$-uncertainty is shown as a shadowed area.

**Table 1.**First levels of parahelium ($S=0$) with orbital angular momentum $L\le 2$. The experimental data are taken from NIST, Ref. [48]—results with the GEM method from this work. All energies are in eV.

Term | J | NIST | GEM | |
---|---|---|---|---|

$1{s}^{2}$ | ${}^{1}S$ | 0 | 0.00000000 | 0.00 |

$1s2s$ | ${}^{1}S$ | 0 | 20.6157751334 | 20.61 |

$1s2p$ | ${}^{1}P$ | 1 | 21.2180230218 | 21.21 |

$1s3s$ | ${}^{1}S$ | 0 | 22.920317682 | 22.91 |

$1s3d$ | ${}^{1}D$ | 2 | 23.07407511941 | 23.07 |

$1s3p$ | ${}^{1}P$ | 1 | 23.0870188528 | 23.08 |

$1s4s$ | ${}^{1}S$ | 0 | 23.6735709133 | 23.67 |

$1s4d$ | ${}^{1}D$ | 2 | 23.73633535786 | 23.73 |

$1s4p$ | ${}^{1}P$ | 1 | 23.7420703918 | 23.74 |

$1s5s$ | ${}^{1}S$ | 0 | 24.0112153129 | 24.00 |

$1s5d$ | ${}^{1}D$ | 2 | 24.042803734930 | 24.04 |

$1s5p$ | ${}^{1}P$ | 1 | 24.0458007297 | 24.04 |

$1s6s$ | ${}^{1}S$ | 0 | 24.1911605982 | 24.18 |

$1s6d$ | ${}^{1}D$ | 2 | 24.209250116893 | 24.20 |

**Table 2.**First levels of orthohelium ($S=1$) with orbital angular momentum $L\le 2$. The experimental data are taken from NIST, Ref. [48]—results with the GEM method from this work. All energies are in eV.

Term | J | NIST | GEM | |
---|---|---|---|---|

$1s2s$ | $3S$ | 1 | 19.81961484203 | 19.82 |

$1s2p$ | $3P$ | 2 | 20.96408720675 | 20.98 |

1 | 20.96409668230 | |||

0 | 20.96421916817 | |||

$1s3s$ | $3S$ | 1 | 22.718466742 | 22.71 |

$1s3p$ | $3P$ | 2 | 23.0070734673 | 23.01 |

1 | 23.0070761918 | |||

0 | 23.0071097475 | |||

$1s3d$ | $3D$ | 3 | 23.07365102990 | 23.07 |

2 | 23.07365134140 | |||

1 | 23.07365682165 | |||

$1s4s$ | $3S$ | 1 | 23.593959036 | 23.59 |

$1s4p$ | $3P$ | 2 | 23.7078915511 | 23.71 |

1 | 23.7078926664 | |||

0 | 23.7079063452 | |||

$1s4d$ | $3D$ | 3 | 23.73609051247 | 23.73 |

2 | 23.73609066143 | |||

1 | 23.73609295768 | |||

$1s5s$ | $3S$ | 1 | 23.9719717413 | 23.97 |

$1s5p$ | $3P$ | 2 | 24.0282253870 | 24.02 |

1 | 24.0282259477 | |||

0 | 24.0282328220 | |||

$1s5d$ | $3D$ | 3 | 24.042662564819 | 24.04 |

2 | 24.042662644310 | |||

1 | 24.042663817021 | |||

$1s6s$ | $3S$ | 1 | 24.1689985463 | 24.16 |

$1s6p$ | $3P$ | 2 | 24.2008157776 | 24.20 |

1 | 24.2008160981 | |||

0 | 24.2008200312 | |||

$1s6d$ | $3D$ | 3 | 24.209163433335 | 24.22 |

2 | 24.209163480258 | |||

1 | 24.209164158016 |

**Table 3.**Results for the Bhaduri potential obtained with a Fadeev calculation (FD) [50] and the GEM method (this work). Masses are given in MeV, while matter radius square ${R}_{m}^{2}$ and charge radius square ${R}_{c}^{2}$ are in fm${}^{2}$.

State | M(FD) | M(GEM) | ${\mathit{R}}_{\mathit{m}}^{2}$(FD) | ${\mathit{R}}_{\mathit{m}}^{2}$(GEM) | ${\mathit{R}}_{\mathit{c}}^{2}$(FD) | ${\mathit{R}}_{\mathit{c}}^{2}$(GEM) |
---|---|---|---|---|---|---|

${\Lambda}_{c}^{+}\left(cud\right)$ | 2300 | 2298.5 | 0.097 | 0.0984 | 0.117 | 0.1180 |

${\Sigma}_{c}^{0}\left(cud\right)$ | 2473 | 2475.0 | 0.111 | 0.1116 | −0.224 | −0.2247 |

${\Sigma}_{c}^{+}\left(cud\right)$ | 0.134 | 0.1347 | ||||

${\Sigma}_{c}^{++}\left(cud\right)$ | 0.494 | 0.4941 | ||||

${\Lambda}_{b}\left(bud\right)$ | 5653 | 5649.6 | 0.043 | 0.0435 | 0.115 | 0.1169 |

${\Sigma}_{b}^{-}\left(bud\right)$ | 5858 | 5859.8 | 0.051 | 0.0509 | −0.280 | −0.2804 |

${\Sigma}_{b}^{0}\left(bud\right)$ | 0.138 | 0.1383 | ||||

${\Sigma}_{b}^{+}\left(bud\right)$ | 0.555 | 0.5571 | ||||

${\Xi}_{c}^{0}\left(nsc\right)$ | 2490 | 2490.9 | 0.097 | 0.0978 | −0.145 | −0.1463 |

${\Xi}_{c}^{+}\left(nsc\right)$ | 0.160 | 0.1617 | ||||

${\Omega}_{c}^{0}\left(css\right)$ | 2700 | 2701.0 | 0.100 | 0.0999 | −0.111 | −0.1111 |

${\Xi}_{b}^{-}\left(nsb\right)$ | 5826 | 5824.8 | 0.045 | 0.0459 | −0.193 | −0.1951 |

${\Xi}_{b}^{0}\left(nsb\right)$ | 0.151 | 0.1517 | ||||

${\Omega}_{b}^{-}\left(bss\right)$ | 6046 | 6046.7 | 0.050 | 0.0505 | −0.164 | −0.1642 |

${\Xi}_{cc}^{+}\left(ncc\right)$ | 3631 | 3632.2 | 0.076 | 0.0766 | −0.034 | −0.0330 |

${\Xi}_{cc}^{++}\left(ncc\right)$ | 0.285 | 0.2852 | ||||

${\Xi}_{bb}^{-}\left(nbb\right)$ | 10,197 | 10,197.4 | 0.031 | 0.0309 | −0.128 | −0.1279 |

${\Xi}_{bb}^{0}\left(nbb\right)$ | 0.215 | 0.2140 | ||||

${\Omega}_{cc}^{+}\left(scc\right)$ | 3739 | 3738.7 | 0.073 | 0.0739 | 0.008 | 0.0091 |

${\Omega}_{cb}^{+}\left(scb\right)$ | 7023 | 7024.2 | 0.043 | 0.0430 | −0.023 | −0.0232 |

${\Omega}_{bb}^{-}\left(sbb\right)$ | 10,271 | 10,271.3 | 0.030 | 0.0304 | −0.083 | −0.0829 |

${\Omega}_{ccc}^{++}\left(ccc\right)$ | 4806 | 4807.2 | 0.062 | 0.0619 | 0.124 | 0.1239 |

${\Omega}_{ccb}^{+}\left(ccb\right)$ | 8032 | 8030.9 | 0.038 | 0.0378 | 0.089 | 0.0891 |

${\Omega}_{cbb}^{0}\left(cbb\right)$ | 11,220 | 11,218.6 | 0.026 | 0.0264 | 0.032 | 0.0318 |

${\Omega}_{bbb}^{-}\left(bbb\right)$ | 14,370 | 14,371.8 | 0.019 | 0.0192 | −0.019 | −0.0192 |

Term | NIST | GEM | |
---|---|---|---|

$2s2p$ | $3P$ | 58.311 [52] | 58.31 |

$2{p}^{2}$ | $3P$ | 59.67378 [53] | 59.66 |

$2p3p$ | $3D$ | 63.120 [52] | |

$2p3d$ | $3D$ | 63.78658 [52] | |

$2p3d$ | $3P$ | 64.0719 [52] |

**Table 5.**The S-matrix pole positions, in ${\mathrm{MeV}/\mathrm{c}}^{2}$, for different coupled-channels calculations [103]. The included channels for each case are shown in the first column. Poles are given in the second and fourth columns by the value of the complex energy in a specific Riemann sheet (RS). The RS columns indicate whether the pole has been found in the first (F) or second (S) Riemann sheet of a given channel. Each channel in the coupled-channels calculation is represented as an array’s element, ordered with increasing energy.

Calculation | ${\mathit{Z}}_{\mathit{c}}\left(3900\right)$ Pole | RS | ${\mathit{Z}}_{\mathit{c}}\left(4020\right)$ Pole | RS |
---|---|---|---|---|

$D{\overline{D}}^{*}$ | $3871.37-2.17\phantom{\rule{0.166667em}{0ex}}i$ | (S) | - | - |

$D{\overline{D}}^{*}+{D}^{*}{\overline{D}}^{*}$ | $3872.27-1.85\phantom{\rule{0.166667em}{0ex}}i$ | (S,F) | $4014.16-0.10\phantom{\rule{0.166667em}{0ex}}i$ | (S,S) |

$\rho {\eta}_{c}+D{\overline{D}}^{*}$ | $3871.32-0.00\phantom{\rule{0.166667em}{0ex}}i$ | (S,S) | - | - |

$\rho {\eta}_{c}+D{\overline{D}}^{*}+{D}^{*}{\overline{D}}^{*}$ | $3872.07-0.00\phantom{\rule{0.166667em}{0ex}}i$ | (S,S,F) | $4013.10-0.00\phantom{\rule{0.166667em}{0ex}}i$ | (S,S,S) |

$\pi J/\psi +\rho {\eta}_{c}+D{\overline{D}}^{*}+{D}^{*}{\overline{D}}^{*}$ | $3871.74-0.00\phantom{\rule{0.166667em}{0ex}}i$ | (S,S,S,F) | $4013.21-0.00\phantom{\rule{0.166667em}{0ex}}i$ | (S,S,S,S) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ortega, P.G.; Entem, D.R.
Coupling Hadron-Hadron Thresholds within a Chiral Quark Model Approach. *Symmetry* **2021**, *13*, 279.
https://doi.org/10.3390/sym13020279

**AMA Style**

Ortega PG, Entem DR.
Coupling Hadron-Hadron Thresholds within a Chiral Quark Model Approach. *Symmetry*. 2021; 13(2):279.
https://doi.org/10.3390/sym13020279

**Chicago/Turabian Style**

Ortega, Pablo G., and David R. Entem.
2021. "Coupling Hadron-Hadron Thresholds within a Chiral Quark Model Approach" *Symmetry* 13, no. 2: 279.
https://doi.org/10.3390/sym13020279