Unveiling the Nature of the Tetraquark Candidate Zc(3900)
YOICHI IKEDA1,2 FOR THE HAL QCD COLLABORATION
1RESEARCH CENTER FOR NUCLEAR PHYSICS,
2NISHINA CENTER, RIKEN
One of the most important subjects in hadron physics is to establish the existence of exotic hadrons different from the standard quark-antiquark mesons and three-quark baryons. Such exotic candidates include the pentaquark candidates Pc(4380) and Pc(4450) observed by the LHCb Collaboration  and the tetraquark state Zc(3900) reported by the BESIII  and the Belle  Collaborations. The Zc(3900), in particular, is observed as a peak in πJ/Ψ and DD* invariant masses of e+e- Y(4260) → ππJ/Ψ and πDD* reactions.
Fig. 1: Decay scheme of the Zc(3900) and the relevant two-meson threshold energy.
Various phenomenological attempts  have been made to understand the nature of Zc(3900) as a compact tetraquark and a s-wave hadronic molecule as well as a threshold cusp when opening the DD* threshold. However, no conclusive result has been achieved due to the lack of information about the coupled-channel interaction relevant to Zc(3900). (See Fig. 1 for the level structure.)
In this circumstance, the first principle lattice QCD calculations with explicit channel couplings is the most desirable method to determine the structure of Zc(3900). The HAL QCD Collaboration  extracts the s-wave diagonal and off-diagonal potentials among the πJ/Ψ, ρηc and DD* channels by the so-called coupled-channel HAL QCD method . The key quantity in the HAL QCD method is the Nambu-Bethe-Salpeter (NBS) wave functions, which are faithful to the QCD S-matrix, and thus we are able to calculate any scattering observables directly based on QCD using the extracted coupled-channel potential from the NBS wave functions.
It has been found that the resulting diagonal elements of the s-wave coupled-channel potential are all weak. This indicates that Zc(3900) is not a state associated with a hadronic molecule. Also, the off-diagonal πJ/Ψ-ρηc potential is weak. This is a consequence of the heavy quark spin symmetry. On the other hand, the off-diagonal elements of the πJ/Ψ and the ρηc are found to be strong.
With the above coupled-channel potential, we have calculated the scattering amplitudes in two-body πJ/Ψ, ρηc and DD* channels by solving the Lippmann-Schwinger equation. In two-body amplitudes, the peak appears around the DD* threshold. We also have examined the complex pole of the amplitudes to understand whether the peak structure is associated with a conventional resonance or not. The result of the pole position is shown in Fig. 2. The pole is far below the DD* threshold and has a large imaginary part; thus, the pole does not contribute to the amplitudes. In addition to the above analyses of the two-body scatterings, we have investigated Y(4260) decay and compared it with the experiments. As shown in Fig. 3, the peak observed in the experiments is well reproduced. Therefore we conclude that the Zc(3900) is not a conventional resonance but a threshold cusp when opening the DD* threshold .
Fig. 2: The pole of the amplitude on the complex energy plane.
In summary, thanks to the coupled-channel HAL QCD method, it turns out that Zc(3900) is not a conventional resonance but a threshold cusp just opening the DD* threshold assisted by the strong πJ/Ψ coupling: the pole position is far below the DD* threshold, and the Y(4260) decay is well reproduced. The novel method developed in this study paves the way to understand the nature of exotic hadron candidates directly based on QCD. Some interesting future targets include Pc's, X(3872) and Zc(4430).
Fig. 3: The πJ/Ψ invariant mass spectrum of the Y(4260) decay. The statistical error of the simulation is included in the shaded area. The dashed curve represents the invariant mass spectrum, when the off-diagonal potential is switched off.
Acknowledgements: This work is supported in part by MEXT as a "Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of the Universe) and SPIRE (Strategic Program for Innovative REsearch).
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 The member of the HAL QCD Collaboration is Sinya Aoki, Takumi Doi, Faisal Etminan, Shinya Gongyo, Tetsuo Hatsuda, Yoichi Ikeda, Takashi Inoue, Takumi Iritani, Noriyoshi Ishii, Daisuke Kawai, Takaya Miyamoto, Keiko Murano, Hidekatsu Nemura and Kenji Sasaki.
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Yoichi Ikeda is a research assistant professor at the Research Center for Nuclear Physics (RCNP), Osaka University. After receiving a PhD. Sci from Osaka University, he worked at the University of Tokyo, RIKEN and Tokyo Institute of Technology before joining RCNP, Osaka University in 2016. His research field is theoretical hadron physics.