In this work, we systematically study the two-body strong decay of the = 3 states, which are observed and grouped into the = 3 meson family. By performing the phenomenological analysis, the underlying properties of these states are obtained and tested. What is more important is that abundant information of their two-body strong decays is predicted, which will be helpful to further and experimentally study these states.

Inspired by the similarity between the mass gaps of the J/ψ and ϒ families, the prediction of a missing higher charmonium with mass 4,263 MeV and very narrow width is made. In addition, the properties of two charmonium-like states, X (3940) and X (4160), and charmonium ψ(4415) are discussed, where our calculation shows that X (3940) as η c (3S) is established, while the explanation of X (4160) to be η c (4S) is fully excluded and that η c (4S) is typically a very narrow state. These predictions might be accessible at BESIII, Belle, and BelleII in near future.Since the observation of J/ψ in 1974 [1,2], the charmonium family has become abundant with more and more such states announced by the experiments [3]. Especially in the past decade, a series of charmonium-like states have been observed, which have further stimulated theorists' extensive interest in revealing their underlying properties (see a recent review in Ref.[4]), since these novel phenomena reflect non-perturbative behavior of quantum chromodynamics (QCD). Among the studies on these states, it is an important research topic for the whole community of particle physics how to identify the exotic states, whose establishment is, of course, tied with our understanding of the charmonium family.When checking the mass spectra of the observed charmonia with spin-parity J PC = 1 −− and comparing them with those of the corresponding bottomonia, we notice that the mass gap between ψ(2S) and J/ψ is almost the same as that between ϒ(2S) and ϒ(1S). There also exist similar mass a e-mail: help08@lzu.edu.cn if ψ(4415) is ψ(4S), such a law is violated since the mass gap of ψ(4415) and ψ(3S) is much larger than that of ϒ(4S) and ϒ(3S).In fact, the properties of the charmonia above 4.1 GeV are still not understood well, which is the possible reason to result in the above puzzling mass gap. In general, compared with the J/ψ family, the bottomonia with the radial quantum numbers n = 1, 2, 3, 4 were well established both by experiment and theory. Thus, the study of the J/ψ family can be borrowed from ϒ family. If this law of mass gap relation still holds for states with n = 3, 4 in J/ψ and ϒ families, we find that the mass of ψ(4S) should be located at 4,263 MeV, where we add the mass gap between ϒ(4S) and ϒ(3S) to the mass of ψ(3S). In Fig. 1, we show the details of the mass gaps for J/ψ and ϒ families and compare the corresponding gaps with each other. After obtaining the above prediction, we notice the results of the mass spectra of charmonium family given in Ref. [5], where the screening potential was adopted in their calculation. They also found that the mass of ψ(4S) is about 4,273 MeV, which is consistent with our estimate of the mass for the missing charmonium ψ(4S). Additionally, another theoretical work [6] also supports existence of a missing ψ(4S). In Ref.[6], the color-screening effect was considered in the calculation of the mass spectrum of charmonium, where the mass of ψ(4S) was obtained as 4,247 MeV.If this predicted state exists in the J/ψ family,...

If the X(3872) is a weakly bound charm-meson molecule, it can be produced in e + e − annihilation by the creation of D * 0D * 0 from a virtual photon followed by the rescattering of the charm-meson pair into X and a photon. A triangle singularity produces a narrow peak in the cross section for e + e − → Xγ about 2.2 MeV above the D * 0D * 0 threshold. We predict the normalized cross section in the region near the peak. The peak from the triangle singularity may be observable by the BESIII detector.