The \(Z_c(3900)\) peak does not come from the “triangle singularity”
Abstract
We compare contributions from the triangle diagram and the \(D\bar{D}^*\) bubble chain with the processes of \(e^{+}e^{}\rightarrow J/\psi \pi ^{+}\pi ^{}\) and \(e^{+}e^{}\rightarrow (D\bar{D}^*)^\mp \pi ^{\pm }\). By fitting the \(J/\psi \pi \) maximum spectrum and the \(D\bar{D}^*\) spectrum, we find that the triangle diagram cannot explain the new experimental results from BESIII Collaboration at center of mass at 4.23 and 4.26 GeV, simultaneously. On the contrary, the molecular assignment of \(Z_c(3900)\) gives a much better description.
1 Introduction
The charged charmoniumlike state \(Z_c(3900)\) was observed in \(J/\psi \pi ^\pm \) mass spectrum by BES III Collaboration in \(e^+e^\rightarrow J/\psi \pi \pi \) process [1], and confirmed by Belle [2] and CLEO [3] Collaborations in the same processes. Afterwards, it was also observed in the \((D\bar{D}^*)^\pm \) invariant mass spectrum in the process of \(e^{+}e^{}\rightarrow D\bar{D}^*\pi ^\mp \), and the quantum number of \(Z_c(3900)\) was determined to be \(\mathrm{I}(J^P)=1(1^+)\) by angular distribution analysis of the \(\pi Z_c(3900)\) system [4]. The experimental discovery has stimulated a lot of discussion because of the unique nature of \(Z_c(3900)\), as it could be (together with \(Z_b\)) an unambiguous candidate of the long wishedfor tetraquark state.
In a recent paper [5], we have made a detailed comparison between the \(D\bar{D}^*\) molecule picture and the “elementary” picture, and we concluded that \(Z_c(3900)\) is of \(D\bar{D}^*\) molecular nature, using the pole counting method [6, 7].
However, there is also found in the literature another possible mechanism, called the anomalous triangle singularity (ATS), to explain the singularity structure at \(Z_c(3900)\). ATS refers to a branch cut in a threepoint loop function other than the normal threshold. The study of ATS can be traced back to about 60 years ago. In Ref. [8] Mandelstam worked out the ATS branch point and discussed its effects on the deuteron electromagnetism form factor, and in Ref. [9] the Landau equations were used to analyze ATS in triangle diagrams. Extensive studies on the triangle singularity using dispersion techniques can also be found in Refs. [10, 11, 12, 13, 14, 15]. Especially in the paper by Lucha, Melikhov and Simula of Refs. [10, 11, 12, 13, 14, 15], a detailed dispersive analysis is given on different variables. Some similar analyses of triangle singularity based on the nonrelativistic expression can be found in Ref. [16].
ATS has attracted renewed interests, recently, because it may contribute to peaks in some certain invariantmass spectra. In other words, some socalled “exotic hadron states” could be just the ATS peak rather than real particles; or even if real exotic hadron states exist, there may be some nonnegligible contributions from ATS. For example, it is suggested in Refs. [17, 18, 19] that the singularity structure of the triangle diagram (see Fig. 1), which contains both the normal threshold effect and the anomalous threshold effect, may lead to the peak at 3900 MeV. In Refs. [20, 21] it is emphasized that the anomalous triangle singularity may have significant impact in understanding the nature of the nearthreshold state. The possible impact of triangle singularity on \(Z_c(3900)\) has also been discussed in Refs. [22].
This paper is devoted to the study of the triangle diagram contribution to the \(Z_c(3900)\) peak. In Sect. 2 we give a pedagogical analysis of general threepoint loop functions using the Feynman parameter representation that can be found in most textbooks, and we discuss the properties of the ATS. In Sect. 3 we calculate the triangle diagram corresponding to \(e^{+}e^{}\rightarrow J/\psi \pi ^{+}\pi ^{}\) and \(e^{+}e^{}\rightarrow (D\bar{D}^*)^\mp \pi ^{\pm }\) processes and fit the experimental data to test whether the \(Z_c(3900)\) peak comes from the triangle diagram. In Sect. 4 the major conclusion of the present paper is reached: basically, it is found that the new experimental results from Refs. [23, 24] play a crucial role in clarifying the issue of the triangle diagram contribution: the experimental data indicate that the peak at 4.23 GeV is higher than that at 4.26 GeV, whereas the triangle diagram predicts an opposite behavior. Our analysis reveals that the \(Z_c(3900)\) peak cannot be explained from the triangle diagram contribution from Fig. 1. Hence, combining with our previous analysis in Ref. [5], the molecular nature of \(Z_c(3900)\) is firmly established.
2 Theoretical framework

when \(s_1<(m_2+m_3)^2+\frac{m_2}{m_1}[(m_3m_1)^2s_2]\), the \(s^\) is located on the second sheet and is below the normal threshold;

when \(s_1=(m_2+m_3)^2+\frac{m_2}{m_1}[(m_3m_1)^2s_2]\), the \(s^\) just rides on the normal threshold;

when \(s_1>(m_2+m_3)^2+\frac{m_2}{m_1}[(m_3m_1)^2s_2]\), the \(s^\) is located on the first sheet and is below the normal threshold.
In addition, to understand the dependence of \(s^\) on \(s_1\), one gives \(s_1\) a small positive imaginary part, \(s_1\rightarrow s_1+i0^+\). Then \(s^\) can be expressed as \(s^(s_1+i0^+)=s^(s_1)+i\frac{\partial s^}{\partial s_1}0^+\) derived from Eq. (14). As \(s_1\) increases, the nearthreshold trajectory of \(s^\) both in the classical stable case (see Fig. 2a) and under the kinematics of Eq. (15) (see Fig. 2b) can thus be drawn.
The aim of this paper is to investigate whether the \(Z_c(3900)\) peak is mainly from the triangle singularity or not, so the processes \(X(4260)\rightarrow \pi ^{+}\pi ^{} J/\psi , D \bar{D}^* \pi \) are considered, with respect to the triangle diagrams shown in Fig. 4 as suggested by Refs. [17, 18, 19, 20, 21]. We set \(s_1\) to be the square of the X(4260) 4momentum, and the pertinent masses to the masses of those particles; then according to Eq. (16), it is found that the ATS lies on the second sheet when \(\sqrt{s_1}\) lies between 4230 and 4260 MeV. Hence the \(Z_c(3900)\) peak cannot be a direct manifestation of the anomalous threshold. We plot the modulussquare of the amplitude in Eq. (8) with different centerofmass energies \(\sqrt{s_1}\), as shown in Fig. 3.
It is, however, found that the location and effect of the anomalous threshold are very sensitive to the energy of X(4260). When the anomalous threshold is on the second sheet as shown in Fig. 3a, the closer it is to normal threshold, the more influence it has on the amplitude. Since the anomalous threshold (on the second sheet) can be rather close to the normal threshold, one still needs to check whether the (anomalous and normal) threshold effects can cause the experimentally observed \(Z_c(3900)\) peak.
The discussions above are only brief qualitative analyses aiming at studying the dependence of ATS peak on \(s_1\) variable, and what we have calculated above is not the whole amplitude of that process to fit the experimental data, since in the full amplitude there exists a X(4260) production process, a X(4260) propagator, spin structures, and derivative couplings. The detailed formulas can be found in Appendices A and B; with these the numerical discussions are presented in the next section.
3 Numerical analyses and discussions
3.1 Fit to the data of Refs. [1, 4]

Fit I: X(4260) decays to final states only through triangle diagram as depicted in Fig. 4.

Fit II: the final states are only produced by \(D\bar{D}^*\) rescattering, and the Feynman diagram is shown in Fig. 5.

Fit III: the mixed situation by combining the triangle diagram and \(D\bar{D}^*\) bubble chains, as shown in Fig. 6.
Note that in reality the \(D_1D^*\pi \) interaction can be either through the Swave or Dwave (of the \(D^* \pi \) system). Equation (18) is in Swave form, while the Dwave vertex can be found in Eq. (5) of Ref. [17]. Here we adopt the Swave form, for the reason that the Dwave interaction would lead to a higher order momentum dependence in the numerator of the amplitude, which suppresses the contribution from the nearthreshold region and increases too fast when the energy goes higher; in other words, it wrecks the line shape of the triangle diagram peak and gives a much worse fit result, as discussed in the following.
We may focus on the dominant contribution \(\Lambda _{22}\) to the \(J/\psi \pi \pi \) process,^{5} which contains no free parameters like the coupling constants; see Eq. (42) in Appendix B. With the \(D_1 D^* \pi \) vertex in Swave and Dwave, this dominant term is depicted in Fig. 7a, b, respectively.
It is found that the Dwave interaction behaves worse when the energy increases—it cannot even give the correct shape of the peak when \(\sqrt{s_1}=4.23\) GeV. As for the \(D D^*\) spectrum, the Dwave interaction of \(D_1D^*\pi \) also makes the line shape worse; see Fig. 8. Furthermore, one may discuss the situation of Swave \(D_1D^*\pi \) interaction with a large \(D_1\) width (\(\simeq 400\) MeV as inspired by PDG), and it is found that the structure of the \(DD^*\) spectrum are totally destroyed. Hence such a scenario is completely ruled out.
With these preparations, it is possible to make a combined fit on the \(J/\psi \pi \) maximum spectrum and \(D\bar{D}^*\) mass distribution [1, 4]. Except for various coupling parameters, two parameters for the \(D\bar{D}^*\) incoherent background and two normalization constants are further introduced. In total, there are 8 and 10 free parameters for Fit I and Fit II, respectively.
Since the value of the centerofmass energy of \(X(4260)/\gamma ^*\) severely influences the ATS contribution as discussed previously, we in the fit also carefully analyze the effect of energy resolution. However, it is found, through numerical tests, that the effect of the energy resolution does not obviously improve the fit quality, since the energy resolution parameters \(\sigma =4.2\) MeV in \(J/\psi \pi \pi \) channel [1], \(\sigma =1.0\) MeV in \(D\bar{D}^*\pi \) channel [4], and \(\sigma =3.0\) MeV for the energy spread of X(4260), are much smaller than the particle widths (\(\sim 30\) MeV).^{6}
The parameters which determine the pole positions of Fit II and Fit III
\(\lambda _1\)  \(c_0\)  Pole position (GeV)  

Fit II  − 345.29  0.002347  \(3.8747\pm 0.0148i\) 
Fit III  − 341.23  0.002342  \(3.8749\pm 0.0145i\) 
3.2 Fit to the new data
Fortunately, the new data from BESIII Collaboration [23, 24] indicates that there are more events in \(Z_c(3900)\) peak at \(\sqrt{s_1}=4.23\) GeV (with an integrated luminosity of \(L=1092\) pb\(^{1}\)) than at \(\sqrt{s_1}=4.26\) GeV (with \(L= 827\) pb\(^{1}\)), after background subtraction (see Fig. 10). On the contrary, the magnitude of the triangle diagram in Fig. 4 at the \(Z_c\) peak is smaller when \(\sqrt{s_1}=4.23\) GeV compared with the magnitude when \(\sqrt{s_1}=4.26\) GeV.^{8} It is noticed that the \(s_1\) dependence of the triangle diagram will be slightly balanced by the \(s_1\) dependence of X(4260) propagator, which takes the standard Breit–Wigner form of the constant width taken from PDG, and the mass of X(4260) is chosen to be 4.23 GeV.^{9} If the mass of X(4260) is set to be 4.26 GeV, the propagator would also suppress the cross section when \(\sqrt{s_1}=4.23\) GeV, making the fit with a pure triangle diagram even worse. On the other side, different from the triangle diagram, the bubble chain amplitude is not sensitive to \(s_1\) in the energy region of interests. New fits to both the \(\sqrt{s_1}=4.23\) GeV and the \(\sqrt{s_1}=4.26\) GeV data are performed.
The results are shown in Fig. 10. The \(\chi ^2_{dof}=5.3\) for a pure triangle diagram (Fit I’) and the \(\chi ^2_{dof}=1.6\) for pure bubble resummation (Fit II’). The pole of Fit II’ is located at \(\sqrt{s}=3.8804\pm 0.0150i\) GeV. Hence the triangle diagram gives a much worse fit compared with the bubble chain diagram, and hence can be ruled out. Apparently, the new data [23, 24] are crucial in supporting of the \(D\bar{D}^*\) molecule explanation of \(Z_c(3900)\).
The parameters which decide the pole position of Fit II’ and Fit III’
\(\lambda _1\)  \(c_0\)  Pole position (GeV)  

Fit II’  − 305.48  0.002871  \(3.8804\pm 0.0150i\) 
Fit III’  − 271.10  0.002859  \(3.8822\pm 0.0119i\) 
Here we should mention that in the above fit using triangle diagrams, the renormalization scale \(\mu \) runs to a ridiculously small number, \(\sim 10^{7}\) GeV. If we fix \(\mu \) at 5 GeV, the fit quality of triangle diagrams gets even worse.
4 Conclusion
To summarize, we have investigated whether the triangle singularity mechanism proposed in the literature can be responsible for the experimentally observed \(Z_c(3900)\) peak. It is found that, though the triangle diagram could barely explain the line shape, up to an arbitrary normalization constant, it fails to explain the dependence of the process \(e^{+}e^{}\rightarrow J/\psi \pi ^{+}\pi ^{}\) on the centerofmass energy, not to mention the weird value of the renormalization scale it requires. Therefore, we conclude that the \(Z_c(3900)\) peak is dominantly contributed by the pole of the \(D \bar{D}^*\) molecular state.
Note added: when this paper was being completed, we became aware of a recent paper [27], where the authors also attacked the same problem using four different amplitude parameterizations. They reached the conclusion that at this stage they cannot have a preference on one of these parameterizations, which contradicts our conclusion. We point out here that they were only able to use the old data (at c.m. energy 4.23 GeV) in the neutral channel of \(J/\psi \pi ^0\) [28], which is much worse in statistics compared with the data of [23, 24]. Hence we urge the authors of Ref. [27] to redo their analysis with the new data incorporated and compare with our result.
Footnotes
 1.
Due to the \(+i \epsilon \) term, poles in the \(l^0\) plane, \(l^0=\pm \sqrt{\Delta +\mathbf {l}^2i \epsilon }\), always are located in the second and fourth quadrant, so the Wick rotation is valid for both \(\Delta \ge 0\) and \(\Delta <0\).
 2.
When \(\Delta (x,y)<0\), the \(i \epsilon \) term guarantees the validity of this integration; see, for example, [25].
 3.
In fact there is another solution corresponding to \(\lambda (s_3,m_1^2,m_2^2)\rightarrow 0^\), that is, \(s_3=(m_1m_2)^2\), which is called a pseudothreshold and it only appears on the unphysical sheet. So it is less relevant to our discussion.
 4.
This fact actually indicates that the triangle diagram does not work in simulating the \(Z_c\) peak.
 5.
The other \(\Lambda _{ij}\)s give rather bad line shapes, hence they are suppressed in the numerical fit.
 6.
G. Y. Tang, private communication.
 7.
Other coupling constants and the normalization constants are multiplied to each other and are not quite interesting physically, so we do not list them here.
 8.
Since the latter is closer to the \(\bar{D}D_1\) threshold.
 9.
Notes
Acknowledgements
The authors are grateful to Changzheng Yuan for bringing Refs. [23, 24] to their attention, and GuangYi Tang for valuable discussions. They also would like to thank Qiang Zhao and FengKun Guo for helpful discussions. This work is supported in part by the National Nature Science Foundations of China (NSFC) under Contract nos. 10925522, 11021092.
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