# \(Z_c(4200)^+\) decay width as a charmonium-like tetraquark state

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## Abstract

To identify the nature of the newly observed charged resonance \(Z_c(4200)^+\), we study its hadronic decays \(Z_c(4200)^+\rightarrow J/\psi \pi ^+, Z_c(4200)^+\rightarrow \eta _c\rho ^+\) and \(Z_c(4200)^+\rightarrow D^+\bar{D}^{*0}\) as a charmonium-like tetraquark state. In the framework of the QCD sum rules, we calculate the three-point functions and extract the coupling constants and decay widths for these interaction vertices. Including all these channels, the full decay width of the \(Z_c(4200)^+\) state is consistent with the experimental value reported by the Belle Collaboration, supporting the tetraquark interpretation of this state.

## Keywords

Decay Width Operator Product Expansion Belle Collaboration Gluon Condensate Tetraquark State## 1 Introduction

Recently, a new charged charmonium-like resonance \(Z_c(4200)^+\) was observed by the Belle Collaboration [1]. It was observed in the \(Z_c(4200)^+\rightarrow J/\psi \pi ^+\) process with the mass and decay width \(M=4196^{+31+17}_{-29-13}\) MeV and \(\varGamma =370^{+70+70}_{-70-132}\) MeV, with a significance of \(6.2\sigma \). Its preferred assignment of the quantum numbers is \(J^P=1^+\). The G-parity of \(Z_c(4200)^+\) is positive. Thus, the quantum numbers of its neutral partner is \(I^GJ^{PC}=1^+1^{+-}\).

The family of the charged charmonium-like states have become more abundant after the discovery of \(Z_c(4200)^+\) [1] and \(Z_c(4050)\) [2]. Before this, the first member \(Z(4430)^+\) was observed in the \(\psi (2S)\pi ^+\) invariant mass spectrum in the process \(\bar{B}^0\rightarrow \psi (2S)\pi ^+K^-\) by the Belle Collaboration [3] and confirmed recently by the LHCb Collaboration [4]. Later, Belle also reported a broad doubly peaked structure in the \(\pi ^+\chi _{c1}\) invariant mass distribution, of which the peaks are called \(Z(4050)^+\) and \(Z(4250)^+\) [5]. Several other similar charged states were observed in last two years. In 2013, the BESIII Collaboration reported \(Z_c(3900)^+\) in \(J/\psi \pi ^+\) final states in the process \(Y(4260)\rightarrow J/\psi \pi ^+\pi ^-\) [6]. \(Z_c(3900)^+\) was also observed by Belle [7] and confirmed in CLEO data [8]. The BESIII Collaboration also observed \(Z_c(4025)^{\pm }\) in the \(\pi ^{\mp }\) recoil mass spectrum in the \(e^+e^-\rightarrow (D^{*}\bar{D}^{*})^{\pm }\pi ^{\mp }\) process [9] and \(Z_c(4020)^{\pm }\) in the \(h_c\pi ^{\pm }\) mass spectrum in the process \(e^+e^-\rightarrow h_c\pi ^+\pi ^-\) [10]. Moreover, the Belle Collaboration also observed two charged bottomonium-like states \(Z_b(10610)\) and \(Z_b(10650)\) in the \(\pi ^{\pm }\varUpsilon (nS)\) and \(h_b\pi ^{\pm }\) mass spectra in the \(\varUpsilon (5S)\) decay [11].

These newly observed charged states have the exotic flavor contents \(c\bar{c}u\bar{d}\) for \(Z_c\) states and \(b\bar{b}u\bar{d}\) for \(Z_b\) states. It is natural to understand them as different manifestations of four-quark states: hadron molecules [12, 13, 14, 15, 16, 17, 18], tetraquark states [19, 20, 21], or many other configurations [22, 23, 24]. For example, \(Z(4430)^+\) was described as a \(D^{*}\bar{D}_1\) molecular state in Refs. [25, 26, 27, 28] and a tetraquark state in Refs. [29, 30, 31]; the \(Z_c(3900)^+\) was speculated to be a molecular state in Refs. [32, 33, 34]; the \(Z_c(4025)^+\) was interpreted as a \(D^*\bar{D}^*\) molecular state in Ref. [35]; the \(Z_b(10610)\) and \(Z_b(10650)\) were studied as \(\bar{B}B^{*}\) and \(\bar{B}^{*}B^{*}\) molecular states in Ref. [36]. One can consult Refs. [37, 38, 39, 40, 41, 42] and references therein for recent reviews of these charged resonances.

Being composed of a diquark and antidiquark pair, a hidden-charm tetraquark state can decay very easily into a pair of open-charm *D* mesons or one charmonium state plus a light meson through quark rearrangement, implying that tetraquark states should be very broad resonances, while the experimental *XYZ* states are usually quite narrow, such as \(Z_c(3900)^+\) [6, 7, 8] and \(Z_c(4025)^+\) [9, 10]. However, the experimental width value of the \(Z_c(4200)^+\) [1] is broad enough to be a good tetraquark candidate. In Ref. [43], \(Z_c(4200)^+\) was studied as a tetraquark state by considering the color-magnetic interaction. In Ref. [44], the authors tried to search for \(Z_c^+\) exotic states in lattice QCD. However, they found no convincing signal for \(Z_c^+\) state below 4.2 GeV.

The hidden-charm tetraquark states with \(J^{PC}=1^{+-}\) has been studied using the method of QCD sum rule in Refs. [24, 45, 46, 47, 48, 49, 50]. We have also done similar QCD sum-rule studies in Refs. [51, 52], in which the extracted mass was found to be consistent with the experimental value of the \(Z_c(4200)^+\) mass. In this work, we will study the hadronic decays of the \(Z_c(4200)^+\) as a tetraquark state in QCD sum rules. The three-point functions for the \(Z_cJ/\psi \pi , Z_c\eta _c\rho \), and \(Z_cDD^{*}\) vertices will be studied to calculate the corresponding coupling constants needed to extract the decay widths.

This paper is organized as follows. In Sect. 2, we study the three-point functions for the \(Z_cJ/\psi \pi \), \(Z_c\eta _c\rho \) and \(Z_cDD^{*}\) vertices. We will calculate the operator product expansion (OPE) series up to dimension five condensates. Then we compute the coupling constants and the decay widths for these channels. Finally, we give a short summary and discuss the possibility of searching for such exotic resonances decaying into \(\eta _c\) charmonium.

## 2 QCD sum rules and three-point correlation function

In the past several decades, QCD sum rule has proven to be a very powerful non-perturbative approach to study hadron properties such as masses, magnetic moments and coupling constants, associated with the low-lying baryons and mesons [53, 54, 55, 56, 57]. Recently, this method was used to yield predictions on the spectroscopy of the new hadron *XYZ* states [24, 39, 45, 46, 47, 48, 49, 50, 51, 52, 58].

*A*,

*B*denote the decay products. In QCD sum rules, we consider the three-point correlation function

*A*and

*B*, respectively. In this paper, we consider the \(Z_c(4200)^+\) meson as a charmonium-like tetraquark state. The corresponding tetraquark current is given by

*a*,

*b*are color indices, and

*u*,

*d*, and

*c*represent up, down, and charm quarks, respectively.

*C*is the charge-conjugation matrix. We have studied this charmoniun-like tetraquark scenario and the extracted mass is around 4.16 GeV [51] consistent with the observed mass of the \(Z_c(4200)^+\) meson [1]. This current can couple to the \(Z_c(4200)^+\) meson via

The \(Z_c(4200)\) meson can decay into several different channels such as hidden-charm decay modes \(J/\psi \pi ^+, \eta _c\rho ^+\), and open-charm decay modes \(D^+\bar{D}^{*0}, \bar{D}^0D^{*+}\). Such decay properties are similar to those for the charmonium-like state \(Z_c(3900)\). Assuming \(Z_c(3900)\) to be a tetraquark state with the same quantum numbers as the \(Z_c(4200)\), the hadronic decay modes of \(Z_c(3900)\) to \(J/\psi \pi ^+, \eta _c\rho ^+, D^+\bar{D}^{*0}\), and \(\bar{D}^0D^{*+}\) were studied in Ref. [47]. Building upon these methods, we will study the same decay channels for the \(Z_c(4200)\) tetraquark state to estimate its decay width.

### 2.1 Decay mode \(Z_c^+(4200)\rightarrow J/\psi \pi ^+\)

*u*,

*d*, and

*c*denote the quark propagators for up, down, and charm quark, respectively. Throughout our evaluation, we use the coordinate-space expression for the light quark propagator and momentum-space expression for the heavy quark propagator [54, 59]: where \(m_c\) is the mass of the charm quark. We neglect the chirally suppressed contributions from the current quark masses (\(m_q=0\) in the chiral limit) because they are numerically insignificant. In Eq. (10), the light quark propagator is defined as \(iS_{ab}^{'q}(x)=C(iS_{ab}^q)^TC\) in which

*T*represents only the transpose operation to the Dirac indices. As indicated above, we will pick out the \(1/q^2\) terms in the OPE series and work at the limit \(q^2\rightarrow 0\). We note that this is the assumption used in Ref. [54], and then we can establish a sum rule by comparing with the three-point function expression of Eq. (8) at the hadron level.

On the phenomenological side in Eq. (8), there are five different tensor structures \(g_{\mu \nu }, q_{\mu }q_{\nu }, q_{\mu }p^{\prime }_{\nu }, q_{\nu }p^{\prime }_{\mu }\), and \(p^{\prime }_{\mu }p^{\prime }_{\nu }\). On the QCD side, we evaluate the three-point function and spectral density up to the dimension five terms. In addition to the perturbative term, we calculate the quark condensate, the gluon condensate and the quark–gluon mixed condensate for the power corrections. The Feynman diagrams for these terms are shown in Fig. 1. In our result for the OPE series, the \(g_{\mu \nu }\) structure contributes to all expansion terms including the perturbative part, quark condensate, gluon condensate and quark–gluon mixed condensate. Other tensor structures contribute just some of these terms in the OPE at leading order. For example, the \(q_{\mu }q_{\nu }\) and \(q_{\nu }p^{\prime }_{\mu }\) structures appear only in the gluon condensate while \(p^{\prime }_{\mu }p^{\prime }_{\nu }\) appears in the perturbative term and the gluon condensate. The structure \(q_{\mu }p^{\prime }_{\nu }\) gives no contributions to the perturbative term.

*ABC*and \(p^*(m_A, m_B, m_C)\) is defined as

### 2.2 Decay mode \(Z_c^+(4200)\rightarrow \eta _c\rho ^+\)

### 2.3 Decay mode \(Z_c^+(4200)\rightarrow D^+\bar{D}^{*0}\)

*D*and \(D^{*}\), respectively. The three-point function can be written at the hadron level

## 3 Summary

In summary, we have studied the three-point functions of the processes \(Z_c(4200)^+\rightarrow J/\psi \pi ^+, Z_c(4200)^+\rightarrow \eta _c\rho ^+\) and \(Z_c(4200)^+\rightarrow D^+\bar{D}^{*0}\), considering \(Z_c(4200)^+\) as a hidden-charm tetraquark state. We calculate the three-point functions by including the perturbative term, quark condensate, gluon condensate and quark–gluon mixed condensate.

The study of the three-point function sum rules gives support to the tetraquark interpretation of the newly observed \(Z_c(4200)^+\) state. This conclusion is consistent with the result obtained from the mass sum rules in Ref. [51]. The branching ratio predictions of \(J/\psi \pi , \eta _c\rho , D^+\bar{D}^{*0}\), and \(\bar{D}^0 D^{*+}\) channels will be helpful for future experimental studies.

## Notes

### Acknowledgments

We thank Hai-Yang Cheng for useful discussion and information. This project was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the National Natural Science Foundation of China under Grants Nos. 11261130311, 11205011, and 11475015.

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