The hidden-charm strong decays of the $Z_c$ states

X iv :1 91 2. 12 78 1v 1 [ he pph ] 3 0 D ec 2 01 9 The hidden-charm strong decays of the Zc states Li-Ye Xiao ∗, Guang-Juan Wang †, and Shi-Lin Zhu ‡ 1)School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 2) School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 3) Center of High Energy Physics, Peking University, Beijing 100871, China and 4) Collaborative Innovation Center of Quantum Matter, Beijing 100871, China


I. INTRODUCTION
Since 2003 the Belle Collaboration observed the first charmonium-like state [1], X(3872), an explosion in the observation of new hadronic states began. Dozens of charmoniumlike states (or XYZ states) [2] have been reported by several major experimental collaborations such as BESIII, LHCb, Belle, BaBar, CDF and so on; see Ref. [3][4][5][6][7][8] for a review. The properties of the charged Z c states cannot be explained by the naive quark model and make them manifestly exotic. It seems that we have a new zoo of exotic hadrons. How to understand their internal structures remains a great challenge.
Before the BESIII's measurement [43], the relative decay rate was predicted either in the molecular or tetraquark scenarios within the framework of a covariant quark model [44], the phenomenological Lagrangian field theory [45], the * E-mail: lyxiao@ustb.edu.cn † E-mail: wgj@pku.edu.cn ‡ E-mail: zhusl@pku.edu.cn nonrelativistic effective field theory [46], the light front model(LFM) [47], QCD sum rules [48][49][50], etc. Later, the author of Ref [51] studied the decay properties of the Z c (3900) as a compact tetraquark state and a hadronic molecular state through the Fierz rearrangement of the Dirac and color indices. We collect the theoretical predictions in Table I, which differ greatly.  [46] 230 +330 −140 [46] 1.78 +0. 41 −0.37 [44] 0.27 +0.40 −0.17 [46] 0.12 [47] 1.86 +0.41 −0.35 [44] 0.007 [45] 1.28 +0.37 −0.30 [44] 0.059 [51] 2.2 [51] 1.08 ± 0.88 [49] 0.95 ± 0.40 [48] 0.66 [35] 0.57 ± 0.17 [50] In the present work we calculate the ratio between the ρη c and πJ/ψ decay modes for the charged states Z c (3900), Z c (4020) and Z c (4430) in two different scenarios. In scenario I, we take the Z c (3900), Z c (4020) and Z c (4430) as the DD * , D * D * and D(2S )D * molecular states with spin-parity J P = 1 + , respectively. In scenario II, we treat the Z c (3900), Z c (4020) and Z c (4430) as the tetraquark states. Our results show that in the molecular and tetraquark scenarios, the ratios for Z c (3900) are R th Z c (3900) ≃ 1.3 and R th Z c (3900) ≃ 1.6, respectively, which are both in the range of experimental result, R exp = 2.2 ± 0.9. For Z c (4020), the ratio is about R th Z c (4020) ∼ 2 in both two scenarios. As to the Z c (4430), we find that the ratio in the molecular scenario (R th Z c (4430) ≃ 1.4) is slightly smaller than that in the tetraquark scenario (R th nario. However both ratios are about 1.2 in the tetraquark scenario. In other words, these ratios are very sensitive to the underlying dynamics of Z c (3900) and Z c (4020), which may be helpful to pin down their inner structures. This paper is organized as follows. In Sec. II, we give an introduction of the quark-exchange model and calculate the transition amplitudes in the molecular and tetraquark scenarios. Then we discuss and compare our results in the two scenarios in Sec. III. We give a short summary in Sec. IV.

A. Decay width
For a four-quark state (F, for short) decaying into two particles labelled as C and D, the decay width in the rest frame of the initial particle has the form Here, M represents the mass of the initial four-quark state F; p c denotes the three-momentum of the meson C in the final state; M(F → C + D) is the transition amplitude of the twobody decay F → C + D, which is related to the T -matrix via where E C and E D denote the energy of the final mesons C and D, respectively. The T -matrix reads Here, ψ CD ( p c ) represents the relative spacial wave function between the final mesons C and D; ψ AB ( k) is the normalized relative spacial wave function between the constituent clusters A and B. In molecular scenario, the constituents represent mesons, while in tetraquark scenario the constituents represent the diquark [cq] and antidiquark [cq]. V eff ( k, p) denotes the effective potential, which is in the general case a function of the initial and final relative momentum k and p c . The four-quark state may be a superposition of terms with different orbital angular momenta 1 . Thus, the relative spacial wave function in the momentum space has the form Then, the Eq. (4) can be written as 1 We assume the orbital excitation is between the diquark and antidiquark. where In this equation, P l (µ) is Legendre function and µ represents the cosine of the angle between the momenta k and p c .
Finally, with the relativistic phase space, the decay width of two-body decay progress reads For the ρη c decay mode, we further consider the decay width of the ρ meson, and get Here, m ρ and Γ ρ stand for the mass and total decay width of the ρ meson, respectively. s denotes the square of the ρ meson invariant mass spectrum.
B. Effective potential

The molecular scenario
The J P quantum number of the Z c (3900), Z c (4020) and Z c (4430) are 1 + . In the molecular scenario, we treat them as the loosely bound S -wave DD * , D * D * and D(2S )D * molecular states according to their mass spectra, respectively. At Born order, the effective potential V eff ( k, p c , µ) is related to the reacting amplitude of the meson-meson scattering process, where 1(3) and 2(4) denote the c(c) quark andq(q) quark, respectively. In the quark interchange model [52][53][54][55][56], the scattering Hamiltonian of the processes D ( * ) /D(2S ) +D * → η c (J/ψ) + ρ(π) is estimated by the sum of the interactions between the inner quarks as illustrated in Fig. 1. Moreover, the short-range interactions are dominant in the scattering processes of two open-charmed mesons into a ground charmonium state plus a light-flavor meson. Thus, the scattering potential can be approximated by the one-gluon-exchange (OGE) potential V i j at quark level 2 , where λ i (λ T i ) represents the quark (antiquark) generator; q is the transferred momentum; b denotes the string tension; σ is the range parameter in the hyperfine spin-spin interaction; m i (m j ) and s i (s j ) correspond to the interacting constituent quark mass and spin operator; α s is the running coupling constant, .
In this equation, Q 2 is the square of the invariant masses of the interacting quarks. The parameters in Eqs. (12)-(13) are fitted by the mass spectra of the observed mesons [57], and their values are listed in Table II.
In the quark model, the color-spin-flavor-space wave func- 2 The interactions in Eq. (12) are the Fourier transformation of the potential in Ref. [57]. In the following. we perform our calculations in the momentum space for the purpose of simplification. The constant potential in the spatial space does not contribute due to the cancelation of the form factors and we just omitted the term in Eq. (12).
tion for a meson is where ω c , φ f , χ s and ψ( p) represent the wave functions in the color, flavor, spin and momentum space, respectively. Here, the wave functions of the mesons are determined by fitting the mass spectra in the Godfrey-Isgur model [58].
According to the decomposition of the meson wave functions, the effective potential can be given as the product of the factors, Here, I with the subscripts color, flavor and spin-space represent the overlaps of the initial and final wave functions in the corresponding space. The color factor I color reads Its value in different diagrams in Fig. 1 is listed in Table III.
For the flavor factor I flavor , its value is simply unity for all diagrams considered in this paper. For the S -wave decay process, the spin and space factors can be decoupled. The spin factor I spin reads where S (S ′ ) stands for the total spin of the initial(final) system. The spin operatorÔ s equals to unitary for the Coulomb and linear interactions, and equals to s i · s j for the spin-spin interaction. We collect the values of color-spin factors I color ·I spin in Table IV.
As to the spatial factor I space , its expression reads In the equations, the spatial operatorÔ q corresponds to 1/q 2 , 1/q 4 and e −q 2 /(4σ 2 ) for the Coulomb, linear and spin-spin interactions, respectively. p 3 denotes the momentum of the third quark. f i (i = 1, 2, 3, 4) is a constituent quark mass dependant function and expressed as Here, m i (i = 1, 2, 3, 4) represents the mass of the i-th quark. Finally, with the obtained effective potential V eff ( k, p c , µ), we can calculate the decay widths by Eqs. (8)- (9) in cases that we know the relative spacial wave function ψ AB ( k) between mesons A and B. In the present work, we adopt an S -wave harmonic ooscillator function to estimate the S -wave component of the relative spacial wave function in Eq. (5), which reads The value of the harmonic oscillator strength α is related to the root mean square radius r mean of the molecular state by Here, we take the r mean in the range of (1.0-3.0) fm, and the corresponding value of α is collected in Table V.  For comparison, we further study the decays of the Z c (3900), Z c (4020) and Z c (4430) as tetraquark states ccqq [18], where Z c (4430) is interpreted as the first radial excitation of the Z c (3900). Similar to the molecular case, the V eff ( k, p c , µ) can be approximated by the interaction between the inner quarks, as shown in Fig. 2. The calculation of the V eff ( k, p c , µ) in the tetraquark scenario is similar to that in the molecular scenario. We can obtain the effective potential V eff ( k, p c , µ) with Eq. (15) as well. The flavor factor I flavor and spin factor I spin are the same as those in the molecular scenario. For the color factor I color , there is a difference between the two scenarios. In the molecular scenario, the initial four-quark state is composed of two mesons, of which the color configurations are 1 c -1 c . However, in the tetraquark scenario, the initial four-quark state is composed of diquark [cq] and antidiquark [cq], of which the color configurations are 3 c -3 c . The difference in color configurations may result in quite different decay properties. The values of color-spin factors are collected in Table VI.
To calculate the space factor I space , we need the wave function of the initial tetraquark state, where k r/R denotes the momentum between the c(c) and u(d) quarks in the diquark (antidiquark), and k X is the one between the diquark [cu] and antidiquark [cd]. The α with the subscripts represents the oscillating parameter along the corresponding Jacobi coordinates. For the Z c (3900) and Z c (4020), the spatial wave function ψ is estimated by the S -wave harmonic oscillating wave func- The α values are taken from Ref. [  As to Z c (4430), the spatial wave function of the diquark [cu] is replaced by that of D, and the spatial wave function of anti-diquark [cd] is replaced by that of D * . The relative spatial wave function between the diquark [cu] and antidiquark [cd] is estimated by an 2S -wave harmonic oscillating space-wave function The value of the harmonic oscillator strength α X is related to the root mean square radius r X of the tetraquark state by We vary the r X in the rang of (0.5-2.0) fm and the corresponding value of α X is listed in Table VIII.

III. RESULTS
Inspired by the recent measurement of the decay Z c (3900) ± → ρ ± η c by the BESIII Collaboration, we calculate the ratios between the ρη c and πJ/ψ decay modes for the charged states Z c (3900), Z c (4020) and Z c (4430) in the molecular and tetraquark scenarios. Our results and theoretical predictions are presented as follows.

A. The molecular scenario
The mass of Z c (3900) (M = 3886.6 ± 2.4 MeV) is slightly higher than the mass threshold of the DD * (∼3872 MeV). In the molecular scenario, we take the Z c (3900) as a DD * resonance molecular state, and calculate its branching fraction ratio between the ρη c and πJ/ψ decay modes. Considering the uncertainty of the effective size for the molecular state, we plot the ratio as a function of the root mean square radius r mean in Fig. 3. The ratio is which roughly accords with the experiment result R exp Z c (3900) = 2.2 ± 0.9 [43] within errors. Meanwhile the ratio is insensitive to r mean in the range of (1.0∼3.0) fm we considered in this work.
With the estimated relative spacial wave function as illustrated in Eq. (24), we further obtain the partial widths of the η c ρ and J/ψπ decay modes and show them in Fig. 4. It is obvious that the partial widths are sensitive to r mean and vary from one MeV to O(10 −2 ) MeV. With r mean increasing, the partial decay widths become smaller, or even close to zero. This can be easily understood since the larger r mean means the freer mesons A and B. It is more difficult to interact with each other and the effective potential tends to vanish. Moreover, for an S -wave molecule composed of two mesons A and B, its size may be estimated by [6,59,60] with the reduced mass µ = m A m B m A +m B . Then, the typical size of Z c (3900) is estimated to be r mean ≃ 1.14 fm. Hence we obtain for Z c (3900) with a mass of M = 3886.6 MeV (see Table IX). For Z c (4020), we take it as the S -wave D * D * resonance molecular state since its mass (M=4024.1 MeV) is slightly about 10 MeV higher than the mass threshold of the D * D * . With the molecular size varying in the range of r mean =(1.0∼3.0) fm, we calculate its partial decay width ratio between the η c ρ and J/ψπ modes, and obtain with the mass being M=4024.1 MeV (see Fig. 3). This value is almost independent of r mean we considered in the present work.
We also plot the partial decay widths of the η c ρ and J/ψπ modes versus the molecular size r mean in Fig. 4. In the figure, we find that the partial widths are about O(10 −1 ∼ 10 −2 ) MeV, and strongly dependent on r mean . Fixing r mean ≃ 1.37 fm estimated by Eq. (30), we obtain The predicted branching ratios are which are quite small. The partial widths of the η c ρ and J/ψπ decay modes for Z c (4020) are smaller than those for Z c (3900), This indicates that the couplings of the DD * to the η c ρ and J/ψπ channels are stronger than those of the D * D * . The main difference between the DD * and D * D * is the spin wave function. Our results show that in the molecular scenario, different spin-spin coupling may have a great impact on the strong decay properties. We take the J/ψπ decay mode as an example.
In Table IV, the spin factor for the coupling with the DD * is three times larger than that of the D * D * in Fig. 1-C1 and Fig. 1-T 1. The hyperfine interaction is expected to be more important for the J/ψπ decay mode of Z c (3900). Moreover, our calculation shows that the hyperfine interaction for the Z c (3900) plays a quite important role in Fig. 1-C1 and even change the sign of its amplitude. As to Z c (4430), in molecular scenario, we take it as an Swave D(2S )D * molecular state. Similarly we change the size of the molecular state in the range of r mean =(1.0∼3.0) fm, and obtain for Z c (4430) with a mass of M=4478 MeV (see Fig. 3). Meanwhile, the partial widths of the η c ρ and J/ψπ modes as the function of r mean for Z c (4430) are shown in Fig. 4 as well.
According to the figure, the decay properties of the D(2S )D * molecular state are similar to the D * D * molecular state. Fixing r mean ≃ 1.00 fm, we further obtain At present, the charged state Z c (4430) was observed both in the ψ ′ π ± and J/ψπ ± channels [9][10][11][12][13][14], and has not been reported in the η c ρ channel. According to our theoretical predictions, if Z c (4430) is a D(2S )D * molecular state, the partial width of η c ρ is comparable to that of J/ψπ, which indicates this state may be observed in the η c ρ channel as well. So far, we have obtained the decay ratios in the molecular scenario. We find that the ratios are not sensitive to the relative

B. The tetraquark scenario
In the tetraquark scenario, we obtain the decay ratio for the Z c (3900) state which agrees with the experimental result (see table X). The predicted partial decay widths of the η c ρ and J/ψπ modes are Via imitating the wave function of Z c (3900), we estimate the wave function of Z c (4020) as listed in Table VII. Similarly we fix the mass of Z c (4020) at M = 4024.1 MeV and obtain Then the predicted partial decay widths ratio is The decay properties of Z c (4020) are similar in the molecular and tetraquark scenarios. Thus, besides the decay ratios, more precise experimental information is required to pin down the inner structure of this state. As shown in table X, the partial widths of the η c ρ and J/ψπ decay modes for Z c (4020) are comparable to those for Z c (3900), The ratios are very different from those in Eqs. (35)- (36). As mentioned earlier, in the molecular scenario the hyperfine interaction plays a quite important role for Z c (3900) in Fig.2-C1 and even changes the sign of its amplitude. Thus the total amplitudes for Z c (3900) are much larger than those for Z c (4020). However, in the tetraquark scenario the Coulomb and linear interactions are dominant for both states Z c (3900) and Z c (4020). There exists good evidence for Z c (3900) in the η c ρ and J/ψπ channels experimentally and no evidence for Z c (4020). Our results support that the two states are more likely to be the molecular states. For the Z c (4430), with the estimated wave function we plot the partial decay width ratio between the η c ρ and J/ψπ decay modes as a function of the effective size r X of the tetraquark state (see Fig. 5). We find that the ratio slightly depends on r X . Varying the r X in the range r X = (0.5 − 2.0) fm, the ratio is which is slightly larger than that as a molecular state. According to our results, the branching fraction ratio between the η c ρ and J/ψπ modes of the Z c (4430) as a molecule or a tetraquark state is larger than one, which indicates that the Z c (4430) is more easier to decay into the η c ρ channel. So far, we have calculated the decay ratios of the Z c (3900), Z c (4020) and Z c (4430) decaying into the η c ρ and J/ψπ channels in the molecular and tetraquark scenarios. Our results show that the decay ratios in both scenarios are similar to each other. We cannot determine the inner structures only with the decay ratios. However, if we look at the partial decay widths, we find that in molecular scenario, the Γ(Z c (3900) → J/ψπ) are much larger than the Γ(Z c (4020) → J/ψπ), while they are similar in the tetraquark scenario. In experiments, the Z c (3900) state is observed in the J/ψπ invariant mass spectrum while no significant Z c (4020) signal is observed. This may support the states Z c (3900) and Z c (4020) as the molecules instead of tightly bound tetraquark states.

IV. SUMMARY
In the present work, we calculate the branching fraction ratios between the η c ρ and J/ψπ decay modes for the charged states Z c (3900), Z c (4020) and Z c (4430) with a quark interchange model. In order to compare the decay properties in different physical scenarios and pin down the inner structure of these three mysterious charmonium-like states, we study the ratios in the molecular and tetraquark scenarios, respectively. Meanwhile, we estimate the absolute partial decay widths for the η c ρ and J/ψπ decay channels. Our main results are summarized as follows.
For Z c (4430), the branching fraction ratio as an S-wave D(2S )D * molecule (R th Z c (4430) ≃ 1.4) is slightly smaller than that in the tetraquark scenario (R th Z c (4430) ≃ 1.7 ∼ 1.4). We notice that the ratios in both two physical scenarios are larger than 1, which indicates that the Z c (4430) prefers to decay into the ρη c channel rather than the πJ/ψ channel. Besides the πJ/ψ channel, the ρη c may be another interesting channel for the observation of Z c (4430) in future experiments. For Z c (3900), we obtain that the ratios are R th Z c (3900) ∼ 1.3 and 1.6 in the molecular and tetraquark scenarios, respectively. Both are comparable with the experimental result. For Z c (4020), the ratios are R th Z c (4020) ∼ 2.4 and 1.6, respectively. The above results show that the ratios in both scenarios are similar to each other. Thus, to investigate the inner structures, considering only the decay ratio R of the Z c itself is not enough.
In the molecular scenario, the partial decay widths of the η c ρ and J/ψπ modes for Z c (4020)(D * D * ) are smaller than those for Z c (3900)(DD * ) by one order. In the molecular scenario, different spin-spin coupling may have a great impact on the strong decay properties. On the other side, the partial decay widths of the η c ρ and J/ψπ modes for Z c (4020)(D * D * ) are comparable to those for Z c (3900) in the tetraquark scenario. At present, there exists good evidence for Z c (3900) in the η c ρ and J/ψπ channels experimentally and no evidence for Z c (4020). Our results indicate that these two states are more likely to be the molecule-like states which arise from the D ( * )D( * ) hadronic interactions.