Aspects of $Z_{cs}(3985)$ and $Z_{cs}(4000)$

In the last twenty years there is an explosion in the observation of hidden-charmed multiquark states [1, 2, 3, 4, 5, 6]. For example: dozens of charmonium-like states have been observed since 2003 [7], most of which are good candidates of four-quark states with quark components $c\bar{c}q\bar{q}$($q$ denotes $u$ or $d$ quark)[1, 2, 3, 4, 5, 6]; The LHCb collaboration reported hidden-charm pentaquark states $P_{c}(4380)$ and $P_{c}(4450)$ in 2015 [8] and updated the data in 2019 [9], which minimally contain four quarks and one antiquark($c\bar{c}qq\bar{q}$) [10]; Last year, the LHCb collaboration observed a narrow structure around 6.9 Gev in the $J/\psi$-pair invariant mass spectrum [11], which may be good candidates of compact tetraquark state $cc\bar{c}\bar{c}$ [12]. All of the experimental progress in the observation of hadrons with anomalous properties has attracted a great deal of attention from the hadron physics community.

Very recently the BESIII collaboration made a great breakthrough in searching for hidden-charmed multiquark states with strangeness and reported a new structure near the $D^{0}D_{s}^{*-}$ and $D^{*0}D_{s}^{-}$ mass thresholds in the $K^{+}$ recoil-mass spectrum [13], whose mass and width are respectively

	$\displaystyle M[Z_{cs}(3985)]$	$\displaystyle=$	$\displaystyle 3982.5^{+1.8}_{-2.6}\pm 2.1~{}\text{MeV},$
	$\displaystyle\Gamma[Z_{cs}(3985)]$	$\displaystyle=$	$\displaystyle 12.8^{+5.3}_{-4.4}\pm 3.0~{}\text{MeV}.$		(1)

From its decay modes, it is reasonable to assign it as the first candidate of the hidden-charm tetraquark with strangeness($c\bar{c}s\bar{q}$). Thus, Most of theoretically references are under the four-quark scenario to exploring its genuine properties of the excess [14, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 17, 18, 19, 20, 21, 22, 23, 24, 25, 15, 16, 36]. There also exist a few discussions that assign this structure as a threshold effect [37] or a threshold cusp [38]. Furthermore, explaining $Z_{cs}(3985)$ as a reflection structure of charmed-strange meson $D_{s2}(2573)$ is also presented [39].

Later the LHCb collaboration further observed a exotic states with quark content $c\bar{c}u\bar{s}$(denoted as $Z_{cs}(4000)$) from an amplitude analysis of the $B^{+}\rightarrow J/\psi\phi K^{+}$ decay [40]. Its mass and width are measured to be

	$\displaystyle M[Z_{cs}(4000)]$	$\displaystyle=$	$\displaystyle 4003\pm 6^{+4}_{-14}~{}\text{MeV},$
	$\displaystyle\Gamma[Z_{cs}(4000)]$	$\displaystyle=$	$\displaystyle 131\pm 15\pm 26~{}\text{MeV}.$		(2)

The mass of this state is comparable to that of $Z_{cs}(3985)$ observed by BESIII [13], while its width is about ten times larger than that of $Z_{cs}(3985)$. Thus, whether they are two different states or not, and how to decode their inner structures are now pretty much the agenda for theorists. So far, there are some discussions of their inner structures with various theoretical methods [41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. The most papular interpretation of $Z_{cs}(3985)/Z_{cs}(4000)$ is explaining them as two different states: $DD_{s}^{*}/D^{*}D_{s}$ molecular states [43, 48] or compacted tetraquark states $c\bar{c}s\bar{u}$ with different spin-parity [44, 41, 49].

The masses of the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ states are comparable and both slightly higher than the mass thresholds of $DD_{s}^{*}$ and $D^{*}D_{s}$. In the molecular scenario, we take the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ states as two $DD_{s}^{*}$/$D^{*}D_{s}$ resonance molecular states. Meanwhile, according to experimental measurement, the $Z_{cs}(4000)$ state is observed in the $J/\psi K$ decay channel and non-observation of the $Z_{cs}(3985)$ signal in this decay mode. Thus, to clarify whether $Z_{cs}(3985)$ and $Z_{cs}(4000)$ are two different states and further show light on the inner structures of the two states, it is critical to study the $J/\psi K$ decay mode for the $DD_{s}^{*}$ and $D^{*}D_{s}$ molecular states. In addition, the authors in Ref. [48] predicted the existence of a tensor $D^{*}D_{s}^{*}$ resonance. As the expansion it is necessary to investigate the decay properties of this tensor resonance. Based on the above considerations, in the present work we conduct a systematically study of the hidden-charm strong decays of the $D^{(*)}D_{s}^{(*)}$ molecular states in the framework of the quark-exchange model. Furthermore, for comparison we also investigate the hidden-charm decay properties of the four-quark system $c\bar{c}s\bar{u}$ in the compact tetraquark scenario. We hope that our theoretical results can provide some reference for the properties of the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ states.

This paper is structured as follows. In Sec.II, we give a brief introduction of the quark-exchange model. In Sec.III, we present our theoretical results and discussions. Finally, we make a short summary in Sec.IV.

In this work we calculate the $J/\psi K$, $\eta_{c}K$, $\eta_{c}K^{*}$ and $J/\psi K^{*}$ decay modes for the four-quark system $c\bar{c}s\bar{u}$ in the molecular and compact tetraquark scenarios using the quark-exchange model [51]. This phenomenological model has been used to study the hidden-charm decay properties for the milt-quark states in our previous works [52, 53, 54]. Here we give a brief presentation for this model and more detail information can refer to our previous work [53].

$\mathbf{A.}~{}\mathbf{Decay~{}width}$   For a two-body decay process($I\rightarrow C+D$), the decay width in the rest frame of the initial particle reads

$\displaystyle d\Gamma=\frac{|\vec{p}_{c}|}{32\pi^{2}M^{2}}|\mathcal{M}(I\rightarrow C+D)|^{2}d\Omega.$

(3)

Here, $\vec{p}_{c}$ represents the three-momentum of the final meson $C$; $M$ is the mass of the initial state $I$; $\mathcal{M}(I\rightarrow C+D)$ denotes the transition amplitude, which is related to the $T$-matrix via

$\displaystyle\mathcal{M}(F\rightarrow C+D)=-(2\pi)^{3/2}\sqrt{2M}\sqrt{2E_{C}}\sqrt{2E_{D}}T,$

(4)

where $E_{C}$ and $E_{D}$ represent the energy of the final mesons $C$ and $D$, respectively. The $T$-matrix has the form

	$\displaystyle T$	$\displaystyle=$	$\displaystyle\langle\psi_{CD}(\vec{p}_{c})|V_{\text{eff}}(\vec{k},\vec{p_{c}})|\psi_{I}(\vec{k})\rangle$		(5)
		$\displaystyle=$	$\displaystyle\langle\psi_{CD}(\vec{p}_{c})|V_{\text{eff}}(\vec{k},\vec{p_{c}})|\psi_{AB}(\vec{k})\rangle.$

Here, the initial state is a four-quark state, which concludes constituent clusters $A$ and $B$. In molecular scenario, the constituents are mesons, while in tetraquark scenario the constituents are the diquark $[cq]$ and antidiquark $[\bar{c}\bar{q}]$. Thus, $\psi_{AB}(\vec{k})$ is the normalized relative spatial wave function between the constituent clusters $A$ and $B$. $\psi_{CD}(\vec{p}_{c})$ denotes the relative spatial wave function between the final mesons $C$ and $D$. $V_{\text{eff}}(\vec{k},\vec{p_{c}})$ represents the effective potential, which is a function of the initial and final relative momentum $\vec{k}$ and $\vec{p}_{c}$.

Considering the four-quark state may be a superposition of terms with different orbital angular momenta, the relative spatial wave function in the momentum space reads

$\displaystyle\psi_{AB}(\vec{k})=\sum_{l}R_{nl}(k)Y_{lm}(\frac{\vec{k}}{k}).$

(6)

Then, the Eq. (4) can be written as

	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{1}{(2\pi)^{3}}\int d\vec{k}\int d\vec{p}\delta(\vec{p}-\vec{p}_{c})V_{\text{eff}}(\vec{k},\vec{p_{c}})\sum_{l}R_{nl}(k)Y_{lm}(\frac{\vec{k}}{k})$		(7)
		$\displaystyle=$	$\displaystyle\frac{1}{(2\pi)^{2}}\sum_{l}M_{ll}Y_{lm}(\frac{\vec{p}_{c}}{p_{c}}),$

where

$\displaystyle M_{ll}=\int_{-1}^{1}P_{l}(\mu)d\mu\int dkV_{\text{eff}}(\vec{k},\vec{p}_{c},\mu)R_{nl}(k)k^{2}.$

(8)

In this equation, $\mu$ represents the cosine of the angle between the momenta $\vec{k}$ and $\vec{p}_{c}$; $P_{l}(\mu)$ is Legendre function.

Finally, the decay width of two-body decay progress with the relativistic phase space has the form

$\displaystyle\Gamma=\frac{E_{C}E_{D}|\vec{p}_{c}|}{(2\pi)^{3}M}|M_{ll}|^{2}.$

(9)

$\mathbf{B.~{}Effective~{}potential}$   In the molecular scenario, we treat the four-quark system as the loosely bound $S$-wave $D^{(*)}D_{s}^{(*)}$ molecular states. At Born order, the effective potential $V_{\text{eff}}(\vec{k},\vec{p}_{c},\mu)$ is related to the reacting amplitude of the meson-meson scattering process, which is estimated by the sum of the interactions between the inner quarks as illustrated in Fig. 1 within the quark-exchange model. Moreover, the short-range interactions are dominant and can be approximated by the one-gluon-exchange (OGE) potential $V_{ij}$ at quark level,

$\displaystyle V_{ij}=\frac{\lambda_{i}}{2}\frac{\lambda_{j}}{2}\left\{\frac{4\pi\alpha_{s}}{q^{2}}+\frac{6\pi b}{q^{4}}-\frac{8\pi\alpha_{s}}{3m_{i}m_{j}}\mathbf{s}_{i}\cdot\mathbf{s}_{j}e^{-\frac{q^{2}}{4\sigma^{2}}}\right\},$

(10)

where $\lambda_{i}(\lambda^{T}_{i})$ is the quark (antiquark) generator; $q$ represents the transferred momentum; $b$ corresponds to the string tension; $m_{i}~{}(m_{j})$ and $\mathbf{s}_{i}~{}(\mathbf{s}_{j})$ are the interacting constituent quark mass and spin operator; $\sigma$ denotes the range parameter in the hyperfine spin-spin interaction; $\alpha_{s}$ represents the running coupling constant,

$\displaystyle\alpha_{s}(Q^{2})=\frac{12\pi}{(33-2n_{f})\text{ln}(A+Q^{2}/B^{2})}.$

(11)

Here, $Q^{2}$ denotes the square of the invariant masses of the interacting quarks.

In the quark model, the wave function for a meson is

$\displaystyle\Psi=\omega_{c}\phi_{f}\chi_{s}\psi(\vec{p}),$

(12)

where $\omega_{c}$, $\phi_{f}$, $\chi_{s}$ and $\psi(\vec{p})$ represent the wave functions in the color, flavor, spin and momentum space, respectively. Thus, the effective potential can be given as the product of the factors,

$\displaystyle V_{\text{eff}}(\vec{k},\vec{p}_{c},\mu)=I_{\text{color}}I_{\text{flavor}}I_{\text{spin-space}}.$

(13)

In this equation, $I$ with the subscripts color, flavor and spin-space represent the overlaps of the initial and final wave functions in the corresponding space.

In addition, to calculate the decay widths by Eq. (9), expecting the obtained effective potential $V_{\text{eff}}(\vec{k},\vec{p}_{c},\mu)$, we also need the relative spatial wave function $\psi_{AB}(\vec{k})$ between mesons $A$ and $B$. In this work, we adopt an $S$-wave harmonic oscillator function to estimate the relative spatial wave function in Eq. (6), which is

$\displaystyle R_{00}(k)=\frac{2\text{exp}^{-\frac{k^{2}}{2\alpha^{2}}}}{\pi^{1/4}\alpha^{3/2}}.$

(14)

The value of the harmonic oscillator strength $\alpha$ is related to the root mean square radius $r_{\text{mean}}$, which varies in the range of (1.0-3.0) fm in the present work.

Similar to the molecular case, the $V_{\text{eff}}(\vec{k},\vec{p}_{c},\mu)$ in the tetraquark scenario can be approximated by the interaction between the inner quarks(shown in Fig. 2), and be obtained with Eq. (13) as well. However there is a difference between the two scenarios for the color factor $I_{\text{color}}$: in the molecular scenario the color configuration is $1_{c}$-$1_{c}$, while in the tetraquark scenario it is $3_{c}$-$\bar{3}_{c}$. The difference in color configurations may result in quite different decay properties.

The spatial wave function of the initial tetraquark states are estimated by the $S$-wave harmonic oscillating wave function,

$\displaystyle\Psi(\vec{k}_{r},\vec{k}_{R},\vec{k}_{X})=\psi_{A}(\vec{k}_{r},\alpha_{r})\psi_{B}(\vec{k}_{R},\alpha_{R})\psi_{AB}(\vec{k}_{X},\alpha_{X}),$

(15)

where $\vec{k}_{r/R}$ represent the momentum between the $c(\bar{c})$ and $s(\bar{u})$ quarks in the diquark (antidiquark), and $\vec{k}_{X}$ denotes the one between the diquark $[cs]$ and antidiquark $[\bar{c}\bar{u}]$. The $\alpha$ with the subscripts is the oscillating parameter along the corresponding Jacobi coordinates, which value is given via imitating the wave function of $Z_{c}(3900)$ in Ref. [55] and listed in Table 1.

The narrow state $Z_{cs}(3985)$ reported by BESIII collaboration [13] is observed in the $D^{0}D_{s}^{*-}$ and $D^{*0}D_{s}^{-}$ final channels, while the wide state $Z_{cs}(4000)$ reported by LHCb collaboration [40] is observed in the $J/\psi K$ decay mode. The mass thresholds of $D^{0}D_{s}^{*-}$($\sim 3977$ MeV) and $D^{*0}D_{s}^{-}$($\sim 3975$ MeV) are slightly lower than the masses of $Z_{cs}(3985)$ and $Z_{cs}(4000)$. Hence it is vital to investigate the decay properties of the molecular states $D^{0}D_{s}^{*-}$ and $D^{*0}D_{s}^{-}$. Meanwhile, the mass degeneracy between the molecular states $D^{0}D_{s}^{*-}$ and $D^{*0}D_{s}^{-}$ indicates that the states may well be mixed. Namely the physical states probably are the mixed states between $D^{0}D_{s}^{*-}$ and $D^{*0}D_{s}^{-}$. It is necessary to introduce a parameter $\theta$ to describe the mixability for a better understanding the decay properties. In addition, we also investigate the decay properties of the molecular state $D^{*0}D_{s}^{*-}$ and hope to give some useful theoretical reference for the future experimental exploring. Moreover, we give a brief discussion of the strong decay properties of the four-quark system $c\bar{c}s\bar{u}$ in the compact tetraquark scenario. Our theoretical results are presented as follows.