Mass spectrum and strong decays of tetraquark $\bar c\bar s qq$ states

We systematically study the mass spectrum and strong decays of the S-wave $\bar c\bar s q q$ states in the compact tetraquark scenario with the quark model. The key ingredients of the model are the Coulomb, the linear confinement, and the hyperfine interactions. The hyperfine potential leads to the mixing between different color configurations, as well as the large mass splitting between the two ground states with $I(J^P)=0(0^+)$ and $I(J^P)=1(0^+)$. We calculate their strong decay amplitudes into the $\bar D^{(*)}K^{(*)}$ channels with the wave functions from the mass spectrum calculation and the quark interchange method. We examine the interpretation of the recently observed $X_0(2900)$ as a tetraquark state. The mass and decay width of the $I(J^P)=1(0^+)$ state are $M=2941$ MeV and $\Gamma_X=26.6$ MeV, respectively, which indicates that it might be a good candidate for the $X_0(2900)$. Meanwhile, we also obtain an isospin partner state $I(J^P)=0(0^+)$ with $M=2649$ MeV and $\Gamma_{X\rightarrow \bar D K}=48.1$ MeV, respectively. Future experimental search for $X(2649)$ will be very helpful.

The nature of the X(2900) states is still controversial. Most of the investigations mentioned above focus on the mass spectroscopy of X(2900), which is a major platform to probe the dynamics of the multi-quark system. However, compared with the conventional baryons and mesons, the tetraquark states are more complicated. For example, the tetraquark states contain much richer color configurations, i.e.,3 c − 3 c and 6 c −6 c components [28]. Therefore, the quark models that succeed in the conventional mesons (qq) and baryons qqq, might have large uncertainties in the multi-quark systems. The other properties such as production and decay patterns become essential, which are very sensitive to the inner structures of the multiquark system. The authors in Ref. [29] discussed the productions and the decays of the X(2900) in different physical scenarios, including the triangle diagrams, the molecules and the tetraquarks.
In this paper, we study the four-quark statecsqq in the tetraquark scenario and examine whether the X 0 (2900) can be accommodated as a compact tetraquark. We first extend our previous work [28] to study the openflavor systemcsqq spectrum within quark model considering the mixing effect of the two color configurations.
The paper is arranged as follows. In Sec. II, we introduce the Hamiltonian for thecsqq system. With the quark model, we calculate the mass spectrum of the Swave tetraquark states and present the results in Sec. III. With the same quark-quark interaction, we investigate their strong decays into theD ( * ) K ( * ) channel using the quark interchange model. The details are elaborated in Sec. IV. We give a summary in Sec. V.

II. HAMILTONIAN
In a tetraquark state, the four-quark Hamiltonian reads, with p j and m j denote the momentum and mass of the inner quark with index j, respectively. T G is the total kinematic energy of the system and vanishes in the center-of-mass frame. The V ij is the quark-quark interaction between the quark pair (ij). In this work, we use the non-relativistic quark model proposed in Ref. [36] to describe the quark-quark interaction, with where r ij is the radius between the ith and jth quarks. The λ i (−λ T i ) is the color generator for the quark (antiquark). The first two terms are the central parts of the model, the Coulomb and linear confinement potentials. The last term in the V ij is the hyperfine interaction with the smearing effect parameterized by r 0 , which is related to the reduced mass of the interacting quarks. The s i is the spin operator of the ith quark. The hyperfine potential contains all the flavor information and is expected to play an important role for the mass spectrum. The values of the parameters are determined by fitting the mass spectra of the mesons in Table I.

A. Wave function
The spatial wave function of the four-body system can be described by three independent Jacobi coordinates. For thecsqq state, there are two sets of Jacobi coordinates as presented in Fig. 1. In general, if we use one set of Jacobi coordinates with complete spatial bases and the complete color configurations, we can describe the four-quark system. In this work, we use the Jacobi coordinate in Fig. 1 (a) to expand the wave function of the tetraquark state. It reads where the (1, 2, 3, 4) denote the four quarks (c,s, q, q), respectively. J (J z ) and I (I z ) are the total angular momentum and the isospin (the third component) of the tetraquark state. The l a , l b , and l ab represent the orbital angular momentum within the (cs) cluster, the (qq) cluster and that between these two clusters, respectively. The l a and l b couples into the orbital angular momentum ℓ. Then ℓ couples with the l ab into the total orbital angular momentum L. The s a and s b are the spin of the (cs) and (qq) clusters. They form the total spin of the tetraquark S. The Y lm is the spherical harmonics function. The sum of the script α represents the superposition of the four-body bases that satisfy the quantum number (JJ z , II z ). A α is the expanding coefficient for the corresponding basis. The χ c , χ s , and ξ are the wave functions in the color, spin and flavor space, respectively. The subscript 1 c denotes that the four quarks form a color singlet state. The φ is the radial wave function in spatial space and reads where ν na is related to the oscillating parameter β a as follows, the n a is the radial quantum number. n max a is the number of the expanding bases along r a . The spatial wave functions φ n b l b (r 34 , β b ) and φ n ab l ab (r, β ab ) have the similar forms. In this work, we concentrate on the S-wave tetraquark state, which should be a superpositions of the states |l a = l b = l = 0 , |l a = l b = 1, l = 0 , |l a = 2, l b = l = 0 etc.. In conventional hadrons, the mass gap between the ground state and its first orbital excitation is about hundreds of MeV. The same pattern is expected in the compact multi-quark states. In addition, the high orbital excitations couple with the ground states through the tensor and spin-orbital potentials, which can be treated as perturbative compared with the Coulomb and linear confinement potentials. Thus, we only expand the tetraquark states by the ground spatial bases.
In the statecsqq, the color-spin-flavor wave configuration is constrained by the Fermi-statistic and the possible wave functions are listed in Table II.

B. Results
By solving the Schrödinger equation with the variational method, we obtain the mass spectrum and present the results in Fig. 2. The explicit values of the mass spectrum and the oscillating parameters are summarized in Table III. In Fig. 2, we find that the tetraquarks with the isospin I = 1 locate higher than those with I = 0. Especially, the ground I(J P ) = 1(0 + ) state is about 300 MeV heavier than the I(J P ) = 0(0 + ) one. This mass difference is about 200 MeV in Ref. [27]. As illustrated in Table II, the I(J P ) = 0(0 + ) and I(J P ) = 1(0 + ) states contain the same color structures but different spin-flavor configurations. Thus, with the color interactions only, i.e., the Coulomb and linear confinement potentials, the mass spectra of the two states should be the same. The mass difference comes from the contribution of the hyperfine potential in view of different spin wave functions. The quark model which we adopted in this work success-fully described the large mass splitting between the π and ρ mesons, which indicates that the hyperfine interaction should be very important for the light quarks. In thē csqq system, the significant hyperfine potential for the diquark qq leads to the large mass splitting between the I(J P ) = 1(0 + ) and I(J P ) = 0(0 + ) states.
In Table II, the I(J P ) = 0(0 + ) and I(J P ) = 1(0 + ) tetraquark states contain two color configurations [(cs) 3c (qq)3 c ] 1c and [(cs) 6c (qq) 6c ] 1c , respectively. We first discuss the mass spectra of the tetraquark states without considering the coupling between the 3 c −3 c and6 c − 6 c states and display the results in Fig. 3. For the I = 1 state, the6 c − 6 c state is located higher than the 3 c −3 c one, while the situation is reversed for the I = 0 state.
With the mixing effect between the 3 c −3 c and6 c − 6 c states, we obtain the mass spectrum and present them in the right panel of Fig. 3. The mixing effect comes from the quark-antiquark interactions between the diquark and the antiquark. With the quark model in Eq. 2, the coupling constants in the Coulomb and linear confinement interactions are flavor-independent. Thus the two interactions cancel with each other exactly and do not contribute to the color configuration mixing [28]. Only the hyperfine interaction contributes. This interaction is inversely proportional to the masses of the interacting quarks. In Fig. 3, one finds that the mixing effects shift the masses at the order of 100 MeV, which is much larger than those in the fully heavy tetraquark states [28].
We also investigate the inner structures of the tetraquark states, including the proportions of different color configurations as listed in Tables IV-VI and the root mean square radii between different interacting quarks in Table VII. The proportions show that the ground S-wave states contain important color configuration [(cq) 8c (sq) 8c ] 1c . As illustrated in Table VII, almost all of the root mean square radii are smaller than 1 fm, which indicates that the four quarks are compactly II. The color-spin-flavor wave functions of the S-wave tetraquark statescsq1q2 with different J P quantum numbers. The subscripts and superscripts denote the color representation and the spin, respectively. The symbol {q1q2} and [q1q2] represent that the two light quarks are symmetric and antisymmetric in the flavor space, respectively. The (βa, β b , β ab ) are the oscillating parameters in the spatial wave functions of the antidiquarkcs, the diquark q1q2 and that between the two clusters. bound within the tetraquark states. One should be careful in identifying the states as the resonances . Since we have used the finite numbers of the bases to expand the tetraquark state, the eigenvectors may correspond to the bound or the resonance states, and the continuum in the finite space. To distinguish the resonance from the continuum, we may include their coupling and investigate whether the tetraquark states may remain as resonances or not by the real scaling method in the future [38,39].
The mass of the I(J P ) = 1(0 + ) state is 2941 MeV, which is 75 MeV larger than that of the X 0 (2900) state. Such a mass deviation is not very significant and conclusive if one considers the uncertainty of the quark model. One notices that the model parameters used in the mass spectrum calculation are obtained from the conventional hadrons. As a phenomenological model, the parameters may change for different systems. For instance, in Refs. [40] and [41], the authors studied the mass spectra of the mesons and the baryons in a unified relativized quark model. The overall constant term in the meson-based potential was −340 MeV , while the baryonbased one was selected as −615 MeV to obtain better solutions for the baryons qqq. When the meson-based quark model is applied to the multi-quark systems, the similar modification might also be essential. Furthermore, the confinement mechanisms in the multi-quark systems are still poorly understood. For example, because of the rich color configurations in the multi-quark system, the three-body interaction arising from the threegluon vertex may also contribute, while it vanishes for the conventional hadrons. Considering the uncertainties of quark model, the X(2900) signal seems a candidate of the I(J P ) = 1(0 + ) state. Besides the X(2941), we also predict the other states such as the I(J P ) = 0(0 + ) state with the mass around 2649 MeV.

IV. DECAY WIDTH
In this section, we calculate the strong decay widths of the teraquark statescsqq. The dominant decay modes of tetraquark statescsqq are X →D ( * ) K ( * ) when the phase spaces are allowed. The tetraquarks may also decay into theDK ( * ) π final state. The three-body decays TABLE III. The mass spectra (in units of MeV) of the tetraquark statescsqq with different J P . The β (a/b) , γ (a/b) and λ (a/b) are the oscillating parameters for different states as listed in Table II. In this work, we use N = 2 3 bases to expand the wave functions of the tetraquark statecsqq. Here, we list the results for the S-wave ground and the first radially excited states.  are largely suppressed compared with the two-body decays due to the phase space suppression. Thus, we do not consider them in this work.

A. Quark interchange model
We first give a brief introduction to the quarkinterchange model for the decaȳ where B and C are two color singlet mesons. The decay process occurs through interchanging the constituent quarks (antiquarks) as illustrated in Fig. 4, followed by hadronization into two mesons. This method has been applied to calculate the I = 3/2 Kπ scattering [42], the short-range NN interaction [43], the strong decays of the Z c states [44], X(3872) [45], and the pentaquark states [46]. In the decay, the interacting Hamiltonian where V ij is the potential as listed in Eq. (2). The Tmatrix element is the sum of the four diagrams in Fig. 4, which are calculated as the overlaps of the wave functions with the Hamiltonian between the initial and final states. The wave functions of the initial tetraquark states and final mesons are obtained from the same quark model in Eq. (3). For each diagram, the T f i is written as the product of factors (as defined in Ref. [44]) The element I flavor is the overlap of wave functions in the flavor space. The color and the spin matrix elements are listed in Table X in Appendix. The calculation details of the spatial matrix element I space are referred to Ref. [44]. The differential decay width is given by where p B is the momentum of the final state. M is the mass of the initial tetraquark state. The decay amplitude M is related to the T-matrix T f i by where the E B and E C are the energies of the two mesons in the final state.

B. Numerical results
We present the decay widths of the tetraquark states in Table VIII. The J P = 0 + states are of special interest because they might be the candidates for the recently observed X 0 (2900) in the LHCb Collaboration. Two Swave decay modes are available to the states,DK and D * K * . For the X(2941) and the X(2649), we predict the partial decay widths into theDK channel as 20.1 MeV and 48.1 MeV, respectively. And the partial width ratio is Γ DK (X(2941)) Γ DK (X(2649)) = 0.4.
The X(2941) also decays into theD * K * with a smaller decay width 6.5 MeV, while the mode is energetically forbidden for the X(2649). In Sec. III B, the mass of X(2941) state is larger than the X 0 (2900) mass by 75 MeV. The predicted decay width 26.6 MeV is reasonably close to the experimental value of 57.2 ± 12.9 MeV. If we regard the X(2941) as the X 0 (2900) state and use the experimental value as the input mass in the decay, theD * K * decay mode is forbidden. It decays into thē DK channel with the decay width 21.7 MeV. The partial width ratio is ΓD K (X(2900)) ΓD K (X(2649)) = 0.5.
The ratio indicates that if the X 0 (2900) is the I(J P ) = 1(0 + ) state, there might exist the other I(J P ) = 0(0 + ) state X(2649) in theDK channel. More experimental study is expected in future to test the tetraquark explanation for the X 0 (2900). For the J P = 0 + states, the X(3222) and X(3013) states are promising to be observed inD * K * mode. Thē DK widths of the two states are much smaller than those of theD * K * ones, despite the larger phase spaces. The Γ DK (X(3222)) is predicted to be 0.01 MeV, as it weekly couples with theDK channel. The J P = 2 + states also decay into theD * K * mode. However, the partial decay widths are rather small and these tetraquark states may be difficult to be observed in this mode.
The J P = 1 + states may decay into theD * K,DK * andD * K * channels. Most of the predicted widths are much smaller than those of the J P = 0 + states except the X(2838) and X(2972). They both have large partial  decay widths into theD * K andDK * , respectively.

V. SUMMARY
In this work, we evaluate the mass spectrum and the decay widths of the system csqq with the quark model. We include the most important ingredients of the quark model, the Coulomb, the linear confinement, and the hyperfine interactions. With the results, we examine whether the X 0 (2900) can be interpreted as a tetraquark state.
We first solve the Schrödinger equation with the variational method. We obtain the mass spectrum of the Swavecsqq states with J P = 0 + , 1 + and 2 + . The J P = 0 + and 1 + states are the mixtures of different color-spin- TABLE VIII. The decay widths (in units of MeV) for the tetraquark statescsqq decaying into theD ( * ) K ( * ) . The script "−" presents that the corresponding decay mode is forbidden by the phase space. flavor configurations. In the quark model, the color interaction does not mix different color-spin configurations.
Only the hyperfine interactions contribute to their mixing effects. Since thecsqq state contains the light quark, the mixing effect is larger than that in the fully heavy tetraquark state. With the Fierz transformation, we obtain the proportions of the (cq) 1c (sq) 1c and (cq) 8c (sq) 8c components. The ground state contains large color component (cq) 8c (sq) 8c . For the J P = 0 +cs qq system, the two isospin partner states have the same color configurations but different color-spin configurations. The mass splittings come from the hyperfine interactions.
With the obtained wave functions from the mass spectrum calculation, we study the strong decays for the tetraquarkscsqq into theD ( * ) K ( * ) channels using the quark interchange model. The partial decay widths might be helpful in the search for tetraquark statescsqq, as they suggest the golden channel to reconstruct the tetraquark states.
Combined the mass spectrum and decay width, the X(2941) with the quantum number I(J P ) = 1(0 + ) seems to be a good candidate for the X 0 (2900). Its mass and decay width are M = 2941 MeV and Γ = 26.6 MeV, respectively. The partial decay widths into theDK and D * K * modes are 20.1 and 6.5 MeV. If the X(2941) is treated as X 0 (2900), theD * K * mode is kinematically forbidden and theDK width is 21.7 MeV, which is close to the experimental data. Remarkably, we also find another state I(J P ) = 0(0 + ) with the M = 2649 MeV and Γ X→DK = 48.1 MeV at the same time. The decay width ratio Γ(X(2941) → DK)/Γ(X(2649) → DK) = 0.4 indicates that if the X(2941) is regarded as the X(2900), there might be another X(2649) state in theDK channel. In addition, one may also expect other states to be observed in theD * K,D * K, andD * K * channels. The possible observation channels in experiments are related to their partial decay widths.
So far, we have evaluated the mass spectrum and the decay width with a meson-based quark model. The potentials in the multi-quark system might be different from the meson-based potentials. For instance, the parameters might change and the different confinement mechanisms might show up, which may bring uncertainties to our results and will be discussed lately. We expect more experimental measurements in future to help test our results and improve the understanding of the interaction in the multi-quark system, especially the confinement mechanisms and dynamics in the strong decay. ACKNOWLEDGMENTS G. J. Wang is very grateful to X. Z. Weng for very helpful discussions. This project is supported by the National Natural Science Foundation of China under Grant 11975033. G.J.Wang is also supported by China Postdoctoral Science Foundation Grant No. 2019M660279.

APPENDIX
The hyperfine interaction plays an important role in the mass spectrum calculation, because it contributes to the mixing effects of different color-spin configurations, as well as inducing the large mass splitting between the I(J P ) = 1(1 + ) and I(J P ) = 0(0 + ) ground states. We list the color-spin factor λi 2 λj 2 s i · s j of the hyperfine interactions in the mass spectrum calculation in Table IX.