Open charm and bottom tetraquarks in an extended relativized quark model

In present work, we systematically calculate the mass spectra of open charm and bottom tetraquarks $q q\bar q \bar Q$ within an extended relativized quark model. The four-body relativized Hamiltonians including the Coulomb potential, confining potential, spin-spin interactions, and relativistic corrections are solved by using the variational method. It is found that the predicted masses of four $0^+$ $ud\bar s \bar c$ states are 2765, 3065, 3152, and 3396 MeV, which disfavor the assignment of the newly observed $X_0(2900)$ as a compact tetraquark. Moreover, the whole mass spectra of the open charm and bottom tetraquarks show quite similar patterns, which preserve the light flavor SU(3) symmetry and heavy quark symmetry well. In addition, our results suggest that the flavor exotic states $nn\bar s \bar c$, $nn\bar s \bar b$, $ss\bar n \bar c$, and $ss\bar n \bar b$ and their antiparticles can be searched in the heavy-light meson plus kaon final states by future experiments. More theoretical and experimental efforts are needed to investigate these singly heavy tetraquarks.


I. INTRODUCTION
In the past years, many new hadronic states have been observed experimentally, and some of them cannot be simply assigned into the conventional mesons or baryons. This significant progress in experiments has triggered plenty of theoretical interests and made the study of those exotic states as one of the intriguing topics in hadronic physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Among those states, the charged resonances Z c(b) [15][16][17][18][19], fully heavy tetraquark X(6900) [20], and pentaquarks P c [21,22], are particularly interesting, since they cannot mix with traditional hadrons in the heavy quark sectors. Besides hidden charm and bottom states, the existence of flavor exotic states, where quarks and antiquarks cannot annihilate though strong and electromagnetic interactions, was also predicted. Therefore, searching for these flavor exotic states have become increasingly important both theoretically and experimentally.
Very recently, the LHCb Collaboration reported the observation of an exotic structure near 2.9 GeV in the D − K + invariant mass spectrum via the B + → D + D − K + decay channel [82]. Then, this peak is modelled according to two resonances, X 0 (2900) and X 1 (2900). Their parameters are fitted to be Given their D − K + decay mode, the quantum numbers of X 0 (2900) and X 1 (2900) should be J P = 0 + and 1 − , respectively. Also, both of them have four different flavors, which indicates their exotic nature. After the observation of the LHCb Collaboration, these two states near 2.9 GeV were discussed within the simple quark model [83]. The X 0 (2900) was interpreted as an isosinglet compact tetraquark state, while the X 1 (2900) may be regarded as an artifact due to rescattering effects or a J P = 2 +D * K * molecule [83]. Until now, no rigorous four-body calculation for these two states does exist, and all sorts of explanations are possible. Therefore, it is essential to investigate the possible compact tetraquark interpretations of X 0 (2900) and X 1 (2900) within realistic potentials.
In Refs. [84,85], we have extended the relativized quark model proposed by Godfrey and Isgur to investigate the doubly and fully heavy tetraquarks with the original model parameters. This extension allows us to describe the tetraquarks and conventional mesons in a uniform frame. Since the relativized potential can give a unified description of different flavor sectors and involve relativistic effects, it is believed to be more suitable to deal with the heavy-light and light-light quark interactions. In present work, we will systematically investigate the open charm and bottom tetraquarks qqqQ in the extended relativized quark model and test the possible assignments of the newly observed resonances. This paper is organized as follows. In Section II, we briefly introduce the formalism of our extended relativized quark model for tetraquarks. Then, the mass spectra and discussions of our numerical results are presented in Section III. Finally, a short summary is given in the last section.

II. EXTENDED RELATIVIZED QUARK MODEL
To investigate the S −wave mass spectra of open charm and bottom tetraquarks q 1 q 2q3Q4 , the extended relativized quark model is employed [84]. This model is a natural generalization of the relativized quark model to deal with the tetraquark states. The relevant Hamiltonian with quark and gluon degrees of freedom for a q 1 q 2q3Q4 state can be written as where is the relativistic kinetic energy, V oge i j is the one gluon exchange potential together with the spin-spin interactions, and V conf i j corresponds to the confinement potential. The explicit formula and parameters of these relativized potentials can be found in Refs. [84,86].
The wave function for a q 1 q 2q3Q4 state is constituted of four different parts: color, flavor, spin, and spatial wave functions. For the color part, there are two types of colorless states with certain permutation properties, Here, the |33 and |66 are antisymmetric and symmetric color wave functions under the exchange of q 1 q 2 orq 3Q4 , respectively. For the flavor part, the combination q 1 q 2 can be either symmetric or antisymmetric, while theq 3 andQ 4 are treated as different particles without symmetry constraint. To distinguish the up, down, and strange quarks clearly, we adopt the notation "n" to stand for the up or down quark, and "s" to represent the strange quark.
In the spin space, the six spin bases can be expressed as where (q 1 q 2 ) 0 and (q 3Q4 ) 0 are antisymmetric for the two fermions under permutations, while the (q 1 q 2 ) 1 and (q 3Q4 ) 1 are symmetric ones. The relevant matrix elements of the color and spin parts for various types of tetraquark states are identical [84].
In the spatial space, the Jacobi coordinates are shown in Fig. 1. For a q 1 q 2q3Q4 state, one can define and Then, other coordinates of this system can be expressed in terms of r 12 , r 34 , and r [84]. A set of Gaussian functions is adopted to approach the S −wave realistic spatial wave function [87] Ψ(r 12 , r 34 , r) = n 12 ,n 34 ,n C n 12 n 34 n ψ n 12 (r 12 )ψ n 34 (r 34 )ψ n (r), (17) where C n 12 n 34 n are the expansion coefficients. The ψ n 12 (r 12 )ψ n 34 (r 34 )ψ n (r) is the position representation of the basis |n 12 n 34 n , where ψ n (r) = 2 7/4 ν 3/4 n π 1/4 e −ν n r 2 Y 00 (r) = 2ν n π The final results are independent with geometric Gaussian size parameters r 1 , a, and N max when adequate bases are chosen [87]. The ψ n 12 (r 12 ) and ψ n 34 (r 34 ) can be expressed in a similar way, and the momentum representation of the basis |n 12 n 34 n can be obtained via Fourier transformation. The numerical error of our approach has been analyzed in Ref. [84], which is sufficient for quark model predictions.
According to the Pauli exclusion principle, the wave function of a tetraquark should be antisymmetric for the identical quarks and antiquarks. All possible configurations for the q 1 q 2q3Q4 systems are listed in Table I. For the notations, the subscripts and superscripts are the spin and color types, respectively. The brackets [ ] and braces { } stand for the antisymmetric and symmetric flavor wave functions, respectively. The parentheses ( ) are adopted for the subsystems without permutation symmetries. With the total wave functions, all the matrix elements involved in the Hamiltonian can be worked out straightforwardly. The masses without mixing mechanism can be obtained by solving the generalized eigenvalue problem where the H i j are the matrix elements of Hamiltonian, N i j are the overlap matrix elements of the Gaussian bases due to their nonorthogonality, E is the mass, and C j is the eigenvector corresponding to the expansion coefficients C n 12 n 34 n for the spatial wave function. For a given system, different configurations with same I(J P ) should mix with each other. The final mass spectra and wave functions are obtained by diagonalizing the mass matrix of these configurations.

III. RESULTS AND DISCUSSIONS
In this work, we adopt N 3 max = 10 3 Gaussian bases to calculate the mass spectra of S −wave q 1 q 2q3Q4 tetraquark states systematacially. Under these large bases, our numerical results are stable enough for quark model estimations. According to the number of strange quarks, one can classify these open charm and bottom tetraquarks into four groups. We will examine the mass spectra of these systems successively.
The predicted masses for the nonstrange tetraquarks nnnc and nnnb are presented in Table II. For the nnnc system, the masses lie in the range of 2570 ∼ 3327MeV, while the masses of nnnb states vary from 5977 to 6621 MeV. It can be noticed that these mass regions have significant overlap with the excited charmed and bottom mesons. From the Review of Particle Physics [88], there exist several higher charmed and bottom states, which may correspond to the nnnc and nnnb tetraquark states. However, these observed resonances can be described under the conventional interpretations well. In fact, the physical resonances may be the admixtures of the conventional mesons and tetraquarks, which disturbs our understanding. The more efficient way is to hunt for the flavor exotic states uudc, ddūc, uudb, and ddūb. With large phase space, these flavor exotic states and their antiparticles can easily decay into the conventional charmed or bottom mesons by emitting one or more pions, which can be searched in future experiments.
There exist several types of flavor contents for the tetraquark states including one strange quark, and the calculated mass spectra are shown in Table. III. Given the D − K + decay mode, the newly observed X 0 (2900) and X 1 (2900) should belong to the udsc states, and their isospins can be either 0 or 1. From Table. III, it can be seen that the predicted masses of 0(0 + ) states are 2765 and 3125 MeV, where the large splitting arises from the significant mixing scheme of pure |33 and |66 states. Also, the mass of the lowest 1(0 + ) nnsc state is 3065 MeV, which is larger than the experimental data. Our results disfavor the observed X 0 (2900) as a compact udsc tetraquark. Since the parity of X 1 (2900) is negative, it has one orbital excitation at least. From our calculations of S −wave states, the P−wave nnsc states should have rather large masses, which excludes the assignment of X 1 (2900) as a nnsc compact tetraquark state. Other interpretations, such as molecules and kinematic effects, are possible for these two states. In Ref. [81], the authors adopt the colormagnetic interaction model to obtain four 0 + csūd states with masses of 2320, 2607, 2850, and 3129 MeV, respectively. It seems that the X 0 (2900) may be assigned as a higher 0 + compact tetraquark through it mass, but the predicted decay width is significant smaller than experimental data. In Ref. [83], the lowest 0 + udsc state is estimated to be 2754 MeV in baryonic-quark picture or 2863 MeV in the string-junction picture within the simple quark model, where the X 0 (2900) can be regarded as a compact tetraquark. The color-magnetic interaction model and simple quark model also disfavor the X 1 (2900) as a compact tetraquark state, which is consistent with our calculations. The differences among these works arise from the different choices of interactions. It should be mentioned that a unified treatment of mesons and tetraquarks are essential to obtain the reliable mass spectra of open charm and bottom tetraquarks, and further investigations with various approaches are encouraged.
The predicted color proportions and root mean square radii of the lower nnsc states are listed in Table IV. Our results    · · · · · · {ss}(nc) · · · · · · {ss}(nb) · · · · · · {ss}(sb) 0(0 + ) |{ss}3 1 (sb) 3 1 0 show that these states have relatively small root mean square radii, which indicates that all of them have compact inner structures. The sketch of these udsc tetraquarks is also plotted in Fig. 2. It can be seen that the four quarks are separated from each other in a compact tetraquark, which is quite different with the diquark-antidiquark or loosely bound molecular picture.
The nnsb states are the bottom partners of nnsc states, and all the predicted masses lie above the respective thresholds. In Ref. [75], the author proposed a possible stable bsūd state, while the calculations under potential model indicate that no stable diquark-antidiquark bsūd state exists [77]. Also, the color-magnetic interaction model suggest that the lowest 0 + and 1 + compact bsūd states should be near the relevant thresholds [81]. More theoretical and experimental efforts are needed to clarify this problem.
Unlike the nnsc and nnsb states, some of the nsnc and nsnb states can mix with the conventional charmed-strange and bottom-strange mesons. At the early stage, the investigations on nsnc and nsnb states mainly focus on the possible tetraquark interpretation of the D * s0 (2317). Since the light quark pair can create or annihilate easily, it is difficult to distinguish various explanations of D * s0 (2317). Hence, searching for the flavor exotic states usdc, dsūc, usdb, and dsūb seems to be more worthwhile. After the observation of X(5568), A plenty of studies on the nsnb system have been performed. Our results on nsnc and nsnb states are much higher than the masses of D * s0 (2317) and X(5568), which exclude their tetraquark interpretations.
For the tetraquarks nnsc and nnsb, they can decay into thē D ( * ) K ( * ) and B ( * ) K ( * ) final states via fall apart mechanism, respectively. For the nsnc and nsnb states, the possible decay channels areD ( * )K( * ) , D ( * )− s π and B ( * )K( * ) and B ( * ) s π. Certainly, the antiparticles of these tetraquarks decay into the similar final states under charge conjugate transformation. We hope our results can provide helpful information for hunting for these flavor exotic tetraquarks in future LHCb and BelleII experiments.
The masses of tetraquark states including two strange quarks are listed in Table. V. For the nssc and nssb states, the predicted masses are much higher than the relevant thresholds, which may fall apart easily. Also, these states can mix with   The calculated masses of sssc and sssb systems are presented in Table. VI, which lie above 3300 and 6600 MeV, respectively. These states can mix with the conventional charmed-strange and bottom-strange mesons via the strange quark pair annihilation. Current experiments have not investigated these energy regions, and future experimental searches can test our calculations.
Finally, we plot the full mass spectra of open charm and bottom tetraquarks in Fig. 3. It can be seen that the spectra for various systems show quite similar patterns, which indicates that the approximate light flavor SU(3) symmetry and heavy quark symmetry are preserved well for the ground states of singly heavy tetraquarks. These two symmetries have achieved great successes for the traditional hadrons, which will also provide a powerful tool for us to investigate the singly heavy tetraquarks. Compared with the prosperities of conventional heavy-light mesons and singly heavy baryons, the studies on singly heavy tetraquarks are far from enough. More theoretical and experimental efforts are encouraged to increase our understanding on these systems.

IV. SUMMARY
In this work, we systematically investigate the mass spectra of open charm and bottom tetraquarks qqqQ within an extended relativized quark model. By using the variational method, the four-body relativized Hamiltonian including the Coulomb potential, confining potential, spin-spin interactions, and relativistic corrections are solved. The predicted masses of four 0 + udsc states are 2765, 3065, 3152, and 3396 MeV, which disfavors the assignment of X 0 (2900) as a compact tetraquark.
The whole mass spectra of open charm and bottom tetraquark show quite similar patterns, which preserves the light flavor SU(3) symmetry and heavy quark symmetry well. Besides the mass spectra, the possible decay modes are also discussed. Our results suggest that the future experiments can search for the flavor exotic states nnsc, nnsb, ssnc, and ssnb in the heavy-light meson plus kaon final states. More theoretical and experimental efforts are needed to investigate these singly heavy tetraquarks.