Investigating Tetraquarks Composed of Us ¯D ¯B and Ud¯S¯B

Eur. Phys. J. C (2019) 79:556 https://doi.org/10.1140/epjc/s10052-019-7065-0

Regular Article - Theoretical Physics

Investigating tetraquarks composed of usd¯b¯ and uds¯b¯

Hongxia Huanga, Jialun Pingb Department of Physics and Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, People’s Republic of China

Received: 17 February 2019 / Accepted: 22 June 2019 / Published online: 2 July 2019 © The Author(s) 2019

Abstract In the framework of the quark delocalization charge-conjugated ones. Unfortunately, this state was not color screening model, we investigate tetraquarks composed confirmed by other collaborations. The LHCb collaboration of usd¯b¯ and uds¯b¯ in two structures: meson-meson structure [2], the CMS collaboration of LHC [3], the CDF collabora- and diquark–antidiquark structure. Neither bound state nor tion of Fermilab [4] and the ATLAS Collaboration of LHC resonance state is found in the system composed of usd¯b¯.The [5] all claimed that no evidence for this state was found. Nev- reported X(5568) cannot be explained as a molecular state or ertheless, the D0 collaboration’s new result still insists on the a diquark–antidiquark resonance of usd¯b¯ in present calcula- existence of this tetraquark X(5568) [6]. Clearly, more other tion. However, two bound states of the diquark–antidiquark measurements are needed. structure are obtained in the tetraquarks system composed of The discovery of this exotic state X(5568) also stimulates uds¯b¯:anIJ = 00 state with the mass of 5701 MeV, and an the theoretical interest. Many approaches have been applied IJ = 01 state with the mass of 5756 MeV, which maybe the to interpret this state, such as the QCD sum rules [7–13], better tetraquark states with four different flavors. Our results quark models [14–16], the extended light front model [17], indicate that the diquark–antidiquark configuration would be rescattering effects [18], and so on. However, several theoret- a good choice for the tetraquarks uds¯b¯ with quantum num- ical calculations gave the negative results [19–22]. For exam- bers IJ = 00 and IJ = 01. The tetraquarks composed of ple, in Ref. [19], authors investigated two structures, diquark– uds¯b¯ is more possible to form bound states than the one com- antidiquark and meson-meson, with all possible color config- posed of usd¯b¯. These bound states are worth investigating in urations by using the Gaussian expansion method, and they future experiments. cannot obtain the reported X(5568).Ref.[20] examined the various interpretations of the state X(5568) and found that the threshold, cusp, molecular, and tetraquark models were 1 Introduction all unfavored the existence of the X(5568). To search for the tetraquark states with four different fla- In the past few decades, the discovery of numbers of exotic vors, a better state is uds¯b¯ (or its charge-conjugated one) states stimulated extensive interest in understanding the with replacing the d/s¯ in X(5568) by d¯/s [23]. Obviously, structures of the multiquark hadrons. So far, most tetraquark such state is a partner of X(5568) under the SU(3) flavor and pentaquark candidates are composed of hidden charm symmetry, and their masses are close to each other. But the or bottom quarks. However, the new state X(5568) observed threshold of uds¯b¯ is BK, 270 MeV higher than the thresh- ¯ ¯ by the D0 collaboration in 2016 [1] was an exception. The old Bsπ of X(5568) with usdb. So there is large mass ( ) = . ± . ( )+0.9( ) ¯ ¯ X 5568 has a mass m 5567 8 2 9 stat −1.9 syst MeV region for this udsb state below threshold and being stable. Γ = . ± . ( )+5.0( ) Besides, Ref. [23] pointed out that if the lowest-lying uds¯b¯ and width 21 9 6 4 stat −2.5 syst MeV [1]. The decay mode is X(5568) → B0π ±, which indicates that the state exists below threshold, it can be definitely observed s /ψ − −π + quark component of the X(5568) should be four different via the weak decay mode J K K , with the expecta- flavors: u, d, s, b. Therefore, the claimed X(5568), if con- tion of hundreds of events in the current LHCb data sample firmed, would differ from any of the previous observations, but rejecting backgrounds due to its long lifetime. There- ¯ ¯ as it must be a tetraquark state with usd¯b¯ or dsu¯b¯ and their fore, the udsb state would be a more promising detectable tetraquark state. Ref. [24] investigated such state composed ¯ ¯ a e-mail: [email protected] of udsb within the chiral quark model, and found the bound state with IJP = 00+ was possible. Liu et al. also proposed b e-mail: [email protected] 123 556 Page 2 of 8 Eur. Phys. J. C (2019) 79 :556 ⎧ ( ) ⎨ 2 , , several partner states of X 5568 and estimated the mass dif- rij i j in the same baryon orbit f (r ) = −μ r2 (3) ference of these partner states based on the color-magnetic ij ⎩ 1−e ij ij μ , otherwise interaction [25], which can provide valuable information on ij the future experimental search of these states. π OGE = 1α λc · λc 1 − δ ) 1 + 1 It is generally known that quantum chromodynamics Vij s i j rij 2 2 4 rij 2 mi m j (QCD) is the fundamental theory of the strong interaction. Understanding the low-energy behavior of QCD and the 4σ i · σ j 3 + − S (4) nature of the strong interacting matter, however, remains 3 ij 3mi m j 4mi m j rij a challenge due to the complexity of QCD. Lattice QCD 3 7 has provided numerical results describing quark confinement OBE = ( ) λa · λa + ( ) λa · λa Vij Vπ rij i j VK rij i j between two static colorful quarks, a preliminary picture of a=1 a=4 the QCD vacuum and the internal structure of hadrons in + ( ) λ8 · λ8 θ − λ0 · λ0 θ addition to a phase transition of strongly interacting mat- Vη rij i j cos P i j sin P (5) ter. But a satisfying description of multiquark system is out 2 2 Λ2 gch mχ χ Vχ (r ) = mχ σ · σ of reach of the present calculation. The QCD-inspired mod- ij 2 2 i j 4π 12m m Λχ − mχ els, incorporating the properties of low-energy QCD: color i j 3 confinement and chiral symmetry breaking, are also power- Λχ Y (mχ r ) − Y (Λχ r ) ij 3 ij ful tools to obtain physical insights into many phenomena of mχ the hadronic world. Among many phenomenological models, 3 Λχ the quark delocalization color screening model (QDCSM), + H(mχ r ) − H(Λχ r ) S , ij 3 ij ij which was developed in the 1990s with the aim of explain- mχ ing the similarities between nuclear (hadronic clusters of χ = π, ,η, K (6) quarks) and molecular forces [26], has been quite successful (σ i · rij)(σ j · rij) in reproducing the energies of the baryon ground states, the Sij = 3 − σ i · σ j , (7) r 2 properties of deuteron, the nucleon-nucleon (NN) and the ij − hyperon-nucleon (YN) interactions [27–31]. Recently, this H(x) = (1 + 3/x + 3/x2)Y (x), Y (x) = e x /x. (8) model has been used to study the pentaquarks with hidden- ( ) ( ) strange [32], hidden-charm and hidden-bottom [33]. There- where Sij is quark tensor operator; Y x and H x are stan- fore, it is interesting to extend this model to the tetraquark dard Yukawa functions; Tc is the kinetic energy of the center α system. In present work, the tetraquark state X(5568) with of mass; s is the quark-gluon coupling constant; gch is the quark contents usd¯b¯ and its partner state with uds¯b¯ are inves- coupling constant for chiral field, which is determined from π tigated. Besides, two structures, meson-meson and diquark– the NN coupling constant through antidiquark, are considered in this work. 2 2 2 2 gch 3 gπ NN mu,d The structure of this paper is as follows. A brief intro- = . (9) 4π 5 4π m2 duction of the quark model and wave functions is given in N section 2. Section 3 is devoted to the numerical results and The other symbols in the above expressions have their usual discussions. The summary is shown in the last section. meanings. All model parameters are determined by fitting the meson spectrum we used in this work and shown in Table 1. The calculated masses of the mesons in comparison with experimental values are shown in Table 2. Besides, a phe- 2 Model and wave functions nomenological color screening confinement potential is used here, and μij is the color screening parameter, which is deter- QDCSM has been described in detail in the literatures [26– mined by fitting the deuteron properties, NNscattering phase 31]. Here, we just present the salient features of the model. shifts, NΛ and NΣ scattering phase shifts, respectively, with −2 −2 −2 The Hamiltonian for the tetraquark states is shown below: μuu = 0.45 fm , μus = 0.19 fm and μss = 0.08 fm , μ2 = μ μ satisfying the relation, us uu ss [34]. When extend- 4 2 ing to the heavy bottom quark case, there is no experimen- pi H = mi + − TCM tal data available, so we take it as a adjustable parameter = 2mi −2 i 1 μbb = 0.001 ∼ 0.0001 fm . We find the results are insen- 4 sitive to the value of μ . So in the present work, we take CON OGE OBE bb + V + V + V , (1) −2 ij ij ij μbb = 0.001 fm . j>i=1 The quark delocalization in QDCSM is realized by spec- CON =− λc · λc ( ( ) + 0 ), Vij ac i j f rij aij (2) ifying the single particle orbital wave function of QDCSM 123 Eur. Phys. J. C (2019) 79 :556 Page 3 of 8 556

−1 −1 Table 1 Model parameters: mπ = 0.7fm , m K = 2.51 fm , mη = states. The wave function of the four-quark system is of the . −1 Λ = . −1 Λ = Λ = . −1 2 /( π) = . 2 77 fm , π 4 2fm , K η 5 2fm , gch 4 0 54, form 0 θp =−15 σ Ψ = A [ψ L ψ ] ψ f ψc . (11) bmu md ms mb JM (fm) (MeV) (MeV) (MeV) (MeV) where ψ L , ψσ , ψ f , and ψc are the orbital, spin, flavor and 0.518 313 313 470 4500 color wave functions, respectively, which are given below. a a0 a0 a0 a0 c uu us ub sb The symbol A is the anti-symmetrization operator. For the −2 2 2 2 2 (MeVfm ) (fm ) (fm ) (fm )(fm) meson-meson structure, A is defined as 58.03 −0.733 −0.309 1.701 1.808 α α α α A = − . suu sus sub ssb 1 P13 (12) 1.50 1.46 1.41 1.40 where 1 and 3 stand for the light quarks in two meson clusters respectively; for the diquark–antidiquark structure, A = 1. Table 2 The masses (in MeV) of the mesons obtained from QDCSM. The orbital wave function is in the form of Experimental values are taken from the Particle Data Group (PDG) [35] L ψ = ψ1(R1)ψ2(R2)χL (R). (13) Meson Mtheo Mexp where R1 and R2 are the internal coordinates for the cluster π 140 140 1 and cluster 2. R = R1 − R2 is the relative coordinate ρ 772 770 between the two clusters 1 and 2. The ψ1 and ψ2 are the K 495 495 internal cluster orbital wave functions of the clusters 1 and K ∗ 892 892 2, and χL (R) is the relative motion wave function between B 5280 5280 two clusters, which is expanded by gaussian bases B∗ 5319 5325 n Bs 5367 5367 1 3 χ (R) = √ C ∗ L 2 i Bs 5393 5415 4π 2πb i=1 3 2 × exp − (R − si ) YLM(sî )dsî . (14) 4b2 as a linear combination of left and right Gaussians, the single particle orbital wave functions used in the ordinary quark where si is called the generate coordinate, n is the number cluster model, of the gaussian bases, which is determined by the stability of the results. By doing this, the integro-differential equa- ψα(s ,) = (φα(s ) + φα(−s )) /N(), i i i tion of RGM can be reduced to an algebraic equation, gen- ψβ (−s ,) = φβ (−s ) + φβ (s ) /N(), i i i eralized eigen-equation. Then the energy of the system can −s2/4b2 be obtained by solving this generalized eigen-equation. The N() = 1 + 2 + 2e i . details of solving the RGM equation can be found in Ref. 3/4 1 − 1 ( − / )2 2 rα si 2 [37]. In our calculation, the maximum generating coordi- φα(si ) = e 2b πb2 nate sn is fixed by the stability of the results. The calculated / 3 4 1 2 results are stable when the distance between the two clusters 1 − (rβ +si /2) φβ (−s ) = e 2b2 . (10) is larger than 6 fm. To keep the dimensions of matrix man- i πb2 ageably small, the two clusters’ separation is taken to be less Here si , i = 1, 2,...,n are the generating coordinates, than 6 fm. which are introduced to expand the relative motion wave- The flavor, spin, and color wave functions are constructed function [27–31]. The mixing parameter (si ) is not an in two steps. First constructing the wave functions for clus- adjusted one but determined variationally by the dynamics ters 1 and 2, then coupling the two wave functions of two of the multi-quark system itself. In this way, the multi-quark clusters to form the wave function for tetraquark system. For system chooses its favorable configuration in the interact- the meson-meson structure, as the first step, we give the wave ing process. This mechanism has been used to explain the functions of the meson cluster. The flavor wave functions of cross-over transition between hadron phase and quark-gluon the meson cluster are shown below. plasma phase [36]. χ 1 = ud¯,χ2 = sd¯,χ3 = ub¯,χ4 = sb¯, In this work, the resonating group method (RGM) [37], I11 I 1 1 I 1 1 I00 a well-established method for studying a bound-state or a 2 2 2 2 χ 5 = us¯,χ6 = ds¯,χ7 = db¯. (15) scattering problem, is used to calculate the energy of all these I 1 1 I 1 − 1 I 1 − 1 2 2 2 2 2 2 123 556 Page 4 of 8 Eur. Phys. J. C (2019) 79 :556 where the superscript of the χ is the index of the flavor wave χ 4 = √1 ( + ), χ 5 = √1 ( + ), [2] rb br [2] gb bg function for a meson, and the subscript stands for the isospin 2 2 I and the third component I . The spin wave functions of the 1 z χ 6 = bb,χ7 = √ (rg − gr), meson cluster are: [2] [11] 2 8 1 9 1 1 2 1 χ = √ (rb − br), χ = √ (gb − bg). (22) χσ = αα, χσ = (αβ + βα), [11] [11] 11 10 2 2 2 3 4 1 and the color wave functions of the antidiquark clusters are: χσ = ββ, χσ = (αβ − βα). (16) 1−1 00 2 χ 1 =¯¯,χ2 =−√1 (¯ ¯ +¯¯), χ 3 =¯¯, and the color wave function of a meson is: [22] rr [22] rg gr [22] gg 2 1 1 1 1 ¯ χ 4 = √ (¯ ¯ + ¯ ¯), χ 5 =−√ ( ¯ ¯ + ¯ ¯), χ[ ] = (rr¯ + gg¯ + bb). (17) [22] rb br [22] gb bg 111 3 22 2 Then, the wave functions for the four-quark system with the χ 6 = ¯ ¯,χ7 = √1 (¯ ¯ −¯¯), [22] bb [211] rg gr meson-meson structure can be obtained by coupling the wave 2 functions of two meson clusters. Every part of wave functions 8 1 ¯ ¯ 9 1 ¯ ¯ χ[ ] =−√ (r¯b − br¯), χ[ ] = √ (g¯b − bg¯). (23) are shown below. The flavor wave functions are: 211 2 211 2 ψ f1 = χ 4 χ 1 ,ψf2 = χ 3 χ 2 , After that, the wave functions for the four-quark system 11 I00 I11 11 I 1 1 I 1 1 2 2 2 2 with the diquark–antidiquark structure can be obtained by 1 ψ f3 = χ 5 χ 7 − χ 7 χ 5 , coupling the wave functions of two clusters. Every part of 00 2 I 1 1 I 1 − 1 I 1 − 1 I 1 1 wave functions are shown below. The flavor wave functions 2 2 2 2 2 2 2 2 1 are: ψ f4 = χ 5 χ 7 + χ 7 χ 5 . (18) 11 I 1 1 I 1 − 1 I 1 − 1 I 1 1 f1 3 5 f2 4 5 2 2 2 2 2 2 2 2 2 ψ = χ χ ,ψ= χ χ , 11 I 1 1 I 1 1 11 I 1 1 I 1 1 The spin wave functions are: 2 2 2 2 2 2 2 2 ψ f3 = χ 1 χ 6 ,ψf4 = χ 2 χ 6 . σ 11 I I 00 I I (24) ψ 1 = χ 4 χ 4 , 11 00 00 00 00 σ00 σ00 The color wave functions are: σ2 1 1 3 2 2 3 1 ψ = χσ χσ − χσ χσ + χσ χσ , 00 11 1−1 10 10 1−1 11 1 3 ψc1 = [χ 1 χ 1 − χ 2 χ 2 + χ 3 χ 3 σ σ [2] [22] [2] [22] [2] [22] ψ 3 = χ 4 χ 1 ,ψ4 = χ 1 χ 4 , 6 11 σ σ 11 σ σ 00 11 11 00 +χ 4 χ 4 − χ 5 χ 5 + χ 6 χ 6 ], [2] [22] [2] [22] [2] [22] σ5 1 1 2 2 1 ψ = χσ χσ − χσ χσ . (19) 11 11 10 10 11 1 2 ψc2 = χ 7 χ 7 − χ 8 χ 8 + χ 9 χ 9 . [11] [211] [11] [211] [11] [211] (25) The color wave function is: 3 Finally, we can acquire the total wave functions by sub- ψc1 = χ 1 χ 1 . (20) [111] [111] stituting the wave functions of the orbital, the spin, the flavor Finally, multiplying the wave functions ψ L , ψσ , ψ f , and ψc and the color parts into the Eq. (10) according to the given according to the definite quantum number of the system, we quantum number of the system. can acquire the total wave functions of the system. For the diquark–antidiquark structure, the orbital and the spin wave functions are the same with those of the meson- 3 The results and discussions meson structure. For the flavor wave functions, we give the functions of the diquark and antidiquark clusters firstly. In present work, we investigate tetraquarks with quark components: usd¯b¯ and uds¯b¯ in two structures, meson-meson χ 1 = √1 ( + ), χ 2 = √1 ( − ), I ud du I ud du and diquark–antidiquark. The quantum numbers of the 10 2 00 2 tetraquarks we study here are I = 0, 1, J = 0, 1 and 1 1 χ 3 = √ (us + su), χ 4 = √ (us − su), the parity is P =+. The orbital angular momenta are set to I 1 1 I 1 1 2 2 2 2 2 2 zero because we are interested in the ground states. To check χ 5 = d¯b¯,χ6 =¯sb¯. (21) I 1 1 I00 whether or not there is any bound state in such tetraquark 2 2 system, we do a dynamic bound-state calculation. Both the Then, the color wave functions of the diquark clusters are: single-channel and channel-coupling calculations are carried

1 2 1 3 out in this work. All the general features of the calculated χ[ ] = rr,χ[ ] = √ (rg + gr), χ[ ] = gg, 2 2 2 2 results are as follows. 123 Eur. Phys. J. C (2019) 79 :556 Page 5 of 8 556

Table 3 The energies (in MeV) of the meson-meson structure for tetraquarks usd¯b¯

f σ c IJ [ψ i ψ j ψ k ] Channel Eth Esc Ecc

σ + [ψ f1 ψ 1 ψc1 ] 0π . . . 10 Bs 5506 9 5514 0 5513 1 σ ∗ + [ψ f1 ψ 2 ψc1 ] 0ρ . . Bs 6165 4 6169 2 σ + [ψ f2 ψ 1 ψc1 ] B K¯ 0 5774.9 5782.6 σ ∗+ ∗ [ψ f2 ψ 2 ψc1 ] B K¯ 0 6212.6 6217.5 σ + [ψ f1 ψ 3 ψc1 ] 0ρ . . . 11 Bs 6139 4 6143 9 5539 3 σ ∗ + [ψ f1 ψ 4 ψc1 ] 0π . . Bs 5532 9 5539 7 σ + ∗ [ψ f2 ψ 3 ψc1 ] B K¯ 0 6172.3 6179.1 σ ∗+ [ψ f2 ψ 4 ψc1 ] B K¯ 0 5814.2 5821.5

Table 4 The energies (in MeV) of the diquark–antidiquark structure Fig. 1 The effective potentials as a function of the distance between ¯ ¯ for tetraquarks usdb the diquark (qq) and antidiquark (q¯q¯)fortheIJ = 10 and IJ = 11 ¯ ¯ f σ c states of usdb IJ [ψ i ψ j ψ k ] Eth Esc Ecc

σ 10 [ψ f1 ψ 1 ψc1 ] 5506.9 6283.1 5551.8 σ structure. However, the energy of the IJ = 10 state is still [ψ f1 ψ 2 ψc2 ] 6186.9 σ higher than the theoretical threshold of the lowest channel [ψ f2 ψ 1 ψc2 ] . 6096 1 0 + σ B π , 5506.9 MeV. Similarly, the energy of the IJ = 11 [ψ f2 ψ 2 ψc1 ] 5846.6 s σ state is higher than the theoretical threshold of the lowest [ψ f1 ψ 4 ψc2 ] . . . 11 5532 9 6308 8 5613 4 ∗0π + . σ channel Bs , 5532 9 MeV. Thus, there is no bound state [ψ f1 ψ 5 ψc2 ] . 6261 9 with diquark–antidiquark structure in the present calculation. f σ c [ψ 1 ψ 3 ψ 1 ] 6276.0 Nevertheless, the colorful subclusters diquark (qq) and σ [ψ f2 ψ 4 ψc1 ] 6216.5 antidiquark (q¯q¯) cannot fall apart because of the color con- σ [ψ f2 ψ 5 ψc1 ] 6059.4 finement, so there may be a resonance state with diquark– σ [ψ f2 ψ 3 ψc2 ] 6118.7 antidiquark structure. To check the possibility, we perform an adiabatic calculation of the effective potentials for both the IJ = 10 and IJ = 11 states. The effective potentials are obtained by V (S) = E(S)− E , where E is the threshold 3.1 Tetraquarks composed of usd¯b¯ E th th of the corresponding lowest channel. E(S) is the energy at each S, which is the distance between two subclusters, and For tetraquarks composed of usd¯b¯, the isospin is I = 1. E(S) can be obtained by the formula: The energies of the states with J = 0, 1 are calculated and the results are listed in Tables 3 and 4. In the tables, the Ψ(S)|H|Ψ(S) E(S) = . second column gives the index of the wave functions of each Ψ(S)|Ψ(S) channel. The columns headed with Esc and Ecc represent the energies of the single-channel and channel-coupling calcula- where Ψ(S)|H|Ψ(S) and Ψ(S)|Ψ(S) are the Hamilto- tion respectively. Eth denotes the theoretical threshold of the nian matrix element and the overlap of the state. The results channel. For meson-meson structure, there is an additional are shown in Fig. 1. It is obvious in Fig. 1 that the effec- column headed with “Channel” which denotes the physical tive potentials of both the IJ = 10 and IJ = 11 states are contents of the channel. From the Table 3, we can see that the increasing when the two subclusters fall apart, which indi- energies of every single channel approach to the correspond- cates that the two subclusters tend to clump together. In this ing theoretical threshold. The channel-coupling cannot help case, the odds are the same for the states being meson-meson too much. Energies are still above the threshold of the low- structure, diquark–antidiquark structure or other structures. 0π + = ∗0π + = est channel (Bs for IJ 10 and Bs for IJ 11), Besides, from the Tables 3 and 4, the energy of the diquark– which indicates that no bound usd¯b¯ state with meson-meson antidiquark structure is always higher than the one of the structure is formed in our quark model calculation. meson-meson structure. So the unstable states will decay With regard to the diquark–antidiquark structure, the ener- through the corresponding meson-meson channel. Then the gies are listed in Table 4. The channels with different flavor- system prefers to be meson-meson structure, which can fall spin-color configurations have different energies and the cou- apart. Therefore neither the state of IJ=10 nor the state of pling of them is rather stronger than that of the meson-meson IJ=11 is a resonance state in QDCSM. 123 556 Page 6 of 8 Eur. Phys. J. C (2019) 79 :556

Table 5 The energies (in MeV) of the meson-meson structure for tetraquarks uds¯b¯

f σ c IJ [ψ i ψ j ψ k ] Channel Eth Esc Ecc

σ + 00 [ψ f3 ψ 1 ψc1 ] B0 K 5774.9 5781.4 5779.9 σ ∗ ∗+ [ψ f3 ψ 2 ψc1 ] B 0 K 6212.6 6218.1 σ ∗+ 01 [ψ f3 ψ 3 ψc1 ] B0 K 6172.3 6176.2 5813.1 σ ∗ + [ψ f3 ψ 4 ψc1 ] B 0 K 5814.2 5820.5 σ ∗ ∗+ [ψ f3 ψ 5 ψc1 ] B 0 K 6212.6 6216.1 σ + 10 [ψ f4 ψ 1 ψc1 ] B0 K 5774.9 5783.2 5783.0 σ ∗ ∗+ [ψ f4 ψ 2 ψc1 ] B 0 K 6212.6 6218.1 σ ∗+ 11 [ψ f4 ψ 3 ψc1 ] B0 K 6172.3 6180.3 5821.5 σ ∗ + [ψ f4 ψ 4 ψc1 ] B 0 K 5814.2 5822.0 σ ∗ ∗+ [ψ f4 ψ 5 ψc1 ] B 0 K 6212.6 6219.1 Fig. 2 The effective potentials as a function of the distance between two mesons for the IJ = 10 and IJ = 11 states of usd¯b¯ Table 6 The energies (in MeV) of the diquark–antidiquark structure for tetraquarks uds¯b¯

Besides, we can also calculate the effective potentials for f σ c IJ [ψ i ψ j ψ k ] Eth Esc Ecc the meson-meson structure, which are shown in Fig. 2.We σ ﬁnd the potentials for both the IJ = 10 and IJ = 11 states 00 [ψ f1 ψ 1 ψc1 ] 5774.9 5867.2 5701.1 σ are repulsive, which further shows that it is difﬁcult to form [ψ f1 ψ 2 ψc2 ] 6058.2 σ a molecular bound state for either IJ = 10 or IJ = 11 01 [ψ f1 ψ 1 ψc1 ] 5814.2 6334.7 5756.3 σ system. [ψ f2 ψ 1 ψc2 ] 6213.3 ( ) σ Based on the discussion above, the X 5568 cannot be [ψ f2 ψ 2 ψc1 ] 5881.4 σ explained as a molecular state or a diquark–antidiquark res- [ψ f1 ψ 1 ψc1 ] . . . ¯ ¯ 10 5774 9 6452 1 6103 8 onance of usdb in the present calculation. Our results are σ [ψ f1 ψ 2 ψc2 ] 6200.8 consistent with the analysis of Refs. [19,20,22]. In Ref. [19], σ ¯ ¯ 11 [ψ f1 ψ 1 ψc1 ] 5814.2 6289.1 6130.1 the four-quark system usdb with both meson-meson struc- σ [ψ f2 ψ 1 ψc2 ] 6253.4 ture and diquark–antidiquark structure was studied in the σ [ψ f2 ψ 2 ψc1 ] . framework of the chiral quark model by using the Gaus- 6447 9 sian expansion method, and no candidate of X(5568) was found. In Ref. [20], Burns and Swanson explored a lot of possible explanations of the X(5568) signal, a tetraquark, a tion. While for the state with IJ = 01, the energy is about 1.0 hadronic molecule or a threshold effect and found that none MeV lower than the threshold of the lowest channel B∗0 K + of them can be a candidate of the observed state. In Ref. [22], after channel-coupling. However, the binding energy is not they concluded that the X(5568) cannot be generated by the very large, so there maybe a weak molecular bound state of ¯ ¯ ¯ Bsπ − BK rescattering. uds¯b with quantum numbers of IJ = 01, and the mass of this state is about 5813 MeV. 3.2 Tetraquarks composed of uds¯b¯ For the diquark–antidiquark structure, all the possible channels are shown in Table 6. One can see that the energy of For tetraquarks composed of uds¯b¯, four states with the quan- each single channel is higher than the theoretical threshold tum numbers IJ = 00, 01, 10 and 11 are studied. The of the corresponding channel, which are shown in Table 5. energies of the meson-meson structure and the diquark– Although the effect of the channel-coupling is much stronger antidiquark structure are listed in Tables 5 and 6, respectively. than that of the meson-meson structure, the energy of the For the meson-meson structure, the results are similar to that IJ = 10 and IJ = 11 states are still above the theoretical of the tetraquarks of usd¯b¯. Table 5 shows that the energies of threshold of the corresponding channel. So there is no any every single channel are above the corresponding theoretical bound states for the IJ = 10 or IJ = 11 state. In order to threshold. The effect of channel-coupling is very small except check if there is any resonance state, we also perform the adi- for the IJ = 01 state. For the states with IJ = 00, IJ = 10, abatic calculation of effective potentials for both the IJ = 10 and IJ = 11, all energies are above the threshold of the low- and IJ = 11 states. The results are shown in Fig. 3. The case est channel (B0 K + for IJ = 00, B0 K + for IJ = 10, and is similar to the tetraquarks composed of usd¯b¯. The effec- B∗0 K + for IJ = 11) even by the channel-coupling calcula- tive potentials of both the IJ = 10 and IJ = 11 states are 123 Eur. Phys. J. C (2019) 79 :556 Page 7 of 8 556

antidiquark open-string configuration. The picture combined the advantages of diquark-based models, which can accom- modate much of the known XYZ spectrum, with the experimental fact that such states are both relatively narrow and are produced promptly. Thus both the IJ = 00 and IJ = 01 states of the diquark–antidiquark structure we obtain here maybe the narrow resonance states. The study of the decay width of these states is our further work. Contrasting with the tetraquarks composed of usd¯b¯,we find the tetraquarks composed of uds¯b¯ is more likely to form bound state. The reasons are as follows. First, the diquark pair of two light quarks (ud) or two heavier quarks (sb)is usually more stable than the one of two quarks with larger mass difference like us or db pair. Our results show that the ¯ Fig. 3 The effective potentials as a function of the distance between tetraquarks composed of uds¯b of the diquark–antidiquark the diquark (qq) and antidiquark (q¯q¯)fortheIJ = 10 and IJ = 11 structure is most possible to form bound states, which just ¯ ¯ states of udsb supports this point. Secondly, the lowest threshold of uds¯b¯ ¯ ¯ is BK, 270 MeV higher than the threshold Bsπ of usdb.So there is large mass region for this uds¯b¯ state below threshold increasing when the two subclusters diquark (qq) and antidi- and being stable. This conclusion also supports the assump- quark (q¯q¯) fall apart, which indicates that the two subclusters tion of Ref. [23], which proposed such particle with the quark tend to clump together. Hence, the IJ = 10 and IJ = 11 component of uds¯b¯ (or its charge-conjugated one) as a part- states is not stable within the diquark–antidiquark configura- ner of X(5568) of usd¯b¯ under the SU(3) flavor symmetry. tion, and no resonance state with quantum numbers IJ = 10 Besides, Ref. [40] also stressed the possible existence of a and IJ = 11 exists. near-threshold bound or virtual state both for J P = 0+ and However, things are different for the IJ = 00 state and 1+ sectors, with quark content sbu¯d¯. the IJ = 01 state. The energy of the IJ = 00 state is about Because of the heavy flavor symmetry, we also extend 5701 MeV, 74 MeV lower than the theoretical threshold of the study to the above tetraquarks by considering a c quark the B0 K +, which indicates that the IJ = 00 state of the instead of a b quark, which are composed of usd¯c¯ or uds¯c¯. diquark–antidiquark structure can be a bound state. Ref. [24] No any bound state or resonance state is found in these sys- also found that the bound state with IJ = 00 was possible. tems. This is possible in our quark model calculation. In the Meanwhile, the energy of the IJ = 01 state is 58 MeV lower heavy-quark sector, the large masses of the heavy quarks than the theoretical threshold of the B∗0 K +,sotheIJ = 01 reduce the kinetic energy of the system, which makes them state is also bound here. Thus, both the IJ = 00 state and easier to form bound states. So it is more difficult for the the IJ = 01 state of diquark–antidiquark structure can form tetraquark in charm sector than the one in bottom sector to bound states. form bound state. However, some other tetraquarks in charm By comparing with the results of the meson-meson struc- sector were studied before. Ref. [40] found two-pole structure ture, we note that the energy of the IJ = 01 state of the of the D∗(2400), with the largest couplings to the Dπ and ¯ 0 diquark–antidiquark structure is about 5756 MeV, which is Ds K channels. They showed the higher pole was a thresh- ¯ much lower than that of the meson-meson structure shown old enhancement in the Ds K invariant mass distribution, and in Table 5. This shows that the IJ = 01 state prefers to be a the lower pole belongs to the same SU(3) multiplet as the ∗ ( ) bound state of the diquark–antidiquark structure. Moreover, Ds0 2317 state. Such tetraquark system is also worth inves- the IJ = 00 state of the diquark–antidiquark structure is eas- tigating in our quark model calculation, which will be our ier to form the bound state than the one of the meson-meson further work. structure. All these indicate that the diquark–antidiquark configuration maybe a good choice for some tetraquarks. Some work have been done to explain the exotic XYZ states 4 Summary depending on the diquark–antidiquark configuration. Ref- erence [38] proposed the hypothesis that the diquarks and In summary, we investigate tetraquarks composed of usd¯b¯ antidiquarks in tetraquarks were separated by a potential bar- and uds¯b¯ in the framework of QDCSM. Two structures, rier to explain the properties of exotic resonances such as X meson-meson and diquark–antidiquark, are considered. Our and Z.Ref.[39] presented a dynamical picture to explain results show that there is no bound state or resonance the nature of some exotic XYZ states based on a diquark– state composed of usd¯b¯. The reported X(5568) cannot be 123 556 Page 8 of 8 Eur. Phys. J. C (2019) 79 :556 explained as a molecular state or a diquark–antidiquark res- References onance of usd¯b¯ in present calculation. In contrast, two bound states are obtained for the tetraquarks system composed of 1. V.M. Abazov et al., (D0 Collaboration). Phys. Rev. Lett. 117, uds¯b¯:anIJ = 00 state with the mass of 5701 MeV, and 022003 (2016) 117 = 2. R. Aaij et al., (LHCb Collaboration). Phys. Rev. Lett. , 152003 an IJ 01 state with the mass of 5756 MeV, which maybe (2016) the better tetraquark candidates with foure different flavors. 3. A.M. Sirunyan et al., (CMS Collaboration). Phys. Rev. Lett. 120, These two bound states are of the diquark–antidiquark struc- 202005 (2018) ture. For the system with IJ = 00 and IJ = 01, it is obvious 4. T.A. Aaltonen et al., (CDF Collaboration). Phys. Rev. Lett. 120, 202006 (2018) that the state of the diquark–antidiquark structure is more 5. M. Aaboud et al., (ATLAS Collaboration). Phys. Rev. Lett. 120, likely to form bound state than that of the meson-meson 202007 (2018) structure, which indicates that the diquark–antidiquark con- 6. V.M. Abazov et al., (D0 Collaboration). Phys. Rev. D 97, 092004 figuration would be a good choice for the tetraquarks uds¯b¯ (2018) 131 = = 7. S.S. Agaev, K. Azizi, H. Sundu, Eur. Phys. J. Plus , 351 (2016) with IJ 00 and IJ 01. During the calculation, we 8. Z.G. Wang, Commun. Theor. Phys. 66, 335 (2016) find that the effect of the channel-coupling in the diquark– 9. Z.G. Wang, Eur. Phys. J. C 76, 279 (2016) antidiquark structure is much stronger than that in the meson- 10. C.M. Zanetti, M. Nielsen, K.P. Khemchandani, Phys. Rev. D 93, meson structure, and the channel-coupling plays an impor- 096011 (2016) 11. W. Chen, H.X. Chen, X. Liu, T.G. Steele, S.L. Zhu, Phys. Rev. Lett. tant role in forming bound states in the diquark–antidiquark 117, 022002 (2016) structure. 12. J.M. Dias, K.P. Khemchandani, A.M. Torres, M. Nielsen, C.M. Meanwhile, our results also show that the tetraquarks com- Zanetti, Phys. Lett. B 758, 235 (2016) posed of uds¯b¯ is more possible to form bound states than the 13. L. Tang, C.F. Qiao, Eur. Phys. J. C 76, 558 (2016) 40 ¯ ¯ ( ) 14. W. Wang, R.L. Zhu, Chin. Phys. C , 093101 (2016) one composed of usdb. 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B 757, 515 (2016) below threshold, it can be definitely observed via the weak 23. F.S. Yu, arXiv:1709.02571 [hep-ph] − − + decay mode J/ψ K K π , with the expectation of hun- 24. X.Y. Chen, J.L. Ping, Phys. Rev. D 98, 054022 (2018) dreds of events in the current LHCb data sample but reject- 25. Y.R. Liu, X. Liu, S.L. Zhu, Phys. Rev. D 93, 074023 (2016) ing backgrounds due to its long lifetime. Therefore, the uds¯b¯ 26. F. Wang, G.H. Wu, L.J. Teng, T. Goldman, Phys. Rev. Lett. 69, 2901 (1992) state would be a promising detectable tetraquark state. We 27. J.L. Ping, F. Wang, T. Goldman, Nucl. Phys. A 657, 95 (1999) hope that experiments will help to discover these interesting 28. G.H. Wu, J.L. Ping, L.J. Teng, F. Wang, T. Goldman, Nucl. Phys. tetraquark states. A 673, 279 (2000) 29. H.R. Pang, J.L. Ping, F. Wang, T. Goldman, Phys. Rev. C 65, 014003 (2001) Acknowledgements This work is supported partly by the National 30. J.L. Ping, F. Wang, T. Goldman, Nucl. Phys. A 688, 871 (2001) Natural Science Foundation of China under Contract Nos. 11675080, 31. J.L. Ping, H.R. Pang, F. Wang, T. Goldman, Phys. Rev. C 65, 11775118 and 11535005, the Natural Science Foundation of the Jiangsu 044003 (2002) Higher Education Institutions of China (Grant No. 16KJB140006). 32. H.X. Huang, X.M. Zhu, J.L. Ping, Phys. Rev. D 97, 094019 (2018) Data Availability Statement This manuscript has no associated data 33. H.X. Huang, C.R. Deng, J.L. Ping, F. Wang, Eur. Phys. J. C 76, or the data will not be deposited. [Authors’ comment: There is no data 624 (2016) need to be deposited.] 34. H.X. Huang, J.L. Ping, F. Wang, Phys. Rev. C 92, 065202 (2015) 35. Particle Data Group, C. Patrignani et al., Chin. Phys. C 40, 100001 Open Access This article is distributed under the terms of the Creative (2016) Commons Attribution 4.0 International License (http://creativecomm 36. M.M. Xu, M. Yu, L.S. Liu, Phys. Rev. Lett. 100, 092301 (2008) ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 37. M. Kamimura, Suppl. Prog. Theor. Phys. 62, 236 (1977) and reproduction in any medium, provided you give appropriate credit 38. L. Maiani, A.D. Polosa, V. Riquer, Phys. Lett. B 778, 247 (2018) to the original author(s) and the source, provide a link to the Creative 39. S.J. Brodsky, D.S. Hwang, R.F.Lebed, Phys. Rev. Lett. 113, 112001 Commons license, and indicate if changes were made. (2014) Funded by SCOAP3. 40. M. Albaladejo, P. Fernandez-Soler, F. Guo, J. Nieves, Phys. Lett. B 767, 465 (2017)

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