searches for ud\bar{s}\bar{b} in the chiral quark model

Inspired by the report of D0 Collaboration on the state $X(5568)$ with four different flavors, a similar state, $ud\bar{s}\bar{b}$ is investigated in the present work. The advantage of looking for this state over the state $X(5568)$ with quark contents, $bu\bar{d}\bar{s}$ or $bd\bar{u}\bar{s}$, is that the $BK$ threshold is 270 MeV higher than that of $B_s \pi$, and it allows a large mass region for $ud\bar{s}\bar{b}$ to be stable. The chiral quark model and Gaussian expansion method are employed to do the calculations of four-quark states $ud\bar{s}\bar{b}$ with quantum numbers $IJ^{P}$($I=0,1;~ J=0,1,2;~P=+$). Two structures, diquark-antidiquark and meson-meson, with all possible color configurations are considered. The results indicate that energies of the tetraquark with diquark-antiquark configuration are all higher than the threshold of $BK$, but for the state of $IJ^P=00^+$ in the meson-meson structure, the energies are below the corresponding thresholds, where the color channel coupling plays an important role. The distances between two objects (quark/antiquark) show that the state is a molecular one.


I. INTRODUCTION
Since the first exotic resonance X(3872) was observed by the Bell collaboration in 2003 [1], many other exotic sates, so called "XYZ" states, have emerged from the reports of Belle, BaBar, BESIII, LHCb, CDF, D0 and other collaborations.In the traditional quark models, the meson is consist of quark and antiquark and the baryon is made up of three quarks.How to explain these exotic states is a big challenge for quark models.The study of the exotic states is helpful for improving the quark model and deepening our understanding of the nonperturbative quantum chromodynamics (QCD).
Although, these "XYZ" states are difficult to be explained as the ordinary hadrons, their quantum numbers can be constructed by quark-antiquark combinations.To find an unambiguous tetraquark state, the particle with four different flavors are expected.Not so long ago, the D0 Collaboration announced a new resonance named X(5568) with the mass M = 5567.8± 2.9 +0.9 −1.9 MeV and narrow width Γ = 21.9 ± 6.4 +5.0 −2.5 MeV, respectively [2].Recently, the D0 Collaboration reported a further evidence about this state in the weak decay of B with a significance of 6.7σ [3], which is consistent with their previous measurement [2].Subsequently, searches for X(5568) in decays to B 0 s π ± , B 0 s → J/ψφ are performed by the LHCb [4], CMS [5] and ATLAS [6] Collaborations in pp collisions and by the CDF Collaboration [7] at the Tevatron, and all of experiments revealed no signal.Clearly, more measurements are needed.
On the theoretical side, the early work based on the QCD sum rule supported the existence of the state X(5568) [8][9][10][11][12].Some quark model calculations also claimed the possible explanation of the results of the D0 Collaboration [13][14][15].However, the detailed examination of the various interpretations of the state X(5568) shown that the threshold, cusp, molecular and tetraquark models are all unfavored [16].Based on the general properties of QCD, F. K. Guo et al. argued that the QCD does not support the existence of the state X(5568) [17].Our previous quark model calculation of tetraquark and molecule also obtained negative results [18].
In this work, we intend to study a particle with also four different flavors, uds b, which is different from the X(5568) consist of us db .For simplicity, we denote the particle uds b as T bs .The reasons for searching this particle are as follows.Firstly, the breaking up of T bs is BK, which the threshold is higher than the B s π threshold of X(5568), and it leads to a large mass region for T bs to be stable.Secondly, for diquark-antidiquark configuration, the ud quark pair is more stable than the us owing to the lower mass of d quark than s quark.In others words, if X(5568) does exist, the T bs must be a stable state.If X(5568) is proved to be nonexistent, there's still probability for T bs to be stable.F. S. Yu also suggested that if uds b exists here, the most favorable decay mode to observe it will be J/ψK − K − π + experimentally [19].
In order to search for the particle of T bs theoretically, we calculate the masses of the states with quantum numbers IJ P (I = 0, 1; J = 0, 1, 2; P = +) including two different structures, diquark-antidiquark and meson-meson in the chiral quark model, and all possible color configurations are investigated by using the Gaussian expansion method (GEM) [20].In the calculation, two ways of using σ meson exchange are adopted.One is that the σ meson exchange only occurs between u quark and/or d quark.Another is the effective σ is exchanged between u, d and s quarks.If a bound state is obtained, the average distances between quarks or antiquarks are calculated, which can be used to clarify the structures of the states, a compact tetraquark or a molecular state.
The chiral quark model, the wave functions of T bs and the method for solving the four quark states are detailed in Sec.II; and Sec.III is devoted to a discussion of the results.Sec.IV is a summary.
The chiral quark model has acquired great achievements both in describing the hadron spectra and hadronhadron interactions.The details of the model can be found in Ref. [21].Here only the Hamiltonian of the chiral quark model is given as follows, The potential energy is constituted of pieces describing quark confinement (C); one-gluon-exchange (G); one Goldstone boson exchange (χ = π, K, η) and σ exchange.
Their form for the low-lying four-quark states is [21], λ a i λ a j , (3e) where Y (x) = e −x /x; {m i } is the constituent mass of i-th quarks/antiquarks, and µ ij is the reduced mass of the two interacting particles and  28.17 running coupling [21], All the parameters are determined by fitting the meson spectrum, from light to heavy; and the resulting values are listed in Table I.

B. The wave functions of T bs
We will introduce the wave functions for the two structures, diquark-antidiquark and meson-meson, respectively.For each degree of freedom, first we construct the wave functions for two-body clusters, then coupling the wave functions of two clusters to the wave functions of the tetraquark states.
For spin, the wave functions for two-body clusters are, then the wave functions for four-quark states are obtained, where the subscript of χ represents the total spin of T bs , it takes the values S = 0, 1, 2.
For flavor, the wave functions for four-quark states are Analogously, the subscript of χ d represents the isospin of T bs , I = 0, 1.
For spin, the wave functions are the same as those of the diquark-antidiquark structure, Eq. ( 7).
The flavor wave functions of T bs take as follows, the subscript of χ m represents the isospin of T bs , I = 0, 1.
As for the spatial wave functions, the total orbital wave functions can be constructed by coupling the orbital wave function for each relative motion of the system, where L is the total orbital angular momentum of T bs and Ψ Lr (r 1234 ) is the wave function of the relative motion between two sub-clusters with orbital angular momentum L r , and the Jacobi coordinates are defined as, for diquark-antidiquark structure, the quarks are numbered as 1, 2, and the antiquarks are numbered as 3, 4; for meson-meson structure, one cluster with antiquark and quark is marked as 1, 2, the other cluster with antiquark and quark is marked as 3, 4. In GEM, the spatial wave function is expanded by Gaussians [20]: where N nl are normalization constants, c n are the variational parameters, which are determined dynamically.The Gaussian size parameters are chosen according to the following geometric progression This procedure enables optimization of the ranges using just a small number of Gaussians.Finally, the complete channel wave function for the four-quark system for diquark-antidiquark structure is written as , 2; S = 0, 1, 2; I = 0, 1).(23) For meson-meson structure, the complete wave function is written as Here, A is the antisymmetrization operator: if all quarks (antiquarks) are taken as identical particles, then (1 − P 13 − P 24 + P 13 P 24 ).
In the present work, for T bs system, only two quarks are the identical particles, so the antisymmetrization operator used is The eigenenergy of the T bs system is obtained by solving a Schrödinger equation: where Ψ MI MJ IJ is the wave function of the T bs , which is the linear combinations of the above channel wave functions, Eq. ( 23) in the diquark-antidiquark structure or Eq. ( 24) in the meson-meson structure, respectively.
The calculation of Hamiltonian matrix elements is complicated if any one of the relative orbital angular momenta is nonzero.In this case, it is useful to employ the method of infinitesimally shifted Gaussians [20], wherewith the spherical harmonics are absorbed into the Gaussians: where, plainly, the quantities C lm,k , D lm,k are fixed by the particular spherical harmonic under consideration and their values ensure the limit ǫ → 0 exists.

III. NUMERICAL RESULTS AND DISCUSSIONS
In the present work, we try to search the particle with quantum numbers IJ P (I = 0, 1; J = 0, 1, 2; P = +) consist of four different flavors uds b, denoted as T bs , in the chiral quark model.All the orbital angular momenta are set to zero because we are interested in the lowlying states.The chiral quark model gives a good description of the meson spectrum which can seen in the comparison of theoretical thresholds and experimental thresholds in Table II.Two structures of T bs , diquarkantidiquark and meson-meson, are investigated.In each structure, all possible states are considered.For diquarkantidiquark structure, two color configurations, color antitriplet-triplet ( 3×3) and sexet-antisextet (6× 6) are examined.And for meson-meson structure, color singletsinglet (1×1) and octet-octet (8×8) are taken into account.
In SU (2) flavor symmetry, the σ meson exchange only occurs between u quark and d quark.In this situation, the results of T bs for diquark-antidiquark and mesonmeson structure, are given in Tables II and III, respectively.
In Table II, the second column gives the index of the antisymmetry wave functions of T bs .E s is the single channel eigenenergy for the different channels; E cc represents the eigenenergy with the effect of channel-coupling of different spin-color configurations.From the table, we can see that the channels with different spin-color configurations have similar energies and the coupling of them is rather strong.However, the energies are all higher than the threshold of BK, 5773MeV, which indicates that cannot fall apart, there may be a resonance even though the higher energy of the state.To check the possibility, we calculate the variety of the eigenenergy of 00 + state with the distance between the diquark and antidiquark sub-clusters and the results are shown in Fig 1 .In this case the number of gaussians used for the relative motion of the two sub-clusters is set to 1. we can see that when the two sub-clusters approach closely or fall apart, the energy is increasing, the minimum of the energy occurs around the separation 0.6 fm.The results indicate that the two sub-clusters cannot fall apart or get too close.So that the state turned to be two mesons B and K is hindered because of the separation.00 + state may be a resonance state in our present calculation.With regard to meson-meson structure, the results are shown in Table III.In our calculations, the color singletsinglet configurations always have the lower energies than those of color octet-octet ones.E cc1 is the eigenenergy from the channel coupling of the two color configurations, which is close to that of single channel (color singletsinglet) result, E s .This indicates that the effect of the hidden color is very small.E cc2 gives the eigen-energy from the channel coupling of all the color singlet-singlet ones, and the results show that the coupling is also very small.E cc3 represents the eigen-energy from the channel coupling of all channels with the same quantum numbers.Naturally, the coupling tends to be small.The obtained energies E cc3 are all higher and approach to the theoretical thresholds in all case except the state, 00 + .For IJ P = 00 + state, the eigen-energies from single channel calculation are higher than their theoretical thresholds.With the help of channel coupling to the color octetoctet configuration, the energies of the states are lower than their corresponding thresholds.For the first channel with spin 0 × 0 → 0, the calculated energy is 5774.4TABLE IV.The contributions of each term of the Hamiltonian for 00 + state in meson-meson structure in SU (2) flavor symmetry (unit:MeV).∆i(i = 1, 2) is the difference between the contributions in four-quark state and the sum of the contributions of two mesons.MeV, which is lower than the theoretical threshold 5774.9 MeV, and The binding energy is −0.5 MeV.For the second channel with spin 1 × 1 → 0, the obtained energy is 6222.9MeV, a little smaller than the theoretical threshold 6233.2MeV, and the binding energy is −10.3MeV.All channels coupling obtain the lowest state with binding energy −0.6 MeV and push 1 × 1 → 0 state above its corresponding threshold.To identify which terms in the Hamiltonian making the state be bound, the contributions from each term of Hamiltonian for the four-quark state and the sum of two mesons are given in Table IV for both two 00 + states.From the table, we can see that the color confinement, one gluon exchange and σ-meson exchange contribute the binding of the states.For spin 1 × 1 → 0 state, π-meson exchange makes a considerable binding due to the compact structure of the state (see Table V).For spin 0 × 0 → 0 state, π-meson exchange makes no contribution because of the large separation between two mesons.F. Close et al. also found that the pion exchange between hadrons can lead to deeply bound quasimolecular states [22,23].Furthermore, the root-mean-square (RMS) distances between quarks and antiquarks in meson-meson structure for 00 + state are calculated and shown in Table V.For 0 × 0 → 0 channel, the distances between the two meson clusters are much larger than those of u − s or d − b within one cluster and it tends to be a molecular state; for 1 × 1 → 0 channel, the distances between the two meson clusters are about twice of that between the quark and antiquark in one cluster which indicates that it may be a little compact molecular state in our present calculation.When the coupling of two channels is considered, the dominant component of the lowest state is 0 × 0 → 0 color singlet-singlet state, and the distances between the two meson clusters are a little smaller but still far larger than that between the quark and antiquark in one cluster, so the 00 + state must be a molecular state in the present work.
The Salamanca version of the chiral quark model can describe the meson spectrum well, where the σ meson exchange is considered between u, d and s quark.So it is interesting to calculate the T bs in SU (3) flavor symmetry, and the results for both diquark-antidiquark structure and meson-meson structure are demonstrated in Tables VI and VII, respectively.From the tables, we found that the energies are much lower than those in the SU (2) flavor symmetry no matter in diquark-antidiquark structure or meson-meson structure.In the diquark-antidiquark structure, the energies are still all higher than the threshold of BK.In the meson-meson structure, the energies are all below the corresponding thresholds.For comparison, for 00 + state, the binding energies are −70.2MeV for spin 0×0 → 0 state and −64.5 MeV for spin 1×1 → 0 state, which are much deeper than those (−0.5 MeV and −10.3 MeV) in the SU (2) symmetry.From the contributions of each term of the Hamiltonian for 00 + state in the SU (3) symmetry (see Table VIII), we can see that the σ meson exchange leads to the deeper binding energies for two channels.Furthermore, the RMS distances between four particles for 00 + states in the SU (3) flavor symmetry are demonstrated in Table IX.From the table, we found that the distances between the two meson clusters are much closer than those in the SU (2) flavor symmetry due to the σ meson exchange.For 0 × 0 → 0 channel, it still tends to be a molecular one and for 1 × 1 → 0 channel, it may be a compact tetraquark state.The effect of the channel coupling is still tiny.

IV. SUMMARY
Benefited from uds b's higher threshold than B s π, it has a larger mass region than X(5568) to be stable and it may be a promising detectable tetraquark state.In this paper we try to calculate the state uds b (T bs ) with quantum numbers IJ P (I = 0, 1; J = 0, 1, 2; P = +) by using GEM.The constituent chiral quark model with flavor symmetries SU (2) and SU (3), which describes the light and heavy meson spectra well, is employed in the calculation.Two structures: diquark-antidiquark and meson-meson, are investigated.We found that the energies of T bs with diquark-antidiquark structure are all higher than the threshold of BK, leaving no space for the bound state, but for the lowest energy 00 + state it may be a resonance state in the SU (2) flavor symmetry in our calculation.Besides, in the SU (2) flavor symmetry with the meson-meson strucure, the mass of 00 + state is just below the threshold of BK with a small binding energy, −0.6MeV, which can be a molecular state in the present work.As for the SU (3) flavor symmetry, the results for diquark-antidiquak structure are unaltered, and the energies with the meson-meson structure are much lower owing to the medium-range attraction supplied by the σ meson exchange between u, d and s quark.From experimental side, it is not expected that there exist so many states, so using σ-meson exchange between u, d and s quark is not a proper way in SU (3) flavor symmetry.A better way is to employ scalar nonet in the SU (3) flavor symmetry instead of one σ meson exchange in the SU (2) flavor symmetry.

TABLE I .
Model parameters, determined by fitting the meson spectrum.

TABLE II .
The eigenenergies of T bs in SU (2) flavor symmetry for diquark-antidiquark structure (unit: MeV).E th1 represents the theoretical threshold and E th2 denotes the experimental threshold.

TABLE III .
The eigenenergies of T bs in SU (2) flavor symmetry for meson-meson structure (unit:MeV).FIG.1.The eigenenergy of 00 + state as a function of the distance between the diquark and antidiquark.

TABLE VII .
The eigenenergies of T bs in SU (3) flavor symmetry for meson-meson structure (unit: MeV).E b represents the binding energy of states.

TABLE VIII .
The contributions of each terms of the Hamiltonian for 00 + state in meson-meson structure in SU (3) flavor symmetry (unit: MeV).

TABLE IX .
the root-mean-square(RMS) radiuses of quarks and antiquarks of the state 00 + in meson-meson structure in SU (3) flavor symmetry(unit:fm).