searches for ud\bar{s}\bar{b} in the chiral quark model
Xiaoyun Chen Jialun Ping




xychen@jit.edu.cn Department of Basic
Courses, Jinling Institute of Technology, Nanjing 211169, P. R.
China


[]jlping@njnu.edu.cn Department of Physics,
Nanjing Normal University, Nanjing 210023, P. R. China




Inspired by the report of D0 Collaboration on the state X(5568)
with four different flavors, a similar state, uds̅b̅
is investigated in the present work. The advantage of looking for
this state over the state X(5568) with quark contents,
bud̅s̅ or bdu̅s̅, is that the BK
threshold is 270 MeV higher than that of B_s π, and it allows
a large mass region for uds̅b̅ to be stable. The
chiral quark model and Gaussian expansion method are employed to
do the calculations of four-quark states uds̅b̅ 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.




searches for uds̅b̅ in the chiral quark model
    Jialun Ping
    December 30, 2023
=============================================







§ INTRODUCTION
 
Since the first exotic resonance X(3872) was observed by the
Bell collaboration in 2003 <cit.>, 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 <cit.>. Recently, the D0 Collaboration
reported a further evidence about this state in the weak decay of
B with a significance of 6.7σ <cit.>, which is
consistent with their previous measurement <cit.>.
Subsequently, searches for X(5568) in decays to
B_s^0π^±, B_s^0→ J/ψϕ are performed
by the LHCb <cit.>, CMS <cit.> and
ATLAS <cit.> Collaborations in pp collisions and by
the CDF Collaboration <cit.> 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) <cit.>.
Some quark model calculations also claimed the possible
explanation of the results of the D0
Collaboration <cit.>. 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 <cit.>. 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) <cit.>. Our previous
quark model calculation of tetraquark and molecule also obtained
negative results <cit.>.

In this work, we intend to study a particle with also four
different flavors, uds̅b̅, which is different from the
X(5568) consist of usd̅b̅. For simplicity, we denote
the particle uds̅b̅ 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 uds̅b̅ exists here, the most favorable decay
mode to observe it will be J/ψ K^-K^-π^+
experimentally <cit.>.

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) <cit.>. 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. <ref>; and Sec. <ref> is devoted to a
discussion of the results. Sec. <ref> is a summary.





§ CHIRAL QUARK MODEL AND WAVE FUNCTIONS OF T_BS




 §.§ The chiral quark model

The chiral quark model has acquired great achievements both in
describing the hadron spectra and hadron-hadron interactions. The
details of the model can be found in Ref. <cit.>.
Here only the Hamiltonian of the chiral quark model is given as
follows,

    H     =    ∑_i=1^4 m_i  +p_12^2/2μ_12+p_34^2/2μ_34
      +p_1234^2/2μ_1234 +∑_i<j=1^4 V_ij,   
    
     V_ij    =     V_ij^C+V_ij^G+∑_χ=π,K,η V_ij^χ
       +V_ij^σ.

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 <cit.>,


    V_ij^C   = ( -a_c r_ij^2-Δ ) λ_i^c
    ·λ_j^c ,  
    
     V_ij^G   = α_s/4λ_i^c ·λ_j^c
    [1/r_ij-2π/3m_im_jσ_i·σ_j
      δ(r_ij)],  
    
    δ(r_ij)    =  e^-r_ij/r_0(μ_ij)/4π r_ijr_0^2(μ_ij), 
    
    V_ij^π   = g_ch^2/4πm_π^2/12m_im_jΛ_π^2/Λ_π^2-m_π^2m_π v_ij^π∑_a=1^3 λ_i^a λ_j^a,  
    
    V_ij^K   = g_ch^2/4πm_K^2/12m_im_jΛ_K^2/Λ_K^2-m_K^2m_K v_ij^K∑_a=4^7 λ_i^a λ_j^a,   
    
     V_ij^η    =
    g_ch^2/4πm_η^2/12m_im_jΛ_η^2/Λ_η^2-m_η^2m_η
    v_ij^η
    
          ×[λ_i^8 λ_j^8 cosθ_P
     - λ_i^0 λ_j^0 sinθ_P ],   
    
     v_ij^χ    =  [ Y(m_χ r_ij)-
    Λ_χ^3/m_χ^3Y(Λ_χ r_ij)
    ]
    σ_i ·σ_j,
    
    V_ij^σ   = -g_ch^2/4πΛ_σ^2/Λ_σ^2-m_σ^2m_σ
    
         ×[
     Y(m_σ r_ij)-Λ_σ/m_σY(Λ_σ r_ij)]  ,



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

    μ_1234=(m_1+m_2)(m_3+m_4)/m_1+m_2+m_3+m_4;


𝐩_ij=(𝐩_i-𝐩_j)/2,
𝐩_1234= (𝐩_12-𝐩_34)/2;

r_0(μ_ij) =s_0/μ_ij; σ are the SU(2) Pauli matrices; λ,
λ^c are SU(3) flavor, color Gell-Mann matrices, respectively;
g^2_ch/4π is the chiral coupling constant, determined from the π-nucleon coupling;
and α_s is an effective scale-dependent running coupling
<cit.>,

    α_s(μ_ij)=α_0/ln[(μ_ij^2+μ_0^2)/Λ_0^2].

All the parameters are determined by fitting the meson spectrum, from light to heavy; and the resulting values
are listed in Table <ref>.





 §.§ 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.

(1) Diqaurk-antidiquark structure.

For spin, the wave functions for two-body clusters are,

    χ_11=αα,  χ_10=1/√(2)(αβ+βα),  χ_1-1=ββ,
       χ_00=1/√(2)(αβ-βα),

then the wave functions for four-quark states are obtained,
 

    χ_0^σ
    1   =χ_00χ_00,
    χ_0^σ
    2   =√(1/3)(χ_11χ_1-1-χ_10χ_10+χ_1-1χ_11),
    χ_1^σ
    3   =χ_00χ_11,
    χ_1^σ
    4   =χ_11χ_00,
    χ_1^σ
    5   =1/√(2)(χ_11χ_10-χ_10χ_11),
    χ_2^σ
    6   =χ_11χ_11,


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
 

    χ_d0^f1   =1/√(2)(ud-du)s̅b̅,
    χ_d1^f2   =1/√(2)(ud+du)s̅b̅.


Analogously, the subscript of χ_d represents the isospin of T_bs, I=0, 1.

For color, there are two color configurations for the quarks pair, [11] and
[2], respectively.


    χ^[11]_1=1/√(2)(rg-gr),
    χ^[11]_2=1/√(2)(rb-br),
    χ^[11]_3=1/√(2)(gb-bg),




    χ^[2]_1   =rr,     χ^[2]_2=gg,
         χ^[2]_3=bb,
    χ^[2]_4   =1/√(2)(rg+gr),
    χ^[2]_5   =1/√(2)(rb+br),
    χ^[2]_6   =1/√(2)(gb+bg).


For the antiquarks pair, the two color configurations are [211] and [22], respectively,


    χ^[211]_1   =1/√(2)(g̅b̅-b̅g̅),
    χ^[211]_2   =1/√(2)(r̅b̅-b̅r̅),
    χ^[211]_3   =1/√(2)(r̅g̅-g̅r̅).




    χ^[22]_1   =r̅r̅,     χ^[22]_2=g̅g̅,     χ^[22]_3=b̅b̅,
    χ^[22]_4   =-1/√(2)(g̅b̅+b̅g̅),
    χ^[22]_5   =1/√(2)(r̅b̅+b̅r̅),
    χ^[22]_6   =-1/√(2)(r̅g̅+g̅r̅).


The color wave functions of T_bs in the diquark-antidiquark
structure should be color singlet [222] and it can be obtained
by using SU(3) Clebsh-Gordan coefficients,


    χ^c_d1    =
    √(3)/6(rgr̅g̅-rgg̅r̅+grg̅r̅-grr̅g̅
          +rbr̅b̅-rbb̅r̅+brb̅r̅-brr̅b̅
          +gbg̅b̅-gbb̅g̅+bgb̅g̅-bgg̅b̅).  
    χ^c_d2   =√(6)/12(2rrr̅r̅+2ggg̅g̅+2bbb̅b̅
        +rgr̅g̅+rgg̅r̅
          +grg̅r̅+grr̅g̅+rbr̅b̅+rbb̅r̅+brb̅r̅
          +brr̅b̅+gbg̅b̅+gbb̅g̅+bgb̅g̅+bgg̅b̅).


Where, χ_d1^c and χ_d2^c represents the color
antitriplet-triplet (3̅×3) and sextet-antisextet
(6×6̅), respectively.

(2)Meson-meson structure.

For spin, the wave functions are the same as those of the
diquark-antidiquark structure, Eq. (<ref>).

The flavor wave functions of T_bs take as follows,


    χ_m0^f1   =1/√(2)(s̅ub̅d-s̅db̅u),
    χ_m1^f2   =1/√(2)(s̅ub̅d+s̅db̅u),


the subscript of χ_m represents the isospin of T_bs, I=0, 1.

For color wave functions, the possible color configurations of a cluster consist of a quark and
an antiquark are [111], and [21],

    χ^[111]_1=1/√(3)(r̅r+g̅g+b̅b).


    χ^[21]_1=b̅r,     χ^[21]_2=b̅g,     χ^[21]_3=-g̅r, 
       χ^[21]_4=1/√(2)(r̅r-g̅g),    χ^[21]_5=1/√(6)(2b̅b-r̅r-g̅g),
       χ^[21]_6=r̅g,     χ^[21]_7=-g̅b,     χ^[21]_8=r̅b.

At last, the color singlet wave functions of T_bs in the meson-meson structure are,


    χ_m1^c   =1/3(r̅r+g̅g+b̅b)(r̅r+g̅g+b̅b),
    χ_m2^c   =√(2)/12(3b̅rr̅b+3g̅rr̅g+3b̅gg̅b+3g̅bb̅g+3r̅gg̅r 
          +3r̅bb̅r+2r̅rr̅r+2g̅gg̅g+2b̅bb̅b-r̅rg̅g 
          -g̅gr̅r-b̅bg̅g-b̅br̅r-g̅gb̅b-r̅rb̅b).


Where, χ_m1^c and χ_m2^c represents the color
singlet-singlet(1×1) and color octet-octet(8×8),
respectively.

 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,

    Ψ_L^M_L=[[Ψ_l_1( r_12)Ψ_l_2( r_34)]_l_12Ψ_L_r( r_1234)
    ]_L^M_L,

where L is the total orbital angular momentum of T_bs and
Ψ_L_r(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,

    r_12   = r_1- r_2, 
     r_34   = r_3- r_4, 
     r_1234   =m_1 r_1+m_2
    r_2/m_1+m_2-m_3 r_3+m_4 r_4/m_3+m_4,

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 <cit.>:



    Ψ_l^m(𝐫)     = ∑_n=1^n_ max c_nψ^G_nlm(𝐫),
    
    ψ^G_nlm(𝐫)     = N_nlr^l
    e^-ν_nr^2Y_lm(𝐫̂),


where N_nl are normalization constants,

    N_nl=[2^l+2(2ν_n)^l+3/2/√(π)(2l+1)]^1/2.

c_n are the variational parameters, which are determined
dynamically. The Gaussian size parameters are chosen according to
the following geometric progression

    ν_n=1/r^2_n,    r_n=r_1a^n-1,   
    a=(r_n_ max/r_1)^1/n_
    max-1.

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

    Ψ_IJ,i,j,k^M_IM_J=[Ψ_Lχ_S^σ i]_J^M_Jχ_dI^fjχ^c_dk,
       (i=1∼6, j=1,2, k=1,2; S=0,1,2; I=0,1).

For meson-meson structure, the complete wave function is written as

    Ψ_IJ,i,j,k^M_IM_J=  A[Ψ_Lχ_S^σ
    i]_J^M_Jχ_mI^fjχ^c_mk,
       (i=1∼6, j=1,2, k=1,2; S=0,1,2; I=0,1).

Here, A is the antisymmetrization operator: if all quarks (antiquarks) are taken as
identical particles, then

    A=1/2(1-P_13-P_24+P_13P_24).

In the present work, for T_bs system, only two quarks are the
identical particles, so the antisymmetrization operator used is

    A=1/√(2)(1-P_13).


The eigenenergy of the T_bs  system is obtained by solving a Schrödinger equation:

    H  Ψ^ M_IM_J_IJ=E^IJΨ^ M_IM_J_IJ,

where Ψ^ M_IM_J_IJ is the wave function of the T_bs, which is the linear combinations
of the above channel wave functions, Eq. (<ref>) in the diquark-antidiquark
structure or Eq. (<ref>) 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 <cit.>, wherewith the spherical harmonics are absorbed into the Gaussians:

    ψ^G_nlm(𝐫)    =    N_nlr^l
     e^-ν_nr^2Y_lm(𝐫̂) 
        =   N_nllim_ϵ→
     01/ϵ^l∑_k^k_ maxC_lm,k  e^-ν_n(𝐫-ϵ𝐃_𝐥𝐦,𝐤)^2,

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 ϵ→ 0 exists.






§ 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 uds̅b̅, denoted as T_bs, in the chiral
quark model. All the orbital angular momenta are set to zero because we are
interested in the low-lying 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 <ref>.
Two structures of T_bs, diquark-antidiquark and
meson-meson, are investigated. In each structure, all possible
states are considered. For diquark-antidiquark structure, two
color configurations, color
antitriplet-triplet (3̅×3) and
sexet-antisextet (6×6̅) are examined. And for
meson-meson structure, color singlet-singlet (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 meson-meson
structure, are given in Tables <ref> and
<ref>, respectively.





In Table <ref>, 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 there are no bound states
with the diquark-antidiquark structure in our model calculation.


Because of the color structure, colorful sub-clusters 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 <ref>. 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
<ref>.  In our calculations, the color
singlet-singlet 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 singlet-singlet) 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 octet-octet 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 5774.4 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 6222.9 MeV, a little smaller than the
theoretical threshold 6233.2 MeV, and the binding energy is
-10.3 MeV. 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 <ref> 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 <ref>). 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 <cit.>.














Furthermore, the root-mean-square (RMS) distances between quarks
and antiquarks in meson-meson structure for 00^+ state are
calculated and shown in Table <ref>. For 0×0
→ 0 channel, the distances between the two meson
clusters are much larger than those of u-s̅ or d-b̅
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 <ref> and
<ref>, 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 -70.2 MeV 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
<ref>), 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
<ref>. 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.







§ SUMMARY



Benefited from uds̅b̅'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 uds̅b̅ (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.

  

This work is supported partly by the National Science Foundation
of China under Contract No. 11775118.







10


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