Investigating tetraquarks composed of us\bar{d}\bar{b} and ud\bar{s}\bar{b}

# Investigating tetraquarks composed of us¯d¯b and ud¯s¯b

## Abstract

In the framework of the quark delocalization color screening model, we investigate tetraquarks composed of and in two structures: meson-meson structure and diquark-antidiquark structure. Neither bound state nor resonance state is found in the system composed of . The reported cannot be explained as a molecular state or a diquark-antidiquark resonance of in present calculation. However, two bound states of the diquark-antidiquark structure are obtained in the tetraquarks system composed of : an state with the mass of MeV, and an state with the mass of MeV, which maybe the partners of states. Our results indicate that the diquark-antidiquark configuration would be a good choice for the tetraquarks with quantum numbers and . The tetraquarks composed of is more possible to form bound states than the one composed of . These bound states are worth investigating in future experiments.

###### pacs:
13.75.Cs, 12.39.Pn, 12.39.Jh

## I Introduction

In the past few decades, the discovery of numbers of exotic states stimulated extensive interest in understanding the structures of the multiquark hadrons. So far, most tetraquark and pentaquark candidates are composed of hidden charm or bottom quarks. However, the new state observed by the D0 collaboration in 2016 D0 () was an exception. The has a mass MeV and width MeV D0 (). The decay mode is , which indicates that the quark component of the should be four different flavors: . Therefore, the claimed , if confirmed, would differ from any of the previous observations, as it must be a tetraquark state with or and their charge-conjugated ones. Unfortunately, this state was not confirmed by other collaborations. The LHCb collaboration LHCb (), the CMS collaboration of LHC CMS (), the CDF collaboration of Fermilab CDF () and the ATLAS Collaboration of LHC ATLAS () all claimed that no evidence for this state was found. Nevertheless, the D0 collaboration’s new result still insists on the existence of this tetraquark  D02 (). Clearly, more other measurements are needed.

The discovery of this exotic state also stimulates the theoretical interest. Many approaches have been applied to interpret this state, such as the QCD sum rules Agaev (); WangZG1 (); WangZG2 (); Zanetti (); ChenW (); Dias (); TangL (), quark models WangW (); XiaoCJ (); Stancu (), the extended light front model KeHW (), rescattering effects LiuXH (), and so on. However, several theoretical calculations gave the negative results ChenXY (); Burns (); GuoFK (); Albaladejo (). For example, in Ref. ChenXY (), authors investigated two structures, diquark-antidiquark and meson-meson, with all possible color configurations by using the Gaussian expansion method, and they cannot obtain the reported . Ref. Burns () examined the various interpretations of the state and found that the threshold, cusp, molecular, and tetraquark models were all unfavored the existence of the .

To search for the tetraquark states with four different flavors, a better state is (or its charge-conjugated one) with replacing the in by  YuFS (). Obviously, such state is a partner of under the flavor symmetry, and their masses are close to each other. But the threshold of is , MeV higher than the threshold of with . So there is large mass region for this state below threshold and being stable. Besides, Ref. YuFS () pointed out that if the lowest-lying state exists below threshold, it can be definitely observed via the weak decay mode , with the expectation of hundreds of events in the current LHCb data sample but rejecting backgrounds due to its long lifetime. Therefore, the state would be a more promising detectable tetraquark state. Ref. ChenXY2 () investigated such state composed of within the chiral quark model, and found the bound state with was possible. Liu also proposed several partner states of and estimated the mass difference of these partner states based on the color-magnetic interaction LiuYR (), which can provide valuable information on the future experimental search of these states.

It is generally known that quantum chromodynamics (QCD) is the fundamental theory of the strong interaction. Understanding the low-energy behavior of QCD and the nature of the strong interacting matter, however, remains a challenge due to the complexity of QCD. Lattice QCD has provided numerical results describing quark confinement between two static colorful quarks, a preliminary picture of the QCD vacuum and the internal structure of hadrons in addition to a phase transition of strongly interacting matter. But a satisfying description of multiquark system is out of reach of the present calculation. The QCD-inspired models, incorporating the properties of low-energy QCD: color confinement and chiral symmetry breaking, are also powerful tools to obtain physical insights into many phenomena of the hadronic world. Among many phenomenological models, the quark delocalization color screening model (QDCSM), which was developed in the 1990s with the aim of explaining the similarities between nuclear (hadronic clusters of quarks) and molecular forces QDCSM0 (), has been quite successful in reproducing the energies of the baryon ground states, the properties of deuteron, the nucleon-nucleon () and the hyperon-nucleon () interactions QDCSM1 (). Recently, this model has been used to study the pentaquarks with hidden-strange HuangHX_Ps (), hidden-charm and hidden-bottom HuangHX_Pc (). Therefore, it is interesting to extend this model to the tetraquark system. In present work, the tetraquark state with quark contents and its partner state with are investigated. Besides, two structures, meson-meson and diquark-antidiquark, are considered in this work.

The structure of this paper is as follows. A brief introduction of the quark model and wave functions is given in section II. Section III is devoted to the numerical results and discussions. The summary is shown in the last section.

## Ii Model and Wave Functions

QDCSM has been described in detail in the literatures QDCSM0 (); QDCSM1 (). Here, we just present the salient features of the model. The Hamiltonian for the tetraquark states is shown below:

 H = 4∑i=1(mi+p2i2mi)−TCM+4∑j>i=1(VCONij+VOGEij+VOBEij), (1) VCONij = ⎧⎪⎨⎪⎩−acλci⋅λcj (r2ij+a0ij),if {i},{j} in the same baryon orbit−acλci⋅λcj (1−e−μijr2ijμij+a0ij),otherwise (2) VOGEij = 14αsλci⋅λcj[1rij−π2δ(rij)(1m2i+1m2j+4σi⋅σj3mimj)−34mimjr3ijSij] (3) VOBEij = Vπ(rij)3∑a=1λai⋅λaj+VK(rij)7∑a=4λai⋅λaj+Vη(rij)[(λ8i⋅λ8j)cosθP−(λ0i⋅λ0j)sinθP] (4) Vχ(rij) = g2ch4πm2χ 12mimjΛ2χΛ2χ−m2χmχ{(σi⋅σj)[Y(mχrij)−Λ3χm3χY(Λχrij)] (5) +[H(mχrij)−Λ3χm3χH(Λχrij)]Sij},      χ=π,K,η, Sij = {3(σi⋅rij)(σj⋅rij)r2ij−σi⋅σj}, (6) H(x) = (1+3/x+3/x2)Y(x),      Y(x)=e−x/x. (7)

Where is quark tensor operator; and are standard Yukawa functions; is the kinetic energy of the center of mass; is the quark-gluon coupling constant; is the coupling constant for chiral field, which is determined from the coupling constant through

 g2ch4π=(35)2g2πNN4πm2u,dm2N. (8)

The other symbols in the above expressions have their usual 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 phenomenological color screening confinement potential is used here, and is the color screening parameter, which is determined by fitting the deuteron properties, scattering phase shifts, and scattering phase shifts, respectively, with fm, fm and fm, satisfying the relation,  HuangHX3 (). When extending to the heavy bottom quark case, there is no experimental data available, so we take it as a adjustable parameter fm. We find the results are insensitive to the value of . So in the present work, we take fm.

The quark delocalization in QDCSM is realized by specifying the single particle orbital wave function of QDCSM as a linear combination of left and right Gaussians, the single particle orbital wave functions used in the ordinary quark cluster model,

 ψα(si,ϵ) = (ϕα(si)+ϵϕα(−si))/N(ϵ), ψβ(−si,ϵ) = (ϕβ(−si)+ϵϕβ(si))/N(ϵ), N(ϵ) = √1+ϵ2+2ϵe−s2i/4b2. (9) ϕα(si) = (1πb2)3/4e−12b2(rα−si/2)2 ϕβ(−si) = (1πb2)3/4e−12b2(rβ+si/2)2.

Here , are the generating coordinates, which are introduced to expand the relative motion wavefunction QDCSM1 (). The mixing parameter is not an adjusted one but determined variationally by the dynamics of the multi-quark system itself. In this way, the multi-quark system chooses its favorable configuration in the interacting process. This mechanism has been used to explain the cross-over transition between hadron phase and quark-gluon plasma phase Xu ().

In this work, the resonating group method (RGM) RGM (), a well-established method for studying a bound-state or a scattering problem, is used to calculate the energy of all these states. The wave function of the four-quark system is of the form

 Ψ=A[[ψLψσ]JMψfψc]. (10)

where , , , and are the orbital, spin, flavor and color wave functions, respectively, which are given below. The symbol is the anti-symmetrization operator. For the meson-meson structure, is defined as

 A=1−P13. (11)

where 1 and 3 stand for the quarks in two meson clusters respectively; for the diquark-antidiquark structure, .

The orbital wave function is in the form of

 ψL=ψ1(R1)ψ2(R2)χL(R). (12)

where and are the internal coordinates for the cluster 1 and cluster 2. is the relative coordinate between the two clusters 1 and 2. The and are the internal cluster orbital wave functions of the clusters 1 and 2, and is the relative motion wave function between two clusters, which is expanded by gaussian bases

 χL(R)=1√4π(32πb2)n∑i=1Ci ×∫exp[−34b2(R−si)2]YLM(^si)d^si. (13)

where is called the generate coordinate, is the number of the gaussian bases, which is determined by the stability of the results. By doing this, the integro-differential equation of RGM can be reduced to an algebraic equation, generalized eigen-equation. Then the energy of the system can be obtained by solving this generalized eigen-equation. The details of solving the RGM equation can be found in Ref. RGM (). In our calculation, the maximum generating coordinate is fixed by the stability of the results. The calculated results are stable when the distance between the two clusters is larger than 6 fm. To keep the dimensions of matrix manageably small, the two clusters’ separation is taken to be less than 6 fm.

The flavor, spin, and color wave functions are constructed in two steps. First constructing the wave functions for clusters 1 and 2, then coupling the two wave functions of two clusters to form the wave function for tetraquark system. For the meson-meson structure, as the first step, we give the wave functions of the meson cluster. The flavor wave functions of the meson cluster are shown below.

 χ1I11 = u¯d,    χ2I1212=s¯d,    χ3I1212=u¯b,    χ4I00=s¯b, χ5I1212 = u¯s,    χ6I12−12=d¯s,    χ7I12−12=d¯b. (14)

where the superscript of the is the index of the flavor wave function for a meson, and the subscript stands for the isospin and the third component . The spin wave functions of the meson cluster are:

 χ1σ11 = αα,    χ2σ10=√12(αβ+βα), χ3σ1−1 = ββ,    χ4σ00=√12(αβ−βα). (15)

and the color wave function of a meson is:

 χ1[111] = √13(r¯r+g¯g+b¯b). (16)

Then, the wave functions for the four-quark system with the meson-meson structure can be obtained by coupling the wave functions of two meson clusters. Every part of wave functions are shown below. The flavor wave functions are:

 ψf111 = χ4I00χ1I11,    ψf211=χ3I1212χ2I1212, ψf300 = √12[χ5I1212χ7I12−12−χ7I12−12χ5I1212], ψf411 = √12[χ5I1212χ7I12−12+χ7I12−12χ5I1212]. (17)

The spin wave functions are:

 ψσ100 = χ4σ00χ4σ00, ψσ200 = √13[χ1σ11χ3σ1−1−χ2σ10χ2σ10+χ3σ1−1χ1σ11], ψσ311 = χ4σ00χ1σ11,    ψσ411=χ1σ11χ4σ00, ψσ511 = √12[χ1σ11χ2σ10−χ2σ10χ1σ11]. (18)

The color wave function is:

 ψc1 = χ1[111]χ1[111]. (19)

Finally, multiplying the wave functions , , , and according to the definite quantum number of the system, we 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-meson structure. For the flavor wave functions, we give the functions of the diquark and antidiquark clusters firstly.

 χ1I10 = 1√2(ud+du),    χ2I00=1√2(ud−du), χ3I1212 = 1√2(us+su),    χ4I1212=1√2(us−su), χ5I1212 = ¯d¯b,    χ6I00=¯s¯b. (20)

Then, the color wave functions of the diquark clusters are:

 χ1[2] = rr,   χ2[2]=1√2(rg+gr),   χ3[2]=gg, χ4[2] = 1√2(rb+br),   χ5[2]=1√2(gb+bg),   χ6[2]=bb, χ7[11] = 1√2(rg−gr),   χ8[11]=1√2(rb−br), χ9[11] = 1√2(gb−bg). (21)

and the color wave functions of the antidiquark clusters are:

 χ1[22] = ¯r¯r,   χ2[22]=−1√2(¯r¯g+¯g¯r),    χ3[22]=¯g¯g, χ4[22] = 1√22(¯r¯b+¯b¯r), χ5[22]=−1√2(¯g¯b+¯b¯g), χ6[22]=¯b¯b, χ7[211] = 1√2(¯r¯g−¯g¯r),    χ8[211]=−1√2(¯r¯b−¯b¯r), χ9[211] = 1√2(¯g¯b−¯b¯g). (22)

After that, the wave functions for the four-quark system with the diquark-antidiquark structure can be obtained by coupling the wave functions of two clusters. Every part of wave functions are shown below. The flavor wave functions are:

 ψf111 = χ3I1212χ5I1212,    ψf211=χ4I1212χ5I1212, ψf311 = χ1I11χ6I00,    ψf400=χ2I00χ6I00. (23)

The color wave functions are:

 ψc1 = √16[χ1[2]χ1[22]−χ2[2]χ2[22]+χ3[2]χ3[22] +χ4[2]χ4[22]−χ5[2]χ5[22]+χ6[2]χ6[22]], ψc2 = √13[χ7[11]χ7[211]−χ8[11]χ8[211]+χ9[11]χ9[211]]. (24)

Finally, we can acquire the total wave functions by substituting the wave functions of the orbital, the spin, the flavor and the color parts into the Eq. (10) according to the given quantum number of the system.

## Iii The results and discussions

In present work, we investigate tetraquarks with quark components: and in two structures, meson-meson and diquark-antidiquark. The quantum numbers of the tetraquarks we study here are , and the parity is . The orbital angular momenta are set to zero because we are interested in the ground states. To check whether or not there is any bound state in such tetraquark system, we do a dynamic bound-state calculation. Both the single-channel and channel-coupling calculations are carried out in this work. All the general features of the calculated results are as follows.

### iii.1 Tetraquarks composed of us¯d¯b

For tetraquarks composed of , the isospin is . The energies of the states with are calculated and the results are listed in Table 3 and 4. In the tables, the second column gives the index of the wave functions of each channel. The columns headed with and represent the energies of the single-channel and channel-coupling calculation respectively. For meson-meson structure, there are two additional columns, the column headed with “Channel” denotes the physical contents of the channel and the coulmn headed with denotes the theoretical threshold of the channel. From the Table 3, we can see that the energies of every single channel approach to the corresponding theoretical threshold. The channel-coupling cannot help too much. Energies are still above the threshold of the lowest channel ( for and for ), which indicates that no bound state with meson-meson structure is formed in our quark model calculation.

With regard to the diquark-antidiquark structure, the energies are listed in Table 4. The channels with different flavor-spin-color configurations have different energies and the coupling of them is rather stronger than that of the meson-meson structure. However, the energy of the state is still higher than the theoretical threshold of the lowest channel , MeV. Similarly, the energy of the state is higher than the theoretical threshold of the lowest channel , MeV. Thus, there is no bound state with diquark-antidiquark structure in the present calculation.

Nevertheless, the colorful subclusters diquark () and antidiquark () cannot fall apart because of the color confinement, so there may be a resonance state with diquark-antidiquark structure. To check the possibility, we perform an adiabatic calculation of energy for both the and states. The results are shown in Fig. 1, where the horizontal axis is the distance between two subclusters and the vertical axis stands for the energy of the system at the corresponding distance . It is obvious in Fig. 1 that the energy of both the and states is increasing when the two subclusters fall apart, which indicates that the two subclusters tend to clump together. In other words, the odds are the same for the states being meson-meson structure, diquark-antidiquark structure or other structures. As mentioned above, the energy of the state is higher than the theoretical threshold of the lowest channel, so neither the state of nor the state of is a resonance state in QDCSM.

Therefore, the cannot be explained as a molecular state or a diquark-antidiquark resonance of in the present calculation. Our results are consistent with the analysis of Ref. ChenXY () and Ref. Burns (). In Ref ChenXY (), the four-quark system with both meson-meson structure and diquark-antidiquark structure was studied in the framework of the chiral quark model by using the Gaussian expansion method, and no candidate of was found. In Ref. Burns (), Burns and Swanson explored a lot of possible explanations of the signal, a tetraquark, a hadronic molecule or a threshold effect and found that none of them can be a candidate of the observed state.

### iii.2 Tetraquarks composed of ud¯s¯b

For tetraquarks composed of , four states with the quantum numbers and are studied. The energies of the meson-meson structure and the diquark-antidiquark structure are listed in Tables 5 and 6, respectively. For the meson-meson structure, the results are similar to that of the tetraquarks of . Table 5 shows that the energies of every single channel are above the corresponding theoretical threshold. The effect of channel-coupling is very small except for the state. For the states with