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Recently, the LHCb collaboration [1] reported the observation of a very narrow state, called
T+cc , in theD0D0π+ invariant mass spectrum. The binding energy and the decay width is:δmBW=−273±61±5+11−14keV/c2,ΓBW=410±165±43+18−38keV.
(1) The LHCb collaboration also released a decay analysis, in which the unitarised Breit–Wigner profile was used [2]. The mass with respect to the
D∗+D0 threshold and width reads,δmU=−360±40+4−0keV/c2,ΓU=48±2+0−14keV.
(2) This observation has two points worth our attention. Firstly, different from the hidden charm or hidden bottom exotic hadron states previously observed experimentally, this is the first observation of an exotic state with open double charm. Dating back to the year of 2002, the SELEX collaboration first reported the observation of the doubly-charmed baryon
Ξ+cc(ccd) in the channelsΛ+cK−π+ andpD+K− [3, 4]. Fifteen years later, the LHCb collaboration also found the doubly-charmed baryonΞ++cc(ccu) in theΛ+cK−π+π+ mass distribution [5]. The value of the mass is 100 MeV higher than theΞ+cc observed by the SELEX collaboration. Secondly, the mass of the observedTcc is just a litter lower thanM(D0)+M(D∗+) , with a very small binding energy. Undoubtedly, the observation ofT+cc will open a brand new window to search for new hadron states beyond the traditional hadrons, both experimentally and theoretically.The discoveries of
T+cc andΞ++cc have an important impact, since they indicate that two identical charm quarks can exist in a hadronic state, which inspires some theoretical studies on possible doubly-charmed tetraquarks and their partner states, doubly-bottomed tetraquarks. Historically, the first study ofQQˉqˉq was made in early 1980s [6], with the observation that the system will be bound, below theQˉq+Qˉq threshold, if the mass ratioM/m becomes large enough. This was confirmed by Heller et al. [7, 8] and Zouzou et al. [9]. The first phenomenological attempt to estimate doubly-heavy tetraquark mass was carried out by Lipkin using the nonrelativistic quark model in 1986 [10]. The author pointed outMTcc≤3935 MeV, 60 MeV above the threshold, andTbb was a bound state with the binding energy of 224 MeV.Until now, there have been many articles published about the doubly-heavy tetraquarks
QQˉqˉq (Q=c,b; q=u,d,s) , such as the color–magnetic interaction (CMI) model [11–16], quark models [17–35], the QCD sum rule approach [36–46], lattice QCD simulations [47–55], effective field theory [56–58] and others [59–64]. One of the controversies is whetherQQˉqˉq tetraquarks with two heavy quarks Q and two light antidiquarksˉq are stable or not against the decay into twoQˉq mesons. Actually, this dispute has a long history, due to the lack of experimental information about the strength of the interaction between two heavy quarks. The other important question is, if theTQQ is bound, is it tightly bound or loosely bound?Most theoretical calculations predict that the double-bottom tetraquark states, at least the
1+ states, lie below the open-bottom threshold. Conversely, for doubly-charmed tetraquarks, some works suggest they are above the open-charm threshold [18, 25, 36, 37, 39, 48, 65]. In Ref. [65], the authors stated thatTcc was aI(JP)=0(1+) state around 3929 MeV (53 MeV above theDD∗ threshold) and all the double-charm tetraquarks were not stable. Karliner et al. [18] predicted the mass ofT(ccˉuˉd) withJP=1+ to be 3882 MeV, 7 MeV above theD0D∗+ threshold and 148 MeV above theD0D+γ threshold against the strong and weak decays. In lattice QCD simulations, the authors [48] showed that the phase shifts in the isospin triplet (I=1 ) channels indicated repulsive interactions, while those in theI=0 channels suggested attraction, although neither bound states nor resonance states were found in theTcc(IJP=01+) . Some works are in favor of them as tightly bound states [11–13, 23, 35]. The common feature of research obtaining a deeply bound state is that the diquark–antidiquark structure is employed. For example, in Ref. [12], the mass splitting indicated that the mass ofTcc with color structure6⊗ˉ6 lay above theDD∗ threshold, but the mass ofTcc with color structureˉ3⊗3 lay at 71 MeV below theDD∗ threshold. On the other hand, in Ref. [14], Li et al. obtained a loosely-bound molecule state with a 470-keV binding energy, which was consistent with the recent experimental data [1, 2]. In their work, only the meson–meson structure was considered. These theoretical works suggest that color structures and quark–quark interactions may play an important role in theTQQ states.The discovery of the
Tcc state provides a chance to check the quark–quark interactions for various theoretical approaches based on quark degrees of freedom. In the quark model, quark–quark interactions within the confinement scale (~ 1 fm) have undergone a wide check in the hadron spectrum, where the unique color structure, singlet, is accepted. When we apply quark–quark interactions to multiquark systems, Casimir scaling is employed for generalization [66], although this generalization may cause anti-confinement in multiquark systems [67]. In the Casimir scaling scheme, the two-body interactions used in color singletqqq andqˉq systems are directly extended to quark-pairs with various color structures, and the effects of color structure are taken care of by the Casimir operatorλi⋅λj .With the accumulation of experimental data on multiquark systems, it is time to check Casimir scaling in detail. Here, we apply the chiral quark model (ChQM) to the tetraquark system
TQQ with meson–meson and diquark–antidiquark structures, and generalize the quark–quark interactions used in color-singlet baryons and mesons to multiquark systems using Casimir scaling. The contributions of each term in the Hamiltonian for different color structures are extracted, and used to study the effects of color structure. In this way, we are trying to make clear why the diquark–antiquark structure leads to deeply bound states, whereas the meson–meson structure brings about weakly bound states. In the present work, we investigate the doubly-heavy tetraquarksccˉuˉd andbbˉuˉd with the quantum numbersI(JP)=1(0+),1(1+),1(2+), 0(1+) constrained by the Pauli principle in the framework of the ChQM. Single-channel and various channel-coupling calculations are performed to show the influence of color structure. Meanwhile, the possible resonance states are also searched with a real scaling method [68] in the complete coupled channels.The paper is organized as follows. In the next section, we present the chiral quark model (ChQM) and the accurate few-body computing method—the Gaussian expansion method (GEM) [69]—as well as the wave functions of the four-body
TQQ system. In Sec. III, we present and analyze our results. Finally, a summary is given in the last section. -
Many theoretical methods have been used to uncover the properties of multiquark candidates observed in experiments since 2003. One of them, the QCD-inspired quark model, is still the most effective and simple tool to describe the hadron spectra and hadron–hadron interactions, and has produced great achievements. It has been used in our previous work to investigate the tetraquark systems and obtain some helpful information [70-72]. Here, the application of the chiral quark model in doubly-heavy tetraquark states
TQQ is quite expected.The Hamiltonian of the chiral quark model can be written as follows for a four-body system,
H=4∑i=1mi+p2122μ12+p2342μ34+p212342μ1234+4∑i<j=1[VCij+VGij+∑χ=π,K,ηVχij+Vσij].
(3) The potential energy terms
VC,G,χ,σij represent confinement, one-gluon exchange (OGE), Goldstone boson exchange and scalar σ-meson exchange, respectively. According to the Casimir scheme, the forms of these potentials can be directly extended to multiqaurk systems with the Casimir factorλi⋅λj [66]. Their forms are:VCij=(−acr2ij−Δ)λci⋅λcj,
(4) VGij=αs4λci⋅λcj[1rij−2π3mimjσi⋅σjδ(rij)],
(5) δ(rij)=e−rij/r0(μij)4πrijr20(μij),
(6) Vπij=g2ch4πm2π12mimjΛ2πΛ2π−m2πmπvπij3∑a=1λaiλaj,
(7) VKij=g2ch4πm2K12mimjΛ2KΛ2K−m2KmKvKij7∑a=4λaiλaj,
(8) Vηij=g2ch4πm2η12mimjΛ2ηΛ2η−m2ηmηvηij×[λ8iλ8jcosθP−λ0iλ0jsinθP],
(9) vχij=[Y(mχrij)−Λ3χm3χY(Λχrij)]σi⋅σj,
(10) Vσij=−g2ch4πΛ2σΛ2σ−m2σmσ×[Y(mσrij)−ΛσmσY(Λσrij)],
(11) where
Y(x)=e−x/x ;{mi} are the constituent masses of quarks and antiquarks, andμij are their reduced masses;μ1234=(m1+m2)(m3+m4)m1+m2+m3+m4;
(12) pij=(pi−pj)/2 ,p1234=(p12−p34)/2 ;r0(μij)=s0/μij ;σ are theSU(2) Pauli matrices;λ ,λc areSU(3) flavor and color Gell-Mann matrices, respectively;g2ch/4π is the chiral coupling constant, determined from π-nucleon coupling; andαs is an effective scale-dependent running coupling [73],αs(μij)=α0ln[(μ2ij+μ20)/Λ20].
(13) All the parameters are determined by fitting the meson spectra, from light to heavy, and the resulting values are listed in Table 1.
Quark masses /MeV mu=md 313 ms 536 mc 1728 mb 5112 Goldstone bosons mπ 0.70 (fm −1∼200 MeV)mσ 3.42 mη 2.77 mK 2.51 Λπ=Λσ 4.2 Λη=ΛK 5.2 g2ch/(4π) 0.54 θp(∘) −15 Confinement ac /(MeV fm−2 )101 Δ/MeV −78.3 OGE α0 3.67 Λ0/fm−1 0.033 μ0 /MeV36.98 s0 /MeV28.17 Table 1. Model parameters, determined by fitting the meson spectra.
With these model parameters, we obtain the list of relevant meson spectra for
D(∗) andB(∗) in Table 2. By comparison with experiments, we can see that the quark model can successfully describe the hadron spectra.State Meson I(JP) Energy Expt [74] cˉu D0 12(0−) 1862.6 1864.8 D∗0 12(1−) 1980.6 2007.0 cˉd D+ 12(0−) 1862.6 1869.6 D∗+ 12(1−) 1980.6 2010.3 bˉu B− 12(0−) 5280.8 5279.3 B∗− 12(1−) 5319.6 5325.2 bˉd ˉB0 12(0−) 5280.8 5279.6 ¯B∗0 12(1−) 5319.6 5325.2 Table 2. The mass spectra of
cˉu ,cˉd ,bˉu ,bˉd in the chiral quark model in comparison with the experimental data [74] (in unit of MeV). -
For the
TQQ(Q=c,b) system, there are two quark configurations—the meson–meson structure (MM) and the diquark–antidiquark structure (DA)—which are shown in Fig. 1. Both structures and their coupling effects are considered in this work.Figure 1. (color online) Two types of configuration of
TQQ(Q=c,b) . Figure (a) represents the meson–meson structure; figure (b) is the diquark–antidiquark structure.The wave functions of tetraquark states should be a product of spin, flavor, color and space degrees of freedom. For the spin component, we denote α and β as the spin-up and spin-down states of quarks, and the spin wave functions for the two-quark system are
χ11=αα,χ10=1√2(αβ+βα),χ1−1=ββ,χ00=1√2(αβ−βα),
(14) then total six wave functions of the four-body system are obtained, which are shown in Table 3. The subscript of χ represents the spin values
S1 andS2 of two sub-clusters. The superscript of χ stands for the total spin S (S1⊗S2 = S) and the third projectionMS of S for the four-quark system.Spin Flavor Color χ0000=χ00χ00 χ00=−12(cˉdcˉu−cˉucˉd) 1⊗1:χc1=13(ˉrr+ˉgg+ˉbb)(ˉrr+ˉgg+ˉbb) χ0011=√13(χ11χ1−1−χ10χ10+χ1−1χ11) χ10=−12(cˉdcˉu+cˉucˉd) 8⊗8:χc2=√212(3ˉbrˉrb+3ˉgrˉrg+3ˉbgˉgb+3ˉgbˉbg+3ˉrgˉgr+3ˉrbˉbr+2ˉrrˉrr+2ˉggˉgg+2ˉbbˉbb−ˉrrˉgg−ˉggˉrr−ˉbbˉgg−ˉbbˉrr−ˉggˉbb−ˉrrˉbb) χ1101=χ00χ11 ˉ3⊗3:χc3=√36(rgˉrˉg−rgˉgˉr+grˉgˉr−grˉrˉg+rbˉrˉb−rbˉbˉr+brˉbˉr−brˉrˉb+gbˉgˉb−gbˉbˉg+bgˉbˉg−bgˉgˉb) χ1110=χ11χ00 6⊗ˉ6:χc4=√612(2rrˉrˉr+2ggˉgˉg+2bbˉbˉb+rgˉrˉg+rgˉgˉr+grˉgˉr+grˉrˉg+rbˉrˉb+rbˉbˉr+brˉbˉr+brˉrˉb+gbˉgˉb+gbˉbˉg+bgˉbˉg+bgˉgˉb) χ1111=1√2(χ11χ10−χ10χ11) χ2211=χ11χ11 Table 3. Wave functions of spin, flavor and color for
TQQ .For the flavor component, the flavor wave functions with isospin
=0 and1 are also tabulated in Table 3. The subscript00 or10 of χ represents the total isospin I and the third projectionIz of I.For the color component, more richer structures in the four-quark system will be considered than conventional
qˉq mesons andqqq baryons. For the meson–meson structure in Fig. 1, in theSU(3) group, the colorless wave functions can be obtained from1⊗1=1 or8⊗8=1 . For the diquark–antidiquark structure, the colorless wave functions can be obtained fromˉ3⊗3=1 or6⊗ˉ6=1 . The detailed expressions of these functions can be found in Table 3.Because the quark contents of the presently investigated four-quark systems are two identical heavy quarks (
Q=c,b ) and two identical light antiquarks (ˉu,ˉd ), the wave functions ofTQQ should satisfy the antisymmetry requirement. Then we collect all the color–spin bases ofTQQ states for possible quantum numbers according to the constraint from the Pauli principle in Table 4.I(JP) 1(0+) 1(1+) 1(2+) 0(1+) Bases [(cˉu)01(cˉd)01]0 [(cˉu)01(cˉd)11]1 [(cˉu)11(cˉd)11]2 [(cˉu)01(cˉd)11]1 [(cˉu)08(cˉd)08]0 [(cˉu)08(cˉd)18]1 [(cˉu)18(cˉd)18]2 [(cˉu)08(cˉd)18]1 [(cˉu)11(cˉd)11]0 [(cˉu)11(cˉd)01]1 [(cc)1ˉ3(ˉuˉd)13]2 [(cˉu)11(cˉd)01]1 [(cˉu)18(cˉd)18]0 [(cˉu)18(cˉd)08]1 [(cˉu)18(cˉd)08]1 [(cc)06(ˉuˉd)0ˉ6]0 [(cc)1ˉ3(ˉuˉd)13]1 [(cˉu)11(cˉd)11]1 [(cc)1ˉ3(ˉuˉd)13]0 [(cˉu)18(cˉd)18]1 [(cc)06(ˉuˉd)1ˉ6]1 [(cc)1ˉ3(ˉuˉd)03]1 Table 4. Color–spin bases for
Tcc system. The bases can be read as the notation:[(cˉu)S1c1(cˉd)S2c2]S for meson–meson structures,[(cc)S1c1(ˉuˉd)S2c2]S for diquark–antidiquark structures. The superscriptsS1 ,S2 represent the spin for two sub-clusters;S(S1⊗S2=S) is the total spin of the four-quark states. The subscriptsc1 andc2 stand for color. ForTbb states, replace c quark with b quark.Next, we discuss the orbital wave functions for a four-body system. These can be obtained by coupling the orbital wave function for each relative motion of the system:
ΨMLL=[[Ψl1(r12)Ψl2(r34)]l12ΨLr(r1234)]MLL,
(15) where
l1 andl2 are the angular momentum of two sub-clusters.ΨLr(r1234) is the wave function of the relative motion between the two sub-clusters with orbital angular momentumLr . L is the total orbital angular momentum of the four-quark state. In the present work, we only consider the low-lying S-wave double heavy tetraquark states, so it is natural to assume that all the orbital angular momenta are zero. In GEM, the spatial wave functions are expanded in series of Gaussian basis functions:Ψml(r)=∑n=1nmax
(16) \psi^G_{nlm}({\boldsymbol{r}}) = N_{nl}r^{l} {\rm e}^{-\nu_{n}r^2}Y_{lm}(\hat{{\boldsymbol{r}}}),
(17) where
N_{nl} are normalization constants,N_{nl} = \left[\frac{2^{l+2}(2\nu_{n})^{l+\frac{3}{2}}}{\sqrt{\pi}(2l+1)} \right]^{{1}/{2}},
(18) c_n are the variational parameters, which are determined dynamically. The Gaussian size parameters are chosen according to the following geometric progression:\nu_{n} = \dfrac{1}{r^2_n}, \quad r_n = r_1a^{n-1}, \quad a = \left(\dfrac{r_{n_{\max}}}{r_1}\right)^{{1}/({n_{\max}-1})}.
(19) This procedure enables optimization of the expansion using just a small number of Gaussian functions. Then, the complete channel wave function
\Psi^{\,M_IM_J}_{IJ} for the four-quark system is obtained by coupling the orbital and spin, flavor, and color wave functions obtained in Table 4. Finally, the eigenvalues of the four-quark system are obtained by solving the Schrödinger equation:H \, \Psi^{\,M_IM_J}_{IJ} = E^{IJ} \Psi^{\,M_IM_J}_{IJ}.
(20) -
In the present work, we are interested in looking for the bound states of low-lying S-wave
T_{QQ}\; (Q = c, b) system. The allowed quantum numbers areI(J^P) = 1(0^+), 1(1^+), 1(2^+), 0(1^+) under the constraints of the Pauli principle. The possible resonances of these states are searched with the help of the real scaling method (RSM). For bound state calculations, we aim to study the influence of the color structures on the binding energy. -
The low-lying eigenvalues of iso-vector
T_{QQ} states withI(J^P) = 1(0^+), 1(1^+), 1(2^+) are shown in Table 5. From the table, we can see that the low-lying energies are all higher than the corresponding theoretical thresholds, no matter whether in a meson–meson structure, a diquark–antidiquark structure or even considering the coupling of the two structures. No bound states are found.I(J^P) MM DA E_{cc} E_{\rm Theo} E_{\rm Exp} T_{cc} 1(0^+) 3726.8 4086.6 3726.7 3725.2 ( D^0D^+ )3734.4 1(1^+) 3844.8 4133.3 3844.7 3843.2 ( D^0D^{*+} )3871.8 1(2^+) 3962.8 4159.4 3962.7 3961.2 ( D^{*0}D^{*+} )4017.3 T_{bb} 1(0^+) 10562.3 10730.2 10562.3 10561.6 ( B^{-}\bar{B}^0 )10558.5 1(1^+) 10601.2 10741.1 10601.1 10600.4 ( B^{-}\bar{B^*}^0 )10604.5 1(2^+) 10639.9 10751.6 10639.9 10639.2 ( B^{*-}\bar{B^*}^0 )10650.5 Table 5. Low-lying eigenvalues of iso-vector
T_{QQ} states withI(J^P) = 1(0^+), 1(1^+), 1(2^+) (in units of MeV). MM and DA represent the meson–meson structures and diquark–antidiquark structures, respectively.E_{cc} is the energy considering the coupling of MM and DA.E_{\rm Theo} andE_{\rm Exp} is the theoretical and experimental threshold.Table 6 and Table 7 give the mass of iso-scalar
T_{cc} andT_{bb} withI(J^P) = 0(1^+) . The first column is the color–spin channel (listed in Table 4). The second column refers to the color structure including1\times1 ,8\times8 and their mixing, as well as\bar{3}\times3 ,6\times \bar{6} and their mixing. The following two columns represent the theoretical mass (E) and theoretical thresholds (E_{\rm Theo} ). The last column gives the binding energy\Delta E = E-E_{\rm Theo} . In Table 6, we find that there are no bound states in the single-channel calculation except for the diquark–antidiquark\bar{3}\times 3 structure. When considering the coupling of1\times 1 and8 \times 8 color structures in the meson–meson configuration, a loosely bound state with massM = 3841.4 MeV is obtained with the binding energy 1.8 MeV, which is consistent with the recent experiment data from LHCb. In diquark–antidiquark coupling calculations, a tightly bound state with massM = 3700.9 MeV is obtained with the binding energy 142.3 MeV. We obtain the lowest state with massM = 3660.7 MeV, a tightly bound state with binding energy 182.5 MeV, by considering the complete channel coupling effects including both meson–meson and diquark–antidiquark structures.Channel color structure E E_{\rm Theo} \Delta E [(c\bar{u})^0_1(c\bar{d})^1_1]^1 1\times1 3843.8 3843.2 ( D^0D^{*+} )+0.6 [(c\bar{u})^0_8(c\bar{d})^1_8]^1 8\times8 4168.7 +325.5 1\times1 +8\times8 3842.0 −1.2 [(c\bar{u})^1_1(c\bar{d})^0_1]^1 1\times1 3843.8 3843.2 ( D^{*0}D^{+} )+0.6 [(c\bar{u})^1_8(c\bar{d})^0_8]^1 8\times8 4168.7 +325.5 1\times1 +8\times8 3842.0 −1.2 [(c\bar{u})^1_1(c\bar{d})^1_1]^1 1\times1 3961.9 3961.2 ( D^{*0}D^{*+} )+0.7 [(c\bar{u})^1_8(c\bar{d})^1_8]^1 8\times8 4102.3 +141.1 1\times1 +8\times8 3958.7 −2.5 all 1\times 1 mixing3841.7 3843.2 ( D^0D^{*+} )−1.5 MM structure mixing 3841.4 3843.2 ( D^0D^{*+} )−1.8 [(cc)^0_6(\bar{u}\bar{d})^1_{\bar{6}}]^1 6\times \bar{6} 4115.1 3843.2 ( D^0D^{*+} )+271.9 [(cc)^1_{\bar{3}}(\bar{u}\bar{d})^0_3]^1 \bar{3}\times 3 3704.8 −138.4 DA structure mixing 3700.9 −142.3 MM and DA mixing 3660.7 3843.2 ( D^0D^{*+} )−182.5 Table 6. Mass of iso-scalar
T_{cc} states withI(J^P) = 0(1^+) (in units of MeV).Channel color structure E E_{\rm Theo} \Delta E [(b\bar{u})^0_1(b\bar{d})^1_1]^1 1\times1 10590.3 10600.4 ( B^{-}\bar{B^*}^0 )−10.1 [(b\bar{u})^0_8(b\bar{d})^1_8]^1 8\times8 10765.7 +165.3 1\times1 +8\times8 10566.7 −33.7 [(b\bar{u})^1_1(b\bar{d})^0_1]^1 1\times1 10590.3 10600.4 ( B^{*-}\bar{B}^0 )−10.1 [(b\bar{u})^1_8(b\bar{d})^0_8]^1 8\times8 10765.7 +165.3 1\times1 +8\times8 10566.7 −33.7 [(b\bar{u})^1_1(b\bar{d})^1_1]^1 1\times1 10629.8 10639.2 ( B^{*-}\bar{B^*}^0 )−9.4 [(b\bar{u})^1_8(b\bar{d})^1_8]^1 8\times8 10738.5 +99.3 1\times1 +8\times8 10598.8 −40.4 all 1\times 1 mixing10551.8 10600.4 ( B^{-}\bar{B^*}^0 )−48.6 MM structure mixing 10545.9 10600.4 ( B^{-}\bar{B^*}^0 )−54.5 [(bb)^0_6(\bar{u}\bar{d})^1_{\bar{6}}]^1 6\times \bar{6} 10746.8 10600.4 ( B^{-}\bar{B^*}^0 )+146.4 [(bb)^1_{\bar{3}}(\bar{u}\bar{d})^0_3]^1 \bar{3}\times 3 10298.9 −301.5 DA structure mixing 10298.2 −302.2 10576.8 -23.6 MM and DA mixing 10282.7^{1st} 10600.4 ( B^{-}\bar{B^*}^0 )−317.7 10516.7^{2nd} −83.7 Table 7. Mass of iso-scalar
T_{bb} states withI(J^P) = 0(1^+) (in units of MeV).The same calculations are extended to the
T_{bb} system in Table 7. With the increase ofm_{Q} , the bound states are easier to obtain. When considering the single channel, for1 \times 1 and\bar{3} \times 3 color structures, there are bound states, except for8 \times 8 and6 \times \bar{6} color structures. The mixing of the meson–meson structure leads to a bound state at 10545.9 MeV, which has 54.5 MeV binding energy. The mixing of diquark–antidiquark structures obtains a deeper bound state. When considering the coupling of meson–meson and diquark–antidiquark structures, we find two bound states with masses 10282.7 and 10516.7 MeV.We compare our results with some recent calculations in Table 8. From the table, we can see that for
T_{cc} of0(1^+) , there is controversy as to whether there are bound states. Our one shallow bound state with binding energy 1.8 MeV is consistent with Refs. [21, 26, 40], also in the case where only meson-meson structures are considered. In lattice QCD calculations, a few works support a bound state, also with small binding energy [52]. If one considers diquark–antidiquark structures in the calculations, a deeper bound state forT_{cc} of0(1^+) is also obtained, as in Refs. [23, 35]. ForT_{bb} of0(1^+) , the situation becomes much clearer. Almost all works obtained bound results. In the case of considering the meson–meson structures, our result with binding energy 54.5 MeV is consistent with Refs. [21, 24] in the quark model and also Ref. [49] in the lattice QCD calculation. No matter forT_{cc} orT_{bb} , when diquark–antidiquark structures are taken into account, deeper binding energies will appear, which seems to be somewhat misleading. In some other works, the same conclusions are obtained, such as in Refs. [12, 14, 29, 31].T_{cc} This work [11] [13] [15] [18] [19] [21] [23] [24] [25] [26] [31] [33] [34] [35] [40] [48] [52] −1.8 (MM) −79 −96 −98 N N −1 −182.3 N N −1 −23 N N −150 −3 N −23.3±11.4 −142.3 (DA) −182.5 (MM+DA) T_{bb} This work [15] [18] [20] [21] [22] [23] [24] [25] [31] [33] [34] [35] [49] [51] [52] [53] [55] −54.5 (MM) −268.6 −215 −150 −35 −120.9 −317 −54 −102 −173 N −145 −278 −30 ~ -57 −189±13 −143.3±34 −128±24±10 −167±19 −302.2 (DA) -317.7^{1st} (MM+DA)-83.7^{2nd} (MM+DA)Table 8. Results for
T_{cc} andT_{bb} tetraquarks with0(1^+) in comparison with the other theoretical calculations (in units of MeV). N stands for "no bound state". In the second column, (MM) represents only meson–meson structures considered, (DA) stands for only diquark–antidiquark configurations, and (MM+DA) represents the coupling of meson–meson and diquark–antidiquark structures.In order to understand the mechanism of forming the bound states for different color structures and their couplings, the contributions of each term in the Hamiltonian for
T_{cc} with0(1^+) are given in Table 9. In the table,K_1 andK_2 are the kinetic energy of the two sub-clusters, andK_3 represents the relative kinetic energy between these two clusters.V^{\rm{C, Coul, CMI}} refer to confinement (C), color Coulomb (Coul), and color–magnetic interaction (CMI).V^{\eta, \pi, \sigma} represent pseudoscalar meson η, π exchange and scalar meson σ exchange. To understand the mechanism, one has to compare the results in the tetraquark calculation to those of two free mesons which are listed in the last column in Table 9. For color singlet channels (index 1, 3, 5), the attractions provided by π and σ are too weak to overcome the relative kinetic energy to bind two color singlet clusters together, and these channels are scattering states when they stand alone. Coupling these color singlet channels adds more attraction, and a shallow bound state is formed (see the column headed by 1+3+5). The effect of coupling color octet–octet channels is very small, adding0.3 MeV to the binding energy (see the column headed by MM). If only the meson–meson structure is considered, the reported result from the LHCb collaboration can be explained by the molecular state. However, for a multi-quark system, more color structures are possible, in which the diquark–antidiquark structure with color representations\bar{3} \times 3 and6 \times \bar{6} are often invoked. For the\bar{3} \times 3 channel (column headed by 8), the strong attractions from π meson exchange, σ meson exchange and CMI overcome the repulsion from the kinetic energy and bind two "good diquark/antidiquark" [75] to form a deep bound state. The channel coupling between\bar{3} \times 3 and6 \times \bar{6} channels adds a little more binding energy to the system. The attraction from π-meson exchange in "good diquark" is very large because of its compact structure, as does the color magnetic interaction. The calculations without invoking pseudoscalar exchange do not obtain a bound state ofT_{cc} [17], supporting this statement.Channel 1 2 3 4 5 6 7 8 1+3+5 MM DA MM+DA D^0+D^{*+} K_1 516.8 317.2 359.8 322.8 359.6 347.3 119.4 191.2 442.2 440.2 192.2 321.8 516.7 K_2 359.8 322.8 516.8 317.2 359.6 347.3 295.4 1052.2 442.2 440.2 1050.6 929.7 359.3 K_3 3.4 307.3 3.4 307.3 3.0 323.4 395.4 262.3 17.4 25.5 267.0 189.9 0 V^{\rm{C}} −390.4 −264.2 −390.4 −264.2 −335.8 −300.8 −271.2 −412.9 −393.1 −393.6 −415.6 −432.8 −390.0 V^{\rm{Coul}} −612.9 −517.8 −612.9 −517.8 −543.8 −539.1 −506.4 −621.6 −614.8 −614.7 −623.6 −645.8 −612.7 V^{\rm{CMI}} −113.1 −6.6 −113.1 −6.6 38.7 −77.9 −6.8 −350.6 −118.7 −120.1 −356.7 −346.4 −112.1 V^{\eta} 0 5.5 0 5.5 0 6.5 −2.5 86.7 0.9 1.1 86.2 75.4 0 V^{\pi} −0.5 −56.9 −0.5 −56.9 −0.4 −63.9 28.3 −530.6 −10.8 −12.9 −527.4 −464.6 0 V^{\sigma} −1.3 −20.5 −1.3 −20.5 −1.1 −22.2 −18.5 −53.8 −5.6 −6.1 −53.8 −48.8 0 E 3843.8 4168.7 3843.8 4168.7 3961.9 4102.3 4115.1 3704.8 3841.7 3841.4 3700.9 3660.7 3843.2 \Delta E +0.6 +325.5 +0.6 +325.5 +0.7 +141.1 +271.9 −138.4 −1.5 −1.8 −142.3 −182.5 0 Table 9. Contributions of each potential in the system Hamiltonian of
T_{cc} withI(J^P) = 0(1^+) for different color structures and their couplings (in units of MeV). In the first row, to save space, we give labels in the sequence 1 ~ 8 for the eight channels ofT_{cc} with0(1^+) , as listed in Table 4.From quantum mechanics, one knows that the physical states are the linear combinations of all possible channels in various structures. So channel coupling with inclusion of different structures is needed. Clearly, the effect of this channel coupling will push the lowest state down further. A deep bound state with binding energy 182.5 MeV appears. Unfortunately, the shallow bound state in the meson–meson structure is pushed up above the threshold and so disappears. Experimentally, if there is only one weekly bound
T_{cc} state, diquark–antidiquark configurations should be abandoned, or the quark–quark interaction in the diquark structure should be modified with the accumulation of experimental data.In the discussion of hadronic states, the electromagnetic interaction is almost neglected due to the small coupling constant
\alpha \approx 1/137 . However, for the weakly bound state,E_{B}/M << 1 , the electromagnetic interaction may play a role. To investigate the role of the electric Coulomb force, we add the Coulomb potential\alpha\dfrac{q_iq_j}{r_{ij}} to the system Hamiltonian and solve the Schrödinger equation in the same way as above. The results show that the states are stable against the inclusion of the electric Coulomb interaction. The lowest eigen-energies are changed to 3840.7, 3700.2 and 3660.2 MeV in the MM structure, the DA structure and structure mixing respectively.In order to learn about the nature of doubly-heavy tetraquark states, we calculate the distances between any quark and quark/antiquark, which are shown in Table 10. We need to note that, for
T_{QQ} , because the heavy quarks Q are identical particles, as well as the light antiquark\bar{u} and\bar{d} , the distancer_{12} in the Table 10 actually is not the real value between the first quark Q and the second antidiquark\bar{u} . It should be the average:I(J^P) T_{cc} T_{bb} Resonance states Γ Resonance states Γ 1(0^+) ... ... 11309 23 1(1^+) 4639 10 11210 62 11318 21 1(2^+) 4687 9 11231 73 11329 0.4 0(1^+) 4304 38 10641 2 Table 10. The decay widths of predicted resonances of
T_{cc} andT_{bb} systems. (unit: MeV).\begin{aligned}[b] r_{12}^2= &r_{34}^2 = r_{14}^2 = r_{23}^2 \\ =& \frac{1}{4}(r_{Q^1\bar{u}^2}^2+r_{Q^1\bar{d}^4}^2+r_{Q^3\bar{u}^2}^2+ r_{Q^3\bar{d}^4}^2 ). \end{aligned}
(21) The numbers '1,2,3,4' are as shown in Fig. 1. From the table, for the weakly bound state of
T_{cc} with mass3841.4 MeV, with just the meson–meson structure considered, the distancer_{13} andr_{24} between two sub-clusters is larger than the distance within one cluster, which indicates that it is a molecularDD^* state. On the contrary, the tightly bound states 3700.9 and 3660.7 MeV are diquark–antidiquark structures. Furthermore, from the table we can see that although the\bar{u}\bar{d} is a good diquark, its size is sightly larger than that of thecc diquark. This should be from the color–electric interaction, not the color–magnetic interaction. -
In our work, we are also concerned about the possible resonance states of iso-vector and iso-scalar
T_{QQ} . To find the genuine resonances, the dedicated real scaling (stabilization) method is employed. In our previous work [76], this method was used successfully to explain theZ_{cs}(3985) observed by the BESIII collaboration [77]. To realize the real scaling method in our calculation, we need to multiply by a factor α the Gaussian size parameterr_n in Eq. (19) just for the meson–meson structure with color singlet–singlet configuration. In our calculation, α takes values from 1.0 to 3.0. With the increase of α, all states will fall off towards their thresholds, but bound states should be stable and resonances will appear as an avoid-crossing structure as in Fig. 2 [78]. In the figure, at the avoid-crossing structure, there are two lines. The upper line is a scattering state with larger slope, which will fall down to the threshold. The lower line represents the resonance state, which tries to keep stable with a smaller slope. When the resonance state and the scattering state interact with each other, this brings about an avoid-crossing point as in Fig. 2. With the increase of the scaling factor α, the energy of the other higher scattering state will fall down, will interact with this resonance state again and another avoid-crossing point will emerge. In our work, we set the number of repetitions of avoid-crossing points to 2, owing to the large amount of computation required. Theoretically speaking, with the continuous increase of α, avoid-crossing points will appear in succession. Then we have found a resonance state. For bound states, with the increase of α, they always stay stable. We show the results for theT_{QQ} system with all possible quantum numbers in Figs. 3–10 by considering the complete channel couplings.Figure 3. (color online) Stabilization plots of the energies of
T_{cc} states forI(J^P) = 1(0^+) with respect to the scaling factor α.Figure 4. (color online) Stabilization plots of the energies of
T_{cc} states forI(J^P) = 1(1^+) with respect to the scaling factor α.Figure 5. (color online) Stabilization plots of the energies of
T_{cc} states forI(J^P) = 1(2^+) with respect to the scaling factor α.Figure 6. (color online) Stabilization plots of the energies of
T_{cc} states forI(J^P) = 0(1^+) with respect to the scaling factor α.Figure 7. (color online) Stabilization plots of the energies of
T_{bb} states forI(J^P)=1(0^+) with respect to the scaling factor α.Figure 8. (color online) Stabilization plots of the energies of
T_{bb} states forI(J^P)=1(1^+) with respect to the scaling factor α.Figure 9. (color online) Stabilization plots of the energies of
T_{bb} states forI(J^P)=1(2^+) with respect to the scaling factor α.Figure 10. (color online) Stabilization plots of the energies of
T_{bb} states forI(J^P)=0(1^+) with respect to the scaling factor α.Figure 3 represents the
T_{cc} states for1(0^+) . The four horizontal blue lines are the thresholds ofD^0D^+ (0 \otimes 0 \rightarrow 0) ,D^{*0}D^{*+} (1 \otimes 1 \rightarrow 0) and their excited statesD^0D^+(2S) andD^{*0}D^{*+}(2S) . Higher energies are not listed here. We found no resonances for this state. From Fig. 4, two thresholdsD^0D^{*+} and its excited statesD^0D^{*+}(2S) appear clearly. Around an energy of 4639 MeV, we found the avoid-crossing phenomenon, which represents a genuine resonance state. In Fig. 5, we can see that the energy of the lowest resonance state is about 4687 MeV forT_{cc} with1(2^+) . For iso-scalarT_{cc} with0(1^+) , a bound state with mass 3660 MeV is obtained and the lowest resonance state shows with a stable energy of 4304 MeV in Fig. 6.For iso-vector and iso-scalar
T_{bb} states, there are also some resonance states found. They are states with masses of 11309 MeV for1(0^+) , 11210 and 11318 MeV for1(1^+) , 11231 and 11329 MeV for1(2^+) , and 10641 MeV for0(1^+) . There may be more resonance states with higher energies, which may be too wide to observe or too difficult to produce. So in our work, we only give the resonance states with energies as low as possible.Furthermore, we calculated the decay widths of these resonance states using the formula taken from Reference [78]:
\Gamma = 4|V(\alpha)|\frac{\sqrt{|S_r||S_c|}}{|S_c-S_r|},
(22) where
V(\alpha) is the difference between the two energies at the avoid-crossing point with the same value α, andS_r andS_c are the slopes of the scattering line and resonance line, respectively. For each resonance, we obtain the values of the decay width at the first and the second avoid-crossing point, and then we give the average decay width of these two values. The results are shown in Table 10. It should be noted that the decay width is the partial strong two-body decay width to S-wave channels included in the calculation. For example, forT_{cc} of1(1^+) , there is a resonance state with energy 4639 MeV, which has the decay width of 10 MeV in Table 10. This decay value is just the partial width to theD^0D^{*+} channel (see the threshold in Fig. 4). ForT_{cc} of0(1^+) , the decay width of resonance (4304) with 38 MeV is the partial decay toD^{*0}D^{*+} ,D^0D^{*+} andD^{+}D^{*0} channels (see the thresholds in Fig. 6).
Doubly-heavy tetraquark states ccˉuˉd and bbˉuˉd
- Received Date: 2021-11-23
- Available Online: 2022-05-15
Abstract: Inspired by the recent observation of a very narrow state, called