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Doubly-heavy tetraquark states ccˉuˉd and bbˉuˉd

  • Inspired by the recent observation of a very narrow state, called T+cc, by the LHCb collaboration, the possible bound states and low-lying resonance states of the doubly-heavy tetraquark states ccˉuˉd (Tcc) and bbˉuˉd (Tbb) are searched in the framework of a chiral quark model with an accurate few-body method, the Gaussian expansion method. The real scaling method is also applied to identify the genuine resonance states. In the calculation, the meson–meson structure, diquark–antidiquark structure, and their coupling are all considered. The numerical results show: (i) For Tcc and Tbb, only I(JP)=0(1+) states are bound in different quark structures. The binding energy varies from a few MeV for the meson–meson structure to over 100 MeV for the diquark–antidiquark structure. For example, for Tcc, in the meson–meson structure, there exists a weakly bound molecule DD state around 3841.4 MeV, 1.8 MeV below the D0D+, which may be a good candidate for the observed state by LHCb; however in the diquark–antidiquark structure, a deeper bound state with mass 3700.9 MeV is obtained. When considering the structure mixing, the energy of system decreases to 3660.7 MeV and the shallow bound state disappears. (ii) Besides bound states, several resonance states for TQQ(Q=c,b) with I(JP)=1(0+),1(1+),1(2+),0(1+) are proposed.
  • Nuclear clustering has recently attracted great attention within the nuclear structure studies, especially for nuclei in the expanded nuclear chart away from the β-stability line [17]. The cluster structure is often formed and stabilized at highly excited resonant states in the vicinity of the cluster-separation threshold [8]. This structure can be probed experimentally by nuclear reaction tools combined with sequential cluster-decay. The reconstruction of the resonant state from the decay fragments, by the so-called invariant mass (IM) method, allows to select the cluster states with large partial decay widths. This selection significantly reduces the level density at high excitation energies, being in favor of the quantitative extraction of the physical properties of the resonances. Furthermore, a model-independent determination of the spin in the reconstructed resonant state can be achieved though the sensitive angular correlation method [9, 10]. This is particularly important at high excitation energies, where the differential cross-section method becomes almost futile due to the overlap of many close-by states and the uncertainties in many fitting parameters [9].

    Determination of the spin of each resonant state is of particular importance to form the molecular band, which is required to firmly establish the clustering structure [11]. Thus far, the angular correlation analysis has been the most sensitive and reliable tool to determine the spin of the resonant state [10, 1215]. Nevertheless, the angular correlation plot depends on the selection of the coordinate system, which varies for different types of experiments and conventions of data analysis. Although in most cases the z-coordinate axis is fixed in the beam direction, the definition of the spherical angle axis differs in different coordinate systems. To date, no detailed analysis and comparison of these coordinate systems is provided in the literature, which could lead to misunderstanding and erroneous application of the angular correlation method.

    In this study, we systemically investigate three kinds of coordinate systems that have been frequently applied in the literature. The definitions and features of these systems are outlined and compared to each other. The consistency of these systems is demonstrated by the experimental data analysis for the 10.29-MeV resonant state in 18O. Some suggestions for the application of the angular correlation method are provided in the summary.

    For a sequential decay reaction a(A,Bc+C)b, the composite resonant particle B may decay into e.g., two spin-zero fragments. The angular correlation of the latter is a sensitive probe for the spin of the resonant state in the mother nucleus B [10]. In a spherical coordinate system, with its z-axis pointing along the beam direction (Fig. 1), the correlation function can be parameterized in terms of four angles [10]. Namely, the center-of-mass (c.m.) scattering polar and azimuthal angles, θ and ϕ, respectively, of the resonant particle B; the polar and azimuthal angles, ψ and χ, respectively, of the relative velocity vector vrel of the two fragments (the arrow connecting HI and LI in Fig. 1). Both polar angles, θ and ψ, are with respect to the beam direction. The azimuthal angle ϕ is 0 (or 180) in the horizontal plane defined by the center positions of the detectors placed at opposite sides of the beam (the chamber plane or the detection plane). Another azimuthal angle χ is defined to be 0 (or 180) in the reaction plane, fixed by the beam axis and the reaction product B. Because of the limited detector geometry in a typical experiment, the correlation is often approximately constrained in the chamber plane, as shown in Fig. 1. In this case, the azimuthal angles ϕ and χ remain at 0 or 180, depending on the selected coordinate system, and the angular correlation appears as a function of only two polar angles θ and ψ. This is referred to as the in-plane correlation.

    Figure 1

    Figure 1.  (color online) Schematic diagram of the sequential decay following a transfer reaction a(A,Bc+C)b , depicting the polar angles θ and ψ in the detection plane. The two decay fragments, c and C, are specified as light ion (LI) and heavy ion (HI), respectively, in the figure.

    When the azimuthal angle χ is restricted to 0 (or 180), the most striking feature of the angular correlation in the θ-ψ plane appears, as the ridge structures associated with the spin of the mother nucleus ([9, 10, 16]). At relatively small θ angles, the structure is characterized by the locus ψ=αθ in the double differential cross-section, where α depicts a constant for the slope of the ridge and is almost inversely proportional to the spin of the resonant state B [10, 16]. The correlation is oscillatory along the ψ angle for a fixed θ, and vice versa. Generally, this in-plane correlation structure can be projected onto the one-dimensional spectrum W(θ=0,ψ=ψαθ) . Within the strong absorption model (SAM) [1719], α may be related to the orbital angular moment li of the dominant partial wave in the entrance channel through α=liJJ, with J being the spin of B [9, 16]. li can be evaluated simply from li=r0(A1/3p+A1/3t)2μEc.m. [20], with Ap and At being the mass numbers of the beam and target nucleus, respectively. Here, μ is the reduced mass and Ec.m. depicts the center-of-mass energy. If the resonant nucleus is emitted at angles close to θ=0, the projected correlation function W(ψ) is simply proportional to the square of the Legendre polynomial of order J, namely |PJ(cos(ψ))|2. This method has been frequently applied in the literature ([21] and references therein) and will also be demonstrated in the following section 2.3.

    As indicated above, in the application of the angular correlation plot, it is important to enhance the ridge structure, which corresponds to the spin of the resonant mother nucleus. Consequently, the selection of the coordinate system for the plot is meaningful. For a non-polarized experiment, the reaction process satisfies the axial symmetry around the beam axis. Moreover, the decay process should satisfy the parity (space inversion) symmetry with respect to the c.m. of the resonant mother nucleus. In the two-dimensional correlation plot (plot of the double differential cross-section), depicting functions of the polar angles θ and ψ, it is natural to plot these symmetrical events in the same ridge band while placing the non-symmetric events elsewhere, to enhance the sensitivity to the associated spin. This should be applied even for the simple case of restricted detection around the chamber plane. For instance, in Fig. 2 we schematically illustrate two processes that generate the same polar angles θ and ψ but are not symmetric in terms of the resonance-decay. These two processes should be distinguished from the ridge plot using an appropriate coordinate system, such as the one using positive and negative θ, as defined below. In contrast, the coordinate definition should keep the four symmetric processes in the same ridge band, as schematically portrayed in Fig. 3, such that the ridge structure appears continuously and a simple projection can bring them together to enhance the sensitivity to the associated spin. We introduce, in the following, three kinds of coordinate systems and demonstrate their differences with respect to plot-definition and consistency in extraction of the spin.

    Figure 2

    Figure 2.  (color online) Schematic diagram of the two independent reaction processes with the same polar angles. The direction of the relative velocity vector vrel is guided by the light ion in both cases.

    Figure 3

    Figure 3.  (color online) Schematic diagram of the four symmetric reaction-decay processes in the chamber plane. (a) and (b) are parity-symmetric processes, while (c) and (d) are their axial-symmetric processes, respectively. All processes are identified by the angles θ and ψ defined in various coordinate systems, as described in the text.

    In the first coordinate system (hereinafter refered to as the ψfix coordinate system), the relative velocity vector of the decay products always points to the fixed detector at one side of the beam (where ϕ = 0). This is convenient when two decay particles are identical, such as in 24Mg12C+12C [10], or each detector is designed to be sensitive to only one type of the particles, such as 18O14C+α with 14C always detected at one-side, while α is at the other side of the beam [9]. By definition, θ is positive on the opposite side of the beam and negative on the same side, in comparison with the fixed positive ψ. With this definition, processes (a) and (b) are plotted at one position (negative θ), while (c) and (d) are at another position (positive θ) (see present Fig. 4(A) or Fig. 5 in Ref. [10]).

    Figure 4

    Figure 4.  (color online) Angular correlation spectra for the 10.29-MeV state in 18O in the (A) ψfix, (B) ψion and (C) ψfull coordinate systems, respectively. The projection lines in (A), (B), and (C) (the red-solid lines) correspond to a slope parameter α=liJJ, with li = 13.9 (using r0 = 1.2 fm). (a), (b), and (c) show the projections of (A), (B), and (C) onto the θ=0 axis, respectively. The black dot-dashed line indicates the simulated detector efficiency in each coordinate system. All experimental distributions are compared with a squared Legendre polynomial of the 4th order.

    Figure 5

    Figure 5.  (color online) Excitation energy spectrum for 18O emitting to small θ angles. The state at 10.29 MeV is selected to demonstrate the angular correlation analysis.

    In another more “physical” convention [16] (hereinafter denoted as the ψion coordinate system), the relative velocity vector vrel always points to a certain decay particle (usually the lighter one, LI), corresponding to ϕ = 0. The positive θ remains at the opposite side of the positive ψ. Under this convention, the axial-symmetric processes (a) and (c) will be plotted at the same position, whereas (b) and (d) are plotted at another position, as demonstrated in the present Fig. 4(B) or in Fig. 3 of Ref. [16]. Due to the different detection efficiencies for light and heavy particles, the correlation structure in this coordinate system may differ from that in the ψfix coordinate system.

    Additionally, on the basis of the ψion coordinate system, the polar angle ψ could also be assigned a positive or negative sign, depending on the azimuthal angle χ. First, 0 of the azimuthal angle ϕ or χ is defined by a detector in the chamber plane. Then, a positive ψ means that χ is close to 0, while negative ψ means that χ is close to 180. Hence, four intrinsically equivalent cases in Fig. 3 are plotted at four different positions in the θ-ψ plane. This coordinate system is denoted as ψfull. This convention was used in some previous studies, such as in Ref. [14], and also demonstrated in the present Fig. 4(C). Thus, both θ and ψ range across positive as well as negative scales. Since the experimental detection system may not be exactly symmetric with respective to the beam axis, the double differential cross-section in Fig. 4(C) seems not symmetric neither. It is evident that this wider scale distribution would give more consistent information for the ridge structure, however in the meantime, it would require higher statistics.

    The above-introduced three coordinate systems are equally meaningful, since the non-symmetric process, as shown in Fig. 2(b), does not appear in any of the defined ridge bands. Meanwhile, these systems should be consistent with each other in terms of extracting the spin of the resonant mother nucleus. This consistency is demonstrated below by experimental data analysis.

    Recently, a multi-nucleon transfer and cluster-decay experiment [22] 9Be(13C, 18O*14C + α)α was performed at the HI-13 tandem accelerator facility at the China Institute of Atomic Energy (CIAE) in Beijing. Resonant states in 18O can be reconstructed according to the invariant mass method [1, 5], as shown in Fig. 5 for events with small θ angles. The state at 10.29 MeV is a good candidate for angular correlation analysis, owing to its clear peak identification and relatively large ψ-angle coverage. In Fig. 4, we plot the angular correlation spectrum for the 18O 10.29-MeV state in the above described three coordinate systems (Fig. 4(A-C)). Further, these two dimensional spectra in θψ plane are projected onto the θ=0 axis according to the above described ψ=ψliJJθ relation, as exhibited in Fig. 4(a-c), respectively. The projections are compared with the square of the Legendre polynomial of order 4. Only the periodicity of the distribution matters, whereas the absolute peak amplitudes depend on the detection efficiency. Although the distributions behave slightly differently in the three coordinate systems, the periodicities of the experimental spectra all agree with the Legendre polynomial of order 4, corresponding to a spin-parity of 4+ for the 10.29-MeV state in 18O. This consistency between various coordinate systems indicates the reliability of the angular correlation method in determining the spin of a resonant state.

    Based on the consistency exhibited in Fig. 4 and the symmetry property of the Legendre polynomial, we may plot the projected correlation spectrum as a function of |cos(ψ)| [21], in order to increase the statistics in each bin of the distribution. Furthermore, this plot is independent of the above-defined coordinate systems. Moreover, the excitation energy spectrum can be reconstructed, similarly to that in Fig. 5, for each bin of |cos(ψ)| and the corresponding event number can be extracted for the pure 10.29-MeV peak by subtracting the smooth background beneath the peak. The experimental correlation spectrum is plotted in Fig. 6. The theoretical function composed of a squared Legendre polynomial with a constant background, corrected by the detection efficiency, is used to describe the experimental results. Not only the periodicity, but also the magnitude of the function for a spin-parity of 4+ provide an excellent fit to the experimental data, whereas other options of spin-parity can be excluded. A constant background is nevertheless needed in the theoretical function, since the experimental data include some uncorrelated components stemming from events within the 10.29-MeV peak, but away from the exact θ = 0 axis [10].

    Figure 6

    Figure 6.  (color online) Angular correlation spectrum for 10.29-MeV state in 18O, in comparison to the Legendre polynomials of order 4 (the red dotted line) and order 6 (the blue dot-dashed line). A uniformly distributed background is assumed to account for the uncorrelated component (the long dot-dashed line). Theoretical angular distributions have been corrected for the detection efficiency. The corresponding reduced ˉχ2 indicating the efficiency of the theoretical description is also indicated in the plot.

    The fragment angular correlation in a sequential cluster-decay reaction provides a method to determine the spins of the resonant nucleus independent of models. When the correlation spectrum is restricted to angles close to the detection chamber plane, the ridged pattern can be clearly seen in the two-dimensional plot with respective to the two polar angles θ and ψ. According to the ways to deal with symmetrical events, three coordinate systems for different θ and ψ definitions have been adopted in the literature for various experiments and for the spin analysis. In the present work, we outlined these systems and compared them to each other to clarify their differences and consistencies. The systems are examined by the cluster-decay data for the 10.29-MeV state in 18O, measured in our recent experiment. This study provides a better understanding of the angular correlation function, and demonstrates the possible choices for the best extraction of the spin in a resonant state.

    In the case where a resonant nucleus decays into two spin-zero fragments, the two-dimensional correlation spectrum for small θ angles can be projected onto the θ = 0 axis. This projected spectrum may be described by a squared Legendre polynomial with an order corresponding to the spin of the resonant mother nucleus. Using this method, a spin-parity of 4+ is decisively determined for the 10.29-MeV state in 18O.

    Based on the above investigations, we propose the basic procedure for the application of the angular correlation method. Firstly, the experiment should be designed to have good detection for events at small θ angles and with wide ψ coverage. Secondly, a proper coordinate system should be selected according to the detection and data-distribution characteristics. In principle, the best choice is the ψfull coordinate system, owing to its wider angular range of the correlation spectrum, which may help identify the ridge structure assuming the statistics are sufficient. However, when the statistics are low, the coordinate systems ψfix or ψion, depending on the detection arrangement, are more convenient. Thirdly, before the projection onto the θ=0 axis, the ridge structure should be distinctly observed. Otherwise, any small shift in the projection direction may lead to the wrong extraction of the spin, especially in the case of higher spin, where more oscillations in the projected spectrum are expected. The detector efficiency should likewise be carefully examined, since large variations in the efficiency curve may give rise to some non-physical structures in the projected distribution. Finally, the projected spectrum may be obtained by subtracting the background for each bin of |cos(ψ)|. This experimental spectrum can be compared with the theoretical function (Legendre polynomial corrected by the detection efficiency), and the goodness of fit can be examined quantitatively. However, the correct projection parameter should be fixed before this fitting procedure.

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    [2] R. Aaij et al. (LHCb, arXiv: 2109.01056 [hep-ex]
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Xiaoyun Chen and Youchang Yang. Doubly heavy tetraquark states ccˉuˉd and bbˉuˉd[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac4ee8
Xiaoyun Chen and Youchang Yang. Doubly heavy tetraquark states ccˉuˉd and bbˉuˉd[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac4ee8 shu
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Doubly-heavy tetraquark states ccˉuˉd and bbˉuˉd

  • 1. College of Science, Jinling Institute of Technology, Nanjing 211169, China
  • 2. Guizhou University of Engineering Science; Zunyi Normal University, China

Abstract: Inspired by the recent observation of a very narrow state, called T+cc, by the LHCb collaboration, the possible bound states and low-lying resonance states of the doubly-heavy tetraquark states ccˉuˉd (Tcc) and bbˉuˉd (Tbb) are searched in the framework of a chiral quark model with an accurate few-body method, the Gaussian expansion method. The real scaling method is also applied to identify the genuine resonance states. In the calculation, the meson–meson structure, diquark–antidiquark structure, and their coupling are all considered. The numerical results show: (i) For Tcc and Tbb, only I(JP)=0(1+) states are bound in different quark structures. The binding energy varies from a few MeV for the meson–meson structure to over 100 MeV for the diquark–antidiquark structure. For example, for Tcc, in the meson–meson structure, there exists a weakly bound molecule DD state around 3841.4 MeV, 1.8 MeV below the D0D+, which may be a good candidate for the observed state by LHCb; however in the diquark–antidiquark structure, a deeper bound state with mass 3700.9 MeV is obtained. When considering the structure mixing, the energy of system decreases to 3660.7 MeV and the shallow bound state disappears. (ii) Besides bound states, several resonance states for TQQ(Q=c,b) with I(JP)=1(0+),1(1+),1(2+),0(1+) are proposed.

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    I.   INTRODUCTION
    • Recently, the LHCb collaboration [1] reported the observation of a very narrow state, called T+cc, in the D0D0π+ invariant mass spectrum. The binding energy and the decay width is:

      δmBW=273±61±5+1114keV/c2,ΓBW=410±165±43+1838keV.

      (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+40keV/c2,ΓU=48±2+014keV.

      (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 channels Λ+cKπ+ and pD+K [3, 4]. Fifteen years later, the LHCb collaboration also found the doubly-charmed baryon Ξ++cc(ccu) in the Λ+cKπ+π+ mass distribution [5]. The value of the mass is 100 MeV higher than the Ξ+cc observed by the SELEX collaboration. Secondly, the mass of the observed Tcc is just a litter lower than M(D0)+M(D+), with a very small binding energy. Undoubtedly, the observation of T+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 and Ξ++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 of QQˉqˉq was made in early 1980s [6], with the observation that the system will be bound, below the Qˉq+Qˉq threshold, if the mass ratio M/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 out MTcc3935MeV, 60 MeV above the threshold, and Tbb 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 [1116], quark models [1735], the QCD sum rule approach [3646], lattice QCD simulations [4755], effective field theory [5658] and others [5964]. One of the controversies is whether QQˉqˉq tetraquarks with two heavy quarks Q and two light antidiquarks ˉq are stable or not against the decay into two Qˉq 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 the TQQ 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 that Tcc was a I(JP)=0(1+) state around 3929 MeV (53 MeV above the DD threshold) and all the double-charm tetraquarks were not stable. Karliner et al. [18] predicted the mass of T(ccˉuˉd) with JP=1+ to be 3882 MeV, 7 MeV above the D0D+ threshold and 148 MeV above the D0D+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 the I=0 channels suggested attraction, although neither bound states nor resonance states were found in the Tcc(IJP=01+). Some works are in favor of them as tightly bound states [1113, 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 of Tcc with color structure 6ˉ6 lay above the DD threshold, but the mass of Tcc with color structure ˉ33 lay at 71 MeV below the DD 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 the TQQ 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 singlet qqq and qˉq systems are directly extended to quark-pairs with various color structures, and the effects of color structure are taken care of by the Casimir operator λiλj.

      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 tetraquarks ccˉuˉd and bbˉuˉd with the quantum numbers I(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.

    II.   THEORETICAL FRAMEWORK

      A.   The chiral quark model

    • 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=4i=1mi+p2122μ12+p2342μ34+p212342μ1234+4i<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 factor λiλj [66]. Their forms are:

      VCij=(acr2ijΔ)λciλcj,

      (4)

      VGij=αs4λciλcj[1rij2π3mimjσiσjδ(rij)],

      (5)

      δ(rij)=erij/r0(μij)4πrijr20(μij),

      (6)

      Vπij=g2ch4πm2π12mimjΛ2πΛ2πm2πmπvπij3a=1λaiλaj,

      (7)

      VKij=g2ch4πm2K12mimjΛ2KΛ2Km2KmKvKij7a=4λaiλaj,

      (8)

      Vηij=g2ch4πm2η12mimjΛ2ηΛ2ηm2ηmηvηij×[λ8iλ8jcosθPλ0iλ0jsinθP],

      (9)

      vχij=[Y(mχrij)Λ3χm3χY(Λχrij)]σiσj,

      (10)

      Vσij=g2ch4πΛ2σΛ2σm2σmσ×[Y(mσrij)ΛσmσY(Λσrij)],

      (11)

      where Y(x)=ex/x; {mi} are the constituent masses of quarks and antiquarks, and μij are their reduced masses;

      μ1234=(m1+m2)(m3+m4)m1+m2+m3+m4;

      (12)

      pij=(pipj)/2, p1234=(p12p34)/2; r0(μij)=s0/μij; σ are the SU(2) Pauli matrices; λ, λc are SU(3) flavor and color Gell-Mann matrices, respectively; g2ch/4π is the chiral coupling constant, determined from π-nucleon coupling; and αs 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
      (fm1200MeV) 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 fm2) 101
      Δ/MeV −78.3
      OGE α0 3.67
      Λ0/fm1 0.033
      μ0/MeV 36.98
      s0/MeV 28.17

      Table 1.  Model parameters, determined by fitting the meson spectra.

      With these model parameters, we obtain the list of relevant meson spectra for D() and B() 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
      D0 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
      ¯B0 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).

    • B.   Wave functions of TQQ

    • 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=12(αβ+βα),χ11=ββ,χ00=12(αββα),

      (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 and S2 of two sub-clusters. The superscript of χ stands for the total spin S (S1S2 = S) and the third projection MS of S for the four-quark system.

      Spin Flavor Color
      χ0000=χ00χ00 χ00=12(cˉdcˉucˉucˉd) 11:χc1=13(ˉrr+ˉgg+ˉbb)(ˉrr+ˉgg+ˉbb)
      χ0011=13(χ11χ11χ10χ10+χ11χ11) χ10=12(cˉdcˉu+cˉucˉd) 88:χc2=212(3ˉbrˉrb+3ˉgrˉrg+3ˉbgˉgb+3ˉgbˉbg+3ˉrgˉgr+3ˉrbˉbr+2ˉrrˉrr+2ˉggˉgg+2ˉbbˉbbˉrrˉggˉggˉrrˉbbˉggˉbbˉrrˉggˉbbˉrrˉbb)
      χ1101=χ00χ11 ˉ33:χc3=36(rgˉrˉgrgˉgˉr+grˉgˉrgrˉrˉg+rbˉrˉbrbˉbˉr+brˉbˉrbrˉrˉb+gbˉgˉbgbˉbˉg+bgˉbˉgbgˉgˉb)
      χ1110=χ11χ00 6ˉ6:χc4=612(2rrˉrˉr+2ggˉgˉg+2bbˉbˉb+rgˉrˉg+rgˉgˉr+grˉgˉr+grˉrˉg+rbˉrˉb+rbˉbˉr+brˉbˉr+brˉrˉb+gbˉgˉb+gbˉbˉg+bgˉbˉg+bgˉgˉb)
      χ1111=12(χ11χ10χ10χ11)
      χ2211=χ11χ11

      Table 3.  Wave functions of spin, flavor and color for TQQ.

      For the flavor component, the flavor wave functions with isospin =0 and 1 are also tabulated in Table 3. The subscript 00 or 10 of χ represents the total isospin I and the third projection Iz of I.

      For the color component, more richer structures in the four-quark system will be considered than conventional qˉq mesons and qqq baryons. For the meson–meson structure in Fig. 1, in the SU(3) group, the colorless wave functions can be obtained from 11=1 or 88=1. For the diquark–antidiquark structure, the colorless wave functions can be obtained from ˉ33=1 or 6ˉ6=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 of TQQ should satisfy the antisymmetry requirement. Then we collect all the color–spin bases of TQQ 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 superscripts S1, S2 represent the spin for two sub-clusters; S(S1S2=S) is the total spin of the four-quark states. The subscripts c1 and c2 stand for color. For Tbb 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 and l2 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 momentum Lr. 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)=nmaxn=1cnψGnlm(r),

      (16)

      ψGnlm(r)=Nnlrleνnr2Ylm(ˆr),

      (17)

      where Nnl are normalization constants,

      Nnl=[2l+2(2νn)l+32π(2l+1)]1/2,

      (18)

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

      νn=1r2n,rn=r1an1,a=(rnmaxr1)1/(nmax1).

      (19)

      This procedure enables optimization of the expansion using just a small number of Gaussian functions. Then, the complete channel wave function ΨMIMJIJ 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:

      HΨMIMJIJ=EIJΨMIMJIJ.

      (20)
    III.   CALCULATIONS AND ANALYSIS
    • In the present work, we are interested in looking for the bound states of low-lying S-wave TQQ(Q=c,b) system. The allowed quantum numbers are I(JP)=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.

    • A.   Calculation of bound states

    • The low-lying eigenvalues of iso-vector TQQ states with I(JP)=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(JP) MM DA Ecc ETheo EExp
      Tcc 1(0+) 3726.8 4086.6 3726.7 3725.2 (D0D+) 3734.4
      1(1+) 3844.8 4133.3 3844.7 3843.2 (D0D+) 3871.8
      1(2+) 3962.8 4159.4 3962.7 3961.2 (D0D+) 4017.3
      Tbb 1(0+) 10562.3 10730.2 10562.3 10561.6 (BˉB0) 10558.5
      1(1+) 10601.2 10741.1 10601.1 10600.4 (B¯B0) 10604.5
      1(2+) 10639.9 10751.6 10639.9 10639.2 (B¯B0) 10650.5

      Table 5.  Low-lying eigenvalues of iso-vector TQQ states with I(JP)=1(0+),1(1+),1(2+) (in units of MeV). MM and DA represent the meson–meson structures and diquark–antidiquark structures, respectively. Ecc is the energy considering the coupling of MM and DA. ETheo and EExp is the theoretical and experimental threshold.

      Table 6 and Table 7 give the mass of iso-scalar Tcc and Tbb with I(JP)=0(1+). The first column is the color–spin channel (listed in Table 4). The second column refers to the color structure including 1×1, 8×8 and their mixing, as well as ˉ3×3, 6×ˉ6 and their mixing. The following two columns represent the theoretical mass (E) and theoretical thresholds (ETheo). The last column gives the binding energy ΔE=EETheo. In Table 6, we find that there are no bound states in the single-channel calculation except for the diquark–antidiquark ˉ3×3 structure. When considering the coupling of 1×1 and 8×8 color structures in the meson–meson configuration, a loosely bound state with mass M=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 mass M=3700.9 MeV is obtained with the binding energy 142.3 MeV. We obtain the lowest state with mass M=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 ETheo ΔE
      [(cˉu)01(cˉd)11]1 1×1 3843.8 3843.2 (D0D+) +0.6
      [(cˉu)08(cˉd)18]1 8×8 4168.7 +325.5
      1×1 + 8×8 3842.0 −1.2
      [(cˉu)11(cˉd)01]1 1×1 3843.8 3843.2 (D0D+) +0.6
      [(cˉu)18(cˉd)08]1 8×8 4168.7 +325.5
      1×1 + 8×8 3842.0 −1.2
      [(cˉu)11(cˉd)11]1 1×1 3961.9 3961.2 (D0D+) +0.7
      [(cˉu)18(cˉd)18]1 8×8 4102.3 +141.1
      1×1 + 8×8 3958.7 −2.5
      all 1×1 mixing 3841.7 3843.2 (D0D+) −1.5
      MM structure mixing 3841.4 3843.2 (D0D+) −1.8
      [(cc)06(ˉuˉd)1ˉ6]1 6×ˉ6 4115.1 3843.2 (D0D+) +271.9
      [(cc)1ˉ3(ˉuˉd)03]1 ˉ3×3 3704.8 −138.4
      DA structure mixing 3700.9 −142.3
      MM and DA mixing 3660.7 3843.2 (D0D+) −182.5

      Table 6.  Mass of iso-scalar Tcc states with I(JP)=0(1+) (in units of MeV).

      Channel color structure E ETheo ΔE
      [(bˉu)01(bˉd)11]1 1×1 10590.3 10600.4 (B¯B0) −10.1
      [(bˉu)08(bˉd)18]1 8×8 10765.7 +165.3
      1×1 + 8×8 10566.7 −33.7
      [(bˉu)11(bˉd)01]1 1×1 10590.3 10600.4 (BˉB0) −10.1
      [(bˉu)18(bˉd)08]1 8×8 10765.7 +165.3
      1×1 + 8×8 10566.7 −33.7
      [(bˉu)11(bˉd)11]1 1×1 10629.8 10639.2 (B¯B0) −9.4
      [(bˉu)18(bˉd)18]1 8×8 10738.5 +99.3
      1×1 + 8×8 10598.8 −40.4
      all 1×1 mixing 10551.8 10600.4 (B¯B0) −48.6
      MM structure mixing 10545.9 10600.4 (B¯B0) −54.5
      [(bb)06(ˉuˉd)1ˉ6]1 6×ˉ6 10746.8 10600.4 (B¯B0) +146.4
      [(bb)1ˉ3(ˉuˉd)03]1 ˉ3×3 10298.9 −301.5
      DA structure mixing 10298.2 −302.2
      10576.8 -23.6
      MM and DA mixing 10282.71st 10600.4 (B¯B0) −317.7
      10516.72nd −83.7

      Table 7.  Mass of iso-scalar Tbb states with I(JP)=0(1+) (in units of MeV).

      The same calculations are extended to the Tbb system in Table 7. With the increase of mQ, the bound states are easier to obtain. When considering the single channel, for 1×1 and ˉ3×3 color structures, there are bound states, except for 8×8 and 6×ˉ6 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 Tcc of 0(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 for Tcc of 0(1+) is also obtained, as in Refs. [23, 35]. For Tbb of 0(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 for Tcc or Tbb, 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].

      Tcc 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)
      Tbb 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)
      317.71st (MM+DA)
      83.72nd (MM+DA)

      Table 8.  Results for Tcc and Tbb tetraquarks with 0(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 Tcc with 0(1+) are given in Table 9. In the table, K1 and K2 are the kinetic energy of the two sub-clusters, and K3 represents the relative kinetic energy between these two clusters. VC,Coul,CMI refer to confinement (C), color Coulomb (Coul), and color–magnetic interaction (CMI). Vη,π,σ 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, adding 0.3 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 ˉ3×3 and 6×ˉ6 are often invoked. For the ˉ3×3 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 ˉ3×3 and 6×ˉ6 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 of Tcc [17], supporting this statement.

      Channel 1 2 3 4 5 6 7 8 1+3+5 MM DA MM+DA D0+D+
      K1 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
      K2 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
      K3 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
      VC −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
      VCoul −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
      VCMI −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
      Vη 0 5.5 0 5.5 0 6.5 −2.5 86.7 0.9 1.1 86.2 75.4 0
      Vπ −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
      Vσ −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
      ΔE +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 Tcc with I(JP)=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 of Tcc with 0(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 Tcc 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 α1/137. However, for the weakly bound state, EB/M<<1, the electromagnetic interaction may play a role. To investigate the role of the electric Coulomb force, we add the Coulomb potential αqiqjrij 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 TQQ, because the heavy quarks Q are identical particles, as well as the light antiquark ˉu and ˉd, the distance r12 in the Table 10 actually is not the real value between the first quark Q and the second antidiquark ˉu. It should be the average:

      I(JP) Tcc Tbb
      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 Tcc and Tbb systems. (unit: MeV).

      r212=r234=r214=r223=14(r2Q1ˉu2+r2Q1ˉd4+r2Q3ˉu2+r2Q3ˉd4).

      (21)

      The numbers '1,2,3,4' are as shown in Fig. 1. From the table, for the weakly bound state of Tcc with mass 3841.4 MeV, with just the meson–meson structure considered, the distance r13 and r24 between two sub-clusters is larger than the distance within one cluster, which indicates that it is a molecular DD 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 ˉuˉd is a good diquark, its size is sightly larger than that of the cc diquark. This should be from the color–electric interaction, not the color–magnetic interaction.

    • B.   Calculation of resonance states

    • In our work, we are also concerned about the possible resonance states of iso-vector and iso-scalar TQQ. 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 the Zcs(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 parameter rn 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 the TQQ system with all possible quantum numbers in Figs. 310 by considering the complete channel couplings.

      Figure 2.  (color online) Stabilization graph for the resonance.

      Figure 3.  (color online) Stabilization plots of the energies of Tcc states for I(JP)=1(0+) with respect to the scaling factor α.

      Figure 4.  (color online) Stabilization plots of the energies of Tcc states for I(JP)=1(1+) with respect to the scaling factor α.

      Figure 5.  (color online) Stabilization plots of the energies of Tcc states for I(JP)=1(2+) with respect to the scaling factor α.

      Figure 6.  (color online) Stabilization plots of the energies of Tcc states for I(JP)=0(1+) with respect to the scaling factor α.

      Figure 7.  (color online) Stabilization plots of the energies of Tbb states for I(JP)=1(0+) with respect to the scaling factor α.

      Figure 8.  (color online) Stabilization plots of the energies of Tbb states for I(JP)=1(1+) with respect to the scaling factor α.

      Figure 9.  (color online) Stabilization plots of the energies of Tbb states for I(JP)=1(2+) with respect to the scaling factor α.

      Figure 10.  (color online) Stabilization plots of the energies of Tbb states for I(JP)=0(1+) with respect to the scaling factor α.

      Figure 3 represents the Tcc states for 1(0+). The four horizontal blue lines are the thresholds of D0D+(000), D0D+(110) and their excited states D0D+(2S) and D0D+(2S). Higher energies are not listed here. We found no resonances for this state. From Fig. 4, two thresholds D0D+ and its excited states D0D+(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 for Tcc with 1(2+). For iso-scalar Tcc with 0(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 Tbb states, there are also some resonance states found. They are states with masses of 11309 MeV for 1(0+), 11210 and 11318 MeV for 1(1+), 11231 and 11329 MeV for 1(2+), and 10641 MeV for 0(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]:

      Γ=4|V(α)||Sr||Sc||ScSr|,

      (22)

      where V(α) is the difference between the two energies at the avoid-crossing point with the same value α, and Sr and Sc 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, for Tcc of 1(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 the D0D+ channel (see the threshold in Fig. 4). For Tcc of 0(1+), the decay width of resonance (4304) with 38 MeV is the partial decay to D0D+, D0D+ and D+D0 channels (see the thresholds in Fig. 6).

    IV.   SUMMARY
    • In the framework of the chiral quark model, we undertake a systematic calculation for the mass spectra of doubly-heavy TQQ with quantum numbers I(JP)=1(0+),1(1+),1(2+),0(1+), to look for possible bound states. Also, some resonance states are found with the real scaling method.

      In bound state calculations, we analyze the effects of different color structures, such as 1×1, 8×8 for meson–meson structures and ˉ3×3, 6×ˉ6 for diquark–antidiquark structures, on the binding energy of TQQ. The masses of states with isospin I=1 are all above the corresponding thresholds, leaving no space for bound states. For Tcc with 0(1+), we find that in the meson–meson structure, a loosely bound state at 3841.1 MeV is obtained with a 1.8 MeV binding energy, which is consistent with the recent observed experiment at LHCb. But in diquark–antidiquark structures, CMI potential, π-exchange and σ exchange offer more attractions and a tightly bound state with mass 3700 MeV is obtained. The couplings of meson–meson and diquark–antidiquark structures can not be neglected, which leads to more deeper bound states and destroys the loosely bound state. For the heavier Tbb system with 0(1+), the same conclusions are obtained, similar to the case of Tcc, and it looks easier to find more deeper bound states. For example, a bound state 10545.9 MeV with binding energy 54.5 MeV is obtained when only meson–meson structures are considered. If only diquark–antidiquark structures are taken into account, two bound states emerge with binding energy 302.2 MeV and 23.6 MeV. When considering the coupling of meson–meson and diquark–antidiquark configurations, these two states will have deeper binding energies of 317.7 MeV and 83.7 MeV.

      For resonance state calculations, some resonances are found. For Tcc, the energies of the possible resonances are 4639 MeV for 1(1+), 4687 MeV for 1(2+), 4304 MeV for 0(1+). For Tbb, the resonance energies are larger than Tcc, which takes 11309 MeV for 1(0+), 11210 and 11318 MeV for 1(1+), 11231 and 11329 MeV for 1(2+), 10641 MeV for 0(1+). Hopefully, our results in this work by the phenomenological framework of the chiral quark model may be confirmed in future high energy experiments.

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