$QQ\bar{s}\bar{s}$ tetraquarks in the chiral quark model

We are witnessing in the last two decades of a big experimental effort for understanding the heavy-flavor quark sectors of both meson and baryon systems. Many experiments have been settled worldwide such as B-factories (BaBar, Belle, and CLEO), $\tau$-charm facilities (CLEO-c and BES) and hadron-hadron colliders (CDF, D0, LHCb, ATLAS, and CMS) providing a sustained progress in the field with new measurements of conventional and exotic heavy-flavored hadrons.

Within the baryon sector, and attending mostly to the spectrum, five excited $\Omega_{c}$ baryons were announced three years ago by the LHCb collaboration in the $\Xi_{c}^{+}K^{-}$ mass spectrum Aaij et al. (2017) and, very recently, the same collaboration has reported additional four narrow excited states of the $\Omega_{b}$ system in the $\Xi_{b}^{0}K^{-}$ mass spectrum Aaij et al. (2020a). In 2019, two excited bottom baryons, $\Lambda_{b}^{0}(6146)$ and $\Lambda^{0}_{b}(6152)$, were discovered in the LHCb experiment Aaij et al. (2019a). Later on, the LHCb collaboration also announced one more $\Lambda^{0}_{b}$ baryon around 6070 MeV in the $\Lambda^{0}_{b}\pi^{+}\pi^{-}$ invariant mass spectrum Aaij et al. (2020b), which is consistent with the reported one of the CMS collaboration Sirunyan et al. (2020). Additionally, three excited $\Xi_{c}^{0}$ states were announced by the LHCb collaboration in the $\Lambda^{+}_{c}K^{-}$ mass spectrum Aaij et al. (2020c).

All of these newly discovered baryons undoubtedly complement the scarce data on heavy flavor baryons in the Review of Particle Physics (RPP) of the Particle Data Group Tanabashi et al. (2018). Furthermore, these experimental findings trigger a large number of theoretical investigations. The three-quark structure of the new $\Omega_{c}$ baryons has been claimed by QCD sum rules Chen et al. (2017a) and different potential models Karliner and Rosner (2017); K. L. Wang and Zhao (2017); Yang and Ping (2018). Also, the description of the $\Omega_{b}$ signals as $P$-wave conventional baryons is preferred by phenomenological quark model approach Karliner and Rosner (2020); Xiao et al. (2020), heavy quark effective theory Chen et al. (2020) and QCD sum rules Wang (2020a). Meanwhile, the baryon-meson molecular interpretation has been suggested for the excited $\Omega_{b}$ baryons in Ref. Liang and Oset (2020). The $\Lambda^{0}_{b}(6072)$, $\Lambda^{0}_{b}(6146)$ and $\Lambda^{0}_{b}(6152)$ have been identified as radial and angular excitations within QCD sum rules Azizi et al. (2020); Wang et al. (2019a); Chen et al. (2019); Yang et al. (2020a) and chiral quark models Wang et al. (2020); L$\ddot{u}$ (2020). However, the $\bar{D}\Lambda-\bar{D}\Sigma$ molecular configurations have been also suggested for these states in Ref. Zhu et al. (2020).

Apart from conventional heavy flavored baryons, there are limited results on open-bottom mesons and detailed studies of the open-charm ones were not undertaken until large datasets were obtained by CLEO at discrete energy points and by the B-factory experiments using radiative returns to obtain a continuous exposure of the mass region. The picture that has emerged is complex due to the many thresholds in the region. This resembles the experimental situation found in the heavy quarkonium spectrum with the observation of more than two dozens of unconventional charmonium- and bottomonium-like states, the so-called XYZ mesons. However, still successful observations of 6 new conventional heavy quarkonium states (4 $c\bar{c}$ and 2 $b\bar{b}$) have been made.

Exotic states such as tetraquarks and pentaquarks have lastly received considerable attention within the hadron physics community. Related with the first structures, the best known is the $X(3872)$, which was observed in 2003 as an extremely narrow peak in the $B^{+}\to K^{+}(\pi^{+}\pi^{-}J/\psi)$ channel and at exactly the $\bar{D}^{0}D^{\ast 0}$ threshold Choi et al. (2003); Aubert et al. (2005), and it is suspected to be a $cn\bar{c}\bar{n}$ ($n=u$ or $d$ quark) tetraquark state whose features resemble those of a molecule, but some experimental findings forbid to discard a more compact, diquark-antidiquark, component or even some $c\bar{c}$ trace in its wave function. On the other hand, there are no doubts of the tetraquark character of the $Z_{c}$’s Ablikim et al. (2013); Liu et al. (2013) and $Z_{b}$’s Bondar et al. (2012); Adachi et al. (2012) states due to its non-zero charge. The most prominent examples of the second mentioned structures are the hidden-charm pentaquarks $P^{+}_{c}(4312)$, $P^{+}_{c}(4380)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$ reported in 2015 and 2019 by the LHCb collaboration in the $\Lambda_{b}^{0}$ decay, $\Lambda_{b}^{0}\to J/\psi K^{-}p$ Aaij et al. (2019b, 2015).

The discussion about the nature of these exotic signals are carried out by various theoretical approaches. In particular, the three newly announced hidden-charm pentaquarks, $P^{+}_{c}(4312)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$ are favored to be molecular states of $\Sigma_{c}\bar{D}^{*}$ in, for instance, effective field theories Liu et al. (2019a); He (2019), QCD sum rules Wang (2020b), phenomenological potential models Guo and Oller (2019); Huang et al. (2019); Mutuk (2019); Zhu et al. (2019); Eides et al. (2019); Weng et al. (2019), heavy quark spin symmetry formalisms Shimizu et al. (2019); Xiao et al. (2019a) and heavy hadron chiral perturbation theory Meng et al. (2019). Moreover, their photo-production Cao and Dai (2019); Wang et al. (2019b) and decay properties Xiao et al. (2019b) have been also discussed. As for the other types of pentaquarks, bound states of the $\bar{Q}qqqq$ system are not found within a constituent quark model Richard et al. (2019). Using the same approach, several narrow double-heavy pentaquark states are found to be possible in the systematical investigations of Refs. Zhou et al. (2018); Giannuzzi (2019); Yang et al. (2020b). Moreover, within the one-boson-exchange model, possible triple-charm molecular pentaquarks $\Xi_{cc}D^{(*)}$ are suggested Wang et al. (2019c). In the tetraquark sector, double-heavy tetraquarks are studied using QCD sum rules Agaev et al. (2019), quark models Fontoura et al. (2019); Yang et al. (2020c) and even lattice-regularized QCD computations Leskovec et al. (2019). Besides, theoretical techniques such as diffusion Monte Carlo Bai et al. (2019), Bethe-Salpeter equation Heupel et al. (2012), QCD sum rules Wang (2017); Chen et al. (2017b) and effective phenomenological models Anwar et al. (2018); Esposito and Polosa (2018); Chen (2019); Liu et al. (2019b); Wang et al. (2019d) have recently contributed to the investigations of fully heavy tetraquarks $QQ\bar{Q}\bar{Q}$. Some reviews on both tetraquark and pentaquark systems can be found in Refs. Vijande et al. (2019); Liu et al. (2019c).

Our QCD-inspired chiral quark model explained successfully the nature of the $P_{c}^{+}$ states in Ref. Yang and Ping (2017), even before the last updated data reported by the LHCb collaboration Aaij et al. (2019b). Based on such fact, the hidden-bottom Yang et al. (2019) and double-charm pentaquarks Yang et al. (2020b) were systematically investigated within the same theoretical framework, finding several either bound or resonance states. Reference Yang et al. (2020c) reported results on the double-heavy tetraquarks $QQ\bar{q}\bar{q}$ $(Q=c,b$ and $q=u,d)$, its natural extension should be the $QQ\bar{s}\bar{s}$ tetraquark sector with the hope of finding either bound or resonance states. In order to do so, we have recently established a complex scaling range formalism of the chiral quark model which allows us to determine (if exist) simultaneously scattering, resonance and bound states. We shall study herein the $QQ\bar{s}\bar{s}$ tetraquarks in the spin-parity channels $J^{P}=0^{+}$, $1^{+}$ and $2^{+}$, and in the isoscalar sector $I=0$. Another relevant feature of our study is that all configurations: meson-meson, diquark-antidiquark and K-type for four-body systems are considered; moreover, every possible color channel is taken into account, too. Finally, the Rayleigh-Ritz variational method is employed in dealing with the spatial wave functions of tetraquark states, which are expanded by means of the well-known Gaussian expansion method (GEM) of Ref. Hiyama et al. (2003).

The present manuscript is arranged as follows. Section II is devoted to briefly describe our theoretical approach which includes the complex-range formulation of the chiral quark model and the discussion of the $QQ\bar{s}\bar{s}$ wave-functions. Section III is devoted to the analysis and discussion of the obtained results. The summary and some prospects are presented in Sec. IV.

The complex scaling method (CSM) applied to our chiral quark model has been already explained in Refs. Yang et al. (2020c, b). The general form of the four-body complex Hamiltonian is given by

where the center-of-mass kinetic energy $T_{\text{CM}}$ is subtracted without loss of generality since we focus on the internal relative motions of quarks inside the multi-quark system. The interplay is of two-body potential which includes color-confining, $V_{\text{CON}}$, one-gluon exchange, $V_{\text{OGE}}$, and Goldstone-boson exchange, $V_{\chi}$, respectively,

In this work, we focus on the low-lying positive parity $QQ\bar{s}\bar{s}$ tetraquark states of $S$-wave, and hence only the central and spin-spin terms of the potentials shall be considered.

By transforming the coordinates of relative motions between quarks as $\vec{r}_{ij}\rightarrow\vec{r}_{ij}e^{i\theta}$, the complex scaled Schrödinger equation

is solved, giving eigenenergies that can be classified into three kinds of poles: bound, resonance and scattering ones, in a complex energy plane according to the so-called ABC theorem Aguilar and Combes (1971); Balslev and Combes (1971). In particular, the resonance pole is independent of the rotated angle $\theta$, i.e. it is fixed above the continuum cut line with a resonance’s width $\Gamma=-2\,\text{Im}(E)$. The scattering state is just aligned along the cut line with a $2\theta$ rotated angle, whereas a bound state is always located on the real axis below its corresponding threshold.

The two-body potentials in Eq. (2) mimic the most important features of QCD at low and intermediate energies. Firstly, color confinement should be encoded in the non-Abelian character of QCD. It has been demonstrated by lattice-QCD that multi-gluon exchanges produce an attractive linearly rising potential proportional to the distance between infinite-heavy quarks Bali et al. (2005). However, the spontaneous creation of light-quark pairs from the QCD vacuum may give rise at the same scale to a breakup of the created color flux-tube Bali et al. (2005). Therefore, the following expression when $\theta=0^{\circ}$ is used for the confinement potential:

where $a_{c}$, $\mu_{c}$ and $\Delta$ are model parameters, and the SU(3) color Gell-Mann matrices are denoted as $\lambda^{c}$. One can see in Eq. (4) that the potential is linear at short inter-quark distances with an effective confinement strength $\sigma=-a_{c}\,\mu_{c}\,(\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j})$, while $V_{\text{CON}}$ becomes constant $(\Delta-a_{c})(\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j})$ at large distances.

Secondly, the QCD’s asymptotic freedom is expressed phenomenologically by the Fermi-Breit reduction of the one-gluon exchange interaction which, in the case of hadron systems with $\geq 3$ quarks, consists on a Coulomb term supplemented by a chromomagnetic contact interaction given by

where $m_{i}$ and $\vec{\sigma}$ are the quark mass and the Pauli matrices, respectively. The contact term of the central potential in complex range has been regularized as

The QCD-inspired effective scale-dependent strong coupling constant, $\alpha_{s}$, offers a consistent description of mesons and baryons from light to heavy quark sectors in wide energy range, and we use the frozen coupling constant of, for instance, Ref. Segovia et al. (2013)

in which $\alpha_{0}$, $\mu_{0}$ and $\Lambda_{0}$ are parameters of the model.

Thirdly, the Goldstone-boson exchange interactions between light quarks, and constituent quark masses, appear because the breaking of chiral symmetry in a dynamical way. Therefore, the following two terms of the chiral potential must be taken into account between the $(\bar{s}\bar{s})$-pair for $QQ\bar{s}\bar{s}$ tetraquarks:

where $Y(x)=e^{-x}/x$ is the standard Yukawa function. The pion- and kaon-exchange interactions do not appear because no up- and down-quarks are considered herein. Furthermore, the physical $\eta$ meson is taken into account by introducing the angle $\theta_{p}$. The $\lambda^{a}$ are the SU(3) flavor Gell-Mann matrices. Taken from their experimental values, $m_{\pi}$, $m_{K}$ and $m_{\eta}$ are the masses of the SU(3) Goldstone bosons. The value of $m_{\sigma}$ is determined through the PCAC relation $m_{\sigma}^{2}\simeq m_{\pi}^{2}+4m_{u,d}^{2}$ Scadron (1982). Finally, the chiral coupling constant, $g_{ch}$, is determined from the $\pi NN$ coupling constant through

which assumes that flavor SU(3) is an exact symmetry only broken by the different mass of the strange quark.

The model parameters, which are listed in Table 1, have been fixed in advance reproducing hadron Valcarce et al. (1996); Vijande et al. (2005); Segovia et al. (2008a, b, 2009, 2011, 2015); Ortega et al. (2016a); Yang et al. (2018, 2020d), hadron-hadron Fernandez et al. (1993); Valcarce et al. (1994); Ortega et al. (2016b, 2017, 2018) and multiquark Vijande et al. (2006); Yang and Ping (2017, 2018); Yang et al. (2019) phenomenology. Additionally, in order to help on our analysis of the $QQ\bar{s}\bar{s}$ tetraquarks in the following section, Table 2 lists the theoretical and experimental masses of the ground state and its first radial excitation (if available) for the $D^{(*)+}_{s}$ and $\bar{B}^{(*)}_{s}$ mesons. Besides, their mean-square radii are collected in Table 2.

Figure 1 shows six kinds of configurations for double-heavy tetraquarks $QQ\bar{s}\bar{s}$ $(Q=c,b)$. In particular, Fig. 1(a) is the meson-meson (MM) structure, Fig. 1(b) is the diquark-antidiquark (DA) one, and the other K-type configurations are from panels (c) to (f). All of them, and their couplings, are considered in our investigation. However, for the purpose of solving a manageable 4-body problem, the K-type configurations are restricted to the case in which the two heavy quarks of $QQ\bar{s}\bar{s}$ tetraquarks are identical. It is important to note herein that just one configuration would be enough for the calculation, if all radial and orbital excited states were taken into account; however, this is obviously much less efficient and thus an economic way is to combine the different configurations in the ground state to perform the calculation.

Four fundamental degrees of freedom at the quark level: color, spin, flavor and space are generally accepted by QCD theory and the multiquark system’s wave function is an internal product of color, spin, flavor and space terms. Firstly, concerning the color degree-of-freedom, plenty of color structures in multiquark system will be available with respect those of conventional hadrons ($q\bar{q}$ mesons and $qqq$ baryons). The colorless wave function of a 4-quark system in di-meson configuration, i.e. as illustrated in Fig. 1(a), can be obtained by either a color-singlet or a hidden-color channel or both. However, this is not the unique way for the authors of Refs. Harvey (1981); Vijande et al. (2009), who assert that it is enough to consider the color singlet channel when all possible excited states of a system are included.¹¹1After a comparison, a more economical way of computing through considering all the possible color structures and their coupling is preferred. The $SU(3)_{\text{color}}$ wave functions of a color-singlet (two coupled color-singlet clusters, ${\bf 1}_{c}\otimes{\bf 1}_{c}$) and hidden-color (two coupled color-octet clusters, ${\bf 8}_{c}\otimes{\bf 8}_{c}$) channels are given by, respectively,

In addition, the color wave functions of the diquark-antidiquark structure shown in Fig. 1(b) are $\chi^{c}_{3}$ (color triplet-antitriplet clusters, ${\bf 3}_{c}\otimes{\bf\bar{3}}_{c}$) and $\chi^{c}_{4}$ (color sextet-antisextet clusters, ${\bf 6}_{c}\otimes{\bf\bar{6}}_{c}$), respectively:

Meanwhile, the colorless wave functions of the K-type structures shown in Fig. 1(c) to 1(f) are obtained by following standard coupling algebra within the $SU(3)$ color group:²²2The group chain of K-type is obtained in sequence of quark number. Moreover, each quark and antiquark is represented, respectively, with [1] and [11] in the group theory.

• •

  $K_{1}$-type of Fig. 1(c): [$C^{[21]}_{[1],[11]}C^{[221]}_{[21],[11]}C^{[222]}_{[221],[1]}$]₅;

  [$C^{[111]}_{[1],[11]}C^{[221]}_{[111],[11]}C^{[222]}_{[221],[1]}$]₆;

• •

  $K_{2}$-type of Fig. 1(d): [$C^{[111]}_{[1],[11]}C^{[211]}_{[111],[1]}C^{[222]}_{[211],[11]}$]₇;

  [$C^{[21]}_{[1],[11]}C^{[211]}_{[21],[1]}C^{[222]}_{[211],[11]}$]₈;

• •

  $K_{3}$-type of Fig. 1(e): [$C^{[2]}_{[1],[1]}C^{[211]}_{[2],[11]}C^{[222]}_{[211],[11]}$]₉;

  [$C^{[11]}_{[1],[1]}C^{[211]}_{[11],[11]}C^{[222]}_{[211],[11]}$]₁₀;

• •

  $K_{4}$-type of Fig. 1(f): [$C^{[22]}_{[11],[11]}C^{[221]}_{[22],[1]}C^{[222]}_{[221],[1]}$]₁₁;

  [$C^{[211]}_{[11],[11]}C^{[221]}_{[211],[1]}C^{[222]}_{[221],[1]}$]₁₂.

These group chains will generate the following K-type color wave functions whose subscripts correspond to those numbers above:

As for the flavor degree-of-freedom, since the quark content of the tetraquark systems considered herein are two heavy quarks, $(Q=c,b)$, and two strange antiquarks, $\bar{s}$, only the isoscalar sector, $I=0$, will be discussed. The flavor wave functions denoted as $\chi^{fi}_{I,M_{I}}$, with the superscript $i=1,~{}2$ and $3$ referring to $cc\bar{s}\bar{s}$, $bb\bar{s}\bar{s}$ and $cb\bar{s}\bar{s}$ systems, can be written as

where, in this case, the third component of the isospin $M_{I}$ is equal to the value of total one $I$.

The total spin $S$ of tetraquark states ranges from $0$ to $2$. All of them shall be considered and, since there is not any spin-orbit potential, the third component $(M_{S})$ can be set to be equal to the total one without loss of generality. Therefore, our spin wave functions $\chi^{\sigma_{i}}_{S,M_{S}}$ are given by

The superscripts $l_{1},\,\ldots\,,l_{4}$ and $m_{1},\ldots,m_{6}$ are numbering the spin wave function for each configuration of tetraquark states, their specific values are shown in Table 3. Furthermore, these expressions are obtained by considering the coupling of two sub-cluster spin wave functions with SU(2) algebra, and the necessary bases read as

Among the different methods to solve the Schrödinger-like 4-body bound state equation, we use the Rayleigh-Ritz variational principle which is one of the most extended tools to solve eigenvalue problems because its simplicity and flexibility. Meanwhile, the choice of basis to expand the wave function solution is of great importance. Within hte CRM, the spatial wave function can be written as follows

where the internal Jacobi coordinates for the meson-meson configuration (Fig. 1(a)) are defined as

and for the diquark-antdiquark one (Fig. 1(b)) are

The Jacobi coordinates for the remaining K-type configurations shown in Fig. 1, panels (c) to (f), are ($i,j,k,l$ are according to the definitions of each configuration in Fig. 1):

Obviously, the center-of-mass kinetic term $T_{\text{CM}}$ can be completely eliminated for a non-relativistic system when using these sets of coordinates.

A very efficient method to solve the bound-state problem of a few-body system is the Gaussian expansion method (GEM) Hiyama et al. (2003), which has been successfully applied by us in other multiquark systems Yang and Ping (2017); Yang et al. (2019, 2020b, 2020c). The Gaussian basis in each relative coordinate is taken with geometric progression in the size parameter.³³3The details on Gaussian parameters and how they are fixed can be found in Ref. Yang and Ping (2017). Therefore, the form of the orbital wave functions, $\phi$’s, in Eq. (41) is

As one can see, the Jacobi coordinates are all transformed with a common scaling angle $\theta$ in the complex scaling method. In this way, both bound states and resonances can be described simultaneously within one scheme. Moreover, only $S$-wave state of double-heavy tetraquarks are investigated in this work and thus no laborious Racah algebra is needed during matrix elements calculation.

Finally, in order to fulfill the Pauli principle, the complete wave-function is written as

where $\cal{A}$ is the antisymmetry operator of $QQ\bar{s}\bar{s}$ tetraquarks when considering interchange between identical particles ($\bar{s}\bar{s},~{}cc$ and $bb$). This is necessary because the complete wave function of the 4-quark system is constructed from two sub-clusters: meson-meson, diquark-antidiquark and K-type structures. In particular, when the two heavy quarks are of the same flavor ($QQ=cc$ or $bb$), the operator $\cal{A}$ with the quark arrangements $\bar{s}Q\bar{s}Q$ is defined as