The sss̄s̄ tetraquark states and the newly observed structure X ( 2239 ) by BESIII Collaboration

Qi-Fang Lü, 2, ∗ Kai-Lei Wang, 2 and Yu-Bing Dong 4, 5, † Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Theoretical Physics Center for Science Facilities (TPCSF), CAS, Beijing 100049, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 101408, China


I. INTRODUCTION
Recently, the BESIII Collaboration analyzed the cross section of the e + e − → K + K − process at center-of-mass energies varying from 2.00 to 3.08 GeV. A resonant structure is observed in the line shape, which has a mass of 2239.2 ± 7.1 ± 11.3 MeV and a width of 139.8 ± 12.3 ± 20.6 MeV [1]. Given its production process, the quantum number of this resonant structure can be assigned as J PC = 1 −− .
In the conventional quark model, several highly excited 1 −− ρ, ω, and φ states are predicted in this energy region, and their strong decay behaviors have been investigated in the quark pair creation model. Due to the large phase space, the predicted total decay width of these states are rather broad, which suggests that the newly observed state with about 140 MeV width may not be a conventional excited meson. More exotic interpretations, such as the ssss tetraquark state, are needed to be considered to clarify its nature. In the litera-ture, the P−wave ssss system was mostly investigated by the QCD sum rule method [9][10][11][12]25] or the simple quark models [13,14], and their results are inconsistent with each other. Hence, it is essential to study this system in a more realistic potential model.
In this work, we firstly employ a relativized quark model to estimate the masses of ssss tetraquark states. The relativized quark model, proposed by Godfrey, Capstick, and Isgur, has been widely used to study the properties of the conventional hadrons and gives a unified description of the traditional hadron spectra [26][27][28][29][30][31][32][33][34][35][36]. Also, it has been extended to investigate various tetraquark systems, such as QqQq, Qqqq and so on [37][38][39]. Moreover, the relativistic effects are involved in this model, which may be essential for the up, down and strange quarks. Therefore, it is suitable to deal with the ssss tetraquark states, where strange quarks and antiquarks are included. To calculate the tetraquark masses, we restrict our works in the diquark-antidiquark picture [37][38][39][40][41][42][43][44][45][46][47][48]. We first calculate the masses and wave functions of the axialvector and vector ss diquarks, and then obtain the mass spectra and diquark-antidiquark wave functions by solving the Schrödinger-type equation between the diquark and antidiquark. The total wave functions can be expressed as the multiplication of the diquark, antidiquark, and diquark-antidiquark wave functions. The predicted mass of the lowest 1 −− ssss tetraquark is 2227 MeV, which is consistent with the experimental data 2239.2±7.1±11.3 MeV by BESIII Collaboration. It suggests that the newly observed resonant structure X(2239) can be assigned as the lowest J PC = 1 −− ssss tetraquark state. Furthermore, using the wave functions obtained from the relativized quark model and the electromagnetic transition operator, we estimate the radiative decays of the ssss tetraquarks. It is found that the radiative decay width for the lowest 1 −− ssss tetraquark state is 27 keV, which is significant and useful for future experimental searches. This paper is organized as follows. In Sec. II, the relativized quark model is briefly introduced, and the masses of ssss tetraquark states are calculated. In Sec. IV, the radiative transitions of ssss tetraquark states are numerically estimated.
Finally, we give a short summary in the last section.

II. MASS SPECTRUM
The Hamiltonian between the quark and antiquark in the relativized quark model can be expressed as with where theH con f i j includes the spin-independent linear confinement and Coulomb-like interaction, theH cont i j ,H ten i j , and H so i j are the color contact term, the color tensor interaction, and the spin-orbit term, respectively. TheH represents that the operator H has taken account of the relativistic effects via the relativized procedure. The explicit forms of these interactions and the details of this relativization scheme can be found in Ref. [26,27]. In the original GI model, the coupled channel or screening effects are ignored, which may influence on the properties of the excited mesons and tetraquarks [33,37,49,50]. The modified procedure br → b(1 − e −µr )/µ with a new screening parameter µ performs a better description of meson and tetraquark spectra, especially for the strange quark systems [33,37]. Hence, we take the screening effects into account for completeness.
In present work, only the antitriplet diquark [3 c ] ss are considered, while the [6 c ] ss type diquarks can not be formed in the GI quark model. For the quark-quark interaction in the antitriplet diquark and triplet antidiquark systems, the relatioñ V ss (p, r) =Ṽss(p, r) =Ṽ ss (p, r)/2 is employed. The parameters used in our calculations are the same as the ones in the original work [26]. The structures of the ssss tetraquarks are illustrated in Fig. 1. The interaction between diquark and an-tidiquarkṼ ss−ss (p, r) equals to the quark-antiquark interactioñ V ss (p, r). The ground ss diquark lies in S −wave, and has the spin-parity J P = 1 + named as axial-vector diquark. For the excited ones, we only consider the P−wave ss diquark with spin-parity J P = 1 − , which is denoted as vector diquark.
Here, we use the Gaussian expansion method to solve the Hamiltonian (1) withṼ ss (p, r) potential [51]. The obtained masses of the axial-vector and vector ss diquarks are presented in Table. I. Since the µ = 0.02 GeV case can give a rather better description of the strange quark systems [33,37], we would like to use the diquark masses at this value to calculate the masses and wave functions of ssss tetraquarks.
With the diquarks listed in Table. I, one can calculate the masses of the ssss tetraquarks and the wave functions between diquarks and anti-diquarks. Then, the total wave function of the ssss tetraquark can be expressed as the multiplication of the diquark, anti-diquark and relative wave functions. The masses of ssss tetraquark states composed of the A and V diquarks and antidiquarks are presented in Tab. II and Fig. 2.  The predicted mass of the lowest 0 ++ state is 1716 MeV, which is consistent with the f 0 (1710) state. For the 1 +− ssss state, only h 1 (1965) state listed in the PDG book lies in this energy region [2]. Since the h 1 (1965) was observed in ωη and ωππ final states, which disfavors its assignment as 1 +− ssss tetraquark. In Ref. [12,25], the authors suggest that the new structure X(2063) observed in the J/ψ → φηη ′ by BESIII Collaboration [52] is a 1 +− ssss tetraquark candidate within the QCD sum rule method. However, our calculated mass is 100 MeV lower than the experimental mass, which does not support this interpretation. Considering the mass, spin parity, and φφ decay mode of the f 2 (2300), we may assign it as a 2 ++ ssss tetraquark state. For the P−wave ssss tetraquarks, we predict three 1 −− states. The lowest one has the internal excitation in the diquark or antidiquark, while the others have the relative excitations between diquarks and antidiquarks. The three 1 −− states together with other theoretical works are listed in Tab. III for comparisons. It can be seen that our quark model classification is significantly different with the QCD sum rule works [9][10][11]25], and the authors did not consider the internal excitation of the diquark or anti-diquark within the simple quark model [13]. The predicted lowest one has a mass of 2227 MeV, which agrees well with the X(2239) observed by BE-SIII Collaboration [1]. The experimental mass of the φ(2170) is about 50 MeV lower than our calculation, which can not be excluded as the ssss tetraquarks. The evidences of these other II: Masses of the ssss tetraquark states composed of the A and V diquarks and antidiquarks. For the V diquark and A antidiquark case, the linear combinations together with V diquark and A antidiquark are understood to form the eigenstates of charge conjugation [37,48]. The units are in MeV. two higher 1 −− states may have been obsevered in the previous experiments [4,11,[53][54][55], or are waiting to be discovered in future searches. Furthermore, we predict several higher ssss tetraquarks around 2.5 GeV. For the higher 0 −+ state, there exists a candidate X(2500) with mass of 2470 +15+101 −19−23 MeV observed in the J/ψ → γφφ process by BESIII Collaboration [56]. In the conventional quark model, the X(2500) was assigned as the φ(5 1 S 0 ) state given its mass and total width [57,58] , but with a tiny φφ partial decay width. The ssss tetraquark interpretation of the X(2500) may avoid this defect due to its falling apart mechanism into the φφ final state. Other predictions can provide helpful information for future experimental searches.

III. RADIATIVE TRANSITIONS
Besides the mass spectrum, the decay behaviors are also needed to clarify these tetraquark states in future experiments. The strong decays can occur if the tetraquarks lie above the meson-meson threshold via falling apart mechanism, and the detailed discussions can be found in Refs. [13,14]. To treat the radiative transitions between these ssss tetraquarks, one can adopt an EM transition operator which has been successfully applied to study the radiative decays of quarkonium and baryons [59][60][61]. In this model, the quark-photon EM coupling at the tree level is taken as where ψ j stands for the jth quark field with coordinate r j and A µ is the photon field with three-momentum k. To match the wave functions obtained by the Schrödinger-like equation, we adopt this quark-photon EM coupling in a nonrelativistic form. In the initial-hadron-rest system, the approximate form can be written as [59][60][61][62][63][64][65][66][67] h e j e j r j · ǫ − e j 2m j σ j · (ǫ ×k) e −ik·r j , where e j , m j , and σ j stand for the charge, consistent mass, Pauli spin vector for the jth quark, respectively. The ǫ is the polarization vector of the final photon.
One can obtain the helicity amplitude A of the radiative transition [59,60] Then, we can estimate the radiative transitions straightforward [59,60] where J i is the total angular momentum of the initial tetraquarks, and J f z and J f i are the components of the total angular momenta along the z axis of the initial and final tetraquarks, respectively. In present work, the masses and wave functions of the ssss tetraquarks are adopted from our theoretical predictions. The radiative transitions of the ssss tetraquarks are estimated and listed in Tab. IV. Here, we eliminate the notation AA of the three ground states without causing misunderstanding. The predicted radiative transitions of the three ground states 0 ++ , 1 +− , and 2 ++ are respectively, which are significant large. As we assign the f 0 (1710) and f 2 (2300) as the 0 ++ and 2 ++ states respectively, the rather large radiative decay rates are useful to searching for the missing 1 +− ssss tetraquark. Since the 0 ++ state has large branching ratios of KK and ηη, more studies of the ssss(1 +− ) → ssss(0 ++ )γ → KKγ and ssss(1 +− ) → ssss(0 ++ )γ → ηηγ decay processes are suggested in future experiments. For the transitions between VĀ type and ground states, the partial radiative decay widths range from 1 eV to tens keV. The |VĀ, 1 −− → |0 ++ γ process is 26.6 keV, which shows that the newly observed X(2239) state has a significant radiative decay width. The other two 1 −− states with relative excitations between diquarks and anti-diquarks can decay into 0 ++ and 2 ++ ground states, respectively, where the ten times divergence of the partial widths derives from the different phase spaces. The radiative decay of the S = 0 state to 2 ++ final state is highly suppressed, and also for the S = 2 state to 0 ++ final state. These predictions may be helpful for searching and distinguishing the two higher 1 −− ssss tetraquark states. About these radiative transitions of the excited ssss tetraquarks, few discussions are found in the literature, thus, more theoretical and experimental studies are expected to be carried out in future.

IV. SUMMARY
In this work, we investigate the masses of ssss tetraquark states using the relativized quark model proposed by Godfrey and Isgur. Here, only the antitriplet diquark [3 c ] cs is considered. The masses of ssss tetraquark states are obtained by solving the Schrödinger-like equation between diquark and antidiquark. The color screening effects are also added in present calculations. It is found that the newly observed resonant structure X(2239) in the e + e − → K + K − process by BESIII Collaboration can be assigned as a P−wave 1 −− ssss tetraquark state.
Besides the mass spectrum, the wave functions of the ssss tetraquark states are obtained simultaneously. With the wave functions, the radiative transitions between these tetraquarks are also estimated. The P−wave 1 −− ssss tetraquark state radiate to the ground state is 27 keV, which can provide helpful information for future experimental searches. Moreover, other ssss tetraquark candidates f 0 (1710), f 2 (2300), and X(2500) are also discussed here. We hope our assignments can be tested by future experiments.