A possible interpretation of the newly observed $\Omega(2012)$ state

Inspired by the newly observed $\Omega(2012)$ state at Belle II, we investigate the two-body strong decays of $\Omega$ baryons up to $N=2$ shell within the chiral quark model. Our results indicate that: (i) the newly observed $\Omega(2012)$ state could be assigned to the spin-parity $J^P=3/2^-$ state $|70,^210,1,1,\frac{3}{2}^-\rangle$ and the experimental data can be reasonably described. However, the spin-parity $J^P=1/2^-$ state $|70,^210,1,1,\frac{1}{2}^-\rangle$ and spin-parity $J^P=3/2^+$ state $|56,^410,2,0,\frac{3}{2}^+\rangle$ can't be completely excluded. (ii) The $D$-wave states in the $N=2$ shell are most likely to be narrow states with a width of dozens of MeV and have a good potential to be observed in the $\Xi K$ and/or $\Xi(1530)K$ channels in future experiments. The $\Omega(2250)$ resonance listed in PDG may be a good candidate of the $J^P=5/2^+$ $1D$ wave state $|56,^410,2,2,5/2^+\rangle$.


I. INTRODUCTION
Searching for the missing baryon resonances and understanding the baryon spectrum are important topics in hadron physics. In the past years, for the limitations of experimental conditions, our knowledge about Ω spectrum is still scarce. There are only a few data on the Ω resonances. In the review of the Particle Data Group (PDG) [1], only four Ω baryon states are listed: Ω(1672), Ω(2250), Ω(2380), and Ω(2470). Except the ground state Ω(1672) being well established with four-star ratings, the nature of the other three excited states are still rather uncertain with three-or two-star ratings. Fortunately, the Belle II experiments offer a great opportunity for our study of the Ω spectrum.
The chiral quark model [23] is developed and successfully used to study the Okubo-Zweig-Iizuka (OZI) allowed twobody strong decays of the heavy-light mesons [24][25][26][27] and baryons [28][29][30][31][32][33][34][35]. In this framework, the spatial wave functions of heavy baryons are described by harmonic oscillators, and an effective chiral Lagrangian is then introduced to account for the quark-meson coupling at the baryon-meson interaction vertex. The light pseudoscalar mesons, i.e., π, K, and η, are treated as Goldstone bosons. Since the quark-meson coupling is invariant under the chiral transformation, some of the low-energy properties of QCD are retained [23,[36][37][38]. Within the chiral quark model, the OZI allowed twobody strong decays of Ω baryons up to N = 2 shell are analyzed in present work. The quark model classification for the Ω baryons and their theoretical masses are listed in Table  I. According to our calculations, we obtain that (i) the newly observed Ω(2012) resonance could be assigned to the spinparity J P = 3/2 − state |70, 2 10, 1, 1, 3 2 − and the experimental data can be reasonably described. However, the spin-parity J P = 1/2 − state |70, 2 10, 1, 1, 1 2 − and spin-parity J P = 3/2 + state |56, 4 10, 2, 0, 3 2 + can't be completely excluded.(ii) The D-wave states in the N = 2 shell are most likely to be narrow states with a with of dozens of MeV and have a good potential to be observed in their corresponding dominant decay channels. The Ω(2250) resonance listed in PDG may be a good candidate of the J P = 5/2 + 1D wave state |56, 4 10, 2, 2, 5/2 + . This paper is organized as follows. In Sec. II we give a brief introduction of the chiral quark model. we present our I: The theoretical masses (MeV) and spin-flavor-space wavefunctions of baryons, denoted by |N 6 , 2S +1 N 3 , N, L, J P [30]. The Clebsch-Gordan series for the spin and angular-momentum addition |J, J z = Lz+S z=Jz LL z , S S z |J J z Ψ σ NLLz χ S z has been omitted.  4 numerical results and discussions in Sec. III and summarize our results in Sec. IV.

II. THE CHIRAL QUARK MODEL
In the chiral quark model, the effective low energy quarkpseudoscalar-meson coupling in the SU(3) flavor basis at tree level is given by [23] where f m stands for the pseudoscalar meson decay constant. ψ j corresponds to the jth quark field in a baryon and φ m denotes the pseudoscalar meson octet To match the nonrelativistic harmonic oscillator spatial wave function Ψ NLL z in the calculations, we adopt a nonrelativistic form of Eq. (3) and get [36][37][38] where (E i , p i ), (E f , p f ) and (ω m , q) stand for the energy and three-vector momentum of the initial baryon, final baryon and meson, respectively. σ j is the Pauli spin vector on the jth quark, and µ q is a reduced mass expressed as 1 is the internal momentum of the jth quark in the baryon rest frame. ϕ m = e −iq·r j and e iq·r j for emitting and absorbing a meson, respectively. The isospin operator I j associated with the pseudoscalar meson is given by Here, a † j (u, d, s) and a j (u, d, s) are the creation and annihilation operator for the u, d, s quarks on jth quark. θ is the mixing angle of the η meson in the flavor basis [1].
For the decay processes, we select the initial-baryon-rest system in the calculations. Then, p i = 0 and p f = −q. The Eq. 5 can be further simplified and the partial decay amplitudes for B → B ′ M can be calculated by where B ′ and B stand for the final and initial baryon wave functions listed in Table. I. With the derived decay amplitudes, the partial decay width for the emission of a light pseudoscalar meson is calculated by where J iz and J f z represent the third components of the to- tal angular momenta of the initial and final baryons, respectively. δ is a global parameter accounting for the strength of the quark-meson couplings.
In this work, the standard quark model parameters are adopted. Namely, we set m s = 450 MeV for the constituent quark mass. The decay constants for K and η are taken as f K = f η = 160 MeV. The harmonic oscillator parameter α in the wave function Ψ NLL z is adopted as α = 400 MeV. The masses of the final mesons and baryons are taken from the PDG [1]. For the global parameter δ, we fix its value the same as our previous study of the strong decays of Ξ baryons [30], i.e., δ = 0.576.

III. RESULTS AND ANALYSIS
Inspired by the newly observed Ω * − candidate by Belle II Collaboration [22], we carry out a systematic study of the strong decays of Ω baryons up to N = 2 shell with a chiral quark model. Since the predicted mass of the Ω − in the relativistic quark model [7] well agrees with the experimental measurement in PDG [1], we adopt the predicted masses of the Ω resonances from Ref. [7] (see Table I) in our calculation.
A. 1P wave states in the N = 1 shell There are two 1P wave states |70, 2 10, 1, 1, 1/2 − and |70, 2 10, 1, 1, 3/2 − according to the quark model classification (see Table I). Their spin-parity quantum numbers are J P = 1/2 − and J P = 3/2 − , respectively. It is seen that the predicted masses in various quark models for these two 1P wave states are about 2000 MeV, which are close to the mass of the newly observed Ω * − candidate [22]. As the possible assignments of the newly observed Ω(2012) state, it is crucial to study the decay properties of the two states.
While the dominant decay modes of |56, 4 From the point of view of the mass and decay width, we can't excluded the first radially excited Ω state |56, 4 10, 2, 0, 3/2 + as a assignment of the newly observed Ω(2012) state.
In addition, we also plot the decay widths of |70, 2 10, 2, 0, 1/2 + and |56, 4  There are six 1D wave states according to the quark model classification (see Table I). Their masses are estimated to be in the range of 2.2 − 2.3 GeV in various quark models. With the predicted masses from Ref. [7], we further analyze the decay properties of the 1D wave states in the N = 2 shell, and collect their strong decay widths in Table III. The predicted masses of the 1D wave states certainly have a large uncertainty, which may bring uncertainties to our theoretical predictions. To investigate this effect, we plot the total and partial decay widths of these states as functions of the masses in the range of M = (2100 − 2400) MeV in Fig. 2 as well.
It should be mentioned that the Ω(2250) resonance with a width of Γ = 55 ± 18 MeV listed in PDG [1] may be a good candidate of |56, 4 10, 2, 2, 5/2 + . The Ω(2250) was seen in the Ξ(1530)K and Ξ − π + K − channels. The measured mass of Ω(2250) is consistent with the quark model predictions [3,7]. Assigning it as Ω(2250), the total width is predicted to be Its strong decays are dominated by the Ξ(1530)K mode, while the decay rate into the ΞK is sizeable. The partial width ratio between Ξ(1530)K and ΞK is predicted to be Both the decay width and decay mode are consistent with the observations. As a whole, the 1D wave states are relatively narrow states with a typical width of 10s MeV. They mainly decay into ΞK and/or Ξ(1530)K final states. To establish these missing 1D wave states, observations in the both ΞK and Ξ(1530)K channels are expected to be carried out in future experiments.

IV. SUMMARY
In the present work, we carry out a systematic study of the OZI allowed two-body strong decays of Ω resonances up to the N = 2 shell within the chiral quark model. For the newly observed Ω(2012) state, we give a possible interpretation in theory. Meanwhile, we give the predictions for the decay properties of the 1D wave states, and hope to provide helpful information for searching these missing Ω states in the future.
The Ω(2250) resonance listed in PDG may be a good candidate of the J P = 5/2 + 1D wave state |56, 4 10, 2, 2, 5/2 + . Generally, the 1D wave states are relatively narrow states with a typical width of 10s MeV. They mainly decay into ΞK and/or Ξ(1530)K final states. To establish these missing 1D wave states, observations in the both ΞK and Ξ(1530)K channels are expected to be carried out in future experiments.