$\Omega$ baryon spectrum and their decays in a constituent quark model

Combining the recent developments of the observations of $\Omega$ sates we calculate the $\Omega$ spectrum up to the $N=2$ shell within a nonrelativistic constituent quark potential model. Furthermore, the strong and radiative decay properties for the $\Omega$ resonances within the $N=2$ shell are evaluated by using the masses and wave functions obtained from the potential model. It is found that the newly observed $\Omega(2012)$ resonance is most likely to be the spin-parity $J^P=3/2^-$ $1P$-wave state $\Omega(1^{2}P_{3/2^{-}})$, it also has a large potential to be observed in the $\Omega(1672)\gamma$ channel. Our calculation shows that the 1$P$-, 1$D$-, and 2$S$-wave $\Omega$ baryons have a relatively narrow decay width of less than 50 MeV. Based on the obtained decay properties and mass spectrum, we further suggest optimum channels and mass regions to find the missing $\Omega$ resonances via the strong and/or radiative decay processes.


I. INTRODUCTION
Searching for the missing baryon resonances and understanding the baryon spectrum are important topics in hadron physics. Our knowledge about the Ω hyperon spectrum is very poor compared with the other light baryon spectra. Only a few data on the Ω resonances have been reported in experiments since the discovery of the ground state Ω(1672) at BNL in 1964 [1]. Before 2018, except for the ground state Ω(1672) only three Ω resonances Ω(2250), Ω(2380) and Ω(2470) were listed in the Review of Particle Physics (RPP) [2]. Due to the slow development in experiments, most of the theoretical studies are limited in the calculations of the mass spectrum of the Ω baryon with various methods, such as the Skyrme model [3], the relativistic quark models [4][5][6], the nonrelativistic quark model [7][8][9][10][11][12], the lattice gauge theory [13,14], and so on.
Fortunately, the Belle II experiments can offer a great opportunity for our study of the Ω spectrum. In 2018, the Belle Collaboration reported a new resonance denoted by Ω(2012) [15], a candidate of excited Ω state decaying into According to the calculations of the Ω mass spectrum in the various models [4][5][6][7][8][9][10][11][12], the Ω(2012) resonance may be a good candidate for the first orbital excitations of Ω baryon. Stimulated by the newly observed Ω(2012), by using a simple harmonic oscillator (SHO) wave functions, the strong decay properties of the Ω spectrum up to the N = 2 shell were studied within the chiral quark model [16] and quark pair creation model [17], respectively. The results show that Ω(2012) could be assigned to the spin-parity J P = 3/2 − 1P-wave Ω state, which is supported by the QCD Sum rule analysis in Refs. [18,19], and the flavor SU(3) analysis in Ref. [20]. There also exist other interpretations, such as hadronic molecule state, of the newly observed Ω(2012) state in the literature. Considering the mass of Ω(2012) is very close to Ξ(1530)K threshold, the authors in Refs. [21][22][23] interpreted Ω(2012) as the S -wave Ξ(1530)K hadronic molecule state with quantum number J P = 3/2 − . In the Ref. [24], Ω(2012) is assumed to be a dynamically generated state with spin parity J P = 3/2 − from the coupled channel S -wave interactions ofKΞ(1530) and ηΩ. Very recently, the Belle Collaboration searched for the three-body decay of the Ω(2012) baryon to KπΞ [25]. No significant Ω(2012) signals have been observed in the studied channels. The experimental result strongly disfavors the molecular interpretation [25].
In this work, we further study the Ω spectrum. First, combining the recent developments of the observations of Ω sates in experiments at Belle we calculate the mass spectrum up to the N = 2 shell within a nonrelativistic constituent quark potential model. Then, by using the masses and wave functions calculated from the potential model, we give our predictions of the strong and radiative decay properties for the Ω resonances. The strong decay properties for the P-and D-wave states predicted with the realistic wave functions of the potential model are compatible with the results obtained with the SHO wave functions in Ref. [16]. The strong decays of the 2S -wave states show some sensitivities to the details of the wave functions, the strong decay properties of these 2S -wave states predicted in present work have some differences from those calculated with the SHO wave functions. The Ω(2012) resonance is most likely to be the spin-parity J P = 3/2 − 1P-wave state Ω(1 2 P 3/2 − ). Both the mass and decay properties predicted in theory are consistent with the observations. The Ω(2012) state may be observed in the radiative decay channel Ω(1672)γ as well. Furthermore, based on the obtained decay properties and mass spectrum, we suggest optimum channels and mass regions to find the missing 1P-, 1D-, and 2S -wave Ω resonances in the strong and/or radiative decay processes. This paper is organized as follows. In Sec. II, we study the mass spectrum of the Ω baryon in the nonrelativistic constituent quark potential model. Then, in Sec. III, we give a review of the decay models, and calculate the strong and radiative decays of the excited Ω states by using the masses and wave functions obtained from the potential model. In Sec. IV, we give our discussions based on the obtained decay properties and masses of the Ω resonances. Finally, a summary is given in Sec. V.

A. Hamiltonian
To calculate the spectrum of the Ω baryon, we adopt the following nonrelativistic Hamiltonian where m i and T i stand for the constituent quark mass and kinetic energy of the i-th quark, respectively; T G stands for the center-of-mass (c.m.) kinetic energy of the baryon system; r i j ≡ |r i − r j | is the distance between the i-th quark and j-th quark; zero point energy C 0 is a constant, and V i j (r i j ) stands for the effective potential between the i-th and j-th quark. In this work, we adopt a widely used potential form for V i j (r i j ) [26][27][28][29][30][31][32][33][34][35], i.e.
where V con f i j stands for the potential for confinement, and is adopted the standard Coulomb+linear scalar form: while V sd i j (r i j ) stands for the spin-dependent interaction, which is the sum of the spin-spin contact hyperfine potential V S S i j , the tensor term V T i j , and the spin-orbit term V LS i j The spin-spin potential V S S i j and the tensor term V T i j are adopted the often used forms: In this work, a simplified phenomenological spin-orbit potential is adopted the same form as that suggested in the literature [12,36], i.e., In the above equations, the parameter b denotes the strength of the confinement potential. The S i , S and L are the spin operator of the i-th quark, the total spin of the baryon and the total orbital angular momentum of the baryon, respectively.
B. Wave functions in the SU(6)×O(3) symmetry limit The total wave function of a baryon system should include four parts: a color wave function ζ, a flavor wave function φ, a spin wave function χ, and a spatial wave function ψ. The color wave function ζ should be a color singlet under SU(3) symmetry, one can explicitly express it as For a light baryon system, the flavor wave function φ and the spin wave function χ together form an SU(6) symmetry. The SU(6) spin-flavor wave functions can be found in the literature [37]. Without a spin-dependent interaction in the Hamiltonian for three-body systems, the total orbital angular momentum L and the total spin S are conserved. The total angular momentum of the baryon is J = L + S, and thus the spatial wave functions possess O(3) symmetry under a rotation transformation. Meanwhile, the Hamiltonian for a threequark system can be invariant under the permutation group S 3 . One thus can express the spatial wave functions as representations of the S 3 group. To consider the S 3 symmetry, one can express the three coordinates r 1 , r 2 , and r 3 with the Jacobi coordinates R, ρ and λ by the following transformation, λ ≡ 2 3 The symmetric coordinate R describes the usual center of mass motion, and two mixed coordinates ρ and λ describe the internal motions which are antisymmetric and symmetric under the exchange of quark 1 and 2. The corresponding spatial wave function ψ may be generally written as where ψ σ NLM L (ρ, λ) is the spatial wave function, which can be determined by solving the Schrödinger equation; σ = s, ρ, λ, a denotes the representation of the S 3 group.  [2,15] and the theory predictions [3, 5-8, 12, 13] are also listed. Recently, the quantum number of the resonances Ω(2012) [15] and Ω(2250) [2] are not determined. According the Ref. [16,17], we think the resonances Ω(2012) and Ω(2250) as state Ω(1 2 P 3/2 − ) and state Ω(1 4 D 5/2 + ), respectively. n 2S +1 L J P |N 6 , 2S +1 N 3 , N, L, J P Ours Exp. Ref. [3] Ref. [5] Ref. [6] Ref. [7] Ref. [8] Ref. [12] Ref. [ The states in the SU(6)×O(3) representation up to N = 2 shell are given in Table I. We denote the baryon states as |N 6 , 2S +1 N 3 , N, L, J P , where N 6 stands for the irreducible representation of spin-flavor SU(6) group, N 3 stands for the irreducible representation of flavor SU(3) group, and N, S , L, and J P stand for the principal, spin, total orbital angular momentum, and spin-parity quantum numbers, respectively. More details about the SU(6)×O(3) wave functions can found in Ref. [37].

Trial spatial wave functions
To obtain spatial wave functions and the masses for every Ω states in the SU(6)×O(3) representation, one need solve the Schrödinger equation. The spatial wave function ψ σ NLM L (ρ, λ) may be expressed as the linear combination of ψ n ρ l ρ m ρ (ρ)ψ n λ l λ m λ (λ): The ρand λ-mode spatial wave functions ψ n ρ l ρ m ρ (ρ) and ψ n λ l λ m λ (λ) can be written with a unified form: where the Y l ξ m ξ (ξ) is the spherical harmonic function. In the above equations, l ρ and l λ are the quantum numbers of the relative orbital angular momenta l ρ and l λ of the ρand λmode oscillators, respectively, while L is the quantum number of the total momentum L = l λ + l ρ for the system. The n ρ and n λ are the principal quantum numbers of the ρand λmode oscillators, respectively, and N = 2n ρ + 2n λ + l ρ + l λ .
The coefficients C n ρ l ρ m ρ n λ l λ m λ and explicit forms of the spatial wave function ψ σ NLM L (ρ, λ) up to the N = 2 shell have been given in Table II.
The radial wave function R n ξ l ξ (ξ) is adopted a trial form by expanding with a series of harmonic oscillator functions: where The F −n ξ , l ξ + 3 2 , ξ d ξℓ 2 is the confluent hypergeometric function. The parameter d ξℓ can be related to the harmonic oscillator frequency ω ξℓ with 1/d 2 . On the other hand, the harmonic oscillator frequency ω ξℓ can be related to the harmonic oscillator stiffness factor K ℓ with ω ξℓ = 3K ℓ /M ξ [37]. For a sss system, one has d ρℓ = d λℓ = d ℓ = (3m s K ℓ ) −1/4 , where m s stands for the constituent mass of the strange quark. With this relation, the spatial wave function ψ σ NLM L (ρ, λ) can be simply expanded as where ψ σ NLM L (d ℓ , ρ, λ) stands for the trial harmonic oscillator functions,

Matrix elements
The problem of solving the Schrödinger equation is now reduced to one of calculating the matrix elements H αα = α|H|α , where |α stands for the total wave function |N 6 , 2S +1 N 3 , N, L, J P for the Ω baryons. Omitting the color and flavor wave functions, the total wave function |α in the L − S coupling scheme can be expressed as With the Jacobi coordinates ρ and λ, the matrix elements H αα can be expressed as where (20) can be easily worked out in the coordinate space.
Then some calculations of the matrix elements of the potentials between quarks become the main task of present work. Considering the permutation symmetry of the total wave function of the Ω baryons, we can obtain Finally, we can therefore specialize our discussion to techniques for calculating potential matrix elements of the V 12 (r 12 ) terms.
The matrix elements of confining potential V con f 12 , and spinorbit potential V LS 12 can be directly worked out with the total wave function |α in the L − S coupling scheme. The calculations of the matrix elements of tensor potential V T 12 , and spinspin potential V S S 12 , are relatively complicated. We transform |α into the |β = |(s 12 s 3 )S , (l ρ l λ )L, (n ρ n λ )N, JM representation with the following relation: The s 12 is the quantum number of the spin angular momentum S 1 + S 2 . The coefficients c i and explicit quantum numbers of the Ω states up to the N = 2 shell have been given in Table III. Then with the Wigner-Eckart theorem, the matrix elements of tensor potential V T 12 can worked out with the following for- and the matrix elements of spin-spin potential V S S 12 can worked out with the following formula

results
In this work, we adopt the variation principle to solve the Schrödinger equation. Following the method used in Refs. [38,39], the oscillator length d ℓ are set to be where n is the number of Gaussian functions, and a is the ratio coefficient. There are three parameters {d 1 , d n , n} to be determined through variation method. It is found that when we take d 1 = 0.085 fm, d n = 3.399 fm, n = 15, we will obtain stable solutions for the Ω baryons.
When all the matrix elements have been worked out, we can solve the generalized matrix eigenvalue problem [39], where The physical state corresponds to the solution with a minimum energy E m . By solving this generalized matrix eigenvalue problem, the masses of the Ω baryons and its spatial wave functions can be determined. It should be emphasized that there are six parameters m s , α s , σ ss , b, α S O and C 0 in the quark potential model. They are determined by fitting the masses of four Ω resonances: (i) The ground state Ω(1672) [2], which is well established in experiments. (ii) The newly observed Ω(2012) resonance at Belle [15], which is interpreted as the first orbital excited state Ω(1 2 P 3/2 − ) [16,17]. (iii) The other first orbital excited state Ω(1 2 P 1/2 − ) whose mass is predicted to be ∼ 1950 MeV within the Lattice QCD [13] and the relativized quark models [5,6]. The measured ΞK invariant mass distributions from Belle show that there is a weak enhancement around 1950 MeV [15], which may be a weak hint of J P = 1/2 − state Ω(1 2 P 1/2 − ). (iv) The Ω(2250) resonance listed in RPP [2] which is assigned as the Ω(1 4 D 5/2 + ) state according to our previous studies [16]. The determined parameter set is listed in Table IV.
The predicted masses of the Ω baryons up to N = 2 shell have been given in Table I and also shown in Fig 1. For a comparison, some predictions from the other models are listed in Table I as well. It is found that the masses of the first radially excited states Ω(2 2 S 1/2 + ) and Ω(2 4 S 3/2 + ) obtained in present work are compatible with the predictions in Refs. [5,12], however, the mass splitting between them ∆m ≃ 70 MeV predicted in this work are obviously larger than the other model predictions. The mass of the J P = 1/2 + D-wave state Ω(1 4 D 1/2 + ), 2141 MeV, predicted in this work is close to the predictions in Ref. [3,12], however, our prediction is about 60-160 MeV lower than the those predicted in Refs. [5][6][7][8]. The masses of the J P = 3/2 + D-wave states Ω(1 4 D 3/2 + ) and Ω(1 2 D 3/2 + ) and their mass splitting ∆m ≃ 60 MeV predicted in this work are close to those predicted in Refs. [7,8]. The masses of the J P = 5/2 + D-wave states Ω(1 4 D 5/2 + ) and Ω(1 2 D 5/2 + ) and their mass splitting ∆m ≃ 50 MeV predicted in this work are close to those predicted in Refs. [5,7,8]. The mass of the J P = 7/2 + D-wave state Ω(1 4 D 7/2 + ) is close to those predictions in Refs. [5,6], however, about 100 MeV higher than the predictions in Refs. [7,8,12]. Finally, it should be mentioned that if we consider a fairly large mass splitting ∆ ≃ 50 MeV between the two 1P-wave states Ω(1 2 P 3/2 − ) and Ω(1 2 P 1/2 − ) due to the spin-orbital interactions, the mass splitting between two adjacent D-wave spin-quartet states Ω(1 4 D J ) and Ω(1 4 D J+1 ) might reach up to ∼ 50 − 70 MeV which is larger than the value ∼ 0 − 20 MeV predicted in the literature [5-8, 12, 13].

III. STRONG AND RADIATIVE DECAYS
A. Framework

strong decay
In the chiral quark model, the effective low energy quarkpseudoscalar-meson coupling in the SU(3) flavor basis at tree level is given by [40] where f m stands for the pseudoscalar meson decay constant. ψ j corresponds to the jth quark field in a baryon and φ m de-notes the pseudoscalar meson octet To match the nonrelativistic baryon wave functions in the calculations, we adopt a nonrelativistic form of Eq.( 28) for a baryon decay process [41][42][43], i.e., 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; while M i and M f stand for the mass of the initial baryon and final baryon. We select the initialbaryon-rest system in the calculations. Then, P i = 0, E i = M i and P f = −q. In the Eq. (30) σ j is the Pauli spin vector on the jth quark, and µ q is a reduced mass expressed as 1/µ q = 1/m j + 1/m ′ j . P ′ j = P j − (m j /M)P c.m. is the internal momentum of the jth quark in the baryon rest frame. The isospin operator I j associated with the pseudoscalar meson is given by where 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, while θ is the mixing angle of the η meson in the flavor basis. In this work we adopt θ = 41.2 • as that used in Ref. [44].
The decay amplitudes for a strong decay process B → B ′ M can be calculated by where |B ′ and |B stand for the wave functions of the final and initial baryon, respectively. With the derived decay amplitudes, the partial decay width for the B → B ′ M process is calculated by where J iz and J f z represent the third components of the total angular momenta of the initial and final baryons, respectively. δ is a global parameter accounting for the strength of the quark-meson couplings. The relativistic effects become significant when momentum q of final baryon increases. As done in the literature [45][46][47][48], a commonly used Lorentz boost factor γ f ≡ M f /E f is introduced into the decay amplitudes to partly remedy the inadequacy of the nonrelativistic wave function as the momentum q increases. In most decays, the sum of the masses of the final hadron states is not far away from the mass of the initial state, the three momenta q carried by the final states are relatively small, which means the nonrelativistic prescription is reasonable and corrections from the Lorentz boost are not drastic.

radiative decay
The quark-photon EM coupling at the tree level is adopted as The photon field A µ has three momentum k, and the constituent quark ψ j carries a charge e j . While r j stands for the coordinate of the jth quark.
In order to match the nonrelativistic wave functions of the baryons, we should adopt the nonrelativistic form of Eq. (35) in the calculations. Including the effects of the binding potential between quarks [49], for emitting a photon the nonrelativistic expansion of H e may be written as [42,43,50] where m j and σ j stand for the constituent mass and Pauli spin vector for the jth quark. The vector ǫ is the polarization vector of the photon. This nonrelativistic EM transition operator has between widely applied to meson photoproduction reactions [41][42][43][44][45][51][52][53][54][55][56][57]. Then, the standard helicity transition amplitude A λ between the initial baryon state |B and the final baryon state |B can be calculated by where ω γ is the photon energy. Finally, one can calculate the EM decay width by where J i is the total angular momentum of an initial meson, J f z and J iz are the components of the total angular momenta along the z axis of initial and final mesons,respectively. In the calculation, the constituent quark masses for the u, d, and s quarks are taken with m u = m d = 350 MeV and m s = 600 MeV. The masses of the Ω baryon states are adopted the determinations by solving the Schrödinger equation in Sec.II. It should be mentioned that, we do not directly adopt the numerical wave functions of Ω baryons calculated by solving the Schrödinger equation. For simplicity, we first fit them with a single Gaussian (SG) form by reproducing the root-mean-square radius of the ρ-mode excitations. The determined harmonic oscillator strength parameters, α, for corresponding Ω baryons are listed in Table VI. It is found determined parameters α are very close to the value ∼ 400 MeV often adopted for the SHO wave functions in the literature.
Furthermore, in the calculations of the strong decays, for simplicity, the wave functions of Ξ and Ξ(1530) baryons appearing in the final states are adopted the SHO form as adopted in Ref. [37]. The harmonic oscillator strength parameter α ρ for the ρ-oscillator in the spatial wave function is taken as α ρ = 400 MeV, while the parameter α λ for the λoscillator is related to α ρ with α λ = 4 √ 3m u /(2m s + m u )α ρ [37]. The masses of the mesons and baryons in the final states are taken from the RPP [2] and have been collected in Table V. The decay constants for K and η mesons are taken as f K = f η = 160 MeV. For the global parameter δ, we fix its value the same as the previous study of the strong decays of Ξ and Ω baryons [16,37], i.e., δ = 0.576. The strong and radiative decay widths of Ω baryons up to N = 2 shell are listed in Table VI and Table VII, respectively.
Combining this partial width with the measured total width of Ω(2012), we estimate the branching fraction for this radiative decay process: The radiative process Ω(2012) → Ω(1672)γ may be observed in forthcoming experiments at Belle II. It should be mentioned that the Γ[Ω(2012) → Ω(1672)γ] = 9.52 keV predicted in this work is about a factor 2 smaller than the early prediction within a nonrelativistic potential model in Ref. [58].
The J P = 1/2 − state Ω(1 2 P 1/2 − ) (|70, 2 10, 1, 1, 1/2 − ) might have mass of ∼ 1950 MeV, which is about 50 MeV lower than that of the J P = 3/2 − state Ω(1 2 P 3/2 − ) according to the predictions within the Lattice QCD [13] and the relativized quark models [5,6]. The measured ΞK invariant mass distributions from Belle show that there is a weak enhancement around 1950 MeV [15], which may be a weak hint of J P = 1/2 − state Ω(1 2 P 1/2 − ). Thus, in this work, we adjust the potential parameters to determine the mass of Ω(1 2 P 1/2 − ) with ∼ 1957 MeV. By using this mass and the wave function calculated from the potential model, we predict the total width of Ω(1 2 P 1/2 − ) to be which is compatible with the previous result with the SHO wave function in the chiral quark model [16], while about factor of 4 narrower than that of the 3 P 0 model [17]. The total width of Ω(1 2 P 1/2 − ) should be saturated by the Ξ 0 K − and Ξ −K0 channels. The branching fraction ratio between Ξ 0 K − and Ξ −K0 is predicted to be The narrow width and the only dominant ΞK decay mode of the J P = 1/2 − state Ω(1 2 P 1/2 − ) indicate that it has a large potential to be established as future experimental statistics increases. We further study the radiative decays of the J P = 1/2 − state Ω(1 2 P 1/2 − ). The partial decay width of the Ω(1672)γ channel is predicted to be which is about a factor 3 smaller than the early prediction within a nonrelativistic potential model in Ref. [58]. Combining our predictions of the radiative decay and total decay width of Ω(1 2 P 1/2 − ), we obtain a branching fraction which is about an order smaller than Br[Ω(2012) → Ω(1672)γ]. Thus, the radiative decay of the J P = 1/2 − Ω state into Ω(1672)γ may be more difficultly observed than Ω(2012).
The J P = 5/2 + state Ω(1 4 D 5/2 + ) (|56, 4 10, 2, 2, 5/2 + ) has a mass of ∼ 2252 MeV. The previous study [16] suggested that the Ω(1 4 D 5/2 + ) state might be a good candidate of Ω(2250) listed in RPP [2]. Assigning Ω(2250) as Ω(1 4 D 5/2 + ), with the wave function calculated from the potential model, its total width is predicted to be which is close to the lower limit of the measured width Γ = 55 ± 18 MeV. The strong decays of Ω(2250) 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 The decay mode is consistent with the observations that the Ω(2250) was seen in the Ξ(1530)K and Ξ − π + K − channels. Thus, the Ω(2250) favors the assignment of Ω(1 4 D 5/2 + ). This conclusion is in agreement with that obtained with a SHO wave function in Ref. [16]. It should be mentioned that the J P = 5/2 + state Ω(1 4 D 5/2 + ) may highly overlap with the J P = 3/2 + state Ω(1 2 D 3/2 + ), and the mass splitting between them is only several MeV in present calculations. Thus, it may bring some difficulties to distinguish them in experiments. 5. Ω(1 2 D 5/2 + ) The J P = 5/2 + state Ω(1 2 D 5/2 + ) (|70, 2 10, 2, 2, 5/2 + ) has a mass of ∼ 2.3 GeV according to our predictions, which is close to the predictions in Refs. [5,7,8]. Its total width is predicted to be The Ω(1 2 D 5/2 + ) state dominantly decays into Ξ(1530)K channel, the decay rate into the ΞK channel is relatively small. The partial width ratio between these two channels is predicted to be The strong decay properties predicted in this work is close to the previous results obtained with a simple harmonic oscillator wave function in Ref. [16]. Furthermore, it is interesting to find that the radiative decay rate of Ω(1 2 D 5/2 + ) into Ω(2012)γ is large. Its radiative partial width is predicted to be Combining it with the predicted total width of Ω(1 2 D 5/2 + ), we find that the branching fraction can reach up to The radiative decay process Ω(1 2 D 5/2 + ) → Ω(2012)γ might be useful for searching for the missing Ω(1 2 D 5/2 + ) state. 6. Ω(1 4 D 7/2 + ) In this work, the J P = 7/2 + state Ω(1 4 D 7/2 + ) (|56, 4 10, 2, 2, 7/2 + ) is predicted to be the highest 1D-wave state. Its mass is estimated to be ∼ 2321 MeV, which is close to the predictions in Refs. [5,6]. The total width of Ω(1 4 D 7/2 + ) is predicted to be This state dominantly decays into ΞK channel, the decay rate into the Ξ(1530)K channel is relatively small. The partial width ratio between these two channels is predicted to be It should be mentioned that the mass of Ω(1 4 D 7/2 + ) is similar to that of Ω(1 2 D 5/2 + ), the mass splitting between these two states is only ∼ 18 MeV according to our predictions. By observing the Ξ(1530)K and ΞK invariant mass distributions, one may find two largely overlapping states around 2.3 GeV.

V. SUMMARY
In this work, combining the recent developments of the observations of Ω sates in experiments we calculate the Ω spectrum up to the N = 2 shell within a nonrelativistic constituent quark potential model. Furthermore, the strong and radiative decay properties for the Ω resonances within the N = 2 shell are estimated by using the predicted masses and wave functions from the potential model.
The Ω(2012) resonance is most likely to be the spin-parity J P = 3/2 − 1P-wave state Ω(1 2 P 3/2 − ). Both the mass and decay properties predicted in theory are consistent with the observations. The Ω(2012) resonance may be observed in the radiative decay channel Ω(1672)γ, the branching fraction is predicted to be O(10 −3 ). The other 1P-wave state with J P = 1/2 − is also a narrow state with a width of ∼ 12 MeV, which is about a factor 2 − 3 broader than that of Ω(2012). If more data were accumulated, the J P = 1/2 − state may be clearly established in the Ξ −K0 and Ξ 0 K − invariant mass distributions around 1.95 GeV.
The Ω(2250) resonance may be a good candidate for the J P = 5/2 + 1D-wave state Ω(1 4 D 5/2 + ), with this assignment both the mass and strong decay properties of Ω(2250) can be reasonably understood in the quark model. It should be mentioned that the J P = 5/2 + 1D-wave state Ω(1 4 D 5/2 + ) may highly overlap with the J P = 3/2 + 1D-wave state Ω(1 2 D 3/2 + ). This state might be a narrow state with a width of several MeV and mainly decays into ΞK and Ξ(1530)K channels. The measurements of the partial width ratio between these two channels might be helpful to distinguish the Ω(1 2 D 3/2 + ) state from the Ω(1 4 D 5/2 + ) state in experiments. Furthermore, the Ω(1 2 D 3/2 + ) state might be established in the radiative decay channel Ω(2012)γ, the predicted branching fraction can reach up to O(10 −3 ).
For the other 1D-wave states, it is found that both Ω(1 4 D 7/2 + ) and Ω(1 4 D 1/2 + ) dominantly decay into the ΞK channel with a width of ∼ 40 MeV, they may be established in the ΞK invariant mass spectrum around 2.3 GeV and 2.1 GeV, respectively. The Ω(1 2 D 5/2 + ) state dominantly decay into the Ξ(1530)K channel with a narrow width of 19 MeV, it is worth to looking for in the Ξ(1530)K invariant mass spectrum around 2.3 GeV. The Ω(1 4 D 3/2 + ) state has a width of ∼ 30 MeV, it mainly decays into both ΞK and Ξ(1530)K channels. To look for this missing state, both ΞK and Ξ(1530)K invariant mass distributions around 2.2 GeV are worth observing in future experiments.
For the 2S -wave states Ω(2 2 S 1/2 + ) and Ω(2 4 S 3/2 + ) might be very narrow state with a width of several MeV. The mass splitting between these two states is about 70 MeV. They may be established in the Ξ(1530)K invariant mass spectrum around 2.2 GeV. It should be mentioned that the strong decays of the 2S -wave states show some sensitivities to the details of the wave functions, the strong decay properties of these 2S -wave states predicted in this work have some differences from those calculated with the SHO wave functions. The Ω(2 2 S 1/2 + ) state might have relatively large radiative decay rates into the 1P-wave Ω states with a branching fraction O(10 −3 ). The Ω(2 2 S 1/2 + ) state might be established with the Ω(2012)γ final state.