Newly observed $\Omega(2012)$ state and strong decays of the low-lying $\Omega$ excitations

Stimulated by the newly discovered $\Omega(2012)$ resonance at Belle II, in this work we have studied the OZI allowed strong decays of the low-lying $1P$- and $1D$-wave $\Omega$ baryons within the $^3P_0$ model. It is found that $\Omega(2012)$ is most likely to be a $1P$-wave $\Omega$ state with $J^P=3/2^-$. We also find that the $\Omega(2250)$ state could be assigned as a $1D$-wave state with $J^P=5/2^+$. The other missing $1P$- and $1D$-wave $\Omega$ baryons may have large potentials to be observed in their main decay channels.


I. INTRODUCTION
The study of hadron spectrum is an important way for us to understand strong interactions. For the baryon spectra, the classification based on S U(3) f flavor symmetry has been achieved a great success. The Ω hyperon as a member of baryon decuplet in the quark model was unambiguously discovered in both production and decay at BNL about one half century ago [1]. More excited Ω baryons should exist as well according to S U(6) × O(3) symmetry. In theory, the mass spectrum of Ω hyperon has been predicted within many models, such as the Skyrme model [2], various constituent quark models [3][4][5][6][7][8][9][10][11][12], the lattice gauge theory [13,14], and so on. However, in experiments there are only a few information of the excited Ω baryons. In the review of particle physics from the Particle Data Group (PDG), except for the ground state Ω(1672), only three possible excited Ω baryons are listed: Ω(2250), Ω(2380), and Ω(2470) [15]. Their nature is 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.
Very recently, a candidate of excited Ω baryon, Ω(2012), was observed by the Belle II collaboration [16]. The measured mass and width are M = 2012.4 ± 0.7(stat) ± 0.6(syst) MeV, Γ = 6.4 +2. 5 −2.0 (stat) ± 1.6(syst) MeV, respectively. In various quark models [3][4][5][6][7][8][9][10][11][12], the masses of the first orbital (1P) excitations of Ω states are predicted to be ∼ 2.0 GeV. The newly observed state Ω(2012) may be a good candidate of the 1P-wave Ω state. Recently, to study the possible interpretation of Ω(2012), its strong decays were calculated with the chiral quark model [17], where it was shown that Ω(2012) could be assigned to the spin-parity J P = 3/2 − 1P-wave state. However, the spin-parity J P = 1/2 − state can't * Electronic address: guilongcheng@hunnu.edu.cn † Electronic address: lvqifang@hunnu.edu.cn ‡ Electronic address: zhongxh@hunnu.edu.cn be completely ruled out. Furthermore, the mass and strong decay patten of Ω(2012) also were studied by QCD sum rule [18,19]. As its mass is very close to Ξ(1530)K threshold, there is also some work considered it as a J P = 3/2 − KΞ * molecular state [20][21][22], or a dynamically generated state [23]. In Ref. [24], by a flavor S U(3) analysis the authors suggested Ω(2012) to be a number of 3/2 − decuplet baryon if the sum of branching ratios for the decay Ω(2012) → ΞKπ, Ω − ππ is not too large ( 70%). In present work, to further reveal the nature of Ω(2012) and better understand the properties of the excited Ω states, we study the Okubo-Zweig-Iizuka(OZI) allowed two-body strong decays of the 1P-and 1D-wave baryons with the widely used 3 P 0 model. The quark model classification for the 1P-and 1D-wave Ω baryons and their theoretical masses are listed in Table I. The spatial wave functions for the Ω baryons are described by harmonic oscillators. According to our calculations, we find that (i) the newly observed Ω(2012) resonance is most likely to be the 1P-wave Ω state with spinparity J P = 3/2 − and the experimental data can be reasonably described. The other 1P-wave state with J P = 1/2 − might be broader state with a width of dozens of MeV. The 3 P 0 results are consistent with the recent predictions of the chiral quark model [17]. (ii) 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 + or |70, 2 10, 2, 2, 5/2 + . Although the widths of the D-wave states predicted within the 3 P 0 model are systematically larger that those predicted with the chiral quark model [17], these states may be observed in their main decay channels in future experiments for their relatively narrow width.
This paper is organized as follows. Firstly, We give a brief review of the 3 P 0 model in Sec. II. Secondly, we present the numerical results of strong decay of 1P-and 1D-wave Ω baryon in Sec. III. Finally, a summary of our results is given in Sec. IV.

II. THE 3 P 0 MODEL
The 3 P 0 model, is also called the quark pair creation (QPC) model. It was first proposed by Micu [25], Car- I: The theory predicted masses (MeV) and spin-flavor-space wave-functions of Ω baryons under S U(6) quark model classification are listed below. 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 S U(6) group, N 3 stands for the irreducible representation of flavor S U(3) group and N, L, J P as principal quantum number, total orbital angular momentum and spin-parity, respectively [17]. The φ,χ,ψ denote flavor, spin and spatial wave function, respectively. The Clebsch-Gorden coefficients of spin-orbital coupling have been omitted. litz and Kislinger [26], and developed by the Orsay group and Yaouanc et al [27][28][29][30][31][32]. In the model, it assumes that a pair of quarks qq is created from the vacuum with J PC = 0 ++ ( 2S +1 L J = 3 P 0 ) when the initial hadron A decays, and the quarks from the hadron A regroups with the created quarks form two daughter hadrons B and C. For baryon decays, two quarks of the initial baryon regroups with the created quark to form a daughter baryon, and the remaining one quark regroup with the created antiquark to form a meson. The process of baryon decays is shown in Fig. 1. The transition operator under the 3 P 0 model is written as where the pair-strength γ is a dimensionless parameter, i and j are the color indices of the created quark q 4 and antiquarkq 5 . ϕ 45 0 = (uū + dd + ss)/ √ 3 and ω 45 0 = δ i j stand for the flavor and color singlet, respectively. χ 45 1,−m is the spin triplet state and According to the definition of the mock state [11], the baryons A and B, and meson C are defined as follows : where the subscripts 1,2,3 denote the quarks of the initial baryon A. The decay amplitude of the Ω baryon in 3 P 0 is written as In the equation Eq. The overlap integral in the momentum space is written as where ) stand for the momentum corresponding to ρ and λ Jacobi coordinates in the center mass frame of baryon A, and |J| stand for the Jacobi determinant which determined by the definition of p ρ , p λ . m i denote the mass of ith quark and m q denote the mass of the created quark pairs. the n ρ and L ρ denote the nodal and orbital angular momentum between the 1,2 quarks (see Fig. 1), while the n λ and L λ stand for the nodal and orbital angular momentum between the 1,2 quarks system and the 3 quark (see Fig. 1).
In the calculations, simple harmonic oscillator (SHO) wave functions is employed as the hadron wave function. The momentum space wave function of baryon is where N stand for a normalization coefficient of the wave function and L l+1/2 n ( p 2 β 2 ) is the Laguerre polynomial function. The Clebsch-Gorden coefficients of l ρ , l λ coupling are equal to 1 in our case.
The ground state wave function of a meson in the momentum space is where the p ab stands for the relative momentum between the quark and antiquark in the meson. As all hadrons in the final states are S -wave in this work, Eq. (7) can be further expressed as follows the expressions of Π(L ρ A , M L A ρ , L λ A , M L A λ , m) and harmonic oscillator wave function for the S -wave, P-wave, D-wave Ω baryons are collected in the appendix.
The decay width Γ of the process A → B + C is where J A are the total angular momentum of the initial baryon A. p is the momentum of the final baryon in the center of mass frame of the initial baryon A p = where m A , m B and m C are the mass of the initial and final hadrons.
In order to partly remedy the inadequacy of the nonrelativistic wave function as the relative momentum p increases [33][34][35][36][37], the decay amplitude is written as where γ f denotes a commonly Lorentz boost factor, γ f = m B /E B . In most decays, the three momenta carried by the final state baryons are relatively small, which means the nonrelativistic prescription is reasonable and the corrections from the Lorentz boost is not drastic. States J n ρ l ρ n λ l λ L S ρ S 70, 2 Table I, the masses of K mesons and Ξ baryons are taken from the PDG [15]. The quantum numbers involved in the calculations are listed in Tab II. Due to the orthogonal relationship of the wave functions, only the λ excited mode contributes. There are three harmonic oscillator parameters, the β ρ and β λ and R in baryon and meson wave functions, respectively. We adopt R = 2.5 GeV −1 for K mesons [38]. The parameter β ρ of the ρ-mode excitation between the 1,2 quarks (see Fig. 1) is taken as β ρ = 0.4 GeV [39]. The β λ is obtained with the relation [40]: For the quark pair creation strength from the vacuum, we take as those in Ref. [38], γ = 6.95.
A. The 1P-wave states According to the S U(6) supermultiplet classification (see table I Assuming Ω(2012) as a candidate of the 1P-wave Ω baryons, we calculate the OZI-allowed two body strong decays in the 3 P 0 model, and list our results in table III.
It is found that if one assigns Ω(2012) as the J P = 1 2 − state 70, 2 10, 1, 1, 1/2 − , the width is predicted to be which is too large to be comparable with the width of Ω(2012). This width predicted in the 3 P 0 model is about a factor 3 larger than that predicted within the chiral quark model [17]. On the other hand, assigning Ω(2012) as the J P = 3 2 − state 70, 2 10, 1, 1, 3/2 − , we find that the width and the branching fraction ratio are consistent with the measured width Γ exp = 6.4 +2.5 −2.0 (stat) ± 1.6(syst) and ratio R exp = 1.2 ± 0.3 for Ω(2012). These 3 P 0 model predictions are compatible with those predicted within the chiral quark model [17].
As a whole most of the 1D-wave states has a relatively narrow width, they has potentials to be observed in their dominant decay modes. The Ω(2250) resonance may be assigned to the J P = 5/2 + state |56, 4 10, 2, 2, 5/2 + or |70, 2 10, 2, 2, 5/2 + . Although the decay widths predicted for the D-wave states within the chiral quark model and 3 P 0 model show some differences, the partial width ratios of Γ[ΞK]/Γ[Ξ(1530)K] predicted within these two models are in a reasonable agreement with each other.
In the 1D-wave Ω states, it is found that the Ω(2250) state may favor the J P = 5/2 + state |56, 4 10, 2, 2, 5/2 + or |70, 2 10, 2, 2, 5/2 + . These two J P = 5/2 + states dominantly decay into the Ξ(1530)K channel, and have a similar decay width to that of Ω(2250). Due to a large uncertainty of the width of Ω(2250), we can't distinguish it belongs to the 70 multiplet or 56 multiplet. Future experiment information will help to clarify this issue. For the other 1D-wave Ω baryons, we recommend looking for the J P = 1/2 + and J P = 7/2 + states in the ΞK decay channel and looking for the J P = 3/2 + states in both ΞK and Ξ(1530)K decay channels in future experiments.