State and Strong Decays of the Low-Lying Ω Excitations

PHYSICAL REVIEW D 98, 114023 (2018)

Newly observed Ωð2012Þ state and strong decays of the low-lying Ω excitations

† ‡ Zuo-Yun Wang,1 Long-Cheng Gui,1,2,3,* Qi-Fang Lü,1,2,3, Li-Ye Xiao,4,5 and Xian-Hui Zhong1,2,3, 1Department of Physics, Hunan Normal University, Changsha 410081, China 2Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 3Synergetic Innovation Center for Quantum Effects and Applications(SICQEA), Hunan Normal University, Changsha 410081,China 4Center of High Energy Physics, Peking University, Beijing 100871, China 5School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China

(Received 31 October 2018; published 20 December 2018)

Stimulated by the newly discovered Ωð2012Þ resonance at Belle II, in this work we have studied the OZI 3 allowed strong decays of the low-lying 1P- and 1D-wave Ω baryons within the P0 model. It is found that Ωð2012Þ is most likely to be a 1P-wave Ω state with JP ¼ 3=2−. We also find that the Ωð2250Þ state could be assigned as a 1D-wave state with JP ¼ 5=2þ. The other missing 1P- and 1D-wave Ω baryons may have large potentials to be observed in their main decay channels.

DOI: 10.1103/PhysRevD.98.114023

I. INTRODUCTION Very recently, a candidate of excited Ω baryon, Ωð2012Þ, The study of the hadron spectrum is an important way for was observed by the Belle II collaboration [16]. The us to understand strong interactions. For the baryon spectra, measured mass and width are the classification based on SUð3Þf flavor symmetry has been achieved a great success. The Ω hyperon as a member M ¼ 2012.4 0.7ðstatÞ0.6ðsystÞ MeV; of baryon decuplet in the quark model was unambiguously Γ ¼ 6.4þ2.5ðstatÞ1.6ðsystÞ MeV; discovered in both production and decay at BNL about one −2.0 half century ago [1]. More excited Ω baryons should exist – as well according to SUð6Þ × Oð3Þ symmetry. In theory, respectively. In various quark models [3 12], the masses 1P Ω the mass spectrum of Ω hyperon has been predicted within of the first orbital ( ) excitations of states are predicted ∼2 0 Ωð2012Þ many models, such as the Skyrme model [2], various to be . GeV. The newly observed state may 1P Ω constituent quark models [3–12], the lattice gauge theory be a good candidate of the -wave state. Recently, to Ωð2012Þ [13,14], and so on. However, in experiments there are only study the possible interpretation of , its strong a few information of the excited Ω baryons. In the review of decays were calculated with the chiral quark model [17], where it was shown that Ωð2012Þ could be assigned to the particle physics from the Particle Data Group (PDG), P − Ωð1672Þ spin-parity J ¼ 3=2 1P-wave state. However, the spin- except for the ground state , only three possible P − excited Ω baryons are listed: Ωð2250Þ, Ωð2380Þ, and parity J ¼ 1=2 state can’t be completely ruled out. Ωð2470Þ [15]. Their nature is still rather uncertain with Furthermore, the mass and strong decay patten of three- or two-star ratings. Fortunately, the Belle II experi- Ωð2012Þ also were studied by QCD sum rule [18,19]. ments offer a great opportunity for our study of the Ω AsitsmassisveryclosetoΞð1530ÞK threshold, there is P − spectrum. also some work that considered it as a J ¼ 3=2 KΞ molecular state [20–22], or a dynamically generated state [23].InRef.[24], by a flavor SUð3Þ analysis the authors *[email protected] † Ωð2012Þ 3=2− [email protected] suggested to be a number of decuplet ‡ [email protected] baryon if the sum of branching ratios for the decay Ωð2012Þ → ΞK¯ π, Ω−ππ is not too large (≤70%). Published by the American Physical Society under the terms of In the present work, to further reveal the nature of the Creative Commons Attribution 4.0 International license. Ωð2012Þ Further distribution of this work must maintain attribution to and better understand the properties of the excited the author(s) and the published article’s title, journal citation, Ω states, we study the Okubo-Zweig-Iizuka (OZI) allowed and DOI. Funded by SCOAP3. two-body strong decays of the 1P- and 1D-wave baryons

2470-0010=2018=98(11)=114023(8) 114023-1 Published by the American Physical Society WANG, GUI, LÜ, XIAO, and ZHONG PHYS. REV. D 98, 114023 (2018)

TABLE I. The theory predicted masses (MeV) and spin-flavor-space wave functions of Ω baryons under SUð6Þ quark model 2Sþ1 P classification are listed below. We denote the baryon states as jN6; N3;N;L;J i where N6 stands for the irreducible representation of P spin-flavor SUð6Þ group, N3 stands for the irreducible representation of flavor SUð3Þ group and N, L, J 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.

Theory States Wave function [4] [3] [2] [7] [13] 4 þ s s s 1S j56; 10; 0; 0; 3=2 i ϕ χ ψ 000 1678 1635 1694 1675 1642(17) ρ 1P j70; 210; 1; 1; 1=2−i p1ffiffi ðϕsχρψ þ ϕsχλψ λ Þ 1941 1950 1837 2020 1944(56) 2 11Lz 11Lz ρ j70; 210; 1; 1; 3=2−i p1ffiffi ðϕsχρψ þ ϕsχλψ λ Þ 2038 2000 1978 2020 2049(32) 2 11Lz 11Lz j56; 410; 2; 2; 1=2þi ϕsχsψ s 2350ð63Þ=2481ð51Þ 1D 22Lz 2301 2255 2140 2210 j56; 410; 2; 2; 3=2þi ϕsχsψ s 2173=2304 22Lz 2280 2282 2215 2 þ 1 ρ 2470(49) j70; 10; 2; 2; 3=2 i pffiffi ðϕsχρψ þ ϕsχλψ λ Þ 2173=2304 2345 2282 2265 2 22Lz 22Lz j56; 410; 2; 2; 5=2þi ϕsχsψ s 22Lz 2401 2280 2225 ρ j70; 210; 2; 2; 5=2þi p1ffiffi ðϕsχρψ þ ϕsχλψ λ Þ 2401 2345 2265 2 22Lz 22Lz j56; 410; 2; 2; 7=2þi ϕsχsψ s 22Lz 2332 2295 2210

3 with the widely used P0 model. The quark model quarks form two daughter hadrons B and C. For baryon classification for the 1P- and 1D-wave Ω baryons and decays, two quarks of the initial baryon regroup with the their theoretical masses are listed in Table I. The spatial created quark to form a daughter baryon, and the remaining wave functions for the Ω baryons are described by one quark regroup with the created antiquark to form a harmonic oscillators. According to our calculations, we meson. The process of baryon decays is shown in Fig. 1. 3 find that (i) the newly observed Ωð2012Þ resonance is most The transition operator under the P0 model is written as likely to be the 1P-wave Ω state with spin-parity JP ¼ Z − X 3=2 and the experimental data can be reasonably T ¼ −3γ h1m;1− mj00i d3p⃗d3p⃗δ3ðp⃗ þ p⃗Þ P − 4 5 4 5 described. The other 1P-wave state with J ¼ 1=2 might m 3 be broader state with a width of dozens of MeV. The P0 p⃗ − p⃗ Ym 4 5 χ45 φ45ω45b† ðp⃗Þd† ðp⃗Þ; ð Þ results are consistent with the recent predictions of the × 1 2 1;−m 0 0 4i 4 5j 5 1 chiral quark model [17]. (ii) The Ωð2250Þ resonance listed JP ¼ 5=2þ 1D in PDG may be a good candidate of the where the pair-strength γ is a dimensionless parameter, i wave state j56; 410; 2; 2; 5=2þi or j70; 210; 2; 2; 5=2þi. and j are the color indices of the createdpffiffiffi quark q4 and Although the widths of the D-wave states predicted within q¯ φ45 ¼ðuu¯ þ dd¯ þ ss¯Þ= 3 ω45 ¼ δ 3 antiquark 5. 0 and 0 ij the P0 model are systematically larger that those predicted 45 stand for the flavor and color singlet, respectively. χ1;−m with the chiral quark model [17], these states may be Ymðp⃗Þ ≡ jp⃗jYmðθ ; ϕ Þ observed in their main decay channels in future experi- is the spin triplet state and 1 1 p p is a P ments for their relatively narrow width. solid harmonic polynomial corresponding to the -wave b† ðp⃗Þd† ðp⃗Þ This paper is organized as follows. First, we give a brief quark pair. 4i 4 5j 5 denotes the creation operator in 3 review of the P0 model in Sec. II. Second, we present the the vacuum. 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. 1 2 4 II. THE 3P MODEL 0 1 3 The P0 model, is also called the quark pair creation 2

(QPC) model. It was first proposed by Micu [25], Carlitz 3 and Kislinger [26], and developed by the Orsay group and 5 Yaouanc et al. [27–32]. In the model, it assumes that a pair of quarks qq¯ is created from the vacuum with JPC ¼ 3 þþ 2Sþ1 3 0 ð LJ ¼ P0Þ when the initial hadron A decays, and 3 the quarks from the hadron A regroups with the created FIG. 1. The decay process of A → B þ C in the P0 model.

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According to the definition of the mock state [11], the jCðn2SCþ1L ÞðP⃗ Þi C CðJC;MJ Þ C baryons A and B, and meson C are defined as follows: pffiffiffiffiffiffiffiffiffi X C ¼ 2E hL M ; S M jJ M i C C LC C SC C JC 2S þ1 A ⃗ ML ;MS jAðnA LAðJ ;M ÞÞðPAÞi C C pffiffiffiffiffiffiffiffi XA JA Z ¼ 2E hL M ; S M jJ M i δ3ðp⃗ þ p⃗ − P⃗ Þd3p⃗d3p⃗ A A LA A SA A JA × 3 5 C 3 5 M ;M LA SA Z Ψ ðp⃗; p⃗Þχ35 × nCLCML 3 5 S M δ3ðp⃗ þ p⃗ þ p⃗ − P⃗ Þd3p⃗d3p⃗d3p⃗ C C SC × 1 2 3 A 1 2 3 35 35 × φC ωC jq3ðp⃗3Þq5ðp⃗5Þi; ð4Þ 123 123 123 × Ψn L M ðp⃗1; p⃗2; p⃗3ÞχS M φA ωA A A LA A SA jq ðp⃗Þq ðp⃗Þq ðp⃗Þi; ð Þ where the subscripts 1, 2, 3 denote the quarks of the initial × 1 1 2 2 3 3 2 baryon A. 1,2, and 4 denote the quarks of the final baryon B, 3 and 5 stand for the quark and antiquark of the final jBðn2SBþ1L ÞðP⃗ Þi B BðJBMJ Þ B C p⃗ði ¼ 1; 2; 3Þ pffiffiffiffiffiffiffiffi X B meson , respectively. i are the momentum of ¼ 2E hL M ; S M jJ M i quarks in baryon A. p⃗iði ¼ 1; 2; 4Þ are the momentum of B B LB B SB B JB M M quarks in baryon B. p⃗3 and p⃗5 are the momentum of the Z LB SB ⃗ ⃗ ⃗ quark and antiquark in meson C. PA; PB; PC denotes the 3 ⃗ 3 3 3 × δ ðp⃗1 þ p⃗2 þ p⃗4 − PBÞd p⃗1d p⃗2d p⃗4 momentum of state A, B, C. SA, SB, SC and JA, JB, JC represent the total spin and the total angular momentum of 124 124 124 × Ψn L M ðp⃗1; p⃗2; p⃗4ÞχS M φB ωB state A, B, C. Ψn L M ,Ψn L M , and Ψn L M denotes B B LB B SB A A LA A A LA C C LC the spatial wave functions of the baryon and meson in × jq1ðp⃗1Þq2ðp⃗2Þq4ðp⃗4Þi; ð3Þ momentum space, respectively. 3 The decay amplitude of the Ω baryon in P0 is written as

MJ MJ MJ 2SBþ1 2SCþ1 2SAþ1 M A B C ¼hBðnB LBðJ ;M ÞÞ;CðnC LCðJ ;M ÞÞjTjAðnA LAðJ ;M Þi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB JBX X XC JC A JA ¼ −3γ 8E E E hL M ; S M jJ M ih1m;1− mj00i A B C A LA A SA A JA M ;M M ;M M ;M ;m LA SA LB SB LC SC 124 35 123 45 × hLBML ; SBMS jJBMJ ihLCML ; SCMS jJCMJ ihχS M χS M jχS M χ1−mi B B B C C C B SB C SC A SA X nALA;M ;nALA;M ;1m ρ ρ LA λ λ LA hφ124φ35jφ123φ45i I ρ λ ðP⃗ Þ; ð Þ × B C A 0 × nBLBM nBLBM n L M B 5 ρ ρ LB λ λ LB C C LC M ;M ;M ρ λ LA LB LC λ λ M A ;M B Lρ Lρ

124 35 123 45 hχS M χS M jχS M χ1−mi¼hS1MS ; S2MS jS12MS ihS12MS ; S3MS jSAMS i B SB C SC A SA 1 2 12 12 3 A hS M ; S M jS M ihS M ; S M jS M i × 1 S1 2 S2 12 S12 12 S12 4 S4 B SB hS M ; S M jS M ihS M ; S M j10i: ð Þ × 3 S3 5 S5 C SC 4 S4 5 S5 6

124 35 123 45 124 35 123 45 In the Eq. (5), hχS M χS M jχS M χ1−mi and hφB φC jφA φ0 i stand for the spin matrix and the flavor matrix, B SB C SC A SA respectively. Si stand for the spin of ith quark and S12 stand for the total spin of 1 and 2 quarks. The prefactor 3 in front of γ 3 arises from the fact that the three decay processes of the Ω baryon in P0 model are equivalent.The summed magnetic quantum numbers are not completely independent of each other. The overlap integral in the momentum space is written as

Z nALA;M ;nALA;M ;1m ρ ρ LA λ λ LA I ρ λ ðP⃗ Þ¼ d3p⃗d3p⃗d3p⃗d3p⃗d3p⃗δ3ðp⃗ þ p⃗ þ p⃗ − P⃗ Þδ3ðp⃗ þ p⃗Þ nBLBM nBLBM n L M B 1 2 3 4 5 1 2 3 A 4 5 ρ ρ LB λ λ LB C C LC ρ λ 3 ⃗ 3 ⃗ × δ ðp⃗1 þ p⃗2 þ p⃗4 − PBÞδ ðp⃗3 þ p⃗5 − PCÞΨAðp⃗1; p⃗2; p⃗3ÞΨ ðp⃗1; p⃗2; p⃗4ÞΨ ðp⃗3; p⃗5Þ Z B C A A ⃗ A A ⃗ A A ¼ dp⃗ρdp⃗λjJjΨAðp⃗λ ; p⃗ρ ÞΨBðPB; p⃗λ ; p⃗ρ ÞΨCð−PB; p⃗λ ; p⃗ρ Þ

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Z A A A A A A A A ¼ dp⃗ dp⃗ dΩ dΩ jJjϕ A A ðp⃗ ÞY A ðΩ Þϕ A A ðp⃗ ÞY A ðΩ Þ ρ λ ρ λ nρ Lρ ρ Lρ M A ρ n L λ L M A λ Lρ λ λ λ L rffiffiffi λ 3 2m A B A 3 ⃗ B × ϕnBLB ðp⃗ρ ÞYLBM ðΩρ ÞϕnBLB p⃗ − PB YLBM ðΩ Þ ρ ρ ρ LB λ λ λ λ LB λ ρ 2 2m3 þ mq λ rffiffiffi rffiffiffi m 2 2 ϕ 3 P⃗ − p⃗A Y ðΩCÞYm P⃗ − p⃗A ; ð7Þ × nCLC B λ LCML 1 B λ m3 þ mq 3 C 3

qffiffi 1 m p⃗−m p⃗ 2 m ðp⃗þp⃗Þ−ðm þm Þp⃗ ΠðL M A ;L M A ;mÞ p⃗¼pffiffið 2 1 1 2Þ p⃗¼ ð 3 1 2 1 2 3Þ the expressions of ρA; Lρ λA; L and harmonic where ρ 2 m þm and λ 3 m þm þm λ 1 2 1 2 3 oscillator wave function for the S-wave, P-wave, D-wave stand for the momentum corresponding to ρ and λ Jacobi Ω baryons are collected in the Appendix A and B. coordinates in the center mass frame of baryon A, and jJj The decay width Γ of the process A → B þ C is stand for the Jacobi determinant which determined by the definition of pρ, pλ. mi denote the mass of ith quark, and X 2 jp⃗j 1 M M M 2 Γ ¼ π jM JA JB JC j ; ð Þ mq denote the mass of the created quark pairs. The nρ and 2 11 m 2JA þ 1 A MJ ;MJ ;MJ Lρ denote the nodal and orbital angular momentum A B C n between the 1, 2 quarks (see Fig. 1), while the λ and where JA are the total angular momentum of the initial L λ stand for the nodal and orbital angular momentum baryon A. p⃗is the momentum of the final baryon in the between the 1, 2 quarks system and the 3 quark (see Fig. 1). center of mass frame of the initial baryon A In the calculations, simple harmonic oscillator (SHO) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wave functions is employed as the hadron wave function. 2 2 2 2 ½mA − ðmB − mCÞ ½mA − ðmB þ mCÞ The momentum space wave function of baryon is jp⃗j¼ ; ð12Þ 2mA Ψ ðP⃗ Þ¼Nψ ðp⃗ Þψ ðp⃗ Þ m m m A A nρ lρ Ml ρA nλ lλ Ml λA where A, B, and C are the mass of the initial and final A A ρA A A λ A hadrons. lþ2 1 2 2 ψ ðp⃗Þ¼ð−1Þnð−iÞl pffiffiffi In order to partly remedy the inadequacy of the non- nlMl πð2l þ 1Þ!! relativistic wave function as the relative momentum p⃗ lþ3 2 2 increases [33–37], the decay amplitude is written as 1 2 p⃗ p⃗ − Llþ1=2 Ymðp⃗Þ; × β exp 2β2 n β2 l Mðp⃗Þ → γfMðγfp⃗Þ; ð13Þ ð8Þ where γf denotes a commonly Lorentz boost factor, γf ¼ mB=EB. In most decays, the three momenta carried N where stands for a normalization coefficient of the wave by the final state baryons are relatively small, which means Llþ1=2ðp⃗2Þ function and n β2 is the Laguerre polynomial func- the nonrelativistic prescription is reasonable and the cor- tion. The Clebsch-Gorden coefficients of lρ, lλ coupling are rections from the Lorentz boost is not drastic. equal to 1 in our case. The ground state wave function of a meson in the III. NUMERICAL RESULTS AND ANALYSIS momentum space is In our calculations, we adopt mu ¼ md ¼ 350 MeV, m ¼ 450 s MeV for the constituent quark masses. The 2 3 2 2 R 4 R p⃗ masses of the Ω baryons listed in Table I, the masses of Ψ ¼ − ab ; ð Þ 0;0 π exp 2 9 K mesons and Ξ baryons are taken from the PDG [15]. The quantum numbers involved in the calculations are listed in Table II. Due to the orthogonal relationship of the wave where the p⃗ab stands for the relative momentum between functions, only the λ excited mode contributes. There are the quark and antiquark in the meson. As all hadrons in the three harmonic oscillator parameters, the βρ and βλ and R in final states are S-wave in this work, Eq. (7) can be further baryon and meson wave functions, respectively. We adopt expressed as follows −1 R ¼ 2.5 GeV for K mesons [38]. The parameter βρ of the ρ nALA;M ;nALA;M ;1m -mode excitation between the 1, 2 quarks (see Fig. 1)is ρ ρ LA λ λ LA A A ρ λ β ¼ 0 4 β ΠðL ;M A ;L ;M A ;mÞ ≡ I ð Þ taken as ρ . GeV [39]. The λ is obtained with the ρ Lρ λ L nBLBM nBLBM n L M 10 λ ρ ρ LB λ λ LB C C LC ρ λ relation [40]:

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TABLE II. The Quantum numbers of 1P-wave and 1D-wave Ω baryons. Due to the orthogonal relationship of the wave functions, only the λ excited mode contributes.

States Jnρ lρ nλ lλ LSρ S j70; 210; 1; 1; 1=2−i 1 1 2 0001112 j70; 210; 1; 1; 3=2−i 3 1 2 0001112 j56; 410; 2; 2; 1=2þi 1 3 2 0002212 j56; 410; 2; 2; 3=2þi 3 3 2 0002212 j70; 210; 2; 2; 3=2þi 3 1 2 0002212 j56; 410; 2; 2; 5=2þi 5 3 2 0002212 j70; 210; 2; 2; 5=2þi 5 0002211 2 2 FIG. 2. The decay width of the JP ¼ 1− Ω baryon as functions j56; 410; 2; 2; 7=2þi 7 3 2 2 0002212 of the initial state mass.

1− 1 – Ω 3m 4 1.94 2.04 GeV. It is found that the decay width of 2 β ¼ 3 β : ð Þ Γ ∼ 40–50 λ 2m þ m ρ 14 baryon is about MeV and dominantly decays to 1 3 ΞK channel. As future experimental statistics increases, it is 1− Ω For the quark pair creation strength from the vacuum, we possible to find the 2 baryon in this channel. Ωð2012Þ JP ¼ 3− take as those in Ref. [38], γ ¼ 6.95. On the other hand, assigning as the 2 state j70; 210; 1; 1; 3=2−i, we find that the width A. The 1P-wave states Γth ≃ 8 MeV; ð16Þ According to the SUð6Þ supermultiplet classification (see total 1P JP ¼ 1− Table I), there are two -wave states with 2 and and the branching fraction ratio P 3− 2 − 2 − J ¼ 2 ,i.e.,j70; 10; 1; 1; 1=2 i and j70; 10; 1; 1; 3=2 i. 2 − 0 − The mass of the newly observed Ωð2012Þ state is close to the Γ½j70; 10; 1; 1; 3=2 i → Ξ K Rth ¼ ≃ 1.1; ð17Þ 1P-wave Ω baryon predicted in various quark models (see Γ½j70; 210; 1; 1; 3=2−i → Ξ−K¯ 0 Table I). Assuming Ωð2012Þ as a candidate of the 1P-wave Ω baryons, we calculate the OZI-allowed two body strong are consistent with the measured width Γexp ¼ 3 decays in the P0 model, and list our results in Table III. 6 4þ2.5ð Þ1 6ð Þ Rexp ¼ 1 2 0 3 P 1− . −2.0 stat . syst and ratio . . for It is found that if one assigns Ωð2012Þ as the J ¼ 3 2 Ωð2012Þ. These P0 model predictions are compatible with j70; 210; 1; 1; 1=2−i state , the width is predicted to be those predicted within the chiral quark model [17]. Γth ≃ 43 MeV; ð15Þ total B. The 1D-wave states which is too large to be comparable with the width of According to the quark model classification, there are six 3 Ωð2012Þ. This width predicted in the P0 model is about a 1D-wave states. Their masses are predicted to be in the factor 3 larger than that predicted within the chiral quark range of 2.2–2.3 GeV in various models (see Table I). Their 1− ΞK model [17]. As the mass of 2 Ω baryon are predicted about OZI allowed two-body strong decay channels are and 1.95 GeV in some models [3,4,13]. In Fig. 2 we shows its Ξð1530ÞK. With the masses predicted in Ref. [7], we study strong decay properties as functions of mass in range of the strong decay processes of 1D-wave states into both ΞK and Ξð1530ÞK channels, and collect their partial decay

2 widths in Table IV. TABLE III. The total decay widths (MeV) of the j70; 10; 1; 1; JP ¼ 5=2þ − 2 − It is interesting to find that the two states 1=2 i and j70; 10; 1; 1; 3=2 i states with mass M ¼ 2012 MeV. j56; 410; 2; 2; 5=2þi j70; 210; 2; 2; 5=2þi Γth B and may be good total denotes the total decay width and stands for the radio of Ωð2250Þ Γ½Ξ0K−=Γ½Ξ−K¯ 0 candidates of listed in the review book of PDG the branching fraction . The results of Ref. [17] Ωð2250Þ are also listed for a comparison. The units of widths is MeV. [15]. (i) The mass of is close the predictions of j56; 410; 2; 2; 5=2þi and j70; 210; 2; 2; 5=2þi in the quark Γth Γth BB Ωð2250Þ Γ ¼ States total total [17] [17] model [7]. (ii) The measured width of , 55 18 Γ ∼ j70; 210; 1; 1; 1=2−i 43.0 15.2 0.96 0.95 MeV, is close to the theoretical predictions, 80=50MeV for j56;410;2;2;5=2þi and j70;210;2;2;5=2þi, j70; 210; 1; 1; 3=2−i 8.19 6.64 1.11 1.12 respectively. (iii) The decays modes of j56; 410; 2; 2; 5=2þi

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1D Γth B TABLE IV. The partial and total decay widths (MeV) of the -wave states. total denotes the total decay width and stands for the radio of the branching fraction Γ½ΞK=Γ½Ξð1530ÞK. The results of Ref. [17] are also shown for a comparison. The widths and masses are in units of MeV.

Γ½ΞK Γ½ΞK Γ½Ξð1530ÞK Γ½Ξð1530ÞK Γth Γth BB States Mass [7] [17] [17] total total [17] [17] j56; 410; 2; 2; 1=2þi 2210 154 51.8 16.5 4.53 171 56.3 9.36 11.4 j56; 410; 2; 2; 3=2þi 2215 76.8 25.8 55.4 15.7 132 41.5 1.39 1.64 j56; 410; 2; 2; 5=2þi 2225 7.78 6.58 77.2 22.6 84.9 29.2 0.10 0.29 j56; 410; 2; 2; 7=2þi 2210 31.7 26.2 2.94 1.51 34.7 27.7 10.8 17.4 j70; 210; 2; 2; 3=2þi 2265 9.04 7.40 14.3 11.9 23.4 20.9 0.63 0.62 j70; 210; 2; 2; 5=2þi 2265 4.34 0.99 50.0 11.6 54.4 13.4 0.09 0.08 and j70; 210; 2; 2; 5=2þi are dominated by Ξð1530ÞK, IV. SUMMARY Ωð2250Þ which is also consistent with the fact that the Stimulated by the newly discovered Ωð2012Þ resonance was seen in the Ξð1530ÞK and Ξ−πþK− channels. In [17], P at Belle II, in this work we have studied the OZI allowed we predicted that Ωð2250Þ is more likely to be the J ¼ 1P 1D Ω þ 4 þ strong decays of - and -wave baryons within the 5=2 j56; 10; 2; 2; 5=2 i with the chiral quark model, 3P 2 þ 0 model. and the width of j70; 10; 2; 2; 5=2 i is predicted to be It is found that the newly observed state Ωð2012Þ favors ∼12 P − 2 − MeV, which is about a factor 4 smaller than the width the 1P-wave Ω state with J ¼ 3=2 , j70; 10; 1; 1; 3=2 i. of Ωð2250Þ. P þ 2 þ Both the decay mode and decay width are consistent The J ¼ 3=2 state j70; 10; 2; 2; 3=2 i is a narrow with the observations. The other 1P-wave Ω state with Γ ≃ 23 P − 2 − state with a width of MeV, and has comparable J ¼ 1=2 , j70; 10; 1; 1; 1=2 i, might have a relatively decay rates into ΞK and Ξð1530ÞK channels. These Oð10Þ JP ¼ 1=2− 3 broad width of MeV. This state should predictions of the P0 model are consistent with those in Ξ0K− Ξ−K¯ 0 P þ be observed in the and channels as well. the chiral quark model [17]. While the other J ¼ 3=2 In the 1D-wave Ω states, it is found that the Ωð2250Þ 4 þ state j56; 10; 2; 2; 3=2 i is found to be a broad state with a state may favor the JP ¼ 5=2þ state j56; 410; 2; 2; 5=2þi or width of Γ ≃ 130 MeV, the partial width ratio between ΞK j70; 210; 2; 2; 5=2þi. These two JP ¼ 5=2þ states domi- and Ξð1530ÞK is predicted to be nantly decay into the Ξð1530ÞK channel, and have a similar decay width to that of Ωð2250Þ. Due to a large uncertainty Γ½ΞK Ωð2250Þ ≃ 1.4: ð18Þ of the width of , we cannot distinguish it belongs Γ½Ξð1530ÞK to the 70 multiplet or 56 multiplet. Future experiment information will help to clarify this issue. For the other 4 þ The predicted width for j56; 10; 2; 2; 3=2 i in this work is 1D-wave Ω baryons, we recommend looking for the JP ¼ about a factor 3 larger than that predicted with the chiral 1=2þ and JP ¼ 7=2þ states in the ΞK decay channel and quark model [17]. looking for the JP ¼ 3=2þ states in both ΞK and Ξð1530ÞK P þ 4 þ The J ¼ 1=2 state j56; 10; 2; 2; 1=2 i is the broadest decay channels in future experiments. state in the 1D-wave states. It has a width of Γ ∼ 170 MeV, and mainly decays into the ΞK channel. However, in the ACKNOWLEDGMENTS Γ ∼ chiral quark model a relatively narrower width This work is supported by the National Natural Science 56 j56; 410; 2; 2; 1=2þi MeV is given for [17]. Foundation of China under Grants No. 11405053, JP ¼ 7=2þ j56; 410; 2; 2; 7=2þi The state may be a No. 11775078, No. U1832173, and No. 11705056. This Γ ∼ 30 narrow state with a width of MeV. This state work is also in part supported by China Postdoctoral ΞK mainly decays into the channel. The decay properties Science Foundation under Grant No. 2017M620492. of j56; 410; 2; 2; 7=2þi predicted in this work are consistent with those of chiral quark model in Ref. [17]. APPENDIX A: THE HARMONIC OSCILLATOR As a whole most of the 1D-wave states has a relatively WAVE FUNCTIONS narrow width, they has potentials to be observed in their S dominant decay modes. The Ωð2250Þ resonance may be For the -wave omega baryon, the harmonic oscillator assigned to the JP ¼ 5=2þ state j56; 410; 2; 2; 5=2þi or wave function is p⃗2 p⃗2 2 þ 1 3 ρ λ j70; 10; 2; 2; 5=2 i 2 2 − − . Although the decay widths predicted 4 1 2β2 2β2 3 Ψð0; 0; 0; 0Þ¼ pffiffiffi y ðp⃗Þe ρ λ D P0 0;0 ρ for the -wave states within the chiral quark model and π βρ model show some differences, the partial width ratios of 1 3 Γ½ΞK=Γ½Ξð1530ÞK predicted within these two models are 4 2 1 2 × pffiffiffi y0;0ðp⃗λÞðA1Þ in a reasonable agreement with each other. π βλ

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For the P-wave omega baryon, the harmonic oscillator 1 Πð0; 0; 1; 1; −1Þ¼− pffiffiffi Δ ¼ Πð0; 0; 1; −1; 1Þ: wave function is 0;1 6λ2

p⃗2 p⃗2 1 3 ρ λ ðB3Þ 2 2 − − 4 1 2β2 2β2 Ψð0; 0; 1;m Þ¼−i pffiffiffi y ðp⃗Þe ρ λ lλ π β 0;0 ρ ρ For the D-wave omega baryon decay, 1 5 8 2 1 2 pffiffiffi pffiffiffi y ðp⃗Þ; ð Þ 2 × 1;mlλ λ A2 λ 6λ 3 π βλ Πð0; 0; 2; 0; 0Þ¼ 3 βjp⃗j3 − 3 jp⃗j Δ ; ð Þ 2 2 0;2 B4 2λ2 3λ2 p⃗2 p⃗2 1 5 ρ λ 2 2 − − 8 1 2β2 2β2 Ψð1;m ; 0; 0Þ¼−i pffiffiffi y ðp⃗Þe ρ λ λ lρ 1;mlρ ρ 3 3 π βρ Πð0;0;2;1;−1Þ¼pffiffiffi jp⃗jΔ0;2 ¼ Πð0;0;2;−1;1Þ: ðB5Þ 2λ2 1 3 2 4 2 1 2 × pffiffiffi y0;0ðp⃗λÞ; ðA3Þ π βλ Here,

For the D-wave omega baryon, the harmonic oscillator 1 1 1 1 R2 λ ¼ þ ; λ ¼ þ þ ; ð Þ wave function is 1 2 02 2 2 02 B6 2βρ 2βρ 2βλ 2βλ 3

p⃗2 p⃗2 1 3 ρ λ 4 2 1 2 − − pffiffiffi pffiffiffi 2β2 2β2 2 Ψð0; 0; 2;m Þ¼− pffiffiffi y ðp⃗Þe ρ λ 6m m 6R lλ 0;0 ρ λ ¼ 3 þ 3 ; ð Þ π βρ 3 02 B7 ð2m3 þ m5Þβ m3 þ m5 3 1 7 λ 16 2 1 2 pffiffiffi y ðp⃗Þ; ð Þ × 2;mlλ λ A4 15 π βλ 3m2 m2 R2 λ ¼ 3 þ 3 ; ð Þ 4 ð2m þ m Þ2β02 ðm þ m Þ2 2 B8 p⃗2 p⃗2 3 5 λ 3 5 1 7 ρ λ 2 2 − − 16 1 2β2 2β2 Ψð2;m ; 0; 0Þ¼− pffiffiffi y ðp⃗Þe ρ λ lρ 2;mlρ ρ 15 π βρ λ3 β ¼ 1 − pffiffiffi ; ðB9Þ 1 3 4 2 1 2 6λ2 pffiffiffi y ðp⃗Þ; ð Þ × π β 0;0 λ A5 λ and qffiffi p1ffiffi 1 Where p⃗ρ ¼ ðp⃗1 − p⃗2Þ and p⃗λ ¼ ðp⃗1 þp⃗2 −2p⃗3Þ 3 3 2 3 2 3 λ2 2 6 1 4 1 4 R 4 π 2 3 2 −ðλ4− Þjp⃗j Δ ¼ e 4λ2 are obtained in the relative Jacobi coordinates. The yl;m ðp⃗Þ 0;0 02 02 l πβρ πβ π λ1λ2 rffiffiffiffiffiffi λ is the solid harmonic polynomial. 3 3 3 1 4 1 4 The wave function of the meson in our calculation is − ; ð Þ × 2 2 B10 4π πβρ πβλ 3 2 2 2 R 4 −ðp⃗3−p⃗5Þ R Ψð0; 0Þ¼ e 8 ðA6Þ π 3 3 2 3 2 3 λ2 1 4 1 4 R 4 π 2 3 2 −ðλ4− Þjp⃗j Δ ¼ e 4λ2 0;1 02 02 p⃗3−p⃗5 1 πβρ πβ π λ1λ2 Here p⃗C ¼ , R ¼ . λ 2 βC 3 1 5 3i 1 4 8 2 1 4 pffiffiffi ; ð Þ × 2 2 B11 4π πβρ 3 π βλ APPENDIX B: THE MOMENTUM SPACE INTEGRATION 3 3 2 3 2 3 λ2 1 4 1 4 R 4 π 2 3 2 −ðλ4− Þjp⃗j A A A A Δ ¼ e 4λ2 The momentum space integration ΠðLρ ;Mρ ;Lλ ;Mλ ;mÞ 0;2 02 02 πβρ πβ π λ1λ2 are presented in the following pffiffiffiffiffi λ 3 1 7 15 1 4 16 2 1 4 × pffiffiffi ; ðB12Þ Πð0; 0; 0; 0; 0Þ¼βjp⃗jΔ0;0: ð Þ 2 2 B1 8π πβρ 15 π βλ

P A A For the -wave omega baryon decay, where the momentum space integration ΠðLρ ;Mρ ; LA;MA;mÞ 1 λ λ λ is introduced in Ref. [41], what makes our Πð0; 0; 1; 0; 0Þ¼ pffiffiffi − 3 βjp⃗j2 Δ ; ð Þ differences from Ref. [41] is that we take into account 2λ 0;1 B2 6λ2 2 difference of the masses of the quarks in baryons.

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