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 Æ 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 − à 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 ÁÁÁ 22Lz 2401 2280 2225 ρ j70; 210; 2; 2; 5=2þi p1ffiffi ðϕsχρψ þ ϕsχλψ λ Þ 2401 2345 ÁÁÁ 2265 2 22Lz 22Lz j56; 410; 2; 2; 7=2þi ϕsχsψ s ÁÁÁ 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. 114023-2 NEWLY OBSERVED Ωð2012Þ STATE AND STRONG … PHYS. REV. D 98, 114023 (2018) 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.