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l , 2 indicate that some of λ-mode Ξbb(1P) and Ωbb(1P) states are In the heavy quark limit, the heavy diquark spin Jρ and the narrow, which have good potentials to be observed by future light quark spin j should be preserved well. With finite bottom experiments. However, the strong decays of the low-lying ρ- quark masses, the states with same JP can be mixed with each mode and ρ-λ hybrid states are highly suppressed owing to the other. However, the physical states should be much closer to orthogonality of spatial wave functions. Moreover, the masses these coupling bases to preserve the heavy diquark symme- and strong decay behaviors of doubly bottom baryons show try approximately, especially for the doubly bottom systems. similar patterns to the bottom/bottom-strange mesons, which Hence, we adopt these coupling bases to investigate the strong suggests that the diquark correlation should play an essential decay behaviors for doubly bottom baryons in present work. role in these systems. The above coupling scheme differs from the j j coupling This paper is organized as follows. The framework of adopted in the singly bottom baryons, because the− light quark strong decays for doubly bottom baryons are introduced in spins for the singly and doubly heavy systems are quite differ- Sec II. We present the numerical results and discussions for ent. In singly heavy baryons, the ρ is the relative coordinate the low-lying doubly bottom baryons in Sec III. A summary between two light quarks, and the λ is the relative coordi- is presented in the last section. nate between the heavy quark and light quark subsystem. For singly heavy baryons, the corresponding coupling basis that preserving the light quark spin j′ can be written as II. MODEL P J , j′ = (lρlλ)LS ρ s3 , (2) A. Coupling scheme | i j′ JP + h i  where the L and j represents the quantum numbers of total To calculate the strong decays of doubly heavy baryons, one ′ orbital angular momentum and light quark spin, respectively. should label the low-lying states with certain quantum num- It can be noticed that there is a linear relationship between bers in a coupling scheme firstly. The Jacobi coordinates of these two j j coupling bases doubly bottom baryons are presented in Figure 1. With this − definition, we can classify the excitations of doubly bottom JP, j = ( 1)S ρ+L+s3+J+2Jρ+2lλ baryons into three types: ρ-mode, λ-mode, and ρ-λ hybrid ex- | i L j − citations. X′ + + + + ρ mode (2 j 1)(2 j′ 1)(2Jρ 1)(2L 1) b b × 1 2 q Jρ S ρ j′ lρ S ρ Jρ P J , j′ . (3) × s3 J j J lλ L | i ( )( ) mode λ Furthermore, the L S coupling scheme is also commonly used in the literature,− and the coupling bases are

2S +1 q3 LJ = (lρlλ)L(S ρs3)S . (4) | i JP + h i FIG. 1: The Jacobi coordinates of doubly bottom baryons. The b1 Then the relations among the j j and L S coupling scheme and b stand for bottom quarks, and q corresponds to a light quark. 2 3 can be written as − − The ρ = (r1 r2)/ √2 is the relative coordinate between two bottom − quarks, and the λ = (r1 + r2 2r3)/ √6 is the relative coordinate P S +L+s +J J , j = ( 1) ρ 3 (2 j + 1)(2S + 1) between the light quark and bottom− quark subsystem. ′ ′ | i S − X p L J j′ 2S +1 With the Jacobi coordinates, we can use a series of quan- LJ . (5) × s3 S ρ S | i tum numbers to characterize the excitations of spatial wave ( ) functions. Here, the nρ and lρ are the radial and orbital quan- and tum numbers between the two bottom quarks, respectively; P similarly, the nλ and lλ correspond to the radial and orbital J , j = (2 j + 1)(2S + 1)(2L + 1)(2Jρ + 1) quantum numbers between the light quark and bottom quark | i L S X q subsystem, respectively. The N = 2nλ + 2nρ + lρ + lλ is the lρ S ρ Jρ principle quantum number, and the low-lying doubly bottom 2S +1 lλ s3 j LJ . (6) baryons denote the states up to N = 2 shell. The S ρ and Jρ ×   | i  LS J  stands for the total spin and total angular momentum of the   two bottom quarks, respectively. The j = lλ + s3 represents   The tree representations for three different coupling schemes the light quark spin, and JP is the spin-parity of a doubly bot- are shown in Figure 2 for reference. tom baryon. Then, the coupling bases can be expressed as It should be mentioned that the a physical state can be the mixture of theoretical states with same JP, and the final re- P J , j = (lρS ρ)Jρ (lλ s3) j . (1) sults for strong decays are independentwith different coupling | i JP h i E

3 schemes. However, because of the heavy diquark symme- m¯ [Bs(1P)] m¯ [Bs(1S )] preserve the heavy diquark symme- try, doubly bottom baryons favor the JP, j bases, that is, the try approximately.− Hence, we have mixing angles between physical and| theoreticali states JP, j | i should be small and the mixing effects can be ignored here. m¯ [Ξbb(1D)] m¯ [Ξbb(1S )] = 791 MeV, (15) P P − Moreover, we employ the notations Ξbb(J , j) and Ωbb(J , j) to represent the orbital excited states JP, j in present work. | i m¯ [Ωbb(1D)] m¯ [Ωbb(1S )] = 796 MeV. (16) −

Then, the predicted average masses of Ξbb(1D) and Ωbb(1D) B. Mass states from the heavy diquark symmetry are about 11016 MeV and 11175 MeV, respectively. Also, the approximate SU(3) For the masses of low-lying Ξbb and Ωbb states, we flavor symmetry for light quarks is preserved well. adopt the theoretical predictions from the relativistic quark With the above two approaches, we obtain the mass ranges model [19]. The masses of λ-mode S-wave and P-wave dou- for Ξbb(1D) and Ωbb(1D) states, which are listed in Table I. bly bottom baryons are listed in Table I. Here, we treat the Due to the small mass splittings, the estimated average masses light quark spin j as a good quantum number and neglect the of Ξbb(1D) and Ωbb(1D) states are enough to investigate their small mixtures among the P-wave states. Also, the masses strong decays. of low-lying ρ-mode states are not listed, as the their strong decays are highly suppressed under the spectator assumption 3 3 of P0 model. For the λ-mode Ξbb(1D) and Ωbb(1D) states, C. P0 Model the authors did not perform their masses within the relativis- tic . Fortunately, we can estimate their average 3 In this work, we adopt the P0 model to calculate the masses with the help of Regge trajectory and heavy diquark Okubo-Zweig-Iizuka-allowed two-body strong decays of the symmetry. low-lying Ξbb and Ωbb states. In this model, a quark-antiquark From Table I, the average masses of the Ξbb(1S ), Ξbb(1P), pair with the quantum number JPC =0++ is created from Ωbb(1S ) and Ωbb(1P) states are the vacuum and then groups into the final states [103]. This model has been employed to study the strong decays m¯ [Ξbb(1S )] = 2m[Ξbb] + 4m[Ξ∗ ] /6 = 10225 MeV, (7) { bb } for different kinds of hadron systems with considerable suc- cesses [104–119]. In the nonrelativistic limit, to describe the 3 m¯ [Ξbb(1P)] = 10664 MeV, (8) decay process A BC, the transition operator T in the P0 model can be written→ as

3 3 3 m¯ [Ωbb(1S )] = 2m[Ωbb] + 4m[Ωbb∗ ] /6 = 10379 MeV, (9) T = 3γ 1m1 m 00 d p d p δ (p + p ) { } − h − | i 4 5 4 5 m Z Xp p Ω = m 4 5 45 45 45 m¯ [ bb(1P)] 10798 MeV. (10) 1 − χ1, mφ0 ω0 b4†i(p4)d4†j(p5), (17) ×Y 2 −   Their Regge trajectories of Ξbb(1D) and Ωbb(1D) between where γ is a dimensionless q4q¯5 quark pair production mass and orbital angular momentum are plotted in Figure. 3. strength, and p4 and p5 are the momenta of the created quark From these trajectories, we can obtain the estimated average q4 and antiquarkq ¯5, respectively. The i and j are the color in- masses of Ξbb(1D) and Ωbb(1D) states are about 11086 MeV dices of the created quark and antiquark. φ45 = (uu¯ + dd¯ + and 11201 MeV, respectively. 0 ss¯)/ √3, ω45 = δ , and χ45 are the flavor singlet, color Meanwhile, because of the doubly diquark symmetry, the i j 1, m singlet, and spin triplet wave− functions of the q q¯ , respec- mass gaps of doubly bottom baryons should be similar to that 4 5 tively. The m(p) p Ym(θ ,φ ) is the solid harmonic poly- of bottom/bottom-strange mesons. With the same relativistic 1 1 p p nomial reflectingY the≡|P-wave| momentum-space distribution of quark model, the mass gaps of bottom/bottom-strange mesons the created quark pair. are estimated as [102] For an initial baryon A, the definition of the mock state is m¯ [B(1P)] m¯ [B(1S )] = 430 MeV, (11) employed in present work, and it can be taken as − 2S A+1 A(n LA J M )(PA) | A A JA i≡ m¯ [B(1D)] m¯ [B(1S )] = 791 MeV, (12) 3 3 3 − 2EA LA MLA S A MS A JA MJA d p1d p2d p3 M ,M h | i p LXA S A Z m¯ [Bs(1P)] m¯ [Bs(1S )] = 440 MeV, (13) 3 123 123 123 − δ (p1 + p2 + p3 PA)ψnA LA ML (p1, p2, p3)χS M φA ωA × − A A S A q1(p1)q2(p2)q3(p3) , (18) m¯ [Bs(1D)] m¯ [Bs(1S )] = 796 MeV. (14) ×| i − which satisfies the normalization condition It can be seen that the mass gapsm ¯ [Ξbb(1P)] m¯ [Ξbb(1S )] − ≃ 3 m¯ [B(1P)] m¯ [B(1S )] andm ¯ [Ωbb(1P)] m¯ [Ωbb(1S )] A(PA) A(P ′) = 2EAδ (PA P ′ ). (19) − − ≃ h | A i − A 4

Sρ lρ lλ s3 Sρ lρ lλ s3 Sρ s3 lρ lλ

L Jρ j S L j′

J J J

(a) (b) (c)

FIG. 2: Three coupling scheme in tree representations.

M M M 0 1 2 3 JA JB JC is defined as M 140 140 3 MJ MJ MJ BC T A = δ (PA PB PC) A B C , (23) Ωbb(1D) h | | i − − M M M M where the JA JB JC is the helicity amplitude of a decay Ξbb(1D) 130 130 process A MB+C. For the low-lying doubly bottom baryons, → only the process A(b1, b2, q3) + P (q4, q¯5) B(b1, b2, q4) + )

2 → C(q3, q¯5) is allowed owing to limited phase space. The helicity M M M 120 120 amplitude JA JB JC can be expressed as, GeV M (

2 3 MJ MJ MJ δ (pB + pC pA) A B C = M − M γ 8EAEBEC 110 110 − MJ ,M j ,Ml ,MS ,Ml p ρA A XρA ρA λA

100 100 MJρ ,M j ,Mlρ ,MS ρ ,Ml ML ,MS Ms ,Ms ,Ms ,Ms ,Ms ,m B B XB B λB XC C 1 2 X3 4 5 Jρ MJ jA M j JA MJ lρ Ml S ρ MS Jρ MJ 0 1 2 3 ×h A ρA A | A ih A ρA A ρA | A ρA i lλ Ml s3 Ms jA M j s1 Ms s2 Ms S ρ MS L ×h A λA 3 | A ih 1 2 | A ρA i Jρ MJ jB M j JBMJ lρ Ml S ρ MS Jρ M j ×h B ρB B | B ih B ρB B ρB | B B i FIG. 3: The Regge trajectories of Ξ (1D) and Ω (1D) states. lλ Ml s4 Ms jBM j s1 Ms s2 Ms S ρ MS bb bb ×h B λB 4 | B ih 1 2 | B ρB i 1m1 m 00 s Ms s Ms 1 m ×h − | ih 4 4 5 5 | − i LC MLC S C MS C JC MJC s3 Ms3 s5 Ms5 S C MS C The p1, p2, and p3 are the momenta of the quarks q1, q2, and ×h | ih | i 124 35 123 45 MLA m q3, respectively. PA denotes the momentum of the initial state φB φC φA φ0 IM M (p), (24) ×h | i LB LC A. χ123 , φ123, ω123, and ψ (p , p , p ) are the spin, S M A A nA LA MLA 1 2 3 A S A 124 35 123 45 flavor, color, and space wave functions of the baryon A. The where φB φC φA φ0 is the overlap of the flavor wave func- h M |m i definitions of the final baryon B and meson C are similar to tions. The I LA (p) are the spatial overlaps of the initial and MLB MLC the initial state A. final states, which can be written as For the strong decays of doubly bottom baryons, there are three possible rearrangements, MLA m 3 3 3 3 3 I (p) = d p1d p2d p3d p4d p5 MLB MLC A(b , b , q ) + P(q , q¯ ) B(b , q , q ) + C(b , q¯ ), (20) Z 3 3 1 2 3 4 5 2 4 3 1 5 δ (p + p + p PA)δ (p + p ) → × 1 2 3 − 4 5 A(b1, b2, q3) + P(q4, q¯5) B(b1, q4, q3) + C(b2, q¯5), (21) 3 3 → δ (p1 + p4 + p2 PB)δ (p3 + p5 PC) A(b , b , q ) + P(q , q¯ ) B(b , b , q ) + C(q , q¯ ). (22) × − − 1 2 3 4 5 1 2 4 3 5 ψ∗ (p , p , p )ψ∗ (p , p ) → × B 1 4 2 C 3 5 m p4 p5 These three possible rearrangements are shown in the Fig- ψA(p1, p2, p3) − . (25) × Y1 2 ure 4. It can be seen that the first and second ones stand for   the singly bottom baryon plus heavy meson channels, while It should be mentioned that the spatial overlap also relies on the last one denotes the doubly bottom baryon plus light me- the radial and orbital quantum numbers that are commonly son decay mode. omitted for simplicity. The relevant spatial wave functions With the transition operator T, the helicity amplitude and overlaps are presented in Appendix. 5

C C C 1 2 3 5 5 5 A 2 4 A 1 4 A 1 4

3 B 3 B 2 B

(a) (b) (c)

FIG. 4: The baryon decay process A B + C in the 3P model. → 0

In this work, we adopt the simplest vertex which assumes Unlike the broad j = 1/2 states, the three j = 3/2 states a spatially constant quark pair creation strength γ, the rela- 1 − 3 are relatively narrow. The strong decay of Ξbb( 2 , 2 ) state is tivistic phase space, and the simple harmonic oscillator wave governed by the Ξbb∗ π channel with a width of 27 MeV.Mean- functions. Then, the decay width Γ(A BC) can be calcu- Ξ 3 − 3 Ξ 5 − 3 → while, the total decay widths of bb( 2 , 2 ) and bb( 2 , 2 ) lated directly states are about 37 and 53 MeV, respectively. The branching P ratios for J = 3/2− and 5/2− states are predicted to be 2 p 1 M M M 2 Γ= π JA JB JC , 2 (26) M 2JA + 1 |M | A M ,M ,M Br(Ξbbπ, Ξbb∗ π) = 28.4%, 71.6%, (27) JA XJB JC

2 2 2 2 and √[MA (MB+MC ) ][MA (MB MC ) ] where p = p = − − − , and MA, MB, | | 2MA and MC are the masses of the hadrons A, B, and C, respec- Br(Ξbbπ, Ξbb∗ π) = 64.0%, 36.0%. (28) tively. Because there is no experimental information on doubly These ratios are independent with the overall quark pair cre- bottom baryons, we can not obtain the overall γ and harmonic ation strength γ and can be tested in future. oscillator parameters by fitting the known decay processes. It can be seen that the total decay widths are quite different Fortunately, the doubly bottom baryons are similar with the for different light quark spins j, and the partial decay widths conventional bottom/bottom-strange mesons. Then, we can are crucial to determine the spin-parities in a certain multi- adopt the same parameters of mesons to estimate the strong plet. Also, one can find the similarity between Ξbb(1P) and decays of doubly bottom baryons. In the literature [108, 109], B(1P). Indeed, the P-wave bottom mesons can be categorized the γ = 0.4 and α = 0.4 GeV are widely used to investigate into two groups: one belongs to the j = 1/2 doublet, and the strong decays of conventional mesons, where a factor of the other is the j = 3/2 doublet. The two narrow j = 3/2 √96π should be added here by considering different filed con- bottom mesons have been observed experimentally, while the ventions and the results of decay widths are unaffected cer- two j = 1/2 states are predicted to be broad if they lie above the relevant threshold [113]. tainly. Hence, the λ-mode harmonic oscillator parameter αλ can be chosen as 0.4 GeV, and the strong decay widths of In the Ref [101], the authors calculated the strong decays low-lying states are independent with the ρ-mode harmonic for these P-wave states within the potential model, and gives j = / j = oscillator parameter αρ that is absent in the overlaps of spatial larger total decay widths for the 1 2 doublet and wave functions. 3/2 triplet. Although the exact values for these decays are model dependent, their features are similar to ours. Also, the strong decays for Ξbb(1P) states are investigated in the L S − III. STRONG DECAYS coupling scheme within the chiral quark model [80], which can not be compared with present results in j j coupling scheme directly. − A. Ξbb(1P) states

In the constituent quark model, there are five λ-mode B. Ξbb(2S ) states Ξbb(1P) states, and their predicted masses are about 10632 10694 MeV. According to approximately conserved light∼ In the Ξbb family, there are two λ-mode first radially excited quark spin j, they belong to two groups: j = 1/2 doublet states, which can be denoted as Ξbb(2S ) and Ξ∗ (2S ). The and j = 3/2 triplet. Their strong decay behaviors are shown bb total decay widths of Ξbb(2S ) and Ξbb∗ (2S ) states are about in Table II. For the two j = 1/2 states, the total decay widths 93 and 79 MeV, respectively, which are shown in Table III. are about 200 MeV, and the slight difference of total widths Clearly, the masses, total widths, and main decay modes of Ξ Ξ arises from the small mass splitting. The bbπ and bb∗ π decay these two states are similar, but the partial decay widths are 1 − 1 3 − 1 channel saturate the total widths for Ξbb( 2 , 2 ) and Ξbb( 2 , 2 ) different. The branching ratios are predicted to be states, respectively, which provides a good criterion to distin- guish them. Br(Ξbbπ, Ξbb∗ π) = 9.9%, 90.1%. (29) 6

TABLE I: Notations, quantum numbers, and masses of the doubly TABLE II: Theoretical predictions of the strong decays for the bottom baryons. The values are in MeV. Ξbb(1P) states in MeV.

P State nρ nλ lρ lλ S ρ Jρ j J Mass 1 − 1 3 − 1 1 − 3 3 − 3 5 − 3 State Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) 1 1 + Ξbb(1S )000011 2 2 10202 Ξbbπ 195.97 10.75 34.01 1 3 + ······ Ξbb∗ (1S )000011 2 2 10237 Ξbb∗ π 205.64 27.21 27.06 19.14 1 1 + ··· Ξbb(2S )010011 2 2 10832 Total width 195.97 205.64 27.21 37.81 53.15 1 3 + Ξbb∗ (2S )010011 2 2 10860 1 − 1 1 1 − Ξbb( 2 , 2 )000111 2 2 10675 3 − 1 1 3 − Ξbb( 2 , 2 )000111 2 2 10694 mated widths may be sensitive to the node of radially excited 3 1 − 3 3 1 − wave functions, which also appeared in previous P0 model Ξbb( 2 , 2 )000111 2 2 10632 calculations [105]. Moreover, the relativistic corrections may Ξ ( 3 −, 3 )000111 3 3 − 10647 bb 2 2 2 2 be important and provide significant contributions for these 5 − 3 3 5 − Ξbb( 2 , 2 )000111 2 2 10661 radially excited baryons [120]. 1 + 3 3 1 + Ξbb( 2 , 2 )000211 2 2 11016-11086 3 + 3 3 3 + Ξbb( 2 , 2 )000211 2 2 11016-11086 TABLE III: Theoretical predictions of the strong decays for the + + 5 3 3 5 Ξbb(2S ) and Ξbb∗ (2S ) states in MeV. Ξbb( 2 , 2 )000211 2 2 11016-11086 3 + 5 5 3 + Ξbb( 2 , 2 )000211 2 2 11016-11086 State Ξbb(2S ) Ξbb∗ (2S ) 5 + 5 5 5 + Ξbb( , )000211 11016-11086 2 2 2 2 Ξbbπ 9.16 31.16 7 + 5 5 7 + Ξbb( , )000211 11016-11086 2 2 2 2 Ξbb∗ π 83.59 47.29 1 1 + Ωbb(1S )000011 10359 Ωbb K 0.63 2 2 ··· 1 3 + Total width 92.75 79.08 Ωbb∗ (1S )000011 2 2 10389 1 1 + Ωbb(2S )010011 2 2 10970 1 3 + Ωbb∗ (2S )010011 2 2 10992 1 − 1 1 1 − Ωbb( 2 , 2 )000111 2 2 10804 3 − 1 1 3 − C. Ξbb(1D) states Ωbb( 2 , 2 )000111 2 2 10821 Ω ( 1 −, 3 )000111 3 1 − 10771 bb 2 2 2 2 In the constituent quark model, there are six states belong- Ω ( 3 −, 3 )000111 3 3 − 10785 bb 2 2 2 2 ing to the λ-mode Ξbb(1D) states, and they can be classified 5 − 3 3 5 − into j = 3/2 triplet and j = 5/2 triplet. These states lie in Ωbb( 2 , 2 )000111 2 2 10798 + + the range of 11016 11086 MeV, which are estimated by the Ω ( 1 , 3 )000211 3 1 11175-11201 bb 2 2 2 2 Regge trajectory and∼ heavy diquark symmetry. With this mass 3 + 3 3 3 + Ωbb( 2 , 2 )000211 2 2 11175-11201 range, their strong decay behaviors are calculated and listed in 5 + 3 3 5 + Table IV. Ωbb( 2 , 2 )000211 2 2 11175-11201 + + It can be seen that the strong decay widths for these states Ω ( 3 , 5 )000211 5 3 11175-11201 bb 2 2 2 2 are large, especially for the j = 5/2 triplet. In Ref. [101], only 5 + 5 5 5 + Ωbb( 2 , 2 )000211 2 2 11175-11201 the emission processes were considered and falsely nar- 7 + 5 5 7 + row states were obtained. From our results, the K, ρ, and ω Ωbb( , )000211 11175-11201 2 2 2 2 meson emissions may be also important. Experimentally, the broad states can be hardly observed. The total decay widths 5 + 3 of Ξbb( , ) state is about 130 156 MeV, which is rela- 2 2 ∼ for the Ξbb(2S ) state, and tively narrower than others. Its main decay modes are Ξbb∗ π and Ωbb∗ K, which can be tested by future experiments. Br(Ξbbπ, Ξbb∗ π, ΩbbK) = 39.4%, 59.8%, 0.8%. (30) for the Ξ (2S ) state, which can help us to distinguish these bb∗ D. Ω (1P) states two states. Hopefully, future experiments can search for them bb in the Ξbbπ and Ξbb∗ π final states. Our results are quite different with the previous work in In the quark model, there are five λ-mode Ωbb(1P) states, 1 − 1 3 − 1 1 − 3 Ref. [101], where significantly small total widths are pre- which are named as Ωbb( 2 , 2 ), Ωbb( 2 , 2 ), Ωbb( 2 , 2 ), 3 − 3 5 − 3 dicted. The different initial masses and shapes of spatial wave Ωbb( 2 , 2 ) and Ωbb( 2 , 2 ), respectively. The estimated functions could lead to this divergence. Actually, the esti- masses for these Ωbb(1P) states are around 10800 MeV. With 7

TABLE IV: Theoretical predictions of the strong decays for the Ξbb(1D) states in MeV.

1 + 3 3 + 3 5 + 3 3 + 5 5 + 5 7 + 5 State Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 ) Ξbb( 2 , 2 )

Ξbbπ 15.97-42.60 9.98-26.62 61.78-93.18 148.27-223.63 ······ Ξbb∗ π 3.54-7.16 14.18-28.64 31.92-64.44 205.29-321.96 156.42-245.31 87.98-137.99

ΩbbK 67.58 -84.47 42.24-52.80 1.24-4.66 2.97-11.17 ······ Ωbb∗ K 6.92-9.94 27.69-39.76 62.31-89.47 2.44-11.75 1.86-8.95 1.05-5.04

Ξbbρ 20.71-64.50 13.99 -47.29 2.80-18.61 59.67-186.22 34.88-111.32 0.18-6.47

Ξbb∗ ρ 0.77-44.76 0.51-31.06 0.09-8.24 2.21-129.04 1.28-75.99 0-1.71

Ξbbω 5.31 -20.56 3.58-14.88 0.69-5.41 15.29-59.33 8.93-35.34 0.03-1.75

Ξbb∗ ω 0-13.50 0-9.30 0-2.30 0-38.90 0-22.86 0-0.41 Total width 151.05-257.24 143.27-219.25 130.33-155.95 284.90-747.20 266.39-597.61 240.48-388.17

these initial masses, the total widths of two j = 1/2 Ωbb states widths of Ωbb(2S ) and Ωbb∗ (2S ) states are predicted to be 184 are rather broad, which can not be observed experimentally. and 175 MeV,respectively, which are listed in Table VI. These For the three j = 3/2 states, the total widths are narrow, which two states have similar masses, total widths, and dominant ¯ ¯ have good potentials to be observed in the ΞbbK and Ξbb∗ K decay modes, but the branching ratios are different. From Ta- modes. ble VI, the predicted branching ratios are From the heavy diquark symmetry, these Ωbb(1P) states can ¯ ¯ be related to the Bs(1P) mesons, where the broad j = 1/2 Br(ΞbbK, Ξbb∗ K, Ωbbη, Ωbb∗ η) = 9.5%, 85.5%, 1.1%, 3.9%. Ωbb(1P) doublet correspond to the j = 1/2 Bs(1P) states and (31) narrow j = 3/2 Ωbb(1P) triplet correspond to the j = 3/2 for the Ωbb(2S ) state, and Bs(1P) states. Also, the two narrow j = 3/2 Bs(1P) states, Br(ΞbbK¯ , Ξ∗ K¯ , Ωbbη, Ω∗ η) = 35.5%, 52.7%, 6.8%, 5.0%. Bs1(5830) and B∗s2(5840), have been observed experimen- bb bb tally [121], which supports our calculations. Meanwhile, the (32) ¯ ¯ two predicted j = 1/2 Bs(1P) states are not found until now. for the Ωbb∗ (2S ) state. The branching ratios of ΞbbK and Ξbb∗ K However, if we go further to consider the heavy quark channels are significantly different, which can help us to dis- flavor symmetry, the two j = 1/2 Bs(1P) states may have tinguish Ωbb(2S ) and Ωbb∗ (2S ) states in future. similar properties with D∗s0(2317) and Ds1(2460). Then, the j = 1/2 Ωbb(1P) doublet are also related with D∗s0(2317) and D (2460) resonances. In this situation, these two j = 1/2 TABLE VI: Theoretical predictions of the strong decays for the s1 Ω Ω ¯ ¯ bb(2S ) and bb∗ (2S ) states in MeV. Ωbb(1P) states should lie below the ΞbbK and Ξbb∗ K thresh- olds, and the broken and radiative modes may domi- Ω Ω nate the decay behaviors. Future experiments can help us to Mode bb(2S ) bb∗ (2S ) disentangle this puzzle and deepen our understandings of the ΞbbK¯ 17.55 62.24 mysterious D∗ (2317) state. ¯ s0 Ξbb∗ K 157.26 92.49

Ωbbη 2.08 11.89 TABLE V: Theoretical predictions of the strong decays for the Ωbb∗ η 7.10 8.79 Ω (1P) states in MeV. bb Total width 183.99 175.41

1 − 1 3 − 1 1 − 3 3 − 3 5 − 3 Mode Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 )

ΞbbK¯ 485.99 4.39 16.79 ······ Ξ K¯ 492.92 2.62 4.66 4.82 bb∗ ··· Total width 485.99 492.92 2.62 9.05 21.61 F. Ωbb(1D) states

The strong decays for the λ-mode Ωbb(1D) states are listed in Table VII. It can be found that all these states have large total widths, especially for the j = 5/2 triplet, which can E. Ωbb(2S ) states be hardly observed in experiments. Owing to the higher ini- tial masses, the Ωbb(1D) states may also decay into the ΞbB¯ In the quark model, there are two λ-mode 2S states, which final states. This singly heavy baryon plus heavy meson can be denoted as Ωbb(2S ) and Ωbb∗ (2S ). The total decay decay mode may be important for the higher excited states 8 but can be neglected for the low-lying states due to limited In the heavy quark limit, the two heavy quark subsystem phase space. With the light quark SU(3) flavor symmetry, in a doubly heavy baryon seems like a antiquark and a heavy these Ωbb(1D) states should have similar decay behaviors with diquark symmetry emerges. Actually, it can be noticed that 3 Ξbb(1D) states, which agrees with our present calculations. the P0 model is a spectator model, where the quarks in the initial state carry their color, flavor, spin, and momenta into the final states, and the change for degrees of freedom arises G. Low-lying ρ-mode and ρ-λ hybrid states from the created quark pair. Under this hypothesis, the two heavy quarks as a whole go into the final states, and the heavy diquark symmetry is preserved automatically. Together with the λ-mode excited states, there also exist lots of ρ-mode and ρ-λ hybrid excited states. For the dou- bly heavy baryons, the ρ-mode between two heavy quarks is Appendix A Overlaps of spatial wave functions more easily excited due to its larger reduced masses. Then, ρ ρ λ the masses of low-lying -mode and - hybrid doubly bot- The harmonic oscillator wave functions for doubly bottom tom states with N 2 should be smaller than that of λ-mode ≤ baryons in momentum representation can be expressed as Ξbb(1D) or Ωbb(1D) states. One can see that these low-lying Λ ¯ Ξ ¯ ρ-mode and ρ-λ hybrid states should lie below the bB or b B ψ n , l , m , n , l , m = threshold, and the singly heavy baryon plus heavy meson de- ρ ρ ρ λ λ λ 1 3 cay modes are forbidden due to the insufficient phase space.   2 +lρ 2n ! 2 + 1 nρ lρ lρ ρ 1 lρ 2 2 2 The other possible decay modes for these low-lying ρ-mode ( 1) ( i) pρ Lnρ (pρ/αρ) − − Γ nρ + lρ + 3/2  αρ and ρ-λ hybrid states are the light meson emission processes   !   1 3 as well as the λ-mode states. However, under spectator as-   +lλ   2n !  2 1 2 + 1 3 nλ lλ lλ λ  lλ 2 2 2 sumption of P0 model, the spatial wave functions between ( 1) ( i) p L (p /α ) × − − λ Γ (n + l + 3/2) α nλ λ λ initial ρ-mode and ρ-λ hybrid doubly bottom baryons and fi- " λ λ # λ ! nal ground states are orthogonal, which leads to the vanish- ~p2 ~p2 ρ λ mρ mλ exp Y ~pρ Y ~pλ , (A.1) ing amplitudes and strong decay widths. It can be seen that × −2α2 − 2α2 lρ lλ 3  ρ λ  the P0 model preserves the heavy diquark symmetry auto-      matically, where the heavy quark subsystems with different  1  1 where ~pρ = ~p1 ~p2 , ~pλ = ~p1 + ~p2 2~p3 , and quantum numbers can not transit into each other through the √2 − √6 − Ym(~p) is a three-dimensional spherical harmonic function. transition operator T. Then, these states should be rather nar- l   row, and the radiative and weak decays become crucial, which Similarly, the harmonic oscillator wave function for ground mesons in momentum representation can be written as provides good opportunities to be searched by future experi- ments. 3 2 1 4 ~p ψ (0, 0, 0) = exp rel , (A.2) πα2 −2α2 !   IV. SUMMARY   where ~prel represents the relative momentum  between the quark and antiquark in the final mesons. In this work, we investigate the strong decays of low-lying In this paper, all the final states are ground states, that is, 3 doubly bottom baryons within the P0 model systematically. nρB =lρB =nλB =lλB =LC=0. Here, we denote the spatial overlap 3 M ,m The relevant formulas of P model are constructed in the j j LA 0 integrals I (~p) as Π(nρ , lρ , mρ , nλ , lλ , mλ , m), and − ML ,ML A A A A A A coupling scheme, where the heavy diquark symmetry are pre- B C the relevant formulas for the low-lying states are present as served. The masses of S - and P-wave doubly bottom states follows. are taken form the relativistic quark model, and the average Define masses of λ-mode Ξbb(1D) and Ωbb(1D) states are estimated with the help of Regge trajectory and heavy diquark symme- 1 1 1 f1 = + + , (A.3) try. Then, the strong decays of these low-lying doubly bottom 2α2 2α′2 3α2 baryons are calculated. λ λ Our results show that some of λ-mode Ξbb(1P) and Ωbb(1P) states are rather narrow, which have good potentials to be ob- 2 1 f2 = + , (A.4) served by future experiments. The other λ-mode states are √6α′2 √6α2 relatively broad, which makes it difficult to search for. For the λ low-lying ρ-mode and ρ-λ hybrid states, the Okubo-Zweig- Iizuka-allowed strong decays are highly suppressed owing to 1 1 f3 = + , (A.5) the orthogonality of spatial wave functions between initial and ′2 2 3αλ 8α final doubly bottom baryons. These ρ-mode and ρ-λ hybrid states should be extremely narrow and the radiative and weak f decays become crucial, and future experiments can test our β = 1 2 , (A.6) phenomenological predictions at the quark level. − √6 f1 9

TABLE VII: Theoretical predictions of the strong decays for the Ωbb(1D) states in MeV.

1 + 3 3 + 3 5 + 3 3 + 5 5 + 5 7 + 5 Mode Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 ) Ωbb( 2 , 2 )

ΞbbK¯ 29.47-46.48 1.43-29.05 113.47-133.38 272.31-320.11 ······ ¯ Ξbb∗ K 6.61-9.08 26.42-36.32 59.48-81.71 374.95-449.56 285.68-342.52 160.69-192.67

Ωbbη 41.73-44.03 26.08-27.52 5.14-7.10 12.33-17.03 ······ Ωbb∗ η 5.52-5.53 22.14-22.47 49.81-50.56 14.19-20.48 10.81-15.61 6.08-8.78

ΞbbK∗ 42.15-73.28 65.44-98.85 104.25-141.47 65.17-93.78 158.55-209.78 289.29-372.18

Ξbb∗ K∗ 16.2234.09 29.39-55.40 51.34-90.93 30.59-55.96 79.61-139.18 148.24-255.70 Total width 172.29-193.00 215.63-248.74 287.86-341.69 484.90-619.78 661.85-861.53 909.54-1199.97

and then we can obtain the spatial overlaps integrals straight- with forwardly 3 √3 1 2 Π (0, 0, 0, 0, 0, 0, 0) = β ~p ∆ , (A.7) ∆00 = 00 5 α α f α 2π 4  λ′ λ 1 !   f 2   2 2 2 3 exp f3 ~p , (A.13) 3 3 β ~p f β ~p × − − 4 f1 Π (0, 0, 0, 1, 0, 0, 0) = β ~p + + 2     2 2 2     − 2 2 α f1 2 √6α f     r r λ λ 1    

f2 ~p 3 5 ∆ , (A.8) 3 1 2 1 2 2 2 00 ∆ = i −3αλ f1 01 5 ! 2 f αα α  s2π  1 λ′ λ   ! !   2   f2 2 1 f 2  exp  f3 ~p , (A.14) 2 4 f Π (0, 0, 0, 0, 1, 0, 0) = β ~p ∆01, (A.9) × − − 31 7 2 f   2  2 √6 f1 − 1 1   1  1    ∆ =       02 5       4 f αα α   2π ! 1 λ′ ! λ ! Π (0, 0, 0, 0, 1, 1, 1) = Π (0, 0, 0, 0, 1, 1, 1) f 2 − − 2 2 1 exp f3 ~p . (A.15) = ∆ , (A.10) × − − 4 f1 − 01     √6 f1        

2 f2 3 √6 f2 Π (0, 0, 0, 0, 2, 0, 0) = β ~p ~p ∆02, (A.11) ACKNOWLEDGEMENTS 2 f 2 − 3 f 2  1 1 

  This work is supported by the National Natural Science   Π (0, 0, 0, 0, 2, 1, 1) = Π (0, 0, 0, 0, 2, 1, 1) Foundation of China under Grants No. 11705056 and No. − − U1832173,and by the State Scholarship Fund of China Schol- f2 = ~p ∆02, (A.12) arship Council under Grant No. 202006725011. √ 2 2 f1

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