Strong and radiative decays of the low-lying D -wave singly heavy baryons
aa r X i v : . [ h e p - ph ] O c t Strong and radiative decays of the low-lying D -wave singly heavy baryons Ya-Xiong Yao , Kai-Lei Wang , Xian-Hui Zhong , ∗
1) Department of Physics, Hunan Normal University, and Key Laboratory of Low-DimensionalQuantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China and2) Synergetic Innovation Center for Quantum E ff ects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China The strong and radiative decays of the low-lying λ -mode D -wave Λ c ( b ) , Σ c ( b ) , Ξ c ( b ) , Ξ ′ c ( b ) , and Ω c ( b ) baryonsare studied in a constituent quark model. Our calculation shows the following: (i) The missing λ -mode D -wave Ω c ( b ) , Λ b , and Ξ b baryons have a relatively narrow decay width of a few MeV or a few tens of MeVand their dominant strong and radiative decay channels can be ideal for searching for their signals in futureexperiments. (ii) The λ -mode 1 D -wave excitations in the Σ c ( b ) and Ξ ′ c ( b ) families appear to have a relativelybroad width of ∼ −
200 MeV. Most of the 1 D -wave states have large decay rates into the 1 P -wave heavybaryons via the pionic or kaonic strong decay processes, which should be taken seriously in future observations.(iii) Both Λ c (2860) and Ξ c (3050) seem to favor the J P = / + excitation | D λλ + i of ¯ F , while both Λ c (2880)and Ξ c (3080) may be assigned as the J P = / + excitation | D λλ + i of ¯ F . The nature of Ξ c (3050) and Ξ c (3080)could be tested by the radiative transitions Ξ c (3055) → Ξ c (2790) γ and Ξ c (3080) → Ξ c (2815) γ , respectively. PACS numbers:
I. INTRODUCTION
The LHC facility provides good opportunities for us to dis-cover some of the missing heavy baryons. Recently, fiveextremely narrow Ω c states, Ω c (3000), Ω c (3050), Ω c (3066), Ω c (3090), and Ω c (3119), were observed in the Ξ + c K − chan-nel by the LHCb Collaboration [1]. Most of them may beinterpreted as the P -wave excited states of Ω c [2–9]. Lately,the LHCb Collaboration observed a new structure Ξ b (6227) − in both the Λ b K − and Ξ b π − invariant mass spectra [10]. Themass of this structure and the observed decay modes are con-sistent with expectations of a P -wave excited state in the Ξ ′ b family [11–23]. Besides the missing P -wave heavy baryons,some low-lying D -wave singly heavy baryons should be alsoobserved at the LHC in forthcoming experiments. Further-more, the Belle II experiments will also o ff er the possibilityof studying excited heavy baryons. Thus, the theoretical stud-ies of the low-lying P - and D -wave singly heavy baryons willprovide very useful references for searching for them in futureexperiments. Considering that the decay properties of heavybaryons should be sensitive to its inner structure, one may bet-ter understand the nature of the heavy baryons by studyingtheir decays. In our recent work [11], we systematically stud-ied the strong and radiative decay properties of the P -wavesingly heavy baryons. As a continuation of Ref. [11], we studythe strong and radiative decays of the low-lying D -wave singlyheavy baryons in the present work.In the heavy baryon resonances listed in the Review ofParticle Physics (RPP) [24], there are several good D -wavecandidates, such as Λ c (2880) + , Ξ c (3055) , + , Ξ c (3080) , + , and Ξ c (3123) + . Recently, a new D -wave candidate in the Λ c fam-ily, i.e., Λ c (2860) + , was observed in the D p channel by theLHCb Collaboration [25]. However, no candidates of the D -wave bottom baryons have been found in experiments. Tolook for the missing D -wave singly heavy baryons, and to ∗ E-mail: [email protected] identify these possible D -wave heavy baryons observed inexperiments, many theoretical studies were carried out withvarious phenomenological methods. For example, the massspectra were calculated in various quark models [18–23, 26–33], the Faddeev method [34], the lattice QCD [35, 36], theQCD sum rules [37–39], and so on. Furthermore, the strongdecay properties of the low-lying D -wave charmed baryonswere also studied within some methods, such as the P model [20, 33, 40–43], the heavy hadron chiral perturbationtheory [44–46], the chiral quark model (ChQM) [47–49], andso on. It should be pointed out that there are few discussionsof the radiative decays of the D -wave charmed and bottombaryons and the strong decays of the D -wave bottom baryons,although there are many discussions about the radiative de-cays [50–68] and strong decays [7–9, 69–88] for the low-lying S - and / or P -wave singly heavy baryons. More details aboutthe status for the studies of the heavy baryons can be foundin Refs. [88–91]. A a whole, it is necessary to carry out asystematical study of the strong and radiative decays for the D -wave singly charmed and bottom baryons.In this work, we apply a nonrelativistic constituent quarkmodel to study the strong decays with emission of one lightpseudoscalar meson and the radiative decays with emission ofone photon for the low-lying D -wave singly heavy baryons.By an analysis of the decay properties for the D -wave states,we will suggest ideal decay channels to observe missing statesin future experiments. For a simplicity, the harmonic oscilla-tor wave functions of the heavy baryons are adopted in ourcalculations. To deal with the strong decays of a hadron,an e ff ective chiral Lagrangian at the tree level [92] is in-troduced. In this interaction, the emitted light pseudoscalarmesons are treated as Goldstone bosons, which only cou-ple with the light constituent quarks. Since the quark-mesoncoupling is invariant under the chiral transformation, someof the low-energy properties of QCD are retained [92–94].This method ( i.e., ChQM) has been successfully applied tostudy the strong decays of heavy-light mesons and charmedand strange baryons [2, 47–49, 95–100]. The chiral quarkmodel used in this work is di ff erent from the often-used P model [101–103]; the di ff erences between them have beenpointed out in Ref. [11]. Meanwhile, to treat the radiativedecay of a hadron, we apply an e ff ective quark-photon elec-tromagnetic (EM) coupling at the tree level. The higher EMmultipole contributions are included by a multipole expansionof the EM interactions. This approach has been successfullyapplied to deal with the radiative decays of c ¯ c and b ¯ b sys-tems [104, 105], and recently it has been extended to studythe radiative transitions of heavy baryons [2, 99, 106].The paper is organized as follows. Section II is our frame-work, in which we give a brief review of the quark modelclassification of the singly heavy baryons and the quark modeldescription of the strong and radiative decays. Then, the nu-merical results for the heavy baryons belonging to ¯ F and F are presented and discussed in Secs. III and IV, respectively.Finally, a summary is given in Sec. V. II. FRAMEWORKA. Spectra
The heavy baryon containing a heavy quark violates theSU(4) symmetry. However, the SU(3) symmetry betweenthe other two light quarks ( u , d , or s ) is approximately kept.The heavy baryons containing a single heavy quark belongto two di ff erent SU(3) flavor representations: the symmetricsextet F and antisymmetric antitriplet ¯ F [11]. In the singlycharmed (bottom) baryons, there are two families, Λ c and Ξ c ( Λ b and Ξ b ) belonging to ¯ F , while there are three families, Σ c , Ξ ′ c , and Ω c ( Σ b , Ξ ′ b , and Ω b ), belonging to F [11].The spatial wave function of a heavy baryon is adopted theharmonic oscillator form in the constituent quark model [47].For a q q Q basis state, it contains two light quarks q and q with equal mass m and a heavy quark Q with mass m ′ . Thebasis states are generated by the oscillator Hamiltonian H = P cm M + m ρ p ρ + m λ p λ + K ( ρ + λ ) . (1)The constituent quarks are confined in an oscillator potentialwith the potential parameter K independent of the flavor quan-tum number. The Jacobi coordinates ρ and λ and c.m. coordi-nate R c . m . can be related to the coordinate r j of the j th quark.The momenta p ρ , p λ , and P c . m . are defined by p ρ = m ρ ˙ ρ , p λ = m λ ˙ λ , P c . m . = M ˙R c . m . , with M = m + m ′ , m ρ = m ,and m λ = mm ′ m + m ′ . The wave function of an oscillator is give by ψ n σ l σ m ( σ ) = R n σ l σ ( σ ) Y l σ m ( σ ) , (2)where σ ≡ ρ, λ . In the wave functions, there are two oscil-lator parameters, i.e., the potential strengths α ρ and α λ . Theparameters α ρ and α λ satisfy the following relation [47]: α λ = r m ′ m + m ′ α ρ . (3)The details of the classifications of the heavy baryons inthe constituent quark model can be found in Refs. [47]. Since the bottom and charm quark masses are much larger than thelight quark mass ( m Q > m q ), the λ -mode excitations of singlyheavy baryons should be easily formed than the ρ -mode ex-citations [27]. Thus, in the present work, we only study the λ -mode excitations. The mass spectra of the single heavybaryons up to the 1 D -wave excitations predicted within var-ious quark models are summarized in Tables I and II. B. Decays
In this work, strong decays of the D -wave singly heavybaryons with emission of one light pseudoscalar meson arestudied within ChQM [92]. This model has been success-fully applied to study the strong decays of heavy-light mesonsand charmed and strange baryons [2, 47–49, 95–100]. Inthis model, the light pseudoscalar mesons, i.e., π , K , and η ,are treated as fundamental states, which only couple with thelight constituent quarks of a hadron via the simple chiral La-grangian [92] H m = X j f m ¯ ψ j γ j µ γ j ψ j ∂ µ φ m , (4)where ψ j represents the j th quark field in the hadron, φ m is thepseudoscalar meson field, and f m is the pseudoscalar mesondecay constant.Meanwhile, to treat the radiative decay of a hadron, weapply the constituent quark model, which has been success-fully applied to study the radiative decays of c ¯ c and b ¯ b sys-tems [104, 105]. In this model, the quark-photon EM couplingat the tree level is adopted as H e = − X j e j ¯ ψ j γ j µ A µ ( k , r j ) ψ j , (5)where A µ represents the photon field with 3-momenta k . e j and r j stand for the charge and coordinate of the constituentquark ψ j , respectively.To match the nonrelativistic harmonic oscillator wave func-tions, in the calculations, one should adopt the nonrelativisticforms for the quark-pseudoscalar and quark-photon EM cou-plings listed in Eqs. (4) and (5), which have been given in theprevious works [2, 47, 48, 93–100, 104–112].For a strong decay process, the partial decay width can becalculated with [47] Γ m = δ f m ! ( E f + M f ) | q | π M i (2 J i + X J fz , J iz |M J fz , J iz | , (6)while for a radiative decay process, the partial decay widthcan be calculated with [104, 105] Γ γ = | k | π J i + M f M i X J fz , J iz |A J fz , J iz | , (7)where M J fz , J iz and A J fz , J iz correspond to the strong and radia-tive transition amplitudes, respectively. The quantum num-bers J iz and J fz stand for the third components of the total TABLE I: Mass spectra of the singly heavy baryons of ¯ F up to D wave from various quark models [21, 22, 27, 33] compared with the datafrom the Particle Data Group [24]. Λ c Λ b State RQM [21] NQM [27] NQM [33] PDG [24] RQM [21] NQM [27] PDG [24]1 S + P λ − P λ − D λλ + D λλ + Ξ c Ξ b State RQM [21] NQM [22] NQM [33] PDG [24] RQM [21] NQM [22] PDG [24]1 S + P λ − P λ − D λλ + D λλ + angular momenta of the initial and final heavy baryons, re-spectively. E f and M f are the energy and mass of the finalheavy baryon, and M i is the mass of the initial heavy baryon. δ as a global parameter accounts for the strength of the quark-meson couplings. It has been determined in our previous studyof the strong decays of the charmed baryons and heavy-lightmesons [47, 96]. Here, we fix its value the same as that inRefs. [47, 96], i.e., δ = . III. RESULTS FOR SINGLY HEAVY BARYONS OF ¯3 F A. Λ c states In the Λ c family, there are two λ -mode 1 D -wave excita-tions | Λ c D λλ + i and | Λ c D λλ + i according to the quarkmodel classification. The masses for the λ -mode 1 D -wave Λ c excitations are predicted to be ∼ . Λ c (2860) with J P = / + and Λ c (2880) with J P = / + listed in RPP [24] most likely corre-spond to the two λ -mode 1 D -wave Λ c excitations | Λ c D λλ + i and | Λ c D λλ + i , respectively.
1. J P = / + state and Λ c (2880) The Λ c (2880) state was first observed in Λ + c π + π − by CLEO[113]. It was confirmed in Σ c π and Σ c (2520) π channels byBelle [114] and in the D p channel by BABAR [115] andLHCb [25]. It has a narrow decay width of Γ ≃ . J P = / + byBelle [114] and were confirmed by LHCb [25] recently. The Λ c (2880) state may be classified as the 1 D -wavecharmed baryons [23, 30, 34, 40, 44]. If Λ c (2880) is a con-ventional λ -mode 1 D -wave excitation, it should be assignedto | Λ c D λλ + i . With this assignment, the width of Λ c (2880)can be reasonably understood by ChQM [47]. It is found thatthe main decay channel of Λ c (2880) should be Σ c (2520) π (seeTable IV). The partial width ratio, R = Γ [ Σ c (2520) π ] Γ [ Σ c (2455) π ] ≃ .
3, pre-dicted by us is too large to compare with the measured value
R ≃ .
225 at Belle [114]. The recent P analysis of thestrong decays of Λ c (2880) in Ref. [41] is consistent with ourpredictions.It should be mentioned that the measured ratio R = Γ [ Σ c (2520) π ] Γ [ Σ c (2455) π ] ≃ .
225 of Λ c (2880) may be strongly a ff ected byits nearby state Λ c (2860) + newly observed in the D p chan-nel at LHCb [25]. Thus, the measured ratio from Belle [114]should not be a genuine ratio for Λ c (2880). This situa-tion is very similar to that of D sJ (2860) before two largelyoverlapping states D s (2860) and D s (2860) were found byLHCb [116]. Considering D sJ (2860) as the J P = − state1 D , the measured partial width ratio R = Γ [ D ∗ K ] Γ [ DK ] ≃ . Λ c (2880), consider-ing it as the 1 D -wave state | Λ c D λλ + i , we further study itsradiative decays into the 1 P -wave charmed baryon states. Ourresults are listed in Table V. It is found that most of the par-tial radiative widths of Λ c (2880) into the 1 P -wave states are O (100) eV. Combining these partial widths with the total de-cay width of Λ c (2880), we find the branching fractions for themain radiative decay channels are O (10 − ). The small decayrates indicate that the radiative decays of Λ c (2880) into the1 P -wave states might be hard to observe in experiments. TABLE II: Mass spectra of the singly heavy baryons of F up to D wave from various quark models [21, 22, 27, 33] compared with the datafrom the Particle Data Group [24]. Σ c Σ b State RQM [21] NQM [27] NQM [33] PDG [24] RQM [21] NQM [27] PDG [24]1 S + S + P λ − P λ − P λ − P λ − P λ − D λλ + D λλ + D λλ + D λλ + D λλ + D λλ + · · · · · · ? Ξ ′ c Ξ ′ b State RQM [21] NQM [22] NQM [33] PDG [24] RQM [21] NQM [22] PDG [24]1 S + S + P λ − P λ − P λ − · · · · · · ?1 P λ − · · · · · · ?1 P λ − D λλ + · · · · · · ?1 D λλ + · · · D λλ + · · · · · · ?1 D λλ + · · · · · · ?1 D λλ + · · · ?1 D λλ + Ω c Ω b State RQM [21] NQM [27] NQM [22] PDG [24] RQM [21] NQM [27] PDG [24]1 S + S + P λ − P λ − P λ − P λ − P λ − D λλ + D λλ + D λλ + D λλ + D λλ + · · · ? 6529 6566 ?1 D λλ + · · · · · · ? TABLE III: Quark model parameters adopted in present work.Parameter m u ( d ) m s m c m b α ρ (for Λ c ( b ) and Σ c ( b ) ) α ρ (for Ξ ( ′ ) c ( b ) ) α ρ (for Ω c ( b ) ) f π f K Value (MeV) 330 450 1480 5000 400 420 440 132 160TABLE IV: Partial widths of strong decays for the λ -mode D -wave Λ c and Λ b baryons. The masses of the D -wave Λ c ( Λ b ) states | D λλ + i and | D λλ + i are taken as 2856 and 2881 (6190 and 6196)MeV, respectively. The superscript (subscript) stands for the uncer-tainty of a prediction with a +
10% ( − α ρ .Decay mode | Λ c D λλ + i (2856) | Λ c D λλ + i (2881) Γ i (MeV) Γ i (MeV) Σ c π . + . − . . − . + . Σ ∗ c π . − . + . . + . − . Sum 5 . + . − . . + . − . Decay mode | Λ b D λλ + i (6190) | Λ b D λλ + i (6196) Γ i (MeV) Γ i (MeV) Σ b π . + . − . . − . + . Σ ∗ b π . − . + . . + . − . Sum 9 . + . − . . + . − .
2. J P = / + state and Λ c (2860) Recently, besides the confirmation of Λ c (2880) in the D p channel, the LHCb Collaboration observed a new charmedbaryon state, Λ c (2860), with a broad width of Γ ≃ . + . − . MeV in the same channel [25]. The determined spin-parityquantum numbers are J P = / + [25]. Both the massand decay modes of Λ c (2860) indicate that it might be as-signed to the λ -mode excited 1 D -wave charmed baryon state | Λ c D λλ + i [21, 33, 41]. Considering Λ c (2860) as the | Λ c D λλ + i state, we predict its partial widths into the Σ c (2455) π and Σ c (2520) π channels, Γ [ Σ c (2455) π ] ≃ . , Γ [ Σ c (2520) π ] ≃ . , (8)which roughly agree with the predictions in Ref. [41]. Com-bining these predicted partial widths with the measured widthof Λ c (2860), we further estimate that the branching fractionsof the Σ c (2455) π and Σ c (2520) π channels can reach up to 7%and 2%, respectively. The relatively large branching fractionsindicate that Λ c (2860) might be observed in the Σ c (2455) π and Σ c (2520) π channels as well.Considering Λ c (2860) as the 1 D -wave state | Λ c D λλ + i ,we also study its radiative decays into the 1 P -wave states. Ourresults are listed in Table V as well. It is found that the ra-diative decay rates into the 1 P -wave states are small. Theirpartial decay widths are O (10) eV. Combining these partialwidths with the total decay width of Λ c (2860), we find thebranching fractions, B [ Λ c (2860) → P γ ], are O (10 − ), whichindicates the radiative decays of Λ c (2860) into the 1 P -wavestates might be hard to observe in experiments. b b b ( P 3/2 ) b ( P 1/2 ) b ( P 3/2 ) b ( P 5/2 ) b 2 D 3/2 + ( M e V )( M e V ) M (MeV) b 2
D 5/2 + FIG. 1: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Λ b states | Λ b D λλ / + i and | Λ b D λλ / + i as functions of their mass. The solid curves stand for the total widths.The masses of the P -wave heavy baryons in final states are adoptedfrom the quark model predictions in Ref. [21] (see Table II). B. Λ b states In the Λ b family, there are two λ -mode 1 D -wave excitations | Λ b D λλ + i and | Λ b D λλ + i according to the quark modelclassification. The masses for the λ -mode 1 D -wave Λ b exci-tations are predicted to be ∼ . D -wave Λ b ex-citations, we study their strong decay properties, which havebeen shown in Fig. 1. To be more specific, taking the massesof the 1 D -wave states as predicted in the relativistic quark-diquark picture [21] we further present the results in Table IV.
1. J P = / + state From Fig. 1, it is found that if the mass of | Λ b D λλ + i is ∼ TABLE V: Partial widths of radiative decays for the λ -mode D -wave Λ c and Λ b baryons. The masses of the D -wave Λ c ( Λ b )states | D λλ + i and | D λλ + i are taken as 2856 and 2881 (6190and 6196) MeV, respectively. M f stands for the masses of P -waveheavy baryons (MeV) in the final states, which are adopted from theRPP [24] and Ref. [21]. The superscript (subscript) stands for theuncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Λ c D λλ + i (2856) | Λ c D λλ + i (2881) Γ i (keV) Γ i (keV) | Λ + c P λ − i γ − . + . | Λ + c P λ − i γ − . + . | Σ + c P λ − i γ − . + . − . + . | Σ + c P λ − i γ | Σ + c P λ − i γ < . < . | Σ + c P λ − i γ − . + . | Σ + c P λ − i γ < .
01 0.19 − . + . Decay mode M f | Λ b D λλ + i (6190) | Λ b D λλ + i (6196) Γ i (keV) Γ i (keV) | Λ b P λ − i γ − . + . − . + . | Λ b P λ − i γ − . + . − . + . | Σ b P λ − i γ | Σ b P λ − i γ − . + . − . + . | Σ b P λ − i γ | Σ b P λ − i γ − . + . − . + . | Σ b P λ − i γ − . + . − . + . a narrow state with a width of Γ total ≃
10 MeV. The decaysmay be saturated by the Σ b π and Σ ∗ b π channels, and the partialwidth ratio between them is predicted to be Γ ( Σ b π ) Γ ( Σ ∗ b π ) ≃ , (9)which is less sensitive to the mass of | Λ b D λλ + i . On theother hand, if the mass of | Λ b D λλ + i is larger than 6240MeV, more strong decay channels may open. The | Λ b D λλ + i state may have a large decay rate into the | Σ b P λ − i π channelas well (see Fig. 1). To establish the missing D -wave state | Λ b D λλ + i , the decay channel Σ b π might be the ideal channelto be observed in future experiments.We further estimate the radiative decays of | Λ b D λλ + i into the 1 P -wave states. Our results are listed in Table V.It is found that | Λ b D λλ + i has a relatively large decay rateinto Λ b (5912) − γ , and the partial width of Γ [ | Λ b D λλ + i → Λ b (5912) − γ ] can reach up to ∼
20 keV. Combining it withour predicted total width, we find the branching fraction of B [ | Λ b D λλ + i → Λ b (5912) γ ] is O (10 − ), which indicatesthat | Λ b D λλ + i has the possibility of being observed in the Λ b (5912) − γ channel.
2. J P = / + state If the mass of | Λ b D λλ + i is less than 6200 MeV aspredicted in various quark models [21, 27], the decays of | Λ b D λλ + i may be saturated by the Σ b π and Σ ∗ b π channels.The | Λ b D λλ + i state has a narrow width of Γ total ≃
10 MeV,which is comparable with that of | Λ b D λλ + i (see Fig. 1).However, the strong decays of | Λ b D λλ + i are governed bythe Σ ∗ b π channel. The partial width ratio between Σ b π and Σ ∗ b π is predicted to be Γ ( Σ b π ) Γ ( Σ ∗ b π ) ≃ . , (10)which shows few sensibilities to the mass of | Λ b D λλ + i . Onthe other hand, if the mass of | Λ b D λλ + i is larger than 6240MeV, more strong decay channels may open. The | Σ b P λ − i π decay mode may play an important role in the decays as well.To establish the missing D -wave state | Λ b D λλ + i , the decaychannel Σ ∗ b π should be the ideal channel to be observed infuture experiments.To know more properties of | Λ b D λλ + i , we further es-timate its radiative decays into the 1 P -wave states. Our re-sults are listed in Table V. It is found that the radiative pro-cess | Λ b D λλ + i → Λ b (5920) − γ has a relatively large par-tial width ∼
24 keV. Combining it with our predicted totalwidth, we find the branching fraction of B [ | Λ b D λλ + i → Λ b (5920) − γ ] is O (10 − ), which indicates that | Λ b D λλ + i has the possibility of being observed in the Λ b (5920) − γ channel. C. Ξ c states In the Ξ c family, there are two λ -mode 1 D -wave excita-tions | Ξ c D λλ + i and | Ξ c D λλ + i . The typical masses ofthe λ -mode 1 D -wave Ξ c excitations are ∼ .
05 GeV withinvarious quark model predictions (see Table I). From the pointof view of mass, the charmed-strange baryons Ξ c (3055) + and Ξ c (3080) + observed in the Λ c ¯ K π final state by the Belle [121]and BABAR [122] Collaborations are good candidates of the λ -mode 1 D -wave states. Recently, a new decay mode D + Λ for both Ξ c (3055) + and Ξ c (3080) + was observed by the BelleCollaboration [123]. They first reported the following partialwidth ratios: Γ [ Ξ c (3055) + → Λ D + ] Γ [ Ξ c (3055) + → Σ c (2455) ++ K − ] = . ± . ± . , (11) Γ [ Ξ c (3080) + → Λ D + ] Γ [ Ξ c (3080) + → Σ c (2455) ++ K − ] = . ± . ± . , (12)and Γ [ Ξ c (3080) + → Σ c (2520) ++ K − ] Γ [ Ξ c (3080) + → Σ c (2455) ++ K − ] = . ± . ± . . (13) b (5935) bb (5955) b 2 D 5/2 + M (MeV) b b (5955) b (5935) b D 3/2 + ( M e V ) ( M e V ) FIG. 2: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Ξ b states | Ξ b D λλ / + i and | Ξ b D λλ / + i as functions of their mass. The solid curves stand for the sum of thestrong decay partial widths. Furthermore, more accurate widths for both Ξ c (3055) + and Ξ c (3080) + were obtained by the Belle Collaboration, i.e., Γ Ξ c (3055) + = . ± . ± . Γ Ξ c (3080) + = . ± . ± .
1. J P = / + state and Ξ c (3055) In Ref. [48], the strong decay properties of the 1 D -wavestates were studied within ChQM. It is found that Ξ c (3055)seems to favor the J P = / + state | Ξ c D λλ + i , whichis consistent with the predictions in Refs. [21, 41]. Basedon the SU(4) symmetry we estimated the partial width of Γ [ Ξ c (3055) + → Λ D + ], which is too small to compare withthe observation at Belle [123]. The serious SU(4) symmetrybreaking might lead to our failed description of the decaysinto the D + Λ channel. Assigning Ξ c (3055) as | Ξ c D λλ + i ,it should have relatively large decay rates into Ξ ′ c π and Σ c (2455) K channels (see Table VI). The predicted partialwidth ratio between these two channels is Γ [ Ξ ′ c π + ] Γ [ Σ c (2455) ++ K − ] ≃ . . (14)Combining the predicted partial width of Γ [ Ξ c (3055) + → Σ c (2455) ++ K − ] ≃ . TABLE VI: Partial widths of strong decays for the λ -mode D -wave Ξ c and Ξ b baryons. The masses of the D -wave Ξ c ( Ξ b ) states | D λλ + i and | D λλ + i are taken as 3055 and 3080 (6373 and 6366) MeV,respectively. M f stands for the masses of P -wave heavy baryons(MeV) in the final states, which are adopted from the RPP [24] andRef. [21]. The superscript (subscript) stands for the uncertainty of aprediction with a +
10% ( − α ρ .Decay mode M f | Ξ c D λλ + i (3055) | Ξ c D λλ + i (3080) Γ i (MeV) Γ i (MeV) Ξ ′ c π + . − . − . + . Ξ ′∗ c π − . + . + . − . Σ c K + . − . < − . + . Σ ∗ c K < − . < + . + . − . | Ξ ′ c P λ − i π < + . − . | Ξ ′ c P λ − i π + . − . | Ξ ′ c P λ − i π + . − . Sum 5.20 + . − . + . − . Decay mode M f | Ξ b D λλ + i (6366) | Ξ b D λλ + i (6373) Γ i (MeV) Γ i (MeV) Ξ ′ b π + . − . − . + . Ξ ′∗ b π − . + . + . − . Σ b K + . − . Σ ∗ b K + . − . + . − . | Ξ ′ b P λ − i π < + . − . | Ξ ′ b P λ − i π + . − . + . − . | Ξ ′ b P λ − i π + . − . Sum 6.8 + . − . + . − . partial width into the Λ D channel: Γ [ Ξ c (3055) + → Λ D + ] ≃ . ± . Ξ c (3055) is es-timated to be Γ ≃ . ± . Ξ c (3055) also can be found in the literature [42, 43, 45].To further confirm the nature of Ξ c (3055), the ratio of Γ [ Ξ ′ c π + ] / Γ [ Σ c (2455) ++ K − ] is worth observing in future ex-periments.Furthermore, the nature of Ξ c (3055) can be tested by its ra-diative decays. Assigning Ξ c (3055) as the J P = / + state | Ξ c D λλ + i , we study its radiative decays into the 1 P -wavecharmed baryon states. Our results are listed in Table VII.It is found that the Ξ c (3055) → Ξ c (2790) γ process has arelatively large partial decay width, ∼
80 keV. Combining itwith the measured width of Ξ c (3055), we predict the branch-ing fraction B [ Ξ c (3055) → Ξ c (2790) γ ] ≃ . Ξ c (3055) is most likely to be observed in the Ξ c (2790) γ channel if it corresponds to the J P = / + state | Ξ c D λλ + i indeed. TABLE VII: Partial widths of radiative decays for the λ -mode D -wave Ξ c and Ξ b baryons. The masses of the D -wave Ξ c ( Ξ b )states | D λλ + i and | D λλ + i are taken as 3055 and 3080 (6366and 6373) MeV, respectively. M f stands for the masses of P -waveheavy baryons (MeV) in the final states, which are adopted from theRPP [24] and Ref. [21]. The superscript (subscript) stands for theuncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Ξ c D λλ + i (3055) | Ξ c D λλ + i (3080) Γ i (keV) Γ i (keV) | Ξ + c P λ − i γ − . + . − . + . | Ξ c P λ − i γ − . + . − . + . | Ξ + c P λ − i γ − . + . − . + . | Ξ c P λ − i γ − . + . − . + . | Ξ ′ + c P λ − i γ − . + . | Ξ ′ c P λ − i γ | Ξ ′ + c P λ − i γ − . + . − . + . | Ξ ′ c P λ − i γ | Ξ ′ + c P λ − i γ − . + . − . + . | Ξ ′ c P λ − i γ | Ξ ′ + c P λ − i γ − . + . − . + . | Ξ ′ c P λ − i γ | Ξ ′ + c P λ − i γ − . + . | Ξ ′ c P λ − i γ M f | Ξ b D λλ + i (6366) | Ξ b D λλ + i (6373) Γ i (keV) Γ i (keV) | Ξ b P λ − i γ − . + . − . + . | Ξ − b P λ − i γ − . + . − . + . | Ξ b P λ − i γ − . + . − . + . | Ξ − b P λ − i γ − . + . − . + . | Ξ ′ b P λ − i γ − . + . − . + . | Ξ ′ − b P λ − i γ | Ξ ′ b P λ − i γ − . + . − . + . | Ξ ′ − b P λ − i γ | Ξ ′ b P λ − i γ − . + . | Ξ ′ − b P λ − i γ | Ξ ′ b P λ − i γ − . + . − . + . | Ξ ′ − b P λ − i γ | Ξ ′ b P λ − i γ − . + . − . + . | Ξ ′ − b P λ − i γ
2. J P = / + state and Ξ c (3080) The Ξ c (3080) resonance is suggested to be the ρ -mode 2 S -wave state with J P = / − in Ref. [48]. The observation of Ξ c (3080) + in the D + Λ channel excludes this assignment be-cause the D + Λ decay mode should be forbidden [48]. Themass and decay modes observed in experiments indicate that Ξ c (3080) is most likely to be the λ -mode 1 D excitation of Ξ c with J P = / + (i.e., | Ξ c D λλ + i ) [21, 41, 44]. Consid-ering Ξ c (3080) + as the | Ξ c D λλ + i state, we find that it hasrelatively large decay rates into the Σ ∗ c (2520) K and Ξ ′∗ c (2645) π (see Table VI). The partial width ratio between these two mainchannels is predicted to be Γ [ Ξ ∗ c (2645) π + ] Γ [ Σ c (2520) ++ K − ] ≃ . . (15)Combining it with the predicted partial width of Γ [ Ξ c (3080) + → Σ c (2455) ++ K − ] ∼ .
22 MeV, we esti-mate that Γ [ Ξ c (3080) + → Λ D + ] ≃ . Ξ c (3080) is estimated to be Γ ≃ . Σ c K and Σ ∗ c K channels Γ [ Σ ++ c (2520) K − ] Γ [ Σ c (2455) ++ K − ] ≃ . , (16)is about an order of magnitude larger than the observed ratiolisted in Eq. (13), and a similar phenomenon is found by Chen et al. within their P analysis [41]. It should be mentionedthat the measured ratio R = Γ [ Σ ++ c (2520) K − ] Γ [ Σ c (2455) ++ K − ] ≃ . ± .
27 of Ξ c (3080) may be strongly a ff ected by its nearby states, suchas Ξ c (3055). Thus, the measured ratio from Belle [123] maynot be a genuine ratio for Ξ c (3080).Furthermore, assigning Ξ c (3080) as the J P = / + state | Ξ c D λλ + i , we study its radiative decays. Our results arelisted in Table VII. It is found that the Ξ c (3080) should have arelatively large decay rate into Ξ c (2815) − γ . The partial de-cay width is predicted to be Γ [ Ξ c (3080) → Ξ c (2815) γ ] ≃ Ξ c (3080), wepredict the branching fraction B [ Ξ c (3055) → Ξ c (2815) γ ] ≃ Ξ c (3080) is most likely to be observedin the Ξ c (2815) γ channel if it corresponds to the J P = / + state | Ξ c D λλ + i indeed. D. Ξ b states In the Ξ b family, there are two λ -mode 1 D -wave excita-tions | Ξ b D λλ + i and | Ξ b D λλ + i . The typical masses ofthe λ -mode 1 D -wave Ξ b excitations are 6 . − . D -wave states arestudied with ChQM. Our results have been shown in Fig. 2.To be more specific, taking the masses of the 1 D -wave statesobtained in the relativistic quark-diquark picture [21], we givethe predicted widths in Table VI.
1. J P = / + state The J P = / + state | Ξ b D λλ + i might be a narrow statewith a width of a few MeV. It mainly decays into Ξ ′ b π , Ξ ′∗ b π and Σ b K channels. The partial widths of Ξ ′ b π , Ξ ′∗ b π are lesssensitive to the mass of | Ξ b D λλ + i ; however, the partialwidth for the Σ b K channel shows a significant linear depen-dence on the mass (see Fig. 2). If the mass of | Ξ b D λλ + i takes the predicted value ∼ .
37 GeV in Ref. [21], the branch-ing fractions for the main channels are predicted to be Γ [ Ξ ′ b π, Ξ ′∗ b π, Σ b K ] Γ total = , , . (17)The Ξ ′ b π and Σ b K decay channels may be ideal channels forour search for this missing 1 D -wave Ξ b baryon in future ex-periments.Furthermore, we study the radiative decays of | Ξ b D λλ + i into the 1 P -wave bottom baryon states. Our results are listedin Table VII. It is found that the charged state | Ξ − b D λλ + i might have a relatively large decay rate into Ξ − b ( − ) γ . Thepartial decay width can reach up to ∼
30 keV if the mass for | Ξ b D λλ + i is taken to be ∼ | Ξ b D λλ + i , we estimate the branching fraction B [ | Ξ − b D λλ + i → Ξ − b ( − ) γ ] ≃ O (10 − ).
2. J P = / + state The J P = / + state | Ξ b D λλ + i may be a narrow statewith a width comparable to the J P = / + state | Ξ b D λλ + i (i.e., a few MeV). The decays of | Ξ b D λλ + i are governedby Ξ ′∗ b π , which is less sensitive to its mass. If the mass of | Ξ b D λλ + i is taken to be ∼ .
37 GeV as the prediction inRef. [21], the decay channel Σ ∗ b K becomes important as well(see Table VI). In this case, the branching fractions for the Ξ ′ b π , Ξ ′∗ b π and Σ b K channels are predicted to be Γ [ Ξ ′ b π, Ξ ′∗ b π, Σ b K ] Γ total = , , . (18)To establish this missing 1 D -wave Ξ b baryon with J P = / + ,its dominant decay modes Ξ ′∗ b π and Σ b K are worth observingin future experiments.We also study the radiative decays of | Ξ b D λλ + i into the1 P -wave bottom baryon states. Our results are listed in Ta-ble VII as well. It is found that the charged state | Ξ − b D λλ + i might have a relatively large decay rate into Ξ − b ( − ) γ . Thepartial decay width can reach up to ∼
40 keV. If the massfor | Ξ b D λλ + i is taken to be ∼ | Ξ b D λλ + i , we estimate the branching fraction B [ | Ξ − b D λλ + i → Ξ − b ( − ) γ ] ≃ O (10 − ). IV. RESULTS FOR SINGLY HEAVY BARYONS OF F A. Σ c In the Σ c family, according to the quark model classifica-tion, there are six λ -mode 1 D -wave excitations: | Σ c D λλ + i , | Σ c D λλ + i , | Σ c D λλ + i , | Σ c D λλ + i , | Σ c D λλ + i , and | Σ c D λλ + i . However, no D -wave states have been estab-lished. The typical masses of the λ -mode 1 D -wave Σ c exci-tations are predicted to be ∼ . D -wave states as predicted in the relativistic quark-diquarkpicture [21], we further present the results in Table VIII.
1. J P = / + state The J P = / + state | Σ c D λλ + i might be a broad state. Ifits mass is taken as the prediction 3041 MeV in Ref. [21], thesum of the partial widths for the pionic and kaonic decays canreach up to Γ Sum ∼
160 MeV (see Table VIII). This state haslarge decay rates into Λ c (2595) π and Λ c (2625) π final states.The ratios between the partial decay widths for the Λ c (2595) π and Λ c (2625) π channels and Γ Sum are predicted to be Γ [ Λ c (2595) π ] Γ Sum ≃ , Γ [ Λ c (2625) π ] Γ Sum ≃ . (19)Both Λ c (2595) π and Λ c (2625) π may be ideal channels for oursearch for | Σ c D λλ + i in future experiments.We also estimate its radiative transitions into the 1 P -wavecharmed baryon states. Our results are listed in Table IX. It isfound that | Σ ++ (0) c D λλ + i might have relatively large decayrates into | Σ ++ (0) c P λ − i γ and | Σ ++ (0) c P λ − i γ , and their par-tial radiative decay widths are estimated to be O (10) − O (100)keV. The branching fractions for these main radiative decayprocesses may reach up to O (10 − ) − O (10 − ).
2. J P = / + states The J P = / + state | Σ c D λλ + i dominantly decays intothe P -wave states through the poinic decay modes Λ c (2625) π and | Σ c P λ − i π , while its decay rate into Σ c π is sizable. It hasa width of O (10) − O (100) MeV, which significantly dependson its mass. If the mass is taken as the prediction 3043 MeVin Ref. [21], the sum of the partial widths of the pionic decaysis Γ Sum ∼
100 MeV (see Table VIII), and the ratios betweenthe partial decay widths for the Λ c (2625) π and Σ c π channelsand Γ Sum are predicted to be Γ [ Λ c (2625) π ] Γ Sum ≃ , Γ [ Σ c π ] Γ Sum ≃ . (20)The Σ c π and Λ c (2625) π decay channels may be ideal channelsfor our search for | Σ c D λλ + i in future experiments.For the other J P = / + state | Σ c D λλ + i , one finds thatit has large decay rates into the P -wave states through thepionic decay modes Λ c (2625) π , | Σ c P λ − i π , | Σ c P λ − i π ,and | Σ c P λ − i π . Its width should be broader than that of | Σ c D λλ + i . Furthermore, the decay rate into Σ c (2520) π issizable as well. If its mass is taken as the prediction 3040MeV in Ref. [21], the sum of the partial widths of the pionic0 c 2 D 3/2 + c cc cc 2
D 5/2 + c ( P 3/2 - ) c ( P 5/2 - ) c (2625)3/2 -c 4 D 1/2 + c ( P 3/2 - ) c 4 D 3/2 + c (2595)1/2 - c ( P 1/2 - ) c 4 D 5/2 + e V ( M e V ) c 4 D 7/2 + M (MeV) M (MeV)M (MeV)
FIG. 3: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Σ c states as functions of their mass. The boldsolid curves stand for the sum of the partial widths. The masses of the P -wave heavy baryons in final states are adopted from the quark modelpredictions in Ref. [21] (see Table II). decays can reach up to Γ Sum ∼
120 MeV (see Table VIII),while the ratios between the partial widths of Λ c (2625) π and Σ c (2520) π and Γ Sum are predicted to be Γ [ Λ c (2625) π ] Γ Sum ≃ , Γ [ Σ c (2520) π ] Γ Sum ≃ . (21)The Λ c (2625) π and Σ c (2520) π may be ideal channels for oursearch for | Σ c D λλ + i in future experiments.We also estimate the radiative transitions of these J P = / + states into the 1 P -wave states. Our results are listed inTable IX. It is found that | Σ ++ (0) c D λλ + i might have relativelylarge decay rates into | Σ ++ (0) c P λ − i γ , while | Σ + c D λλ + i might have a relatively large decay rates into Λ c (2595) + γ ,and their partial radiative decay widths are estimated to be O (10) − O (100) keV. The branching fractions for these mainradiative decay processes may reach up to O (10 − ) − O (10 − ).The | Σ + c D λλ + i may have the possibility of being observedin the Λ c (2595) + γ channel.
3. J P = / + states The J P = / + state | Σ c D λλ + i might be a narrow statewith a width of O (10) MeV (see Fig. 3). It has large decayrates into Λ c π , Σ c π , Λ c (2595) π , and Λ c (2625) π with compara-ble partial decay widths. If its mass is taken as the prediction 3038 MeV in Ref. [21], the sum of the partial widths of the pi-onic decays is about Γ Sum ∼
60 MeV (see Table VIII), and theratios between the partial widths for the main decay modes, Λ c π , Σ c π , Λ c (2595) π , and Λ c (2625) π , and Γ Sum are predictedto be Γ [ Λ c π, Σ c π, Λ c (2595) π, Λ c (2625) π ] Γ Sum ≃ , , , . (22)The Λ c π , Σ c π , Λ c (2595) π , and Λ c (2625) π decay channels maybe ideal channels for our search for the missing J P = / + state | Σ c D λλ + i .The other J P = / + state | Σ c D λλ + i might also be a nar-row state with a width of O (10) MeV. This state has relativelylarge decay rates into Σ c (2520) π , Λ c (2595) π , | Σ c P λ − i π , and | Σ c P λ − i π . If its mass is taken as the prediction 3023 MeVin Ref. [21], the sum of the partial widths of the pionic decaysis about Γ Sum ∼
70 MeV (see Table VIII), while the ratios be-tween the partial widths for the main decay modes, Σ c (2520) π and Λ c (2595) π , and Γ Sum are predicted to be Γ [ Σ c (2520) π, Λ c (2595) π ] Γ Sum ≃ , . (23)The Σ c (2520) π and Λ c (2595) π may be ideal channels for asearch for | Σ c D λλ + i in future experiments.1 M e V b (5920)3/2 - b ( P 3/2 - ) b 2 D 3/2 + b bb b 2 D 5/2 + b b b - b 4 D 1/2 + b M e V M (MeV) b ( P 1/2 - ) b 4 D 3/2 + b b M (MeV) b 4
D 5/2 + b ( P 5/2 - ) b M (MeV) b 4
D 7/2 + FIG. 4: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Σ b states as functions of their mass. The boldsolid curves stand for the sum of the partial widths. The masses of the P -wave heavy baryons in final states are adopted from the quark modelpredictions in Ref. [21], if there are no observations (see Table II). The radiative transitions of these J P = / + states intothe 1 P -wave charmed baryon states are estimated as well.Our results are listed in Table IX. It is found that for the | Σ c D λλ + i state the main radiative decay processes are | Σ ++ (0) c D λλ + i → | Σ ++ (0) c P λ − i γ , | Σ ++ (0) c P λ − i γ , and | Σ + c D λλ + i → Λ c (2595 , + γ , while for the other J P = / + state | Σ c D λλ + i , the main radiative decay processesare | Σ ++ (0) c D λλ + i → | Σ ++ (0) c P λ − i γ , | Σ ++ (0) c P λ − i γ , and | Σ + c D λλ + i → Λ c (2595 , + γ . The partial radiative de-cay widths for these processes are estimated to be O (10) keV,while the branching fractions may be O (10 − ). These radiativetransitions may be hard to observe in experiments.
4. J P = / + state The J P = / + state | Σ c D λλ + i might have a relativelynarrow width of O (10) MeV. It mainly decays into Λ c π , Λ c (2595) π , and Λ c (2625) π channels. If one adopts the pre-dicted mass 3013 MeV in Ref. [21], the sum of the partialwidths of the pionic decays is estimated to be Γ Sum ∼ Λ c π , Λ c (2595) π , and Λ c (2625) π , and Γ Sum are predicted to be Γ [ Λ c π, Λ c (2595) π, Λ c (2625) π ] Γ Sum ≃ , , . (24)The Λ c π , Λ c (2595) π , and Λ c (2625) π may be ideal channelsfor a search for | Σ c D λλ + i in future experiments.The radiative transitions into the 1 P -wave states are esti-mated as well. Our results are listed in Table IX. It is foundthat the main radiative decay processes are | Σ ++ (0) c D λλ + i →| Σ ++ (0) c P λ − i γ and | Σ + c D λλ + i → Λ c (2595) + γ . Their partialradiative decay widths are estimated to be O (10) keV, whilethe branching fractions are O (10 − ). These radiative transi-tions may be hard to observe in experiments. B. Σ b In the Σ b family, according to the quark model classifica-tion, there are six λ -mode 1 D -wave excitations: | Σ b D λλ + i , | Σ b D λλ + i , | Σ b D λλ + i , | Σ b D λλ + i , | Σ b D λλ + i , and | Σ b D λλ + i . However, no 1 D -wave states have been estab-lished. The typical masses of the λ -mode 1 D -wave Σ b exci-tations are predicted to be ∼ . TABLE VIII: Strong decay partial widths of the main decay modes for the λ -mode D -wave Σ c and Σ b baryons, the masses (MeV) of whichare taken from the quark model predictions of Ref. [21]. M f stands for the masses of P -wave heavy baryons (MeV) in the final states, whichare adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Σ c D λλ + i (3043) | Σ c D λλ + i (3038) | Σ c D λλ + i (3041) | Σ c D λλ + i (3040) | Σ c D λλ + i (3023) | Σ c D λλ + i (3013) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Λ c π + . − . − . + . + . − . + . − . − . + . − . + . Σ c π + . − . − . + . + . − . + . − . − . + . − . + . Σ ∗ c π − . + . + . − . + . − . + . − . + . − . + . − . Ξ c K + . − . + . − . + . − . < | Λ c P λ − i π + . − . − . + . − . + . + . − . − . + . + . − . | Λ c P λ − i π + . − . + . − . + . − . + . − . + . < + . + . − . | Σ c P λ − i π + . − . − . + . − . + . + . − . − . + . + . − . | Σ c P λ − i π + . − . − . + . + . − . + . − . + . − . | Σ c P λ − i π + . − . − . + . + . − . + . − . < | Σ c P λ − i π + . − . + . − . + . + . + . − . + . − . − . + . | Σ c P λ − i π + . < + . + . − . + . − . + . − . + . − . − . + . Sum 105 + . − . − . + . + . − . + . − . + . − . + . − . Decay mode M f | Σ b D λλ + i (6326) | Σ b D λλ + i (6284) | Σ b D λλ + i (6311) | Σ b D λλ + i (6285) | Σ b D λλ + i (6270) | Σ b D λλ + i (6260) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Λ b π + . − . − . + . + . − . + . − . − . + . − . + . Σ b π + . − . − . + . + . − . + . − . − . + . − . + . Σ ∗ b π − . + . + . − . + . − . + . − . + . − . + . − . Ξ b K + . − . · · · + . − . · · · · · · · · ·| Λ b P λ − i π + . − . − . + . − . + . + . − . − . + . + . − . | Λ b P λ − i π + . − . − . + . + . − . + . − . − . + . + . − . | Σ b P λ − i π + . − . − . + . − . + . + . − . + . − . | Σ b P λ − i π + . − . + . − . + . − . + . − . < + . − . | Σ b P λ − i π − . + . + . − . − . + . + . − . + . − . < | Σ b P λ − i π + . − . + . − . + . − . + . − . + . − . + . < − . | Σ b P λ − i π + . + . + . − . + . − . + . − . + . − . + . − . Sum 121.4 + . − . − . + . + . − . + . − . + . − . + . − . D -wave states obtained in the relativistic quark-diquark pic-ture [21], we present the results in Table VIII.
1. J P = / + state The J P = / + state | Σ b D λλ + i might be a broad statewith a width of O (100) MeV. If its mass is taken as the pre-diction in Ref. [21], the sum of the partial widths for thepionic decays can reach up to Γ Sum ∼
150 MeV (see Ta-ble VIII). This state might mainly decay into the P -wave Λ b states Λ b (5912) − and Λ b (5920) − via pionic decay modes Λ b (5912) π and Λ b (5920) π . The ratios between the partial de-cay widths for the Λ b (5912) π and Λ b (5920) π channels and Γ Sum are predicted to be Γ [ Λ b (5912) π ] Γ Sum ≃ , Γ [ Λ b (5920) π ] Γ Sum ≃ . (25) Both Λ b (5912) π and Λ b (5920) π may be ideal channels for oursearch for | Σ b D λλ + i in future experiments.We also estimate its radiative transitions. Our results arelisted in Table IX. It is found that | Σ + b D λλ + i might haverelatively large decay rates into | Σ + b P λ − i γ and | Σ + b P λ − i γ ,and their partial radiative decay widths are estimated to be O (100) keV. The branching fractions for these main radiativedecay processes may reach up to O (10 − ).
2. J P = / + states The J P = / + state | Σ b D λλ + i has large decay ratesinto the P -wave states through the pionic decay modes Λ b (5920) − π and | Σ b P λ − i π . It has a width of O (10) −O (100) MeV, which obviously depends on its mass (seeFig. 4). If the mass is taken as the prediction 6326 MeV inRef. [21], the sum of the partial widths of the pionic decayscan reach up to Γ Sum ∼
120 MeV (see Table VIII), and the ra-3
TABLE IX: Partial widths of radiative decays for the λ -mode D -wave Σ c and Σ b baryons, the masses (MeV) of which are taken from the quarkmodel predictions of Ref. [21]. M f stands for the masses of P -wave heavy baryons (MeV) in the final states, which are adopted from theRPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Σ c D λλ + i (3043) | Σ c D λλ + i (3038) | Σ c D λλ + i (3041) | Σ c D λλ + i (3040) | Σ c D λλ + i (3023) | Σ c D λλ + i (3013) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) | Σ ++ c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ + c P λ − i γ − . − . − . + . − . + . − . + . − . + . | Σ c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ ++ c P λ − i γ + . − . − . + . − . + . − . + . − . + . − . + . | Σ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ ++ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ + c P λ − i γ − . + . − . + . − . + . − . + . | Σ c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ ++ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ ++ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Λ + c P λ − i γ − . + . − . − . − . + . − . + . − . + . − . + . | Λ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . Decay mode M f | Σ b D λλ + i (6326) | Σ b D λλ + i (6284) | Σ b D λλ + i (6311) | Σ b D λλ + i (6285) | Σ b D λλ + i (6270) | Σ b D λλ + i (6260) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) | Σ + b P λ − i γ − . + . − . + . − . + . − . + . − . + . < . | Σ b P λ − i γ − . + . − . + . − . + . − . + . < . | Σ − b P λ − i γ − . + . − . + . − . + . − . + . − . + . < . | Σ + b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ b P λ − i γ − . + . − . + . − . + . | Σ − b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ + b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ − b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Σ + b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ − b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ + b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Σ − b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Λ b P λ − i γ + . − . + . − . − . + . − . + . − . + . − . + . | Λ b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . tio between the partial decay width for the Λ b (5920) π channeland Γ Sum is predicted to be Γ [ Λ b (5920) π ] Γ Sum ≃ . (26)The Λ b (5920) π decay channel may be an ideal channel for oursearch for | Σ b D λλ + i in future experiments.For the other J P = / + state | Σ b D λλ + i , one finds that this state mainly decays into the P -wave states through the pi-onic decay modes Λ b (5920) π , | Σ b P λ − i π , and | Σ b P λ − i π .Furthermore, the decay rate into Σ ∗ b (5832) π is sizable as well.Its width should be narrower than that of | Σ b D λλ + i . If itsmass is taken as the predictions in Ref. [21], the sum of thepartial widths of the pionic decays can reach up to Γ Sum ∼ Γ [ Σ ∗ b (5832) π ] and Γ [ Λ b (5920) π ] and Γ Sum are pre-4 c (2815)3/2 - ccc D + c (2625)3/2 - c ’ ( P 3/2 - ) cc D + cc D + c (2790)1/2 - cc D + c ’ ( P 5/2 - ) c ( P 1/2 - ) ccc D + c M (MeV) ( M e V ) c (2595)1/2 - c D + M (MeV) M (MeV) ( M e V ) FIG. 5: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Ξ ′ c states as functions of their mass. The boldsolid curves stand for the sum of the partial widths. The masses of the P -wave heavy baryons in final states are adopted from the quark modelpredictions in Ref. [21], if there are no observations (see Table II). dicted to be Γ [ Σ ∗ b (5832) π ] Γ Sum ≃ , Γ [ Λ b (5920) π ] Γ Sum ≃ . (27)Both Σ ∗ b (5832) π and Λ b (5920) π may be ideal channels for asearch for | Σ b D λλ + i in future experiments.The radiative decays of the J P = / + states into the 1 P -wave bottom baryon states are estimated as well. Our resultsare listed in Table IX. It is found that | Σ + b D λλ + i might haverelatively large decay rates into | Σ + b P λ − i γ and | Σ + b P λ − i γ ,while | Σ + b D λλ + i might have a relatively large decay rateinto | Σ + b P λ − i γ . Their partial radiative decay widths are es-timated to be O (100) keV, while the branching fractions mayreach up to O (10 − ).
3. J P = / + states The | Σ b D λλ + i might be a relatively narrow state with awidth of O (10) MeV. It has large decay rates into Λ b π , Σ b π , Λ b (5912) π , and Λ b (5920) π . If its mass is taken as the predic-tion 6284 MeV in Ref. [21], the sum of the partial widths ofthe pionic decays is about Γ Sum ∼
50 MeV (see Table VIII),and the ratios between the partial decay widths for these mainchannels, Λ b π , Σ b π , Λ b (5912) π , and Λ b (5920) π , and Γ Sum are predicted to be Γ [ Λ b π, Σ b π, Λ b (5912) π, Λ b (5920) π ] Γ Sum ≃ , , , . (28)The Λ b π , Σ b π , Λ b (5912) π , and Λ b (5920) π decay channelsmay be ideal channels for our search for | Σ b D λλ + i in fu-ture experiments.For the other J P = / + state | Σ b D λλ + i , it has a width of O (10) −O (100) MeV, which significantly depends on the mass.This state has large decay rates into Λ b π , Σ ∗ b π , Λ b (5912) π , | Σ b P λ − i π , and | Σ b P λ − i π . If its mass is taken as the pre-diction 6270 MeV in Ref. [21], the sum of the partial widths ofthe pionic decays is about Γ Sum ∼
50 MeV (see Table VIII),and the ratios between the partial decay widths for the Λ b π , Σ ∗ b π , and Λ b (5912) π final states and Γ Sum are predicted to be Γ [ Λ b π, Σ ∗ b π, Λ b (5912) π ] Γ Sum ≃ , , . (29)The Λ b π , Σ ∗ b π , and Λ b (5912) π decay channels might be idealchannels for our search for | Σ b D λλ + i in future experiments.The radiative decays of these J P = / + states intothe 1 P -wave bottom baryon states are estimated as well.Our results are listed in Table IX. It is found that forthe | Σ b D λλ + i state the main radiative decay processes5 bb bb D + b bbb D + bb M (MeV)M (MeV)M (MeV) b D + M e V M e V bb D + bb D + b (5912) K b D + b (5920) b ( P 3/2 - ) b ( P 5/2 - ) b ( P 1/2 - FIG. 6: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Ξ ′ b states as functions of their mass. The boldsolid curves stand for the sum of the partial widths. The masses of the P -wave heavy baryons in final states are adopted from the quark modelpredictions in Ref. [21], if there are no observations (see Table II). are | Σ + b D λλ + i → | Σ + b P λ − i γ , | Σ + b P λ − i γ , and | Σ b D λλ + i → Λ b (5912 , + γ , while for the other J P = / + state | Σ b D λλ + i , the main radiative decay pro-cesses are | Σ + b D λλ + i → | Σ + b P λ − i γ , | Σ + b P λ − i γ , and | Σ b D λλ + i → Λ b (5912 , + γ . The partial radiative de-cay widths for these processes are estimated to be O (10) keV,while the branching fractions may be O (10 − ) − O (10 − ).
4. J P = / + state The J P = / + state | Σ b D λλ + i might have a width of O (10) − O (100) MeV, which strongly depends on its mass (seeFig. 4). This state has large decay rates into Λ b π , Λ b (5912) π ,and Λ b (5920) π channels. If one adopts the predicted mass6260 MeV in Ref. [21], the sum of the partial widths of thepionic decays is estimated to be Γ Sum ∼
50 MeV (see Ta-ble VIII), and the ratios between the partial decay widths forthese main channels, Λ b π , Λ b (5912) π , and Λ b (5920) π , and Γ Sum are predicted to be Γ [ Λ b π, Λ b (5912) π, Λ b (5920) π ] Γ Sum ≃ , , . (30)The Λ b π , Λ b (5912) π , and Λ b (5920) π decay channels may beideal channels for our search for | Σ b D λλ + i in future exper-iments. We also estimate its radiative decays into the 1 P -wavebottom baryon states. Our results are listed in Table IX.It is found that the main radiative decay processes are | Σ + b D λλ + i → | Σ + b P λ − i γ and | Σ b D λλ + i → Λ b (5920) γ .Their partial radiative decay widths are estimated to be O (10)keV, while the branching fractions may be O (10 − ) − O (10 − ).The neutral J P = / + state | Σ b D λλ + i may have the possi-bility of being observed in the Λ b (5920) γ channel. C. Ξ ′ c In the Ξ ′ c family, according to the quark model classifica-tion, there are six λ -mode 1 D -wave excitations: | Ξ ′ c D λλ + i , | Ξ ′ c D λλ + i , | Ξ ′ c D λλ + i , | Ξ ′ c D λλ + i , | Ξ ′ c D λλ + i , and | Ξ ′ c D λλ + i . However, no 1 D -wave states have been estab-lished. The typical masses of the λ -mode 1 D -wave Ξ ′ c excita-tions are ∼ .
14 within various quark model predictions (seeTable II).In Ref. [48], the strong decay properties of the D -wave ex-cited Ξ ′ c states were studied in their possible mass ranges.However, we did not give correct predictions of the partialwidths of the D -wave excited Ξ ′ c states decaying into the P -wave charmed baryons. In this work, we update our predic-tions, which have been shown in Fig. 5. To be more specific,taking the masses of the 1 D -wave states predicted within the6 ( M e V ) c D 1/2 + c D 3/2 + c K c ’ K c ’*(2645) K c (2790) K ( M e V ) c D 3/2 + ( M e V ) c D 5/2 + c D 5/2 + M (MeV) c D 7/2 + M (MeV) 010203001020 6480 6540 66000102030 6480 6540 6600 ( M e V ) ( M e V ) b D 1/2 + b D 3/2 +b D 3/2 + ( M e V ) b D 5/2 +b D 5/2 + b b b * M (MeV) b D 7/2 + M (MeV)
FIG. 7: Strong decay partial widths of the main decay modes for the λ -mode 1 D -wave excited Ω c and Ω b states as functions of their mass.The bold solid curves stand for the total widths. relativistic quark-diquark picture [21], we give our predictionsin Table X.
1. J P = / + state If the mass of the J P = / + state | Ξ ′ c D λλ + i is less than3.12 GeV, this state should be a rather narrow state with awidth of ∼
10 MeV; its main decay modes are Ξ c π , Λ c K , and Σ c K . However, if the mass of | Ξ ′ c D λλ + i is taken as theprediction 3163 MeV in Ref. [21], the decay channels intothe P -wave charmed baryon final states should open, and thedecay channels Λ c (2625) K and Ξ c (2815) π will play dominantroles. In this case, the J P = / + state | Ξ ′ c D λλ + i might be abroad state, and the sum of the partial widths for the pionic andkaonic decays can reach up to Γ Sum ∼
120 MeV (see Table X),while the ratios between the partial widths for the Λ c (2625) K and Ξ c (2815) π channels and Γ Sum are predicted to be Γ [ Λ c (2625) K ] Γ Sum ≃ . , Γ [ Ξ c (2815) π ] Γ Sum ≃ . . (31)The decay channels Λ c (2625) K and Ξ c (2815) π may be idealchannels for our search for | Ξ ′ c D λλ + i in future experiments.We also estimate its radiative decays into the 1 P -wave bot-tom baryon states. Our results are listed in Table XI. It is found that | Ξ ′ c D λλ + i might have relatively large decay ratesinto | Ξ ′ c P λ − i γ , and the partial radiative decay width is esti-mated to be O (100) keV. The branching fractions for this mainradiative decay process may reach up to O (10 − ).
2. J P = / + states For | Ξ ′ c D λλ + i , if its mass is less than 3.12 GeV, it shouldbe a rather narrow state with a width of ∼
10 MeV, and itsmain decay modes are Ξ c π , Ξ ′ c π , and Σ c K . However, if themass of | Ξ ′ c D λλ + i is taken as the prediction 3167 MeV inRef. [21], the decay channels into the P -wave charmed baryonfinal states should open, and the decay channels Λ c (2625) K and Ξ c (2815) π will play dominant roles. In this case, the J P = / + state | Ξ ′ c D λλ + i might be a broad state, and the sum ofthe partial widths for the pionic and kaonic decays can reachup to Γ Sum ∼
90 MeV (see Table X), while the ratios betweenthe partial widths for the Λ c (2625) K and Ξ c (2815) π channelsand Γ Sum are predicted to be Γ [ Λ c (2625) K ] Γ Sum ≃ , Γ [ Ξ c (2815) π ] Γ Sum ≃ . (32)The Λ c (2625) K and Ξ c (2815) π decay channels may be idealchannels for our search for | Ξ ′ c D λλ + i in future experiments.7 TABLE X: Partial widths of strong decays for the λ -mode D -wave Ξ ′ c and Ξ ′ b baryons, the masses (MeV) of which are taken from the quarkmodel predictions of Ref. [21]. M f stands for the masses of P -wave heavy baryons (MeV) in the final states, which are adopted from theRPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Ξ ′ c D λλ + i (3167) | Ξ ′ c D λλ + i (3166) | Ξ ′ c D λλ + i (3163) | Ξ ′ c D λλ + i (3160) | Ξ ′ c D λλ + i (3153) | Ξ ′ c D λλ + i (3147) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Ξ c π + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′ c π + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′∗ c π − . + . + . − . + . − . + . − . + . − . + . − . Λ c K + . − . − . + . + . − . + . − . − . + . − . + . Σ c K + . − . − . + . + . − . + . − . − . + . − . + . Σ ∗ c K − . + . + . − . + . − . + . − . + . − . + . − . | Λ c P λ − i K + . − . + . − . − . + . + . − . − . + . + . − . | Λ c P λ − i K + . − . + . − . + . − . + . − . + . − . + . − . | Ξ c P λ − i π + . − . + . − . − . + . + . − . − . + . + . − . | Ξ c P λ − i π + . − . − . + . + . − . + . − . + . − . + . − . | Ξ ′ c P λ − i π + . − . − . + . − . < + . + . − . + . − . | Ξ ′ c P λ − i π + . − . − . + . + . − . + . − . < + . − . | Ξ ′ c P λ − i π − . + . + . − . − . + . + . − . + . − . < | Ξ ′ c P λ − i π + . − . + . − . + . − . + . − . + . − . < − . < + . | Ξ ′ c P λ − i π + . − . + . − . + . − . + . − . + . − . < − . + . Sum 86.4 + . − . − . + . + . − . + . − . + . − . + . − . Decay mode M f | Ξ ′ b D λλ + i (6459) | Ξ ′ b D λλ + i (6432) | Ξ ′ b D λλ + i (6447) | Ξ ′ b D λλ + i (6431) | Ξ ′ b D λλ + i (6420) | Ξ ′ b D λλ + i (6414) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Γ i (MeV) Ξ b π + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′ b π + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′∗ b π − . + . + . − . + . − . + . − . + . − . + . − . Λ b K + . − . − . + . + . − . + . − . − . + . − . + . Σ b K + . − . − . + . + . − . + . − . − . + . Σ ∗ b K − . + . + . − . + . − . + . − . + . − . + . − . | Λ b P λ − i K + . − . − . + . + . − . + . − . | Λ b P λ − i K + . − . + . − . + . − . + . − . · · ·| Ξ b P λ − i π + . − . + . − . − . + . + . − . − . + . + . − . | Ξ b P λ − i π + . − . − . + . + . − . + . − . + . < − . + . < − . | Ξ ′ b P λ − i π + . − . − . + . − . + . + . − . + . − . | Ξ ′ b P λ − i π + . − . + . − . + . − . + . − . + . − . | Ξ ′ b P λ − i π + . − . − . + . + . − . + . − . < | Ξ ′ b P λ − i π + . − . + . − . + . − . + . − . + . − . < + . < − . | Ξ ′ b P λ − i π + . < − . + . − . + . − . + . − . + . − . + . < − . Sum 100.8 + . − . − . + . + . − . + . − . + . − . + . − . For the other J P = / + state | Ξ ′ c D λλ + i , if its mass isless than 3.12 GeV, it should be a rather narrow state with awidth of ∼
10 MeV as well, and its main decay modes are Ξ c π , Ξ ′∗ c π , and Σ ∗ c K . However, if the mass of | Ξ ′ c D λλ + i istaken as the prediction 3160 MeV in Ref. [21], the decay chan-nels into the P -wave charmed baryon final states should open,and the decay channels Λ c (2625) K , Ξ c (2790) π , Ξ c (2815) π ,and | Ξ ′ c P λ − i π will dominate the decays of | Ξ ′ c D λλ + i .In this case, the sum of the partial widths for the pionic andkaonic decays can reach up to Γ Sum ∼
80 MeV (see Table X), while the ratios between the partial widths for the Λ c (2625) K , Ξ c (2790) π , and Ξ c (2815) π channels and Γ Sum are predicted tobe Γ [ Λ c (2625) K , Ξ c (2790) π, Ξ c (2815) π ] Γ Sum ≃ , , . (33)The Λ c (2625) K , Ξ c (2790) π , and Ξ c (2815) π decay channelsmay be ideal channels for a search for | Ξ ′ c D λλ + i in futureexperiments.The radiative decays of these J P = / + states into the 1 P -wave charmed baryon states are also estimated. Our results8 TABLE XI: Partial widths of radiative decays for the λ -mode D -wave Ξ ′ c and Ξ ′ b baryons, the masses (MeV) of which are taken from the quarkmodel predictions of Ref. [21]. M f stands for the masses of P -wave heavy baryons (MeV) in the final states, which are adopted from theRPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Ξ ′ c D λλ + i (3167) | Ξ ′ c D λλ + i (3166) | Ξ ′ c D λλ + i (3163) | Ξ ′ c D λλ + i (3160) | Ξ ′ c D λλ + i (3153) | Ξ ′ c D λλ + i (3147) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) | Ξ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ c P λ − i γ | Ξ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Ξ c P λ − i γ | Ξ ′ + c P λ − i γ − . + . − . + . − . + . − . + . < . | Ξ ′ c P λ − i γ − . + . − . + . − . + . − . + . < . | Ξ ′ + c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ ′ c P λ − i γ − . + . − . + . − . + . − . + . | Ξ ′ + c P λ − i γ − . + . < .
01 0.54 − . + . < . | Ξ ′ c P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ ′ + c P λ − i γ − . + . − . + . − . + . | Ξ ′ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Ξ ′ + c P λ − i γ − . + . − . + . − . + . | Ξ ′ c P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . Decay mode M f | Ξ ′ b D λλ + i (6459) | Ξ ′ b D λλ + i (6432) | Ξ ′ b D λλ + i (6447) | Ξ ′ b D λλ + i (6431) | Ξ ′ b D λλ + i (6420) | Ξ ′ b D λλ + i (6414) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) Γ i (keV) | Ξ b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ − b P λ − i γ | Ξ b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Ξ − b P λ − i γ | Ξ ′ b P λ − i γ − . + . − . + . − . + . < . | Ξ ′− b P λ − i γ − . + . − . + . − . + . − . + . − . + . < . | Ξ ′ b P λ − i γ − . + . − . + . − . + . | Ξ ′− b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ ′ b P λ − i γ − . + . < .
01 16.7 − . + . − . + . − . + . − . + . | Ξ ′− b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ ′ b P λ − i γ − . + . − . + . − . + . − . + . − . + . | Ξ ′− b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Ξ ′ b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . | Ξ ′− b P λ − i γ − . + . − . + . − . + . − . + . − . + . − . + . are listed in Table XI. It is found that for the | Ξ ′ c D λλ + i state the main radiative decay processes are | Ξ ′ + c D λλ + i → Ξ + c (2815) γ , | Ξ ′ c D λλ + i → | Ξ ′ c P λ − i γ, | Ξ ′ c P λ − i γ ,while for the | Ξ ′ c D λλ + i state, the main radiative decayprocesses are | Ξ ′ + c D λλ + i → Ξ + c (2790) γ , | Ξ ′ c D λλ + i →| Ξ ′ c P λ − i γ, | Ξ ′ c P λ − i γ . Their partial radiative decaywidths are estimated to be O (10) keV, while the branchingfractions may reach up to O (10 − ) − O (10 − ).
3. J P = / + states For | Ξ ′ c D λλ + i , if its mass is less than 3.12 GeV, it shouldbe a rather narrow state with a width of ∼
10 MeV, and its main decay modes are Ξ c π , Ξ ′ c π . However, if the mass of | Ξ ′ c D λλ + i is taken as the prediction 3166 MeV in Ref. [21],the decay channels into the P -wave charmed baryon finalstates should open, and the Ξ c (2815) π decay mode togetherwith Ξ c π and Ξ ′ c π dominates its decays. In this case, the widthof | Ξ ′ c D λλ + i might be still fairly narrow, and the sum of thepartial widths for the pionic and kaonic decays is Γ Sum ∼ Ξ c π , Ξ ′ c π and Ξ c (2815) π channels and Γ Sum are pre-dicted to be Γ [ Ξ c π, Ξ ′ c π, Ξ c (2815) π ] Γ Sum ≃ , , . (34)The Ξ c π , Ξ ′ c π , and Ξ c (2815) π decay modes may be idealmodes for our search for | Ξ ′ c D λλ + i in future experiments.9For the other J P = / + state | Ξ ′ c D λλ + i , if its mass is lessthan 3.12 GeV, it should be a rather narrow states with a widthof ∼
10 MeV as well, its main decay modes are Ξ ′∗ c π and Σ ∗ c K .However, if the mass of | Ξ ′ c D λλ + i is taken as the prediction3153 MeV in Ref. [21], the decay channels into the P -wavecharmed baryon final states should open, and the Ξ c (2790) π , | Ξ ′ c P λ − i π , and | Ξ ′ c P λ − i π decay modes together with Ξ ′∗ c π and Σ ∗ c K will dominate the decays of | Ξ ′ c D λλ + i . In this case,the sum of the partial widths for the pionic and kaonic decayscan reach up to Γ Sum ∼
40 MeV (see Table X), while the ratiosbetween the partial widths for the Ξ ′∗ c π , Σ ∗ c K , and Ξ c (2790) π channels and Γ Sum are predicted to be Γ [ Ξ ′∗ c π, Σ ∗ c K , Ξ c (2790) π ] Γ Sum ≃ , , . (35)The decay modes Ξ ′∗ c π , Σ ∗ c K , and Ξ c (2790) π may be idealmodes for our search for | Ξ ′ c D λλ + i in future experiments.The radiative decays of these J P = / + states into the 1 P -wave charmed baryon states are also estimated. Our resultsare listed in Table XI. It is found that for the | Ξ ′ c D λλ + i state the main radiative decay processes are | Ξ ′ + c D λλ + i → Ξ + c (2790 , γ , | Ξ ′ c D λλ + i → | Ξ ′ c P λ − i γ , while forthe | Ξ ′ c D λλ + i state, the main radiative decay processesare | Ξ ′ + c D λλ + i → Ξ + c (2790 , γ , | Ξ ′ c D λλ + i →| Ξ ′ c P λ − i γ, | Ξ ′ c P λ − i γ, | Ξ ′ c P λ − i γ . Their partial radia-tive decay widths are estimated to be O (10) keV, while thebranching fractions may reach up to O (10 − ) − O (10 − ).
4. J P = / + state If the mass of the J P = / + state | Ξ ′ c D λλ + i is less than3.12 GeV, it should be a rather narrow state with a widthof a few MeV, and its main decay modes are Ξ c π and Λ c K .However, if the mass of | Ξ ′ c D λλ + i is taken to be 3147MeV as predicted in Ref. [21], the decay channels into the P -wave charmed baryon final states, Λ c (2595) K , Ξ c (2790) π ,and Ξ c (2815) π , become dominant. In this case, the sum of thepartial widths for the pionic and kaonic decays can reach upto Γ Sum ∼
50 MeV (see Table X), while the ratios between thepartial widths for the main channels and Γ Sum are predicted tobe Γ [ Ξ c π, Λ c (2595) K , Ξ c (2790) π, Ξ c (2815) π ] Γ Sum ≃ , , , . (36)The Ξ c π , Λ c K , Λ c (2595) K , Ξ c (2790) π , and Ξ c (2815) π decaymodes may be ideal modes for our search for | Ξ ′ c D λλ + i infuture experiments.The radiative decays of | Ξ ′ c D λλ + i into the 1 P -wavecharmed baryon states are also estimated. Our results arelisted in Table XI. It is found that the main radiative decayprocesses are | Ξ ′ + c D λλ + i → Ξ + c (2815) γ , | Ξ ′ c D λλ + i →| Ξ ′ c P λ − i γ . Their partial radiative decay widths are esti-mated to be O (10) keV, while the branching fractions may be O (10 − ). D. Ξ ′ b In the Ξ ′ b family, there are six λ -mode 1 D -wave excita-tions: | Ξ ′ b D λλ + i , | Ξ ′ b D λλ + i , | Ξ ′ b D λλ + i , | Ξ ′ b D λλ + i , | Ξ ′ b D λλ + i and | Ξ ′ b D λλ + i . However, no 1 D -wave stateshave been established. The typical masses of the λ -mode 1 D -wave Ξ ′ b excitations are ∼ .
44 GeV within various quarkmodel predictions (see Table II). In the possible mass ranges,we study their strong decay transitions by emitting one lightpseudoscalar meson within the ChQM. Our results are shownin Fig. 6. To be more specific, taking the masses of the1 D -wave states obtained in the relativistic quark-diquark pic-ture [21], we give the predicted widths in Table X.
1. J P = / + state The J P = / + state | Ξ ′ b D λλ + i might be a broad statewith a width of O (100) MeV, which obviously depends on itsmass. This state might mainly decay into the P -wave bottombaryons via pionic decay modes | Ξ b P λ − i π and | Ξ b P λ − i π and kaonic decay mode Λ b (5920) K . If its mass is taken asthe prediction 6447 MeV in Ref. [21], the sum of the par-tial widths for the pionic and kaonic decays can reach up to Γ Sum ∼
110 MeV (see Table VIII), while the ratio between thepartial width for the Λ b (5920) K channel and Γ Sum is predictedto be Γ [ Λ b (5920) K ] Γ Sum ≃ . (37)The Λ b (5920) K decay channel may be an ideal channel forour search for | Ξ ′ b D λλ + i in future experiments.We also estimate its radiative decays into the 1 P -wave bot-tom baryon states. Our results are listed in Table XI. It isfound that | Ξ ′− b D λλ + i might have relatively large decayrates into | Ξ ′− b P λ − i γ and | Ξ ′− b P λ − i γ , and the partial ra-diative decay width is estimated to be O (10) keV. The branch-ing fractions for these main radiative decay processes may be O (10 − ).
2. J P = / + states The J P = / + state | Ξ ′ b D λλ + i has a width of O (10) − O (100) MeV, which significantly depends on the mass. Itmight mainly decay into the P -wave bottom baryons via the | Ξ b P λ − i π and Λ b (5920) K decay modes. The decay rateinto Σ b K is also sizeable. If its mass is taken as the predic-tion in Ref. [21], the sum of the partial widths for the pionicand kaonic decays can reach up to Γ Sum ∼
100 MeV (see Ta-ble X), while the ratios between the partial widths for the Σ b K and Λ b (5920) K channels and Γ Sum are predicted to be Γ [ Σ b K ] Γ Sum ≃ , Γ [ Λ b (5920) K ] Γ Sum ≃ . (38)The Σ b K and Λ b (5920) K decay modes may be ideal modesfor our search for | Ξ ′ b D λλ + i in future experiments.0For the other J P = / + state | Ξ ′ b D λλ + i , one finds that ithas large decay rates into the P -wave states through the decaymodes Λ b (5912) K , Λ b (5920) K , | Ξ b P λ − i π , | Ξ b P λ − i π , | Ξ ′ b P λ − i π , and | Ξ ′ b P λ − i π . Furthermore, the decay ratesinto Ξ ′∗ b π and Σ ∗ b K are sizable as well. Its width should beabout a factor of 2 narrower than that of | Ξ ′ b D λλ + i . If itsmass is taken as the predictions in Ref. [21], the sum of thepartial widths of the poinic and kaonic decays can reach up to Γ Sum ∼
60 MeV (see Table VIII), while the ratios between thepartial widths for the Ξ ′∗ b π , Σ ∗ b K , Λ b (5912) K , and Λ b (5920) K decay modes and Γ Sum are predicted to be Γ [ Ξ ′∗ b π, Σ ∗ b K , Λ b (5912) K , Λ b (5920) K ] Γ Sum ≃ , , , . (39)The Ξ ′∗ b π , Σ ∗ b K , Λ b (5912) K , and Λ b (5920) K decay modes maybe ideal modes for our search for | Ξ ′ b D λλ + i in future exper-iments.We also estimate the radiative decays of these J P = / + states into the 1 P -wave bottom baryon states. Our resultsare listed in Table XI. It is found that for | Ξ ′ b D λλ + i themain radiative processes are | Ξ ′ b D λλ + i → | Ξ b P λ − i γ and | Ξ ′− b D λλ + i → | Ξ ′− b P λ − i γ, | Ξ ′− b P λ − i γ , while for | Ξ ′ b D λλ + i , the main radiative processes are | Ξ ′ b D λλ + i →| Ξ b P λ − i γ and | Ξ ′− b D λλ + i → | Ξ ′− b P λ − i γ, | Ξ ′− b P λ − i γ .Their partial radiative decay widths are estimated to be O (10)keV, while the branching fractions may be O (10 − ).
3. J P = / + states The J P = / + state | Ξ ′ b D λλ + i might be a narrow statewith a width of a few tens of MeV. Its dominant decay modeis Ξ b π . If the mass of | Ξ ′ b D λλ + i is taken as the prediction6432 MeV in Ref. [21], the sum of the partial widths for thepionic and kaonic decays is Γ Sum ∼
25 MeV (see Table X),while the ratio between the partial width for the Ξ b π channeland Γ Sum is predicted to be Γ [ Ξ b π ] Γ Sum ≃ . (40)To look for this state, the Ξ b π decay mode is worth observingin future experiments.The other J P = / + state | Ξ ′ b D λλ + i might be also anarrow state with a width of a few tens of MeV. It has largedecay rates into Ξ ′∗ b π and Σ ∗ b K . If its mass is taken as thepredictions in Ref. [21], the sum of the partial widths of thepoinic and kaonic decays is Γ Sum ∼
30 MeV (see Table VIII),while the ratios between the partial widths for the Ξ ′∗ b π and Σ ∗ b K decay modes and Γ Sum are predicted to be Γ [ Ξ ′∗ b π ] Γ Sum ≃ , Γ [ Σ ∗ b K ] Γ Sum ≃ . (41)To look for this state, the Ξ ′∗ b π and Σ ∗ b K decay modes are worthobserving in future experiments. We also estimate the radiative decays of these J P = / + states into the 1 P -wave bottom baryon states. Our resultsare listed in Table XI. It is found that for | Ξ ′ b D λλ + i themain radiative processes are | Ξ ′ b D λλ + i → | Ξ b P λ − i γ and | Ξ ′− b D λλ + i → | Ξ ′− b P λ − i γ , while for | Ξ ′ b D λλ + i , themain radiative processes are | Ξ ′ b D λλ + i → | Ξ b P λ − i γ and | Ξ ′− b D λλ + i → | Ξ ′− b P λ − i γ, | Ξ ′− b P λ − i γ . Their partial ra-diative decay widths are estimated to be ∼ −
20 keV, whilethe branching fractions is no more than O (10 − ).
4. J P = / + state The J P = / + state | Ξ ′ b D λλ + i might have a relativelynarrow width of O (10) MeV. This state mainly decays into Ξ b π , Λ b K , Λ b (5912) K , | Ξ b P λ − i π , and | Ξ b P λ − i π chan-nels. If one adopts the predicted mass in Ref. [21], the sumof the partial widths of the pionic and kaonic decays is esti-mated to be Γ Sum ∼
40 MeV (see Table VIII), and the ratiosbetween the partial widths for the Ξ b π , Λ b K , and Λ b (5912) K decay modes and Γ Sum are predicted to be Γ [ Ξ b π, Λ b K , Λ b (5912) K ] Γ Sum ≃ , , . (42)The Ξ b π , Λ b K , and Λ b (5912) K decay modes may be idealmodes for our search for | Ξ ′ b D λλ + i in future experiments.We also estimate its radiative decays into the 1 P -wave bot-tom baryon states. Our results are listed in Table XI. Itis found that the decay rates for these radiative transitionsare small. Their branching fractions may be no more than O (10 − ). E. Ω c In the Ω c family, there are six λ -mode 1 D -wave excita-tions: | Ω c D λλ + i , | Ω c D λλ + i , | Ω c D λλ + i , | Ω c D λλ + i , | Ω c D λλ + i , and | Ω c D λλ + i . However, no D -wave stateshave been established. The typical masses of the λ -mode 1 D -wave Ω c excitations are ∼ . D -wave states obtained in the relativistic quark-diquarkpicture [21], we give the predicted widths in Table XII.
1. J P = / + state The J P = / + state | Ω c D λλ + i may be a narrow statewith a width of Γ total ∼
20 MeV. Its decays are dominatedby the Ξ c (2470) K . The decay rates into Ξ ′ c (2575) K and Ξ ′∗ c (2645) K are sizeable as well. The branching fractions forthe Ξ c (2470) K , Ξ ′ c (2575) K , and Ξ ′∗ c (2645) K modes are pre-1 TABLE XII: Partial widths of strong and radiative decays for the λ -mode D -wave Ω c and Ω b baryons, the masses (MeV) of which are takenfrom the quark model predictions of Ref. [21]. M f stands for the masses of P -wave heavy baryons (MeV) in the final states, which are adoptedfrom the RPP [24] and Refs. [1, 21]. The units for the partial widths of radiative and strong decays are keV and MeV, respectively. Thesuperscript (subscript) stands for the uncertainty of a prediction with a +
10% ( − α ρ .Decay mode M f | Ω c D λλ + i (3282) | Ω c D λλ + i (3286) | Ω c D λλ + i (3287) | Ω c D λλ + i (3298) | Ω c D λλ + i (3297) | Ω c D λλ + i (3283) Γ i B i (%) Γ i B i (%) Γ i B i (%) Γ B i (%) Γ i B i (%) Γ i B i (%) Ξ c K + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′ c K + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′∗ c K − . + . + . − . + . − . + . − . − . + . + . − . | Ω c P λ − i γ − . + . − . + . < − . + . < − . + . < ≃ . < | Ω c P λ − i γ − . + . − . + . < < − . + . < − . + . < | Ω c P λ − i γ − . + . < < − . + . − . + . − . + . − . + . | Ω c P λ − i γ − . + . < < − . + . − . + . − . + . − . + . | Ω c P λ − i γ < − . + . < − . + . < − . + . − . + . − . + . + . − . − . + . + . − . + . − . − . + . − . + . Decay mode M f | Ω b D λλ + i (6530) | Ω b D λλ + i (6520) | Ω b D λλ + i (6540) | Ω b D λλ + i (6549) | Ω b D λλ + i (6529) | Ω b D λλ + i (6517) Γ i B i (%) Γ i B i (%) Γ i B i (%) Γ B i (%) Γ i B i (%) Γ i B i (%) Ξ b K + . − . − . + . + . − . + . − . − . + . − . + . Ξ ′ b K + . − . − . + . + . − . + . − . Ξ ′∗ b K − . − . + . − . + . − . + . − . + . − . + . − . | Ω b P λ − i γ − . + . − . + . − . + . < − . + . < − . + . < ≃ . < | Ω b P λ − i γ − . + . − . + . − . + . < − . + . < − . + . < − . + . < | Ω b P λ − i γ − . + . < < − . + . − . + . − . + . − . + . < | Ω b P λ − i γ − . + . < − . + . < − . + . − . + . − . + . − . + . | Ω b P λ − i γ − . + . < − . + . < − . + . − . + . − . + . − . + . + . − . − . + . + . − . + . − . − . + . − . + . dicted to be Γ [ Ξ c K , Ξ ′ c K , Ξ ′∗ c K ] Γ total ≃ , , . (43)From Fig. 7, it is found that the strong decay properties of | Ω c D λλ + i are less sensitive to its mass. The Ξ c (2470) K and Ξ ′ c (2575) K may be optimal channels for us to search for thismissing J P = / + state | Ω c D λλ + i .We also estimate its radiative decays into the 1 P -wavecharmed baryon states. Our results are listed in Table XII.It is found that | Ω c D λλ + i might have relatively large decayrates into | Ω c P λ − i γ and | Ω c P λ − i γ , and the partial ra-diative decay width is estimated to be O (10) keV. The branch-ing fractions for these main radiative decay processes may be O (10 − ).
2. J P = / + states For the J P = / + state | Ω c D λλ + i , the width is predictedto be Γ total ∼
18 MeV, which is less sensitive to the phasespace of strong decays (see Fig. 7). It is found that Ξ c (2470) K together with Ξ ′ c (2575) K governs the decays of | Ω c D λλ + i . Their branching fractions are predicted to be Γ [ Ξ c K , Ξ ′ c K ] Γ total ≃ , . (44)The Ξ c (2470) K and Ξ ′ c (2575) K may be ideal channels for usto search for this missing J P = / + state | Ω c D λλ + i .For the other J P = / + state | Ω c D λλ + i , if its mass isbelow the threshold of Ξ c (2790) K , the width is predicted tobe Γ total ∼
16 MeV, which is less sensitive to the phase spaceof strong decays (see Fig. 7). In this case, both Ξ c (2470) K and Ξ ′∗ c (2645) K are the dominant decay channels of | Ω c D λλ + i ,while the decay rate into Ξ ′ c (2575) K is sizeable as well. Theirbranching fractions are predicted to be Γ [ Ξ c K , Ξ ′∗ c K , Ξ ′ c K ] Γ total ≃ , , . (45)However, if the Ξ c (2790) K decay channel is open, this decaymode should be the dominant decay mode. The decay widthof | Ω c D λλ + i can reach up to 30 −
40 MeV. The Ξ c (2470) K and Ξ ′∗ c (2645) K may be ideal channels for us to search for thismissing J P = / + state | Ω c D λλ + i .We also estimate the radiative decays of these J P = / + Ω c states into the 1 P -wave charmed baryon states. Our re-sults are listed in Table XII. It is found that | Ω c D λλ + i → | Ω c P λ − i γ and | Ω c D λλ + i → | Ω c P λ − i γ, | Ω c P λ − i γ might have relatively large decay rates. The partial radia-tive decay widths are estimated to be O (10) keV, while theirbranching fractions may reach up to O (10 − ).
3. J P = / + states For the J P = / + state | Ω c D λλ + i , if we take its mass asthe prediction 3286 MeV in Ref. [21], the width is predictedto be Γ total ∼ | Ω c D λλ + i are governed by the Ξ c (2470) K channel, while the decay ratesinto Ξ ′ c (2575) K and Ξ ′∗ c (2645) K are sizeable as well. Theirbranching fractions are predicted to be Γ [ Ξ c K , Ξ ′∗ c K , Ξ ′ c K ] Γ total ≃ , , . (46)The Ξ c (2470) K , Ξ ′ c (2575) K , and Ξ ′∗ c (2645) K may be idealchannels for us to search for this missing J P = / + state | Ω c D λλ + i .For the other J P = / + state | Ω c D λλ + i , if we take itsmass as the prediction 3297 MeV in Ref. [21], the width ispredicted to be Γ total ∼ Ξ ′∗ c (2645) K are the dominant decay channels of | Ω c D λλ + i , while thedecay rate into Ξ c (2470) K is sizeable as well. Their branchingfractions are predicted to be Γ [ Ξ ′∗ c K , Ξ c K ] Γ total ≃ , . (47)The Ξ c (2470) K and Ξ ′∗ c (2645) K may be ideal channels for usto search for this missing J P = / + state | Ω c D λλ + i .We also estimate the radiative decays of the J P = / + Ω c states into the 1 P -wave charmed baryon states. Our re-sults are listed in Table XII. It is found that | Ω c D λλ + i →| Ω c P λ − i γ and | Ω c D λλ + i → | Ω c P λ − i γ might have rel-atively large decay rates. The partial radiative decay widthsare estimated to be O (10) keV, while their branching fractionsmay reach up to O (10 − ).
4. J P = / + state The J P = / + state | Ω c D λλ + i may be a narrow statewith a width of a few MeV. If its mass is taken to be 3283 MeVas predicted in Ref. [21], the width is predicted to be Γ total ∼ | Ω c D λλ + i aregoverned by the Ξ c (2470) K channel. The branching fractionis predicted to be Γ [ Ξ c K ] Γ total ≃ . (48)The Ξ c (2470) K may be an ideal channel for our search for thismissing J P = / + state | Ω c D λλ + i . We also estimate its radiative decays into the 1 P -wavecharmed baryon states. Our results are listed in Table XII. Itis found that | Ω c D λλ + i might have a relatively large decayrate into | Ω c P λ − i γ , and the partial radiative decay widthis estimated to be O (10) keV. The branching fraction for thismain radiative decay process may be O (10 − ). F. Ω b In the Ω b family, according to the quark model classifica-tion, there are six λ -mode 1 D -wave excitations: | Ω b D λλ + i , | Ω b D λλ + i , | Ω b D λλ + i , | Ω b D λλ + i , | Ω b D λλ + i , and | Ω b D λλ + i . However, no D -wave states have been estab-lished. The typical masses of the λ -mode 1 D -wave Ω b exci-tations are ∼ . D -wave states ob-tained in the relativistic quark-diquark picture [21], we givethe predicted widths in Table XII.
1. J P = / + state The J P = / + state | Ω b D λλ + i may be a narrow statewith a width of Γ total ∼
30 MeV. Its decays are dominatedby the Ξ b (5795) K . The decay rates into Ξ ′ b (5935) K and Ξ ′∗ b (5955) K are sizeable as well. The branching fractions forthe Ξ b (5795) K , Ξ ′ b (5935) K , and Ξ ′∗ b (5955) K modes are pre-dicted to be Γ [ Ξ b K , Ξ ′ b K , Ξ ′∗ b K ] Γ total ≃ , , . (49)From Fig. 7, it is found that the strong decay properties of | Ω b D λλ + i are less sensitive to its mass. Ξ b (5795) K and Ξ ′ b (5935) K may be ideal channels for us to search for thismissing J P = / + state | Ω b D λλ + i .We also estimate its radiative decays into the 1 P -wave bot-tom baryon states. Our results are listed in Table XII. It isfound that | Ω − b D λλ + i might have a relatively large decayrate into | Ω − b P λ − i γ . The partial radiative decay width ofthis process is estimated to be O (10) keV, while the branchingfraction may be O (10 − ).
2. J P = / + states For the J P = / + state | Ω b D λλ + i , the width is predictedto be Γ total ∼ ± Ξ b (5795) K together with Ξ ′ b (5935) K governs the decays of | Ω b D λλ + i . If its mass is taken to be 6530 MeV as predictedin Ref. [21], the branching fractions for the Ξ b (5795) K and3 Ξ ′ b (5935) K modes are predicted to be Γ [ Ξ b K , Ξ ′ b K ] Γ total ≃ , , (50)The Ξ b (5795) K and Ξ ′ b (5935) K may be ideal channels for usto search for this missing J P = / + state | Ω b D λλ + i .For the other J P = / + state | Ω b D λλ + i , if its mass istaken to be 6549 MeV as predicted in Ref. [21], the width ispredicted to be Γ total ∼
20 MeV. The decays are dominated bythe Ξ b (5795) K and Ξ ′∗ b (5955) K channels, while the decay rateinto Ξ ′ b (5935) K is sizeable as well. Their branching fractionsare predicted to be Γ [ Ξ b K , Ξ ′∗ b K , Ξ ′ b K ] Γ total ≃ , , , (51)which is sensitive to the phase space of strong decays (seeFig. 7). Ξ b (5795) K and Ξ ′∗ b (5955) K may be ideal channels forour search for this missing J P = / + state | Ω b D λλ + i .We also estimate the radiative decays of these J P = / + Ω b states into the 1 P -wave bottom baryon states. Our re-sults are listed in Table XII. It is found that | Ω − b D λλ + i →| Ω − b P λ − i γ and | Ω − b D λλ + i → | Ω − b P λ − i γ might haverelatively large decay rates. The partial radiative decay widthsare estimated to be O (10) keV, while their branching fractionsmay reach up to O (10 − ).
3. J P = / + states For the J P = / + state | Ω b D λλ + i , if we take its mass asthe prediction 6520 MeV in Ref. [21], the width is predicted tobe Γ total ∼ | Ω b D λλ + i are governed bythe Ξ b (5795) K channel, while the decay rate into Ξ ′∗ b (5955) K are sizeable as well. Their branching fractions is predicted tobe Γ [ Ξ b K , Ξ ′∗ b K ] Γ total ≃ , . (52)It should be pointed out that the decay properties of | Ω b D λλ + i show some sensitivities to its mass (see Fig. 7). Ξ b (5795) K and Ξ ′∗ b (5955) K may be ideal channels for us tosearch for this missing J P = / + state | Ω b D λλ + i .For the other J P = / + state | Ω b D λλ + i , if we takeits mass as the prediction 6549 MeV in Ref. [21], the widthis predicted to be Γ total ∼ Ξ ′∗ b (5955) K is the domi-nant decay channel of | Ω b D λλ + i , while the decay rate into Ξ b (5790) K is sizeable as well. Their branching fractions arepredicted to be Γ [ Ξ ′∗ b K , Ξ b K ] Γ total ≃ , . (53)The decay properties of | Ω b D λλ + i show some uncertain-ties with its mass changes (see Fig. 7). Ξ b (5790) K and Ξ ′∗ b (5955) K may be ideal channels for us to search for thismissing J P = / + state | Ω b D λλ + i .We also estimate the radiative decays of the J P = / + Ω b states into the 1 P -wave bottom baryon states. Our re-sults are listed in Table XII. It is found that | Ω − b D λλ + i →| Ω − b P λ − i γ and | Ω − b D λλ + i → | Ω − b P λ − i γ, | Ω − b P λ − i γ might have relatively large decay rates. The partial radia-tive decay widths are estimated to be O (10) keV, while theirbranching fractions may reach up to O (10 − ).
4. J P = / + state The J P = / + state | Ω b D λλ + i may be a narrow statewith a width of a few MeV or a few tens of MeV. If its mass istaken to be 6517 MeV as predicted in Ref. [21], the width ispredicted to be Γ total ∼ | Ω b D λλ + i are governed by the Ξ b (5970) K channel. Thebranching fraction is predicted to be Γ [ Ξ b K ] Γ total ≃ . (54) Ξ b (5970) K may be an ideal channel for us to search for thismissing J P = / + state | Ω b D λλ + i .We also estimate its radiative decays into the 1 P -wave bot-tom baryon states. Our results are listed in Table XII. It isfound that | Ω − b D λλ + i might have a relatively large decayrate into | Ω − b P λ − i γ , and the partial radiative decay widthis estimated to be O (10) keV. The branching fraction for thismain radiative decay process may be O (10 − ). V. SUMMARY
In this work, we carry out a systematic study of the strongdecays with emission of one light pseudoscalar meson and theradiative decays with emission one photon of the low-lying D -wave singly heavy baryons in a constituent quark model.Our results may provide helpful references to establish thesemissing D -wave singly heavy baryons in future experiments.Several key results from this study can be learned as follows.The λ -mode 1 D -wave J P = / + and J P = / + ex-citations of ¯ F (i.e., | D λλ + i and | D λλ + i ) in the Λ c and Ξ c families might have been observed in experiments. Both Λ c (2860) and Ξ c (3050) seem to favor the J P = / + excitation | D λλ + i of ¯ F , while Λ c (2880) and Ξ c (3080) seem to favorassigning the J P = / + excitation | D λλ + i . The nature of Ξ c (3050) and Ξ c (3080) may be tested by the radiative transi-tions Ξ c (3055) → Ξ c (2790) γ and Ξ c (3080) → Ξ c (2815) γ ,respectively.The missing λ -mode 1 D -wave J P = / + and J P = / + excitations of ¯ F in the Λ b and Ξ b families have a large po-tential to be found in forthcoming experiments. They mightbe narrow states with a width of ∼
10 MeV. In the Λ b fam-ily, the J P = / + state might be established in the Σ b π and4 Λ b (5912) γ final states, while the J P = / + state might beestablished in the Σ ∗ b π and Λ b (5920) γ final states. In the Ξ b family, the J P = / + state might be established in the Ξ ′ b π , Ξ ′∗ b π and Σ b K final states, while the J P = / + state might beestablished in the Σ ∗ b K and Ξ ′∗ b π final states.The λ -mode 1 D -wave excitations of F in the Ω c and Ω b families have a large potential to be found in forthcoming ex-periments as well. They are fairly narrow states with a widthof a few MeV or a few tens of MeV. The kaonic decay chan-nels Ξ c (2470) K , Ξ ′ c (2575) K , and Ξ ′∗ c (2645) K may be idealchannels for us to search for these missing 1 D -wave excited Ω c states, while the kaonic decay channels Ξ b K , Ξ ′ b K , and Ξ ′∗ b K may be ideal channels for us to search for these missing1 D -wave excited Ω b states.The λ -mode 1 D -wave excitations in the Σ c ( b ) and Ξ ′ c ( b ) fam-ilies appear to have relatively broad widths. The sum of thepartial widths with emission of a one-pion meson and one-kaon meson is about 50 ∼
200 MeV. These 1 D -wave statesmight have large decay rates into the 1 P -wave heavy baryonstates via the pionic and / or kaonic decays. The Λ c (2595) π and Λ c (2625) π channels may be ideal channels for looking for themissing 1 D -wave excitations in the Σ c family. The Ξ c (2790) π , Ξ c (2815) π , Λ c (2595) K , and Λ c (2625) K decay channels maybe ideal channels for looking for the missing 1 D -wave excita- tions in the Ξ ′ c family. The Λ b (5912) π , and Λ b (5920) π decaychannels may be ideal channels for looking for the missing D -wave excitations in the Σ b family.Finally, it should be pointed out that some of our predic-tions bear a fairly large uncertainty from the nonrelativisticharmonic oscillator wave functions adopted in the calcula-tions. Considering a 10% uncertainty of the oscillator param-eter, one finds that the uncertainty of our predictions can reachup to ∼ Acknowledgments
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