Strong and radiative decays of the low-lying $D$-wave singly heavy baryons

The strong and radiative decays of the low-lying $\lambda$-mode $D$-wave $\Lambda_{c(b)}$, $\Sigma_{c(b)}$, $\Xi_{c(b)}$, $\Xi_{c(b)}'$, and $\Omega_{c(b)}$ baryons are studied in a constituent quark model. Our calculation shows the following: (i) The missing $\lambda$-mode $D$-wave $\Omega_{c(b)}$, $\Lambda_{b}$, and $\Xi_{b}$ baryons have a relatively narrow decay width of a few MeV or a few tens of MeV and their dominant strong and radiative decay channels can be ideal for searching for their signals in future experiments. (ii) The $\lambda$-mode $1D$-wave excitations in the $\Sigma_{c(b)}$ and $\Xi_{c(b)}'$ families appear to have a relatively broad width of $\sim 50-200$ MeV.Most of the $1D$-wave states have large decay rates into the $1P$-wave heavy baryons via the pionic or kaonic strong decay processes, which should be taken seriously in future observations. (iii) Both $\Lambda_c(2860)$ and $\Xi_c(3050)$ seem to favor the $J^P=3/2^+$ excitation $|^2D_{\lambda\lambda} \frac{3}{2}^+ \rangle$ of $\bar{\mathbf{3}}_F$, while both $\Lambda_c(2880)$ and $\Xi_c(3080)$ may be assigned as the $J^P=5/2^+$ excitation $|^2D_{\lambda\lambda} \frac{5}{2}^+ \rangle$ of $\bar{\mathbf{3}}_F$. The nature of $\Xi_c(3050)$ and $\Xi_c(3080)$ could be tested by the radiative transitions $\Xi_c(3055)^0\to \Xi_c(2790)^0 \gamma$ and $\Xi_c(3080)^0 \to \Xi_c(2815)^0 \gamma$, respectively.


I. INTRODUCTION
The LHC facility provides good opportunities for us to discover some of the missing heavy baryons. Recently, five extremely 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 − channel by the LHCb Collaboration [1]. Most of them may be interpreted as the P-wave excited states of Ω c [2][3][4][5][6][7][8][9]. Lately, the LHCb Collaboration observed a new structure Ξ b (6227) − in both the Λ 0 b K − and Ξ 0 b π − invariant mass spectra [10]. The mass of this structure and the observed decay modes are consistent with expectations of a P-wave excited state in the Ξ ′ b family [11][12][13][14][15][16][17][18][19][20][21][22][23]. Besides the missing P-wave heavy baryons, some low-lying D-wave singly heavy baryons should be also observed at the LHC in forthcoming experiments. Furthermore, the Belle II experiments will also offer the possibility of studying excited heavy baryons. Thus, the theoretical studies of the low-lying P-and D-wave singly heavy baryons will provide very useful references for searching for them in future experiments. Considering that the decay properties of heavy baryons should be sensitive to its inner structure, one may better understand the nature of the heavy baryons by studying their decays. In our recent work [11], we systematically studied the strong and radiative decay properties of the P-wave singly heavy baryons. As a continuation of Ref. [11], we study the strong and radiative decays of the low-lying D-wave singly heavy baryons in the present work.
In this work, we apply a nonrelativistic constituent quark model to study the strong decays with emission of one light pseudoscalar meson and the radiative decays with emission of one 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 states in future experiments. For a simplicity, the harmonic oscillator wave functions of the heavy baryons are adopted in our calculations. To deal with the strong decays of a hadron, an effective chiral Lagrangian at the tree level [92] is introduced. In this interaction, the emitted light pseudoscalar mesons are treated as Goldstone bosons, which only couple with the light constituent quarks. Since the quark-meson coupling is invariant under the chiral transformation, some of the low-energy properties of QCD are retained [92][93][94]. This method ( i.e., ChQM) has been successfully applied to study the strong decays of heavy-light mesons and charmed and strange baryons [2,[47][48][49][95][96][97][98][99][100]. The chiral quark model used in this work is different from the often-used 3 P 0 model [101][102][103]; the differences between them have been pointed out in Ref. [11]. Meanwhile, to treat the radiative decay of a hadron, we apply an effective quark-photon electromagnetic (EM) coupling at the tree level. The higher EM multipole contributions are included by a multipole expansion of the EM interactions. This approach has been successfully applied to deal with the radiative decays of cc and bb systems [104,105], and recently it has been extended to study the radiative transitions of heavy baryons [2,99,106].
The paper is organized as follows. Section II is our framework, in which we give a brief review of the quark model classification of the singly heavy baryons and the quark model description of the strong and radiative decays. Then, the numerical results for the heavy baryons belonging to3 F and 6 F are presented and discussed in Secs. III and IV, respectively. Finally, a summary is given in Sec. V.

A. Spectra
The heavy baryon containing a heavy quark violates the SU(4) symmetry. However, the SU(3) symmetry between the other two light quarks (u, d, or s) is approximately kept. The heavy baryons containing a single heavy quark belong to two different SU(3) flavor representations: the symmetric sextet 6 F and antisymmetric antitriplet3 F [11]. In the singly charmed (bottom) baryons, there are two families, Λ c and Ξ c (Λ b and Ξ b ) belonging to3 F , while there are three families, Σ c , Ξ ′ c , and Ω c (Σ b , Ξ ′ b , and Ω b ), belonging to 6 F [11]. The spatial wave function of a heavy baryon is adopted the harmonic oscillator form in the constituent quark model [47]. For a q 1 q 2 Q basis state, it contains two light quarks q 1 and q 2 with equal mass m and a heavy quark Q with mass m ′ . The basis states are generated by the oscillator Hamiltonian The constituent quarks are confined in an oscillator potential with the potential parameter K independent of the flavor quantum number. The Jacobi coordinates ρ and λ and c.m. coordinate R c.m. can be related to the coordinate r j of the jth quark. The momenta p ρ , p λ , and P c.m. are defined by p ρ = m ρρ , p λ = m λλ , P c.m. = MṘ c.m. , with M = 2m + m ′ , m ρ = m, and m λ = 3mm ′ 2m+m ′ . The wave function of an oscillator is give by where σ ≡ ρ, λ. In the wave functions, there are two oscillator parameters, i.e., the potential strengths α ρ and α λ . The parameters α ρ and α λ satisfy the following relation [47]: The details of the classifications of the heavy baryons in the constituent quark model can be found in Refs. [47]. Since the bottom and charm quark masses are much larger than the light quark mass (m Q > m q ), the λ-mode excitations of singly heavy baryons should be easily formed than the ρ-mode excitations [27]. Thus, in the present work, we only study the λ-mode excitations. The mass spectra of the single heavy baryons up to the 1D-wave excitations predicted within various quark models are summarized in Tables I and II.

B. Decays
In this work, strong decays of the D-wave singly heavy baryons with emission of one light pseudoscalar meson are studied within ChQM [92]. This model has been successfully applied to study the strong decays of heavy-light mesons and charmed and strange baryons [2,[47][48][49][95][96][97][98][99][100]. In this model, the light pseudoscalar mesons, i.e., π, K, and η, are treated as fundamental states, which only couple with the light constituent quarks of a hadron via the simple chiral Lagrangian [92] where ψ j represents the jth quark field in the hadron, φ m is the pseudoscalar meson field, and f m is the pseudoscalar meson decay constant. Meanwhile, to treat the radiative decay of a hadron, we apply the constituent quark model, which has been successfully applied to study the radiative decays of cc and bb systems [104,105]. In this model, the quark-photon EM coupling at the tree level is adopted as where A µ represents the photon field with 3-momenta k. e j and r j stand for the charge and coordinate of the constituent quark ψ j , respectively. To match the nonrelativistic harmonic oscillator wave functions, in the calculations, one should adopt the nonrelativistic forms for the quark-pseudoscalar and quark-photon EM couplings listed in Eqs. (4) and (5), which have been given in the previous works [2,47,48,[93][94][95][96][97][98][99][100][104][105][106][107][108][109][110][111][112].
For a strong decay process, the partial decay width can be calculated with [47] Γ m = δ f m angular momenta of the initial and final heavy baryons, respectively. E f and M f are the energy and mass of the final heavy baryon, and M i is the mass of the initial heavy baryon. δ as a global parameter accounts for the strength of the quarkmeson couplings. It has been determined in our previous study of the strong decays of the charmed baryons and heavy-light mesons [47,96]. Here, we fix its value the same as that in Refs. [47,96], i.e., δ = 0.557. In the calculation, we adopt the same quark model parameter set as that in Ref. [11], which has been collected in Table III. The masses of the well-established hadrons used in the calculations are adopted from the RPP [24].
To better understand the properties of Λ c (2880), considering it as the 1D-wave state |Λ c 2 D λλ 5 2 + , we further study its radiative decays into the 1P-wave charmed baryon states. Our results are listed in Table V. It is found that most of the partial radiative widths of Λ c (2880) into the 1P-wave states are O(100) eV. Combining these partial widths with the total decay width of Λ c (2880), we find the branching fractions for the main radiative decay channels are O(10 −5 ). The small decay rates indicate that the radiative decays of Λ c (2880) into the 1P-wave states might be hard to observe in experiments.  Parameter ) 330 450 1480 5000  400  420  440  132 160   TABLE IV: Partial widths of strong decays for the λ-mode Dwave Λ c and Λ b baryons. The masses of the D-wave Λ c (Λ b ) states | 2 D λλ which roughly agree with the predictions in Ref. [41]. Combining these predicted partial widths with the measured width of Λ c (2860), we further estimate that the branching fractions of the Σ c (2455)π and Σ c (2520)π channels can reach up to 7% and 2%, respectively. The relatively large branching fractions indicate 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 1D-wave state |Λ c 2 D λλ we also study its radiative decays into the 1P-wave states. Our results are listed in Table V   The masses of the P-wave heavy baryons in final states are adopted from the quark model predictions in Ref. [21] (see Table II).

B. Λ b states
In the Λ b family, there are two λ-mode 1D-wave excitations |Λ b 2 D λλ The masses for the λ-mode 1D-wave Λ b excitations are predicted to be ∼ 6.2 GeV in various models (see Table I). In the possible mass region of the 1D-wave Λ b excitations, we study their strong decay properties, which have been shown in Fig. 1. To be more specific, taking the masses of the 1D-wave states as predicted in the relativistic quarkdiquark picture [21] we further present the results in Table IV. 1. J P = 3/2 + state From Fig. 1, it is found that if the mass of |Λ b 2 D λλ 3 2 + is ∼ 6200 MeV as predicted in theory [21,27] it should be  In Ref. [48], the strong decay properties of the 1D-wave states were studied within ChQM. It is found that Ξ c (3055) seems to favor the J P = 3/2 + state |Ξ c 2 D λλ is consistent with the predictions in Refs. [21,41]. Based on the SU(4) symmetry we estimated the partial width of Γ[Ξ c (3055) + → ΛD + ], which is too small to compare with the observation at Belle [123]. The serious SU(4) symmetry breaking might lead to our failed description of the decays into the D + Λ channel. Assigning Ξ c (3055) as |Ξ c 2 D λλ it should have relatively large decay rates into Ξ ′ c π and Σ c (2455)K channels (see Table VI). The predicted partial width ratio between these two channels is Combining the predicted partial width of Γ[Ξ c (3055) + → Σ c (2455) ++ K − ] ≃ 2.4 MeV with Eq. (11), we estimate the respectively. M f stands for the masses of P-wave heavy baryons (MeV) in the final states, which are adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode Decay mode  and 6373) 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] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Ξ c 2 D λλ The Ξ c (3080) resonance is suggested to be the ρ-mode 2Swave state with J P = 1/2 − in Ref. [48]. The observation of Ξ c (3080) + in the D + Λ channel excludes this assignment because the D + Λ decay mode should be forbidden [48]. The mass and decay modes observed in experiments indicate that Ξ c (3080) is most likely to be the λ-mode 1D excitation of Ξ c with J P = 5/2 + (i.e., |Ξ c 2 D λλ [21,41,44]. Considering Ξ c (3080) + as the |Ξ c 2 D λλ 5 2 + state, we find that it has relatively large decay rates into the Σ * c (2520)K and Ξ ′ * c (2645)π (see Table VI). The partial width ratio between these two main channels is predicted to be Combining it with the predicted partial width of Finally, the total width of Ξ c (3080) is estimated to be Γ ≃ 6.9 MeV, which is close to the upper limit of the observation from the Belle Collaboration [123]. However, our predicted partial width ratio between the Σ c K and Σ * c K channels is about an order of magnitude larger than the observed ratio listed in Eq. (13), and a similar phenomenon is found by Chen et al. within their 3 P 0 analysis [41]. It should be mentioned that the measured ratio R = 27 of Ξ c (3080) may be strongly affected by its nearby states, such as Ξ c (3055). Thus, the measured ratio from Belle [123] may not be a genuine ratio for Ξ c (3080).
Furthermore, assigning Ξ c (3080) as the J P = 5/2 + state |Ξ c 2 D λλ 5 2 + , we study its radiative decays. Our results are listed in Table VII. It is found that the Ξ c (3080) 0 should have a relatively large decay rate into Ξ c (2815) 0 3 2 − γ. The partial de- The neutral state Ξ c (3080) 0 is most likely to be observed in the Ξ c (2815) 0 γ channel if it corresponds to the J P = 5/2 + state |Ξ c 2 D λλ In the Ξ b family, there are two λ-mode 1D-wave excitations |Ξ b 2 D λλ  Table I). In the possible mass regions, the strong decays of these 1D-wave states are studied with ChQM. Our results have been shown in Fig. 2.
To be more specific, taking the masses of the 1D-wave states obtained in the relativistic quark-diquark picture [21], we give the predicted widths in Table VI.
with 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 less sensitive to the mass of |Ξ b 2 D λλ 3 2 + ; however, the partial width for the Σ b K channel shows a significant linear dependence on the mass (see Fig. 2). If the mass of |Ξ b 2 D λλ 3 2 + takes the predicted value ∼ 6.37 GeV in Ref. [21], the branching fractions for the main channels are predicted to be The Ξ ′ b π and Σ b K decay channels may be ideal channels for our search for this missing 1D-wave Ξ b baryon in future experiments.
Furthermore, we study the radiative decays of |Ξ b 2 D λλ + is taken to be ∼ 6366 MeV as predicted in the relativistic quark model [21]. Combining the predicted total width of |Ξ b 2 D λλ 3 2 + , we estimate the branching fraction + is taken to be ∼ 6.37 GeV as the prediction in Ref. [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 To establish this missing 1D-wave Ξ b baryon with J P = 5/2 + , its dominant decay modes Ξ ′ * b π and Σ b K are worth observing in future experiments.
We also study the radiative decays of |Ξ b 2 D λλ + is taken to be ∼ 6373 MeV as predicted in the relativistic quark model [21]. Combining the predicted total width of |Ξ b 2 D λλ 5 2 + , we estimate the branching fraction In the Σ c family, according to the quark model classification, there are six λ-mode 1D-wave excitations: |Σ c 4 D λλ + . However, no D-wave states have been established. The typical masses of the λ-mode 1D-wave Σ c excitations are predicted to be ∼ 3.0 GeV within various quark models (see Table II). In the possible mass range, we study their strong decay transitions within ChQM. Our results are shown in Fig. 3. To be more specific, taking the masses of the 1D-wave states as predicted in the relativistic quark-diquark picture [21], we further present the results in Table VIII.
its mass is taken as the prediction 3041 MeV in Ref. [21], the sum of the partial widths for the pionic and kaonic decays can reach up to Γ Sum ∼ 160 MeV (see Table VIII). This state has large 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 Both Λ c (2595)π and Λ c (2625)π may be ideal channels for our search for |Σ c 4 D λλ We also estimate its radiative transitions into the 1P-wave charmed baryon states. Our results are listed in Table IX. It is found that |Σ ++(0) the P-wave states through the poinic decay modes Λ c (2625)π and |Σ c 2 P λ 3 2 − π, while its decay rate into Σ c π is sizable. It has a width of O(10) − O(100) MeV, which significantly depends on its mass. If the mass is taken as the prediction 3043 MeV in Ref. [21], the sum of the partial widths of the pionic decays is Γ Sum ∼ 100 MeV (see Table VIII), and the ratios between the partial decay widths for the Λ c (2625)π and Σ c π channels and Γ Sum are predicted to be The Σ c π and Λ c (2625)π decay channels may be ideal channels for our search for |Σ c 2 D λλ For the other J P = 3/2 + state |Σ c 4 D λλ 3 2 + , one finds that it has large decay rates into the P-wave states through the pionic decay modes Λ c (2625)π, |Σ c 4 P λ and |Σ c 4 P λ  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 The Λ c (2625)π and Σ c (2520)π may be ideal channels for our search for |Σ c 4 D λλ We also estimate the radiative transitions of these J P = 3/2 + states into the 1P-wave states. Our results are listed in Table IX. It is found that |Σ ++(0)   . 3). It has large decay rates into Λ c π, Σ c π, Λ c (2595)π, and Λ c (2625)π with comparable partial decay widths. If its mass is taken as the prediction 3038 MeV in Ref. [21], the sum of the partial widths of the pionic decays is about Γ Sum ∼ 60 MeV (see Table VIII), and the ratios 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 predicted to be The Λ c π, Σ c π, Λ c (2595)π, and Λ c (2625)π decay channels may be ideal channels for our search for the missing The other J P = 5/2 + state |Σ c 4 D λλ 5 2 + might also be a narrow state with a width of O(10) MeV. This state has relatively large decay rates into Σ c (2520)π, Λ c (2595)π, |Σ c 4 P λ If its mass is taken as the prediction 3023 MeV in Ref. [21], the sum of the partial widths of the pionic decays is about Γ Sum ∼ 70 MeV (see Table VIII), while the ratios between the partial widths for the main decay modes, Σ c (2520)π and Λ c (2595)π, and Γ Sum are predicted to be The Σ c (2520)π and Λ c (2595)π may be ideal channels for a search for |Σ c 4 D λλ  [21], if there are no observations (see Table II).
The radiative transitions of these J P = 5/2 + states into the 1P-wave charmed baryon states are estimated as well. Our results are listed in Table IX. It is found that for the |Σ c 2 D λλ It mainly decays into Λ c π, Λ c (2595)π, and Λ c (2625)π channels. If one adopts the predicted mass 3013 MeV in Ref. [21], the sum of the partial widths of the pionic decays is estimated to be Γ Sum ∼ 60 MeV (see Table VIII), and the ratios between the partial decay widths for these main channels, 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 ≃ 22%, 19%, 41%. (24) The Λ c π, Λ c (2595)π, and Λ c (2625)π may be ideal channels for a search for |Σ c 4 D λλ   Table II). In the possible mass ranges, we study their strong decay transitions within ChQM. Our results are shown in Fig. 4. To be more specific, taking the masses of the 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 which are 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, which are adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Σ c 2 D λλ 3 2  1D-wave states obtained in the relativistic quark-diquark picture [21], we present the results in Table VIII. 1. J P = 1/2 + state with a width of O(100) MeV. If its mass is taken as the prediction in Ref. [21], the sum of the partial widths for the pionic decays can reach up to Γ Sum ∼ 150 MeV (see Table VIII). This state might mainly decay into the P-wave Λ b states Λ b (5912) 1 2 − and Λ b (5920) 3 2 − via pionic decay modes Λ b (5912)π and Λ b (5920)π. The ratios between the partial decay widths for the Λ b (5912)π and Λ b (5920)π channels and Γ Sum are predicted to be Both Λ b (5912)π and Λ b (5920)π may be ideal channels for our search for |Σ b 4 D λλ We also estimate its radiative transitions. Our results are listed in Table IX. It is found that |Σ +  MeV, which obviously depends on its mass (see Fig. 4). If the mass is taken as the prediction 6326 MeV in Ref. [21], the sum of the partial widths of the pionic decays can reach up to Γ Sum ∼ 120 MeV (see Table VIII), and the ra- 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 quark model predictions of Ref. [21]. M f stands for the masses of P-wave heavy baryons (MeV) in the final states, which are adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Σ c 2 D λλ 3 2 Decay mode tio between the partial decay width for the Λ b (5920)π channel and Γ Sum is predicted to be The Λ b (5920)π decay channel may be an ideal channel for our search for |Σ b 2 D λλ Furthermore, the decay rate into Σ * b (5832)π is sizable as well. Its width should be narrower than that of |Σ b 2 D λλ 3 2 + . If its mass is taken as the predictions in Ref. [21], the sum of the partial widths of the pionic decays can reach up to Γ Sum ∼ 85 MeV (see Table VIII), while the ratios between the partial widths of Γ[Σ * b (5832)π] and Γ[Λ b (5920)π] and Γ Sum are pre-  Table II). dicted to be Both Σ * b (5832)π and Λ b (5920)π may be ideal channels for a search for |Σ b 4 D λλ The radiative decays of the J P = 3/2 + states into the 1Pwave bottom baryon states are estimated as well. Our results are listed in Table IX. It is found that |Σ +  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 prediction 6284 MeV in Ref. [21], the sum of the partial widths of the pionic decays is about Γ Sum ∼ 50 MeV (see Table VIII), and the ratios between the partial decay widths for these main channels, Λ b π, Σ b π, Λ b (5912)π, and Λ b (5920)π, and Γ Sum are predicted to be The Λ b π, Σ b π, Λ b (5912)π, and Λ b (5920)π decay channels may be ideal channels for our search for |Σ b 2 D λλ If its mass is taken as the prediction 6270 MeV in Ref. [21], the sum of the partial widths of the 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 The Λ b π, Σ * b π, and Λ b (5912)π decay channels might be ideal channels for our search for |Σ b 4 D λλ The radiative decays of these J P = 5/2 + states into the 1P-wave bottom baryon states are estimated as well. Our results are listed in Table IX. It is found that for the |Σ b 2 D λλ  [21], if there are no observations (see Table II MeV, which strongly depends on its mass (see Fig. 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 mass 6260 MeV in Ref. [21], the sum of the partial widths of the pionic decays is estimated to be Γ Sum ∼ 50 MeV (see Table VIII), and the ratios between the partial decay widths for these main channels, Λ b π, Λ b (5912)π, and Λ b (5920)π, and Γ Sum are predicted to be The Λ b π, Λ b (5912)π, and Λ b (5920)π decay channels may be ideal channels for our search for |Σ b 4 D λλ 7 2 + in future experiments.
We also estimate its radiative decays into the 1P-wave bottom baryon states. Our results are listed in Table IX. It is found that the main radiative decay processes are |Σ +  Table II).
In Ref. [48], the strong decay properties of the D-wave excited Ξ ′ c states were studied in their possible mass ranges. However, we did not give correct predictions of the partial widths of the D-wave excited Ξ ′ c states decaying into the Pwave charmed baryons. In this work, we update our predictions, which have been shown in Fig. 5. To be more specific, taking the masses of the 1D-wave states predicted within the relativistic quark-diquark picture [21], we give our predictions in Table X 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 The decay channels Λ c (2625)K and Ξ c (2815)π may be ideal channels for our search for |Ξ ′ We also estimate its radiative decays into the 1P-wave bottom baryon states. Our results are listed in Table XI. It is found that |Ξ ′0  Ref. [21], the decay channels into the P-wave charmed baryon final 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 = 3/2 + state |Ξ ′ c 2 D λλ 3 2 + might be a broad state, and the sum of the partial widths for the pionic and kaonic decays can reach up to Γ Sum ∼ 90 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 The Λ c (2625)K and Ξ c (2815)π decay channels may be ideal channels for our search for |Ξ ′  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 quark model predictions of Ref. [21]. M f stands for the masses of P-wave heavy baryons (MeV) in the final states, which are adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Ξ ′   For the other J P = 3/2 + state |Ξ ′ In this case, the sum of the partial widths for the pionic and kaonic 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 to be The Λ c (2625)K, Ξ c (2790)π, and Ξ c (2815)π decay channels may be ideal channels for a search for |Ξ ′ The radiative decays of these J P = 3/2 + states into the 1Pwave charmed baryon states are also estimated. Our results 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 quark model predictions of Ref. [21]. M f stands for the masses of P-wave heavy baryons (MeV) in the final states, which are adopted from the RPP [24] and Ref. [21]. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Ξ ′  Decay while for the |Ξ ′  Table X), while the ratios between the partial widths for the Ξ c π, Ξ ′ c π and Ξ c (2815)π channels and Γ Sum are predicted to be The Ξ c π, Ξ ′ c π, and Ξ c (2815)π decay modes may be ideal modes for our search for |Ξ ′ MeV 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 the partial widths for the pionic and kaonic decays can reach up to Γ Sum ∼ 50 MeV (see Table X), while the ratios between the partial widths for the main channels and Γ Sum are predicted to be The Ξ c π, Λ c K, Λ c (2595)K, Ξ c (2790)π, and Ξ c (2815)π decay modes may be ideal modes for our search for |Ξ ′   Table II). In the possible mass ranges, we study their strong decay transitions by emitting one light pseudoscalar meson within the ChQM. Our results are shown in Fig. 6. To be more specific, taking the masses of the 1D-wave states obtained in the relativistic quark-diquark picture [21], we give the predicted widths in Table X. and kaonic decay mode Λ b (5920)K. If its mass is taken as the prediction 6447 MeV in Ref. [21], the sum of the partial widths for the pionic and kaonic decays can reach up to Γ Sum ∼ 110 MeV (see Table VIII), while the ratio between the partial width for the Λ b (5920)K channel and Γ Sum is predicted to be The Λ b (5920)K decay channel may be an ideal channel for our search for |Ξ ′ We also estimate its radiative decays into the 1P-wave bottom baryon states. Our results are listed in Table XI. It is found that |Ξ ′−  − π and Λ b (5920)K decay modes. The decay rate into Σ b K is also sizeable. If its mass is taken as the prediction in Ref. [21], the sum of the partial widths for the pionic and kaonic decays can reach up to Γ Sum ∼ 100 MeV (see Table 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 The Σ b K and Λ b (5920)K decay modes may be ideal modes for our search for |Ξ ′ + , one finds that it has large decay rates into the P-wave states through the decay Furthermore, the decay rates into Ξ ′ * b π and Σ * b K are sizable as well. Its width should be about a factor of 2 narrower than that of |Ξ ′ b 2 D λλ 3 2 + . If its mass is taken as the predictions in Ref. [21], the sum of the partial widths of the poinic and kaonic decays can reach up to Γ Sum ∼ 60 MeV (see Table VIII), while the ratios between the partial 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 The Ξ ′ * b π, Σ * b K, Λ b (5912)K, and Λ b (5920)K decay modes may be ideal modes for our search for |Ξ ′ We also estimate the radiative decays of these J P = 3/2 + states into the 1P-wave bottom baryon states. Our results are listed in Their partial radiative decay widths are estimated to be O(10) keV, while the branching fractions may be O(10 −4 ). + is taken as the prediction 6432 MeV in Ref. [21], the sum of the partial widths for the pionic and kaonic decays is Γ Sum ∼ 25 MeV (see Table X), while the ratio between the partial width for the Ξ b π channel and Γ Sum is predicted to be To look for this state, the Ξ b π decay mode is worth observing in future experiments. The other J P = 5/2 + state |Ξ ′ b 4 D λλ 5 2 + might be also a narrow state with a width of a few tens of MeV. It has large decay rates into Ξ ′ * b π and Σ * b K. If its mass is taken as the predictions in Ref. [21], the sum of the partial widths of the poinic 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 To look for this state, the Ξ ′ * b π and Σ * b K decay modes are worth observing in future experiments.
We also estimate the radiative decays of these J P = 5/2 + states into the 1P-wave bottom baryon states. Our results are listed in Table XI. It is found that for |Ξ ′ − π channels. If one adopts the predicted mass in Ref. [21], the sum of the partial widths of the pionic and kaonic decays is estimated to be Γ Sum ∼ 40 MeV (see Table VIII), and the ratios between 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 The Ξ b π, Λ b K, and Λ b (5912)K decay modes may be ideal modes for our search for |Ξ ′ We also estimate its radiative decays into the 1P-wave bottom baryon states. Our results are listed in Table XI. It is found that the decay rates for these radiative transitions are small. Their branching fractions may be no more than O(10 −4 ).

E. Ω c
In the Ω c family, there are six λ-mode 1D-wave excitations: |Ω c 4 D λλ  Table II). In the possible mass ranges, we study their strong decay transitions within the ChQM. Our results are shown in Fig. 7. To be more specific, taking the masses of the 1D-wave states obtained in the relativistic quark-diquark picture [21], we give the predicted widths in Table XII.  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 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, which are adopted from the RPP [24] and Refs. [1,21]. The units for the partial widths of radiative and strong decays are keV and MeV, respectively. The superscript (subscript) stands for the uncertainty of a prediction with a +10% (−10%) uncertainty of the oscillator parameter α ρ .
Decay mode M f |Ω c 2 D λλ  Decay mode M f |Ω b 2 D λλ The Ξ b (5795)K and Ξ ′ b (5935)K may be ideal channels for us to search for this missing J P = 3/2 + state |Ω b 2 D λλ For the other J P = 3/2 + state |Ω b 4 D λλ 3 2 + , if its mass is taken to be 6549 MeV as predicted in Ref. [21], the width is predicted to be Γ total ∼ 20 MeV. The decays are dominated by the Ξ b (5795)K and Ξ ′ * b (5955)K channels, while the decay rate into Ξ ′ b (5935)K is sizeable as well. Their branching fractions are predicted to be which is sensitive to the phase space of strong decays (see Fig. 7). Ξ b (5795)K and Ξ ′ * b (5955)K may be ideal channels for our search for this missing J P = 3/2 + state |Ω b 4 D λλ Λ b (5912)γ final states, while the J P = 5/2 + state might be established in the Σ * b π and Λ b (5920)γ final states. In the Ξ b family, the J P = 3/2 + state might be established in the Ξ ′ b π, Ξ ′ * b π and Σ b K final states, while the J P = 5/2 + state might be established in the Σ * b K and Ξ ′ * b π final states. The λ-mode 1D-wave excitations of 6 F in the Ω c and Ω b families have a large potential to be found in forthcoming experiments as well. They are fairly narrow states with a width of a few MeV or a few tens of MeV. The kaonic decay channels Ξ c (2470)K, Ξ ′ c (2575)K, and Ξ ′ * c (2645)K may be ideal channels for us to search for these missing 1D-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 missing 1D-wave excited Ω b states.
The λ-mode 1D-wave excitations in the Σ c(b) and Ξ ′ c(b) families appear to have relatively broad widths. The sum of the partial widths with emission of a one-pion meson and onekaon meson is about 50 ∼ 200 MeV. These 1D-wave states might have large decay rates into the 1P-wave heavy baryon states 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 the missing 1D-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 may be ideal channels for looking for the missing 1D-wave excita-tions in the Ξ ′ c family. The Λ b (5912)π, and Λ b (5920)π decay channels 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 predictions bear a fairly large uncertainty from the nonrelativistic harmonic oscillator wave functions adopted in the calculations. Considering a 10% uncertainty of the oscillator parameter, one finds that the uncertainty of our predictions can reach up to ∼ 30%. In some senses, our results are only a semiquantitative estimation based on the SU(3) symmetry. Fortunately, it is found that most of the featured results of the singly heavy baryons predicted in the present work and previous work [11] are consistent with other model approaches and observations, which indicates that the constituent quark model can still serve as a useful tool for investigating the heavy baryon mesonic decays and radiative transitions.