Interpretation of the newly observed $\Lambda_b(6146)^{0}$ and $\Lambda_b(6152)^0$ states in a chiral quark model

The strong decays of the low-lying $\lambda$-mode $\Lambda_b(1D,2S)$ and $\Sigma_b(2S)$ states are studied in a chiral quark model. We find that: (i) the newly observed $\Lambda_b(6146/6152)$ resonances in the $\Lambda_b\pi^+\pi^-$ spectrum by the LHCb Collaboration might be explained with the $\lambda$-mode $\Lambda_b(1D)$ states in the quark model. It should be emphasized that whether the structure in the $\Lambda_b\pi^+\pi^-$ spectrum correspond to two states or one state should be further clarified with more observations in future experiments. (ii) The $\Lambda_b(2S)$ $|J^P = \frac{1}{2}^+,0\rangle$ state mainly decays into $\Sigma_b^*\pi$ channel, which may be an ideal channel for searching for this $\Lambda_b(2S)$ state in future experiments. (iii) The $\Sigma_b(2S)$ $|J^P = \frac{1}{2}^+,1 \rangle$ and $|J^P = \frac{3}{2}^+,1 \rangle$ dominantly decay into $\Lambda_b\pi$ with $\Gamma \simeq 3.82$ MeV and $\Gamma \simeq 4.72$ MeV, respectively.

To identify the Λ b (6146) and Λ b (6152) resonances in the quark model, their strong decay behaviors were studied with the 3 P 0 model in Ref. [41]. The results indicated that the Λ b (6146) and Λ b (6152) can be assigned as the Λ b (1D) doublet with J P = 5/2 + and J P = 3/2 + , respectively. In our previous works [21,34], we have studied the strong decays of the singly bottom baryons Σ b (1P) within the chiral quark model. We find that the Σ b (6097) favors the light spin j = 2 states with spin-parity numbers J P = 3/2 − or J P = 5/2 − . If Σ b (6097) corresponds to the |J P = 3 2 − , 2 or |J P = 5 2 − , 2 state, the typical mass for the Σ b (1P) states should be around 6090 MeV since the mass splitting between them is within 10 MeV according to quark model predictions [28]. Thus, one should exclude the Σ b (6146) and Σ b (6152) resonances as the Σ b (1P) states for the masses of Σ b (6146) and Σ b (6152) are obviously larger than the quark model predictions.
In previous works, the decays of the P-and D-wave singly heavy baryons have been studied within the chiral quark model [34,42]. Since the newly observed Λ b (6146) and Λ b (6152) resonances may favor the D-wave Λ b states, to confirm this assignment, in present work we revisit the strong decays of the D-wave Λ b states by adopting the measured masses. Moreover, the 2S -wave Λ b and Σ b states were not investigated in the frame work of chiral quark model. Hence, in this work, as a supplement of Refs. [34,42], we employ the chiral quark model to study the strong decays with emission of one light pseudoscalar meson for the low-lying Λ b (1D, 2S ) and Σ b (2S ) states. Combining the masses, total widths and decay modes, our results suggest that the Λ b (6146) and Λ b (6152) states may favor the Λ b (1D) |J P = 5 2 + , 2 and I: Mass spectra of the Λ b up to D wave and the Σ b up to 2S wave from various quark models [25,27,28,30,37] This paper is organized as follows. The spectrum and notations are presented in Sec. II. The chiral quark model are briefly introduced in Sec. III. The strong decays of the lowlying Λ b (1D, 2S ) and Σ b (2S ) states are estimated in Sec. IV. A short summary is presented in the last section.

II. SPECTROSCOPY
TABLE II: The classifications of the low-lying 1P and 2S -wave states belonging to 6 F in the j − j coupling scheme.
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. According to the symmetry, the heavy baryons containing a single heavy quark belong to two different SU(3) flavor representations: the symmetric sextet 6 F and antisymmetric antitriplet 3 F . In the singly bottom baryons, Λ b and Ξ 0,− b belonging to and Ω b form a 6 F representation.
It should be pointed out that the quark model states that we obtain for the baryons containing a single heavy quark respect the dictates of the heavy quark effective theory (HQET). In the heavy quark effective theory description, the total angular momentum of the two light quarks j = L + s ρ is conserved and coupled to the spin of a heavy quark with spin s Q = 1/2 [3,27]. The total angular momentum can take the values J = j + s Q . In the heavy-quark symmetry limit, the quark model states may more favor the j − j coupling scheme The Σ b 1P-and 2S -wave states belonging to 6 F and the Λ b 1P-, 2S -and 1D-wave states belonging to3 F in the j − j coupling scheme and their corresponding quantum numbers have been collected in Table II and III.
To match the nonrelativistic harmonic oscillator wave functions in this work, one should adopt the quark-pseudoscalarmeson interactions in the nonrelativistic form [15,[44][45][46][47][48][49][50]: with G ≡ −(1 + ω m E f +M f ) and h ≡ ω m 2µ q . In the above equation, ω m and q are the energy and three momenta of the emitted light meson, respectively; µ q stand for a reduced mass given by 1/µ q = 1/m j + 1/m ′ j with m j and m ′ j for the masses of the jth quark in the initial and final hadrons, respectively; σ j and p j are the Pauli spin vector and internal momentum operator for the jth quark of the initial hadron; and I j is the isospin operator associated with the pseudoscalar meson.
For a light pseudoscalar meson emission in a strong decay process, the partial decay width can be calculated with [45,48] Γ m = δ f m where M J f z ,J iz corresponds to the strong amplitudes. The quantum numbers J iz and J f z stand for the third components of the total angular momenta of the initial and final heavy baryons, respectively. M i is the mass of the initial heavy baryon. E f and M f are the energy and mass of the final heavy baryon. δ as a global parameter accounts for the strength of the quark-meson couplings. It has been determined in our previous work of the strong decays of the charmed baryons and heavy-light mesons [45,48]. Here, we fix its value the same as that in Refs. [45,48], i.e. δ = 0.557. In the calculation, the standard quark model parameters are adopted. Namely, we set m u = m d = 330 MeV and m b = 5000 MeV for the constituent quark masses. The harmonic oscillator parameter α ρ in the wave function ψ n lm = R nl Y lm for uu/ud/dd diquark systems is taken as α ρ = 400 MeV. Another harmonic oscillator parameter α λ can be related to α ρ with the relation α 2 λ = √ 3m b /(2m + m b )α 2 ρ . The decay constant for π meson is taken as f π = 132 MeV. The masses of the well-established hadrons used in the calculations are taken from the Particle Data Group (PDG) [1], and the masses of the undiscovered initial states adopt from the predictions in Refs. [25,28].

IV. RESULTS AND DISCUSSIONS
In the Λ b family, there are two λ-mode 1D-wave excitations |J P = 3 2 + , 2 and |J P = 5 2 + , 2 according to the classification of quark models. The masses for the λ-mode 1D-wave Λ b excitations are predicted to be ∼ 6.1-6.2 GeV (see Table I). The measured masses of the Λ b (6146) and Λ b (6152) indicate that they may be good candidates of the λ-mode 1D-wave excitations.
Our calculations are presented in Table IV. The Λ b (6146) resonance is most likely to be the J P = 5/2 + |J P = 5 2 + , 2 state. Assigning the Λ b (6146) as |J P = 5 2 + , 2 , one can find that it has a narrow width of ∼ 5 MeV, and dominantly decays into Σ * b π. The partial widths into Σ b π and Σ * b π channels are predicted to be Both the decay width and decay mode are consistent with the observations of Λ b (6146) considering the model uncertainties. This conclusion is consistent with that of the 3 P 0 model [41].
To further confirm the nature of the Λ b (6146) resonance, the partial width ratio between Σ b π and Σ * b π channels, is worth to observing in future experiments. On the other hand, the Λ b (6152) resonance most likely corresponds to the λ-mode 1D-wave Λ b excitation |J P = 3 2 + , 2 .
When we assign the Λ b (6152) as |J P = 3 2 + , 2 , it has a narrow width of ∼ 6 MeV, and dominantly decays into Σ b π and Σ * b π. We predicted partial widths which are in agreement with the the 3 P 0 model predictions [41]. The partial width ratio between Σ b π and Σ * b π channels is predicted to be This ratio might be crucial to test the nature of Λ b (6152), which is suggested to be measured in future experiments. Finally, it should be mentioned that the Λ c (2860)3/2 + and Λ c (2880)5/2 + are often assigned to 1D doublet in the Λ c family [42,53,54]. It indicates that the mass of the J P = 3/2 + Dwave state should be smaller than that of the J P = 5/2 + state. Then, if assigning Λ b (6146) and Λ b (6152) to be J P = 5/2 + and J P = 3/2 + D-wave states, respectively, one should face a serious problem of mass reverse. Whether the structure in the Λ b π + π − spectrum correspond to two states or one state should be further clarified with more observations in future experiments.  In the Λ b family, there is only one λ-mode 2S -wave excitation |J P = 1 2 + , 0 according to the quark model classification.
The mass for the λ-mode 2S -wave Λ b excitation is predicted to be ∼ 6.1 GeV in various quark models (see Table I). According to the predicted masses in Ref. [37], the measured mass of the Λ b (6146) or Λ b (6152) indicates that it might be a good candidate of the λ-mode 2S -wave excitation. Considering Λ b (6146) or Λ b (6152) as the 2S -wave state, the strong decay properties are studied, our results are listed in Table V. It is found that if assigning Λ b (6146) to the 2S -wave state, both the total width Γ ≃ 2 MeV and the dominant decay mode Σ * b π predicted in theory are consistent with the observations. However, the other resonance Λ b (6152) cannot be understood in the quark model. It cannot be assigned to any D-wave states in the Λ b family, because the mass splitting between the 2Swave and D-wave Λ b states is ∼ 50 − 100 MeV. The D-wave Σ b states should be excluded as well for their typical mass is ∼ 6.3 GeV [25,42]. As a whole if we assign Λ b (6146) as the Λ b (2S ) state, the other state Λ b (6152) cannot be reasonably explained according to the classification of quark models and mass splitting [37].
Since the Λ b (2S ) |J P = 1 2 + , 0 state may not favor Λ b (6146), in this work, we take its mass M = 6045 MeV as predicted in Ref. [25], and estimate the strong decay of Λ b (2S ) into the Σ b π and Σ * b π channels. It is found that Λ b (2S ) mainly decays into Σ b π and Σ * b π modes. The predicted partial widths are and the corresponding total decay width reads The Σ * b π might be an ideal channel for searching for the λmode 2S -wave Λ b state |J P = 1 2 + , 0 in future experiments.
We also study the strong decay properties of two λ-mode 2S -wave excitations |J P = 1 2 + , 1 and |J P = 3 2 + , 1 in the Σ b family according to the quark model classification. The masses for the λ-mode 2S -wave Σ b excitations are predicted to be 6.2 ∼ 6.3 GeV in various models (see Table I).
To study the strong decay properties of the 2S wave Σ b excitations, we adopt the predicted masses in Ref. [28]. Our results are listed in Table VI. One can see that |J P = 1 2 + , 1 and |J P = 3 2 + , 1 dominantly decay into the Λ b π channel. Their partial widths are predicted to be The Λ b π might be an ideal channel to look for these radial excitations in future experiments. Finally, it should be mentioned that we do not consider the Λ b (6146) and Λ b (6152) as the 2S Σ b states, for the mainly decay modes and masses of Σ b (2S ) predicted in the quark model are inconsistent with the observations.

V. SUMMARY
In this work, we study the strong decays of the low-lying λmode Λ b (1D, 2S ) and Σ b (2S ) states. Our results indicate that the newly observed Λ b (6146) might be assigned as the Λ b (1D) |J P = 5 2 + , 2 state, which dominantly decays into Σ * b π channel.
The Λ b (6152) seems to favor the Λ b (1D) |J P = 3 2 + , 2 states, with this assignment, its decay behaviors are dominated by the Σ b π and Σ * b π channels, which are consistent with the experimental observations. However, if we assign Λ b (6146) and Λ b (6152) to the J P = 5/2 + and J P = 3/2 + D-wave states, respectively, one should face a serious problem of mass reverse. Whether the structure in the Λ b π + π − spectrum corresponds to two states or one state should be further clarified with more observations in future experiments. Moreover, we find that the Λ b (2S ) |J P = 1 2 + , 0 state mainly decays into Σ * b π, and the Σ b (2S ) |J P = 1 2 + , 1 and |J P = 3 2 + , 1 dominantly decay into Λ b π with narrow widths Γ ≃ 3.82 MeV and Γ ≃ 4.72 MeV, respectively. These theoretical predictions may provide helpful information for future experimental searches.