Toward establishing the low-lying $P$-wave $\Sigma_b$ states

In the present work, we analyze the $P$-wave singly-heavy baryon spectrum belonging to $\mathbf{6}_F$ by combining the observations of the heavy baryon states, and restudy the strong decays of the $1P$ wave $\Sigma_b$ states within the $j$-$j$ coupling scheme using the chiral quark model. We obtain that: (i) the structure $\Sigma_b(6097)$ observed in the $\Lambda_b\pi$ final state may arise from the overlapping of $\Sigma_b|J^P=\frac{3}{2}^-,2\rangle$ and $\Sigma_b|J^P=\frac{5}{2}^-,2\rangle$. (ii) The broad structure $\Sigma_b(6072)$ observed in the $\Lambda_b\pi\pi$ final state may arise from the overlapping of $\Sigma_b|J^P=\frac{1}{2}^-,1\rangle$ and $\Sigma_b|J^P=\frac{3}{2}^-,1\rangle$. (iii) The missing state $\Sigma_b|J^P=\frac{1}{2}^-,0\rangle$ is most likely to be a narrow state with a width of $\Gamma\sim10$ MeV, and mainly decays into $\Lambda_b\pi$ channel.


I. INTRODUCTION
The LHC experiments have demonstrated their discovery capability of heavy flavored baryons. During the past several years, many missing heavy baryon states have been discovered by LHC experiments. For the charmed baryon sector, in 2017 the LHCb Collaboration observed five extremely narrow Ω c (X) states, Ω c (3000), Ω c (3050), Ω c (3066), Ω c (3090) and Ω c (3119), in the Ξ + c K − channel [1], and the first four of them were confirmed by the subsequent Belle experiments [2]. Very recently, the LHCb Collaboration observed three new states Ξ c (2923) 0 , Ξ c (2939) 0 , and Ξ c (2965) 0 in the Λ + c K − mass spectrum with a large significance [3]. For the bottom baryon sector, a great progress has been achieved as well. In 2018, the LHCb collaboration reported two new bottom baryon states Σ b (6097) ± in the Λ 0 b π ± channels [4], and another new state Ξ b (6227) − in both Λ 0 b K − and Ξ 0 b π decay modes [5]. Very recently, the LHCb Collaboration observed four narrow Ω b (X) states Ω b (6316), Ω b (6330), Ω b (6340), and Ω b (6350) [6], in the Ξ 0 b K − final state.
Recently, the CMS collaboration observed a broad enhancement around 6070 MeV in the Λ 0 b π + π − invariant mass spectrum [31], which was confirmed by the subsequent LHCb experiments with high significance [32]. The mass and width * E-mail: lyxiao@ustb.edu.cn † E-mail: zhongxh@hunnu.edu.cn for the structure are determined to be From the point of view of mass, this structure around 6070 MeV [denoted with Σ b (6072)] may be some signals of the Pwave states in the Σ b family, although it was interpreted as the first radially excited Λ b state, Λ b (2S ), in the literature [32,33]. Together with the other P-wave candidate Σ b (6097) observed in the Λ 0 b π ± channels about two years ago at LHCb [4], the Σ b (6072) provide us a good chance to establish a fairly complete P-wave spectrum in the Σ b family.
In the present work we re-study the 1P wave Σ b states. We hope to understand the nature of the broad structure Σ b (6072) observed by the CMS collaboration [31] and LHCb collaboration [32]. Furthermore, by combining the newest observations of the heavy baryon states, we give our opinion for establishing a complete P-wave Σ b spectrum. This paper is organized as follows. In Sec. II we give an analysis of the 1P wave singly-heavy baryons' spectra. We discuss the strong decay properites of the 1P wave Σ b states within the j-j coupling scheme in Sec. III and summarize our results in Sec. IV.  [26][27][28][29][30].
L-S scheme j-j scheme Observed states/structures belonging to 6 F multiplet. It is interesting to find that the newly observed state Ω c (3000) at LHCb [1] may be explained as a J P = 1/2 − resonance via a | 2 P 1/2 -| 4 P 1/2 mixing with a mixing angle φ ≃ 24 • [30], which is smaller than 35 • from the j-j coupling scheme. This may indicate that the physical states of the singly-heavy baryon states should lie between L-S and j-j coupling schemes. The physical state for the bottom sector may be closer to the j-j coupling scheme than that for the charmed sector because the bottom quark is much heavy than the charm quark.
It is difficult to predict the mass order and mass splittings in theory due to the unclear spin-dependent interactions for quarks. In fact, the mass order and mass splittings between the P-wave spin multiplets are crucial to determine their quantum numbers. Fortunately, the recent observations of the heavy baryons in the Ω c , Ξ ′ c , and Ω b families provide us important information about the mass order and mass splitting. , respectively. The states labeled with "?" are waiting to be discovered in future experiments.
According to our previous studies in Refs. [26,30] we further find that the four Ω b (6316), Ω b (3330), Ω b (6340) and Ω b (6350) are flavor partners of the four Ω c (X) states Ω c (3000), Ω c (3050), Ω c (3065) and Ω c (3090), respectively. They are good candidates of the P-wave singly-heavy baryon states with spin-parity numbers J P = 1/2 − , J P = 3/2 − , J P = 3/2 − , J P = 5/2 − , respectively. The possible quark model classifications of these possible P-wave singly-heavy baryons observed in experiments are summarized in Table I. If our assignments are correct, in the heavy quark symmetry limit, the mass order for the P-wave singly-heavy baryon states might be Until now, we may conclude that the Σ b (6097) and . Furthermore, we also should mention that the Σ b (6097) and Σ b (6072) observed in experiments may be caused by several nearby states due to a rather small mass splitting, ∼ 10 MeV, between two nearby states. Finally, it should be pointed out that there is less knowledge about the P-wave state |J P = 1 2 − , 0 . Only a hint comes from the recent LHCb experiments [3]. It is found a broad structure Ξ c (2880) in the Λ + c K − mass spectrum with a small significance [3]. The Ξ c (2880) may be a candidate of the |J P = 1 2 − , 0 in the Ξ ′ c family. If this is confirmed by future experiments, the two Pwave J P = 1/2 − states |J P = 1 2 − , 1 and |J P = 1 2 − , 0 may be largely overlapping states according to the equal spacing rule [34,35].
As a whole, combining the equal spacing rule with the observations of the heavy baryon states, we can predict a fairly complete P-wave singly-heavy baryon spectrum belonging to 6 F . These predictions may be helpful to looking for the missing state in forthcoming experiments at LHC and/or Belle II.

III. STRONG DECAY ANALYSIS
As a possible explanation of the broad structure below 6100 MeV in the Λ b ππ invariant mass spectrum observed by the CMS collaboration [31] and LHCb collaboration [32], it is crucial to re-study the strong decay properties of the low-lying λ-mode 1P wave Σ b states. In the following we will analyze the strong decays of the P-wave Σ b states with the chiral quark model. This model has been successfully applied to the strong decays of heavy-light mesons, charmed and strange baryons [30,[36][37][38][39][40][41][42][43][44]. The model parameters have been well determined in theses works. In present work, the parameters are adopted the same as those of our previous studies of the singly-heavy baryons [30,36,38,39,44], where the details for the framework can be found as well.
A. J P = 1/2 − states In previous mass spectrum analysis, it is found that the Σ b (6072) may be assigned as the J P = 1/2 − states Σ b |J P = 1 2 − , 1 . We further test this assignment from the sector of strong decay properties.
There are two J P = 1/2 − states Σ b |J P = 1 2 − , 1 and Σ b |J P = 1 2 − , 0 within the j-j coupling scheme (see Table II). They can be expressed as mixed states between | 2 P λ 1 2 − and | 4 P λ with the L-S coupling scheme: Their masses are predicted to about M ≃ (6050 − 6140) MeV (see Table II), and their OZI-allowed two body strong decay channels are Σ ( * ) b π, Λ b π, Λ b (5912)π and Λ b (5920)π. To know about the effects of the uncertainties of the masses on the decay properties of the Σ b |J P = 1 Considering the Σ b (6072) observed in the Λ b π + π − final state at CMS [31] and LHCb [32] as the Σ b |J P = 1 2 − , 1 state, its width is predicted to be Γ ≃ 28 MeV.
The predicted branching fraction of the dominant decay mode The observed Λ b ππ decay mode of Σ b (6072) can be naturally explain with the cascade decay process Σ b (6072) → Σ b π → Λ b ππ. The decay rates into the Σ * b π, Λ b (5912)π and Λ b (5920)π final states are small (see Table III). The decay predicted width of Σ b (6072) is about a factor 2 smaller than the observations from the CMS [31] and LHCb [32], which indicates that the structure Σ b (6072) 0 observed at CMS [31] and LHCb [32] may not be well understood with the Σ b |J P = iments [4], which may be due to a too large energy scanned step, ∼ 6 MeV, adopted in measurements. In previous mass spectrum analysis, it is found that the Σ b (6097) may be assigned as the J P = 3/2 − states Σ b |J P = 3 2 − , 2 . We further test this assignment from the sector of strong decay properties.
In the Σ b family, there are two λ-mode 1P wave J P = 3/2 − states Σ b |J P = 3 2 − , 2 and Σ b |J P = 3 2 − , 1 . They can relate to the states in the L-S coupling scheme: According to the predicted masses collected in Table II,    find it has a relatively broad width of (see Table III) which is slightly larger than the observed width of Γ exp. ≃ 30 MeV at LHCb [4]. Our predicted branching fraction of the dominant decay mode Σ b π is which is consistent with the fact that the Σ b (6097) resonance was first observed in the final state Λ b π [4]. Meanwhile, the decay rate of Σ b (6097) into Σ b π is considerable, and the predicted branching fraction is The decay mode Σ b π may be observed in future experiments. As a whole, the state Σ b (6097) is a good candidate of the The dominant decay mode is Σ * b π with the branching fraction To establish the Σ b |J P = 3 2 − , 1 state, the Σ * b π is worth to observing in future experiments. Since the Σ * b dominantly decay into the Λ b π channel, thus, the Σ b |J P = 3 2 − , 1 state should also have a large decay rate into the Λ b ππ channel via the intermediate Σ * b π process. Considering the mass of Σ b |J P = 3 2 − , 1 , M ≃ 6087 MeV, is very to that of the broad structure Σ b (6072) observed in the Λ b ππ invariant mass spectrum [31,32], thus, we may conclude that the Σ b |J P = 3 2 − , 1 state should contribute to the structure Σ b (6072) as well. Then, we can understand why we cannot explain width of the Σ b (6072) with the Σ b |J P = 3 2 − , 1 state only. As a conclusion, the broad structure may be caused by two highly overlapping states Σ b |J P = 1 2 − , 1 and It should be pointed out that in Refs. [32,33] the Σ b (6072) was suggested to be the Λ b (2S ) resonance. However, as the Λ b (2S ) assignment, the width of Σ b (6072) is predicted to be very narrow (Γ ∼ 1 MeV) with our chiral quark model [45]. Within the quark pair creation model, the authors in Ref. [46] obtained that the total decay widths of Λ b (2S ) was about ∼12 MeV, which was also inconsistent with the measured widths Γ = 72 ± 11 ± 2 from the LHCb [32].
There is only one λ-mode 1P wave Σ b state with J P = 5/2 − , which is no difference between the j-j coupling scheme and L-S coupling scheme, namely The predicted mass of the state Σ b |J P = 5 2 − , 2 is listed in Table II. From the table, and the predicted branching ratio is Meanwhile, the decay rate of Σ b |J P = 5 2 − , 2 into Σ * b π is considerable with the predicted branching fraction We notice that the measured natures of the new peak Σ b (6097) observed in the Λ b π [4] are good consistent with the predicted properties of the state Σ b |J P = 5 2 − , 2 . Thus, the new peak Σ b (6097) can be explained as the J P = 5/2 − state Σ b |J P = 5 2 − , 2 as well. Hence, our predictions indicate that the new peak Σ b (6097) has two candidates, Σ b |J P = 3 2 − , 2 and Σ b |J P = 5 2 − , 2 . Since the similar decay properties and close masses for the two candidates, it is difficult to clearly distinguish them. Thus, the new peak Σ b (6097) most likely arises from the overlapping of Σ b |J P = 3 2 − , 2 and Σ b |J P = 5 2 − , 2 .

IV. SUMMARY
In this paper, we predict a fairly complete P-wave singlyheavy baryon spectrum belonging to 6 F by combining the equal spacing rule with the observations of the heavy baryon states. Further combining the mass spectrum with the strong decay properties of the 1P wave Σ b states, we give possible interpretations for the newly observed structures Σ b (6072) and Σ b (6097).
It is found that the newly observed peak Σ b (6097) ob-served in the Λ b π final state is most likely to arise from the overlapping of Σ b |J P = 3 dominantly decay into Σ b π and Σ * b π, respectively, with a comparable width, Γ ∼ 30 MeV.
The Σ b |J P = 1 2 − , 0 , is most likely to be a quite narrow state with a total decay width of Γ ∼ 10 MeV. The Λ b π decay channel almost saturates its total decay widths. Considering the predicted width of Σ b |J P = 1 2 − , 0 being narrow, this state might be observed in the Λ b π channel when enough data are accumulated in experiments.