Identification of the newly observed $\Sigma_b(6097)^\pm$ baryons from their strong decays

Two bottom $\Sigma_b(6097)^\pm$ baryons were observed in the final states $\Lambda_b^0\pi^-$ and $\Lambda_b^0\pi^+$ in $pp$ collision by LHCb collaboration, whose masses and widths were measured. In a $^{3}P_{0}$ model, the strong decay widths of two ground $S$-wave and seven excited $P$-wave $\Sigma_b$ baryons have been systematically computed. Numerical results indicate that the newly observed $\Sigma_b(6097)^\pm$ are very possibly $\Sigma_{b2}^1({3\over 2}^-)$ with $J^P={3\over 2}^-$ or $\Sigma_{b2}^1({5\over 2}^-)$ with $J^P={5\over 2}^-$. The predicted decay widths of $\Sigma_b(6097)^\pm$ are consistent with experimental measurement from LHCb. In particular, it may be possible to distinguish these two assignments through ratios $\Gamma({\Sigma_b(6097)^\pm\to \Sigma_b^\pm\pi^0})/\Gamma({\Sigma_b(6097)^\pm\to \Sigma_b^{*\pm}\pi^0})$, which can be measured by experiments in the future. In the meantime, our results support the assignments that $\Sigma_b^\pm$ and $\Sigma_b^{*\pm}$ are the ground $S$-wave $\Sigma_b$ baryons with $J^P={1\over 2}^+$ and $J^P={3\over 2}^+$, respectively.


I. INTRODUCTION
There are two light u, d quarks and one heavy b quark in Σ b baryons, and the two light quarks couple to isospin 1 inside. Four Σ ± b and Σ * ± b have been observed by the CDF collaboration [1,2]. Their spins or parities have not been measured by experiment, they are assigned as the ground S-wave Σ b with J P = 1 2 + and J P = 3 2 + , respectively, in quark models. The assignments need confirmation in more ways. The masses and widths of these baryons from Particle Data Group [3] are given in Table I. Recently, the data were precisely improved by LHCb experiment [4].
In the same LHCb experiment, two bottom Σ b (6097) ± baryons were first observed in final states Λ 0 b π − and Λ 0 b π + in pp collision. The masses and widths of the Σ b (6097) ± are measured The identification of heavy baryons provides an excellent way to explore the structure and dynamics in baryons [5][6][7][8][9]. Therefore, the identification of Σ b (6097) ± is an important topic in the quark model. In Ref. [10], Σ b (6097) ± were explained as P -wave baryons with J P = 3 2 − or J P = 5 2 − based on the mass spectrum analysis and the strong decay calculation in a diquark picture. In Ref. [11], Σ b (6097) ± were also explained as P -wave baryons with J P = 3 2 − or J P = 5 2 − based on their strong decay analysis in a chiral quark model.
As a phenomenological method, 3 P 0 model has been employed to compute the OZI-allowed hadronic decay * Electronic address: zhangal@staff.shu.edu.cn widths of hadrons after its appearance [12][13][14][15]. Though the bridge between the phenomenological 3 P 0 model and QCD has not been established, some attempts have been made [16][17][18]. The 3 P 0 model is also capable of exploring the dynamics and structure of baryons or multi-quark systems. Recently, the approach has been employed to study of the structure of charmed baryons through their strong decays [19][20][21][22][23]. In this work, we will study the P -wave possibility of Σ b (6097) ± in detail. By the way, the ground S-wave Σ b possibility of Σ b and Σ * b will be examined.
The work is organized as follows. In Sec.II, the 3 P 0 model is briefly introduced, some notations of heavy baryons and related parameters are indicated. We present our numerical results and analyses in Sec.III. In the last section, we give our conclusions and discussions.
II. 3 P0 MODEL, SOME NOTATIONS AND PARAMETERS 3 P 0 model is also called a Quark Pair Creation (QPC) model. It was first proposed by Micu [12] and further developed by Yaouanc et al [13][14][15]. The basic idea of this model is assumed that a pair of quark qq is created from the QCD vacuum with vacuum quantum numbers J P C = 0 ++ , and then regroup with the quarks from the initial hadron A to form two daughter hadrons B and C [12]. For a bottom baryon, there are three ways of regrouping shown in the following equations [19,23]: where uub (the constituent quark of initial baryon A) could be replaced by ddb, and the created quark pair uū could be replaced by dd.
In the 3 P 0 model, the hadronic decay width Γ of a In the equation, p is the momentum of the daughter baryon in A's center of mass frame, m A and J A are the mass and the total angular momentum of the initial baryon A, respectively. m B and m C are the masses of the final hadrons. M MJ A MJ B MJ C is the helicity amplitude, which reads [19][20][21]23] The factor 2 in front of γ results from the fact that Eq. (2) and Eq. (3) give the same final states. In the equation above, the matrix of the flavor wave functions ϕ i (i = A, B, C, 0) can also be computed in terms of a matrix of the isospins as follows [15,20] where f takes the value of ( 2 3 ) 1/2 or −( 1 3 ) 1/2 for the isospin 1 2 or 0 of the created quarks, respectively. I A , I B and I M represent the isospins of the initial baryon, the final baryon and the final meson. I 12 , I 3 and I 4 denote the isospins of relevant quarks, respectively.
The space integral follows as, with a simple harmonic oscillator (SHO) wave functions for the baryons [20,24,25] where N = 3 3 4 represents a normalization coefficient of the total wave function. Explicitly, where L L+1/2 n p 2 β 2 denotes the Laguerre polynomial function, and Y LML (Ω p ) is a spherical harmonic function. The relation between the solid harmonica polynomial In above equations, Jacobi coordinates ρ and λ [26] were employed.
Notations and internal structures of the heavy baryons in quark model are explained in Refs. [19,22,23,27]. In this model, there are two S-wave and seven P -wave Σ b . In other quark model, the structure and dynamics in baryons may be different. In fact, the difference is an indication of the complexity of baryon. Accordingly, the numbers of P -wave Σ b may be different in different models. For example, there are five P -wave Σ b in Refs. [10,11,[29][30][31]. For a practical calculation, the quantum numbers of two 1S-wave and seven 1P -wave Σ b baryons are presented in Table II.
In this table, L ρ denotes an orbital angular momentum between the two light quarks, L λ denotes the orbital angular momentum between the bottom quark and the two light quark system, and L is the total orbital angular momentum of L ρ and L λ (L =L ρ + L λ ). S ρ denotes the total spin of the two light quarks, J l is the total angular momentum of L and S ρ (J l = L + S ρ ), and J is the total angular momentum of the baryons (J = J l + 1 2 ). For Σ L bJ l , a superscript L is specialized to denote different total angular momentum. The tilde indicates L ρ = 1, and the blank indicates L ρ = 0.

III. NUMERICAL RESULTS
A. Decays of Σ b and Σ * b Σ ± b and Σ * ± b were first observed in the final states Λ 0 b π ± in pp collision by the CDF collaboration [1], and were interpreted as the lowest-lying Σ ± b and Σ * ± b baryons with into Λ 0 b π ± as 1S-wave states or 1P -wave excitations.  and Σ * + b , and Λ 0 b π − is the only decay mode of Σ − b and Σ * − b . In the 3 P 0 model, the hadronic decay widths of these four observed Σ ± b into Λ 0 b π ± in two S-wave and seven P -wave assignments are computed and presented in Table IV, where a '0' indicates a vanish decay channel. In comparison with experimental results (see Table I As pointed out in the second section, there are seven P -wave Σ b baryons. The masses of low-lying bottom baryons have been systemically predicted in many references such as [28][29][30][31]. If Σ b (6097) ± are P -wave Σ b baryons, there exist five possible OZI-allowed hadronic decay modes. The five channels are : Λ 0 b π ± , Σ ± b π 0 , Σ * ± b π 0 , Λ 0 b (5912)π ± and Λ 0 b (5920)π ± . The strong decay widths of Σ b (6097) − into these five channels are calculated in seven different P -wave assignments, and presented in Table V. The results of Σ b (6097) + are presented in Table VI. In the calculation, Σ b and Σ * b are set to the ground S-wave Σ 0 b1 ( 1 2 + ) and Σ 0 b1 ( 3 2 + ) as indicated in previous subsection. Λ 0 b (5912) and Λ 0 b (5920) were observed in Λ 0 b π + π − in pp collision by LHCb [35], and interpreted as the orbitally excited Λ * 0 b (5912) and Λ * 0 b (5920) though their exact assignment as the P -wave Λ b has not been made. For simplicity, Λ 0 b (5912) and Λ 0 b (5920) are set to Λ b ( 1 2 − ) and Λ b ( 3 2 − ), respectively.   Based on numerical results, Σ b (6097) ± are very possi- where there is no ρ-mode excitation inside. Under theoretical uncertainties, the total decay widths (Γ ≈ 19 − 20 MeV ) are consistent with the experimentally measured ones by LHCb. In both assignments, Λ 0 b π ± are their dominant decay channels with branching fraction ratios ≈ 72 − 74%.
, the branching fraction ra- These ratios can be employed by experiment to distin- ). They depend weakly on the parameters in the 3 P 0 model.
There are some uncertainties in the 3 P 0 model. In addition to the masses of the hadron involved in the decay, the strong decay widths depend on some parameters such as γ and β. In principle, β can be derived directly in quark model. Unfortunately, β of baryons have not yet been determined for a complexity of quark dynamics in baryons. β were also set to those for mesons as in existed references. These uncertainties may change some predicted decay widths.