The analysis of the excited bottom and bottom strange states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{s2}^{*}(5840)$, $B_{J}(5840)$ and $B_{J}(5970)$ in B meson family

In order to make a further confirmation about the assignments of the excited bottom and bottom strange mesons $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{s2}^{*}(5840)$ and meanwhile identify the possible assignments of $B_{J}(5840)$, $B_{J}(5970)$, we study the strong decays of these states with the $^{3}P_{0}$ decay model. Our analysis support $B_{1}(5721)$ and $B_{2}^{*}(5747)$ to be the $1P_{1}'$ and $1^{3}P_{2}$ assignments and the $B_{s1}(5830)$, $B_{s2}^{*}(5840)$ to be the strange partner of $B_{1}(5721)$ and $B_{2}^{*}(5747)$. Besides, we tentatively identify the recently observed $B_{J}(5840)$, $B_{J}(5970)$ as the $2^{3}S_{1}$ and $1^{3}D_{3}$ states, respectively. It is noticed that this conclusion needs further confirmation by measuring the decay channel to $B\pi$ of $B_{J}(5840)$ and $B_{J}(5970)$ in experiments.

In our previous work, we studied the two-body strong decays of the B 1 (5721), B * 2 (5747), B(5970), B s1 (5830) and B s2 (5840) with the heavy meson effective theory in the leading order approximation, and assigned states 2S1 − , 1D1 − and 1D3 − as the candidate of B J (5970) [23]. As a continuation of our previous work, we study the strong decay behaviors of more bottom mesons with the 3 P 0 decay model and give a simple discussion about the quantum numbers of these mesons. The calculated strong decay widths in this work will be confronted with the experimental data in the future and will be helpful in determining the nature of these heavy-light mesons. This article is arranged as follows: In section 2, we give a brief review of the 3 P 0 decay model; in Sec.3 we study the strong decays of  To study the strong decay properties of the mesons, the 3 P 0 decay model is an effective and simple method, which can give a good prediction about the decay behaviors of many hadrons [34][35][36][37][38]. This model was first introduced by Micu in 1969 [20] and further developed by Le Yaouanc and other collaborations [21,22]. In Ref. [39] Barnes et al. performed a comprehensive study of light meson strong decays with the 3 P 0 model. Now, this model has been extensively used to describe the strong decays of the heavy mesons in the charmonium [40][41][42][43] and bottommonium systems [44][45][46], the baryons [47] and even the teraquark states [48].
At first, people considered an alternative phenomenological model to study the strong decays, in which quark-antiquark pairs are produced with 3 S 1 quantum numbers. However, this possibility is disfavoured by measuring ratios of partial wave amplitudes [49]. In 3 P 0 decay model, it is now generally accepted that a quark-antiquark pair(q 3 q 4 ) with 0 ++ quantum numbers(in the 3 P 0 state) is created from the vacuum [20][21][22]34]. For a meson decay process A→BC, the quark-antiquark pair(q 3 q 4 ) regroups into final state mesons(BC) with the q 1 q 2 from the initial meson A. This process is illustrated in FIG.1 and its transition operator in the nonrelativistic limit is written as, where q † 3 and q † 4 are the creation operators in the momentum-space for the quark-antiquark q 3 q 4 pair. γ is a dimensionless parameter reflecting the creation strength of the quark-antiquark pair. ϕ 34 0 , ω 34 0 and χ 34 1−m denote its flavor, color and spin wave functions.
In the c.m. frame, the amplitude of a decay process A → BC can be written as, where χ 14 are the spin and flavor matrix elements.
The two terms in the last factor correspond to the two possible diagrams in FIG.1. The momentum space integral I( P , m 1 , m 2 , m 3 ) is given by where P = P B = − P C , p = p 3 , m 3 is the mass of the created quark q 3 . In Eq.(3), ψ is the simple harmonic oscillator (SHO) function which is use to describe the space part of the meson. In momentum space, it is defined as where R is the scale parameter of the SHO. With the Jacob-Wick formula, we can convert the helicity amplitude into the partial wave amplitude The decay width in terms of partial wave amplitudes using the relative phase space is is the three momentum of the daughter mesons in the c.m. frame. M A , M B , and M C are the masses of the mesons A, B, and C, respectively. One can consult references [20][21][22]34] for more details of the decay model.

The results and discussions
The parameters involved in the 3 P 0 model include the light quark pair(qq) creation strength γ, the SHO wave function scale parameter R, and the masses of the mesons and the constituent quarks. First, the masses of the quark are taken as m u = m d = 0.22 GeV, m s = 0.42 GeV and m b = 4.81GeV [7].
Second, as for the factor γ, it describes the strength of quark-antiquark pair creation from the vacuum and its value needs to be fitted according to experimental data. We take the fitted value γ = 6.25 for u/d quark and γ ss = γ/ √ 3 for s quark [34]. This value is higher than that used by Kokoski and Isgur by a factor of √ 96π due to different field theory conventions, constant factors in T , etc [50].
The input parameter R has a significant influence on the shape of the radial wave function, which lead to the spatial integral of Eq.(3) being sensitive to the parameter R. Thus, the decay width based on the 3 P 0 decay model is also sensitive to the parameter R. Taking the strong decay of B * 2 (5747) as an example, we plot the decay width versus the input parameters R in FIG.2. We can clearly see the dependence of the decay widths on the input parameter R. If the R B 0 , R B + , R B * 0 , R B * + and R π are fixed to be 2.5GeV −1 , the decay width of B * 2 (5747) changes several times with the value of R B * 2 (5747) changing from 2.0GeV −1 to 3.0GeV −1 . As for this problem, there are two kinds of choices which are the common value and the effective value. The effective value is fixed to reproduce the realistic root mean square radius by solving the Schrodinger equation with a linear potential. For the common value, H.G. Blunder et al [34] carry out a series of least squares fits of the model predictions to the decay widths of 28 of the best known meson decays. And the common oscillator parameter R with the vaue 2.5GeV −1 is suggested to be optimal. In our previous work, we studied strong decays of some charmed mesons with common value and obtained consistent results with experimental data.
Thus, we still adopt common value as the input of R in this work. Finally, the mass of the meson has also a significant influence on the strong decay of the studied meson. For B * 2 (5747) as an example, if the masses of the daughter mesons are taken to be the standard values in PDG, the decay widths of B * 2 (5747) vary greatly with its mass, which can be seen in FIG.3. We know that the masses of the bottom mesons, especially the newly observed bottom states, have  It is noticed that mixing can occur between states with J = L and S = 1 or S = 0. The relation between the heavy quark symmetric states and the non-relativistic states 3 L L and 1 L L is written as [51], For the states J = L = 1, the mixture angle is θ = −54.7 • or θ = 35.3 • , thus this relation transforms For a decay process A → BC, if the initial states A(l P )are the mixture, the partial wave amplitude can be written as In our calculations, the states B 1 (5721), B s1 (5830) are the 1 + bottom and bottom-strange states and each of them is the mixing of 3 P 1 and 1 P 1 states. In addition, we will study the strong decays of B J (5970) as the 2 − state and it is the mixture of 3 D 2 and 1 D 2 states. Considering the mixture of the initial states, the decay width can be expressed as  The bottom mesons B * + 2 (5747), B * 0 2 (5747) are assigned to be the 2 + state with their total decay widths to be 20 ± 5M eV and 24.2 ± 1.7M eV , respectively. As the 1 3 P 2 (2 + ) states, we calculate their strong decay widths and the results 23.9M eV and 24.7M eV for B * + 2 (5747), B * 0 2 (5747) are consistent well with these experimental data. A further confirmation of this assignment is the predicted versus measured ratio of partial widths to B 0 π + and B * 0 π + . The predicted partial ratio Γ B * + 2 (5747)→B 0 π + Γ B * + 2 (5747)→B * 0 π + = 1.18 is in agreement with the experimental data 1.12, and so does for the B * 0 2 (5747). As for B + 1 (5721), B 0 1 (5721) mesons, each of them is the mixing bottom state of 3 P 1 and 1 P 1 . In TABLE III and TABLE  IV, the 1P 1 , 1P ′ 1 states denote the j q = 1 2 and j q = 3 2 state, respectively. We can see that the results for j q = 3 2 (1P ′ 1 ) bottom states with total decay widths to be 39.8M eV , 37.9M eV , are roughly compatible with the experimental data 31 ± 6M eV and 27.5 ± 3.4M eV . These results favor B 1 (5721) to be the j q = 3 2 spin partner of B * 2 (5747) state After identifying the 1P ′ 1 assignment, the remaining 1P 1 together with 1 3 P 0 state are the spin doublets with j q = 1 2 . The total widths of 1 3 P 0 are predicted to be 231.4M eV , which is broader comparing with those of j q = 3 2 P-wave doublets. This prediction is consistent with that of the heavy quark limit(HQL).  We notice that the PDG only reported the B J (5970) bottom meson and omitted the B J (5840) state from the summary tables, and the spin-parity of B J (5970) was unknown. Thus, we study the strong decay behaviors with the 2 1 S 0 , 2 3 S 1 assignments for B J (5840) state and 2 3 S 1 , 1 3 D 1 , 1 3 D 3 , 1D ′ 2 , 1D 2 assignments for B J (5970) state. The results are showed in TABLE V and TABLE VI. The LHCb collaboration has suggested that the B J (5840), B J (5970) signals should be identified with the 2 1 S 0 and 2 3 S 1 bottom states. We note also that the Bπ decay mode is reported by LHCb as 'possibly seen' for the strong decays of B J (5840) and B J (5970). However, our analysis indicate that the decay mode to Bπ is forbidden for B J (5840) as a 2 1 S 0 assignment. If the decay to Bπ is confirmed in the future, the 2 1 S 0 assignment can be ruled out. As the 2 3 S 1 assignments for B + J (5840) and B 0 J (5840), their total decay widths are 121.9M eV and 117.5M eV , and these values are compatible with the experimental data. Overall, we tentatively identify 2 3 S 1 as the assignment of B J (5840). Certainly, the conclusion about the assignments depend strongly on the accurate measurement of the decay mode Bπ of B J (5840) and B J (5970).

B
The bottom strange mesons B s1 (5830) and B * s2 (5840) are identified as the 1 + and 2 + assignments in PDG, but it is noted that the J P need confirmation [7]. In order to give a further confirmation, we study the strong decay behaviors of B * s2 (5840) as the 1 3 P 2 assignment and B s1 (5830) as the 1P ′ 1 , 1P 1 assignments. The predicted total decay width of B * s2 (5840) is 1.35M eV and it is consistent well with the experimental data 1.40 ± 0.4. In addition, the predicted partial decay ratio This value is roughly compatible with the experimental data 0.093 ± 0.018, which supports 1 3 P 2 to be the assignment of B * s2 (5840). As a 1 + state, B s1 (5830) meson is the mixture between 1 3 P 1 and 1M eV and this value is consistent with the experimental data 0.5 ± 0.4M eV within the predictive power of the model. Thus, the 1P ′ 1 is the optimal assignment for B s1 (5830) and we can conclude that B s1 (5830) and B * s2 (5840) are the j q = 3 2 doublets, (B s1 (5830), B * s2 (5840)) = (1 + , 2 + ) 3 2 n = 1, L = 1 Again, the remaining states 1P 1 and 1 3 P 0 in TABLE VII are the spin doublets with j q = 1 2 and their total decay widths are much broader than those of the spin doublets with j q = 3 2 .