Towards establishing an abundant $B$ and $B_s$ spectrum up to the second orbital excitations

Stimulated by the exciting progress in experiments, we carry out a combined analysis of the masses, and strong and radiative decay properties of the $B$ and $B_s$-meson states up to the second orbital excitations. Based on our good descriptions of the mass and decay properties for the low-lying well-established states $B_1(5721)$, $B_2^*(5747)$, $B_{s1}(5830)$ and $B_{s2}^*(5840)$, we give a quark model classification for the high mass resonances observed in recent years. It is found that (i) the $B_{J}(5840)$ resonance may be explained as the low mass mixed state $B(|SD\rangle_L)$ via $2^3S_1$-$1^3D_1$ mixing, or the pure $B(2^3S_1)$ state, or $B(2^1S_0)$. (ii) The $B_J(5970)$ resonance may be assigned as the $1^3D_3$ state in the $B$ meson family, although it as a pure $2^3S_1$ state cannot be excluded. (iii) The narrow structure around 6064 MeV observed in the $B^+K^-$ mass spectrum at LHCb may be mainly caused by the $B_{sJ}(6109)$ resonance decaying into $B^{*+}K^-$, and favors the assignment of the high mass $1D$-wave mixed state $B_s(1D'_2)$ with $J^P=2^-$, although it as the $1^3D_3$ state cannot be excluded. (iv) The relatively broader $B_{sJ}(6114)$ structure observed at LHCb may be explained with the mixed state $B_s(|SD\rangle_H)$ via $2^3S_1$-$1^3D_1$ mixing, or a pure $1^3D_1$ state. Most of the missing $1P$-, $1D$-, and $2S$-wave $B$- and $B_s$-meson states have a relatively narrow width, they are most likely to be observed in their dominant decay channels with a larger data sample at LHCb.


I. INTRODUCTION
Since 2007, significant progress has been made in the observations of the bottom and bottom-strange mesons [1]. In 2007, two low-lying orbitally excited narrow B mesons B 1 (5721) 0,+ and B * 2 (5747) 0,+ were observed by the D0 experiment [2], and were confirmed by the CDF experiment one year later [3]. Their strange analogues, B s1 (5830) and B * s2 (5840), as the first orbitally excited B s mesons, were also reported by the CDF Collaboration in 2007 [4]. The B s1 (5830) and B * s2 (5840) were confirmed by the D0 and LHCb experiments [5,6]. In 2013, two higher resonances B(5970) 0,+ were observed in the Bπ final states by the CDF Collaboration [7]. In 2015, four higher resonances B J (5840) 0,+ and B J (5960) 0,+ were observed in the Bπ final states by the LHCb Collaboration when they carried out precise measurements of the properties of the B 1 (5721) 0,+ and B * 2 (5747) 0,+ states [8]. The properties of the B J (5960) 0,+ states are consistent with those of B(5970) 0,+ obtained by the CDF Collaboration. Recently, the LHCb Collaboration observed two structures B sJ (6064) and B sJ (6114) in the B + K − mass spectrum [9]. More experimental information about the excited bottom and bottom-strange mesons is collected in Table I. More and more excited bottom and bottom-strange mesons are expected to be observed in future LHCb experiments due to its huge production cross sections of beauty, together with a good reconstruction efficiency, versatile trigger scheme and an excellent momentum and mass resolution [10].
The experimental progress provides us good opportunities to establish an abundant B and B s -meson spectrum up to the second orbital (L = 2) excitations. In this work we deepen our study by carrying out a combined analysis of the masses I: Summary of the experimental information for the excited B-and B s -meson states. The date for the B 1 (5721) +,0 , B 2 (5747) +,0 , B s1 (5830) 0 , and B s2 (5840) 0 resonances are adopted the average values of the Review of Particle Physics (RPP) of Particle Data Group (PDG) [1]. The N and UN stand for the natural spin parity P = (−1) J and unnatural spin parity P = −(−1) J .
This paper is organized as follows. In Sec. II, the mass spectrum is calculated within a non-relativistic linear potential model. In Sec. III, a brief review of the chiral quark model is given. The numerical results are presented and discussed in Sec. IV. Finally, a summary is given in Sec. V.

II. MASS SPECTRUM
To describe the bottom and bottom-strange meson spectra, we adopt a nonrelativistic linear potential model. In this model, the effective potential is adopted as [79][80][81][82] where includes the standard color Coulomb interaction and linear confinement, and zero point energy C 0 . The spin-dependent part V sd (r) can be expressed as [81][82][83] V sd (r) = H S S + H T + H LS ,  where is the spin-spin contact hyperfine potential. Here, we takẽ δ σ (r) = (σ/ √ π) 3 e −σ 2 r 2 as suggested in Ref. [79]. The tensor potential H T is adopted as with S T = 3S q ·rSq·r r 2 − S q · Sq. The spin-orbit interaction H LS can be decomposed into symmetric part H sym and antisymmetric part H anti : with In these equations, L is the relative orbital angular momentum of the qq system; S q and Sq are the spins of the quark q and antiquarkq, respectively, and S ± ≡ S q ± Sq; m q and mq are the masses of quark q and antiquarkq, respectively; α s is the running coupling constant of QCD; and r is the distance between the quark q and antiquarkq. The six parameters in the above potentials (α s , b, σ, m q , mq, C 0 ) are determined by fitting the mass spectrum. It should be emphasized that when m q mq, the antisymmetric part of the spin-orbit potential, H anti , can cause a configuration mixing between spin triplet n 3 L J and spin singlet = cos θ nL sin θ nL − sin θ nL cos θ nL where J = L = 1, 2, 3 · · · , and the θ nL is the mixing angle. In this work nL ′ J corresponds to the higher mass mixed state as often adopted in the literature.
In this work, we solve the radial Schrödinger equation by using the three-point difference central method [84] from central (r = 0) towards outside (r → ∞) point by point. This method was successfully to deal with the spectroscopies of cc, bb, bc and ss [76-78, 85, 86]. To overcome the singular behavior of 1/r 3 in the spin-dependent potentials, following the method of our previous works [76-78, 85, 86], we introduce a cutoff distance r c in the calculation. Within a small range r ∈ (0, r c ), we let 1/r 3 = 1/r 3 c . By introducing the cutoff distance r c , we can nonperturbatively include the corrections from these spin-dependent potentials containing 1/r 3 to both the mass and wave function of a meson state, which are crucial for our predicting the decay properties.
The model parameters adopted in this work are listed in Table II. To be consistent with our previous study [64,78,85,86] It should be pointed out that the zero-point-energy parameter C 0 is taken to be zero for the cc, bb, bc heavy quarkonium systems in the literature [76][77][78]85]. For these heavy quarkonium systems, the zero point energy can be absorbed into the constituent quark masses because it only affects the heavy quark masses slightly. However, if the zero point energy is absorbed into the meson systems containing light quarks, it can significantly change the light constituent quark masses, which play an important role in the spin-dependent potentials. Thus, to obtain a good description of both the masses and the hyperfine/fine splittings for the meson systems containing light quarks, a zero-pointenergy parameter C 0 is usually adopted in the calculations.
Finally, we need determine the cutoff distance r c in our calculations. It is found that the masses of the 1 3 P 0 states are more sensitive to the cutoff distance r c . The reason is that the singular terms of 1/r 3 in the spin-dependent potentials have more effects on the mass of the 1 3 P 0 state due to its relatively larger factors of S + · L and S T than the other excited meson states. Thus, in the present work, we determine the r c by fitting the masses of the B(1 3 P 0 ) and B s (1 3 P 0 ) states. Note that when the other parameters are well determined, the masses of these 1 3 P 0 states can be reliably worked out with the perturbation method without introducing the cutoff distance r c , although the wave functions obtain no corrections from the spin-dependent potentials containing 1/r 3 . To calculate the masses of the B(1 3 P 0 ) and B s (1 3 P 0 ) states with the perturbation method, we let H = H 0 + H ′ , where H 0 is the main part of the Hamiltonian without singular 1/r 3 term contributions, i.e., where P 2 2µ is the kinetic energy term with a reduced mass µ = m q mq/(m q + mq); while H ′ is the perturbation part containing 1/r 3 terms, i.e., For a 1 3 P 0 state, one can easily obtain the matrix elements S q · Sq = 1/4, S + · L = −2, and S T = −1 by using the relations S q · Sq = S (S + 1) By solving the radial Schrödinger equation of H 0 ψ (0) (r) = E 0 ψ (0) (r) with the three-point difference central method, we work out the eigen energies E 0 = 508, 415 MeV for the B(1 3 P 0 ) and B s (1 3 P 0 ) states as well as their zero order radial wave functions ψ (0) (r). Using these zero order wave functions, we further work out the perturbation energies ψ (0) (r)|H ′ |ψ (0) (r) = −88, −79 MeV for the B(1 3 P 0 ) and B s (1 3 P 0 ) states, respectively. Finally, with the relation M = m q + mq + E 0 + ψ (0) (r)|H ′ |ψ (0) (r) we predict the masses 5722 and 5788 MeV for B(1 3 P 0 ) and B s (1 3 P 0 ), respectively. These masses calculated with the perturbation method are in good agreement with the predictions in the literature [12-14, 24, 28]. Within a small range r ∈ (0, r c ), letting 1/r 3 = 1/r 3 c in the perturbation Hamiltonian H ′ , we solve the radial Schrödinger equation (H 0 + H ′ )ψ(r) = Eψ(r) with the three-point difference central method again. By reproducing the masses 5722 and 5788 MeV of the B(1 3 P 0 ) and B s (1 3 P 0 ) states obtained with the perturbation method, we determine the cutoff distance parameters r c = 0.337 and 0.292 fm for the B and B s mass spectra, respectively. The corrections of the perturbation part H ′ to the radial wave function ψ(r) can be conveniently included by introducing the cutoff distance r c . As an example, in Fig. 1 we plot the radial wave functions of the B(1 3 P 0 ) state with and without the corrections from the perturbation part H ′ containing 1/r 3 . It is seen that the perturbation term H ′ has an obvious correction to the radial wave function.
With the determined model parameters listed in Table II, by solving the radial Schrödinger equation we obtain the masses of the bottom and bottom-strange meson states, which have been listed in Tab. III and Tab. IV, respectively. For comparison, some other model predictions in Refs. [12-15, 24, 26-28] and the data from the Review of Particle Physics (RPP) of Particle Data Group (PDG) [1] are listed in the same table as well. Furthermore, for a clarity, the spectra are also shown in Fig. 2. It is shown that the B J (5970) resonance and the structures B sJ (6064) and B sJ (6114) newly observed at LHCb can be explained as the 1D-wave states from the point of view of the mass, while the B J (5840) may be a good candidate of the 2S -wave state.

A. Models
We calculate the strong decays of the bottom and bottomstrange mesons with the chiral quark model [49][50][51][52]. In this model, the light pseudoscalar mesons, i.e. π, K and η, are treated as fundamental states. The low energy quarkpseudoscalar-meson and quark-vector-meson interactions in the SU(3) flavor basis are described by the effective Lagrangian where ψ j represents the jth quark field in the hadron, φ m is the pseudoscalar meson field, f m is the pseudoscalar meson decay constant. The nonrelativistic form of Eq. (12) is given by [50][51][52] in the center-of-mass system of the initial meson, where we have defined A ≡ −(1 + ω m E f +M f ) and h ≡ ω m 2µ q . In Eq. (13), q and ω m are the three-vector momentum and energy of the final-state light meson, respectively; p j is the internal momentum operator of the jth quark in the heavy-light meson rest frame; σ j is the spin operator corresponding to the jth quark of the heavy-light system; and µ q is 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 mesons, respectively. The plane wave part of the emitted light meson is ϕ m = e −iq·r j , and I j is the flavor operator defined for the transitions in the SU(3) flavor space. The chiral quark model has been successfully applied to describe the strong decays of the heavy-light mesons and baryons [24,47,[53][54][55][56][57][58][59]. The details of this model can be found in Refs. [55,56]. It should be mentioned that the chiral quark model is similar to the pseudoscalar emission model in the literature [11,87,88]. The nonrelativistic form of quarkpseudoscalar-meson interactions expressed in Eq. (13) is similar to that of the pseudoscalar emission model, except that the factors A ≡ −(1 + ω m E f +M f ) and h ≡ ω m 2µ q in this work have an explicit dependence on the energies of final hadrons.
Meanwhile, to treat the radiative decay of a hadron we apply the constituent quark model [50][51][52][65][66][67][68][69][70][71]. In this model, the quark-photon EM coupling at the tree level is adopted as where A µ represents the photon field with three-momentum k. e j and r j stand for the charge and coordinate of the constituent quark ψ j , respectively. In the initial-hadron-rest system, including the effects of the binding potential between quarks, the nonrelativistic form of the quark-photon EM coupling can be written as [50-52, 65, 68-71] H nr e = j e j r j · ǫ − e j 2m j σ j · (ǫ ×k) ϕ γ , where σ j stand for the Pauli spin vector for the jth quark. The vector ǫ is the polarization vector of the photon. The plane wave part of the emitted light meson is ϕ γ = e −ik·r j . The first and second terms in Eq.(15) are responsible for the electric and magnetic transitions, respectively. The second term in Eq. (15) is the same as that used in Refs. [11,[87][88][89][90], while the first term in Eq.(15) is different from (1/m j )p j · ǫ of these works due to the binding potential effects are included [65]. This model has been successfully applied to describe the radiative decays of baryon states [60-64, 72, 73] and meson systems [74][75][76][77][78]. For a strong decay process, the partial decay width is calculated with [55,56] while for a radiative decay process, the partial decay width is calculated with [76,77] where M J f z ,J iz and A J f z ,J iz correspond to the strong and radiative transition amplitudes, respectively. The quantum numbers J iz and J f z stand for the third components of the total angular momenta of the initial and final hadron states, respectively. δ as a global parameter accounts for the strength of the quark-meson couplings. It has been determined in our previous study of the strong decays of the charmed baryons and heavy-light mesons [55,56]. Here, we fix its value the same as that in Refs. [55,56], i.e. δ = 0.557.

B. Parameters
In the calculation, the constituent quark masses for the u, d, and s quarks are taken with m u = m d = 450 MeV and m s = 600 MeV to be consistent with the spectrum study in Sec. II. The decay constants for π, K and η mesons are taken as f π = 132 MeV, f K = f η = 160 MeV, respectively. The masses of the well established hadrons involving in the calculations are adopted from the PDG [1]. The masses of the missing B-and B s -meson states are adopted our determinations by solving the Schröinger equation in Sec. II.
It should be mentioned that, we do not directly adopt the numerical wave functions of B-and B s -meson states calculated by solving the Schröinger equation. For simplicity, we first fit them with a simple harmonic oscillator wave function by reproducing the root-mean-square radius r 2 . The obtained effective harmonic oscillator parameters β e f f for the meson states are listed in Tab. III and Tab. IV. It is found that the effective harmonic oscillator parameters β NR e f f obtained from our nonrelativistic quark model calculations are obviously smaller than the parameters β R e f f obtained from the relativized quark models [26]. It indicates that the relativistic effects of the length contraction on the wave function may be important. To take into account the relativistic effects, as that suggested in Ref. [91] we introduce the Lorentz boost factor γ in the spatial wave function, i.e., where γ = M q /E q . M q and E q correspond to the effective mass and energy of the light quark, respectively. According to Ref. [92], the effective mass M q can be estimated by M q = p 2 + m 2 q , while the energy E q is estimated by To realize this transformation, we only need replace β NR e f f in the harmonic oscillator wave function with β C e f f = γβ NR e f f . The effective harmonic oscillator parameters β C e f f including relativistic corrections of the length contraction are given in Tab. III and Tab. IV as well. It is found that the β C e f f values are comparable with those obtained with the relativized quark models [26]. In Refs. [47,55], the strong decays of the heavy-light meson states are studied with the chiral quark model by using the simple harmonic oscillator wave functions with fixed harmonic oscillator parameters β = 468, 466 MeV for the B and B s spectra, respectively, which are close to the parameters β C e f f determined for the 1P-, 1D-and 2S -wave states in present work. The parameters β C e f f of the ground states 1 1 S 0 and 1 3 S 1 are notably lager than that of the excited states, this effect is mainly caused by the strong color Coulomb interaction at the small distance r between two quarks.
The effective parameters β C e f f of the ground states B(1 1 S 0 ) and B * (1 3 S 1 ) are crucial for understanding the decay properties of the excited B and B s states because all of the excited states should decay into these ground states. Considering the uncertainty of the parameters β C e f f of the ground states B(1 1 S 0 ) and B * (1 3 S 1 ), we properly adjust their β C e f f parameters to more reasonably describe the strong decay properties of the well established 1P-wave states B * 2 (5747) 0 and B * s2 (5840). In this work, we determine them to be β B(1 1 S 0 ) = 0.537 GeV and β B * (1 3 S 1 ) = 0.510 GeV for B(1 1 S 0 ) and B * (1 3 S 1 ), respectively. There is about a 10% correction to the effective parameters β C e f f . With above parameters, our calculated decay properties for the 1P-, 2S -, and 1D-wave states are listed in Tables V, VI, and VII, respectively. From Table V, it is found that the decay properties of the well-established 1P-wave state can be successfully described.

1 3 P 2 states
There are no puzzles to assign the B * 2 (5747) +,0 and B * s2 (5840) 0 resonances to the 1 3 P 2 states in the B and B s families, respectively.
In the B meson sector, as the 1 3 P 2 state both the mass and width of B * 2 (5747) can be resonantly reproduced in the quark model. Our fitted mass M ≃ 5727 MeV and width Γ ≃ 31 MeV are compatible with the measurements M exp = 5737 MeV and width Γ exp = (20 ± 5) MeV for B * 2 (5747) + . This state dominantly decays into Bπ and B * π channels with comparable partial widths. The ideal partial width ratios between Bπ and B * π channels for the B * 2 (5747) +,0 are fitted to be which are also consistent with the recent LHCb measurements R exp 1 = 0.71 ± 0.14 ± 0.30 and R exp 2 = 1.0 ± 0.5 ± 0.8 [8] and the predictions in Refs. [25, 27, 28, 31-33, 40, 42].
In the B s meson sector, as the 1 3 P 2 state both the mass and width of B * s2 (5840) can be well reproduced in the quark model as well. Our fitted mass M ≃ 5821 MeV and width Γ ≃ 1.3 MeV are compatible with the measured mass M exp = 5840 MeV and width Γ exp = (1.49 ± 0.27) MeV. There are two OZI allowed two-body strong decay channels BK and B * K. The BK mode governs the decays of B * s2 (5840). The ideal partial width ratio between B * K and BK is fitted to be which is in good agreement with the recent LHCb measured one R exp = (9.3 ± 2.5)% [5] and the predictions in the literature [25,[27][28][29]38]. Furthermore, it is found that B * s2 (5840) 0 has a large decay rate into B * s γ, the partial width and branching fraction are predicted to be Our predicted radiative partial decay width is consistent with those predictions in Refs. [42,43]. In the literature, a larger partial width Γ[B * s2 (5840) 0 → B * 0 s γ] ≃ 100 − 230 keV is predicted [26][27][28]. The B * 0 s γ decay channel of B * s2 (5840) 0 may have good potentials to be observed in future experiments.

1 3 P 0 states
The 1 3 P 0 states in the B and B s families are still missing experimentally. In the B meson sector, we predict that the mass of the B(1 3 P 0 ) state is 5722 MeV, which is consistent with those predictions in Refs. [13,14,17,24,26,28]. The Bπ channel is the only OZI-allowed two body decay channel. Taking the estimated mass M = 5722 MeV, we obtain a broad width Γ ≃ 290 MeV for the B(1 3 P 0 ) state, which is compatible with our previous result Γ ≃ 270 MeV predicted with the SHO wave functions in Ref. [55]. The B(1 3 P 0 ) state is also predicted to be a broad state with a width of ∼ 150 − 250 MeV in the other models [25-27, 29, 31, 41]. We also study the radiative decays of B(1 3 P 0 ) +,0 , our results have been listed in Table V, the predicted partial width for Γ[B(1 3 P 0 ) + → B * + γ] ≃ 477 keV is about a factor 3 larger V: Partial and total decay widths (MeV) for 1P-wave bottom and bottom-strange mesons compared with the the data and some recent model predictions. It should be mention that some masses for the initial states adopted in the literature are slightly different. The total widths inside the square brackets are estimated with the mixing angle θ 1P = −(55 ± 5) • .
than that for Γ[B(1 3 P 0 ) 0 → B * 0 γ] ≃ 149 keV. Our predictions are comparable with those predicted in Refs. [26,28]. It should be mentioned that the radiative decays of B(1 3 P 0 ) +,0 are governed by the E1 transitions. For a heavy-light meson system, the E1 transition amplitude A is mainly contributed by the light quark, it is proportional to e q /m q , where e q and m q stands for the charge and mass of the light quark, respectively. The heavy quark contributions are strongly suppressed by the heavy quark mass m Q (i.e., the factor 1/m Q ). If neglecting the heavy quark contributions, one has Γ[B(1 3 P 0 ) + → B * + γ] : Γ[B(1 3 P 0 ) 0 → B * 0 γ] = 4 : 1, which is slightly larger than the ratio ∼ 3 including heavy quark contributions.
In the B s meson sector, the mass of B s (1 3 P 0 ) is expected to be around 5788 MeV, which is compatible with those predictions in Refs. [12-14, 17, 24, 26-28]. The BK channel is the only OZI-allowed two body strong decay channel. Taking the estimated mass M = 5788 MeV, we obtain a broad width Γ ≃ 270 MeV for the B s (1 3 P 0 ) state. This prediction is compatible with our previous result Γ ≃ 227 MeV predicted with the SHO wave functions in Ref. [55]. The B s (1 3 P 0 ) state is also predicted to be a broad state with a width of ∼ 130 − 230 MeV in the other models [25,26,29]. We also study the radiative decay of B s (1 3 P 0 ), the predicted partial width for Γ[B s (1 3 P 0 ) → B 0 * s γ] ≃ 100 keV is close to the predictions in Refs. [26,28]. Finally, it should be mentioned that in some works [12,27,28], the predicted mass of B s (1 3 P 0 ) is below the BK mass threshold, which will lead to a very narrow width for the B s (1 3 P 0 ) state. The spin-orbit potential causes a strong configuration mixing between 1 3 P 1 and 1 1 P 1 . It is generally believed that B 1 (5721) and B s1 (5830) correspond to the mixed states |1P ′ 1 via the 1 1 P 1 -1 3 P 1 mixing in the B and B s families, respectively. The other two mixed states B 1 (1P 1 ) and B s1 (1P 1 ) in the the B and B s families are waiting to be established in future experiments.
Considering B 1 (5721) as the mixed state |1P ′ 1 defined in Eq.(9), our fitted mass M = 5753 MeV is reasonably comparable with the measured value M exp = 5726 MeV. With the mixing angle θ 1P = −35.2 • determined from our quark model, the width of B 1 (5721) 0 is predicted to be Γ ≃ 41 MeV, which is in good agreement with the predictions in Refs. [27,31], while slightly larger than the observed width Γ exp ≃ 30 MeV at LHC [8] and those predictions in Refs. [25,26,28,30]. The decay width is nearly saturated by the B * π channel. If we taking the mixing angle around the value obtained in the heavyquark symmetry limit, i.e. θ 1P = −(55 ± 5) • [55,[93][94][95][96][97], the decay width is predicted to be in the range of Γ = (24.5 ± 2.5) MeV, which seems to be more comparable with the LHCb observations [8]. We also study the radiative decay processes of B 1 (5721) → B * γ, Bγ, their partial decay widths are predicted to be Our predictions are comparable with those in Ref. [42], however, most of our predictions are notably smaller than the predictions in Refs. [26][27][28]. The radiative decay modes B * γ and Bγ of B 1 (5721) may be observed in future experiments since their branching fractions can reach up to the order of O(10 −3 ).
In the B-meson family, the mass for the other mixed state B 1 (1P 1 ) is about 40 MeV lower than that of B 1 (5721) according to our potential model calculations, which is consistent with the prediction in Ref. [24]. A slightly smaller mass splitting, ∼ (10 − 30) MeV, between B 1 (5721) and B 1 (1P 1 ) is given in Refs. [13,14,[26][27][28]. Thus, the mass of B 1 (1P 1 ) might be in the range of (5700 ± 15) MeV. Considering the mass uncertainties, with the mixing angle θ 1P = −35.2 • we plot the decay width of B 1 (1P 1 ) as a function of its mass in Fig. 3. It is found that B 1 (1P 1 ) is a broad state with a width of Γ ≃ (155±10) MeV. The B * π channel is the only OZI-allowed two body strong decay channel. The B J (5732) listed in the RPP [1] is a good candidate for the B 1 (1P 1 ). With this assignment, both the measured mass M exp = 5698 MeV and width Γ exp = (128 ± 18) MeV for the B J (5732) are in good agreement with the quark model predictions. However, in Ref. [23] the B J (5732) is assigned to the B(1 3 P 0 ) state according to the mass spectrum analysis.
As the mixed state |1P ′ 1 , the mass of B s1 (5830) is consistent with our fitted value M = 5821 MeV and other determinations in the literature (See Table IV). It dominantly decays into the B * K channel. Taking the mixing angle θ 1P = −39.6 • determined by our potential model, we find that the theoretical width, Γ ≃ 7.6 MeV, is too large to be comparable with the measured one Γ exp ≃ (0.5 ± 0.3 ± 0.3) MeV. However, if we taking the mixing angle around the value obtained in the heavy-quark symmetry limit, i.e. θ 1P = −(55 ± 5) • , as that adopted in Ref. [55], the decay width is predicted to be in the range of Γ = 0.08 − 0.8 MeV, which is comparable with the data. It indicates that the mixing angle between 1 1 P 1 and 1 3 P 1 may be close to the value θ 1P = −55 • obtained in the heavy-quark symmetry limit. The B s1 (5830) has large decay rates into B * s γ and B s γ channels. Their partial decay widths are predicted to be The branching fractions of these radiative decays may reach up to O(10 −2 ). Our predictions are comparable with those predicted in Refs. [26,27,42]. The B * s γ and B s γ decay channels of B s1 (5830) may have good potentials to be observed in future experiments.
In the B s meson sector, the mass of the other mixed state B s1 (1P 1 ) is predicted to be about 2 − 40 MeV lower than that of B s1 (5830) in various quark models [12-14, 24, 26-28]. Thus, the mass of B s1 (1P 1 ) is estimated to be ∼ (5808 ± 19) MeV, which is just around the B * + K − and B * 0 K 0 mass thresholds. From Fig. 3, it is seen that the strong decay properties of B s1 (1P 1 ) are very sensitive to the mass threshold. There are three cases to be considered. (i) If the mass of B s1 (1P 1 ) is below the B * + K − mass threshold 5818 MeV, the radiative decay modes B * γ and Bγ may play crucial roles in the decays. The with partial decay widths are estimated to be MeV many year ago [98]. As a candidate of B s1 (1P 1 ), both the measured mass and width of B sJ (5850) are consistent with the predictions. To confirm the B sJ (5850) resonance and established the B s1 (1P 1 ) state, more observations of the B * 0 K 0 and B * + K − final states are suggested to be carried out in future experiments. As a whole the high mass mixed state |1P ′ 1 via the 1 1 P 1 -1 3 P 1 mixing in the B and B s families have been wellestablished, they correspond to two narrow states B 1 (5721) and B s1 (5830) observed in experiments. Some evidence for the low mass mixed states |1P 1 with broad widths predicted in theory may have been observed in experiments. The The 2 1 S 0 states in the B and B s families are still not established. In the B meson sector, our predicted mass for the B(2 1 S 0 ) state is M = 5876 MeV, which is in agreement with the predictions in Refs. [14,24,26,28]. Taking the mass M = 5876 MeV, we calculate the strong and radiative decay properties of B(2 1 S 0 ), our results are listed in Table VI. The decays of B(2 1 S 0 ) are governed by the B * π mode with a fairly large branching fraction ∼ 94%. The width of B(2 1 S 0 ) is predicted to be Γ ≃ 63 MeV, which is about a factor 1.5 − 4 larger than the predictions in Refs. [24,25,28,47], while about a factor 1.5 − 2 smaller than the predictions in Refs. [26,27,31,33].
In 2015, the LHCb Collaboration observed two new resonances B J (5840) 0,+ [8]. Considering B J (5840) 0,+ as unnatural parity states, the relatively accurate measurements of the mass and width for the neutral one are M exp = (5863 ± 9) MeV and Γ exp = (127±51) MeV, respectively [8]. In this case, signals of B J (5840) 0,+ should come from the B * π decay mode other than Bπ. In Refs. [18,[26][27][28]39], the B J (5840) 0,+ resonances were suggested to be the B(2 1 S 0 ) assignment. If B J (5840) 0,+ have an unnatural parity indeed, they strongly favor the B(2 1 S 0 ) assignment. The measured mass and width together with the decay modes of B J (5840) are consistent with the our theoretical predictions. However, a natural parity for B J (5840) 0,+ is also possible according to the LHCb analysis (see Case B listed in Table I). The B J (5840) 0,+ may possibly decay into both the B * π and Bπ channels. If the Bπ decay mode is confirmed in future experiments, the B J (5840) resonance should be other assignments since the Bπ mode is forbidden for the B(2 1 S 0 ) state.
In the B s meson sector, our predicted mass for the B s (2 1 S 0 ) state is M = 5944 MeV, which is comparable with the other quark model predictions [12-14, 24, 26-28]. Taking M = 5944 MeV, we calculate the strong and radiative decay properties of B s (2 1 S 0 ), our results are listed in Table VI. The B * K channel is the only OZI-allowed two body strong decay channel for B s (2 1 S 0 ). Its decay width is predicted to be Γ ≃ 55 MeV, which is comparable with those predictions in Refs. [25,26,28,33,47]. The B s (2 1 S 0 ) state should have large potentials to be seen in the B * + K − channel since it has a fairly narrow width.

2 3 S 1 states
In the B meson sector, our predicted mass for B(2 3 S 1 ) is M = 5899 MeV, which is in good agreement with the predictions in Refs. [14,28]. Taking the mass M = 5899 MeV, we calculate the strong and radiative decay properties of B(2 3 S 1 ), our results are listed in Table VI. It is found that the B(2 3 S 1 ) state is a fairly narrow state with a width of Γ ≃ 40 MeV, which is consistent with the prediction in Refs. [25,47], however, a factor ∼ 2 − 4 smaller than the predictions in Refs. [26,27,31,33]. This state dominantly decays into the B * π channel with a branching fraction ∼ 54%. The partial width ratio between the two typical channels Bπ and B * π is predicted to be which can be used to identify the B(2 3 S 1 ) from its possible candidates observed in future experiments. It should be mentioned that the B J (5840) resonance may be a candidate of the B(2 3 S 1 ) state as suggested in Refs. [29,31]. As this assignment, both the mass and typical decay modes  5747) B * π and Bπ predicted in theory are consistent with the LHCb observations [8]. Our predicted width Γ ≃ 41 MeV is close to the lower limit of the measured width Γ exp = (107 ± 54) MeV assuming a natural parity for B J (5840) 0 [8]. Furthermore, the B J (5970) resonance was also suggested to be the B(2 3 S 1 ) state in the literature [18, 25-27, 33, 35]. The B J (5970) 0,+ resonances were first observed by the CDF Collaboration in the Bπ final states in 2013 [7], and confirmed by the LHCb Collaboration two years later [8]. The central value of the measured width is Γ exp ≃ 60 − 70 MeV with large uncertainties (see Table I). In the LHCb observations, the B * π decay mode has been seen, while the Bπ mode may be possibly seen [1]. If assigning B J (5970) resonance to B(2 3 S 1 ), our predicted mass, M ≃ 5899 MeV, is about 70 MeV larger than the observation, while the predicted width, Γ ≃ 54 MeV, is consistent with the data. As a conclusion, we cannot exclude the possibilities of the B J (5840) and B J (5970) resonances as candidates of the B(2 3 S 1 ) state based on the present experimental information.
In the B s meson sector, our predicted mass for the B s (2 3 S 1 ) state is M = 5966 MeV, which is in good agreement with the predictions in Refs. [12,14,28]. Taking M = 5966 MeV, we calculate the strong and radiative decay properties of B s (2 3 S 1 ), our results are listed in Table VI. It is found that the B s (2 3 S 1 ) state is also a narrow state with a width of Γ ≃ 50 MeV, which is consistent with the predictions in Refs. [25,47], however, a factor ∼ 2 − 4 smaller than the predictions in Refs. [26,27,33]. This state mainly decays into the B * K and BK channel with large branching fractions ∼ 68% and ∼ 32%, respectively. The partial width ratio between the two main channels BK and B * K is predicted to be which may be an important criterion for establishing the B s (2 3 S 1 ). There may exist a strong configuration mixing between the 2 3 S 1 and 1 3 D 1 states, which is to be discussed in the last part of this section.  In Ref. [47], by analyzing the decay properties within the chiral quark model our group found that the B J (5970) is most likely to be the 1 3 D 3 assignment in the B-meson family. The B J (5970) as a candidate of B(1 3 D 3 ) is also suggested in Refs. [27,31]. In present work, we restudy the B J (5970) by combining the decay properties with the mass spectrum. It is found that as the B(1 3 D 3 ) assignment the mass of B(5970) can be well explained with the potential model. Our predicted mass M = 5979 MeV is in good agreement with the observed value M exp = (5971 ± 5) MeV for the neutral state B(5970) 0 [1]. By using the wave function of B(1 3 D 3 ) calculated from the potential model, we further study the decay properties, our results are listed in Table VII. It is found that the predicted decay width, Γ ≃ 39 MeV, is also consistent the measured width Γ exp ≃ (56 ± 16) MeV of B(5970) 0 by assuming P = (−1) J and using three relativistic Breit-Wigner functions in the fit for mass difference at LHCb [8]. The partial width ratio between Bπ and B * π is predicted to be which is waiting to be tested in future experiments. If the B J (5970) 0,+ resonances correspond to the B(1 3 D 3 ) assignment indeed, the charged state B J (5970) + should have a large radiative decay rate into B * 2 (5747) + γ, the partial width and branching fraction are predicted to be The radiative decay mode B * 2 (5747) + γ may be observed in future experiments.
In Ref. [47], considering the B J (5970) as the B(1 3 D 3 ) assignment, our group further predicted that the mass and width of the B s (1 3 D 3 ) state, as a flavor partner of B J (5970), might be M ≃ 6.07 GeV and Γ ≃ 30 MeV, respectively. The predicted mass of B s (1 3 D 3 ) is consistent with the those predicted in Refs. [12,27,28], while a relatively narrow width is also predicted by other works [25-28, 33, 35]. It is interestingly found that the new bottom-strange structure B sJ (6064) with a mass of M exp = (6063.5 ± 2.0) MeV and a very narrow width of Γ exp = (26 ± 8) MeV observed at LHCb [9] is consistent with the predictions. In present work, from the aspects of both mass spectrum and decay properties we further discuss the possibility of the B sJ (6064) structure as the B s (1 3 D 3 ) assignment in the B s -meson family. With this assignment, it is found that the measured mass for the B sJ (6064) structure is consistent with the theoretical mass M = 6067 MeV. Furthermore, the narrow width of B sJ (6064) can also be explained within our chiral quark model. From our predicted decay properties listed in Table VII, it is found that the theoretical width Γ ≃ 13 MeV is close to lower limit of the measured value Γ exp = (26 ± 8) MeV from LHCb [9]. The partial width ratio between BK and B * K is predicted to be Our predicted mass and decay properties of B s (1 3 D 3 ) are compatible with those predicted in Refs. [27,28,47]. It should be pointed out that the large branching fraction for Br[B sJ (6064) → B * K] ≃ 45% seems to be not consistent with the observations naturally. Since B sJ (6064) causes a clear bump structure around 6064 MeV in the B + K − mass spectrum through the B + K − decay, it should also cause another narrow bump structure around 6019 MeV through the B * + K − decay with a missing photon from B * + → B + γ, however, this structure was not observed at LHCb. Thus, it indicates that the B s (1 3 D 3 ) may not be the main contributor to the B sJ (6064) structure observed in the B + K − mass spectrum. As a whole, the B J (5970) may be assigned as the 1 3 D 3 assignment, which can be tested by the partial width ratio between Bπ and B * π. It may be a flavor partner of the D * 3 (2750) and D s3 (2860) resonances listed in RPP [1]. Their masses might be systematically overestimated by a value of ∼ 100 MeV in some quark models [11,14,26,99]. There still exists a puzzle to identify the B sJ (6064) structure as the B s (1 3 D 3 ) state, although both the predicted mass and width seem to be consistent with the observations. To establish the narrow B s (1 3 D 3 ) state finally, the observations of both the B + K − and B * + K − decays and their partial width ratio are crucial in future experiments.

1 3 D 1 states
In the B meson sector, the mass for the B(1 3 D 1 ) state is predicted to be M = 6056 MeV in our potential model calculations, which is comparable with the predictions in Refs. [13,24,27,28]. Our predicted mass for B(1 3 D 1 ) is about 80 MeV larger than that for B(1 3 D 3 ). Taking the mass M = 6056 MeV, we calculate the strong and radiative decay properties of B(1 3 D 1 ), our results are listed in Table VII. It is found that the B(1 3 D 1 ) state is a broad state with a width of Γ ≃ 350 MeV, which is consistent with the predictions in Refs. [25,27]. This state mainly decays into the Bπ, B * π, and B 1 (5721)π channels with branching fractions ∼ 26%, 12%, and 42%, respectively. The partial width ratio between the two typical channels Bπ and B * π is predicted to be which may be helpful to identify the B(1 3 D 1 ) state from future observations. In Refs. [28,29], B J (5970) was suggested to be a candidate for B(1 3 D 1 ). With this assignment we find that the theoretical width Γ ≃ 230 MeV is too broad to be comparable with the measured value measured value Γ exp ≃ 60 − 70 MeV (see Table I). Thus, B J (5970) may not be a good candidate of the 1 3 D 1 state. In the B s meson sector, the mass for the B s (1 3 D 1 ) state is predicted to be M = 6101 MeV in our potential model calculations, which is comparable with the predictions in Refs. [13,24,27]. Our predicted mass for B s (1 3 D 1 ) is about 30 MeV larger than that of B s (1 3 D 3 ). There are large uncertainties in the predictions of the mass splitting between B s (1 3 D 1 ) and B s (1 3 D 3 ). In some works [12,13,28], the B s (1 3 D 1 ) mass is even predicted to be smaller than that of B s (1 3 D 3 ). Taking the mass M = 6101 MeV, we calculate the strong and radiative decay properties of B s (1 3 D 1 ), our results are listed in Table VII. It is found that the B s (1 3 D 1 ) state has a width of Γ ≃ 130 MeV, and mainly decays into the BK and B * K channels with branching fractions ∼ 65% and 27%, respectively. The partial width ratio between the two typical channels BK and B * K is predicted to be which is comparable with the predictions in Refs. [25,27]. From the point of view of mass, the observed resonance B sJ (6114) by the LHCb Collaboration [9] is a good candidate for the B s (1 3 D 1 ) state. While, the theoretical width Γ ≃ 130 MeV is also close the upper limit of the measured width Γ exp = (66 ±39) MeV. There may exist a configuration mixing between the 2 3 S 1 and 1 3 D 1 states, which will be discussed in the last part of this section.
There is a strong configuration mixing between the 1 3 D 2 and 1 1 D 2 states for the heavy-light mesons predicted in the potential models. For the B meson sector, with the mixing scheme defined in Eq. (9) the mixing angle is predicted to be θ 1D = −39.5 • , which is close to the value −50.8 • extracted in the heavy quark symmetry limit [25,95,96]. Our predicted masses for the B(1D 2 ) and B(1D ′ 2 ) states are about 5973 and 6067 MeV, respectively, which are close to the predictions in Refs. [13,24]. A fairly large mass splitting between B(1D 2 ) and B(1D ′ 2 ), ∆M = 94 MeV, is obtained in present work. It is comparable with the predictions of ∆M = 110 − 130 MeV in Refs. [26,27]. With the masses and wave functions obtained from our potential model calculations, the decay properties for these two mixed states B(1D 2 ) and B(1D ′ 2 ) are estimated, the results are listed in Table VII. It is found that the low mass state B(1D 2 ) has a broad width of Γ ≃ 170 MeV, and dominantly decays into B * π and B * 2 (5747)π channels with branching fractions about 54% and 38%, respectively. While the high mass state B(1D ′ 2 ) has a relatively narrow width of Γ ≃ 90 MeV, and dominantly decays into B * π, B 1 (5721)π and B * 2 (5747)π channels with branching fractions about 73%, 11% and 7%, respectively. Our predicted decay properties are roughly comparable with those predictions in Refs. [25,26,47].
For the B s meson sector, we predict the mixing angle θ 1D = −39.9 • . Our predicted masses for the B s (1D 2 ) and B s (1D ′ 2 ) states are about 6061 and 6113 MeV, respectively, which are close to the predictions in Refs. [12,24,27]. An intermediate mass splitting between B s (1D 2 ) and B s (1D ′ 2 ), ∆M = 52 MeV, is obtained in present work, which is comparable with the predictions of ∆M = 45 − 70 MeV in Refs. [24,27]. With the masses and wave functions obtained from our potential model calculations, the decay properties for these two mixed states B s (1D 2 ) and B s (1D ′ 2 ) are estimated, the results are listed in Table VII. It is found that the low mass state B s (1D 2 )(6061) has a intermediate width of and dominantly decays into B * K channel with a branching fraction about 94%. Our predicted width is about a factor 1.5 − 2 smaller than those predictions in Refs. [25][26][27][28]. The B s (1D 2 ) state may be observed around 6016 MeV in the B + K − mass spectrum through the B * + K − decay with a missing photon from B * + → B + γ. While the high mass state B s (1D ′ 2 )(6113) has a very narrow width of and dominantly decays into B * K channel with a branching fraction about 95%. The decay properties predicted in present work are in good agreement with those predictions in Refs. [25,26,28,47]. The narrow mixed state B s (1D ′ 2 ) with a mass of M = 6113 MeV may be the main contributor to the B sJ (6064) structure observed in the B + K − mass spectrum at LHCb [9]. In this case, the signal in the B + K − mass spectrum may mainly come from the B * + K − decay with a missing photon from B * + → B + γ. Including the energy of the missing photon, the mass and width are determined to be M exp = (6109 ± 1.8) MeV and Γ exp = (22 ± 9) MeV for the resonance B sJ (6109) [9]. It is interestingly found that the B sJ (6109) resonance favors the assignment of the 1D-wave mixed state B s (1D ′ 2 ). The predicted mass, width, decay mode are in good agreement with the observations. Finally, it should be mentioned that the B s (1 3 D 3 ) state may have a few contributions to the B sJ (6064) structure through the B + K − decay as well, since this state with a mass of M ≃ 6067 MeV lies just around the peak position.
It should be mentioned that there may exist a configuration mixing between the 2 3 S 1 and 1 3 D 1 states. To explain the strong decay properties of the D * J (2600) and/or D s1 (2700), configuration mixing between 2 3 S 1 and 1 3 D 1 is suggested in the literature [100][101][102][103][104]. In Refs. [53,54], our group also carefully studied the strong decay properties of the D * J (2600) and D s1 (2700). According the analysis, both D * J (2600) and D s1 (2700) could be explained as the mixed state |S D L via the 2 3 S 1 -1 3 D 1 mixing with the following mixing scheme: where the mixed angle is estimated to be θ ≃ −(45 ± 16) • . The D s1 (2860) resonance observed in the BK final state at LHCb [105,106] seems to be the high mass mixed state |S D H as the partner of the low mass state D s1 (2700) [47], from which one can estimate a mass splitting ∆M ≃ 150 MeV between the high and low mass mixed states. Similarly, the 2 3 S 1 -1 3 D 1 mixing might also exist in the B and B s meson families.
The strong decay properties for the mixed states |S D L and |S D H in the B and B s meson families were studied in another work of our group [47]. It is interestingly found that the newly observed resonances B J (5840) and B sJ (6114) at LHCb [8,9] are most likely to be the mixed states B(|S D L ) and B s (|S D H ), respectively, by comparing the measured masses and widths with the theoretical predictions (see Figs. 1 and 2 in Ref. [47]).
Considering the B J (5840) resonance as the low mass mixed state B(|S D L ), we revise the strong decay properties by using the wave functions obtained from our quark potential model calculations. Our results are shown in Fig. 4. With the mixing angle θ ≃ −(45 ± 16) • determined in Refs. [53,54] and the mass M exp ≃ 5890 MeV measured at LHCb [8], the B(|S D L ) state has a width of Γ ≃ (76±20) MeV, and dominantly decays into B * π channel. There may be a sizeable decay rate into the Bπ channel. The partial width ratio between Bπ and B * π is predicted to be which is sensitive to the mixing angle. The predicted width is consistent with the measured width Γ exp = (107 ± 54) MeV by assuming P = (−1) J . Moreover, the predicted decay modes are also consistent with the observations of B J (5840). To better understand the nature of B J (5840), more accurate measurements of the width together with the partial width ratio are expected to be carried out in future experiments. If B J (5840) corresponds to the low mass state B(|S D L ) in- Within the mixing angle range θ ≃ −(45 ± 16) • suggested in [53,54], the B(|S D H ) has a width of Γ ≃ (197 ± 47) MeV, and mainly decay into Bπ and B 1 (5721)π channels. The partial width ratio between Bπ and B 1 (5721)π is predicted to be which is insensitive to the mixing angle. Future observations in the Bπ channel with a larger data sample at LHCb may have a potential to discover this high mass mixed state B 1 (|S D H ).
In the B s meson sector, considering the B sJ (6114) resonance observed in the B + K − mass spectrum [9] as the high mass mixed state B s (|S D H ), we revise the strong decay properties by using the wave functions obtained from our quark potential model calculations. Our results are shown in Fig. 5.
It is found that with the mixing angle θ ≃ −(45 ± 16) • determined in Refs. [53,54], the B s (|S D H ) state has a width of Γ ≃ (95 ± 15) MeV, and dominantly decays into BK channel with a branching fraction ∼ 90%. Both the decay mode B + K − and width Γ exp = (66 ± 39) MeV observed for B sJ (6114) at LHCb [9] can be well understood in our quark model calculations. Thus, the B sJ (6114) may favor the mixed state B s (|S D H ).
The mass for the low mass state B s1 (|S D L ) may be about 150 MeV smaller than that of B sJ (6114). Taking a mass of M ≃ 5960 MeV for the low mass state B s1 (|S D L ), we show its decay properties in Fig. 5 as well. In the mixing angle range θ ≃ −(45 ± 16) • , the B s1 (|S D L ) has a width of Γ ≃ (70 ± 10) MeV, and mainly decay into B * K channel. The partial width ratio between BK and B * K, is sensitive to the mixing angle. The B s1 (|S D L ) is most likely to be observed in the B + K − final state with a larger data sample at LHCb. As a whole the B J (5840) and B sJ (6114) may favor the mixed states B(|S D L ) and B s (|S D H ) via 2 3 S 1 -1 3 D 1 mixing, respectively. Their partners B(|S D H ) and B s (|S D L ) are expected to be observed in their dominant decay channels with a larger data sample at LHCb.

V. SUMMARY
The experimental progress provides us good opportunities to establish an abundant B and B s -meson spectrum up to the second orbital excitations. In this work, combining the newest experimental progress, we carry out a systematical study of the mass spectrum, strong decays and radiative decays of the 1P-, 1D-, and 2S -wave excited B and B s states in the constitute quark model. The mass and strong decay properties for the well established 1P-wave resonances B 1 (5721) +,0 , B * 2 (5747) +,0 , B s1 (5830) and B * s2 (5840) can be consistently explained. The possible assignments for the high mass resonances/structures B J (5840) 0,+ , B(5970) 0,+ , B sJ (6064) and B sJ (6114) are discussed. We hope that our study can provide some useful information towards establishing an abundant B and B s -meson spectrum. Our main results are summarized as follows.
For the P-wave states, several points should be emphasized. (i) Some radiative decay processes, such as B s1 (5830) → B ( * ) s γ and B * s2 (5840) → B * s γ, have good potentials to be found in future experiments due to their fairly branching fractions of O(10 −2 ). (ii) The B J (5732) and B sJ (5850) resonances listed by the PDG [1] may good candidates for the missing |1P 1 state in the B and B s families, respectively. (iii) Both B(1 3 P 0 ) and B s (1 3 P 0 ) may hardly be observed in experiments due to their very broad width of Γ ∼ 300 MeV.
The B J (5840) resonance and the new B sJ (6114) structure observed in the B + K − mass spectrum may be explained with the mixed states B(|S D L ) and B s (|S D H ) via 2 3 S 1 -1 3 D 1 mixing, respectively. To confirm the nature of the B J (5840) and B sJ (6114), the typical ratios Γ(Bπ)/Γ(B * π) and Γ(B * K)/Γ(BK) are suggested to be measured in future experiments. The other two missing states, B(|S D H ) with a mass of M ≃ 6010 MeV and B s (|S D L ) with a mass of M ≃ 5960 MeV, may be observed in the Bπ and B * K final states. On the other hand, if there is little mixing between 2 3 S 1 -1 3 D 1 , the B J (5840) and B sJ (6114) resonances may be candidates for the B(2 3 S 1 ) and B s (1 3 D 1 ) states, respectively.
The B J (5970) resonance may be assigned as the 1 3 D 3 state in the B meson family, although it as a pure 2 3 S 1 state cannot be excluded according to present experimental information. To clarify the nature of B J (5970), further observations of the Bπ and B * π channels and a measurement of their partial width ratio are necessary. In the B s family, the predicted mass M ≃ 6067 MeV and width Γ ≃ 13 MeV for the B s (1 3 D 3 ) are consistent with the B sJ (6064) structure observed in the B + K − mass spectrum. However, the B s (1 3 D 3 ) may not be the main contributor to the B sJ (6064) structure. If B sJ (6064) corresponds to B s (1 3 D 3 ), another narrow structure around 6019 MeV coming from the B * + K − decay should be observed in the B + K − mass spectrum, however, it was not seen at LHCb.
The narrow B sJ (6064) structure observed in the B + K − mass spectrum may mainly come from the resonance B sJ (6109) decaying into B * + K − . The B sJ (6109) resonance favors the assignment of the high mass 1D-wave mixed state B s (1D ′ 2 ) with J P = 2 − . This state dominantly decays into B * K channel with branching fraction about 95%, and the BK decay is forbidden. The other missing mixed state B s (1D 2 ) has a mass of M ≃ 5973 MeV and a width of Γ ≃ 90 MeV. It is most likely to be established in the B + K − mass spectrum through the B * + K − decay with a missing photon from B * + → B + γ. In the B-meson sector, two relatively broad mixed states, B(1D 2 ) with M ≃ 5973 MeV and B(1D ′ 2 ) with M ≃ 6067 MeV, may be observed their main decay channel B * π with a larger data sample at LHCb.
Finally, it should be mentioned that the B J (5840) resonance may be a candidate of the 2S -wave state B(2 1 S 0 ) as well. In this case, the Bπ mode of B J (5840) is forbidden, which should be further confirmed in future experiments. In the B s meson sector, our predicted mass and width for the B s (2 1 S 0 ) state are M = 5944 MeV and Γ ≃ 55 MeV, respectively. The B * K channel is the only OZI-allowed two body strong decay mode. The B s (2 1 S 0 ) state should have large potentials to be seen in the B * + K − channel since it has a fairly narrow width.