Towards establishing an abundant B and B_s spectrum up to the second orbital excitations
aa r X i v : . [ h e p - ph ] F e b Towards establishing an abundant B and B s spectrum up to the second orbital excitations Qi Li, Ru-Hui Ni, Xian-Hui Zhong ∗
1) Department of Physics, Hunan Normal University, Changsha 410081, China2) Synergetic Innovation Center for Quantum E ff ects and Applications (SICQEA), Changsha 410081,China and3) Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China Stimulated by the exciting progress in experiments, we carry out a combined analysis of the masses, andstrong and radiative decay properties of the B and B s -meson states up to the second orbital excitations. Basedon our good descriptions of the mass and decay properties for the low-lying well-established states B (5721), B ∗ (5747), B s (5830) and B ∗ s (5840), we give our quark model classifications for the high mass resonances ob-served in recent years. It is found that (i) the B J (5840) resonance may be explained as the low mass mixed state B ( | S D i L ) via 2 S -1 D mixing, or interpreted as the 2 S -wave state B (2 S ). (ii) The B J (5970) resonance maybe assigned as the 1 D state in the B meson family. (iii) The narrow structure around 6064 MeV observed inthe B + K − mass spectrum at LHCb may be mainly caused by the B sJ (6109) resonance decaying into B ∗ + K − , andfavors the assignment of the high mass 1 D -wave mixed state B s (1 D ′ ) with J P = − . (iv) The relatively broader B sJ (6114) structure observed at LHCb may be explained with the mixed state B s ( | S D i H ) via 2 S -1 D mixing.Most of the missing 1 P -, 1 D -, and 2 S -wave B - and B s -meson states have a relatively narrow width, they aremost likely to be observed in their dominant decay channels with a larger data sample at LHCb. PACS numbers:
I. INTRODUCTION
Since 2007, significant progress has been made in theobservations of the bottom and bottom-strange mesons [1].In 2007, two low-lying orbitally excited narrow B mesons B (5721) , + and B ∗ (5747) , + were observed by the D0 ex-periment [2], and were confirmed by the CDF experimentone year later [3]. Their strange analogues, B s (5830) and B ∗ s (5840), as the first orbitally excited B s mesons, were alsoreported by the CDF Collaboration in 2007 [4]. The B s (5830)and B ∗ s (5840) were confirmed by the D0 and LHCb experi-ments [5, 6]. In 2013, two higher resonances B (5970) , + wereobserved in the B π final states by the CDF Collaboration [7].In 2015, four higher resonances B J (5840) , + and B J (5960) , + were observed in the B π final states by the LHCb Collabora-tion when they carried out precise measurements of the prop-erties of the B (5721) , + and B ∗ (5747) , + states [8]. The prop-erties of the B J (5960) , + states are consistent with those of B (5970) , + obtained by the CDF Collaboration. Recently, theLHCb Collaboration observed two structures B sJ (6064) and B sJ (6114) in the B + K − mass spectrum [9]. More experimen-tal information about the excited bottom and bottom-strangemesons is collected in Table I. More and more excited bot-tom and bottom-strange mesons are expected to be observedin future LHCb experiments due to its huge production crosssections of beauty, together with a good reconstruction e ffi -ciency, versatile trigger scheme and an excellent momentumand mass resolution [10].Stimulated by the exciting progress in experiments, manytheoretical studies of the masses [11–19], strong decays [16–32], radiative decays [17–19, 32, 33], and semileptonic de-cays [34] for the excited bottom and bottom-strange mesonstates have been carried out with di ff erent methods in recent ∗ mail: [email protected] years. For the well established states B (5721), B ∗ (5747), B s (5830) and B ∗ s (5840), there are no puzzles to classify themas the first orbital excitations (i.e. the 1 P -wave states) pre-dicted in the quark model. While for the newly observed res-onances / structures B J (5840) , + , B (5970) , + , B sJ (6064), and B sJ (6114), although they are good candidates for the 2 S and1 D -wave states according to the mass spectrum predictionsin various quark models [13–19, 35–37], their quark modelclassification is not clear. There have been some theoreticalinterpretations of the newly observed B J (5840) and B J (5970)based on the predicted masses and strong decay properties,since they were reported by the CDF and LHCb experi-ments. In the literature, the B J (5840) resonance is explainedwith the B (2 S ) [18–20], the B (2 S ) [23], or the B (1 D )state [21]. While for the resonance B J (5970), there are inter-pretations with the radially excited state B (2 S ) [14, 16, 20],or with the second orbitally excited B -meson states [22] either B (1 D ) [18, 23] or B (1 D ) [19]. It should be mentioned thatin Ref. [38], our group assigned the B J (5970) resonance to bethe B (1 D ) state by analyzing the strong decay properties of B J (5970) within a chiral quark model. With this assignment,the authors further predicted that as the partner of B J (5970),the mass and width for the B s (1 D ) state might be M ≃ . Γ ≃
30 MeV, respectively. Thus, if assigning the B sJ (6064) to be the B s (1 D ) state, both the measured massand width are consistent with the predictions of our group. Nodiscussions about the recently observed structures B sJ (6064)and B sJ (6114) can be obtained in the literature. More informa-tion about the status of the bottom and bottom-strange mesonstudy can be found in the recent review work [39].The experimental progress provides us good opportunitiesto establish an abundant B and B s -meson spectrum up to thesecond orbital ( L =
2) excitations. In this work we deepenour study by carrying out a combined analysis of the massesand decay properties of the B and B s -meson states up to the L = B and B s mesons within a nonrelativistic potential model. TABLE I: Summary of the experimental information for the excited B - and B s -meson states. The date for the B (5721) + , , B (5747) + , , B s (5830) , and B s (5840) 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 = ( − J and unnatural spin parity P = − ( − J .Resonance J P Mass (MeV) Width (MeV) Observed channel Experiment B (5721) + . ± . . ± . B ∗ + π − D0 [2],CDF [3],LHCb [8] B (5721) + + . + . − . ± B ∗ π + D0 [2],CDF [3],LHCb [8] B (5747) + . ± . . ± . B + π − , B ∗ + π − D0 [2],CDF [3],LHCb [8] B (5747) + + . ± . ± B + π − , B ∗ + π − D0 [2],CDF [3],LHCb [8] B J (5970) + ? 5961 ±
17 60 + − ± B π + [or B ∗ π + ] CDF [7] B J (5970) ? 5978 ±
17 70 + − ± B + π − [or B ∗ + π − ] CDF [7]Case A a B J (5960) + UN 5964 . ± . . ± . B ∗ π + LHCb [8] B J (5960) UN 5969 . ± . . ± . B ∗ + π − LHCb [8] B J (5840) + UN 5850 . ± . . ± . B ∗ π + LHCb [8] B J (5840) UN 5862 . ± . . ± . B ∗ + π − LHCb [8]Case B b B J (5960) + UN 6010 . ± . . ± . B ∗ π + LHCb [8] B J (5960) UN 6015 . ± . . ± . B ∗ + π − LHCb [8] B J (5840) + N 5874 . ± . . ± . B ∗ π + , B π + ? LHCb [8] B J (5840) N 5889 . ± . . ± . B ∗ + π − , B + π − ? LHCb [8]Case C c B J (5960) + N 5966 . ± . . ± . B ∗ π + , B π + ? LHCb [8] B J (5960) N 5993 . ± . . ± . B ∗ + π − , B + π − ? LHCb [8] B J (5840) + UN 5889 . ± . . ± . B ∗ π + LHCb [8] B J (5840) UN 5907 . ± . . ± . B ∗ + π − LHCb [8] B s (5830) + . ± .
20 0 . ± . ± . B ∗ + K − , B ∗ K CDF [4], D0 [6], LHCb [5] B s (5840) + . ± .
12 1 . ± . B ∗ K , BK CDF [4], D0 [6],LHCb [5] B sJ (6063) ? 6063 . ± . ± ± B + K − LHCb [9][or B sJ (6109) ] ? [or 6108 . ± .
8] [or 22 ± ±
4] [or B ∗ + K − ] LHCb [9] B sJ (6114) ? 6114 ± ± ± B + K − LHCb [9][or B sJ (6158) ] ? [or 6158 ±
9] [or 72 ± ±
25] [or B ∗ + K − ] LHCb [9] With this model the masses for the observed B and B s -mesonstates can be described successfully. Then, with the avail-able wave functions from the potential model, we calculatethe OZI-allowed two-body strong decays of the excited B and B s mesons with a chiral quark model. This model hasbeen successfully applied to describe the strong decays of theheavy-light mesons and baryons [38, 40–51]. To provide moreknowledge for the excited B and B s meson states, we alsoevaluate the their electromagnetic (EM) transitions within anonrelativistic constituent quark model developed in our pre-vious works [52–54]. Based on our good descriptions of themass and decay properties for the low-lying well-establishedstates B (5721) , + , B ∗ (5747) , + , B s (5830) and B ∗ s (5840), wegive our quark model classifications of the high mass res-onances / structures B J (5840) , + , B (5970) , + , B sJ (6064), and B sJ (6114). Finally, according to our assignments for thenewly observed resonances, we attempt to predict the proper-ties of the missing resonances, which may be useful for futureinvestigations in experiments.This paper is organized as follows. In Sec. II, the massspectrum is calculated within a non-relativistic linear potentialmodel. In Sec. III, a brief review of the chiral quark model isgiven. The numerical results are presented and discussed inSec. IV. Finally, a summary is given in Sec. V. TABLE II: The parameters of the nonrelativistic potential model.
B B s m b (GeV) 4.852 4.852 m u , d (GeV) 0.450 · · · m s (GeV) · · · α σ (GeV) 0.98 1.06 b (GeV ) 0.120 0.120 C (GeV) -0.2537 -0.2318 r c (fm ) 0.337 0.292 II. MASS SPECTRUM
To describe the bottom and bottom-strange meson spec-tra, we adopt a nonrelativistic linear potential model. In thismodel, the e ff ective potential is adopted as [54–58] V ( r ) = V ( r ) + V sd ( r ) , (1)where V ( r ) = − α s r + br + C (2)includes the standard color Coulomb interaction and linearconfinement, and zero point energy C . The spin-dependentpart V sd ( r ) can be expressed as [57–59] V sd ( r ) = H S S + H T + H LS , (3) L J B J (5970) M a ss ( M e V ) J P L J B J (5840) D D’ P D D P P S S P’ - - + - - + + - - + B spectrum J P B s spectrum B SJ (6114)B SJ (6109) D D’ P D D P P S S P’ - - + - - + + - - + FIG. 1: The predicted mass spectra of B and B s mesons. The solid dots indicate the measured masses obtained from the PDG [1]. The data for B sJ (6109) and B sJ (6114) are taken from the recent LHCb measurements [9].TABLE III: The predicted bottom meson masses (MeV) compared with the data and some other model predictions. The mixing angle of1 P − P and 1 D − D obtained in present work are θ P = − . ◦ , and θ D = − . ◦ . In the table, β NRef f and β Ref f stand for thee ff ective harmonic oscillator parameters (GeV) of our nonrelativistic quark model calculations and those with the relativized quark modelcalculations [17], respectively, while β Cef f is our results including relativistic corrections.State J P β NRef f β Cef f β Ref f [17] Ours KDR [12] EFG [13] LPW [18] LL [14] GI [17] AMS [19] PE [37] Exp. [1] B (1 S ) 0 − B (1 S ) 1 − B (1 P ) 0 + · · · B (1 P ) 1 + B (1 P ′ ) 1 + B (1 P ) 2 + B (2 S ) 0 − B (2 S ) 1 − · · · B (1 D ) 1 − · · · B (1 D ) 2 − · · · B (1 D ′ ) 2 − · · · B (1 D ) 3 − where H S S = πα s m q m ¯ q ˜ δ σ ( r ) S q · S ¯ q (4)is the spin-spin contact hyperfine potential. Here, we take˜ δ σ ( r ) = ( σ/ √ π ) e − σ r as suggested in Ref. [55]. The ten- sor potential H T is adopted as H T = α s m q m ¯ q r S q · rS ¯ q · r r − S q · S ¯ q ! . (5) TABLE IV: The predicted bottom-strange meson masses (MeV) compared with the data and some other model predictions. The mixing angleof 1 P − P and 1 D − D obtained in present work are θ P = − . ◦ , and θ D = − . ◦ . In the table, β NRef f and β Ref f stand for the e ff ectiveharmonic oscillator parameters (GeV) obtained from our nonrelativistic quark model calculations and those with the relativized quark modelcalculations [17], respectively, while β Cef f is our results including relativistic corrections.State J P β NRef f β Cef f β Ref f [17] Ours KDR [12] EFG [13] ZVR [36] LPW [18] AMS [19] GI [17] PE [37] Exp. [1] B s (1 S ) 0 − B s (1 S ) 1 − B s (2 S ) 0 − · · · B s (2 S ) 1 − · · · B s (1 P ) 0 + · · · B s (1 P ) 1 + · · · B s (1 P ′ ) 1 + B s (1 P ) 2 + B s (1 D ) 1 − B s (1 D ) 2 − · · · B s (1 D ′ ) 2 − B s (1 D ) 3 − The spin-orbit interaction H LS can be decomposed into sym-metric part H sym and antisymmetric part H anti : H LS = H sym + H anti , (6)with H sym = S + · L m q + m q α s r − br ! + α s m q m ¯ q r , (7) H anti = S − · L m q − m q α s r − br ! . (8)In these equations, L is the relative orbital angular momentumof the q ¯ q system; S q and S ¯ q are the spins of the quark q andantiquark ¯ q , respectively, and S ± ≡ S q ± S ¯ q ; m q and m ¯ q are themasses of quark q and antiquark ¯ q , respectively; α s is the run-ning coupling constant of QCD; and r is the distance betweenthe quark q and antiquark ¯ q . The six parameters in the abovepotentials ( α s , b , σ , m q , m ¯ q , C ) are determined by fitting themass spectrum.It should be emphasized that when m q , m ¯ q , the antisym-metric part of the spin-orbit potential, H anti , can cause a con-figuration mixing between spin triplet n L J and spin singlet n L J . Thus, the physical states nL J and nL ′ J are expressed as nL J nL ′ J ! = cos θ nL sin θ nL − sin θ nL cos θ nL ! n L J n L J ! . (9)where J = L = , , · · · , and the θ nL is the mixing angle. Inthis work nL ′ J corresponds to the higher mass mixed state asoften adopted in the literature.In this work, we solve the radial Schr¨odinger equation byusing the three-point di ff erence central method [60] from cen-tral ( r =
0) towards outside ( r → ∞ ) point by point. Thismethod was successfully to deal with the spectroscopies of c ¯ c , b ¯ b , b ¯ c and s ¯ s [52–54, 61, 62]. To overcome the singularbehavior of 1 / r in the spin-dependent potentials, followingthe method of our previous works [52–54, 61, 62], we intro-duce a cuto ff distance r c in the calculation. Within a smallrange r ∈ (0 , r c ), we let 1 / r = / r c . The model parameters adopted in this work are listed in Ta-ble II. To be consistent with our previous study [54, 61, 62],the bottom quark mass m b , the light up or down quark mass m u / d , the strange quark mass m s are taken from the determi-nations, i.e., m b = .
852 GeV, m u / d = .
45 GeV, m s = . α s , b , σ , C ) for the bottommeson sector, they are determined by fitting the masses ofthe well established states B (5279), B ∗ (5325), B (5721) and B ∗ (5747), while for the bottom-strange meson sector, they aredetermined by fitting the masses of the well established states B s (5367), B ∗ s (5415), B s (5830) and B ∗ s (5840). The cuto ff dis-tance r c is determined by the mass of 1 P state. To deter-mine the masses of the B (1 P ) and B s (1 P ) states, we adopta method of perturbation, i.e., we let H = H + H ′ , where H ′ isa part which contained the term of 1 / r . By solving the equa-tion of H | ψ (0) n i = E | ψ (0) n i , we can get the energy E and wavefunction | ψ (0) n i , then, we obtain the masses of B (1 P ) and B s (1 P ) with the relation M = m q + m ¯ q + E + h ψ (0) n | H ′ | ψ (0) n i .By solving the radial Schr¨odinger equation and with the de-termined model parameters, we obtain the masses of the bot-tom and bottom-strange meson states, which have been listedin Tab. III and Tab. IV, respectively. For comparison, someother model predictions in Refs. [12–14, 17–19, 36, 37] andthe data from the Review of Particle Physics (RPP) of Par-ticle Data Group (PDG) [1] are listed in the same table aswell. Furthermore, for a clarity, the spectra are also shown inFig. 1. It is shown that the B J (5970) resonance and the struc-tures B sJ (6064) and B sJ (6114) newly observed at LHCb canbe explained as the 1 D -wave states from the point of view ofthe mass, while the B J (5840) may be a good candidate of the2 S -wave state. III. STRONG AND RADIATIVE DECAYSA. Models
We use the chiral quark model to calculate the strong decaysof the bottom and bottom-strange mesons. In this model, thelight pseudoscalar mesons, i.e. π , K and η , are treated as fun-damental states. The low energy quark-pseudoscalar-mesonand quark-vector-meson interactions in the SU(3) flavor basisare described by the e ff ective Lagrangian [63–66] H m = X j f m ¯ ψ j γ µ γ ψ j ~τ · ∂ µ ~φ m , (10)where ψ j represents the j th quark field in the hadron, φ m is thepseudoscalar meson field, f m is the pseudoscalar meson decayconstant. The nonrelativistic form of Eq. (10) is given by H m = X j " A σ j · q + ω m µ q σ j · p j I j ϕ m , (11)in the center-of-mass system of the initial meson, where wehave defined A ≡ − (1 + ω m E f + M f ). In Eq. (11), q and ω m are thethree-vector momentum and energy of the final-state light me-son, respectively; p j is the internal momentum operator of the j th quark in the heavy-light meson rest frame; σ j is the spinoperator corresponding to the j th quark of the heavy-light sys-tem; and µ q is a reduced mass given by 1 /µ q = / m j + / m ′ j with m j and m ′ j for the masses of the j th quark in the initial andfinal mesons, respectively. The plane wave part of the emittedlight meson is ϕ m = e − i q · r j , and I j is the flavor operator definedfor the transitions in the SU(3) flavor space. The chiral quarkmodel has been successfully applied to describe the strong de-cays of the heavy-light mesons and baryons [38, 40–46]. Thedetails of this model can be found in Refs. [41, 43].Meanwhile, to treat the radiative decay of a hadron we ap-ply the constituent quark model, which has been successfullyapplied to study the radiative decays of heavy meson sys-tems [52–54] and baryon states [47–51, 70, 71]. In this model,the quark-photon EM coupling at the tree level is adoptedas [64–69] H e = − X j e j ¯ ψ j γ j µ A µ ( k , r j ) ψ j , (12)where A µ represents the photon field with three-momentum k . e j and r j stand for the charge and coordinate of the constituentquark ψ j , respectively. In the initial-hadron-rest system, thenonrelativistic form of the quark-photon EM coupling can bewritten as H nre = X j " e j r j · ǫ − e j m j σ j · ( ǫ × ˆ k ) ϕ γ , (13)where σ j stand for the Pauli spin vector for the j th quark.The vector ǫ is the polarization vector of the photon. Theplane wave part of the emitted light meson is ϕ γ = e − i k · r j . Thedetails of this model can be found in Refs. [52, 53]. For a strong decay process, the partial decay width is cal-culated with [41, 43] Γ m = δ f m ! ( E f + M f ) | q | π M i (2 J i + X J fz , J iz |M J fz , J iz | , (14)while for a radiative decay process, the partial decay width iscalculated with [52, 53] Γ γ = | k | π J i + M f M i X J fz , J iz |A J fz , J iz | , (15)where M J fz , J iz and A J fz , J iz correspond to the strong and radia-tive transition amplitudes, respectively. The quantum num-bers J iz and J fz stand for the third components of the totalangular momenta of the initial and final hadron states, respec-tively. δ as a global parameter accounts for the strength of thequark-meson couplings. It has been determined in our pre-vious study of the strong decays of the charmed baryons andheavy-light mesons [41, 43]. Here, we fix its value the sameas that in Refs. [41, 43], i.e. δ = . 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 ofthe well established hadrons involving in the calculations areadopted from the PDG [1]. The masses of the missing B - and B s -meson states are adopted our determinations by solving theSchr¨oinger equation in Sec. II.It should be mentioned that, we do not directly adopt the nu-merical wave functions of B - and B s -meson states calculatedby solving the Schr¨oinger equation. For simplicity, we first fitthem with a simple harmonic oscillator wave function by re-producing the root-mean-square radius p h r i . The obtainede ff ective harmonic oscillator parameters β e f f for the mesonstates are listed in Tab. III and Tab. IV. It is found that the ef-fective harmonic oscillator parameters β NRe f f obtained from ournonrelativistic quark model calculations are obviously smallerthan the parameters β Re f f obtained from the relativized quarkmodels [17]. It indicates that the relativistic corrections towave function may be important.To take into account the relativistic e ff ects, we introduce theLorentz boost factor γ in the spatial wave function, i.e., ψ nlm ( r ) → ψ nlm ( γ r ) , (16)where γ = M q / E q . M q and E q correspond to the e ff ec-tive mass and energy of the light quark, respectively. Ac-cording to Ref [72], the e ff ective mass M q can be estimatedby M q = q h p i + m q , while the energy E q is estimated by E q = h p i / (2 M q ) + M q . To realize this transformation, we onlyneed replace β NRe f f in the harmonic oscillator wave functionwith β Ce f f = γβ NRe f f . The e ff ective harmonic oscillator parame-ters β Ce f f including relativistic correction are given in Tab. IIIand Tab. IV as well. It is found that the β Ce f f values are com-parable with those obtained with the relativized quark mod-els [17]. In Refs.[38, 41], the strong decays of the heavy-lightmeson states are studied with the chiral quark model by us-ing the simple harmonic oscillator wave functions with fixedharmonic oscillator parameters β = ,
466 MeV for the B and B s spectra, respectively, which are close to the parame-ters β Ce f f determined for the 1 P -, 1 D - and 2 S -wave states inpresent work. The parameters β Ce f f of the ground states 1 S and 1 S are notably lager than that of the excited states, thise ff ect is mainly caused by the strong color Coulomb interac-tion at the small distance r between two quarks.The e ff ective parameters β Ce f f of the ground states B (1 S )and B ∗ (1 S ) are crucial for understanding the decay prop-erties of the excited B and B s states because all of the ex-cited states should decay into these ground states. Consider-ing the uncertainty of the parameters β Ce f f of the ground states B (1 S ) and B ∗ (1 S ), we properly adjust their β Ce f f parame-ters to more reasonably describe the strong decays of the wellestablished 1 P -wave states. In this work, we determine themto be β B (1 S ) = .
537 GeV and β B ∗ (1 S ) = .
510 GeV for B (1 S ) and B ∗ (1 S ), respectively. There is about a 10%correction to the e ff ective parameters β Ce f f .With above parameters, our calculated decay properties forthe 1 P -, 2 S -, and 1 D -wave states are listed in Tables V, VI,and VII, respectively. From Table V, it is found that the de-cay properties of the well-established 1 P -wave state can besuccessfully described. IV. DISCUSSIONA. P -wave states Two 1 P -wave excited B meson states B (5721) and B ∗ (5747) + , together with their flavor partners B s (5830) and B ∗ s (5840) in the B s -meson family have been well-establishedin experiments. However, two resonances with J P = + , B (1 P ) and B s (1 P ), and two resonances with J P = + , B ( P ) and B s ( P ), predicted in the quark model are still miss-ing. P states There are no puzzles to assigned the B ∗ (5747) + , and B ∗ s (5840) resonances to the 1 P states in the B and B s fam-ilies, respectively.In the B meson sector, as the 1 P state both the mass andwidth of B ∗ (5747) can be well understood in the quark model.Our theoretical mass M = Γ =
31 MeVare compatible with the measurements M exp = Γ exp = (20 ±
5) MeV for B ∗ (5747) + . This state dominantly decays into B π and B ∗ π channels with compara-ble partial widths. The predicted partial width ratios for the B ∗ (5747) + , , R = Γ [ B ∗ (5747) → B ∗ + π − ] Γ [ B ∗ (5747) → B + π − ] ≈ . , (17) R = Γ [ B ∗ (5747) + → B ∗ π + ] Γ [ B ∗ (5747) + → B π + ] ≈ . , (18)are also consistent with the recent LHCb measurements R exp = . ± . ± .
30 and R exp = . ± . ± .
8, re-spectively [8]. We also study the radiative decay processes of B ∗ (5747) + , → B ∗ + , γ , their partial decay widths are predictedto be Γ [ B ∗ (5747) → B ∗ γ ] =
51 keV , (19) Γ [ B ∗ (5747) + → B ∗ + γ ] =
146 keV , (20)which are the same magnitude of the predictions in Refs. [17,73]. The radiative decay branching fractions can reach up to O (10 − ), thus radiative decays of B ∗ (5747) + , → B ∗ + , γ mightbe observed in future experiments.In the B s meson sector, as the 1 P state both the mass andwidth of B ∗ s (5840) can be well understood in the quark modelas well. Our theoretical mass M = Γ = . M exp = Γ exp = (1 . ± .
27) MeV. There are two OZIallowed two-body strong decay channels BK and B ∗ K . The BK mode governs the decays of B ∗ s (5840). Our predictedpartial width ratio between B ∗ K and BK , R = Γ [ B ∗ s (5840) → B ∗ + K − ] Γ [ B ∗ s (5840) → B + K − ] ≈ . , (21)is in good agreement with the recent LHCb measured one R exp = (9 . ± . B ∗ s (5840) has a large decay rate into B ∗ s γ , the partial widthand branching fraction are predicted to be Γ [ B ∗ s (5840) → B ∗ s γ ] ≃
51 keV , (22) Br [ B ∗ s (5840) → B ∗ s γ ] ≃ . . (23)Our predicted radiative partial decay width is consistent withthose predictions in Refs. [32, 73]. In the literature, a largerpartial width Γ [ B ∗ s (5840) → B ∗ s γ ] ≃ −
230 keV is pre-dicted [17–19]. The B ∗ s γ decay channel of B ∗ s (5840) mayhave good potentials to be observed in future experiments. P states The 1 P states in the B and B s families are still missingexperimentally. In the B meson sector, we predict that themass of the B (1 P ) state is about 5 MeV smaller than thatof B ∗ (5747). Thus, the mass for B (1 P ) is expected to bearound 5735 MeV, which is consistent with those predictionsin Refs. [12, 37]. The B π channel is the only OZI-allowed twobody decay channel. Taking the estimated mass M = TABLE V: Partial and total decay widths (MeV) for 1 P -wave bottom and bottom-strange mesons compared with the the data and some recentmodel predictions. It should be mention that some masses for the initial states adopted in the literature are slightly di ff erent. The total widthsinside the square brackets are estimated with the mixing angle θ P = − (55 ± ◦ . n S + L J State Channel Ours XZ [41] SSC [16] LPW [18] GI [17] AMS [19] Γ exp P B ∗ (5735) + B + π + B π + + B ∗ + γ × − × − × − Total 306.2 142.08 B ∗ (5735) B π + B + π − + B ∗ γ × − × − × − × − Total 305.7 272 225 230.43 154 141.71 P B ∗ (5747) + B + π + B π + + B ∗ + π + B ∗ π + + B ∗ + γ × − × − × − Total 31 11.71 20.3 20 ± B ∗ (5747) B π + B + π − + B ∗ π + B ∗ + π − + B ∗ γ × − × − × − × − Total 32 47 3.7 24.51 11.40 19.8 24 . ± . P B (5680) + B ∗ + π + B ∗ π + + B ∗ + γ × −
300 448 B + γ × − × − × − Total 140.9 163 126.4 128 ± B (5680) B ∗ π + B ∗ + π − + B ∗ γ × − × − × − × − B γ × − × − × − × − Total 140.8 219 200 199.4 163 125.8 128 ± P ′ B (5721) + B ∗ + π + B ∗ π + + B ∗ + γ × − × − × − B + γ × − × − × − Total 41.4 [24 . ± .
5] 7.27 16.4 31 ± B (5721) B ∗ π + B ∗ + π − + B ∗ γ × − × − × − × − B γ × − × − × − × − Total 41.3 [24 . ± .
5] 30 10 40.63 6.93 15.9 27 . ± . P B ∗ s (5810) B + K − + B ¯ K + B ∗ s γ × − × − × − × − Total 270 227 225 138 135.81 P B ∗ s (5840) B + K − + B ¯ K + B ∗ + K − + B ∗ ¯ K + B ∗ s γ × − × − × − × − Total 1.31 2 0.26 1.66 0.777 1.9 1.49 ± P B s (5820) B ∗ + K − + B ∗ ¯ K · · ·
149 120 B ∗ s γ × − × − × − B s γ × − × − × − Total 0 .
093 149 120 160 0.10511 P ′ B s (5830) B ∗ + K − + B ∗ ¯ K + . − ∼ B ∗ s γ × − × − × − B s γ × − × − × − Total 7.6 [0 . − .
8] 0 . − ∼ ± ± MeV, we obtain a broad width Γ ≃
300 MeV for the B (1 P )state, which is compatible with our previous result Γ ≃ B (1 P ) state is also predicted to be a broad state with a widthof ∼ −
250 MeV in the other models [16–18, 22, 23, 74].We also study the radiative decays of B (1 P ) + , , our resultshave been listed in Table V, the predicted partial width for Γ [ B (1 P ) + → B ∗ + γ ] ≃
477 keV is about a factor 3 largerthan that for Γ [ B (1 P ) → B ∗ γ ] ≃
149 keV. Our predictionsare comparable with those predicted in Ref. [17]. In the B s meson sector, we predict that the mass of the B s (1 P ) state is about 30 MeV smaller than that of B ∗ s (5840).Thus, the mass of B s (1 P ) is expected to be around 5810MeV. Our prediction is consistent with that predicted inRef. [12]. The BK channel is the only OZI-allowed two bodystrong decay channel. Taking the estimated mass M = Γ ≃
270 MeV for the B s (1 P )state. This prediction is compatible with our previous result Γ ≃
227 MeV predicted with the SHO wave functions inRef. [41]. The B s (1 P ) state is also predicted to be a broadstate with a width of ∼ −
230 MeV in the other models[16, 17, 22]. We also study the radiative decay of B s (1 P ),the predicted partial width for Γ [ B s (1 P ) → B ∗ s γ ] ≃ B s (1 P ) is below the BK mass thresh-old, which will lead to a very narrow width for the B s (1 P )state. B *+ K - B*K/TotalB s (1P ) B *0 K ( M e V ) B(1P ) B* /Total ( M e V ) Mass (GeV)
FIG. 2: Decay widths of the 1 P -wave mixed states B s (1 P ) and B (1 P ) as a function of mass. The mixing angles for the B s (1 P )and B (1 P ) states are adopted the potential model predictions θ P = − . ◦ and − . ◦ , respectively. P - P mixing The spin-orbit potential causes a strong configuration mix-ing between 1 P and 1 P . It is generally believed that B (5721) and B s (5830) correspond to the mixed states | P ′ i via the 1 P -1 P mixing in the B and B s families, respec-tively. The other two mixed states B (1 P ) and B s (1 P ) inthe the B and B s families are waiting to be established in fu-ture experiments.Considering B (5721) as the mixed state | P ′ i defined inEq.(9), the theoretical mass M = M exp = θ P = − . ◦ determined from our quark model, the width of B (5721) is predicted to be Γ ≃
41 MeV, which is slightlylarger than the observed width Γ exp ≃
30 MeV [8]. The decaywidth is nearly saturated by the B ∗ π channel. If we taking themixing angle around the value obtained in the heavy-quarksymmetry limit, i.e. θ P = − (55 ± ◦ , as that adopted inRef. [41], the decay width is predicted to be in the range of Γ = (24 . ± .
5) MeV, which seems to be more comparablewith the LHCb observations [8]. We also study the radiativedecay processes of B (5721) → B ∗ γ, B γ , their partial decaywidths are predicted to be Γ [ B (5721) → B ∗ γ/ B γ ] = /
24 keV , (24) Γ [ B (5721) + → B ∗ + γ/ B + γ ] = /
69 keV . (25)Our predictions are comparable with those predicted inRef. [73], however, most of our predictions are notablysmaller than the predictions in Refs. [17–19]. The radiativedecay modes B ∗ γ and B γ of B (5721) may be observed in fu-ture experiments since their branching fractions can reach upto the order of O (10 − ).In the B -meson family, the mass for the other mixed state B (1 P ) is about 40 MeV lower than that of B (5721) accord-ing to our potential model calculations, which is consistentwith the prediction in Ref. [37]. A slightly smaller mass split-ting, ∼ (10 −
30) MeV, between B (5721) and B (1 P ) isgiven in Refs. [12, 13, 17–19]. Thus, the mass of B (1 P )might be in the range of 5700 ±
15 MeV. Considering themass uncertainties, with the mixing angle θ P = − . ◦ weplot the decay width of B (1 P ) as a function of its mass inFig. 2. It is found that B (1 P ) is a broad state with a widthof Γ ≃ (155 ±
10) MeV. The B ∗ π channel is the only OZI-allowed two body strong decay channel. The B J (5732) listedin the RPP [1] is a good candidate for the B (1 P ). With thisassignment, the measured mass M exp = Γ exp = (128 ±
18) MeV for the B J (5732) are in good agreementwith the quark model predictions.As the mixed state | P ′ i , the mass of B s (5830) is consis-tent with the our quark model prediction M = B ∗ K channel. Taking the mixing angle θ P = − . ◦ determined by our potential model, we find thatthe theoretical width, Γ ≃ . Γ exp ≃ (0 . ± . ± .
3) MeV. How-ever, if we taking the mixing angle around the value obtainedin the heavy-quark symmetry limit, i.e. θ P = − (55 ± ◦ , asthat adopted in Ref. [41], the decay width is predicted to be inthe range of Γ = . − . P and1 P may be close to the value θ P = − ◦ obtained in theheavy-quark symmetry limit. The B s (5830) has large decayrates into B ∗ s γ and B s γ channels. Their partial decay widthsare predicted to be Γ [ B s (5830) → B ∗ s γ ] =
53 keV , (26) Γ [ B s (5830) → B s γ ] =
27 keV . (27)The branching fractions of these radiative decays may reachup to O (10 − ). Our predictions are comparable with those pre-dicted in Refs. [17, 18, 73]. The B ∗ s γ and B s γ decay channelsof B s (5830) may have good potentials to be observed in fu-ture experiments.In the B s meson sector, the mass of the other mixed state B s (1 P ) is predicted to be about 2 −
40 MeV lower than thatof B s (5830) in various quark models [12, 36]. Thus the massof B s (1 P ) is estimated to be ∼ (5808 ±
19) MeV, which is justaround the B ∗ + K − and B ∗ K mass thresholds. From Fig. 2,it is seen that the strong decay properties of B s (1 P ) are verysensitive to the mass threshold. There are three cases to beconsidered. (i) If the mass of B s (1 P ) is below the B ∗ + K − mass threshold 5818 MeV, the radiative decay modes B ∗ γ and B γ may play crucial roles in the decays. The with partial de-cay widths are estimated to be Γ [ B s (1 P ) → B ∗ s γ ] ≃
60 keV , (28) Γ [ B s (1 P ) → B s γ ] ≃
40 keV . (29)Then, the B s (1 P ) should has a very narrow width of Γ ∼O (100) keV. (ii) If the mass of B s (1 P ) lies between the B ∗ + K − mass threshold 5818 MeV and B ∗ K mass threshold 5822MeV, the B s (1 P ) dominantly decays into B ∗ K mode, andhas a narrow width of Γ ≃ (20 ±
15) MeV. (iii) If the massof B s (1 P ) is above the B ∗ K mass threshold 5822 MeV,the B s (1 P ) dominantly decays into B ∗ K and B ∗ + K − mode,and has a relatively broad width of Γ ≃ (70 ±
30) MeV. Itshould be mentioned that the OPAL Collaboration observedsome signals of a resonance denoted by B sJ (5850) with a massof M exp = (5853 ±
15) MeV and a width of Γ ≃ (47 ± B s (1 P ), the measured massand width of B sJ (5850) are consistent with the predictions. Toconfirm the B sJ (5850) resonance and established the B s (1 P )state, more observations of the B ∗ K and B ∗ + K − final statesare suggested to be carried out in future experiments.As a whole the high mass mixed state | P ′ i via the1 P -1 P mixing in the B and B s families have been well-established, they are narrow states B (5721) and B s (5830)observed in experiments. Some evidence for the low massmixed states | P i with broad widths predicted in theorymay have been observed in experiments. The B J (5732) and B sJ (5850) resonances listed by the PDG [1] may good candi-dates for the missing | P i in the B and B s families, respec-tively. B. S -wave states S states The 2 S states in the B and B s families are still not es-tablished. In the B meson sector, our predicted mass for the B (2 S ) state is M = M = B (2 S ), our results are listed in Table VI. Thedecays of B (2 S ) are governed by the B ∗ π mode with a fairlylarge branching fraction ∼ B (2 S ) is pre-dicted to be Γ ≃
63 MeV, which is about a factor 1.5 largerthan our previous prediction with SHO wave function [38]. In 2015, the LHCb Collaboration observed two new reso-nances B J (5840) , + [8]. Considering B J (5840) , + as unnaturalparity states, the relatively accurate measurements of the massand width for the neutral one are M exp = (5863 ±
9) MeV and Γ exp = (127 ±
51) MeV, respectively [8]. In this case, signalsof B J (5840) , + should come from the B ∗ π decay mode otherthan B π . In Refs. [18–20], the B J (5840) , + resonances weresuggested to be the B (2 S ) assignment. If B J (5840) , + havean unnatural parity indeed, they strongly favor the B (2 S ) as-signment. The measured mass and width together with the de-cay modes of B J (5840) are consistent with the our theoreticalpredictions. However, a natural parity for B J (5840) , + is alsopossible according to the LHCb analysis (see Case B listed inTable I). The B J (5840) , + may possibly decay into both the B ∗ π and B π channels. If the B π decay mode is confirmed infuture experiments, the B J (5840) resonance should be otherassignments since the B π mode is forbidden for the B (2 S )state.In the B s meson sector, our predicted mass for the B s (2 S )state is M = M = B s (2 S ), our results are listed in Table VI.The B ∗ K channel is the only OZI-allowed two body strong de-cay channel for B s (2 S ). Its decay width is predicted to be Γ ≃
55 MeV, which is slightly larger than our previous predic-tion Γ ≃
40 MeV with SHO wave function [38]. The decaywidth predicted in present work is comparable with those pre-dictions in Refs. [16, 17]. The B s (2 S ) state should havelarge potentials to be seen in the B ∗ + K − channel since it has afairly narrow width. S states In the B meson sector, our predicted mass for B (2 S ) is M = M = B (2 S ),our results are listed in Table VI. It is found that the B (2 S )state is a fairly narrow state with a width of Γ ≃
40 MeV,which is consistent with the prediction in Ref. [16]. This statedominantly decays into the B ∗ π channel with a branching frac-tion ∼ B π and B ∗ π is predicted to be Γ [ B π ] Γ [ B ∗ π ] ≃ . , (30)which can be used to identify the B (2 S ) from its possiblecandidates observed in future experiments. It should be men-tioned that in Ref. [23], the B J (5840) resonance was assignedas the B (2 S ) state. As this assignment, both the mass andtypical decay modes B ∗ π and B π predicted in theory are con-sistent with the LHCb observations [8]. However, our pre-dicted width Γ ≃
41 MeV is notably smaller than the mea-sured value Γ exp = (107 ±
54) MeV assuming a natural parityfor B J (5840) [8]. Thus, the B J (5840) may not favor the pure B (2 S ) assignment.0 TABLE VI: Partial and total decay widths (MeV) for the 2 S -wave B and B s mesons. n S + L J State Channel Γ th (MeV) State Channel Γ th (MeV)( b ¯ u / b ¯ d )2 S B (5876) B ∗ π B s (5944) B ∗ K B ∗ η B ∗ s η · · · B (1 P ) π B ∗ s γ × − B ∗ γ / × − B s (1 P ) γ B (1 P ) γ / B s (5830) γ × − B (5721) γ / × − Total 55Total 632 S B (5899) B π B s (5966) BK B η B s η B s K B ∗ K B ∗ π B ∗ s η B ∗ η B s γ × − B (5747) π < . B s (2 S ) γ × − B (1 P ) π B s (5840) γ × − B (5721) π B s (1 P ) γ × − B γ / B s (5830) γ × − B (2 S ) γ × − / × − B (1 P ) γ × − B (5747) γ / B (1 P ) γ / × − B (5721) γ / × − B (1 P ) γ / × − Total 41
In the B s meson sector, our predicted mass for the B s (2 S )state is M = M = B s (2 S ), our results are listed in Table VI. It is foundthat the B s (2 S ) state is also a narrow state with a width of Γ ≃
50 MeV, which is consistent with our previous predic-tion with SHO wave function [38] and the prediction from the P model in Ref. [16]. This state mainly decays into the B ∗ K and BK channel with large branching fractions ∼
68% and ∼ BK and B ∗ K is predicted to be Γ [ BK ] Γ [ B ∗ K ] ≃ . , (31)which may be an important criterion for establishing the B s (2 S ).There may exist a strong configuration mixing between the2 S and 1 D states, which is to be discussed in the last partof this section. C. D -wave states Some evidence of the 1 D -wave B and B s states may havebeen observed in experiments. The B (5970) , + together withthe new resonances B sJ (6064) and B sJ (6114) observed atLHCb are good candidates of the 1 D -wave states accordingto the mass spectrum predictions in various quark models. D states In a previous work [38], by analyzing the decay proper-ties within the chiral quark model our group found that the B J (5970) is most likely to be the 1 D assignment in the B -meson family. The B J (5970) , + resonances were first ob-served by the CDF Collaboration in the B π final states in2013 [7], and confirmed by the LHCb Collaboration twoyears later [8]. The central value of the measured width is Γ exp ≃ −
70 MeV with large uncertainties (see Table I). Inthe LHCb observations, the B ∗ π decay mode has been seen,while the B π mode may be possibly seen [1].In this work, we study the B J (5970) by combining the de-cay properties with the mass spectrum. It is found that asthe B (1 D ) assignment the mass of B (5970) can be well ex-plained with the potential model. Our predicted mass M = M exp = (5971 ±
5) MeV for the neutral state B (5970) [1]. Byusing the wave function of B (1 D ) calculated from the poten-tial model, we further study the decay properties, our resultsare listed in Table VII. It is found that the predicted decaywidth of B (1 D ), Γ ≃
39 MeV, is also consistent the mea-sured width Γ exp ≃ (56 ±
16) MeV for B (5970) by assuming P = ( − J and using three relativistic Breit-Wigner functionsin the fit for mass di ff erence at LHCb [8]. The partial widthratio between B π and B ∗ π is predicted to be Γ [ B π ] Γ [ B ∗ π ] ≃ . , (32)which is waiting to be tested in future experiments. The B J (5970) as a candidate of B (1 D ) is also suggested in other1works [18, 23]. If the B J (5970) , + resonances correspond tothe B (1 D ) assignment indeed, the charged state B J (5970) + should have a large radiative decay rate into B ∗ (5747) + γ , thepartial width and branching fraction are predicted to be Γ [ B J (5970) + → B ∗ (5747) + γ ] ≃
125 keV , (33) Br [ B J (5970) + → B ∗ (5747) + γ ] ≃ × − . (34)The radiative decay mode B ∗ (5747) + γ may be observed in fu-ture experiments.In a previous work [38], considering the B J (5970) as the B (1 D ) assignment, our group further predicted that themass and width of the B s (1 D ) state, as a flavor partner of B J (5970), might be M ≃ .
07 GeV and Γ ≃
30 MeV, respec-tively. It is interestingly found that the new bottom-strangestructure B sJ (6064) with a mass of M exp = . ± . Γ exp = (26 ±
8) MeV observed atLHCb [9] is consistent with the predictions. In present work,from the aspects of both mass spectrum and decay propertieswe further discuss the possibility of the B sJ (6064) structure asthe B s (1 D ) assignment in the B s -meson family. With this as-signment, it is found that the measured mass for the B sJ (6064)structure is consistent with the theoretical mass M = B sJ (6064) can alsobe explained within our chiral quark model. From our pre-dicted decay properties listed in Table VII, it is found that thetheoretical width Γ ≃
13 MeV is close to lower limit of themeasured value Γ exp = (26 ±
8) MeV from LHCb [9]. Thepartial width ratio between BK and B ∗ K is predicted to be Γ [ BK ] Γ [ B ∗ K ] ≃ . . (35)However, the large branching fraction for Br [ B sJ (6064) → B ∗ K ] ≃
45% seems to be inconsistent the observations.Since B sJ (6064) contributes a clear structure around 6064MeV in the B + K − mass spectrum through the B + K − decay,it should contribute another narrow structure around 6019MeV through the B ∗ + K − decay with a missing photon from B ∗ + → B + γ , which was not observed at LHCb. Thus, the B s (1 D ) 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 D as-signment, which can be tested by the partial width ratio be-tween B π and B ∗ π . It may be a flavor partner of the D ∗ (2750)and D s (2860) resonances listed in RPP [1]. Their mass mightbe systematically overestimated by a value of ∼
100 MeV insome well-known quark models, for example [13, 17, 35, 76].To establish the narrow B s (1 D ) state finally, the observa-tions of both the B + K − and B ∗ + K − decays and their partialwidth ratio are crucial in future experiments. D states In the B meson sector, the mass for the B (1 D ) is predictedto be M = B (1 D ) is about 80 MeV larger than that for B (1 D ). Taking the mass M = B (1 D ),our results are listed in Table VII. It is found that the B (1 D )state is a broad state with a width of Γ ≃
350 MeV, whichis consistent with the predictions in Refs. [16, 18]. This statemainly decays into the B π , B ∗ π , and B (5721) π channels withbranching fractions ∼ B π and B ∗ π is predicted to be Γ [ B π ] Γ [ B ∗ π ] ≃ . , (36)which may be helpful to identify the B (1 D ) state from futureobservations. In Refs. [19, 22], B J (5970) was suggested to bea candidate for B (1 D ) [19]. With this assignment we findthat the theoretical width Γ ≃
230 MeV is too broad to becomparable with the measured value measured value Γ exp ≃ −
70 MeV (see Table I). Thus, B J (5970) may not be a goodcandidate of the 1 D state.In the B s meson sector, the mass for the B s (1 D ) ispredicted to be M = B s (1 D ) is about30 MeV larger than that for B s (1 D ). There are large uncer-tainties in the predictions of the mass splitting of B s (1 D )- B s (1 D ). In some works [12, 19, 36], the B s (1 D ) mass iseven predicted to be smaller than that for B s (1 D ). Takingthe mass M = B s (1 D ), our results are listed inTable VII. It is found that the B s (1 D ) state has a width of Γ ≃
130 MeV, and mainly decays into the BK and B ∗ K chan-nels with branching fractions ∼
65% and 27%, respectively.The partial width ratio between the two typical channels BK and B ∗ K is predicted to be Γ [ BK ] Γ [ B ∗ K ] ≃ . , (37)which is comparable with the predictions in Refs. [16, 18].From the point of view of mass, the observed resonance B sJ (6114) by the LHCb Collaboration [9] is a good candi-date for the B s (1 D ) state. However, the theoretical width Γ ≃
130 MeV is sligtly larger than the upper limit of the mea-sured width Γ exp = (66 ±
39) MeV. There may exist a config-uration mixing between the 2 S and 1 D states, which willbe discussed in the last part of this section. D - D mixing There is a strong configuration mixing between the 1 D and 1 D states for the heavy-light mesons predicted in thepotential models. For the B meson sector, with the mix-ing scheme defined in Eq. (9) we predict the mixing angle θ D = − . ◦ , which is close to the value − . ◦ extractedin the heavy quark symmetry limit [77, 78]. Our predictedmasses for the B (1 D ) and B (1 D ′ ) states are about 5973 and6067 MeV, respectively, which are close to the predictions in2Refs. [12, 37]. A fairly large mass splitting between B (1 D )and B (1 D ′ ), ∆ M =
94 MeV, is obtained in present work. Itis comparable with the predictions of ∆ M = −
130 MeVin Refs. [17, 18]. With the masses and wave functions ob-tained from our potential model calculations, the decay prop-erties for these two mixed states B (1 D ) and B (1 D ′ ) are esti-mated, the results are listed in Table VII. It is found that thelow mass state B (1 D ) has a broad width of Γ ≃
180 MeV,and dominantly decays into B ∗ π and B ∗ (5747) π channels withbranching fractions about 50% and 35%, respectively. Whilethe high mass state B (1 D ′ ) has a relatively narrow width of Γ ≃
110 MeV, and dominantly decays into B ∗ π , B (5721) π and B ∗ (5747) π channels with branching fractions about 60%,15% and 14%, respectively. The decay properties predicted inthis work are roughly comparable with those calculated withthe SHO wave functions in our previous work [38].For the B s meson sector, we predict the mixing angle θ D = − . ◦ . Our predicted masses for the B s (1 D ) and B s (1 D ′ )states are about 6061 and 6113 MeV, respectively, which areclose to the predictions in Refs. [18, 36, 37]. An intermediatemass splitting between B s (1 D ) and B s (1 D ′ ), ∆ M =
52 MeV,is obtained in present work, which is comparable with the pre-dictions of ∆ M = −
70 MeV in Refs. [18, 37]. With themasses and wave functions obtained from our potential modelcalculations, the decay properties for these two mixed states B s (1 D ) and B s (1 D ′ ) are estimated, the results are listed inTable VII. It is found that the low mass state B s (1 D )(6061)has a intermediate width of Γ ≃
90 MeV , (38)and dominantly decays into B ∗ K channel with branching frac-tions about 94%. The B s (1 D ) state may be observed around6016 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 (1 D ′ )(6113) has a very narrowwidth of Γ ≃
23 MeV , (39)and dominantly decays into B ∗ K channel with branching frac-tion about 95%. The decay properties predicted in this workare in reasonably agreement with those calculated with theSHO wave functions in our previous work [38].The narrow mixed state B s (1 D ′ ) with a mass of M = B sJ (6064) struc-ture observed in the B + K − mass spectrum at LHCb [9]. Inthis case, the signal in the B + K − mass spectrum may mainlycome from the B ∗ + K − decay with a missing photon from B ∗ + → B + γ . Including the energy of the missing photon, themass and width are determined to be M exp = (6109 ± .
8) MeVand Γ exp = (22 ±
9) MeV for the resonance B sJ (6109) [9]. It isinterestingly found that the B sJ (6109) resonance favors the as-signment of the 1 D -wave mixed state B s (1 D ′ ). The predictedmass, width, decay mode are in good agreement with the ob-servations. Finally, it should be mentioned that the B s (1 D )state may have a few contributions to the B sJ (6064) structurethrough the B + K − decay as well, since this state with a massof M ≃ -90 -60 -30 0 30 60 9004080120160200-90 -60 -30 0 30 60 9050100150200250300 ( M e V ) B J (5840) (degree) BB*B ( M e V ) TotalB (5721)B* B B(|SD B (1P) Total FIG. 3: The partial decay widths and total decay widths for the mixedstates via 2 S − D mixing in the B -meson family as functions ofthe mixing angle θ . In the horizontal direction, the shaded regionrepresents the possible range of the measured width from LHCb. Inthe vertical direction, shaded region represents the possible range ofthe mixing angle θ ≃ − (45 ± ◦ suggested in Refs. [40, 42]. Themasses for B J (5840) and B ( | S D i H ) are taken to be 5890 and 6040MeV, respectively. D. S - D mixing It should be mentioned that there may exist a configurationmixing between the 2 S and 1 D states. In Refs. [40, 42],our group carefully studied the strong decay properties ofthe D ∗ J (2600) and D s (2700). According the analysis, both D ∗ J (2600) and D s (2700) could be explained as the mixed state | S D i L via the 2 S -1 D mixing with the following mixingscheme: | S D i L | S D i H ! = cos θ sin θ − sin θ cos θ ! S D ! , (40)where the mixed angle is estimated to be θ ≃ − (45 ± ◦ .The D s (2860) resonance observed in the BK final state atLHCb [79, 80] seems to be the high mass mixed state | S D i H as the partner of the low mass state D s (2700) [38]. Then themass splitting between the high and low mass mixed statesis estimated to be about 150 MeV. To explain the strong de-3 -90 -60 -30 0 30 60 9004080120160-90 -60 -30 0 30 60 90020406080100 ( M e V ) (degree) B*KB Bs ( M e V ) (degree) Total B sJ (6114) B B s (|SD L ) B s B*K Total
FIG. 4: The partial decay widths and total decay widths for the mixedstates via 2 S − D mixing in the B s -meson family as functionsof the mixing angle θ . In the horizontal direction, the shaded regionrepresents the possible range of the measured width from LHCb. Inthe vertical direction, shaded region represents the possible range ofthe mixing angle θ ≃ − (45 ± ◦ suggested in Refs. [40, 42]. Themasses for B sJ (6114) and B s ( | S D i L ) are taken to be 6114 and 5964MeV, respectively. cay properties of the D ∗ J (2600) and / or D s (2700), configura-tion mixing between 2 S and 1 D is also suggested in theliterature [81–85]. Thus, the 2 S -1 D mixing might alsoexist in the B and B s meson families.The strong decay properties for the mixed states | S D i L and | S D i H in the B and B s meson families were further studiedin another work of our group. It is interestingly found thatthe newly observed resonances B J (5840) and B sJ (6114) atLHCb [8, 9] are most likely to be the mixed states B ( | S D i L )and B s ( | S D i H ) predicted in Ref. [38], respectively, by com-paring the measured masses and widths with the theoreticalpredictions (see Figs. 1 and 2 in Ref. [38]).Considering the B J (5840) resonance as the low mass mixedstate B ( | S D i L ), we revise the strong decay properties by usingthe wave functions obtained from our quark potential modelcalculations. Our results are shown in Fig. 3. With the mixingangle θ ≃ − (45 ± ◦ determined in Refs. [40, 42] and themass M exp ≃ B ( | S D i L ) state has a width of Γ ≃ (76 ±
20) MeV, and dominantly decaysinto B ∗ π channel. There may be a sizeable decay rate into the B π channel. The partial width ratio between B π and B ∗ π ispredicted to be Γ [ B π ] Γ [ B ∗ π ] ≃ . − . , (41)which is sensitive to the mixing angle. The predicted width isconsistent with the measured width Γ exp = (107 ±
54) MeV byassuming P = ( − J . Moreover, the predicted decay modesare also consistent with the observations of B J (5840). To bet-ter understand the nature of B J (5840), more accurate mea-surements of the width together with the partial width ratioare expected to be carried out in future experiments.If B J (5840) corresponds to the low mass state B ( | S D i L ) in-deed, the mass of B ( | S D i H ) may be about 150 MeV largerthan that of B J (5840). Taking a mass of M ≃ B ( | S D i H ), we show its decay properties in Fig. 3 as well.Within the mixing angle range θ ≃ − (45 ± ◦ suggestedin [40, 42], the B ( | S D i H ) has a width of Γ ≃ (197 ±
47) MeV,and mainly decay into B π and B (5721) π channels. The par-tial width ratio between B π and B (5721) π is predicted to be Γ [ B π ] Γ [ B (5721) π ] ∼ , (42)which is insensitive to the mixing angle. Future observationsin the B π channel with a larger data sample at LHCb may havea potential to discover this high mass mixed state B ( | S D i H ).In the B s meson sector, considering the B sJ (6114) reso-nance observed in the B + K − mass spectrum [9] as the highmass mixed state B s ( | S D i H ), we revise the strong decay prop-erties by using the wave functions obtained from our quarkpotential model calculations. Our results are shown in Fig. 4.It is found that with the mixing angle θ ≃ − (45 ± ◦ deter-mined in Refs. [40, 42], the B s ( | S D i H ) state has a width of Γ ≃ (95 ±
15) MeV, and dominantly decays into BK channelwith a branching fraction ∼ B + K − and width Γ exp = (66 ±
39) MeV observed for B sJ (6114) atLHCb [9] can be well understood in our quark model cal-culations. Thus, the B sJ (6114) may favor the mixed state B s ( | S D i H ).The mass for the low mass state B s ( | S D i L ) may be about150 MeV smaller than that of B sJ (6114). Taking a mass of M ≃ B s ( | S D i L ), we showits decay properties in Fig. 4 as well. In the mixing angle range θ ≃ − (45 ± ◦ , the B s ( | S D i L ) has a width of Γ ≃ (70 ± B ∗ K channel. The partial widthratio between BK and B ∗ K , Γ [ BK ] Γ [ B ∗ K ] < . , (43)is sensitive to the mixing angle. The B s ( | S D i L ) is most likelyto be observed in the B + K − final state with a larger data sam-ple at LHCb.As a whole the B J (5840) and B sJ (6114) may favor themixed states B ( | S D i L ) and B s ( | S D i H ) via 2 S -1 D mixing,respectively. Their partners B ( | S D i H ) and B s ( | S D i L ) are ex-pected to be observed in their dominant decay channels witha larger data sample at LHCb.4 V. SUMMARY
The experimental progress provides us good opportunitiesto establish an abundant B and B s -meson spectrum up to thesecond orbital excitations. In this work, combining the newestexperimental progress, we carry out a systematical study ofthe mass spectrum, strong decays and radiative decays ofthe 1 P -, 1 D -, and 2 S -wave excited B and B s states in theconstitute quark model. The mass and strong decay proper-ties for the well established 1 P -wave resonances B (5721) + , , B ∗ (5747) + , , B s (5830) and B ∗ s (5840) can be consistently ex-plained. The possible assignments for the high mass res-onances / structures B J (5840) , + , B (5970) , + , B sJ (6064) and B sJ (6114) are discussed. We hope that our study can providesome useful information towards establishing an abundant B and B s -meson spectrum. Our main results are summarized asfollows.For the P -wave states, several points should be emphasized.(i) Some radiative decay processes, such as B s (5830) → B ( ∗ ) s γ and B ∗ s (5840) → B ∗ s γ , have good potentials to be foundin future experiments due to their fairly branching fractions of O (10 − ). (ii) The B J (5732) and B sJ (5850) resonances listedby the PDG [1] may good candidates for the missing | P i state in the B and B s families, respectively. (iii) Both B (1 P )and B s (1 P ) may hardly be observed in experiments due totheir very broad width of Γ ∼
300 MeV.The B J (5840) resonance and the new B sJ (6114) struc-ture observed in the B + K − mass spectrum may be explainedwith the mixed states B ( | S D i L ) and B s ( | S D i H ) via 2 S -1 D mixing, respectively. To confirm the nature of the B J (5840) and B sJ (6114), the typical ratios Γ ( B π ) / Γ ( B ∗ π ) and Γ ( B ∗ K ) / Γ ( BK ) are suggested to be measured in future exper-iments. The other two missing states, B ( | S D i H ) with a massof M ≃ B s ( | S D i L ) with a mass of M ≃ B π and B ∗ K final states.The B J (5970) resonance may be assigned as the 1 D statein the B meson family, which is consistent with our previousconclusion. To clarify the nature of B J (5970), further observa-tions of the B π and B ∗ π channels and a measurement of their partial width ratio are necessary. In the B s family, the pre-dicted mass M ≃ Γ ≃
13 MeV for the B s (1 D ) are consistent with the B sJ (6064) structure observedin the B + K − mass spectrum. However, the B s (1 D ) may notbe the main contributor to the B sJ (6064) structure. In this caseanother 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 − massspectrum may mainly come from the resonance B sJ (6109) de-caying into B ∗ + K − . The B sJ (6109) resonance favors the as-signment of the high mass 1 D -wave mixed state B s (1 D ′ ) with J P = − . This state dominantly decays into B ∗ K channel withbranching fraction about 95%, and the BK decay is forbid-den. The other missing mixed state B s (1 D ) has a mass of M ≃ Γ ≃
90 MeV. It is mostlikely to be established in the B + K − mass spectrum throughthe B ∗ + K − decay with a missing photon from B ∗ + → B + γ . Inthe B -meson sector, two relatively broad mixed states, B (1 D )with M ≃ B (1 D ′ ) with M ≃ B ∗ π with a larger datasample at LHCb.Finally, it should be mentioned that the B J (5840) resonancemay be a candidate of the 2 S -wave state B (2 S ) as well. Inthis case, the B π mode of B J (5840) is forbidden, which shouldbe further confirmed in future experiments. In the B s mesonsector, our predicted mass and width for the B s (2 S ) stateare M = Γ ≃
55 MeV, respectively. The B ∗ K channel is the only OZI-allowed two body strong decay mode.The B s (2 S ) state should have large potentials to be seen inthe B ∗ + K − channel since it has a fairly narrow width. Acknowledgement
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20 Total 133 B (5747) γ / B (1 P ) γ / B (5721) γ / B (1 P ) γ / D ′ B (6067) B ∗ π B s (6113) B ∗ K B ∗ η B ∗ S η B ∗ s K B s (5840) γ B (1 P ) π B s (1 P ) γ × − B (5747) π B s (5830) γ B (5721) π B s (1 P ) γ × − B (1 P ) π B (5747) γ / B (1 P ) γ / × − B (5721) γ / B (1 P ) γ × − / × − Total 1111 D B (5973) B ∗ π B s (6061) B ∗ K B ∗ η B ∗ S η B ∗ s K B s (5840) γ × − B (1 P ) π B s (1 P ) γ B (5747) π B s (5830) γ × − B (1 P ) π B s (1 P ) γ × − B (5721) π B (5747) γ / B (1 P ) γ / B (5721) γ × − / × − B (1 P ) γ × − / × −3