The excited bottom-charmed mesons in a nonrelativistic quark model

Using the newly measured masses of $B_c(1S)$ and $B_c(2S)$ from the CMS Collaboration and the $1S$ hyperfine splitting determined from the lattice QCD as constrains, we calculate the $B_c$ mass spectrum up to the $6S$ multiplet with a nonrelativistic linear potential model. Furthermore, using the wave functions from this model we calculate the radiative transitions between the $B_c$ states within a constituent quark model. For the higher mass $B_c$ states lying above $DB$ threshold, we also evaluate the Okubo-Zweig-Iizuka (OZI) allowed two-body strong decays with the $^{3}P_{0}$ model. Our study indicates that besides there are large potentials for the observations of the low-lying $B_c$ states below the $DB$ threshold via their radiative transitions, some higher mass $B_c$ states, such as $B_c(2^3P_2)$, $B_c(2^3D_1)$, $B_c(3^3D_1)$, $B_c(4^3P_0)$, and the $1F$-wave $B_c$ states, might be first observed in their dominant strong decay channels $DB$, $DB^*$ or $D^*B$ at the LHC for their relatively narrow widths.


I. INTRODUCTION
The B c states composed of a bottom-charmed quarkantiquark pair, as an important family of hadron spectra was predicted in theory about 40 years ago [1], however, the experimental progress towards establishing the B c spectrum is not obvious. Except for the ground state B c meson observed in 1998 by the CDF Collaboration at Fermilab [2], until 2018, only the ATLAS Collaboration reported evidence of an excited B c state with a mass of 6842 ± 9 MeV [3] consistent with the values predicted for B c (2S ), while it was not confirmed by the LHCb Collaboration by using their 8 TeV data sample [4]. The poor situation of the observations and measurements of B c spectrum is due to that the production yields are significantly smaller than those of the charmonium and bottomonium (cc and bb) states. Fortunately, the LHC provides good opportunities for our search for the excited B c states with its high collision energies and integrated luminosity. Very recently, two excited B + c states were observed in the B + c π + π − invariant mass spectrum by the CMS Collaboration [5]. Signals are consistent with the B c (2S ) and B * c (2S ) states. These two states are well resolved from each other and are observed with a significance exceeding five standard deviations. The mass of B c (2S ) meson, 6871±2.8 MeV, measured by the CMS Collaboration is inconsistent with the determination 6842 ± 9 MeV by the ATLAS Collaboration. The reason is that the peak observed by ATLAS could be the superposition of the B c (2S ) and B * c (2S ) states, too closely spaced with respect to the resolution of the measurement [5].
The B c states as the only conventional heavy mesons with different flavors have aroused great interests in theory. Compared with the cc and bb spectra, the B c spectrum has several special features for the bottom-charmed quark-antiquark pair. (i) The B c states cannot annihilate into gluons, thus, the low- * E-mail: lvqifang@hunnu.edu.cn † E-mail: guilongcheng@hunnu.edu.cn ‡ E-mail: zhongxh@hunnu.edu.cn lying excited B c states below the DB threshold are more stable with a narrow width less than a few hundred keV, they mainly decay via the electromagnetic or hadronic transitions between two different B c states. (ii) In the B c meson spectrum there are configuration mixings between the states with different total spins but with the same total angular momentum, such as 3 P 1 − 1 P 1 , 3 D 2 − 1 D 2 , and 3 F 3 − 1 F 3 mixings via the antisymmetric part of the spin-orbit potential. (iii) Additionally, the B c states provide a unique window for studying the heavyquark dynamics that is very different from those provided by the cc and bb states. In the past years, the B c mass spectrum has been predicted with various models . Furthermore, a few lattice calculations can be found in Refs. [35][36][37][38][39]. To estimate the production rates in experiments, the production of the excited B c states were often discussed in the literature [40][41][42][43][44][45][46][47][48][49][50][51][52][53]. As the dominant decay modes, the electromagnetic transitions of the low-lying B c states were also widely estimated in the literature [7][8][9][10][11][12][13][14][15][16][54][55][56][57][58]. However, the studies of the OZIallowed strong decays for the high-lying B c states are confined only to a few calculations [17,18,59,60].
The successes of the observations of the radially excited B c states B c (2S ) and B * c (2S ) by the CMS Collaboration [5] have demonstrated that more excited B c states are to be discovered in future LHC experiments. Stimulated by the great discovery potentials of the missing B c states in future experiments, in present work we carry out a systematical study of the B c spectrum. First, using the newly measured masses of B c (1S ) and B c (2S ) from the CMS Collaboration [5] and the 1S hyperfine splitting determined from the lattice QCD [36][37][38] as constrains, we calculate the B c mass spectrum up to the 6S multiplet with a nonrelativistic linear potential model. The slope parameter of the linear potential has been well determined in our previous study of the charmonium states [61]. To involve the spin-dependent corrections of the spatial wave functions, following the method adopted in Refs. [61,62] we treat the spin-dependent interactions as nonperturbative terms in our calculations. With this nonperturbative treatment, we can reasonably include the effect of spin-dependent interactions on the spatial wave functions, which is essential for us to gain reliable predictions of the decay behaviors.
Then, with the available wavefunctions from the potential model, we evaluate the electromagnetic transitions between the B c states within a nonrelativistic constituent quark model developed in our previous works [61,62]. With this approach the possible higher EM multipole contributions to a EM transition process can be included naturally. Considering the fact that the higher B c states lying above the DB threshold may have enough possibilities to be produced at LHC, and they are easy to be established in the D ( * ) B ( * ) hadronic final states, thus to give useful references for the LHC observations, we further calculate the OZI-allowed strong decays of the higher B c states within the widely used 3 P 0 model [63][64][65]. It is found that B c (2 3 P 2 ), B c (2 3 D 1 ), B c (3 3 D 1 ) together with the 1F-wave B c states might be first observed in their dominant strong decay channels DB, DB * or D * B at LHC for their relatively narrow width.
This paper is organized as follows. In Sec. II, the B c mass spectrum is calculated within a nonrelativistic linear potential model. Then, with the obtained B c spectrum the radiative transitions between the B c states are estimated in Sec. III within a nonrelativistic constituent quark model. In Sec. IV, the OZIallowed two-body strong decays of the excited B c state are also studied within the 3 P 0 model. In Sec. V, we focus on the calculation results and discuss some strategies for looking for the B c states in future experiments. Finally, a summary is given in Sec. VI.

II. MASS SPECTRUM
To describe a bottom-charmed meson system, we adopt a nonrelativistic linear potential model. In this model, the effective quark-antiquark potential is written as the sum of the spin-independent term H 0 (r) and spin-dependent term H sd (r), i.e., where includes the standard color Coulomb interaction and the linear confinement. The spin-dependent part H sd (r) can be expressed as [1,9,11] 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. [66]. The tensor potential H T is adopted as For convenience in the calculations, the potential of the spinorbit interaction H LS is 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 five parameters in the above equations (α s , b, σ, m b , m c ) are determined by fitting the spectrum. We can get the masses and wave functions by solving the radial Schrödinger equation with where µ R = m q mq/(m q +mq) is the reduced mass of the system, and E is the binding energy of the system. Then, the mass of a bottom-charmed state is obtained by In this work, to reasonably include the corrections from these spin-dependent potentials to both the mass and wave function of a meson state, we deal with the spindependent interactions nonperturbatively. We solve the radial Schrödinger equation by using the three-point difference central method [67] from central (r = 0) towards outside (r → ∞) point by point. This method was successfully to deal with the spectroscopies of cc and bb [61,62]. To overcome the singular behavior of 1/r 3 in the spin-dependent potentials, following the method of our previous works [61,62], 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 . Finally, it should be mentioned that the antisymmetric part of the spin-orbit potential, H anti , can let the states with different total spins but with the same total angular momentum, such as B c (n 3 L J ) and B c (n 1 L J ), mix with each other. Thus, as mixing states between B c (n 3 L J ) and B c (n 1 L J ), the physical B c states B c (nL) and B c (nL ′ ) are expressed as 42.4 • · · · · · · · · · m c = 1.483 GeV and b = 0.1425 GeV 2 . The other three parameters (m b , α s , σ) are determined by fitting the masses of the B c , B * c and B c (2S ) mesons. The masses of B c and B c (2S ) are taken from the recent measurements of the CMS Collaboration [5]. Although the B * c meson is still not measured in experiments, the mass difference between the B * c and B c is predicted to be around 55 MeV from lattice QCD [36][37][38].
Thus, combining it with the measured mass 6271 MeV for B c , in present work we estimate the mass of B * c as ∼ 6326 MeV. The cutoff distance r c is determined by the mass of B c (1 3 P 0 ). To determined the mass of B c (1 3 P 0 ), we adopt a method of perturbation, i.e., we let H = H 0 + H ′ , where H ′ is a part which contained the term of 1/r 3 . By solving the equation of H 0 |ψ (0) n = E 0 |ψ (0) n , we can get the energy E 0 and wave function |ψ (0) n , then, we obtain the mass of B c (1 3 P 0 ), n |H ′ |ψ (0) n . By solving the radial Schrödinger equation and with the determined parameter set, we obtain the masses of the bottomcharmed states, which have been listed in Tab. 1 and shown in Fig. 1. For comparison, the other model predictions in Refs. [7-11, 15, 16] are listed in the same table as well.

III. RADIATIVE TRANSITIONS
We use the nonrelativistic constituent quark model as adopted in Refs. [61,62,[68][69][70][71][72] to calculate the radiative transitions between the B c states. In this model, the quark-photon EM coupling at the tree level is taken as where A µ represents the photon field with three momenta k; while e j and r j stand for the charge and coordinate of the constituent quark ψ j , respectively. In order to match the nonrelativistic wave functions of the B c states, we adopt the nonrelativistic form of Eq. (13), which is given by [73][74][75][76][77][78], where ǫ is the polarization vector of the final photon, m j and σ j stand for the constituent mass and Pauli spin vector for the jth quark. The helicity amplitude A can be expressed as Finally, we obtain the partial decay width of a radiative transition by where J i is the total angular momentum of an initial meson, and J f z and J iz are the components of the total angular momenta along the z axis of initial and final mesons, respectively. M i and M f correspond to the masses of the initial and final B c states, respectively. The radiative decays properties for the B c states have been listed in Tables III-VII. For a comparison, some other predictions of the low-lying B c states from Refs. [7,[9][10][11] are also given in the tables.

IV. STRONG DECAYS
In this work, we use the 3 P 0 model [63][64][65] to calculate the OZI-allowed strong decays of the bottom-charmed mesons. In this model, it assumes that the vacuum produces a quarkantiquark pair with the quantum number 0 ++ and the heavy meson decay takes place via the rearrangement of the four quarks. The transition operatorT in this model can be written where γ is a dimensionless constant that denotes the strength of the quark-antiquark pair creation with momentum p 3 and p 4 from vacuum; b † 3i (p 3 ) and d † 4 j (p 4 ) are the creation operators for the quark and antiquark, respectively; the subscriptions, i and j, are the SU(3)-color indices of the created quark and anti-quark; φ 34 0 = (uū + dd + ss)/ √ 3 and ω 34 0 = 1 √ 3 δ i j correspond to flavor and color singlets, respectively; χ 34 1,−m is a spin triplet state; and Y ℓm (k) ≡ |k| ℓ Y ℓm (θ k , φ k ) is the ℓ-th solid harmonic polynomial. The factor (−3) is introduced for convenience, which will cancel the color factor.
For an OZI allowed two-body strong decay process A → B + C, the helicity amplitude M M J A M J B M J C (P) can be derived as follow Using the Jacob-Wick formula [79], one can convert the helicity amplitudes In the above equations, (J A , J B and J C ), (L A , L B and L C ) and (S A , S B and S C ) are the quantum numbers of the total angular momenta, orbital angular momenta and total spin for hadrons A, B, C, respectively; In the c.m. frame of hadron A, the momenta P B and P C of mesons B and C satisfy P B = −P C ≡ P.
Then the strong decay partial width for a given decay mode of A → B + C is given by where M A is the mass of the initial hadron A, while E B and E C stand for the energies of final hadrons B and C, respectively. The details of the 3 P 0 model can be found in our recent paper [80].
In the calculations, the wavefunctions of the initial B c states are adopted our quark model predictions. Furthermore, we need the wavefunctions of the final hadrons, i.e., the B ( * ) , B ( * ) s , D ( * ) , D ( * ) s mesons and some of their excitations, which are adopted from the quark model predictions of Refs. [81,82].
In this work, for the masses of the light constituent u, d and s quarks, we set m u = m d = 0.33 GeV, m s = 0.45GeV; while for the heavy b and c quarks, their masses are taken to be m b = 4.852 GeV and m c = 1.483 GeV as the determinations in the calculations of the B c mass spectrum. The masses of the final hadron states in the decay processes are adopted from the    Table V). There is no experimental data which can be used to determine the quark pair creation strength, thus, in this work we adopt a typical value γ = 0.4 that gives a reasonably accurate description of the overall scale of decay widths of both light and heavy mesons [66,[84][85][86][87][88]. The strong de-cays properties for the bottom-charmed states are presented in Tab. X to XV.
V. DISCUSSION

S -wave states
Recently, signals of two excitedbc states B c (2S ) and B * c (2S ) were observed in the B + c π + π − invariant mass spectrum by the CMS Collaboration at LHC. These two states are well resolved from each other and are observed with a significance exceeding five standard deviations. The mass of B c (2S ) meson is measured to be 6871 ± 2.8 MeV. Furthermore, a more precise mass of B c (2S ), M(B + c ) = 6871.1 ± 0.5 MeV, is measured by the CMS Collaboration as well. Combining these newest measurements, we predict that the mass of B c (2S ) might be ∼ 6890 MeV, and the mass hyperfine splitting be-tween B * c (2S ) and B c (2S ), is slightly smaller than 30 − 45 MeV predicted in previous works (see Table 1). The predicted masses for the other higher S -wave states compared with other works are also given in Table 1. Obvious differences can be found in various theoretical predictions.
The M1 transitions of the low-lying S -wave states B * c (2S ) and B ( * ) c (1S ) were often discussed in the literature for these transitions which might be used to establish them in experiments. In this work we also calculate their M1 transitions. Our results compared with the some other predictions are listed Table III. Obvious model dependence can be seen in various calculations. Our predicted partial width for the M1 transition B * c (2S ) → B c γ is about an order of magnitude larger than that predicted in Refs. [7,9,10], and about a factor 2 larger than the value predicted within the GI model [11]. Combining our calculations of the EM transitions B * c (2S ) → 1Pγ and the strong transitions B * c (2S ) → B * c ππ predicted in [11], the total decay width of B * c (2S ) meson is estimated to be Γ total ∼ 75 keV, then the branching fraction for The fairly large branching fraction may give a good opportunity for us to observe the B * c (2S ) via the M1 transition B * c (2S ) → B c γ. This process may be used to determined the mass of B * c (2S ) in future experiments. The masses of 3S -wave states B c (3 1 S 0 ) and B c (3 3 S 1 ) are predicted to be ∼ 7.24 GeV and ∼ 7.25 GeV, respectively, which are just above the DB * threshold. Their radiative and strong decay properties are estimated in this work. The results for the M1 transitions, E1 dominant transitions and strong decays of the 3S -wave states are given in Tables III, VII and X, respectively. There are only a few works about the radiative and strong decay properties of the 3S -wave states [11,18,59,60]. The M1 transitions of the 3S -wave states roughly agree with the predictions in Ref. [11], except that our predicted partial width Γ[3 3 S 1 → 1 1 S 0 + γ] ≃ 510 eV for the M1 transition 3 3 S 1 → 1 1 S 0 + γ is about an order of magnitude smaller than that in Ref. [11]. The strong decay widths of B c (3 1 S 0 ) and B c (3 3 S 1 ) predicted by us are comparable to those predicted in recent works [18,59]. Both B c (3 1 S 0 ) and B c (3 3 S 1 ) might be broad states with a width of ∼ 100 MeV. The B c (3 1 S 0 ) dominantly decay into DB * channel, while B c (3 3 S 1 ) dominantly decay into both DB and DB * channels. The production rates of the 3S -wave B c states in pp collisions at the LHC may be comparable with those of the 2S -wave B c states [18], thus, the 3S -wave B c states may have large potentials to be established in the DB * final states.
The higher S -wave states B c (n 1 S 0 ) and B c (n 3 S 1 ) (n ≥ 4) are far from the DB threshold, thus many OZI-allowed twobody strong decay channels are open. There are few discussions of the decay properties of the higher mass S -wave states in the literature. To know some decay properties of these higher S -wave states, in this work we give our predictions of the M1 transitions and strong decays of B c (nS ) (n = 4, 5, 6), which are listed in Table IV and X, respectively. It is found these higher mass S -wave states are broad states with a width of ∼ 100 − 400 MeV. Combining M1 transitions of higher Swave states with their strong decays, we found that the branching fractions of the M1 transitions B c (nS ) → B c (1S ) + γ may reach up to a sizeable value O(10 −5 ).

P-wave states
The masses of 1P-wave states B c (1P) might lie in the range of (6710, 6790) MeV, which are consistent with the other predictions with potential models [7][8][9][10][11], and the recent lattice calculations [36]. The 1P-wave B c (1P) states mainly decays via the E1 dominate transitions 1P → 1S . We have calculated the partial decay widths for the EM transitions 1P → 1S , our results compared with some other predictions are listed in Table VI. Most of our results are compatible with the predictions in [7,[9][10][11], except our predicted partial decay widths 40 keV are about a factor of 3 − 5 larger than the predictions in Refs. [9][10][11]. The B c (1P 1 ) and B c (1P ′ 1 ) states might be first found in the B c γ final state via their radiative transitions. The branching fractions for B c (1P 1 ) and B c (1P ′ 1 ) decay into B c γ are predicted to be While the B c (1 3 P 0 ) and B c (1 3 P 2 ) states dominantly decay into B * c γ final state with a decay rate ∼ 100%, thus, they have good potentials to be found via the radiative decay chains For the 2P-wave states B c (2P), their masses might lie in the range (7100, 7160) MeV, which are consistent with the other model predictions in the literature [7-11, 15, 16]. The masses for B c (2 3 P 0 ) and B c (2P 1 ) are slightly lower than the DB mass threshold, while B c (2P ′ 1 ) and B c (2 3 P 2 ) slightly lie above the DB mass threshold. The B c (2 3 P 2 ) state mainly decay into the DB channel, while its radiative decay rates into the B c (n 3 S 1 )γ (n = 1, 2) are also sizeable. Their partial widths are predicted to be Thus, the total width of B c (2 3 P 2 ) is Γ total [B c (2 3 P 2 )] ≃ 880 keV. The B c (2 3 P 2 ) state may have potentials to be observed in the DB and B c γ final states. While for B c (2 3 P 0 ), B c (2P 1 ) and B c (2P ′ 1 ) states, their decays are governed by the EM transitions. The radiative decay properties of these states have been given in Table VIII. With these predictions, the total widths for B c (2 3 P 0 ), B c (2P 1 ) and B c (2P ′ 1 ) are estimated to be Γ total [B c (2 3 P 0 )] ≃ 100 keV, Γ total [B c (2P 1 )] ≃ 120 keV, and Γ total [B c (2P ′ 1 )] ≃ 133 keV, respectively. The branching fractions for B c (2P 1 ) → B c γ, B c (2P ′ 1 ) → B c γ and B c (2 3 P 0 ) → B * c γ are predicted to be The large branching fractions indicate that B c (2P 1 ) and B c (2P ′ 1 ) may be established in the B c γ channel, while B c (2 3 P 0 ) may be observed via the radiative decay chain B c (2 3 P 0 ) → B * c γ → B c γγ. It should be pointed out that the B c (2P 1 ), B c (2P ′ 1 ) and B c (2 3 P 2 ) states may lie above the B * D threshold, so they may have fairly large strong decay widths O(10 − 100) MeV into B * D and/or BD channels as predicted in Ref. [17].
For the higher P-wave states B c (nP) (n = 3, 4), many OZI allowed strong decay channels are open (see Table XII), thus, these states usually are broad states with a width of O(100) MeV, except the B c (4 3 P 0 ) state has a relatively narrow width of O(10) MeV. The B c (4 3 P 0 ) state may be first observed in the DB channel, the branching fraction for the process B c (4 3 P 0 ) → DB can reach up to ∼ 20%.

D-wave states
The masses of the 1D-wave states B c (1D) is predicted to be ∼ 7.02 GeV in this work. The mass splitting between the 1Dwave states is no more than 15 MeV. The masses predicted by us are consistent with the results in Refs. [7,8,11]. The 1D-wave states mainly decay via the EM transitions, which have been given in Table VI. It is seen that our main results are in reasonable agreement with the other predictions. Our study indicates that the B c (1 3 D 3 ) state may have a relatively large potential to be observed via the radiative decay chain B c (1 3 D 3 ) → B c (1 3 P 2 )γ → B c (1 3 S 1 )γγ → B c (1 1 S 0 )γγγ, and the branching fraction for this chain is estimated to be ∼ 100%. The optimal decay chain for the observations of B c (1 3 D 1 ) is B c (1 3 D 1 ) → B c (1 3 P 0 )γ → B c (1 3 S 1 )γγ → B c (1 1 S 0 )γγγ, and the branching fraction for this chain is estimated to be ∼ 60%. The optimal decay chains for the observa- (1 1 S 0 )γγ, and the branching fraction for these chains are estimated to be ∼ 50% and ∼ 30%, respectively. While for the observations of B c (1D ′ 2 ), the optimal decay chains are (1 1 S 0 )γγ, and the branching fraction for these chains are estimated to be ∼ 35% and ∼ 47%, respectively.
The masses of the 2D states are predicted to be ∼ 7.34 GeV, which is very close to the D s B s threshold. Their decays are governed by the strong decay modes, such as DB, DB * , BD * or B * D * . Their strong decay properties predicted by us have been listed in Table XIII. There are few discussions about the radiative decays of the 2D-wave B c states in the literature. In this work, we also calculate their radiative decay properties, our results are given in Table VII. It is found that the B c (2 3 D 1 ) state has a relatively narrow width of Γ ∼ 58 MeV. The decays of B c (2 3 D 1 ) are governed by the BD * mode with a branching fraction The other three 2D states B c (2 3 D 3 ), B c (2D 2 ) and B c (2D ′ 2 ) are broad states with a width of ∼ 100 − 200 MeV. The B c (2 3 D 3 ) state mainly decays into DB, DB * , and B * D * channels. While the B c (2D 2 ) and B c (2D ′ 2 ) states dominantly decay into DB * , BD * or B * D * channels. Combing the strong and radiative decay properties with each other, it is found that the branching fractions of the dominant EM decay processes B c (2D) → B c (nP) (n = 1, 2) are O(10 −4 ). The observations of the DB, DB * , BD * or B * D * final states might be useful to search for these missing 2D states in future experiments.
The higher 3D-wave states B c (3D) are also studied in present work. The masses predicted by us are about 7.62 GeV, which are comparable with those predicted in Ref. [8], while about 150 MeV smaller than those predicted in Refs. [15,16]. The strong decay properties are shown in Table XIII. It is found that these higher 3D-wave states have a width of ∼ 100 MeV. These higher states might be observed in their dominant strong decay channels.

F-wave states
The masses of the 1F-wave states B c (1 3 F 4 ), B c (1F 3 ), B c (1F ′ 3 ) and B c (1 3 F 2 ) are predicted to be ∼ 7.23 GeV, which are comparable to those predicted in Refs. [8,11,16]. These 1F wave states lie above the mass threshold of DB and B * D, while below the D * B threshold. From our predictions of the strong decay properties for these 1F wave states (see Table XIV), it is found that the B c (1 3 F 4 ) state might be a very narrow state with a width of ∼ 1 MeV, its decays are governed by the DB mode. Both B c (1F 3 ) and B c (1F ′ 3 ) are narrow states with a width of ∼ 10 MeV, they dominantly decay into the DB * channel. The B c (1 3 F 2 ) should be a relatively broad state with a width of ∼ 70 MeV, it mainly decays into the DB channel with a branching fraction of Br[B c (1 3 F 2 ) → DB] ≃ 85%. To look for the missing 1F-wave B c states, the DB and B * D final states are worth observing.
The predicted masses for the 2F-and 3F-wave B c states are ∼ 7.5 GeV and ∼ 7.8 GeV, respectively, which are comparable with the predictions in Refs. [8,11]. There are many strong decay channels for these higher mass F-wave states. Our predictions of their strong decay properties have been listed in Tables XIV and XV. It is found that the higher mass F-wave states might be broad states with a width of ∼ 100 − 300 MeV.

VI. SUMMARY
In this paper, we have calculated the B c meson spectrum up to the 6S states with a nonrelativistic linear potential model by further constraining the model parameters with the mass of B c (2S ) newly measured by the CMS Collaboration. As important tasks of this work, the radiative transitions between the B c states and the OZI allowed two body strong decays for the higher mass excited B c states are evaluated with the wavefunctions obtained from the linear potential model. Our calculations may provide useful references to search for the excited B c states. The main results are emphasized as follows.
For the S -wave states, the 2S hyperfine splitting is pre- For the P-wave states, it is found that the decays of the 2Pwave states, B c (2 3 P 0 ), B c (2P 1 ) and B c (2P ′ 1 ) together with all of the 1P-wave states are governed by the E1 transitions, their typical decay widths are ∼ 100 keV. It should be possible to observe these P-wave states via their dominant radiative decay processes with the higher statistics of the LHC. The B c (2 3 P 2 ) state is just ∼ 20 MeV above the DB threshold. It mainly decays into the DB channel with a very narrow width of Γ ∼ 1 MeV, so it has a large potential to be first observed in the DB final state. The predicted masses of 3P-wave states are in the range of (7420,7470) MeV. They are broad states with widths of ∼ 200 MeV, and strongly couple to the B * D * final state. It is interesting found that the 4P-wave states B c (4 3 P 0 ), B c (4P 1 ) and B c (4P ′ 1 ) with a mass around 7.7 GeV may have relatively narrow widths O(100) MeV, these higher P-wave states might be first observed in their dominant channel DB or DB * .
The 1D-wave states mainly decay via the EM transitions. Our study indicates that these 1D-wave states may have a relatively large potential to be observed via the radiative decay chains. For example, to look for the B c (1 3 D 3 ) state, the B c (1 3 D 3 ) → B c (1 3 P 2 )γ → B c (1 3 S 1 )γγ → B c (1 1 S 0 )γγγ is worthy to be searched, for the branching fraction of this chain is estimated to be ∼ 100%. The masses of the 2D and 3D states are predicted to be ∼ 7.34 and 7.62 GeV, respectively. Their decays are governed by the strong decay modes, such as DB, DB * , BD * or B * D * . These higher D-wave states usually have a width of O(100) MeV. The observations of the DB, DB * , BD * or B * D * final states might be useful to search for these missing 2D and 3D states in future experiments.
For the F-wave states, one should pay more attention to 1F-wave B c states in future observations. They have a mass of ∼ 7.23 GeV, lie between the DB and B * D mass thresholds. They are narrow states with a width of several MeV to several ten MeV, and dominantly decay into DB or B * D channels. For example the B c (1 3 F 4 ) state might be a very narrow state with a width of ∼ 1 MeV, its decays are governed by the DB mode. To look for the missing 1F-wave B c states, the DB and B * D final states are worth observing.
Finally, it should be pointed out the strong decay widths of the excited B c states predicted in this work may have large uncertainties, for the parameter γ cannot be directly determined by the strong decay processes of B c states. Fortunately, the uncertainties of the total strong decay widths of the excited B c states do not affect the important information, such as the dominant decay modes and corresponding decay rates, for our searching for the excited B c states in future experiments. Furthermore, the mixing angles for 3 P 1 − 1 P 1 , 3 D 2 − 1 D 2 , and 3 F 3 − 1 F 3 have obvious model dependencies. The uncertainties of the mixing angles also affect our predictions of the decay properties of the mixed states. VI: Partial widths of the E1 dominant radiative transitions for the 1P-, 1D-, and 1F-wave B c states. For comparison, the predictions from the relativistic quark model [10], relativized quark model [11], nonrelativistic constituent quark models [7,9] are listed in the table as well.