Semi-leptonic decays of B ∗ , B ∗ s , and B ∗ c with the Bethe-Salpeter method
Tianhong Wang, Yue Jiang, Tian Zhou, Xiao-Ze Tan, Guo-Li Wang
SSemi-leptonic decays of B ∗ , B ∗ s , and B ∗ c with the Bethe-Salpetermethod Tianhong Wang ∗ , Yue Jiang, Tian Zhou, Xiao-Ze Tan, and Guo-Li Wang † Department of Physics, Harbin Institute of Technology, Harbin, 150001, China
Abstract
In this paper we study the semileptonic decays of B ∗ , B ∗ s , and B ∗ c by using the Bethe-Salpetermethod with instantaneous approximation. Both the V → P l − ¯ ν l and V → V l − ¯ ν l cases areconsidered. The largest partial width of these channels is of the order of 10 − GeV. The branchingratios of these semileptonic decays are also estimated by using the partial width of the one photondecay channel to approximate the total width. For B ∗− → D ( ∗ )0 e − ¯ ν e and B ∗ s → D ( ∗ )+ s e − ¯ ν e ,the branching ratios are of the order of 10 − and 10 − , respectively. For B ∗− c , the J/ψe − ¯ ν e and B ∗ s e − ¯ ν e channels have the largest branching ratio, which is of the order of 10 − . ∗ [email protected] † gl [email protected] a r X i v : . [ h e p - ph ] O c t . INTRODUCTION The b -flavored pseudoscalar heavy mesons, namely B , B s , and B c have been studiedextensively both in theories and experiments. The most important reason for this is thatthey can only decay weakly, hence provides an opportunity to do the precision investigationtests on the Standard Model (SM). However, the vector mesons B ∗ , B ∗ s , and B ∗ c still lackenough experimental results (see Ref. [1]), because both their production rates and detectionefficiency are lower than the pseudoscalar partners. This situation will change as LHCbcollecting more and more data, which makes their precise detection be possible. For example,when LHC runs at 14 TeV, the cross section for the hadronic production of B ∗ c is predictedto be 33.1 nb [2]. If the integrated luminosity is taken to be 1 fb − , there are about 10 B ∗ c expected per year. Also, the future B-factories, such as Belle II, will also provide moreinformation for these particles. So the theoretical studies of these vector heavy-light mesonsbecomes more and more necessary.A notable property for these particles is that their masses are not large enough to decayto the corresponding pseudoscalar partner and a light meson, such as π , K , et al. . So theseparticles cannot decay strongly, but can only decay weakly and electromagnetically. As aresult, the partial widths of the electromagnetic decay channels, especially the one-photondecay channel, are dominant, which can be used to estimate the total widths. Theoretically,these channels have partial widths less than 1 keV [3–5]. This makes the branching ratiosof their weak decay models may be within the detection ability of current experiments.Recently, there are some interests of finding new physics in the B ∗ d,s meson decays [6–9],such as the B ∗ s → µ + µ − channel, which has the branching ratio around 10 − in the SM [10].This result is too small to be detected nowadays at the LHC (although there is possibilityby the end of run III of the LHC as Ref. [7] mentioned). However, the smileptonic channelscould have larger branching fractions so that they can be investigated experimentally. Thisis the case for their pseudoscalar partners B d,s , of which the l + l − decays have branchingfractions much smaller than those of the semilptonic decay channels [1].Until now, there are only limited theoretical calculations of such decay channels carriedout. In Ref. [11], the smileptonic decays of the B ∗ d,s with a final pseudoscalar meson werestudied. In their work, the hadronic transition matrix elements are calculated in the Bauer-Stech-Wirbel (BSW) model. However, using a different method to study such channels arenecessary, as by comparing the results of different models can make us to know how largethey are model dependent. In this paper, we will use the instantaneously approximated2ethe-Salpeter (BS) method which also has been applied extensively to deal with weakdecays of B q mesons [12–14]. As the instantaneous approximation is reliable only for theheavy mesons, we will focus on the decay channels with the final meson also being heavy.There are also some approaches to deal with light mesons, such as the Dyson-Schwingerequation (DSE) model [15, 16]. Both in the DSE model and our model, the calculation ofthe transition amplitude contains two main elements, the quark propagator and the mesonamplitude. In the DSE model, the dressed-quark propagator is applied, where the effects ofconfinement and the dynamical chiral symmetry breaking are considered, which are morerelated to QCD; for the heavy meson amplitude, usually a simple function, such as theexponential function is assumed. In our model, a simple form of the quark propagator isapplied, and by solving the instantaneous BS equations, we can get the wave function ofthe heavy meson, which can be used directly to calculate the form factors. Besides the B ∗ d,s mesons, we will also study the semileptonic decays of the B ∗ c meson, which have been studiedby even limited work. For instance, in Ref. [17], the QCD sum rules approach is appliedto study its semileptonic decays, but only the B ∗− c → η c l − ¯ ν l channels are considered. Onereason for this may be that this particle has not been found in experiments. However, LHCbhave made some efforts very recently to find excited B c states [18]. We expect that the B ∗ c state can be found in the near future. So the study of its decay properties is also of interest.The article is organized as follows. In Section II we give the theoretical formalism of thecalculation. The hadronic transition amplitudes both for the V → P and V → V processesare presented. The numerical results of the partial widths, the branching fractions, theleptonic spectra, and corresponding discussions are given in Section III. Finally, we concludein Section IV. II. THEORETICAL FORMALISM
The wave function χ P ( q ) of the two-body bound state fulfills the BS equation S − ( p ) χ P ( q ) S − ( − p ) = i (cid:90) d k (2 π ) V ( P ; q, k ) χ P ( k ) , (1)where p and p are the momenta of quark and antiquark, respectively; S ( p ) and S ( − p )are propagators of quark and antiquark, respectively; P is the momentum the bound state; q is the relative momentum between quark and antiquark; V ( P ; q, k ) is the interactionkernel. By taking the instantaneous approximation V ( P ; q, k ) ≈ V ( P ; q ⊥ , k ⊥ ) and defining3 P ( q ⊥ ) ≡ i (cid:82) dq π χ P ( q ), the BS equation can be reduced to the Salpeter equation [19]( M − ω − ω ) ϕ ++ P ( q ⊥ ) = Λ +1 η P ( q ⊥ )Λ +2 , ( M + ω + ω ) ϕ −− P ( q ⊥ ) = − Λ − η P ( q ⊥ )Λ − ,ϕ + − P ( q ⊥ ) = ϕ − + P ( q ⊥ ) = 0 , (2)where q µ ⊥ = q µ − P · qM P µ , ω = (cid:112) m − q ⊥ , and ω = (cid:112) m − q ⊥ ; m and m are the massesof quark and antiquark, respectively. In the above equation, we have defined η P ( q ⊥ ) = (cid:90) d k ⊥ (2 π ) V ( P ; q ⊥ , k ⊥ ) ϕ P ( k ⊥ ) , (3)and ϕ ±± P ( q ⊥ ) = Λ ± /PM ϕ P ( q ⊥ ) /PM Λ ± , (4)where Λ ± i = ω i (cid:104) /PM ω i ∓ ( − i ( /q ⊥ + m i ) (cid:105) is the projection operator. The expressions for ϕ and ϕ ++ are given in the Appendix.We use the Cornell-like interaction potential, which in the momentum space has theform [19] V ( (cid:126)q ) = V s ( (cid:126)q ) + γ ⊗ γ V v ( (cid:126)q ) , (5)where V s ( (cid:126)q ) = − (cid:18) λα + V (cid:19) δ ( (cid:126)q ) + λπ (cid:126)q + α ) ,V v ( (cid:126)q ) = − π α s ( (cid:126)q ) (cid:126)q + α ,α s ( (cid:126)q ) = 12 π
27 1ln (cid:16) a + (cid:126)q Λ QCD (cid:17) . (6)The parameters involved are a = e = 2 . α = 0 .
06 GeV, λ = 0 .
21 GeV , Λ QCD = 0 . m b = 4 .
96 GeV, m c = 1 .
62 GeV, m s = 0 . m u = 0 .
305 GeV, m d = 0 .
311 GeV; V is decided by fitting the mass of the ground state.The Feynman diagram for the semileptonic decay is presented in Figure 1. The amplitudeof this process can be written as the product of the leptonic part and the hadronic transitionmatrix element M = G F √ V Qq ¯ u l γ µ (1 − γ ) v ¯ ν l (cid:104) P f | J µ | P, (cid:15) (cid:105) , (7)where G F is the Fermi constant; V Qq is the Cabibbo-Kobayashi-Maskawa (CKM) matrixelement; P and P f are the momenta of the initial and final meson, respectively; (cid:15) is thepolarization vector of the initial meson; J µ = ¯ qγ µ (1 − γ ) Q = V µ − A µ is the weak current.4 p m p ′ m ′ P f p m p ′ m ′ γ µ (1 − γ ) l − ¯ ν l FIG. 1: The Feynman diagram of the semileptonic decay of the vector meson.
Within Mandelstam formalism, the hadronic transition matrix element can be written asthe overlap integral of the instantaneous BS wave functions of the initial and final heavymesons [20], (cid:104) P f | J µ | P, (cid:15) (cid:105) = (cid:90) d(cid:126)q (2 π ) Tr (cid:20) /PM ϕ ++ Pf ( (cid:126)q f ) γ µ (1 − γ ) ϕ ++ P ( (cid:126)q ) (cid:21) , (8)where (cid:126)q and (cid:126)q f are the relative three-momenta between quark and antiquark within theinitial and final mesons, respectively; ϕ ++ P ( (cid:126)q ) and ϕ ++ Pf ( (cid:126)q f ) are the positive energy parts ofthe wave functions of initial and final mesons, respectively, whose explicit expressions canbe found in the Appendix. The final meson can be a pseudoscalar or a vector, and we givethe expressions of hadronic transition matrix elements for both cases.For the 1 − → − channel [21] (cid:104) P f | V µ | P, (cid:15) (cid:105) = 2 s M + M f i(cid:15) µνρσ (cid:15) ν P ρ P fσ , (cid:104) P f | A µ | P, (cid:15) (cid:105) = s ( M + M f ) (cid:15) µ − ( s P µ − s P µf ) (cid:15) · P f M , (9)where M and M f are the masses of the initial and final mesons, respectively; s ∼ s areform factors which are the integrals of (cid:126)q .For the 1 − → − channel [21] (cid:104) P f , (cid:15) f | V µ | P, (cid:15) (cid:105) = ( t P µ + t P µf ) (cid:15) · P f (cid:15) f · PM − t (cid:15) µ (cid:15) f · P − t (cid:15) µf (cid:15) · P f + ( t P µ + t P µf ) (cid:15) · (cid:15) f , (cid:104) P f , (cid:15) f | A µ | P, (cid:15) (cid:105) = i(cid:15) µαγδ P γ P fδ M ( h (cid:15) α (cid:15) f · P + h (cid:15) fα (cid:15) · P f ) + i(cid:15) µαβγ (cid:15) α (cid:15) fβ × ( h P γ + h P fγ ) , (10)5here (cid:15) f is the polarization vector of the final meson; t ∼ t and h ∼ h are the formfactors.The partial decay width is achieved by finishing the phase space integral Γ = 13 18 M (2 π ) (cid:90) dE l dE f (cid:88) λ |M| , (11)where E l and E f are the energy of charged lepton and final meson, respectively; λ representsthe polarization indexes of both initial and final mesons. From this, one can also easilycalculate the differential partial widths. III. RESULTS AND DISCUSSIONS
The B ∗− and B ∗ s mesons have been found experimentally [1] with masses M ( B ∗− ) =5 .
325 GeV and M ( B ∗ s ) = 5 .
415 GeV, respectively. However, there is still not enoughexperimental data about both their total and partial widths. As the strong decays areforbidden by the phase space, the total decay widths of these vector b -flavored mesons canbe estimated by the partial width of the single-photon decay channel [3, 4]Γ B ∗− (cid:39) Γ( B ∗ + → B + γ ) = 468 +73 − eV , Γ B ∗ s (cid:39) Γ( B ∗ s → B s γ ) = 68 ±
17 eV . (12)The B ∗− c meson has not been found experimentally. Here we take the value M ( B ∗− c ) = 6.333GeV predicted by the quark petential model [22]. The one photon decay width is calculatedrecently in Ref. [5], which can be used to approximate the total widthΓ B ∗− c (cid:39) Γ( B ∗− c → B − c γ ) = 23 eV . (13)One notices that the one photon decay widths of B ∗ s and B ∗− c are about one order smallerthan that of the B ∗− meson. These results surely are model dependent, however, the orderof magnitude should be affirmatory.The partial widths of the V → P channels are presented in Table I. All the cases when l − = e − , µ − , and τ − are considered. For B ∗− and B ∗ s , the decay channels with the samecharged lepton have close decay widths. This is the reflection of chiral symmetry. For B ∗− c ,both the b → c ( u ) and ¯ c → ¯ d (¯ s ) are calculated. The channel B ∗− c → ¯ D l − ¯ ν l is muchsmaller than those of other channels, the reason of which is that the CKM matrix elementin this case is V ub = 4 . × − which is much smaller. With the total width estimated6n Eq. (6), the branching ratios of these channels are presented in the third column. Forthe decay channels of B ∗− and B ∗ s , our results are little larger than those of Ref. [11].There are two main reasons for this. First, the wave functions in Ref. [11] are solutions of arelativistic scalar harmonic oscillator potential, while we get the wave functions by solvingthe instantaneous BS equation with a Cornell-like potential. Second, in Ref. [11], the formfactors at Q ≡ ( P − P f ) = 0 are calculated, and the explicit expressions are achieved byusing the assumption of the pole structure. While in our calculation, the numerical resultsof the form factors at all the physical-allowed Q can be achieved by applying Eq. (2). For B ∗− c , the η c l − ¯ ν l channels were studied in Ref. [17] by using the QCD sum rules. Therethe authors got the partial widths 6 . × − GeV, 6 . × − GeV, and 2 . × − GeV for l − = e − , µ − , and τ − , respectively, which are close to ours. The largest branchingratio comes from the channel B s e − ¯ ν e , which is the order of 10 − . The partial widths of the V → V channels are presented in Table II. Compared with the V → P case, the results are2 ∼ − . e (GeV)00.20.40.60.8 d Γ / ( Γ d E e )( G e V - ) B*->D+e+ ν B*->D*+e+ ν (a) B ∗ → D ( ∗ ) eν e (GeV)00.511.52 d Γ / ( Γ d E e )( G e V - ) B*->D+ τ+ν
B*->D*+ τ+ν (b) B ∗ → D ( ∗ ) τ ν FIG. 2: The energy spectra of final charged lepton in the B ∗ → D ( ∗ ) processes. The energy spectra of the final charged lepton are presented in Figure 2 ∼
6. Forcomparison, the results of V → P and V → V with the same final charged lepton areplotted in the same figure. For example, in Figure 2, the spectra of B ∗− → D l − ¯ ν l and B ∗− → D ∗ l − ¯ ν l are presented. One can see that for l − = e − , when E l less (more) thanabout 1.6 GeV, the spectrum of the pseudoscalar case is smaller (larger) than that of thevector case, and the peak value of the former is larger than that of the later. For l − = τ − ,the dividing point is at E τ (cid:39) B ∗ s → D ( ∗ )+ s l − ¯ ν l (Figure 3) and B ∗− c → η c ( J/ψ ) l − ¯ ν l (Figure4) channels. The spectra of these three cases are quite similar to each other. The reasonfor this is that these decay channels have close phase space, which can be estimated by themass difference of initial and final mesons: M ( B ∗− ) − M ( D ( ∗ )0 ) (cid:39) M ( B ∗ s ) − M ( D ( ∗ )+ s ) (cid:39) M ( B ∗− c ) − M ( η c ( J/ψ )). For the B ∗− c → ¯ D ( ∗ ) l − ¯ ν l channels, M ( B ∗− c ) − M ( ¯ D ( ∗ )0 ) is morethan 1 GeV larger than the former three cases, which makes the spectra (see Figure 5) havedifferent forms. And the peak value for the ¯ D τ − ¯ ν τ channel gets larger than that of the¯ D ∗ τ − ¯ ν τ channel. For the B ∗− c → B ( ∗ ) d,s e − ¯ ν e cases (Figure 6), when E e is around 0.45 GeV,the spectra reach the maximum, which is larger for the pseudoscalar channels. e (GeV)00.20.40.60.81 d Γ / ( Γ d E e )( G e V - ) Bs*->Ds+e+ ν Bs*->Ds*+e+ ν (a) B ∗ s → D ( ∗ ) s eν e (GeV)00.511.52 d Γ / ( Γ d E e )( G e V - ) Bs*->Ds+ τ+ν
Bs*->Ds*+ τ+ν (b) B ∗ s → D ( ∗ ) s τ ν FIG. 3: The energy spectra of final charged lepton in the B ∗ s → D ( ∗ ) s processes. The ratio of the branching fractions is an interested quantity in experiments. Recently,the experimental results of this value for B , B s , and B c sates have attracted more attentionsas they deviate from the SM predictions by several standard deviations [23] (although thelatest results from Belle [24] are consistent with the SM prediction), which may indicatepossible new physics beyond the SM [25]. If this is confirmed, similar results should alsoexist in their vector partners. In Table III, we present the ratios of the branching fractionsfor the vector cases. We define the following quantities R = Br ( V → P τ ¯ ν τ ) Br ( V → P e ¯ ν e ) , R ∗ = Br ( V → V τ ¯ ν τ ) Br ( V → V e ¯ ν e ) . (14)One can see that for B ∗− → D ( D ∗ ), B ∗ s → D + s ( D ∗ + s ), and B ∗− c → η c ( J/ψ ), the resultsof R ( R ∗ ) are close to each other. This is also the reflection of similar phase space. Besidesthat, one also notices that R is larger than R ∗ for these channels. For B ∗− c → ¯ D ( ¯ D ∗ ), R ( R ∗ ) is 2 ∼ R and R ∗ reversed compared with former three cases. Asthe numerator and denominator in Eq. 8 share the same CKM matrix elements and part ofuncertainties of the form factors which are canceled in the calculation, the two ratios areless model dependent and more robust, and can be compared with the future experimentalresults. e (GeV)00.20.40.60.81 d Γ / ( Γ d E e )( G e V - ) Bc*-> η c+e+ ν Bc*->J/ ψ +e+ ν (a) B ∗ c → η c ( J/ψ ) eν e (GeV)00.511.52 d Γ / ( Γ d E e )( G e V - ) Bc*-> η c+ τ+ν Bc*->J/ ψ + τ+ν (b) B ∗ c → η c ( J/ψ ) τ ν FIG. 4: The energy spectra of final charged lepton in the B ∗ c → η c ( J ψ ) processes. e (GeV)00.511.52 d Γ / ( Γ d E e )( G e V - ) Bc*->D0+e+ ν Bc*->D0*+e+ ν (a) B ∗ c → ¯ D ( ∗ ) eν e (GeV)00.511.52 d Γ / ( Γ d E e )( G e V - ) Bc*->D0+ τ+ν
Bc*->D0*+ τ+ν (b) B ∗ c → ¯ D ( ∗ ) τ ν FIG. 5: The energy spectra of final charged lepton in the B ∗ c → ¯ D ( ∗ ) processes. IV. CONCLUSIONS
As a conclusion, we have studied the semileptonic decays of the b -flavored vector heavymesons. Both cases for the final meson being a pseudoscalar or vector are considered. The9 e (GeV)00.511.522.53 d Γ / ( Γ d E e )( G e V - ) Bc*->B+e+ ν Bc*->B*+e+ ν (a) B ∗ c → B ( ∗ ) eν e (GeV)00.511.522.53 d Γ / ( Γ d E e )( G e V - ) Bc*->Bs+e+ ν Bc*->Bs*+e+ ν (b) B ∗ c → B ( ∗ ) s eν FIG. 6: The energy spectra of final charged lepton in the B ∗ c → B ( ∗ )( d,s ) processes. partial widths of these channels are of the order of 10 − ∼ − GeV. As the single-photondecay channel is dominant, its partial width is used to estimate the total width of theinitial meson. As a result, for B ∗− , the D ∗ e − ¯ ν e channel has the largest branching ratio9 . × − ; for B ∗ s , the D ∗ + s e − ¯ ν e channel has the largest branching ratio 5 . × − ; for B ∗− c , the B ∗ s e − ¯ ν e channel has the largest branching ratio 2 . × − . Experimental resultsfor these channels at LHCb and future B-factories are expected, which will be helpful to setmore stringent constraint on the SM parameters and clarify the possible anomalies observedin the semileptonic decays of b -flavored pseudoscalar mesons. V. ACKNOWLEDGMENTS
This paper was supported in part by the National Natural Science Foundation of China(NSFC) under Grant No. 11405037, No. 11505039 and No. 11575048.
Appendix
The quantity ϕ is constructed with momenta and gamma matrices by considering corre-sponding spin and parity properties. For the 1 − initial state, it has the form ϕ − ( q ⊥ ) = ( q ⊥ · (cid:15) ) (cid:34) f ( q ⊥ ) + /PM f ( q ⊥ ) + /q ⊥ M f ( q ⊥ ) + /P /q ⊥ M f ( q ⊥ ) (cid:35) + M /(cid:15) (cid:34) f ( q ⊥ ) + /PM f ( q ⊥ ) + /q ⊥ M f ( q ⊥ ) + /P /q ⊥ M f ( q ⊥ ) (cid:35) , (15)10 ABLE I: The partial decay widths (in units of GeV) and branching ratios of B ∗ u,s,c → P l − ¯ ν l . Theerrors come from varying the parameters in our model by ± B ∗− → D e − ¯ ν e . +0 . − . × − . +0 . − . × − . × − B ∗− → D µ − ¯ ν µ . +0 . − . × − . +0 . − . × − . × − B ∗− → D τ − ¯ ν τ . +0 . − . × − . +0 . − . × − . × − B ∗ s → D + s e − ¯ ν e . +0 . − . × − . +0 . − . × − . × − B ∗ s → D + s µ − ¯ ν µ . +0 . − . × − . +0 . − . × − . × − B ∗ s → D + s τ − ¯ ν τ . +0 . − . × − . +0 . − . × − . × − B ∗− c → η c e − ¯ ν e . +0 . − . × − . +0 . − . × − . × − B ∗− c → η c µ − ¯ ν µ . +0 . − . × − . +0 . − . × − . × − B ∗− c → η c τ − ¯ ν τ . +0 . − . × − . +0 . − . × − . × − B ∗− c → ¯ D e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → ¯ D µ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗− c → ¯ D τ − ¯ ν τ . +0 . − . × − . +0 . − . × − B ∗− c → B e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → B µ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗− c → B s e − ¯ ν e . +0 . − . × − . +1 . − . × − B ∗− c → B s µ − ¯ ν µ . +0 . − . × − . +1 . − . × − where f i s are functions of q ⊥ . For the 0 − final state, it can be written as ϕ − ( q f ⊥ ) = (cid:34) g ( q f ⊥ ) + /P f M f g ( q f ⊥ ) + /q f ⊥ M f g ( q f ⊥ ) + /P f /q f ⊥ M f g ( q f ⊥ ) (cid:35) γ , (16)where we have used q µf ⊥ = q µf − P · q f M P µ ; g i s are functions of q f ⊥ . The numerical results of f i and g i can be achieved by solving Eq. (2). In the calculation, not all the f i s or g i s areindependent, as the last two equations in Eq. (2) provide the constraint conditions. For the1 − state, we choose f , f , f , f as the independent variables, and for the 0 − state, wechoose g and g .The positive energy part of ϕ are kept in the calculation. For the 1 − state, it has the11 ABLE II: The partial decay widths (in units of GeV) and branching ratios of B ∗ u,s,c → V l − ¯ ν l .The errors come from varying the parameters in our model by ± B ∗− → D ∗ e − ¯ ν e . +0 . − . × − . +1 . − . × − B ∗− → D ∗ µ − ¯ ν µ . +0 . − . × − . +1 . − . × − B ∗− → D ∗ τ − ¯ ν τ . +1 . − . × − . +0 . − . × − B ∗ s → D ∗ + s e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗ s → D ∗ + s µ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗ s → D ∗ + s τ − ¯ ν τ . +1 . − . × − . +0 . − . × − B ∗− c → J/ψe − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → J/ψµ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗− c → J/ψτ − ¯ ν τ . +0 . − . × − . +0 . − . × − B ∗− c → ¯ D ∗ e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → ¯ D ∗ µ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗− c → ¯ D ∗ τ − ¯ ν τ . +1 . − . × − . +0 . − . × − B ∗− c → B ∗ e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → B ∗ µ − ¯ ν µ . +0 . − . × − . +0 . − . × − B ∗− c → B ∗ s e − ¯ ν e . +0 . − . × − . +0 . − . × − B ∗− c → B ∗ s µ − ¯ ν µ . +0 . − . × − . +0 . − . × − form ϕ ++1 − ( q ⊥ ) = ( q ⊥ · (cid:15) ) (cid:34) A ( q ⊥ ) + /PM A ( q ⊥ ) + /q ⊥ M A ( q ⊥ ) + /P /q ⊥ M A ( q ⊥ ) (cid:35) + M /(cid:15) (cid:34) A ( q ⊥ ) + /PM A ( q ⊥ ) + /q ⊥ M A ( q ⊥ ) + /P /q ⊥ M A ( q ⊥ ) (cid:35) , (17)12 ABLE III: The ratios of branching fractions of different decay channels of B ∗ u,s,c .Channel R Channel R ∗ B ∗− → D B ∗− → D ∗ B ∗ s → D + s B ∗ s → D ∗ + s B ∗− c → η c B ∗− c → J/ψ B ∗− c → ¯ D B ∗− c → ¯ D ∗ where A i s are defined as A = ( ω + ω ) q ⊥ f + ( m + m ) q ⊥ f + 2 M ω f − M m f M ( m ω + m ω ) ,A = ( m − m ) q ⊥ f + ( ω − ω ) q ⊥ f − M m f + 2 M ω f M ( m ω + m ω ) ,A = 12 ( f + m + m ω + ω f − M m ω + m ω f ) ,A = 12 ( ω + ω m + m f + f − M m ω + m ω f ) ,A = 12 ( f − ω + ω m + m f ) , A = 12 ( − m + m ω + ω f + f ) ,A = A M ( ω − ω ) m ω + m ω , A = A M ( ω + ω ) m ω + m ω . (18)And for the 0 − final state, the positive energy part of ϕ has the form ϕ ++0 − ( q f ⊥ ) = (cid:34) B ( q f ⊥ ) + /P f M f B ( q f ⊥ ) + /q f ⊥ M f B ( q f ⊥ ) + /P f /q f ⊥ M f B ( q f ⊥ ) (cid:35) γ , (19)where B = M f ω (cid:48) + ω (cid:48) m (cid:48) + m (cid:48) g + g ) ,B = M f g + m (cid:48) + m (cid:48) ω (cid:48) + ω (cid:48) g ) ,B = − M f ( ω (cid:48) − ω (cid:48) ) m (cid:48) ω (cid:48) + m (cid:48) ω (cid:48) B ,B = − M f ( ω (cid:48) + ω (cid:48) ) m (cid:48) ω (cid:48) + m (cid:48) ω (cid:48) B . (20)Here we have used the definitions ω (cid:48) = (cid:113) m (cid:48) − q f ⊥ and ω (cid:48) = (cid:113) m (cid:48) − q f ⊥ , where m (cid:48) and13 (cid:48) are respectively the masses of quark and antiquark in the final meson. [1] C. Patrignani et al . (Particle Data Group), Chin. Phys. C , 100001 (2016).[2] C.-H. Chang, C.-F. Qiao, J.-X. Wang, and X.-G. Wu, Phys. Rev. D , 114009 (2005).[3] C.-Y. Cheung and C.-W. Hwang, JHEP , 177 (2014).[4] H.M. Choi, Phys. Rev. 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