# Study ofb⟶c Induced B¯∗⟶Vℓν¯ℓ Decays

Authors: Qin Chang; Xiao-Lin Wang; Jie Zhu; Xiao-Nan Li
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

In this paper, we investigate the tree-dominatedB¯u,d,s,c∗⟶Vℓ−ν¯ℓ (V=Du,d∗,Ds∗,J/ψ and ℓ=e,μ,τ) decays in the Standard Model with the relevant form factors obtained in the light-front quark model. These decays involve much more helicity states relative to the corresponding B¯∗⟶Pℓ−ν¯ℓ and B¯⟶Vℓ−ν¯ℓ decays, and moreover, the contribution of longitudinal polarization mode (V meson) is relatively small, ~30%, compared with the corresponding B meson decays. We have also computed the branching fraction, lepton spin asymmetry, forward-backward asymmetry, and ratio RV∗L≡BB¯∗⟶Vτ−ν¯τ/BB¯∗⟶Vℓ′−ν¯ℓ′ℓ′=e,μ. Numerically, the branching fractions of B¯∗⟶Vℓ′−ν¯ℓ′ decays are at the level of O10−7 and are hopeful to be observed by LHC and Belle-II experiments. The ratios RD∗,Ds∗,J/ψ∗L have relatively small theoretical uncertainties and are close to each other, RD∗∗L≃RDs∗∗L≃RJ/ψ∗L≃0.26,0.270.27,0.29, which are a bit different from the predictions in some previous works. The future measurements are expected to make tests on these predictions.


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## Body

## 1. Introduction

In the past years, a large amount ofBB¯ events have been accumulated by Babar, Belle, Tevatron and LHCb experiments, and most of B-meson decays having branching fractions ≳O10−7 have been measured [1]. Moreover, some deviations between the standard model (SM) predictions and the experimental data have been observed, for instance, the angular observable P5′ of B⟶K∗μ+μ− decay with 2.6σ discrepancy [2–6], the differential branching fraction of Bs⟶ϕμ+μ− decay with 3.3σ discrepancy [7, 8], and the well-known “πK CP puzzle” [9, 10]. Besides the flavor-changing-neutral-current precesses mentioned above, the B-meson semileptonic decays induced by b⟶cℓν¯ℓ transition also play an important role in testing the SM and probing the hints of possible new physics (NP). For instance, the well-known “RD∗ anomaly” reported by BaBar [11, 12], Belle [13–15], and LHCb [16, 17] collaborations exhibits a significant deviation between the SM prediction and experimental data [1, 18, 19]. Many studies have been done within the model-independent frameworks [20–27], as well as in some specific NP models, for instance Refs. [28–48]. One can refer to Refs. [49, 50] for recent reviews.The spin-triplet vectorBq∗ meson with a quantum number of n2s+1LJ=13S1 and JP=1− [51–54] has the same flavor components as the spin-singlet pseudoscalar Bq (q=u,d,s and c) meson and can also decay through the b⟶cℓν¯ℓ transition at quark-level; therefore, its b⟶c-induced semileptonic decays can play a similar role as B meson decays for testing the SM and probing possible hints of NP.TheBq∗ meson is an unstable particle, it cannot decay via strong interaction due to that mBq∗−mBq.≤50 MeV<mπ [55]; Bq∗ meson decay is dominated by the radiative process [55], Bq∗⟶Bqγ; the weak decay modes via the bottom-changing transition (for instance, the b⟶c induced semileptonic Bq∗ decays considered in this work) are generally very rare, and their branching fractions are expected to be very small within the SM. Until now, there is no experimental information and few theoretical works concentrating on the Bq∗ weak decays. Fortunately, thanks to the high luminosity and large production cross section at the running LHC and SuperKEKB/Belle-II experiments, a huge amount of the Bq∗ meson data samples would be accumulated. At Belle-II experiment, the B∗ and Bs∗ mesons are produced mainly via Y5S decays. With the target annual integrated luminosity, ~13ab−1 [56], and the cross section of Y5S production in e+e− collisions, σe+e−⟶Y5S=0.301±0.002±0.039nb [57], it is expected that about 4×109 Y5S samples could be produced per year by Belle-II. Further considering that Y5S meson mainly decays to final states with a pair of Bs∗ mesons and using the branching fractions of Y5S decays given by PDG [55], it can be estimated that about NB∗+B¯∗/year~4×109 and NBs∗+B¯s∗/year~2×109 samples can be accumulated by Belle-II per year. Unfortunately, the Bc∗ meson and its decays are out of the scope of Belle-II experiment. In addition, a lot of Bq∗ samples can also be produced via pp collision and be accumulated in the future by LHC with high collision energy, high luminosity and rather large production cross section [58–60], and some Bq∗ weak decays are hopeful to be observed, such as the leptonic Bs∗⟶ℓ+ℓ− decay with branching fraction ~O10−11 [61].Encouraged by the abundantBq∗ data samples at future heavy-flavor experiments, some interesting theoretical studies for the Bq∗ weak decays have been made within the SM, for instance, the pure leptonic B¯s∗⟶ℓ+ℓ− and B¯u,c∗⟶ℓ−ν¯ℓ decays [61, 62], the impact of B¯s,d∗⟶μ+μ− on B¯s,d⟶μ+μ− decays [63], the studies of the semileptonic Bc∗ decays within the QCD sum rules [64–66], the semileptonic Bu,d,c,s∗⟶P,Vℓ−ν¯ℓ with P=D,Ds,ηc,V=D∗,Ds∗,J/ψ decays within the Bethe-Salpeter (BS) method [67], and an approach under the assumption of heavy quark symmetry (HQS) [68], B¯∗⟶Pℓ−ν¯ℓ with P=D,Ds,π,K [69] and the nonleptonic B¯d,s∗0⟶Dd,s+M− (M=π,K,ρ and K∗) [70, 71], B¯d,s∗⟶Dd,sV [72], Bc∗⟶Bu,d,sV,Bu,d,sP [73], Bc∗⟶ηcV [74], B∗⟶D¯D [75], and Bc∗⟶ψ1S,2SP,ηc1S,2SP [76] decays. Moreover, the NP effects on the semileptonic B¯∗⟶Pℓ−ν¯ℓ with P=D,Ds,π,K decays have been investigated in a model-independent scheme [77] and the vector leptoquark model [78]. In this paper, we pay our attention to the CKM-favored and tree-dominated semileptonic B¯u,d,s,c∗⟶Vℓν¯ℓV=Du,d∗,Ds∗,J/ψ weak decays, which are generally much more complicated than the corresponding B decay modes because they involve much more allowed helicity states.Our paper is organized as follows. In Section2, the helicity amplitudes and observables of B¯∗⟶Vℓν¯ℓ decays are calculated. Section 3 is devoted to the numerical results and discussions, and the B¯∗⟶V transition form factors obtained within the covariant light-front quark model are used in the computation. Finally, we give our summary in Section 4.


## 2. Theoretical Framework and Results

### 2.1. Effective Lagrangian and Amplitude

In the SM,B¯u,d,s,c∗⟶Vℓν¯ℓ V=Du,d∗,Ds∗,J/ψ decays are induced by b⟶cℓν¯ℓ transition at quark level via W-exchange and can be described by the effective Lagrangian
(1)Leff=−22GFVcbc¯LγμbLℓ¯LγμνL+h.c.,at low energy scale μ=Omb, where GF is the Fermi coupling constant and Vcb denotes the CKM matrix element. Using Eq. (1), the amplitude of B¯∗⟶Vℓν¯ℓ decay can be written as the product of the hadronic matrix element and leptonic current. Then, in terms of leptonic (Lμν) and hadronic Hμν tensors built from the respective products of the leptonic and hadronic currents, the square amplitude can be expressed as
(2)MB¯∗⟶Vℓ−ν¯ℓ2=Vℓ−ν¯ℓLeffB¯∗2=GF2Vcb22LμνHμν.Inserting the completeness relation of the polarization vector of virtualW∗ boson,
(3)∑m,nε¯μmε¯ν∗ngmn=gμν,the product of Lμν and Hμν can be rewritten as
(4)LμνHμν=∑m,m′,n,n′Lm,nHm′,n′gmm′gnn′,where Lm,n≡Lμνε¯μmε¯ν∗n and Hm,n≡Hμνε¯μ∗mε¯νn are Lorentz invariant and therefore can be evaluated in different reference frames. In our following evaluation, Hm,n and Lm,n will be calculated in the B∗-meson rest frame and the ℓ−ν¯ℓ center-of-mass frame, respectively.


### 2.2. Kinematics

In the rest frame ofB∗ meson, assuming the final state V-meson moving along with positive z-direction, the momenta of B∗, V, and W∗ could be written as
(5)pB∗μ=mB∗,0,0,0,pVμ=EV,0,0,p→,qμ=q0,0,0,−p→,respectively, where q0=mB∗2−mV2+q2/2mB∗ and p→=λ1/2mB∗2,mV2,q2/2mB∗, with λa,b,c≡a2+b2+c2−2ab+bc+ca and q2=pB∗−pV2 being the momentum transfer squared, are the energy and momentum of virtual W∗. The polarization vectors of the initial B∗-meson and daughter V-meson, ε1μ0,± and ε2μ0,±, can be written as
(6)ε1μ0=0,0,0,1,ε1μ±=120,∓1,−i,0;ε2μ0=1mVp→,0,0,EV,ε2μ±=120,∓1,−i,0,respectively. For the four polarization vectors of virtual W∗, ε¯μt,0,±, one can conveniently choose [79, 80]
(7)ε¯μt=1q2q0,0,0,−p→,ε¯μ0=1q2p→,0,0,−q0,ε¯μ±=120,±1,−i,0,in which, λW∗=t has to be understood as λW∗=0 and J=0.Turning to theℓ−ν¯ℓ center-of-mass frame, the four-momenta of lepton and antineutrino are given as
(8)pℓμ=Eℓ,p→ℓsinθ,0,p→ℓcosθ,pνℓμ=p→ℓ,−p→ℓsinθ,0,−p→ℓcosθ,where Eℓ=q2+mℓ2/2q2, p→ℓ=q2−mℓ2/2q2, and θ is the angle between V and ℓ three-momenta. In this frame, the polarization vectors ε¯μλW∗ have the form
(9)ε¯μt=1,0,0,0,(10)ε¯μ0=0,0,0,1,(11)ε¯μ±=120,∓1,−i,0.


### 2.3. Hadronic Helicity Amplitudes

For hadronic part, one has to calculate the hadronic helicity amplitudesHλW∗λB∗λV of B¯∗⟶Vℓ−ν¯ℓ decay defined by
(12)HλW∗λB∗λVq2=VpV,λV∣c¯γμ1−γ5b∣B¯∗pB∗,λB∗ε¯∗μλW∗,which describes the decay of three helicity states of B∗ meson into the three helicity states of daughter V meson and the four helicity states of virtual W∗. For the B∗⟶V transition, the matrix elements VpV,λV∣c¯γμ1−γ5b∣B¯∗pB∗,λB∗ can be factorized in terms of ten form factors V1,2,3,4,5,6q2 and A1,2,3,4q2 as [81, 82]
(13)Vε2,pV∣c¯γμb∣B¯∗ε1,pB∗=ε1·ε2∗−PμV1q2+qμV2q2+ε1·qε2∗·qmB∗2−mV2PμV3q2−qμV4q2−ε1·qε2μ∗V5q2+ε2∗·qε1μV6q2,Vε2,pV∣c¯γ5γμb∣B¯∗ε1,pB∗=−iεμναβε1αε2∗βPνA1q2−qνA2q2−iε2∗·qmB∗2−mV2εμναβε1νPαqβA3q2+iε1·qmB∗2−mV2εμναβε2∗νPαqβA4q2with the sign convention ε0123=−1.Then, by contracting these hadronic matrix elements with the polarization vector of virtualW∗ boson, we can finally obtain the nonvanishing hadronic helicity amplitudes, HλW∗λB∗λV, given as
(14)H0++q2=−mB∗2−mV2q2A1q2+q2A2q2+2mB∗p→q2V1q2,Ht++q2=−2mB∗p→q2A1q2+mB∗2−mV2q2V1q2−q2V2q2,H−+0q2=−mB∗2+3mV2−q22mVA1q2+mB∗2−mV2−q22mVA2q2−2mB∗2p→2mVmB∗2−mV2A3q2−mB∗p→mVV6q2,H0−−q2=mB∗2−mV2q2A1q2−q2A2q2+2mB∗p→q2V1q2,Ht−−q2=2mB∗p→q2A1q2+mB∗2−mV2q2V1q2−q2V2q2,H+−0q2=mB∗2+3mV2−q22mVA1q2−mB∗2−mV2−q22mVA2q2+2mB∗2p→2mVmB∗2−mV2A3q2−mB∗p→mVV6q2,H+0+q2=3mB∗2+mV2−q22mB∗A1q2−mB∗2−mV2+q22mB∗A2q2+2mB∗p→2mB∗2−mV2A4q2−p→V5q2,H−0−q2=−3mB∗2+mV2−q22mB∗A1q2+mB∗2−mV2+q22mB∗A2q2−2mB∗p→2mB∗2−mV2A4q2−p→V5q2,H000q2=p→mB∗2+mV2−q2q2mVV1q2+2mB∗2p→3q2mVmB∗2−mV2V3q2−p→mB∗2−mV2−q22q2mVV5q2+p→mB∗2−mV2+q22q2mVV6q2,Ht00q2=mB∗2−mV2mB∗2+mV2−q22q2mB∗mVV1q2−q2mB∗2+mV2−q22mB∗mVV2q2+mB∗p→2q2mVV3q2−mB∗p→2q2mVmB∗2−mV2V4q2−mB∗p→2q2mVV5q2+mB∗p→2q2mVV6q2.Obviously, only the amplitudes withλB∗=λV−λW∗ survive due to the helicity conservation.


### 2.4. Helicity Amplitudes and Observables

For the leptonic part, the leptonic tensor could be expanded in terms of a complete set of Wigner'sdJ-functions, which have been widely used in the study of hadron semileptonic [79, 83, 84]. As a result, LμνHμν can be reduced to a very compact form
(15)LμνHμν=18∑λℓ,λν¯ℓ,J′,JλW∗,λW∗′−1J+J′hλℓ,λν¯ℓ2δλB∗,λV−λW∗δλB∗,λV−λ′W∗×dλW∗,λℓ−1/2Jdλ′W∗,λℓ−1/2J′HλW∗λB∗λVHλ′W∗λB∗λVwhere J and J′ run over 1 and 0, λW∗′, and λℓ run over their components. For the standard expression of dJ function, we take their value from PDG [55]. The leptonic helicity amplitude hλℓ,λν¯ℓ in Eq. (15) defined as
(16)hλℓ,λν¯ℓ=u¯ℓλℓγμ1−γ5νν¯12ε¯μλW∗,Taking the exact forms of spinors andW∗ polarization vectors given in Eq. (11), we obtain
(17)h−1/2,122=8q2−mℓ2,h1/2,1/22=8mℓ22q2q2−mℓ2,which are the same as the results obtained in semileptonic B and hyperon decays [83, 84].Using the amplitudes obtained above, we can then further evaluate the observables ofB¯∗⟶Vℓ−ν¯ℓ decays. The double differential decay rate is written as
(18)d2Γdq2dcosθ=GF2Vcb22π3p→8mB∗2131−mℓ2q2LμνHμν,where the factor 1/3 is caused by averaging over the spins of initial B¯∗ meson. The double differential decay rate with a given helicity state of lepton (λℓ=±1/2) is written as
(19)d2Γλℓ=−1/2dq2dcosθ=GF2Vcb2p→256π3mB∗213q21−mℓ2q22×1−cosθ2H+0+2+H+−02+1+cosθ2H−0−2+H−+02+2sin2θH0++2+H0−−2+H0002,(20)d2Γλℓ=1/2dq2dcosθ=GF2Vcb2p→256π3mB∗213q21−mℓ2q22mℓ2q2sin2θH+0+2+H+−02+H−0−2+H−+02+2Ht++−cosθH0++2+2Ht−−−cosθH0−−2+2Ht00−cosθH0002.Integrating overcosθ and summing over the lepton helicity, we can obtain the differential decay rate written as
(21)dΓdq2=GF2Vcb2p→96π3mB∗213q21−mℓ2q22×3mℓ22q2Ht++2+Ht−−2+Ht002+H+0+2+H+−02+H−0−2+H−+02+H0002+H0−−2+H0++21+mℓ22q2,where the three nondiagonal interference terms in Eq. (20) vanish. In addition, paying attention to the polarization states of V meson, one can obtain the longitudinal differential decay width dΓL/dq2 by picking out Ht002, H+−02, H−+02, and H0002 terms in Eq. (21).Using Eqs. (19) and (20) given above, we can also construct some useful observables as follows. The q2-dependent ratios is defined as
(22)RV∗Lq2≡dΓLB¯∗⟶Vτ−ν¯τ/dq2dΓLB¯∗⟶Vℓ′−ν¯ℓ′/dq2,where ℓ′ denotes the light leptons μ and e (in the following calculations, we take me,μ=0). The lepton spin asymmetry and forward-backward asymmetry are defined as
(23)Aλ∗Vq2=dΓλℓ=−1/2/dq2−dΓλℓ=1/2/dq2dΓλℓ=−1/2/dq2+dΓλℓ=1/2/dq2,and
(24)Aθ∗Vq2=∫−10dcosθd2Γ/dq2dcosθ−∫01dcosθd2Γ/dq2dcosθdΓ/dq2,respectively. These observables are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent, therefore, they can be predicted with a rather high accuracy.


## 2.1. Effective Lagrangian and Amplitude

In the SM,B¯u,d,s,c∗⟶Vℓν¯ℓ V=Du,d∗,Ds∗,J/ψ decays are induced by b⟶cℓν¯ℓ transition at quark level via W-exchange and can be described by the effective Lagrangian
(1)Leff=−22GFVcbc¯LγμbLℓ¯LγμνL+h.c.,at low energy scale μ=Omb, where GF is the Fermi coupling constant and Vcb denotes the CKM matrix element. Using Eq. (1), the amplitude of B¯∗⟶Vℓν¯ℓ decay can be written as the product of the hadronic matrix element and leptonic current. Then, in terms of leptonic (Lμν) and hadronic Hμν tensors built from the respective products of the leptonic and hadronic currents, the square amplitude can be expressed as
(2)MB¯∗⟶Vℓ−ν¯ℓ2=Vℓ−ν¯ℓLeffB¯∗2=GF2Vcb22LμνHμν.Inserting the completeness relation of the polarization vector of virtualW∗ boson,
(3)∑m,nε¯μmε¯ν∗ngmn=gμν,the product of Lμν and Hμν can be rewritten as
(4)LμνHμν=∑m,m′,n,n′Lm,nHm′,n′gmm′gnn′,where Lm,n≡Lμνε¯μmε¯ν∗n and Hm,n≡Hμνε¯μ∗mε¯νn are Lorentz invariant and therefore can be evaluated in different reference frames. In our following evaluation, Hm,n and Lm,n will be calculated in the B∗-meson rest frame and the ℓ−ν¯ℓ center-of-mass frame, respectively.


## 2.2. Kinematics

In the rest frame ofB∗ meson, assuming the final state V-meson moving along with positive z-direction, the momenta of B∗, V, and W∗ could be written as
(5)pB∗μ=mB∗,0,0,0,pVμ=EV,0,0,p→,qμ=q0,0,0,−p→,respectively, where q0=mB∗2−mV2+q2/2mB∗ and p→=λ1/2mB∗2,mV2,q2/2mB∗, with λa,b,c≡a2+b2+c2−2ab+bc+ca and q2=pB∗−pV2 being the momentum transfer squared, are the energy and momentum of virtual W∗. The polarization vectors of the initial B∗-meson and daughter V-meson, ε1μ0,± and ε2μ0,±, can be written as
(6)ε1μ0=0,0,0,1,ε1μ±=120,∓1,−i,0;ε2μ0=1mVp→,0,0,EV,ε2μ±=120,∓1,−i,0,respectively. For the four polarization vectors of virtual W∗, ε¯μt,0,±, one can conveniently choose [79, 80]
(7)ε¯μt=1q2q0,0,0,−p→,ε¯μ0=1q2p→,0,0,−q0,ε¯μ±=120,±1,−i,0,in which, λW∗=t has to be understood as λW∗=0 and J=0.Turning to theℓ−ν¯ℓ center-of-mass frame, the four-momenta of lepton and antineutrino are given as
(8)pℓμ=Eℓ,p→ℓsinθ,0,p→ℓcosθ,pνℓμ=p→ℓ,−p→ℓsinθ,0,−p→ℓcosθ,where Eℓ=q2+mℓ2/2q2, p→ℓ=q2−mℓ2/2q2, and θ is the angle between V and ℓ three-momenta. In this frame, the polarization vectors ε¯μλW∗ have the form
(9)ε¯μt=1,0,0,0,(10)ε¯μ0=0,0,0,1,(11)ε¯μ±=120,∓1,−i,0.


## 2.3. Hadronic Helicity Amplitudes

For hadronic part, one has to calculate the hadronic helicity amplitudesHλW∗λB∗λV of B¯∗⟶Vℓ−ν¯ℓ decay defined by
(12)HλW∗λB∗λVq2=VpV,λV∣c¯γμ1−γ5b∣B¯∗pB∗,λB∗ε¯∗μλW∗,which describes the decay of three helicity states of B∗ meson into the three helicity states of daughter V meson and the four helicity states of virtual W∗. For the B∗⟶V transition, the matrix elements VpV,λV∣c¯γμ1−γ5b∣B¯∗pB∗,λB∗ can be factorized in terms of ten form factors V1,2,3,4,5,6q2 and A1,2,3,4q2 as [81, 82]
(13)Vε2,pV∣c¯γμb∣B¯∗ε1,pB∗=ε1·ε2∗−PμV1q2+qμV2q2+ε1·qε2∗·qmB∗2−mV2PμV3q2−qμV4q2−ε1·qε2μ∗V5q2+ε2∗·qε1μV6q2,Vε2,pV∣c¯γ5γμb∣B¯∗ε1,pB∗=−iεμναβε1αε2∗βPνA1q2−qνA2q2−iε2∗·qmB∗2−mV2εμναβε1νPαqβA3q2+iε1·qmB∗2−mV2εμναβε2∗νPαqβA4q2with the sign convention ε0123=−1.Then, by contracting these hadronic matrix elements with the polarization vector of virtualW∗ boson, we can finally obtain the nonvanishing hadronic helicity amplitudes, HλW∗λB∗λV, given as
(14)H0++q2=−mB∗2−mV2q2A1q2+q2A2q2+2mB∗p→q2V1q2,Ht++q2=−2mB∗p→q2A1q2+mB∗2−mV2q2V1q2−q2V2q2,H−+0q2=−mB∗2+3mV2−q22mVA1q2+mB∗2−mV2−q22mVA2q2−2mB∗2p→2mVmB∗2−mV2A3q2−mB∗p→mVV6q2,H0−−q2=mB∗2−mV2q2A1q2−q2A2q2+2mB∗p→q2V1q2,Ht−−q2=2mB∗p→q2A1q2+mB∗2−mV2q2V1q2−q2V2q2,H+−0q2=mB∗2+3mV2−q22mVA1q2−mB∗2−mV2−q22mVA2q2+2mB∗2p→2mVmB∗2−mV2A3q2−mB∗p→mVV6q2,H+0+q2=3mB∗2+mV2−q22mB∗A1q2−mB∗2−mV2+q22mB∗A2q2+2mB∗p→2mB∗2−mV2A4q2−p→V5q2,H−0−q2=−3mB∗2+mV2−q22mB∗A1q2+mB∗2−mV2+q22mB∗A2q2−2mB∗p→2mB∗2−mV2A4q2−p→V5q2,H000q2=p→mB∗2+mV2−q2q2mVV1q2+2mB∗2p→3q2mVmB∗2−mV2V3q2−p→mB∗2−mV2−q22q2mVV5q2+p→mB∗2−mV2+q22q2mVV6q2,Ht00q2=mB∗2−mV2mB∗2+mV2−q22q2mB∗mVV1q2−q2mB∗2+mV2−q22mB∗mVV2q2+mB∗p→2q2mVV3q2−mB∗p→2q2mVmB∗2−mV2V4q2−mB∗p→2q2mVV5q2+mB∗p→2q2mVV6q2.Obviously, only the amplitudes withλB∗=λV−λW∗ survive due to the helicity conservation.


## 2.4. Helicity Amplitudes and Observables

For the leptonic part, the leptonic tensor could be expanded in terms of a complete set of Wigner'sdJ-functions, which have been widely used in the study of hadron semileptonic [79, 83, 84]. As a result, LμνHμν can be reduced to a very compact form
(15)LμνHμν=18∑λℓ,λν¯ℓ,J′,JλW∗,λW∗′−1J+J′hλℓ,λν¯ℓ2δλB∗,λV−λW∗δλB∗,λV−λ′W∗×dλW∗,λℓ−1/2Jdλ′W∗,λℓ−1/2J′HλW∗λB∗λVHλ′W∗λB∗λVwhere J and J′ run over 1 and 0, λW∗′, and λℓ run over their components. For the standard expression of dJ function, we take their value from PDG [55]. The leptonic helicity amplitude hλℓ,λν¯ℓ in Eq. (15) defined as
(16)hλℓ,λν¯ℓ=u¯ℓλℓγμ1−γ5νν¯12ε¯μλW∗,Taking the exact forms of spinors andW∗ polarization vectors given in Eq. (11), we obtain
(17)h−1/2,122=8q2−mℓ2,h1/2,1/22=8mℓ22q2q2−mℓ2,which are the same as the results obtained in semileptonic B and hyperon decays [83, 84].Using the amplitudes obtained above, we can then further evaluate the observables ofB¯∗⟶Vℓ−ν¯ℓ decays. The double differential decay rate is written as
(18)d2Γdq2dcosθ=GF2Vcb22π3p→8mB∗2131−mℓ2q2LμνHμν,where the factor 1/3 is caused by averaging over the spins of initial B¯∗ meson. The double differential decay rate with a given helicity state of lepton (λℓ=±1/2) is written as
(19)d2Γλℓ=−1/2dq2dcosθ=GF2Vcb2p→256π3mB∗213q21−mℓ2q22×1−cosθ2H+0+2+H+−02+1+cosθ2H−0−2+H−+02+2sin2θH0++2+H0−−2+H0002,(20)d2Γλℓ=1/2dq2dcosθ=GF2Vcb2p→256π3mB∗213q21−mℓ2q22mℓ2q2sin2θH+0+2+H+−02+H−0−2+H−+02+2Ht++−cosθH0++2+2Ht−−−cosθH0−−2+2Ht00−cosθH0002.Integrating overcosθ and summing over the lepton helicity, we can obtain the differential decay rate written as
(21)dΓdq2=GF2Vcb2p→96π3mB∗213q21−mℓ2q22×3mℓ22q2Ht++2+Ht−−2+Ht002+H+0+2+H+−02+H−0−2+H−+02+H0002+H0−−2+H0++21+mℓ22q2,where the three nondiagonal interference terms in Eq. (20) vanish. In addition, paying attention to the polarization states of V meson, one can obtain the longitudinal differential decay width dΓL/dq2 by picking out Ht002, H+−02, H−+02, and H0002 terms in Eq. (21).Using Eqs. (19) and (20) given above, we can also construct some useful observables as follows. The q2-dependent ratios is defined as
(22)RV∗Lq2≡dΓLB¯∗⟶Vτ−ν¯τ/dq2dΓLB¯∗⟶Vℓ′−ν¯ℓ′/dq2,where ℓ′ denotes the light leptons μ and e (in the following calculations, we take me,μ=0). The lepton spin asymmetry and forward-backward asymmetry are defined as
(23)Aλ∗Vq2=dΓλℓ=−1/2/dq2−dΓλℓ=1/2/dq2dΓλℓ=−1/2/dq2+dΓλℓ=1/2/dq2,and
(24)Aθ∗Vq2=∫−10dcosθd2Γ/dq2dcosθ−∫01dcosθd2Γ/dq2dcosθdΓ/dq2,respectively. These observables are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent, therefore, they can be predicted with a rather high accuracy.


## 3. Numerical Results and Discussions

In our numerical calculation, for the well-known Fermi coupling constantGF and the masses of mesons and τ, we take their central values given by PDG [55]. For the CKM element, we take Vcb=41.80−0.60+0.28×10−3 given by CKMFitter Group [85]. In order to evaluate the branching fractions, the total decay widths (or lifetimes), ΓtotBu,d,s,c∗, are also essential inputs. However, there is no available experimental or theoretical information until now. While, due to the fact that the electromagnetic processes B∗⟶Bγ dominates B∗ decays, we can take the approximation ΓtotB∗≃ΓB∗⟶Bγ. In the light-front quark model (LFQM), the decay width of B∗⟶Bγ decay is given by [86]
(25)ΓB∗⟶Bγ=α3e1Im1,m2,0+e2Im2,m1,02κγ3,Im1,m2,q2=∫01dx8π3∫d2k⊥ψx,k′⊥ψx,k⊥xM~0M~′0×A+2M0k⊥2−k⊥·q⊥2q⊥2,where A=x¯m1+xm2 with x¯=1−x, M0=M0+m1+m2 with M0 being the invariant mass of bound-state, α is the fine-structure constant, κγ=mB∗2−mB22/2mB∗ is the kinematically allowed energy of the outgoing photon. The radial wavefunction (WF) ψx,k⊥ of bound-state is responsible for describing the momentum distribution of the constituent quarks. In this paper, we shall use the Gaussian-type WF
(26)ψx,k⊥=4π3/4β3/2∂kz∂xexp−kz2+k⊥22β2,where kz is the relative momentum in z-direction and has the form kz=x−1/2M0+m22−m12/2M0. One can refer to Ref. [86] for more details. Using the constituent quark masses and the Gaussian parameter β given in Table 1, we obtain the numerical results for ΓB∗⟶Bγ as follows,
(27)ΓtotB∗+≃ΓB∗+⟶B+γ=349±18eV,(28)ΓtotB∗0≃ΓB∗0⟶B0γ=116±6eV,(29)ΓtotBs∗0≃ΓBs∗0⟶Bs0γ=84−9+11eV,(30)ΓtotBc∗+≃ΓBc∗+⟶Bc0γ=49−21+28eV.Table 1
The values of constituent quark masses and Gaussian parameters (in units of MeV) obtained by fitting to the data of decay constants [101, 102], where q=u,d.
mq=250,
ms=450,
mc=1400,
mb=4640;
βbq¯=540.7±9.6,
βbs¯=601.9±7.4,
βbc¯=933.9±11.1,
forP-meson
βcq¯=413.0±12.0,
βcs¯=514.1±18.5,
βcc¯=684.4±6.7,
βbq¯=504.4±14.2,
βbs¯=556.4±10.1,
βbc¯=863.4±32.8,
forV-mesonThese theoretical predictions are generally in agreement with the ones obtained in the previous work based on different theoretical models [86–92].Besides the inputs given above, theB∗⟶V transition form factors are also crucial inputs for evaluating observables, especially for the branching fraction. In this work, we adopt the covariant light-front quark model (CLFQM) [93–95] to evaluate their values. The theoretical formulas for the form factors of V′⟶V′′ have been given in our previous work (see Eqs. (39–48) in the appendix of Ref. [96]). These theoretical results are obtained within Drell-Yan-West frame, q+=0, which implies that the form factors are known only for space-like momentum transfer, q2=−q⊥20, and the ones in the physical time-like region need an additional q2 extrapolation. Following the strategy employed in Refs. [82, 93–95], one can parameterize the form factors as functions of q2 by using dipole model in the space-like region and then extend them to the whole physical region 0≤q2≤mB∗−mV2. The form factors in the dipole model have the form
(31)Fq2=F01−aq2/mB∗2+bq2/mB∗22,where F denotes A1−4 and V1−6. Using the inputs given in Table 1, we then present our theoretical prediction for the form factors of B¯∗⟶D∗, B¯s∗⟶Ds∗, and B¯c∗⟶J/ψ transitions in Table 2. Their q2 dependences are shown in Figure 1.Table 2
The numerical results of form factors forB¯∗⟶D∗, B¯s∗⟶Ds∗, and B¯c∗⟶J/ψ transitions within the CLFQM. The uncertainties are caused by the Gaussian parameters listed in Table 1.
B¯∗⟶D∗
A1
A2
A3
A4
V1
V2
V3
V4
V5
V6
F0
0.66−0.01+0.01
0.36−0.00+0.00
0.07−0.00+0.00
0.08−0.00+0.00
0.67−0.01+0.01
0.36−0.00+0.00
0.13−0.00+0.00
0.00−0.00+0.00
1.17−0.01+0.01
0.48−0.01+0.01
a
1.31−0.02+0.02
1.32−0.02+0.02
1.79−0.02+0.02
1.81−0.02+0.02
1.30−0.02+0.02
1.32−0.02+0.02
1.72−0.02+0.02
−0.09−0.40+0.45
1.30−0.02+0.02
1.29−0.02+0.02
b
0.42−0.02+0.02
0.42−0.02+0.02
1.10−0.03+0.03
1.15−0.04+0.04
0.43−0.02+0.02
0.42−0.02+0.02
1.01−0.04+0.03
1.27−0.28+0.38
0.41−0.02+0.02
0.40−0.02+0.02
B¯s∗⟶Ds∗
F0
0.65−0.01+0.01
0.38−0.01+0.01
0.10−0.00+0.00
0.09−0.00+0.00
0.66−0.01+0.01
0.38−0.01+0.01
0.15−0.00+0.00
−0.02−0.00+0.00
1.19−0.02+0.02
0.53−0.01+0.01
a
1.42−0.04+0.03
1.47−0.03+0.03
1.89−0.03+0.03
1.88−0.03+0.02
1.43−0.04+0.03
1.48−0.03+0.03
1.79−0.03+0.03
2.22−0.03+0.04
1.41−0.03+0.03
1.35−0.04+0.04
b
0.64−0.05+0.04
0.67−0.04+0.04
1.33−0.06+0.05
1.36−0.07+0.09
0.64−0.05+0.04
0.67−0.05+0.04
1.20−0.06+0.06
1.92−0.12+0.08
0.61−0.05+0.04
0.56−0.05+0.04
B¯c∗⟶J/ψ
F0
0.55−0.01+0.01
0.35−0.00+0.00
0.14−0.00+0.00
0.15−0.01+0.01
0.57−0.01+0.01
0.35−0.00+0.00
0.21−0.01+0.00
−0.01−0.01+0.01
1.19−0.02+0.02
0.64−0.01+0.01
a
2.48−0.07+0.07
2.65−0.08+0.08
2.88−0.09+0.09
2.88−0.08+0.08
2.48−0.07+0.07
2.56−0.08+0.08
2.75−0.09+0.08
3.58−0.12+0.17
2.42−0.07+0.07
2.32−0.06+0.06
b
2.71−0.22+0.20
2.87−0.26+0.23
3.88−0.34+0.31
3.90−0.33+0.30
2.73−0.22+0.20
2.88−0.26+0.23
3.51−0.32+0.29
6.37−0.13+0.23
2.54−0.22+0.20
2.33−0.19+0.17Figure 1
Theq2-dependences of form factors for B¯∗⟶D∗, B¯s∗⟶Ds∗, and B¯c∗⟶J/ψ transitions.
(a)
(b)
(c)
(d)
(e)
(f)Using the formulas given in the last section and inputs given above, we then present our numerical results for theq2-integrated observables of B¯∗⟶Vℓ−ν¯ℓ decays in Tables 3 and 4. For the branching fractions, the three errors in Table 3 are caused by the uncertainties of form factors, Vcb and ΓtotB∗, respectively. For the other observables listed in Table 4, the theoretical uncertainties are caused only by the form factors. Besides, the q2 dependence of differential decay rates dΓL/q2 and Aλ,θ∗V, RV∗L are shown in Figures 2 and 3. The following are some analyses and discussions:
(1)
From Table3, one can find a clear relation B¯∗−⟶D∗0ℓ−ν¯ℓ: BB¯∗0⟶D∗+ℓ−ν¯ℓ:BB¯s∗0⟶Ds∗+ℓ−ν¯ℓ:BB¯c∗−⟶J/ψℓ−ν¯ℓ≈1:3:4:6, which is caused mainly by their total decay widths ΓtotB∗ illustrated by Eqs. (27), (28), (29), and (30).Table 3
The SM predictions for the branching fractions ofB¯∗⟶Vℓ−ν¯ℓ decays.
Decay mode
This work
BS method [67]
HQS [68]
B¯∗−⟶D∗0ℓ′−ν¯ℓ′
8.42−0.24+0.23−0.24+0.11−0.42+0.45×10−8
1.26×10−7
6.41×10−8
B¯∗−⟶D∗0τ−ν¯τ
2.26−0.08+0.08−0.06+0.03−0.11+0.12×10−8
2.74×10−8
1.29×10−8
B¯∗0⟶D∗+ℓ′−ν¯ℓ′
2.51−0.07+0.08−0.07+0.03−0.12+0.13×10−7
−
1.92×10−7
B¯∗0⟶D∗+τ−ν¯τ
6.73−0.25+0.24−0.19+0.09−0.32+0.35×10−8
−
3.88×10−8
B¯s∗0⟶Ds∗+ℓ′−ν¯ℓ′
3.46−0.17+0.17−0.10+0.05−0.41+0.41×10−7
4.63×10−7
2.53×10−7
B¯s∗0⟶Ds∗+τ−ν¯τ
9.10−0.59+0.60−0.26+0.12−1.07+1.07×10−8
1.05×10−7
5.05×10−8
B¯c∗−⟶J/ψℓ′−ν¯ℓ′
5.44−0.33+0.33−0.16+0.07−2.00+4.06×10−7
5.37×10−7
2.91×10−7
B¯c∗−⟶J/ψτ−ν¯τ
1.43−0.12+0.13−0.04+0.02−0.52+1.07×10−7
1.49×10−7
5.65×10−8Table 4
Predictions forq2-integrated observables Aλ,θ∗Vℓ=τ, RV∗L, and FL∗V.
Obs.
Prediction
Obs.
Prediction
Obs.
Prediction
Aλ∗D∗
0.237−0.016+0.017
Aλ∗Ds∗
0.231−0.030+0.029
Aλ∗J/ψ
0.214−0.040+0.043
Aθ∗D∗
0.070−0.006+0.007
Aθ∗Ds∗
0.071−0.011+0.011
Aθ∗J/ψ
0.078−0.013+0.014
RD∗∗
0.269−0.003+0.003
RDs∗∗
0.263−0.005+0.005
RJ/ψ∗
0.262−0.007+0.009
RD∗∗L
0.285−0.003+0.004
RDs∗∗L
0.277−0.007+0.006
RJ/ψ∗L
0.278−0.009+0.009
FL∗D∗
0.304−0.004+0.003
FL∗Ds∗
0.306−0.006+0.005
FL∗J/ψ
0.303−0.007+0.006Figure 2
Theq2-dependences of differential decay rates dΓ/dq2 (solid lines) and dΓL/dq2 (dashed lines).
(a)
(b)
(c)Figure 3
Theq2-dependences of RV∗Lq2, Aθ∗Vq2, and Aθ∗Vq2.
(a)
(b)
(c)
(d)In Table3, the previous predictions based on the Bethe-Salpeter (BS) method [67] and the assumption of heavy quark symmetry (HQS) [68] are also listed for comparison. It can be found that the results based on the BS method and the assumption of HQS are a little bit smaller and larger respectively than our results, but they are also in agreement in the order of magnitude. These b⟶cℓ−ν¯ℓ induced B∗ weak decays have the branching fractions of the order O10−8−10−7>10−9, and therefore are in the scope of Belle-II or LHC experiments. In addtion, due to the fact that Vub/Vcb≈0.088, the b⟶uℓ−ν¯ℓ induced B∗ weak decays should have much smaller branching fractions, which are at the level of O10−10−10−9, and thus are hard to be observed in the near future.
(2)
Deviations from the SM predictions inB¯⟶D∗ℓν¯ℓ decay modes have been observed by the BaBar [11, 12], Belle [13–15], and LHCb [16, 17] collaborations in the ratios RD∗≡BB¯⟶D∗τ−ν¯τ/BB¯⟶D∗ℓ′−ν¯ℓ′ℓ′=μ,e. The combination of these measurements performed by the Heavy Flavour Averaging Group (HFLAV) [1] reads
(32)RDHFLAV=0.407±0.039±0.024,RD∗HFLAV=0.306±0.013±0.007,which show tensions of about 2 and 4σ, respectively, with the SM predictions [97]. Very recent measurement of RD∗ by Belle [98] results in values more compatible with the SM and yields a downward shift in the average. However, even though such measurement is included in the global average, the deviation is still larger than 3σ [97]. If this “RD∗ anomaly” is the truth, it possibly exists also in the b⟶c induced B¯∗⟶Vℓν¯ℓ decays, which therefore can provide another useful test on the lepton flavor universality and the various method based on the SM and NP for resolving “RD∗ anomaly.” Our numerical results for RV∗L are summarized in Table 4, and the q2-spectra of RV∗L are shown in Figure 3. It can be found that
(33)RD∗∗L≃RDs∗∗L≃RJψ∗L,within theoretical uncertainties. Moreover, their q2-spectra almost overlap with each other as shown in Figures 3(a) and 3(b). Using the results summarized in Table 3, we can also obtain the predictions based on the BS method and HQS,
(34)RD∗∗=0.217,RDs∗∗=0.227,RJ/ψ∗=0.277,BSmethodRD∗∗=0.202,RDs∗∗=0.200,RJ/ψ∗=0.194.HQSIt can be found that these results are different from our predictions more or less because different models and parameterizations are used for evaluating form factors, which has been observed in the case ofRD∗ [18]. Future measurement will make a judgement on these results.
(3)
Besides, the lepton spin asymmetry and the forward-backward asymmetry are also important observables for testing the SM and NP scenarios, for instance, two-Higgs-doublet models, and R-parity violating supersymmetry models [42–47], because their theoretical uncertainties can be well controlled and the zero-crossing points of their q2-spectra are sensitive to the NP effects [42]. Our numerical results for q2-integrated Aλ∗V and Aθ∗V are collected in Table 4, and the q2 dependences of Aλ∗Vq2 and Aθ∗V are shown by Figures 3(c) and 3(d). One can easily find that Aλ,θ∗D∗≃Aλ,θ∗Ds∗≃Aλ,θ∗J/ψ. Moreover, the q2-spectra of Aλ∗V are almost overlapping with each other as shown in Figure 3(c), and the case of Aθ∗V is similar except that the q2-spectrum of Aθ∗J/ψ deviates from the ones of Aθ∗D∗ and Aθ∗Ds∗ at large q2. In addition, Aλ∗V crosses the zero point at q2≈5GeV2, however, Aθ∗V does not have the zero point in all q2 region.
(4)
TheD∗ longitudinal polarization fraction in semileptonic B0⟶D∗−τ+ντ decay, defined as FLD∗=ΓλD∗=0B0⟶D∗−τ+ντ/ΓB0⟶D∗−τ+ντ, has been measured by Belle experiment with FLD∗=0.60±0.08stat.±0.04syst. [99], which deviates from the SM prediction FLD∗SM=0.457±0.010 [100] by 1.6σ. Similarly, we can define the longitudinal polarization fraction
(35)FL∗V=ΓλD∗=0B¯∗⟶Vτ−ν¯τΓB¯∗⟶Vτ−ν¯τfor B¯∗⟶Vτ−ν¯τ decay modes. From the numerical results given in the last row of Table 4, one can easily find that
(36)FL∗D∗≃FL∗Ds∗≃FL∗J/ψ≃30%,which implies that B¯∗⟶Vτ−ν¯τ decay is dominated by the transverse polarization. It is obviously different from the corresponding B¯⟶Vτ−ν¯τ decay mode, which is dominated by the longitudinal polarization state.


## 4. Summary

In this paper, motivated by abundantB∗ data samples at high-luminosity heavy-flavor experiments in the future, we have studied the b⟶c induced B¯u,d,s,c∗⟶Vℓ−ν¯ℓV=Du,d∗,Ds∗,J/ψandℓ=e,μ,τ decays within the SM. The helicity amplitudes are investigated in detail, and the form factors of B¯∗⟶V transitions are computed within the covariant light-front quark model. After that, we present our predictions for the observables including branching fraction (decay width), leptonic spin asymmetry, forward-backward asymmetry, ratio RV∗L, and longitudinal polarization fraction in Tables 3 and 4 and Figures 2 and 3. It is found that all these semileptonic B∗ decays have relatively large branching fractions of O10−8~O10−7, in which BB¯c∗⟶J/ψℓ′−ν¯ℓ′~5×10−7 is the largest one, and are hopeful to be observed at running LHC and SuperKEKB/Belle-II experiments; in addition, for the B¯∗⟶Vτ−ν¯τ decay, the longitudinal polarization state of V meson presents only about 30% contribution to the integrated decay width, which is obviously different from the corresponding B¯⟶Vτ−ν¯τ decay. All of results and findings in this paper are waiting for the experimental test in the future.



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