# Probing the Effects of New Physics inB¯⁎→Plν¯l Decays

Authors: Qin Chang; Jie Zhu; Na Wang; Ru-Min Wang
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

The significant divergence between the SM predictions and experimental measurements for the ratios,RD(⁎)≡B(B¯→D(⁎)τ-ν¯τ)/B(B¯→D(⁎)l′-ν¯l′) with (l′=e,μ), implies possible hint of new physics in the flavor sector. In this paper, motivated by the “RD(⁎) puzzle” and abundant B⁎ data samples at high-luminosity heavy-flavor experiments in the future, we try to probe possible effects of new physics in the semileptonic B¯u,d,s⁎→Pl-ν¯l(P=D,Ds,π,K) decays induced by b→(u,c)l-ν¯l transitions in the model-independent vector and scalar scenarios. Using the spaces of NP parameters obtained by fitting to the data of RD and RD⁎, the NP effects on the observables including branching fraction, ratio RP⁎, lepton spin asymmetry, and lepton forward-backward asymmetry are studied in detail. We find that the vector type couplings have large effects on the branching fraction and ratio RP⁎. Meanwhile, the scalar type couplings provide significant contributions to all of the observables. The future measurements of these observables in the B¯⁎→Pl-ν¯l decays at the LHCb and Belle-II could provide a way to crosscheck the various NP solutions to the “RD(⁎) puzzle”.


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## Body

## 1. Introduction

Thanks to the fruitful running of theB factories and Large Hadron Collider (LHC) in the past years, most of the Bu,d mesons decays with branching fractions ≳O(10-7) have been measured. The rare B-meson decays play an important role in testing the standard model (SM) and probing possible hints of new physics (NP). Although most of the experimental measurements are in good agreement with the SM predictions, several indirect hints for NP, the tensions or the so-called puzzles, have been observed in the flavor sector.The semileptonicB¯→D(∗)lν¯l decays are induced by the CKM favored tree-level charged current, and therefore, their physical observables could be rather reliably predicted in the SM and the effects of NP are expected to be tiny. In particular, the ratios defined by RD(∗)≡B(B¯→D(∗)τ-ν¯τ)/B(B¯→D(∗)l′-ν¯l′)(l′=e,μ) are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent; thus they could be predicted with a rather high accuracy. However, the BaBar [1, 2], Belle [3–5], and LHCb [6] collaborations have recently observed some anomalies in these ratios. The latest experimental average values for RD(∗) reported by the Heavy-Flavor Average Group (HFAG) are [7](1)RDExp=0.403±0.040±0.024,RD∗Exp=0.310±0.015±0.008,which deviate from the SM predictions(2)RDSM=0.300±0.008,RD∗SM=0.252±0.003,(see [8] in the first equation of (2) and [9] in the second equation of (2)) at the levels of 2.2σ and 3.4σ errors, respectively. Moreover, when the correlations between RD and RD∗ are taken into account, the tension would reach up to 3.9σ level [7]. Besides, the ratio RJ/ψ≡B(Bc→J/ψτ-ν¯τ)/B(Bc→J/ψμ-ν¯μ) has recently been measured by the LHCb collaboration [10], which also shows an excess of about 2σ from the central value range of the corresponding SM predictions [0.25,0.28]. In addition, another mild hint of NP in the b→ulν¯ induced B→τν¯ decay has been observed by the BaBar and Belle collaborations [11–14]; the deviation is at the level of 1.4σ [15].The large deviations inRD(∗) and possible anomalies in the other decay channels mentioned above imply possible hints of NP relevant to the lepton flavor violation (LFV) [15]. The investigations for these anomalies have been made extensively both within model-independent frameworks [16–37] and in some specific NP models where the b→cτν¯τ transition is mediated by leptoquarks [16, 17, 38–46], charged Higgses [16, 47–59], charged vector bosons [16, 60, 61], and sparticles [62–65].In addition toB mesons, the vector ground states of bq¯ system, B∗ mesons, with quantum number of n2s+1LJ=13S1 and JP=1- [66–69], also can decay through the b→(u,c)lν¯l transitions at quark-level. Therefore, in principle, the corresponding NP effects might enter into the semileptonic B∗ decays as well. The B∗ decay occurs mainly through the electromagnetic process B¯∗→B¯γ, and the weak decay modes are very rare. Fortunately, thanks to the rapid development of heavy-flavor experiments instruments and techniques, the B∗ weak decays are hopeful to be observed by the running LHC and forthcoming SuperKEK/Belle-II experiments [70–72] in the near future. For instance, the annual integrated luminosity of Belle-II is expected to reach up to ~13 ab-1 and the B∗ weak decays with branching fractions >O(10-9) are hopeful to be observed [70, 73, 74]. Moreover, the LHC experiment also will provide a lot of experimental information for B∗ weak decays due to the much larger beauty production cross-section of pp collision relative to e+e- collision [75].Recently, some interesting theoretical studies for theB∗ weak decays have been made within the SM in [73, 74, 76–82]. In this paper, motivated by the possible NP explanation for the RD(∗) puzzles, the corresponding NP effects on the semileptonic B∗ decays will be studied in a model-independent way. In the investigation, the scenarios of vector and scalar NP interactions are studied, respectively; their effects on the branching fraction, differential branching fraction, lepton spin asymmetry, forward-backward asymmetry, and ratio RP∗ (P=D,π,K) of semileptonic B∗ decays are explored by using the spaces of various NP couplings obtained through the measured RD(∗).Our paper is organized as follows. In Section2, after a brief description of the effective Lagrangian for the b→(u,c)lν¯l transitions, the theoretical framework and calculations for the B¯∗→Plν¯l decays in the presence of various NP couplings are presented. Section 3 is devoted to the numerical results and discussions for the effects of various NP couplings. Finally, we give our conclusions in Section 4.


## 2. Theoretical Framework and Calculation

### 2.1. Effective Lagrangian and Amplitudes

We employ the effective field theory approach to compute the amplitudes ofB¯∗→Plν¯l decays in a model-independent scheme. The most general effective Lagrangian at μ=O(mb) for the b→pl-ν¯l (p=u,c) transition can be written as [19, 21, 40, 46](3)Leff=-22GF∑p=u,cVpb1+VLp¯LγμbLl¯LγμνL+VRp¯RγμbRl¯LγμνL+V~Lp¯LγμbLl¯RγμνR+V~Rp¯RγμbRl¯RγμνR+SLp¯RbLl¯RνL+SRp¯LbRl¯RνL+S~Lp¯RbLl¯LνR+S~Rp¯LbRl¯LνR+TLp¯RσμνbLl¯RσμννL+T~Lp¯LσμνbRl¯LσμννR+h.c.,where GF is the Fermi coupling constant, Vpb denotes the CKM matrix elements, and PL,R=(1±γ5)/2 is the negative/positive projection operator. Assuming the neutrinos are left-handed and neglecting the tensor couplings, the effective Lagrangian can be simplified as(4)Leff=-GF2∑p=u,cVpb1+VLp¯γμ1-γ5bl¯γμ1-γ5ν+VRp¯γμ1+γ5bl¯γμ1-γ5ν+SLp¯1-γ5bl¯1-γ5ν+SRp¯1+γ5bl¯1-γ5ν+h.c.,where VL,R and SL,R are the effective NP couplings (Wilson coefficients) defined at μ=O(mb). In the SM, all the NP couplings will be zero.We use the method of [83–87] to calculate the helicity amplitudes. The square of amplitudes for the B¯∗→Pl-ν¯l decay can be written as the product of leptonic (Lμν) and hadronic (Hμν) tensors,(5)MB¯∗→Pl-ν¯l2=Pl-ν¯lLeffB¯∗2=∑i,jLμνijHij,μν,where the superscripts i and j refer to four operators in the effective Lagrangian given by (4) (the tensors related to the scalar and pseudoscalar operators can be understood through the relations given by (21) and (22)); in the SM, i=j corresponds to the operator p¯γμ(1-γ5)bl¯γμ(1-γ5)ν. For convenience in writing, these superscripts are omitted below. Inserting the completeness relation(6)∑m,nϵ¯μmϵ¯ν∗ngmn=gμν,the product of Lμν and Hμν can be further expressed as(7)LμνHμν=∑m,m′,n,n′Lm,nHm′,n′gmm′gnn′.Here, ϵ¯μ is the polarization vector of the virtual intermediate states, which is W∗ boson in the SM and named as ω in this paper for convenience of expression. The quantities L(m,n)≡Lμνϵ¯μ(m)ϵ¯ν∗(n) and H(m,n)≡Hμνϵ¯μ∗(m)ϵ¯ν(n) are Lorentz invariant and therefore can be evaluated in different reference frames. In the following evaluation, H(m,n) and L(m,n) will be calculated in the B∗-meson rest frame and the l-ν¯l center-of-mass frame, respectively.


### 2.2. Kinematics forB¯∗→Pl-ν¯l Decays

In theB∗-meson rest frame with daughter P-meson moving in the positive z-direction, the momenta of particles B∗ and P are(8)pB∗μ=mB∗,0,0,0,pPμ=EP,0,0,p→.For the four polarization vectors, ϵ¯μ(λω=t,0,±), one can conveniently choose [83, 84](9)ϵ¯μt=1q2q0,0,0,-p→,ϵ¯μ0=1q2p→,0,0,-q0,ϵ¯μ±=120,±1,-i,0,where q0=(mB∗2-mP2+q2)/2mB∗ and |p→|=λ1/2(mB∗2,mP2,q2)/2mB∗, with λ(a,b,c)≡a2+b2+c2-2(ab+bc+ca) and q2=(pB∗-pP)2 being the momentum transfer squared, are the energy and momentum of the virtual ω. The polarization vectors of the initial B∗-meson can be written as(10)ϵμ0=0,0,0,1,ϵμ±=120,∓1,-i,0.In thel-ν¯l center-of-mass frame, the four momenta of lepton and antineutrino pair are given as(11)plμ=El,p→lsin⁡θ,0,p→lcos⁡θ,pνlμ=p→l,-p→lsin⁡θ,0,-p→lcos⁡θ,where El=(q2+ml2)/2q2, |p→l|=(q2-ml2)/2q2, and θ is the angle between the P and l three-momenta. In this frame, the polarization vector ϵ¯μ takes the form(12)ϵ¯μt=1,0,0,0,ϵ¯μ0=0,0,0,1,ϵ¯μ±=120,∓1,-i,0.


### 2.3. Hadronic Helicity Amplitudes

For theB¯∗→Pl-ν¯l decay, the hadronic helicity amplitudes HλB∗λωVL,R and HλB∗λωSL,R are defined by(13)HλB∗λωVLq2=ϵ¯∗μλωPpPp¯γμ1-γ5bB¯∗pB∗,λB∗,(14)HλB∗λωVRq2=ϵ¯∗μλωPpPp¯γμ1+γ5bB¯∗pB∗,λB∗,(15)HλB∗λωSLq2=PpPp¯1-γ5bB¯∗pB∗,λB∗,(16)HλB∗λωSRq2=PpPp¯1+γ5bB¯∗pB∗,λB∗,which describe the decay of three helicity states of B∗ meson into a pseudoscalar P meson and the four helicity states of virtual ω. It should be noted that λω in HλB∗λωSL,R(q2), (15) and (16), should always be equal to t.ForB∗→P transition, the matrix elements of the vector and axial-vector currents can be written in terms of form factors V(q2) and A0,1,2(q2) as(17)PpPp¯γμbB¯∗ϵ,pB∗=-2iVq2mB∗+mPεμνρσϵνpPρpB∗σ,(18)PpPp¯γμγ5bB¯∗ϵ,pB∗=2mB∗A0q2ϵ·qq2qμ+mP+mB∗A1q2ϵμ-ϵ·qq2qμ+A2q2ϵ·qmP+mB∗pB∗+pPμ-mB∗2-mP2q2qμ,with the sign convention ϵ0123=-1. Furthermore, using the equations of motion,(19)i∂μp¯γμb=mbμ-mpμp¯b,(20)i∂μp¯γμγ5b=-mbμ+mpμp¯γ5b,one can write the matrix elements of scalar and pseudoscalars currents as(21)PpPp¯bB¯∗ϵ,pB∗=1mbμ-mpμqμPpPp¯γμbB¯∗ϵ,pB∗=0,(22)PpPp¯γ5bB¯∗ϵ,pB∗=-1mbμ+mpμqμPpPp¯γμγ5bB¯∗ϵ,pB∗=-ϵ·q2mB∗mbμ+mpμA0q2,in which mb(μ) and mp(μ) are the running quark masses.Then, by contracting above hadronic matrix elements with the polarization vectors in theB∗-meson rest frame, we obtain five nonvanishing helicity amplitudes(23)H0tq2=H0tVLq2=-H0tVRq2=2mB∗p→q2A0q2,(24)H00q2=H00VLq2=-H00VRq2=12mB∗q2mB∗+mPmB∗2-mP2+q2A1q2+4mB∗2p→2mB∗+mPA2q2,(25)H±∓q2=H±∓VLq2=-H∓±VRq2=-mB∗+mPA1q2∓2mB∗p→mB∗+mPVq2,(26)H0t′q2=H0tSLq2=-H0tSRq2=-2mB∗p→mbμ+mcμA0q2.It is obvious that only the amplitudes with λB∗=λP-λω=-λω survive.


### 2.4. Leptonic Helicity Amplitudes

Expanding the leptonic tensor in terms of a complete set of Wigner’sdJ-functions [9, 83, 87], LμνHμν can be rewritten as a compact form(27)LμνHμν=18∑λl,λν¯l,λω,λω′,J,J′-1J+J′hλl,λν¯lihλl,λν¯lj∗δλB∗,-λωδλB∗,-λω′×dλω,λl-1/2Jdλω′,λl-1/2J′HλB∗λωiHλB∗λω′j∗,in which J and J′ run over 1 and 0, λω(′) and λl run over their components, and massless right-handed antineutrinos with λν¯l=1/2. In (27), hλl,λν¯li,j are the leptonic helicity amplitudes defined as(28)hλl,λν¯lVL,R=u¯lλlγμ1-γ5νν¯12ϵ¯μλω,(29)hλl,λν¯lSL,R=u¯lλl1-γ5νν¯12.In the l-ν¯l center-of-mass frame, taking the exact forms of the spinors and polarization vectors, we finally obtain four nonvanishing contributions(30)h-1/2,1/2VL,R2=8q2-ml2,(31)h1/2,1/2VL,R2=8ml22q2q2-ml2,(32)h1/2,1/2SL,R2=8q2-ml22,(33)h1/2,1/2VL,R×h1/2,1/2SL,R=8ml2q2q2-ml2.


### 2.5. Observables ofB¯∗→Pl-ν¯l Decays

With the amplitudes obtained in above subsections, we then present the observables considered in our following evaluations. The double differential decay rate ofB¯∗→Pl-ν¯l decay is written as(34)dΓdq2dcos⁡θ=GF2Vpb22π3p→8mB∗2131-ml2q2MB¯∗→Pl-ν¯l2,where the factor 1/3 is caused by averaging over the spins of initial state B¯∗. Using the standard convention for dJ-function [88], we finally obtain the double differential decay rates with a given leptonic helicity state (λl=±1/2), which are(35)d2Γλl=-1/2dq2dcos⁡θ=GF2Vpb2p→256π3mB∗213q21-ml2q22×1+VL21-cos⁡θ2H-+2+1+cos⁡θ2H-+2+2sin2θH002+VR21-cos⁡θ2H+-2+1+cos⁡θ2H-+2+2sin2θH002-4Re1+VLVR∗1+cos⁡θ2H+-H-++sin2θH002,(36)d2Γλl=1/2dq2dcos⁡θ=GF2Vpb2p→256π3mB∗213q21-ml2q22ml2q2×1+VL2+VR2sin2θH-+2+H+-2+2H0t-cos⁡θH002-4Re1+VLVR∗sin2θH-+H+-+H0t-cos⁡θH002+4Re1+VL-VRSL∗-SR∗q2mlH0t′H0t-cos⁡θH00+2SL-SR2q2ml2H0t′2.Using (35) and (36), one can get the explicit forms of various observables of B¯∗→Pl-ν¯l decays as follows: (i)
The differential decay rate is(37)dΓdq2=GF2Vpb2p→96π3mB∗213q21-ml2q22×1+VL2+VR2H-+2+H+-2+H0021+ml22q2+3ml22q2H0t2-2Re1+VLVR∗2H-+H+-+H0021+ml22q2+3ml22q2H0t2+3Re1+VL-VRSL∗-SR∗H0t′H0tmlq2+32SL-SR2H0t′2.(ii)
Theq2 dependent ratio is(38)RP∗q2≡dΓB¯∗→Pτ-ν¯τ/dq2dΓB¯∗→Pl′-ν¯l′/dq2,wherel′ denotes the light lepton.(iii)
The lepton spin asymmetry is(39)AλPq2=dΓλl=-1/2/dq2-dΓλl=1/2/dq2dΓλl=-1/2/dq2+dΓλl=1/2/dq2.(iv)
The forward-backward asymmetry is(40)AθPq2=∫-10dcos⁡θd2Γ/dq2dcos⁡θ-∫01dcos⁡θd2Γ/dq2dcos⁡θdΓ/dq2.The SM results can be obtained from above formulae by takingVL=VR=SL=SR=0.In the following evaluations, in order to fit the NP spaces, we also need the observables ofB¯→D(∗)l-ν¯l decays, which have been fully calculated in the past years. In this paper, we adopt the relevant theoretical formulae given in [46].


## 2.1. Effective Lagrangian and Amplitudes

We employ the effective field theory approach to compute the amplitudes ofB¯∗→Plν¯l decays in a model-independent scheme. The most general effective Lagrangian at μ=O(mb) for the b→pl-ν¯l (p=u,c) transition can be written as [19, 21, 40, 46](3)Leff=-22GF∑p=u,cVpb1+VLp¯LγμbLl¯LγμνL+VRp¯RγμbRl¯LγμνL+V~Lp¯LγμbLl¯RγμνR+V~Rp¯RγμbRl¯RγμνR+SLp¯RbLl¯RνL+SRp¯LbRl¯RνL+S~Lp¯RbLl¯LνR+S~Rp¯LbRl¯LνR+TLp¯RσμνbLl¯RσμννL+T~Lp¯LσμνbRl¯LσμννR+h.c.,where GF is the Fermi coupling constant, Vpb denotes the CKM matrix elements, and PL,R=(1±γ5)/2 is the negative/positive projection operator. Assuming the neutrinos are left-handed and neglecting the tensor couplings, the effective Lagrangian can be simplified as(4)Leff=-GF2∑p=u,cVpb1+VLp¯γμ1-γ5bl¯γμ1-γ5ν+VRp¯γμ1+γ5bl¯γμ1-γ5ν+SLp¯1-γ5bl¯1-γ5ν+SRp¯1+γ5bl¯1-γ5ν+h.c.,where VL,R and SL,R are the effective NP couplings (Wilson coefficients) defined at μ=O(mb). In the SM, all the NP couplings will be zero.We use the method of [83–87] to calculate the helicity amplitudes. The square of amplitudes for the B¯∗→Pl-ν¯l decay can be written as the product of leptonic (Lμν) and hadronic (Hμν) tensors,(5)MB¯∗→Pl-ν¯l2=Pl-ν¯lLeffB¯∗2=∑i,jLμνijHij,μν,where the superscripts i and j refer to four operators in the effective Lagrangian given by (4) (the tensors related to the scalar and pseudoscalar operators can be understood through the relations given by (21) and (22)); in the SM, i=j corresponds to the operator p¯γμ(1-γ5)bl¯γμ(1-γ5)ν. For convenience in writing, these superscripts are omitted below. Inserting the completeness relation(6)∑m,nϵ¯μmϵ¯ν∗ngmn=gμν,the product of Lμν and Hμν can be further expressed as(7)LμνHμν=∑m,m′,n,n′Lm,nHm′,n′gmm′gnn′.Here, ϵ¯μ is the polarization vector of the virtual intermediate states, which is W∗ boson in the SM and named as ω in this paper for convenience of expression. The quantities L(m,n)≡Lμνϵ¯μ(m)ϵ¯ν∗(n) and H(m,n)≡Hμνϵ¯μ∗(m)ϵ¯ν(n) are Lorentz invariant and therefore can be evaluated in different reference frames. In the following evaluation, H(m,n) and L(m,n) will be calculated in the B∗-meson rest frame and the l-ν¯l center-of-mass frame, respectively.


## 2.2. Kinematics forB¯∗→Pl-ν¯l Decays

In theB∗-meson rest frame with daughter P-meson moving in the positive z-direction, the momenta of particles B∗ and P are(8)pB∗μ=mB∗,0,0,0,pPμ=EP,0,0,p→.For the four polarization vectors, ϵ¯μ(λω=t,0,±), one can conveniently choose [83, 84](9)ϵ¯μt=1q2q0,0,0,-p→,ϵ¯μ0=1q2p→,0,0,-q0,ϵ¯μ±=120,±1,-i,0,where q0=(mB∗2-mP2+q2)/2mB∗ and |p→|=λ1/2(mB∗2,mP2,q2)/2mB∗, with λ(a,b,c)≡a2+b2+c2-2(ab+bc+ca) and q2=(pB∗-pP)2 being the momentum transfer squared, are the energy and momentum of the virtual ω. The polarization vectors of the initial B∗-meson can be written as(10)ϵμ0=0,0,0,1,ϵμ±=120,∓1,-i,0.In thel-ν¯l center-of-mass frame, the four momenta of lepton and antineutrino pair are given as(11)plμ=El,p→lsin⁡θ,0,p→lcos⁡θ,pνlμ=p→l,-p→lsin⁡θ,0,-p→lcos⁡θ,where El=(q2+ml2)/2q2, |p→l|=(q2-ml2)/2q2, and θ is the angle between the P and l three-momenta. In this frame, the polarization vector ϵ¯μ takes the form(12)ϵ¯μt=1,0,0,0,ϵ¯μ0=0,0,0,1,ϵ¯μ±=120,∓1,-i,0.


## 2.3. Hadronic Helicity Amplitudes

For theB¯∗→Pl-ν¯l decay, the hadronic helicity amplitudes HλB∗λωVL,R and HλB∗λωSL,R are defined by(13)HλB∗λωVLq2=ϵ¯∗μλωPpPp¯γμ1-γ5bB¯∗pB∗,λB∗,(14)HλB∗λωVRq2=ϵ¯∗μλωPpPp¯γμ1+γ5bB¯∗pB∗,λB∗,(15)HλB∗λωSLq2=PpPp¯1-γ5bB¯∗pB∗,λB∗,(16)HλB∗λωSRq2=PpPp¯1+γ5bB¯∗pB∗,λB∗,which describe the decay of three helicity states of B∗ meson into a pseudoscalar P meson and the four helicity states of virtual ω. It should be noted that λω in HλB∗λωSL,R(q2), (15) and (16), should always be equal to t.ForB∗→P transition, the matrix elements of the vector and axial-vector currents can be written in terms of form factors V(q2) and A0,1,2(q2) as(17)PpPp¯γμbB¯∗ϵ,pB∗=-2iVq2mB∗+mPεμνρσϵνpPρpB∗σ,(18)PpPp¯γμγ5bB¯∗ϵ,pB∗=2mB∗A0q2ϵ·qq2qμ+mP+mB∗A1q2ϵμ-ϵ·qq2qμ+A2q2ϵ·qmP+mB∗pB∗+pPμ-mB∗2-mP2q2qμ,with the sign convention ϵ0123=-1. Furthermore, using the equations of motion,(19)i∂μp¯γμb=mbμ-mpμp¯b,(20)i∂μp¯γμγ5b=-mbμ+mpμp¯γ5b,one can write the matrix elements of scalar and pseudoscalars currents as(21)PpPp¯bB¯∗ϵ,pB∗=1mbμ-mpμqμPpPp¯γμbB¯∗ϵ,pB∗=0,(22)PpPp¯γ5bB¯∗ϵ,pB∗=-1mbμ+mpμqμPpPp¯γμγ5bB¯∗ϵ,pB∗=-ϵ·q2mB∗mbμ+mpμA0q2,in which mb(μ) and mp(μ) are the running quark masses.Then, by contracting above hadronic matrix elements with the polarization vectors in theB∗-meson rest frame, we obtain five nonvanishing helicity amplitudes(23)H0tq2=H0tVLq2=-H0tVRq2=2mB∗p→q2A0q2,(24)H00q2=H00VLq2=-H00VRq2=12mB∗q2mB∗+mPmB∗2-mP2+q2A1q2+4mB∗2p→2mB∗+mPA2q2,(25)H±∓q2=H±∓VLq2=-H∓±VRq2=-mB∗+mPA1q2∓2mB∗p→mB∗+mPVq2,(26)H0t′q2=H0tSLq2=-H0tSRq2=-2mB∗p→mbμ+mcμA0q2.It is obvious that only the amplitudes with λB∗=λP-λω=-λω survive.


## 2.4. Leptonic Helicity Amplitudes

Expanding the leptonic tensor in terms of a complete set of Wigner’sdJ-functions [9, 83, 87], LμνHμν can be rewritten as a compact form(27)LμνHμν=18∑λl,λν¯l,λω,λω′,J,J′-1J+J′hλl,λν¯lihλl,λν¯lj∗δλB∗,-λωδλB∗,-λω′×dλω,λl-1/2Jdλω′,λl-1/2J′HλB∗λωiHλB∗λω′j∗,in which J and J′ run over 1 and 0, λω(′) and λl run over their components, and massless right-handed antineutrinos with λν¯l=1/2. In (27), hλl,λν¯li,j are the leptonic helicity amplitudes defined as(28)hλl,λν¯lVL,R=u¯lλlγμ1-γ5νν¯12ϵ¯μλω,(29)hλl,λν¯lSL,R=u¯lλl1-γ5νν¯12.In the l-ν¯l center-of-mass frame, taking the exact forms of the spinors and polarization vectors, we finally obtain four nonvanishing contributions(30)h-1/2,1/2VL,R2=8q2-ml2,(31)h1/2,1/2VL,R2=8ml22q2q2-ml2,(32)h1/2,1/2SL,R2=8q2-ml22,(33)h1/2,1/2VL,R×h1/2,1/2SL,R=8ml2q2q2-ml2.


## 2.5. Observables ofB¯∗→Pl-ν¯l Decays

With the amplitudes obtained in above subsections, we then present the observables considered in our following evaluations. The double differential decay rate ofB¯∗→Pl-ν¯l decay is written as(34)dΓdq2dcos⁡θ=GF2Vpb22π3p→8mB∗2131-ml2q2MB¯∗→Pl-ν¯l2,where the factor 1/3 is caused by averaging over the spins of initial state B¯∗. Using the standard convention for dJ-function [88], we finally obtain the double differential decay rates with a given leptonic helicity state (λl=±1/2), which are(35)d2Γλl=-1/2dq2dcos⁡θ=GF2Vpb2p→256π3mB∗213q21-ml2q22×1+VL21-cos⁡θ2H-+2+1+cos⁡θ2H-+2+2sin2θH002+VR21-cos⁡θ2H+-2+1+cos⁡θ2H-+2+2sin2θH002-4Re1+VLVR∗1+cos⁡θ2H+-H-++sin2θH002,(36)d2Γλl=1/2dq2dcos⁡θ=GF2Vpb2p→256π3mB∗213q21-ml2q22ml2q2×1+VL2+VR2sin2θH-+2+H+-2+2H0t-cos⁡θH002-4Re1+VLVR∗sin2θH-+H+-+H0t-cos⁡θH002+4Re1+VL-VRSL∗-SR∗q2mlH0t′H0t-cos⁡θH00+2SL-SR2q2ml2H0t′2.Using (35) and (36), one can get the explicit forms of various observables of B¯∗→Pl-ν¯l decays as follows: (i)
The differential decay rate is(37)dΓdq2=GF2Vpb2p→96π3mB∗213q21-ml2q22×1+VL2+VR2H-+2+H+-2+H0021+ml22q2+3ml22q2H0t2-2Re1+VLVR∗2H-+H+-+H0021+ml22q2+3ml22q2H0t2+3Re1+VL-VRSL∗-SR∗H0t′H0tmlq2+32SL-SR2H0t′2.(ii)
Theq2 dependent ratio is(38)RP∗q2≡dΓB¯∗→Pτ-ν¯τ/dq2dΓB¯∗→Pl′-ν¯l′/dq2,wherel′ denotes the light lepton.(iii)
The lepton spin asymmetry is(39)AλPq2=dΓλl=-1/2/dq2-dΓλl=1/2/dq2dΓλl=-1/2/dq2+dΓλl=1/2/dq2.(iv)
The forward-backward asymmetry is(40)AθPq2=∫-10dcos⁡θd2Γ/dq2dcos⁡θ-∫01dcos⁡θd2Γ/dq2dcos⁡θdΓ/dq2.The SM results can be obtained from above formulae by takingVL=VR=SL=SR=0.In the following evaluations, in order to fit the NP spaces, we also need the observables ofB¯→D(∗)l-ν¯l decays, which have been fully calculated in the past years. In this paper, we adopt the relevant theoretical formulae given in [46].


## 3. Numerical Results and Discussions

### 3.1. Input Parameters

Before presenting our numerical results and analyses, we would like to clarify the values of input parameters used in the calculation. For the CKM matrix elements, we use [89](41)Vcb=4.181-0.060+0.028×10-2,Vub=3.715-0.060+0.060×10-3.For the well-measured Fermi coupling constant GF, the masses of mesons and leptons, and the running masses of quarks at μ=mb, we take their central values given by PDG [88]. The total decay widths (or lifetimes) of B∗ mesons are essential for estimating the branching fraction; however there is no available experimental data until now. According to the fact that the electromagnetic process B∗→Bγ dominates the decays of B∗ meson, we take the approximation Γtot(B∗)≃Γ(B∗→Bγ); the latter has been evaluated within different theoretical models [90–96]. In this paper, we adopt the most recent results [95, 96](42)ΓtotB∗+≃ΓB∗+→B+γ=468-75+73eV,(43)ΓtotB∗0≃ΓB∗0→B0γ=148±20eV,(44)ΓtotBs∗0≃ΓBs∗0→Bs0γ=68±17eV.Then the residual inputs are the transition form factors, which are crucial for evaluating the observables ofB¯∗→Pl-ν¯l and B¯→D(∗)l-ν¯l decays. For the B→D(∗) transitions, the scheme of Caprini, Lellouch, and Neubert (CLN) parametrization [97] is widely used, and the CLN parameters can be precisely extracted from the well-measured B¯→D(∗)l-ν¯l decays; numerically, their values read [7](45)ρD2=1.128±0.033,V11Vcb=41.30±0.99×10-3;(46)ρD∗2=1.205±0.026,hA11Vcb=35.38±0.43×10-3,R11=1.404±0.032,R21=0.854±0.020.However, for the B¯u,d,s∗→Pu,d,s transition, there is no experimental data and ready-made theoretical results to use at present. Here, we employ the Bauer-Stech-Wirbel (BSW) model [98, 99] to evaluate the form factors for both B¯∗→P and B¯→D(∗) transitions. Using the inputs mu=md=0.35GeV, ms=0.55GeV, mc=1.7GeV, mb=4.9GeV, and ω=〈p→⊥2〉=0.4GeV, we obtain the results at q2=0,(47)A0B¯∗→D0=0.71,A1B¯∗→D0=0.75,A2B¯∗→D0=0.62,VB¯∗→D0=0.76;(48)A0B¯s∗→Ds0=0.66,A1B¯s∗→Ds0=0.69,A2B¯s∗→Ds0=0.59,VB¯s∗→Ds0=0.72;(49)A0B¯∗→π0=0.34,A1B¯∗→π0=0.38,A2B¯∗→π0=0.30,VB¯∗→π0=0.35;(50)A0B¯s∗→K0=0.28,A1B¯s∗→K0=0.29,A2B¯s∗→K0=0.26,VB¯s∗→K0=0.30;(51)F0B¯→D0=F1B¯→D0=0.70;(52)A0B¯→D∗0=0.63,A1B¯→D∗0=0.66,A2B¯→D∗0=0.69,VB¯→D∗0=0.71.To be conservative, 15% uncertainties are assigned to these values in our following evaluation. Moreover, with the assumption of nearest pole dominance, the dependences of form factors on q2 read [98, 99](53)F0q2≃F001-q2/mBq0+2,F1q2≃F101-q2/mBq1-2,A0q2≃A001-q2/mBq0-2,A1q2≃A101-q2/mBq1+2,A2q2≃A201-q2/mBq1+2,Vq2≃V01-q2/mBq1-2,where Bq(JP) is the state of Bq with quantum number of JP (J and P are the quantum numbers of total angular momenta and parity, respectively).With the theoretical formulae and inputs given above, we then proceed to present our numerical results and discussion, which are divided into two scenarios with different simplification for our attention to the types of NP couplings as follows:(i)
Scenario I: takingSL=SR=0,i.e., only considering the NP effects of VL,R couplings(ii)
Scenario II: takingVL=VR=0,i.e., only considering the NP effects of SL,R couplingsIn these two scenarios, we consider all the NP parameters to be real for our analysis. In addition, we assume that only the third generation leptons get corrections from the NP in theb→(u,c)lν¯l processes and for l=e,μ the NP is absent. In the following discussion, the allowed spaces of NP couplings are obtained by fitting to RD and RD∗ (1), with the data varying randomly within their 1σ error, while the theoretical uncertainties are also considered and obtained by varying the inputs randomly within their ranges specified above.


### 3.2. Scenario I: Effects ofVL and VR Type Couplings

In this subsection, we vary couplingsVL and VR while keeping all other NP couplings to zero. Under the constraints from the data of RD and RD∗, the allowed spaces of new physics parameters, VL and VR, are shown in Figure 1. In the fit, the B→D(∗) form factors based on CLN parametrization and BSW model are used, respectively; it can be seen from Figure 1 that their corresponding fitting results are consistent with each other, but the constraint with the former is much stronger due to the relatively small theoretical error. Therefore, in the following evaluations and discussions, the results obtained by using CLN parametrization are used. In addition, our fitting result in Figure 1 agrees well with the ones obtained in the previous works, for instance, [26, 35].Figure 1
The allowed spaces ofVL and VR obtained by fitting to RD and RD∗. The red and green regions are obtained by using the form factors of CLN parametrization and BSW model, respectively. (b) shows the minimal result (solution A) of the four solutions shown in (a).
(a) (b)From Figure1, we find that (i) the allowed spaces of (VL,VR) are bounded into four separate regions, namely, solutions A-D. (ii) Except for solution A, the other solutions are all far from the zero point (0,0) and result in very large NP contributions. Taking solution C (D) as an example, the SM contribution is completely canceled out by the NP contribution related to VL, and the VR coupling presents sizable positive (negative) NP contribution to fit data. The situation of solution B is similar, but only VL coupling presents sizable NP contribution. Numerically, one can easily conclude that the NP contributions of solutions B-D are about two times larger than the SM, which seriously exceeds our general expectation that the amplitudes should be dominated by the SM and the NP only presents minor corrections. In this point of view, the minimal solution (solution A) is much favored than solutions B-D. So, in our following discussions, we pay attention only to solution A, which is replotted in Figure 1(b) and numerical result is(54)VL=0.14-0.06+0.06,VR=0.05-0.07+0.06.solution  AUsing the values of NP couplings given by (54), we then present our theoretical predictions for B(B¯∗→Pτ-ν¯τ) and q2-integrated RP∗ in Table 1, in which the SM results are also listed for comparison. The q2-dependence of differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays is shown in Figure 2; the case of B¯∗0→D+τ-ν¯τ and B¯s∗0→Ds+τ-ν¯τ (B¯∗0→π+τ-ν¯τ and B¯s∗0→K+τ-ν¯τ) is similar to the one of B∗-→D0τ-ν¯τ (B∗-→π0τ-ν¯τ) decay, not shown here. The following are some discussions and comments:(1)
From Table1, it can be seen that the branching fractions of b→cτν¯τ induced B¯u,d,s∗ decays are at the level of O(10-8-10-7), while the b→uτν¯τ induced decays are relatively rare due to the suppression caused by the CKM factor. In addition, the difference between the branching fractions of three decay modes induced by b→cτν¯τ (or b→uτν¯τ) transition is mainly attributed to the relation of total decay widths, Γtot(B∗-):Γtot(B¯∗0):Γtot(B¯s∗0)~1:2:6, illustrated by (42), (43), and (44).(2)
Comparing with the SM results, one can easily find from Table1 that B(B¯∗→Pτ-ν¯τ) are enhanced about 20% by the NP contributions of VL and VR. It can also be clearly seen from Figures 2(a) and 2(b). However, as shown in Figures 2(a) and 2(b), due to the large theoretical uncertainties caused by the form factors, the NP hints are hard to be totally distinguished from the SM results.(3)
The theoretical uncertainties can be well controlled by using the ratioRP∗ instead of decay rate due to the cancelation of nonperturbative errors; therefore RP∗ is much suitable for probing the NP hints. From the last three rows of Table 1, it can be found that the NP prediction for RP∗ significantly deviates from the SM result. In particular, as Figures 2(c) and 2(d) show, the NP effects can be totally distinguished from the SM at q2≳7GeV2 even though the theoretical errors are considered. So, future measurements on B¯∗→Pτ-ν¯τ decays can make further test on the NP models which provide possible solutions to the RD and RD∗ problems.(4)
From Figures2(e)–2(h) it can be found that the NP contribution of solution A has little effect on the observables AλP and AθP in the whole q2 region, which can be understood from the following analyses. Because the NP contribution of solution A is dominated by the left-handed coupling VL, we can find that |M(B¯∗→Pl-ν¯l)|∝|(1+VL)|2 in the limit of (1+VL)≫VR. As a result, the NP contributions (solution A) to the numerator and denominator of AλP and AθP cancel each other out to a large extent. For AλP, the cases of solutions B, C, and D are similar to solution A.Table 1
The theoretical predictions for the branching fractions ofB-∗→Pτ-ν-τ decays and RP∗ within the SM and the two scenarios. The first error is caused by the uncertainties of form factors, CKM factors, and Γtot(B∗); and the second error given in the last two columns is caused by the NP couplings.
Obs. SM Prediction Scenario I Scenario II B(B∗-→D0τ-ν-τ) 0.8 7 - 0.32 + 0.46 × 1 0 - 8 1.0 4 - 0.38 - 0.05 + 0.54 + 0.06 × 1 0 - 8 1.0 0 - 0.36 - 0.04 + 0.51 + 0.03 × 1 0 - 8 B(B-∗0→D+τ-ν-τ) 2.7 4 - 0.94 + 1.29 × 1 0 - 8 3.2 7 - 1.14 - 0.15 + 1.66 + 0.19 × 1 0 - 8 3.1 3 - 1.13 - 0.11 + 1.52 + 0.10 × 1 0 - 8 B(B-s∗0→Ds+τ-ν-τ) 5.1 3 - 2.13 + 3.67 × 1 0 - 7 6.1 3 - 2.48 - 0.28 + 4.51 + 0.35 × 1 0 - 7 5.8 9 - 2.39 - 0.22 + 3.93 + 0.20 × 1 0 - 7  B(B∗-→π0τ-ν-τ) 1.4 2 - 0.50 + 0.79 × 1 0 - 10 1.7 1 - 0.63 - 0.07 + 0.91 + 0.09 × 1 0 - 10 1.7 4 - 0.62 - 0.10 + 0.94 + 0.10 × 1 0 - 10 B(B-∗0→π+τ-ν-τ) 0.9 9 - 0.41 + 0.38 × 1 0 - 9 1.0 8 - 0.37 - 0.05 + 0.55 + 0.06 × 1 0 - 9 1.0 9 - 0.39 - 0.06 + 0.52 + 0.06 × 1 0 - 9 B(B-s∗0→K+τ-ν-τ) 0.9 5 - 0.40 + 0.65 × 1 0 - 9 1.1 4 - 0.46 - 0.05 + 0.78 + 0.06 × 1 0 - 9 1.2 0 - 0.47 - 0.08 + 0.87 + 0.08 × 1 0 - 9  R D ∗ 0.29 8 - 0.010 + 0.012 0.35 5 - 0.011 - 0.016 + 0.015 + 0.020 0.34 1 - 0.026 - 0.012 + 0.048 + 0.011 R π ∗ 0.67 7 - 0.014 + 0.013 0.81 6 - 0.012 - 0.035 + 0.017 + 0.044 0.82 7 - 0.073 - 0.048 + 0.126 + 0.046 R K ∗ 0.63 8 - 0.015 + 0.017 0.77 0 - 0.015 - 0.034 + 0.021 + 0.042 0.81 0 - 0.084 - 0.054 + 0.144 + 0.052Figure 2
Theq2-dependence of the differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays within the SM and scenario I.
(a) (b) (c) (d) (e) (f) (g) (h)


### 3.3. Scenario II: Effects ofSL and SR Type Couplings

In this subsection, we only consider the effects of scalar interactionsSL and SR and take the other NP couplings to be zero. Under the 1σ constraint from the data of RD and RD∗, the allowed spaces of SL and SR are shown in Figure 3. Similar to scenario I, four solutions for SL and SR are found in scenario II, which can be seen from Figure 3(a); and the fitting results obtained by using form factors in CLN parametrization and BSW model are consistent with each other. Solutions B-D result in so large NP contributions; therefore, in the following discussion, we pay our attention to solution A, which are replotted in Figure 3(b). The numerical result of solution A is(55)SL=-0.46-0.24+0.24,SR=0.70-0.24+0.23.Using these values, we present in Table 1 our numerical predictions of scenario II for the observables, B(B¯∗→Pτ-ν¯τ) and q2-integrated RP∗. Moreover, the q2 distributions of differential observables dΓ/dq2, RP∗, AλP, and AθP are shown in Figure 4. The following are some discussions for these results: (1)
From Table1 and Figures 4(a) and 4(b), it can be found that the B(B¯∗→Pτ-ν¯τ) and RP∗ can be enhanced about 15% compared with the SM results by the NP contributions. Similar to the situation of scenario I, the NP effect of SL and SR on RP∗ is much more significant than the one on branching fraction due to the theoretical uncertainties of RP∗ which can be well controlled. In particular, as Figures 4(a) and 4(b) show, the spectra of the SM and NP for RP∗ can be clearly distinguished at middle q2 region.(2)
The main difference between the effects of scalar and vector couplings on theB¯∗→Pτ-ν¯τ decays is that the former only contributes to the longitudinal amplitude, which can be found from (37). As a result, their effects on B(B¯∗→Pτ-ν¯τ) and RP∗ are a little different, which can be seen by comparing Figures 2(a)–2(d) with Figures 4(a)–4(d).(3)
Another significant difference between the scalar and vector couplings is that only the leptonic helicity amplitudes of scalar type withλl=1/2 survive, which can be easily found from (35) and (36). Therefore, as Figures 4(e) and 4(f) show, the scalar couplings lead to significant NP effects on AλP, which is obviously different from predictions of vector couplings in scenario I (Figures 2(e) and 2(f)). Besides, as Figures 4(e) and 4(f) show, SL and SR couplings also have large contributions to the AθP at all q2 region, which is another difference with the vector couplings (Figures 2(g) and 2(h)). Therefore, the future measurements on these observables will provide strict tests on the SM and various NP models.Figure 3
The allowed spaces ofSL and SR obtained by fitting to the data of RD and RD∗. The other captions are the same as in Figure 1.
(a) (b)Figure 4
Theq2-dependence of the differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays within the SM and scenario II.
(a) (b) (c) (d) (e) (f) (g) (h)


## 3.1. Input Parameters

Before presenting our numerical results and analyses, we would like to clarify the values of input parameters used in the calculation. For the CKM matrix elements, we use [89](41)Vcb=4.181-0.060+0.028×10-2,Vub=3.715-0.060+0.060×10-3.For the well-measured Fermi coupling constant GF, the masses of mesons and leptons, and the running masses of quarks at μ=mb, we take their central values given by PDG [88]. The total decay widths (or lifetimes) of B∗ mesons are essential for estimating the branching fraction; however there is no available experimental data until now. According to the fact that the electromagnetic process B∗→Bγ dominates the decays of B∗ meson, we take the approximation Γtot(B∗)≃Γ(B∗→Bγ); the latter has been evaluated within different theoretical models [90–96]. In this paper, we adopt the most recent results [95, 96](42)ΓtotB∗+≃ΓB∗+→B+γ=468-75+73eV,(43)ΓtotB∗0≃ΓB∗0→B0γ=148±20eV,(44)ΓtotBs∗0≃ΓBs∗0→Bs0γ=68±17eV.Then the residual inputs are the transition form factors, which are crucial for evaluating the observables ofB¯∗→Pl-ν¯l and B¯→D(∗)l-ν¯l decays. For the B→D(∗) transitions, the scheme of Caprini, Lellouch, and Neubert (CLN) parametrization [97] is widely used, and the CLN parameters can be precisely extracted from the well-measured B¯→D(∗)l-ν¯l decays; numerically, their values read [7](45)ρD2=1.128±0.033,V11Vcb=41.30±0.99×10-3;(46)ρD∗2=1.205±0.026,hA11Vcb=35.38±0.43×10-3,R11=1.404±0.032,R21=0.854±0.020.However, for the B¯u,d,s∗→Pu,d,s transition, there is no experimental data and ready-made theoretical results to use at present. Here, we employ the Bauer-Stech-Wirbel (BSW) model [98, 99] to evaluate the form factors for both B¯∗→P and B¯→D(∗) transitions. Using the inputs mu=md=0.35GeV, ms=0.55GeV, mc=1.7GeV, mb=4.9GeV, and ω=〈p→⊥2〉=0.4GeV, we obtain the results at q2=0,(47)A0B¯∗→D0=0.71,A1B¯∗→D0=0.75,A2B¯∗→D0=0.62,VB¯∗→D0=0.76;(48)A0B¯s∗→Ds0=0.66,A1B¯s∗→Ds0=0.69,A2B¯s∗→Ds0=0.59,VB¯s∗→Ds0=0.72;(49)A0B¯∗→π0=0.34,A1B¯∗→π0=0.38,A2B¯∗→π0=0.30,VB¯∗→π0=0.35;(50)A0B¯s∗→K0=0.28,A1B¯s∗→K0=0.29,A2B¯s∗→K0=0.26,VB¯s∗→K0=0.30;(51)F0B¯→D0=F1B¯→D0=0.70;(52)A0B¯→D∗0=0.63,A1B¯→D∗0=0.66,A2B¯→D∗0=0.69,VB¯→D∗0=0.71.To be conservative, 15% uncertainties are assigned to these values in our following evaluation. Moreover, with the assumption of nearest pole dominance, the dependences of form factors on q2 read [98, 99](53)F0q2≃F001-q2/mBq0+2,F1q2≃F101-q2/mBq1-2,A0q2≃A001-q2/mBq0-2,A1q2≃A101-q2/mBq1+2,A2q2≃A201-q2/mBq1+2,Vq2≃V01-q2/mBq1-2,where Bq(JP) is the state of Bq with quantum number of JP (J and P are the quantum numbers of total angular momenta and parity, respectively).With the theoretical formulae and inputs given above, we then proceed to present our numerical results and discussion, which are divided into two scenarios with different simplification for our attention to the types of NP couplings as follows:(i)
Scenario I: takingSL=SR=0,i.e., only considering the NP effects of VL,R couplings(ii)
Scenario II: takingVL=VR=0,i.e., only considering the NP effects of SL,R couplingsIn these two scenarios, we consider all the NP parameters to be real for our analysis. In addition, we assume that only the third generation leptons get corrections from the NP in theb→(u,c)lν¯l processes and for l=e,μ the NP is absent. In the following discussion, the allowed spaces of NP couplings are obtained by fitting to RD and RD∗ (1), with the data varying randomly within their 1σ error, while the theoretical uncertainties are also considered and obtained by varying the inputs randomly within their ranges specified above.


## 3.2. Scenario I: Effects ofVL and VR Type Couplings

In this subsection, we vary couplingsVL and VR while keeping all other NP couplings to zero. Under the constraints from the data of RD and RD∗, the allowed spaces of new physics parameters, VL and VR, are shown in Figure 1. In the fit, the B→D(∗) form factors based on CLN parametrization and BSW model are used, respectively; it can be seen from Figure 1 that their corresponding fitting results are consistent with each other, but the constraint with the former is much stronger due to the relatively small theoretical error. Therefore, in the following evaluations and discussions, the results obtained by using CLN parametrization are used. In addition, our fitting result in Figure 1 agrees well with the ones obtained in the previous works, for instance, [26, 35].Figure 1
The allowed spaces ofVL and VR obtained by fitting to RD and RD∗. The red and green regions are obtained by using the form factors of CLN parametrization and BSW model, respectively. (b) shows the minimal result (solution A) of the four solutions shown in (a).
(a) (b)From Figure1, we find that (i) the allowed spaces of (VL,VR) are bounded into four separate regions, namely, solutions A-D. (ii) Except for solution A, the other solutions are all far from the zero point (0,0) and result in very large NP contributions. Taking solution C (D) as an example, the SM contribution is completely canceled out by the NP contribution related to VL, and the VR coupling presents sizable positive (negative) NP contribution to fit data. The situation of solution B is similar, but only VL coupling presents sizable NP contribution. Numerically, one can easily conclude that the NP contributions of solutions B-D are about two times larger than the SM, which seriously exceeds our general expectation that the amplitudes should be dominated by the SM and the NP only presents minor corrections. In this point of view, the minimal solution (solution A) is much favored than solutions B-D. So, in our following discussions, we pay attention only to solution A, which is replotted in Figure 1(b) and numerical result is(54)VL=0.14-0.06+0.06,VR=0.05-0.07+0.06.solution  AUsing the values of NP couplings given by (54), we then present our theoretical predictions for B(B¯∗→Pτ-ν¯τ) and q2-integrated RP∗ in Table 1, in which the SM results are also listed for comparison. The q2-dependence of differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays is shown in Figure 2; the case of B¯∗0→D+τ-ν¯τ and B¯s∗0→Ds+τ-ν¯τ (B¯∗0→π+τ-ν¯τ and B¯s∗0→K+τ-ν¯τ) is similar to the one of B∗-→D0τ-ν¯τ (B∗-→π0τ-ν¯τ) decay, not shown here. The following are some discussions and comments:(1)
From Table1, it can be seen that the branching fractions of b→cτν¯τ induced B¯u,d,s∗ decays are at the level of O(10-8-10-7), while the b→uτν¯τ induced decays are relatively rare due to the suppression caused by the CKM factor. In addition, the difference between the branching fractions of three decay modes induced by b→cτν¯τ (or b→uτν¯τ) transition is mainly attributed to the relation of total decay widths, Γtot(B∗-):Γtot(B¯∗0):Γtot(B¯s∗0)~1:2:6, illustrated by (42), (43), and (44).(2)
Comparing with the SM results, one can easily find from Table1 that B(B¯∗→Pτ-ν¯τ) are enhanced about 20% by the NP contributions of VL and VR. It can also be clearly seen from Figures 2(a) and 2(b). However, as shown in Figures 2(a) and 2(b), due to the large theoretical uncertainties caused by the form factors, the NP hints are hard to be totally distinguished from the SM results.(3)
The theoretical uncertainties can be well controlled by using the ratioRP∗ instead of decay rate due to the cancelation of nonperturbative errors; therefore RP∗ is much suitable for probing the NP hints. From the last three rows of Table 1, it can be found that the NP prediction for RP∗ significantly deviates from the SM result. In particular, as Figures 2(c) and 2(d) show, the NP effects can be totally distinguished from the SM at q2≳7GeV2 even though the theoretical errors are considered. So, future measurements on B¯∗→Pτ-ν¯τ decays can make further test on the NP models which provide possible solutions to the RD and RD∗ problems.(4)
From Figures2(e)–2(h) it can be found that the NP contribution of solution A has little effect on the observables AλP and AθP in the whole q2 region, which can be understood from the following analyses. Because the NP contribution of solution A is dominated by the left-handed coupling VL, we can find that |M(B¯∗→Pl-ν¯l)|∝|(1+VL)|2 in the limit of (1+VL)≫VR. As a result, the NP contributions (solution A) to the numerator and denominator of AλP and AθP cancel each other out to a large extent. For AλP, the cases of solutions B, C, and D are similar to solution A.Table 1
The theoretical predictions for the branching fractions ofB-∗→Pτ-ν-τ decays and RP∗ within the SM and the two scenarios. The first error is caused by the uncertainties of form factors, CKM factors, and Γtot(B∗); and the second error given in the last two columns is caused by the NP couplings.
Obs. SM Prediction Scenario I Scenario II B(B∗-→D0τ-ν-τ) 0.8 7 - 0.32 + 0.46 × 1 0 - 8 1.0 4 - 0.38 - 0.05 + 0.54 + 0.06 × 1 0 - 8 1.0 0 - 0.36 - 0.04 + 0.51 + 0.03 × 1 0 - 8 B(B-∗0→D+τ-ν-τ) 2.7 4 - 0.94 + 1.29 × 1 0 - 8 3.2 7 - 1.14 - 0.15 + 1.66 + 0.19 × 1 0 - 8 3.1 3 - 1.13 - 0.11 + 1.52 + 0.10 × 1 0 - 8 B(B-s∗0→Ds+τ-ν-τ) 5.1 3 - 2.13 + 3.67 × 1 0 - 7 6.1 3 - 2.48 - 0.28 + 4.51 + 0.35 × 1 0 - 7 5.8 9 - 2.39 - 0.22 + 3.93 + 0.20 × 1 0 - 7  B(B∗-→π0τ-ν-τ) 1.4 2 - 0.50 + 0.79 × 1 0 - 10 1.7 1 - 0.63 - 0.07 + 0.91 + 0.09 × 1 0 - 10 1.7 4 - 0.62 - 0.10 + 0.94 + 0.10 × 1 0 - 10 B(B-∗0→π+τ-ν-τ) 0.9 9 - 0.41 + 0.38 × 1 0 - 9 1.0 8 - 0.37 - 0.05 + 0.55 + 0.06 × 1 0 - 9 1.0 9 - 0.39 - 0.06 + 0.52 + 0.06 × 1 0 - 9 B(B-s∗0→K+τ-ν-τ) 0.9 5 - 0.40 + 0.65 × 1 0 - 9 1.1 4 - 0.46 - 0.05 + 0.78 + 0.06 × 1 0 - 9 1.2 0 - 0.47 - 0.08 + 0.87 + 0.08 × 1 0 - 9  R D ∗ 0.29 8 - 0.010 + 0.012 0.35 5 - 0.011 - 0.016 + 0.015 + 0.020 0.34 1 - 0.026 - 0.012 + 0.048 + 0.011 R π ∗ 0.67 7 - 0.014 + 0.013 0.81 6 - 0.012 - 0.035 + 0.017 + 0.044 0.82 7 - 0.073 - 0.048 + 0.126 + 0.046 R K ∗ 0.63 8 - 0.015 + 0.017 0.77 0 - 0.015 - 0.034 + 0.021 + 0.042 0.81 0 - 0.084 - 0.054 + 0.144 + 0.052Figure 2
Theq2-dependence of the differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays within the SM and scenario I.
(a) (b) (c) (d) (e) (f) (g) (h)


## 3.3. Scenario II: Effects ofSL and SR Type Couplings

In this subsection, we only consider the effects of scalar interactionsSL and SR and take the other NP couplings to be zero. Under the 1σ constraint from the data of RD and RD∗, the allowed spaces of SL and SR are shown in Figure 3. Similar to scenario I, four solutions for SL and SR are found in scenario II, which can be seen from Figure 3(a); and the fitting results obtained by using form factors in CLN parametrization and BSW model are consistent with each other. Solutions B-D result in so large NP contributions; therefore, in the following discussion, we pay our attention to solution A, which are replotted in Figure 3(b). The numerical result of solution A is(55)SL=-0.46-0.24+0.24,SR=0.70-0.24+0.23.Using these values, we present in Table 1 our numerical predictions of scenario II for the observables, B(B¯∗→Pτ-ν¯τ) and q2-integrated RP∗. Moreover, the q2 distributions of differential observables dΓ/dq2, RP∗, AλP, and AθP are shown in Figure 4. The following are some discussions for these results: (1)
From Table1 and Figures 4(a) and 4(b), it can be found that the B(B¯∗→Pτ-ν¯τ) and RP∗ can be enhanced about 15% compared with the SM results by the NP contributions. Similar to the situation of scenario I, the NP effect of SL and SR on RP∗ is much more significant than the one on branching fraction due to the theoretical uncertainties of RP∗ which can be well controlled. In particular, as Figures 4(a) and 4(b) show, the spectra of the SM and NP for RP∗ can be clearly distinguished at middle q2 region.(2)
The main difference between the effects of scalar and vector couplings on theB¯∗→Pτ-ν¯τ decays is that the former only contributes to the longitudinal amplitude, which can be found from (37). As a result, their effects on B(B¯∗→Pτ-ν¯τ) and RP∗ are a little different, which can be seen by comparing Figures 2(a)–2(d) with Figures 4(a)–4(d).(3)
Another significant difference between the scalar and vector couplings is that only the leptonic helicity amplitudes of scalar type withλl=1/2 survive, which can be easily found from (35) and (36). Therefore, as Figures 4(e) and 4(f) show, the scalar couplings lead to significant NP effects on AλP, which is obviously different from predictions of vector couplings in scenario I (Figures 2(e) and 2(f)). Besides, as Figures 4(e) and 4(f) show, SL and SR couplings also have large contributions to the AθP at all q2 region, which is another difference with the vector couplings (Figures 2(g) and 2(h)). Therefore, the future measurements on these observables will provide strict tests on the SM and various NP models.Figure 3
The allowed spaces ofSL and SR obtained by fitting to the data of RD and RD∗. The other captions are the same as in Figure 1.
(a) (b)Figure 4
Theq2-dependence of the differential observables dΓ/dq2, RP∗, AλP, and AθP for B∗-→D0τ-ν¯τ and π0τ-ν¯τ decays within the SM and scenario II.
(a) (b) (c) (d) (e) (f) (g) (h)


## 4. Summary

In this paper, motivated by the observed “RD∗ and RD puzzles" and its implication of NP, we have studied the NP effects on the b→(c,u)l-ν¯l induced semileptonic B¯u,d,s∗→Pl-ν¯l (P=D,Ds,π,K) decays in a model-independent scheme. Using the allowed spaces of vector and scalar couplings obtained by fitting to the data of RD∗ and RD, the NP effects on the decay rate, ratio RP∗, lepton spin asymmetry, and forward-backward asymmetry are studied in vector and scalar scenarios, respectively. It is found that the vector couplings present large contributions to the decay rate and RP∗, but their effects on AλP and AθP are very tiny. Different from the vector couplings, the scalar couplings present significant effects not only on the decay rate and RP∗ but also on AλP and AθP. The future measurements on the B¯u,d,s∗→Pl-ν¯l decays will further test the predictions of the SM and NP and confirm or refute possible NP solutions to RD∗ and RD.



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