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To date, the LHC has not provided any direct evidence for new physics (NP) particles beyond the standard model (SM). However, several hints referring to the lepton flavor university (LFU) violation emerge in the measurements of semileptonic b-hadron decays, which, if confirmed with more precise experimental data and theoretical predictions, depict unambiguous signs of NP [1, 2].
The charged-current decays
B→D(∗)lˉν , withℓ=e ,μ , orτ , have been measured by the BaBar [3, 4], Belle [5–8], and LHCb [9–11] collaborations. The ratios of the branching fractions①,RD(∗)≡B(B→D(∗)τˉν)/B(B→D(∗)ℓˉν) , withℓ=e and/orμ , obtained by the latest experimental averages by the heavy flavor averaging group read [12]RexpD=0.407±0.039(stat.)±0.024(syst.),
(1) RexpD∗=0.306±0.013(stat.)±0.007(syst.),
(2) both of which exceed their respective SM predictions [12]①
RSMD=0.299±0.003,RSMD∗=0.258±0.005,
(3) by
2.3σ and3.0σ , respectively. Considering the experimental correlation of −0.203 betweenRD andRD∗ , the combined results exhibit a ~3.78σ deviation from the SM predictions [12]. This discrepancy, referred to as theRD(∗) anomaly, may provide a hint of LFU violating NP [1, 2]. For theBc→J/ψℓˉν decay, a ratioRJ/ψ can be similarly defined. The recent LHCb measurement,RexpJ/ψ=0.71± 0.17(stat.)±0.18(syst.) [17], lies about2σ above the SM prediction,RSMJ/ψ=0.248±0.006 [18]. In addition, the LHCb measurements of the ratiosRK(∗)≡B(B→K(∗)μ+μ−)/ B(B→K(∗)e+e−) ,RexpK=0.745+0.090−0.074±0.036 for1.0GeV2⩽ q2⩽6.0GeV2 [19] andRexpK∗=0.69+0.11−0.07±0.05 for1.1GeV2⩽ q2⩽6.0GeV2 [20], are found to be about2.6σ and ~2.5σ lower than the SM expectation,RSMK(∗)≃1 [21, 22], respectively. These anomalies motivated numerous studies both in the effective field theory approach [23–28] and in specific NP models [29–45]. We refer to Refs. [1, 2] for recent reviews.Recently, the Belle collaboration reported the first preliminary result of the
D∗ longitudinal polarization fraction in theB→D∗τˉν decay [46, 47]PD∗L=0.60±0.08(stat.)±0.04(syst.),
(4) which is consistent with the SM prediction
PD∗L=0.46±0.04 [48] at1.5σ . Together with the measurements of theτ polarization,PτL=−0.38±0.51(stat.)+0.21−0.16(syst.) [7, 8], these results provide valuable information on the spin structure of the interaction involved inB→D(∗)τˉν decays and are good observables for testing of various NP scenarios [48–53]. The measurement of angular observables in these decays will be considerably improved in the future [54, 55]. For example, the Belle II experiment with50ab−1 data can measurePτL with an expected precision of±0.07 [54].In this work, motivated by these experimental progresses and future prospects, we study five
b→cτˉν decays,B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν , in the leptoquark (LQ) model proposed in Ref. [56]. Models with one or more LQ states, which are colored bosons that couple to both quarks and leptons, depict some of the most popular scenarios employed to explain theRD(∗) andRK(∗) anomalies [57–74]. In Ref. [56], the SM is extended with two scalar LQs, one of which is aSU(2)L singlet, whereas the other is aSU(2)L triplet. This model is also featured by the fact that these two LQs have the same mass and hypercharge, and their couplings to fermions are related by a discrete symmetry. In this manner, the anomalies inb→cτˉν andb→sμ+μ− transitions can be explained simultaneously, while avoiding potentially dangerous contributions tob→sνˉν decays. By taking into account recent developments on transition form factors [13, 14, 18, 75–77], we derive constraints on LQ couplings in this model. Subsequently, predictions in the LQ model are made for the fiveb→cτˉν decays, focusing on theq2 distributions of the branching fractions, LFU ratios, and various angular observables. Implications for future research at the High-Luminosity LHC (HL-LHC) [78] and SuperKEKB [54] are also briefly discussed.This paper is organized as follows: in Section 2, we provide a brief review of the LQ model proposed in Ref. [56]. In Section 3, we recapitulate the theoretical formulae for the various flavor processes and discuss the LQ effects on these decays. In Section 4, we present our detailed numerical analysis and discussions. Our conclusions are given in Section 5. The relevant transition form factors and helicity amplitudes are presented in the appendices.
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In this section, we recapitulate the LQ model proposed in Ref. [56], where a scalar LQ singlet
Φ1 and a tripletΦ3 are added to the SM field content, to explain the observed flavor anomalies. Under the SM gauge group(SU(3)C,SU(2)L,U(1)Y) , the LQ statesΦ1 andΦ3 transform as(3,1,−2/3) and(3,ˉ3,−2/3) , respectively. Their interactions with the SM fermions are described by the Lagrangian [56]L=λ1LjkˉQcjiτ2LkΦ†1+λ3LjkˉQcjiτ2(τ⋅Φ3)†Lk+h.c.,
(5) where
Qj andLk denote the left-handed quark and lepton doublet with generation indices j and k, respectively. The couplingsλ1Ljk andλ3Ljk are generally complex, however assumed to be real throughout this work. It is further assumed that these two scalar LQs have the same mass M, and their couplings to the SM fermions satisfy the following discrete symmetry [56]:λLjk≡λ1Ljk,λ3Ljk=eiπjλLjk.
(6) With these two assumptions, the tree-level LQ contributions to the
b→sνˉν decays are canceled. After rotating to the mass eigenstate basis, the LQ couplings to the left-handed quarks involve the CKM elements asλLdjk=λLjk,λLujk=V∗jiλLik,
(7) where
Vij is the CKM matrix element. -
In this section, we introduce the theoretical framework for the relevant flavor processes and discuss the LQ effects on these decays.
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Including the LQ contributions, the effective Hamiltonian responsible for
b→cℓiˉνj transitions is given by [56]Heff=4GF√2VcbCijL(ˉcγμPLb)(ˉℓiγμPLνj),
(8) with the Wilson coefficient
CijL=CSM,ijL+CNP,ijL . The W-exchange contribution within the SM yieldsCSM,ijL=δij , and the LQ contributions result inCNP,ijL=√28GFM2VckVcbλL3jλL∗ki[1+(−1)k].
(9) This Wilson coefficient is given at the matching scale
μNP∼M . However, as the corresponding current is conserved, we can obtain the low-energy Wilson coefficient,CNP,ijL(μb)=CNP,ijL , without considering the renormalization group evolution (RGE) effect.In this study, we consider five processes mediated by the quark-level
b→cℓˉν transition, includingB→D(∗)ℓˉν ,Bc→ηcℓˉν ,Bc→J/ψℓˉν , andΛb→Λcℓˉν decays. All these processes can be uniformly represented byM(pM,λM)→N(pN,λN)+ℓ−(pℓ,λℓ)+ˉνℓ(pˉνℓ),
(10) where
(M,N)=(B,D(∗)),(Bc,ηc),(Bc,J/ψ) , and(Λb,Λc) , and(ℓ,ˉν)=(e,ˉνe),(μ,ˉνμ) , and(τ,ˉντ) . For each particle i in the above decay, its momentum and helicity are denoted bypi andλi , respectively. In particular, the helicity of a pseudoscalar meson is zero, i.e.,λB(c),D,ηc=0 . After averaging over the non-zero helicity of the hadron M, the differential decay rate of this process can be written as [53, 79]dΓλN,λℓ(M→Nℓ−ˉνℓ)=12mM12|λM|+1∑λM|MλMλN,λℓ|2dΦ3,
(11) with the phase space
dΦ3=√Q+Q−256π3m2M√1−m2ℓq2dq2dcosθℓ,
(12) where
Q±=m2±−q2 , withm±=mM±mN andq2 depicts the dilepton invariant mass squared.θℓ∈[0,π] denotes the angle between the three-momentum ofℓ and that of N in theℓ -ˉν center-of-mass frame. The helicity amplitudesMλMλN,λℓ≡⟨Nℓˉνℓ|Heff|M⟩ can be written as [76]MλMλN,λτ=GFVcb√2(HSPλM,λNLSPλτ+∑λWηλWHVAλM,λN,λWLVAλτ,λW+∑λW1,λW2ηλW1ηλW2HTλM,λN,λW1,λW2LTλτ,λW1λW2),
(13) where
λWi denotes the helicity of the virtual vector bosons W,W1 andW2 . The coefficientηλWi=1 forλλWi=t , andηλWi=−1 forλλWi=0,±1 . Explicit analytical expressions of the leptonic and hadronic helicity amplitudes H and L are given in appendices A and C.From Eq. (11), we can derive the following observables:
• The differential decay width and branching fraction
dB(M→Nℓˉνℓ)dq2=1ΓMdΓ(M→Nℓˉνℓ)dq2=1ΓM∑λN,λℓdΓλN,λℓ(M→Nℓˉνℓ)dq2,
(14) where
ΓM=1/τM is the total width of the hadron M.• The
q2 -dependent LFU ratioRN(q2)=dΓ(M→Nτˉντ)/dq2dΓ(M→Nlˉνl)/dq2,
(15) where
dΓ(M→Nlˉνl)/dq2 denotes the average of the different decay widths of the electronic and muonic modes.• The lepton forward-backward asymmetry
AFB(q2)=∫10dcosθℓ(d2Γ/dq2dcosθℓ)−∫0−1dcosθℓ(d2Γ/dq2dcosθℓ)dΓ/dq2.
(16) • The
q2 -dependent polarization fractionsPτL(q2)=dΓλτ=+1/2/dq2−dΓλτ=−1/2/dq2dΓ/dq2,PNL(q2)=dΓλN=+1/2/dq2−dΓλN=−1/2/dq2dΓ/dq2,forN=Λc,PNL(q2)=dΓλN=0/dq2dΓ/dq2,forN=D∗,J/ψ,PNT(q2)=dΓλN=1/dq2−dΓλN=−1/dq2dΓ/dq2,forN=D∗,J/ψ.
(17) Analytical expressions of all the above observables are given in Appendix C. As these angular observables depict ratios of the decay widths, they are largely free of hadronic uncertainties and thus provide excellent tests of the NP effects.
As shown in Eq. (8), LQ effects generate an operator with the same chirality structure as in the SM. Therefore, it is straightforward to derive the following relation:
RNRSMN=3∑i=1|δ3i+C3iL|2,
(18) with
N=D(∗),ηc,J/ψ andΛc . Here, vanishing contributions to the electronic and muonic channels are already assumed.One of the main inputs in our calculations are the transition form factors. In this respect, notable progresses have been achieved in recent years [13–16, 75–77, 80–87]. This study adopts the Boyd-Grinstein-Lebed (BGL) [13, 88] and Caprini-Lellouch-Neubert (CLN) [14, 89] parametrization for the
B→D andB→D∗ transition form factors, respectively. In these approaches, both the transition form factors and the CKM matrix element|Vcb| are simultaneously extracted from the experimental data. In addition, we use theBc→ηc,J/ψ transition form factors obtained in the covariant light-front approach [18]. For theΛb→Λc transition form factor, we adopt the recent Lattice QCD results in Refs. [75, 76]. Explicit expressions of all the relevant transition form factors are recapitulated in Appendix B. -
With the LQ effects considered, the effective Hamiltonian for the
b→sℓ+iℓ−j transition can be written as [90]Heff=−4GF√2VtbV∗ts∑aCijaOija+h.c.,
(19) where the operators relevant to our study are
Oij9=αe4π(ˉsγμPLb)(ˉℓiγμℓj),Oij10=αe4π(ˉsγμPLb)(ˉℓiγμγ5ℓj).
(20) The LQ contributions result in [56]
CNP,ij9=−CNP,ij10=−√22GFVtbV∗tsπαe1M2λL3jλL∗2i.
(21) In the model-independent approach, the current
b→sμ+μ− anomalies can be explained by aCNP,229= −CNP,2210 -like contribution, with the permitted range given by [91–93]−0.91(−0.71)⩽CNP,229=−CNP,2210⩽−0.18(−0.35),
(22) at the
2σ (1σ ) level, which in turn sets a constraint onλL∗22λL32 . Furthermore, the LQ contributions tob→sτ+τ− andb→cτˉντ transitions depend on the same productλL∗23λL33 , therefore making a direct correlation between the branching fractionB(Bs→τ+τ−) andRD(∗) .For the
b→sνˉν transitions, both the LQsΦ1 andΦ3 generate tree-level contributions. However, assuming that they have the same mass, their effects are canceled out due to the discrete symmetry in Eq. (6). In addition, this LQ scenario can accommodate the(g−2)μ anomaly [94, 95], once the right-handed interaction termλRfiˉucfℓiΦ†1 is introduced to Eq. (5) [56]. We do not consider such a term in this study. Further details can be found in Ref. [56], where various lepton flavor violating decays of leptons and B meson have also been discussed.Finally, we provide brief comments on direct searches for the LQs at high-energy colliders. Because the LQ contributions to
b→cτˉν transitions only involve the productλL∗23λL33 , searches for the LQs with couplings to the second and third generations are more relevant to our work. At the LHC, both the CMS and ATLAS collaborations have performed searches for such LQs in several channels, e.g.,LQ→tμ [96],LQ→tτ [97],LQ→bτ [97], etc. Current results from the LHC have excluded the LQs with masses below about1TeV [95]. For example, searches for pair-produced scalar LQs decaying into t quark andμ lepton have been performed by the CMS Collaboration, in which a scalar LQ with mass below1420GeV was excluded at 95% CL under the assumption ofB(LQ→tμ)=1 [96]. All these collider constraints depend on the assumption of the total width of the LQ, which involves all the LQ couplingsλLij . To apply the collider constraints to our scenario, one needs to perform a global fit on all the LQ couplings and derive bounds on the total width. Such analysis is beyond the scope of this study. Regarding the scenario with one singlet and one triplet LQ, we refer to Ref. [72] for a more detailed collider analysis. Furthermore, our analysis does not depend on the mass of the LQ, because LQ couplings always appear in the form ofλL∗23λL33/M2 inb→cτˉν transitions, as in Eq. (9). -
In this section, we present our numerical analysis of the LQ effects on the decays considered. After deriving the constraints of the model parameters, we concentrate on the LQ effects on the five
b→cτˉν decays, i.e.,B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν . -
In Table 1, we list the relevant input parameters used in our numerical analysis. Using the theoretical framework described in Section 3, the SM predictions for
B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν decays are given in Table 2. To obtain the theoretical uncertainties, we vary each input parameter within their respective1σ range and add each individual uncertainty in quadrature. Correlations among fit parameters were considered to obtain uncertainties of the transition form factors. In particular, for theΛb→Λcτˉν decay, we follow the treatment of Ref. [75] to obtain the statistical and systematic uncertainties induced by theΛb→Λc transition form factors. From Table 2, the experimental data on the ratiosRD ,RD∗ , andRJ/ψ are found to deviate from the SM predictions by2.31σ ,2.85σ and1.83σ , respectively.Table 1. Input parameters used in our numerical analysis.
observable SM NP exp Ref B(B→Dτˉν) 0.711+0.042−0.041 [0.702,0.991] 0.90±0.24 [95] RD 0.301+0.003−0.003 [0.313,0.400] 0.407±0.039±0.024 [12] B(Bc→ηcτˉν) 0.204+0.024−0.024 [0.188,0.299] – Rηc 0.281+0.035−0.031 [0.263,0.416] – B(B→D∗τˉν) 1.261+0.087−0.085 [1.234,1.788] 1.78±0.16 [95] RD∗ 0.258±0.008 [0.263,0.351] 0.306±0.013±0.007 [12] PτL −0.503±0.013 [−0.516,−0.490] −0.38±0.51+0.21−0.16 [7, 8] PD∗L 0.453±0.012 [0.441,0.465] 0.60±0.08±0.04 [46, 47] B(Bc→J/ψτˉν) 0.398+0.045−0.049 [0.366,0.583] – RJ/ψ 0.248+0.006−0.005 [0.255,0.335] 0.71±0.17±0.18 [17] B(Λb→Λcτˉν) 1.762+0.105−0.104 [1.737,2.457] – RΛc 0.333+0.010−0.010 [0.339,0.451] – Table 2. Predictions for branching fractions (in units of
10−2 ) and ratiosRN of the fiveb→cτˉν decay modes in the SM and the LQ scenario. The entry "––" indicates that no measurement is yet available for the corresponding observable. -
To obtain the permitted ranges of LQ parameters, we impose the experimental constraints in the same manner as in Refs. [99, 100]; i.e., for each point in the parameter space, if the difference between the corresponding theoretical prediction and experimental data is less than
2σ (3σ ) error bar, which is calculated by adding the theoretical and experimental uncertainties in quadrature, this point is regarded as permitted at2σ (3σ ) level.In the LQ scenario introduced in Section 2, the LQ contributions to
b→cτˉν transitions are all controlled by the productλL∗23λL33 . In the following analysis, the couplingsλL23 andλL33 are assumed to be real. After considering the current experimental measurements ofRD(∗) ,RJ/ψ ,PτL(D∗) , andPD∗L , we find that the constraints onλL∗23λL33 are dominated byRD andRD∗ . The permitted ranges ofλL∗23λL33 at2σ level are obtained as follows−2.90<λL∗23λL33<−2.74,or0.03<λL∗23λL33<0.20,
(23) where a common LQ mass
M=1TeV is assumed. The solution with negativeλL∗23λL33 corresponds to the case in which the LQ interactions dominate over the SM contributions. We do not pursue this possibility in the following analysis. For the solution with positiveλL∗23λL33 , the permitted regions of(λL23,λL33) at both2σ and3σ levels are shown in Fig. 1. In this figure, we also show the individual constraint from theD∗ polarization fractionPD∗L , which remains weaker than the ones fromRD(∗) . In addition, the current measurement of theτ polarization fractionPτL inB→D∗τν decay cannot provide any relevant constraint.Figure 1. (color online) Combined constraints on
(λL23,λL33) by allb→cτˉν processes at2σ (black) and3σ (gray) levels. The dark (light) green area indicates the allowed region byPD∗L only at2σ (3σ ).As mentioned in Section 3, the LQ contributions to
b→sτ+τ− andb→cτˉντ depend on the same productλL∗23λL33 . In the case of positiveλL∗23λL33 , we show in Fig. 2 the correlation betweenRD(∗)/RSMD(∗) andB(Bs→τ+τ−) . The LQ effects enhance the branching fraction ofBs→τ+τ− in most of the parameter space. At present, the experimental upper limit6.8×10−3 [101] is far above the SM prediction(7.73±0.49)×10−7 [102]. However, to obtain the2σ experimental range ofRD(∗) , the LQ contributions enhanceB(Bs→τ+τ−) by about 2–3 orders of magnitude compared to the SM prediction, which reaches the expected LHCb sensitivity5×10−4 by the end of Upgrade II [55, 103]. TheB→K(∗)τ+τ− decay may also play an important role in probing the LQ effects. Although the Belle II experiment would improve the current upper limit2.25×10−3 at a 90% confidence level by no more than two orders of magnitude, the proposed FCC-ee collider can yield a few thousand ofB0→K∗0τ+τ− events fromO(1013) Z decays [104].Figure 2. (color online) Correlation between
RD(∗)/RSMD(∗) andB(Bs→τ+τ−) . The black (gray) region denotes the2σ (3σ ) experimental ranges ofRD(∗)/RSMD(∗) . The horizontal dashed and dotted lines correspond to the current LHCb upper limit and the expected sensitivity by the end of LHCb Upgrade II, respectively. The black point indicates the SM prediction. -
Using the constrained parameter space at the
2σ level derived in the last subsection, we present predictions for the fiveb→cτˉν processes. Table 2 shows the SM and LQ predictions for the branching fractionsB and LFU ratios R ofB→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν decays. The LQ predictions include the uncertainties induced by the transition form factors and CKM matrix elements. Considering that the polarization fractionsPτL andPD∗L have already been measured, their SM and LQ predictions are also shown in Table 2. Although the LQ predictions for the branching fractionsB and the LFU ratios R of theBc→ηcτˉν andBc→J/ψτˉν decays lie within the1σ range of their respective SM values, they can be significantly enhanced by LQ effects.We set out to analyze the
q2 distributions of the branching fractionB , LFU ratio R, polarization fractions of theτ lepton (PτL ), and daughter hadron (PD∗L,T ,PJ/ψL,T ,PΛcL ), as well as the lepton forward-backward asymmetryAFB . TheB→Dτν andBc→ηcτˉν decays, both part of the "B→P " transition, have differential observables in the SM and the LQ scenario as shown in Fig. 3. All differential observables of theB→Dτˉν andBc→ηcτˉν decays are similar to each other, while the observables in the latter have larger theoretical uncertainties due to the less preciseBc→ηc transition form factors. Therefore, theB→Dτˉν decay is more sensitive to the LQ effects, with the differential branching fraction largely enhanced, especially nearq2∼7GeV2 . The large difference between the SM and LQ predictions in this kinematic region could, therefore, provide a testable signature of the LQ effects. More interestingly, theq2 distribution of the ratio R in the LQ model is enhanced in the entire kinematic region and does not have overlap with the1σ SM range. In the future, more precise measurements of these distributions are of importance to confirm the existence of a possible NP effect in theB→Dτˉν decay. With regard to the forward-backward asymmetryAFB and theτ -lepton polarization fractionPτL in bothB→Dτˉν andBc→ηcτˉν decays, the LQ predictions are indistinguishable from the ones in SM, because the LQ effects only modify the Wilson coefficientCℓνℓL , which is canceled out exactly in the definitions of these observables (see Eqs. (16) and (17),Fig. 3). This feature is different from the NP scenarios that use scalar or tensor operators to explain theRD(∗) anomaly [58–60].Figure 3. (color online)
q2 distributions of observables inB→Dτˉν (left) andBc→ηcτˉν (right) decays. The black curves (gray band) indicate the SM (LQ) central values with1σ theoretical uncertainty.The
q2 distributions of the observables inB→D∗τˉν andBc→J/ψτˉν decays are shown in Fig. 4. Because both of these two decays belong to "B→V " transition, their differential observables are similar. While the differential branching fractions of these two decays are enhanced in the LQ model, their theoretical uncertainties are larger than the ones in theB→Dτˉν decay. For theq2 distributions of the ratiosRD∗ andRJ/ψ , they are largely enhanced in the entire kinematic region, especially in the largeq2 region. More importantly, although the ranges of theq2 -integrated ratioRD∗,J/ψ in the SM and the LQ scenario overlap at the1σ level, the1σ ranges of the differential ratioRD∗,J/ψ(q2) at largeq2 in the SM and LQ exhiibt significant differences. The increases ofRD∗ andRJ/ψ in the largeq2 region are larger than the one observed inRD . Measurements of the differential ratios in the large dilepton invariant mass region are, therefore, crucial to confirm theRD(∗) anomaly and test the LQ model considered. Similarly to the ones inB→Dτˉν andBc→ηcτˉν decays, the angular distributionsAFB ,PD∗,J/ψL,T , andPτL are likewise not affected by the LQ effects, as shown in Fig. 4.Figure 4. (color online)
q2 distributions of observables inB→D∗τˉν (left) andBc→J/ψτˉν (right) decays. Other captions are the same as in Fig. 3.For the
Λb→Λcτˉν decay, theq2 distributions of the observables are shown in Fig. 5. The situation is similar in theB→D∗τˉν andBc→J/ψτˉν decays. Theq2 distributions of the branching fractionB and the ratioRΛc are greatly enhanced by the LQ effects. In the largeq2 region, the differential ratioRΛc exhibits a deviation between the1σ permitted ranges of the SM and the LQ scenario. With the large numbers ofΛb obtained at the HL-LHC [78], we expect that this prediction could provide helpful information on the LQ effects. For the angular distributions, the LQ effects vanish due to the same reason as in the mesonic decays.Figure 5. (color online)
q2 distributions of observables inΛb→Λcτˉν decay. Other captions are the same as in Fig. 3. -
During the past few years, intriguing hints pointing towards an LFU violation have emerged in the
B→D(∗)τˉν data. Motivated by the recent measurements ofRJ/ψ ,PτL , andPD∗L , we revisited the LQ model proposed in Ref. [56], where two scalar LQs, one of which is aSU(2)L singlet, whereas the other is aSU(2)L triplet, are introduced simultaneously. Taking into account the recent progress on the transition form factors and the most recent experimental data, we obtained constraints on the LQ couplingsλL23 andλL33 . Subsequently, we systematically investigated the LQ effects on the fiveb→cτˉν decays,B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν . In particular, we focused on theq2 distributions of the branching fractions, LFU ratios, and various angular observables. The main results of this study can be summarized as follows:• After considering the
RD andRD∗ data, we obtain the bound on the LQ couplings,0.03<λL∗23λL33<0.20 , at the2σ level. The current measurements ofRJ/ψ ,PτL andPD∗L cannot provide further constraints on the LQ couplings.• The
Bs→τ+τ− decay is strongly correlated withB→D(∗)τˉν . To reproduce the2σ experimental range ofRD(∗) , the LQ effects enhanceB(Bs→τ+τ−) by about 2–3 orders of magnitude compared to the SM prediction and hence reach the expected sensitivity of the LHCb Upgrade II.• The differential branching fractions and LFU ratios are largely enhanced by the LQ effects. Due to their small theoretical uncertainties, the latter provide testable signatures of the LQ model considered, especially in the large dilepton invariant mass squared region. Moreover,
RΛc in the baryonic decayΛb→Λcτˉν has the potential to shed new light on theRD(∗) anomalies.• Because no new operators are generated by the LQ effects, all angular distributions in the LQ model are the same as in the SM. We provide the most recent SM predictions for the
τ -lepton forward-backward asymmetry, theτ , and meson polarization fractions of the fiveb→cτˉν modes. Although precision measurements of these angular distributions are very challenging at the HL-LHC and SuperKEKB, they are crucial for the verification of the LQ scenario investigated in this work.The
q2 distributions of the branching fractions, the LFU ratios, and the various angular observables inb→cτˉν transitions can help to confirm possible NP resolutions of theRD(∗) anomalies and distinguish among the various NP candidates. With the experimental progress expected from the SuperKEKB [54] and the future HL-LHC [78], our predictions for these observables can be further probed in the near future.Note Added. After the completion of this work, the Belle Collaboration announced their results of
RD andRD∗ with a semileptonic tagging method [105,106]. The measured values areRexpD=0.307±0.037(stat.)±0.016(syst.) andRexpD∗=0.283±0.018(stat.)±0.014(syst.) . After including this new measurement, the world averages becomeRavg,2019D=0.337±0.030 andRavg,2019D∗=0.299±0.013 [107]. The deviation of the current world averages from the SM predictions decreases from3.8σ to3.1σ [105]. Because the difference between the new and previous averages is small, our numerical results are expected to remain qualitatively unchanged. For example, the updated bounds onλL∗23λL33 in Eq. (23) becomes−2.88<λL∗23λL33<−2.73 and0.02<λL∗23λL33<0.17 .We thank Xin-Qiang Li for useful discussions.
-
In the presence of NP, the most general effective Hamiltonian for the
b→cτˉν transition can be written as [23, 76]Heff=2√2GFVcb[(1+gL)(ˉcγμPLb)(ˉτγμPLντ)+gR(ˉcγμPRb)(ˉτγμPLντ)+12gS(ˉcb)(ˉτPLντ)+12gP(ˉcγ5b)(ˉτPLντ)+gT(ˉcσμνPLb)(ˉτσμνPLντ)]+h.c..
(24) In this appendix, for completeness, we consider the most general case of NP and provide the helicity amplitudes in the five
b→cτˉν decays,B→D(∗)τˉν ,Bc→ηcτˉν ,Bc→J/ψτˉν , andΛb→Λcτˉν . Explicit expressions of the spinors and polarization vectors used to calculate the helicity amplitudes are also presented. -
To calculate the hadronic helicity amplitudes of
M→Nτˉν in Eq. (13), we work in the M rest frame and follow the notation of Ref. [79]:pμM=(mM,0,0,0),pμN=(EN,0,0,|→pN|),qμ=(q0,0,0,−|→q|),
(25) where
qμ is the four-momentum of the virtual vector boson in the M rest frame, andq0=12mM(m2M−m2N+q2),EN=12mM(m2M+m2N−q2),|→q|=|→pN|=12mM√Q+Q−,Q±=(mM±mN)2−q2.
(26) Subsequently, substituting the momentum into Eq. (A12), the Dirac spinors in the
Λb→Λcτντ decay can be written asuΛb(→pΛb,λΛb)=√2mΛb(χ(→pΛb,λΛb)0),uΛc(→pΛc,λΛc)=(√E+mΛcχ(→pΛc,λΛc)2λΛc√E−mΛcχ(→pΛc,λΛc)),
(27) where
χ(→pΛb,1/2)=χ(→pΛc,1/2)=(1,0)T,χ(→pΛb,−1/2)=χ(→pΛc,−1/2)= (0,1)T. .In the
B→D∗τˉν decay, the polarization vectors of theD∗ meson are given byϵμ(→pD∗,0)=1mD∗(|→pD∗|,0,0,ED∗),ϵμ(→pD∗,±)=1√2(0,±1,i,0).
(28) In all the five
b→cτˉν decays, the polarization vectors for the virtual vector boson W can be written asϵμ(t)=1√q2(q0,0,0,−|→q|),ϵμ(0)=1√q2(|→q|,0,0,−q0),ϵμ(±)=1√2(0,∓1,i,0),
(29) and the orthonormality and completeness relation [108]
∑μϵ∗μ(m)ϵμ(n)=gmn,∑m,nϵμ(m)ϵ∗ν(n)gmn=gμν,m,n∈{t,±,0},
(30) where
gmn=diag(+1,−1,−1,−1) .In the calculation of the leptonic helicity amplitudes, we work in the rest frame of the virtual vector boson W, which is equivalent to the rest frame of the
τ -ˉντ system. Following Ref. [79], we haveqμ=(√q2,0,0,0),pμτ=(Eτ,|→pτ|sinθτ,0,|→pτ|cosθτ),pμˉν=|→pτ|(1,−sinθτ,0,−cosθτ),
(31) where
|→pτ|=√q2v2/2 ,Eτ=|→pτ|+m2τ/√q2 ,v=√1−m2τ/q2 , andθτ denotes the angle between the three-momenta of theτ and the N.The Dirac spinors for
τ andˉντ readuτ(→pτ,λτ)=(√Eτ+mτχ(→pτ,λτ)2λτ√Eτ−mτχ(→pτ,λτ)),vˉντ(−→pτ,12)=√Eν(ξ(−→pτ,12)−ξ(−→pτ,12)),
(32) respectively. Further details are given in Appendix A.2
The polarization vectors of the virtual vector boson in the W rest frame are written as
\tag{A10} \bar{\epsilon}^{\mu}(t) = (1,0,0,0), \quad \bar{\epsilon}^{\mu}(0) = (0,0,0,-1), \quad \bar{\epsilon}^{\mu}(\pm) = \frac{1}{\sqrt{2}}(0,\mp 1,i,0),
(33) which can also be obtained from Eq. (A6) by a Lorentz transformation and satisfy the orthonormality and completeness relation in Eq. (A7).
-
The definitions of the helicity operator
h_{\vec{p}} and its eigenstates are given as follows [109]h_{\vec{p}}\equiv\frac{1}{2}\hat{\vec{p}}\cdot\vec{\sigma},\quad\hat{\vec{p}}\equiv\frac{\vec{p}}{|\vec{p}|},\quad h_{\vec{p}}\; \chi(\vec{p},s) = s\; \chi(\vec{p},s),
where
\vec{p} denotes the momentum of the particle, and\vec{\sigma} = \{\sigma^1,\sigma^2,\sigma^3\} are the Pauli matrices. Eigenstates of the helicity operatorh_{\vec{p}} read\tag{A11} \begin{split} \chi \left(\vec p,\frac{1}{2}\right) =& {\left( \begin{array}{l} \cos \frac{\theta }{2}\\ {{\rm e}^{{\rm i}\phi }}\sin \frac{\theta }{2} \end{array} \right),}\quad{\chi \left(\vec p, - \frac{1}{2}\right) = \left( \begin{array}{l} - {{\rm e}^{ - {\rm i}\phi }}\sin \frac{\theta }{2}\\ \;\;\;\;\cos \frac{\theta }{2} \end{array} \right),}\\ \chi \left( - \vec p,\frac{1}{2}\right) =& {\left( \begin{array}{l} \;\;\;\sin \frac{\theta }{2}\\ - {{\rm e}^{{\rm i}\phi }}\cos \frac{\theta }{2} \end{array} \right),}\quad{\chi \left( - \vec p, - \frac{1}{2}\right) = \left( \begin{array}{l} {{\rm e}^{ - {\rm i}\phi }}\cos \frac{\theta }{2}\\ \;\;\;\;\sin \frac{\theta }{2} \end{array} \right),} \end{split}
(34) for the normalized momentum
\hat{\vec{p}} = \{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\} .Using these eigenstates, the solution of Dirac equation
(\gamma^{\mu} p_{\mu}-m)u(\vec{p},s) = 0 in Dirac representation can be written as\tag{A12} u(\vec p,s) = \left( \begin{array}{l} \;\sqrt {E + m} \;\chi (\vec p,s)\\ 2s\sqrt {E - m} \;\chi (\vec p,s) \end{array} \right).
(35) Further, the spinor for the antiparticle can be obtained by
v(\vec{p},s)\equiv C\bar{u}(\vec{p},s)^{T} = i\gamma^0\gamma^2 \bar{u}(\vec{p},s)^{T} ②, whose explicit expression reads\tag{A13}v(\vec p,s) = \left( \begin{array}{l} \; \sqrt {E - m} \;\xi (\vec p,s)\\ - 2s\sqrt {E + m} \;\xi (\vec p,s) \end{array} \right),
(36) where
\xi(\vec{p},s) = \chi(\vec{p},-s) and\xi(\vec{p},s) satisfiesh_{\vec{p}}\; \xi(\vec{p},s) = -s\; \xi(\vec{p},s) .The spinors in Weyl representation read
\tag{A14} \begin{split} u_W(\vec{p},s) =& \left( \begin{array}{l} \sqrt{E-2s\left\vert{{\vec{p}}}\right\vert}\; \chi(\vec{p},s)\\ \sqrt{E+2s\left\vert{{\vec{p}}}\right\vert} \; \chi(\vec{p},s) \end{array} \right), \\ v_W(\vec{p},s) =& \left( \begin{array}{l} -2s\sqrt{E+2s\left\vert{{\vec{p}}}\right\vert}\; \xi(\vec{p},s)\\ \hphantom{-} 2s\sqrt{E-2s\left\vert{{\vec{p}}}\right\vert} \; \xi(\vec{p},s) \end{array} \right). \end{split}
(37) They can also be obtained from the Dirac representation by the relation
u_{W}(\vec{p},s) = X u(\vec{p},s) with the transformation matrixX = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{l}} 1&{ - 1}\\ 1&1 \end{array}} \right).
In the
\tau -\bar\nu_\tau center-of-mass frame, we emphasize that if the\tau spinor is specified asu(\vec{p},s) in the leptonic helicity amplitude, then the\bar\nu_\tau spinor has the formv(-\vec{p},s) , as in Eq. (A9). All calculations in our work are depictec in Dirac representation. -
The leptonic helicity amplitudes in Eq. (13) are defined as [79]
\tag{A15} \begin{split} L^{SP}_{\lambda_\tau} = &\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau} (1-\gamma_5)\nu_\tau\left| 0\right\rangle = \bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})(1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau,1/2), \\ L^{VA}_{\lambda_\tau,\lambda_W} = &\bar{\epsilon}^{\mu} (\lambda_W)\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau}\gamma_\mu (1-\gamma_5)\nu_\tau\left| 0\right\rangle \\=& \bar{\epsilon}^{\mu}(\lambda_W)\bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})\gamma_\mu (1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau, 1/2) , \\ L^{T}_{\lambda_\tau,\lambda_{W_1} ,\lambda_{W_2}} = &-i\bar{\epsilon}^{\mu} (\lambda_{W_1})\bar{\epsilon}^{\nu} (\lambda_{W_2})\left\langle \tau\bar{\nu}_\tau\right|\bar{\tau}\sigma_{\mu \nu} (1-\gamma_5)\nu_\tau\left| 0\right\rangle \\ = &-i\bar{\epsilon}^{\mu} (\lambda_{W_1})\bar{\epsilon}^{\nu} (\lambda_{W_2})\bar{u}_\tau(\vec{p}_\tau,\lambda_{\tau})\sigma_{\mu \nu} (1-\gamma_5)v_{\bar{\nu}_\tau}(-\vec{p}_\tau, 1/2), \end{split}
(38) Obtaining
L^T_{\lambda_\tau,\lambda_{W_1} ,\lambda_{W_2}} = -L^T_{\lambda_\tau ,\lambda_{W_2},\lambda_{W_1}} is straightforward. The non-zero leptonic helicity amplitudes read\tag{A16} \begin{split} L^{SP}_{1/2}& = 2\sqrt{q^2}v,\\ L^{VA}_{1/2,t}& = 2m_\tau v,\\ L^{VA}_{1/2,0}& = -2m_\tau v \cos\theta_\tau ,\\ L^{VA}_{-1/2,0}& = 2\sqrt{q^2}v \sin\theta_\tau,\\ L^{VA}_{1/2,\pm}& = \mp\sqrt{2} m_\tau v \sin\theta_\tau,\\ L^{VA}_{-1/2,\pm}& = \sqrt{2 q^2}v (-1 \mp \cos\theta_\tau),\\ L^{T}_{1/2,0,\pm} & = \pm L^{T}_{1/2,\pm,t} = \sqrt{2 q^2}v\sin\theta_\tau, \\ L^{T}_{1/2,t,0}& = L^{T}_{1/2,+,-} = -2\sqrt{q^2}v \cos\theta_\tau,\\ L^{T}_{-1/2,0,\pm}& = \pm L^{T}_{-1/2,\pm,t} = \sqrt{2}m_\tau v (\pm 1+ \cos\theta_\tau), \\ L^{T}_{-1/2,t,0}& = L^{T}_{-1/2,+,-} = 2m_\tau v \sin\theta_\tau. \end{split}
(39) -
The hadronic helicity amplitudes
M \to N are defined as\tag{A17} \begin{split} H^S_{\lambda_M,\lambda_N}& = \left\langle N(\lambda_N)\right|\bar{c} b\left| M(\lambda_M)\right\rangle,\\ H^P_{\lambda_M,\lambda_N}& = \left\langle N(\lambda_N)\right|\bar{c}\gamma_5 b\left| M(\lambda_M)\right\rangle,\\ H^V_{\lambda_M,\lambda_N,\lambda_{W}}& = \,\epsilon_\mu^{*}(\lambda_{W})\left\langle N(\lambda_N)\right|\bar{c}\gamma^\mu b\left| M(\lambda_M)\right\rangle, \\ H^A_{\lambda_M,\lambda_N,\lambda_{W}}& = \,\epsilon_\mu^{*}(\lambda_{W})\left\langle N(\lambda_N)\right|\bar{c}\gamma^\mu\gamma_5 b\left| M(\lambda_M)\right\rangle,\\ H^{T_1,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = i\epsilon_\mu^{*}(\lambda_{W_1})\epsilon_\nu^{*}(\lambda_{W_2})\left\langle N(\lambda_N)\right|\bar{c}\sigma^{\mu \nu} b\left| M(\lambda_M)\right\rangle, \\ H^{T_2,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = i\epsilon_\mu^{*}(\lambda_{W_1})\epsilon_\nu^{*}(\lambda_{W_2})\left\langle N(\lambda_N)\right|\bar{c}\sigma_{\mu \nu}\gamma_5 b\left| M(\lambda_M)\right\rangle, \end{split}
(40) and
\tag{A18} \begin{split} H^{SP}_{\lambda_M,\lambda_N}& = g_S H^S_{\lambda_M,\lambda_N}+g_P H^P_{\lambda_M,\lambda_N}, \\ H^{VA}_{\lambda_M,\lambda_N,\lambda_{W}}& = (1+g_L+g_R)H^V_{\lambda_N,\lambda_{W}}-(1+g_L-g_R)H^A_{\lambda_M,\lambda_N,\lambda_{W}}, \\ H^{T,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}& = g_TH^{T_1,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}}-g_T H^{T_2,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}} , \end{split}
(41) H^{T,\lambda_M}_{\lambda_N,\lambda_{W_1} ,\lambda_{W_2}} = -H^{T,\lambda_M}_{\lambda_N,\lambda_{W_2} ,\lambda_{W_1}} is easily obtained. The amplitudesH_{\lambda_N,\lambda_{W_1},\lambda_{W_2}}^{T_1,\lambda_M} andH_{\lambda_N,\lambda_{W_1},\lambda_{W_2}}^{T_2,\lambda_M} are connected by the relation\sigma_{\mu \nu}\gamma_{5} = -(i/2)\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta} , where\epsilon^{0123} = -1 . -
The hadronic matrix elements for the
B\to D transition can be parameterized in terms of form factorsF_{+,0,T} [110, 111]. In the BGL parametrization, the form factorsF_{+,0} can be written as expressions ofa_n^+ anda_n^0 [13],\tag{B1} \begin{split}F_{+}(z) =& \frac{1}{P_{+}(z)\phi_{+}(z,{\cal N})}\sum\limits_{n = 0}^{\infty} a_{n}^{+}z^n(w,{\cal N}), \\ F_{0}(z) =& \frac{1}{P_{0}(z)\phi_{0}(z,{\cal N})}\sum\limits_{n = 0}^{\infty} a_{n}^{0}z^n(w,{\cal N}), \end{split}
(42) where
r = m_D / m_B ,{\cal N} = (1+r)/(2\sqrt{r}) ,w = (m_{B}^2+m_{D}^2-q^{2})/(2m_{B}m_{D}) ,z(w,{\cal N}) = (\sqrt{1+w}-\sqrt{2{\cal N}})/(\sqrt{1+w}+\sqrt{2{\cal N}}) , andF_+(0) = F_0(0) . The values of the fit parameters are taken from Ref. [13]. Expressions of the tensor form factorF_T can be found in Ref. [110].For
B\to D^* transition, the relevant form factors\{V,A_{0,1,2}\} can be written in terms of the form factors\{h_V,h_{A_{1,2,3}}\} in the heavy quark effective theory (HQET) [110],\tag{B2} \begin{split}V(q^2) =& { m_+ \over 2\sqrt{m_B m_{D^{*}}} } \, h_V(w), \\ A_0(q^2) =& { 1 \over 2\sqrt{m_B m_{D^{*}}} } \left[ { m_+^2 \!-\! q^2 \over 2m_{D^{*}} } \, h_{A_1}(w) -\! { m_+m_- + q^2 \over 2m_B } \, h_{A_2}(w) -\! {m_+m_– q^2 \over 2m_{D^{*}} } \, h_{A_3}(w) \right] , \\ A_1(q^2) =& { m_+^2 - q^2 \over 2\sqrt{m_B m_{D^{*}}} m_+ } \, h_{A_1}(w), \\ A_2(q^2) =& {m_+ \over 2\sqrt{m_B m_{D^{*}}} } \left[ h_{A_3}(w) + { m_{D^{*}} \over m_B } h_{A_2}(w) \right] , \end{split}
(43) where
m_\pm = m_B \pm m_{D^*} andw = (m_B^2+m_{D^*}^2-q^2)/2m_Bm_{D^*} . In the CLN parametrization, the HQET form factors can be expressed as [89]\begin{split} \frac{h_V(w)}{h_{A_1}(w) } = R_1(w) , \quad \frac{h_{A_2}(w)}{h_{A_1}(w)} = { R_2(w)-R_3(w) \over 2\,r_{D^{*}} }, \end{split}
\tag{B3}\begin{split} \frac{h_{A_3}(w)}{h_{A_1}(w)} = { R_2(w)+R_3(w) \over 2 } , \end{split}
(44) with
r = m_{D^*}/m_B . Numerically we obtain,\tag{B4} \begin{split} h_{A_1}(w) =& h_{A_1}(1)[1-8\rho_{D^*}^2 z + (53 \rho_{D^*}^2 -15)z^2 - (231 \rho_{D^*}^2 -91)z^3],\\ R_1(w) =& R_1(1)-0.12(w-1)+0.05(w-1)^2,\\ R_2(w) =& R_2(1)+0.11(w-1)-0.06(w-1)^2,\\ R_3(w) =& 1.22 -0.052(w-1) +0.026(w-1)^2, \end{split}
(45) with
z = (\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2}) . The fit parametersR_{1}(1) ,R_{2}(1) ,h_{A_1}(1) and\rho_{D^{*}}^2 are taken from Ref. [14]. Expressions of the tensor form factorsT_{1,2,3} can be found in Ref. [110].The
\Lambda_b\rightarrow\Lambda_c hadronic matrix elements can be written in terms of ten helicity form factors\{F_{0,+,\perp},G_{0,+,\perp},h_{+,\perp},\widetilde{h}_{+,\perp}\} [75, 76]. Following Ref. [75], the lattice calculations are fitted to two (Bourrely-Caprini-Lellouch) BCL z-parametrizations. In the so called "nominal" fit, a form factor f reduces to the form\tag{B5} f(q^2) = \frac{1}{1-q^2/(m_{\rm pole}^f)^2} \big[ a_0^f + a_1^f\:z^f(q^2) \big],
(46) while a form factor f in the higher-order fit is given by
\tag{B6} \begin{split} f_{\rm HO}(q^2) =& \frac{1}{1-q^2/(m_{\rm pole}^f)^2} \left\{ a_{0,{\rm HO}}^f + a_{1,{\rm HO}}^f\:z^f(q^2) \right.\\&\left.+ a_{2,{\rm HO}}^f\:[z^f(q^2)]^2 \right\},\end{split}
(47) where
t_0 = (m_{\Lambda_b} - m_{\Lambda_c})^2 ,t_+^f = (m_{\rm pole}^f)^2 , andz^f(q^2) = \left.\left(\sqrt{t_+^f-q^2}-\sqrt{t_+^f-t_0}\right)\right/ \left(\sqrt{t_+^f-q^2}+\sqrt{t_+^f-t_0}\right) .The values of the fit parameters and all the pole masses are taken from Ref. [76].In addition, the form factors for
B_c \to J/\psi\ell\bar{\nu_\ell} andB_c \to \eta_c\ell\bar{\nu_\ell} decays are taken form the results in the Covariant Light-Front Approach in Ref. [18]. -
Because similar expressions hold for the
B \to D \tau \bar\nu andB_c \to \eta_c \tau \bar\nu decays, we only provide the theoretical formulae of the former. Using the form factors in Appendix B, the non-zero helicity amplitudes for theB \to D \tau \bar\nu decay in Eq. (A18) can be written as\begin{split} H_{0}^{VA}(q^2) & = (1+g_{\rm L}+g_{\rm R})\sqrt{\frac{Q_+Q_-}{q^2}} F_{+}(q^2) , \end{split}
\tag{C1} \begin{split} H_{t}^{VA}(q^2) & = (1+g_{\rm L}+g_{\rm R})\frac{m_B^2-m_D^2}{\sqrt{q^2}} F_0(q^2) ,\\ H^{SP}(q^2) & = g_S{m_B^2-m_D^2 \over m_b-m_c} F_0(q^2) , \\ H_{-,+}^T(q^2) & = H_{t,0}^T(q^2) = g_T{\sqrt{Q_+Q_-} \over m_B+m_D} F_T(q^2) . \end{split}
(48) Subsequently, the differential decay width in Eq. (11) and angular observables in Eq. (16) and (17) are obtained
\tag{C2} \begin{split} \frac{{\rm d}\Gamma}{{\rm d}q^2} = &\frac{N_D}{2} \biggl[\frac{3m_{\tau}^2}{q^2}|H^{VA}_{t}|^2+ \Bigl(2+\frac{m_\tau^2}{q^2}\Bigr) |H^{VA}_{0}|^2+3|H^{SP}|^2 +16\Big(1+\frac{2m_\tau^2}{q^2}\Big) \\ &\times|H^{T}_{t,0}|^2 +\frac{6m_\tau}{\sqrt{q^2}} \Re[H^{SP}H^{VA*}_{t}]+\frac{24m_\tau}{\sqrt{q^2}} \Re[H^{T}_{t,0}H^{VA*}_{0}] \biggr], \end{split}
(49) \tag{C3} \frac{{\rm d}A_{\rm FB}}{{\rm d} q^2} = \frac{3N_D}{2}\Re \biggl[\biggl(4H^{T*}_{t,0}+ \frac{m_\tau}{\sqrt{q^2}}H_0^{VA*}\biggr) \biggl(H^{SP} + \frac{m_\tau}{\sqrt{q^2}}H_t^{VA} \biggr) \biggr],
(50) \tag{C4} \begin{split} \frac{{\rm d}P_{L}^{\tau}}{{\rm d} q^2} = &\frac{1}{\rm{d}\Gamma/{\rm d}q^2}\frac{N_D}{2} \biggl[\frac{3m_\tau^2}{q^2}|H^{VA}_{t}|^2+ \left(\frac{m_\tau^2}{q^2}-2 \right)|H^{VA}_{0}|^2 +3|H^{SP}|^2\\&+16 \left(1-\frac{2m_\tau^2}{q^2} \right)|H^{T}_{t,0}|^2 +\frac{6 m_\tau}{\sqrt{q^2}} \Re[H^{SP}H^{VA*}_{t}] \\ &-\frac{8 m_\tau}{\sqrt{q^2}}\Re[H^{T}_{t,0}H^{VA*}_{0}] \biggr] , \end{split}
(51) with
\tag{C5} N_D = \frac{G_{F}^{2}|V_{cb}|^2}{192\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{B}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2.
(52) -
Because similar expressions hold for the
B \to D^* \tau \bar\nu andB_c \to J/\psi \tau \bar\nu decays, only theoretical formulae of the former are provided in this subsection. Using the form factors in Appendix B, the non-zero helicity amplitudes for theB \to D^* \tau \bar\nu decay in Eq. (A18) can be written as\tag{C6} \begin{split} H_0^{SP}(q^2) & = -g_P{ \sqrt{Q_+Q_-} \over m_b+m_c } A_0(q^2) , \\ H^{VA}_{\pm,\pm}(q^2) & = -(1+g_L-g_R) m_+ A_1(q^2) \pm (1+g_L+g_R){ \sqrt{Q_+Q_-} \over m_+ } V(q^2) , \\ H^{VA}_{0,t}(q^2) & = -(1+g_L-g_R){ \sqrt{Q_+Q_-} \over \sqrt{q^2} } A_0(q^2) , \\ H^{VA}_{0,0}(q^2) & = \frac{(1+g_L-g_R)}{2m_{D^*}\sqrt{q^2}}\biggl[- m_+(m_+ m_- -q^2) A_1(q^2) +{Q_+Q_- \over m_+ } A_2(q^2)\biggr] , \\ H^{T}_{\pm,\pm,t}(q^2) = &\pm H^{T}_{\pm,\pm,0}(q^2) = \frac{g_{T}}{\sqrt{q^{2}}} \biggl[\mp\sqrt{Q_+Q_-} T_1(q^2)-m_+m_- T_2(q^2) \biggr] , \\ H^{T}_{0,t,0}(q^2) = &\; H^{T}_{0,-,+}(q^2) = \frac{g_T}{2m_{D^{*}}}\biggl[-(m_B^2+3m_{D^{*}}^2-q^2) T_2(q^2)\\&+\frac{Q_+Q_-}{m_+ m_-}T_3(q^2) \biggr] , \end{split}
(53) with
m_\pm = m_B \pm m_{D^*} . Then, the differential decay width in Eq. (11) and the angular observables in Eq. (16) and (17) are obtained, respectively, as\tag{C7} \begin{split} \frac{{\rm d}\Gamma}{{\rm d} q^2} = & N_{D^*} \biggl[\frac{3m_\tau^2}{2q^2} |H^{VA}_{0,t}|^2+ \Bigl( 1+\frac{m_\tau^2}{2q^2} \Bigr)(|H^{VA}_{-,-}|^2+|H^{VA}_{0,0}|^2+|H^{VA}_{+,+}|^2) \\ &+\frac{3}{2}|H^{SP}_{0}|^2 +8 \Bigl( 1+\frac{2 m_\tau^2}{q^2} \Bigr)(|H^{T}_{0,t,0}|^2+|H^{T}_{+,+,t}|^2+|H^{T}_{-,-,t}|^2) \\ &+ \frac{3m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0} H^{VA*}_{0,t}]+\frac{12 m_\tau}{\sqrt{q^2}}(\Re[H^{T}_{0,t,0}H^{VA*}_{0,0}-H^{T}_{+,+,t}H^{VA*}_{+,+}\\&-H^{T}_{-,-,t}H^{VA*}_{-,-}])\biggr], \end{split}
(54) \tag{C8} \begin{split} \frac{{\rm d}A_{\rm FB}}{{\rm d} q^2} = & \frac{3N_{D^*}}{4} \biggl[\frac{2m_\tau^2}{q^2} \Re[H^{VA}_{0,0}H^{VA*}_{0,t}]-|H^{VA}_{-,-}|^2+|H^{VA}_{+,+}|^2+8\Re[H_0^{SP} H_{0,t,0}^{T*}] \\ & +\frac{16m_\tau^2}{q^2}(|H^{T}_{+,+,t}|^2-|H^{T}_{-,-,t}|^2)+\frac{2m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,0}] \\ &+\frac{8m_\tau}{\sqrt{q^2}}\Re[H^{T}_{0,t,0}H^{VA*}_{0,t}+H^{T}_{-,-,t}H^{VA*}_{-,-}-H^{T}_{+,+,t}H^{VA*}_{+,+}] \biggr], \end{split}
(55) \tag{C9} \begin{split} \frac{{\rm d}P_{L}^{{D^*}}}{{\rm d} q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d}q^2}\frac{N_{D^*}}{2}\left[\frac{3m_\tau^2}{q^2}|H^{VA}_{0,t}|^2+ \left( 2 + \frac{m_\tau^2}{q^2} \right) |H^{VA}_{0,0}|^2 +3|H^{SP}_{0}|^2\right. \\&+16 \left(1+\frac{2m_\tau^2}{q^2} \right)|H^{T}_{0,t,0}|^2 +\frac{6 m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,t}]\\ &\left.+\frac{24 m_\tau}{\sqrt{q^2}}\Re[H^{T}_{0,t,0}H^{VA*}_{0,0}] \right], \end{split}
(56) \tag{C10} \begin{split} \frac{{\rm d} P_L^\tau}{{\rm d}q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d}q^2}\frac{N_{D^*}}{2}\biggl[\frac{3m_\tau^2}{q^2}|H^{VA}_{0,t}|^2+\Bigl(\frac{m_\tau^2}{q^2}-2\Bigr)(|H^{VA}_{+,+}|^2+|H^{VA}_{0,0}|^2+|H^{VA}_{-,-}|^2 ) \\ &+3|H^{SP}_{0}|^2+\frac{6 m_\tau}{\sqrt{q^2}}\Re[H^{SP}_{0}H^{VA*}_{0,t}]+16 \Bigl(1-\frac{2m_\tau^2}{q^2} \Bigr)(|H^{T}_{0,t,0}|^2\\ &+|H^{T}_{-,-,t}|^2+|H^{T}_{+,+,t}|^2) +\frac{8m_\tau}{\sqrt{q^2}}\Re[H^{T}_{-,-,t}H^{VA*}_{-,-}\\ &+H^{T}_{+,+,t}H^{VA*}_{+,+}-H^{T}_{0,t,0}H^{VA*}_{0,0}] \biggr], \end{split}
(57) \tag{C11} \begin{split} \frac{{\rm d} P_{T}^{{D^*}}}{ {\rm d} q^2} =& \frac{1}{{\rm d}\Gamma/{\rm d} q^2}\frac{N_{D^*}}{2}\biggl[\Bigl(2 + \frac{m_\tau^2}{q^2} \Bigr)(|H^{VA}_{+,+}|^2-|H^{VA}_{-,-}|^2) \\ &+16(1+\frac{2m_\tau^2}{q^2})(|H^{T}_{+,+,t}|^2-|H^{T}_{-,-,t}|^2)\\ &+\frac{24m_\tau}{\sqrt{q^2}}\Re[H^{T}_{-,-,t}H^{VA*}_{-,-}-H^{T}_{+,+,t}H^{VA*}_{+,+}] \biggr], \end{split}
(58) with
\tag{C12} N_{D^*} = \frac{G_{F}^{2}|V_{cb}|^2}{192\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{B}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2.
(59) -
Using the transition form factors in Appendix B, the helicity amplitudes for the
\Lambda_b \to \Lambda_c decay in Eq. (A18) can be written as\tag{C13} \begin{split} H^{SP}_{\pm 1/2, \pm 1/2} = &F_0g_S \frac{\sqrt{Q_+}}{m_b-m_c} m_- \mp G_0g_P\frac{\sqrt{Q_-}}{m_b+m_c} m_+, \\ H^{VA}_{\pm 1/2, \pm 1/2,t} = &F_0(1+g_L+g_R)\frac{\sqrt{Q_+}}{\sqrt{q^2}} m_- \mp G_0(1+g_L-g_R)\frac{\sqrt{Q_-}}{\sqrt{q^2}} m_+ , \\ H^{VA}_{\pm 1/2, \pm 1/2,0} = &F_+ (1+g_L+g_R)\frac{\sqrt{Q_-}}{\sqrt{q^2}} m_+ \mp G_+ (1+g_L-g_R)\frac{\sqrt{Q_+}}{\sqrt{q^2}} m_-, \\ H^{VA}_{\mp 1/2,\pm 1/2,\pm} = &F_\perp (1+g_L+g_R)\sqrt{2Q_-} \mp G_\perp (1+g_L-g_R)\sqrt{2Q_+} , \\ H^{T,\pm 1/2}_{\pm 1/2,t,0} = &H^{T,\pm 1/2}_{\pm 1/2,-,+} = g_T\Big[h_+\sqrt{Q_-}\pm \widetilde{h}_+\sqrt{Q_+}\Big] , \\ H^{T,\pm 1/2}_{\mp 1/2,t,\mp} = &\mp H^{T,\pm1/2}_{\mp 1/2,0,\mp} = g_T\frac{\sqrt{2}}{\sqrt{q^2}}\Big[h_\perp m_+\sqrt{Q_-}\mp\widetilde{h}_\perp m_- \sqrt{Q_+}\Big] , \end{split}
(60) with
m_\pm = m_{\Lambda_b}\pm m_{\Lambda_c} . Thus, the differential decay width in Eq. (11) can be written as\tag{C14} \begin{split} \frac{{ \rm d}\Gamma}{{\rm d} q^2} = & N_{\Lambda_c} \biggl[ A_1^{VA}+\frac{m_\tau^2}{2q^2}A_2^{VA} +\frac{3}{2}A_3^{SP}+8\Big(1+\frac{2m_\tau^2}{q^2}\Big)A_4^{T}\\&+\frac{3m_\tau}{\sqrt{q^2}} (A_5^{VA-SP}+ 4 A_6^{VA-T}) \biggr], \end{split}
(61) with
\tag{C15} \begin{split} N_{\Lambda_c} & = \frac{G_{F}^{2}|V_{cb}|^2}{384\pi^{3}}\frac{q^2\sqrt{Q_+ Q_-}}{m_{\Lambda_b}^3}\Big(1-\frac{m_\tau^2}{q^2}\Big)^2, \\ A_1^{VA} = &|H^{VA}_{-1/2,1/2,+}|^2+ \sum |H^{VA}_{s,s,0}|^2+|H^{VA}_{1/2,-1/2,-}|^2 , \\ A_2^{VA} = & A_1^{VA}+3 \sum |H^{VA}_{s,s,t}|^2, \\ A_3^{SP} = &\sum |H^{SP}_{s,s}|^2 , \\ A_4^{T} = &\sum |H^{T,s}_{s,t,0}|^2+|H^{T,1/2}_{-1/2,t,-}|^2+|H^{T,-1/2}_{1/2,t,+}|^2, \\ A_5^{VA-SP} = &\sum \Re[H^{SP*}_{s,s} H^{VA}_{s,s,t}] , \\ A_6^{VA-T} = &\sum \Re[H^{VA*}_{s,s,0}H^{T,s}_{s,t,0}] + \Re[H^{VA*}_{-1/2,1/2,+}H^{T,-1/2}_{1/2,t,+}]\\&+ \Re[H^{VA*}_{1/2,-1/2,-}H^{T,1/2}_{-1/2,t,-}], \end{split}
(62) where
\sum depicts the summation overs = \pm 1/2 . For the forward-backward asymmetry in Eq. (16), we obtain\tag{C16} \begin{split} \frac{ {\rm d} A_{\rm FB}} { {\rm d} q^2} =& \frac{ N_{\Lambda_c} }{{\rm d} \Gamma / {\rm d} q^2}\frac{3}{4} \biggl[ B_1^{VA}+\frac{2m_\tau^2}{q^2} \bigl(B_2^{VA}+8 B_3^T \bigr)\\&+ \frac{2m_\tau}{\sqrt{q^2}} \bigl(B_4^{VA-SP} +4 B_5^{VA-T} \bigr) + 8B_6^{SP-T} \biggr], \end{split}
(63) where
\tag{C17} \begin{split} B_1^{VA} = &\; |H^{VA}_{-1/2,1/2,+}|^2-|H^{VA}_{1/2,-1/2,-}|^2 , \\ B_2^{VA} = &\; \sum \Re[H^{VA*}_{s,s,t}H^{VA}_{s,s,0}] , \\ B_3^{T} = &\; |H^{T,-1/2}_{1/2,t,+}|^2-|H^{T,1/2}_{-1/2,t,-}|^2, \\ B_4^{VA-SP} = & \sum \Re[H^{SP*}_{s,s}H^{VA}_{s,s,0}], \\ B_5^{VA-T} = & \sum\Re[H^{VA*}_{s,s,t} H^{T,s}_{s,t,0}]+\Re[H^{VA*}_{-1/2,1/2,+} H^{T,-1/2}_{1/2,t,+}]\\&-\Re[H^{VA*}_{1/2,-1/2,-} H^{T,1/2}_{-1/2,t,-}], \\ B_6^{SP-T} = &\; \sum\Re[H^{SP*}_{s,s}H^{T,s}_{s,t,0}]. \end{split}
(64) For the
\Lambda_c longitudinal polarization fraction in Eq. (17), we obtain\tag{C18} \begin{split} \frac{{\rm d}P_{L}^{\Lambda_c}}{{\rm d} q^2} = &\frac{N_{\Lambda_c}}{{\rm d}\Gamma/{\rm d}q^2}\frac{1}{2}\biggl[2 C_1^{VA}+\frac{m_\tau^2}{q^2}C_2^{VA}+3C_3^{SP} \\ &+16 \Bigl(1 + \frac{2m_\tau^2}{q^2} \Bigr)C_4^{T} +6\frac{m_\tau}{\sqrt{q^2}} \bigl(C_5^{VA-SP}+ 4C_6^{VA-T} \bigr) \biggr], \end{split}
(65) where
\tag{C19} \begin{split} C_1^{VA} = &|H^{VA}_{1/2,1/2,0}|^2-|H^{VA}_{-1/2,-1/2,0}|^2+|H^{VA}_{-1/2,1/2,+}|^2-|H^{VA}_{1/2,-1/2,-}|^2, \\ C_2^{VA} = &C_1^{VA} -3|H^{VA}_{-1/2,-1/2,t}|^2 +3|H^{VA}_{1/2,1/2,t}|^2, \\ C_3^{SP} = &|H^{SP}_{1/2,1/2}|^2-|H^{SP}_{-1/2,-1/2}|^2, \\ C_4^{T} = &\sum 2s|H^{T,s}_{s,t,0}|^2+|H^{T,-1/2}_{1/2,t,+}|^2 -|H^{T,1/2}_{-1/2,t,-}|^2, \\ C_5^{VA-SP} = & \sum 2s\Re[H^{SP*}_{s,s}H^{VA}_{s,s,t}], \\ C_6^{VA-T} = &\sum 2s\Re\big[H^{T,s}_{s,t,0}H^{VA*}_{s,s,0}\big]+\Re\big[H^{T,-1/2}_{1/2,t,+}H^{VA*}_{-1/2,1/2,+}\big] \\&-\Re\big[H^{T,1/2}_{-1/2,t,-}H^{VA*}_{1/2,-1/2,-}\big]. \end{split}
(66) For the
\tau -lepton longitudinal polarization fraction, we obtain\tag{C20} \begin{split} \frac{{\rm d} P_L^\tau}{{\rm d} q^2} = &\frac{N_{\Lambda_c}}{{\rm d}\Gamma/{\rm d}q^2}\frac{1}{2}\biggl[-2 D_1^{VA}+\frac{m_\tau^2}{q^2}D_2^{VA}+3D_3^{SP} \\ &+16 \Bigl(1-\frac{2m_\tau^2}{q^2} \Bigr)D_4^{T}+\frac{m_\tau}{\sqrt{q^2}} \bigl( 6D_5^{VA-SP}-8D_6^{VA-T} \bigr) \biggr], \end{split}
(67) where
\tag{C21}\begin{split} D_1^{VA} = & \sum |H^{VA}_{s,s,0}|^2 + |H^{VA}_{-1/2,1/2,+}|^2 + |H^{VA}_{1/2,-1/2,-}|^2, \\ D_2^{VA} = & D_1^{VA}+3\sum |H^{VA}_{s,s,t}|^2, \\ D_3^{SP} = & \sum |H^{SP}_{s,s}|^2, \\ D_4^{T} = & \sum |H^{T,s}_{s,t,0}|^2+|H^{T,-1/2}_{1/2,t,+}|^2 + |H^{T,1/2}_{-1/2,t,-}|^2 , \\ D_5^{VA-SP} = & \sum \Re[H^{SP*}_{s, s}H^{VA}_{ s, s,t}], \\ D_6^{VA-T} = &\sum \Re[H^{T,s}_{s,t,0}H^{VA*}_{s,s,0}] +\Re[H^{T,-1/2}_{1/2,t,+}H^{VA*}_{-1/2,1/2,+}]\\&+\Re[H^{T,1/2}_{-1/2,t,-}H^{VA*}_{1/2,-1/2,-}]. \end{split}
(68)
Phenomenology of b→cτˉν decays in a scalar leptoquark model
- Received Date: 2019-03-21
- Available Online: 2019-08-01
Abstract: During the past few years, signs of lepton flavor universality (LFU) violation have been observed in