$b\to c\tau\nu$ Transitions in the Standard Model Effective Field Theory

The $R(D^{(\ast)})$ anomalies observed in $B\to D^{(\ast)}\tau\nu$ decays have attracted much attention in recent years. In this paper, we study the $B\to D^{(\ast)}\tau\nu$, $\Lambda_b\to\Lambda_c\tau\nu$, $B_c\to (J/\psi,\,\eta_c)\tau\nu$, $B\to X_c\tau\nu$, and $B_c\to\tau\nu$ decays, all being mediated by the same quark-level $b\to c\tau\nu$ transition, in the Standard Model Effective Field Theory. The most relevant dimension-six operators for these processes are $Q_{lq}^{(3)}$, $Q_{ledq}$, $Q^{(1)}_{lequ}$, and $Q^{(3)}_{lequ}$ in the Warsaw basis. Evolution of the corresponding Wilson coefficients from the new physics scale $\Lambda=1$~TeV down to the characteristic scale $\mu_b\simeq m_b$ is performed at three-loop in QCD and one-loop in EW/QED. It is found that, after taking into account the constraint ${\cal B}(B_c\to\tau\nu)\lesssim 10\%$, a single $\left[C_{lq}^{(3)}\right]_{3323}(\Lambda)$ or $\left[C^{(3)}_{lequ}\right]_{3332}(\Lambda)$ can still be used to resolve the $R(D^{(\ast)})$ anomalies at $1\sigma$, while a single $\left[C^{(1)}_{lequ}\right]_{3332}(\Lambda)$ is already ruled out by the measured $R(D^{(\ast)})$ at more than $3\sigma$. By minimizing the $\chi^2(C_i)$ function constructed based on the current data on $R(D)$, $R(D^\ast)$, $P_\tau(D^\ast)$, $R(J/\psi)$, and $R(X_c)$, we obtain eleven most trustworthy scenarios, each of which can provide a good explanation of the $R(D^{(\ast)})$ anomalies at $1\sigma$. To further discriminate these different scenarios, we predict thirty-one observables associated with the processes considered under each NP scenario. It is found that most of the scenarios can be differentiated from each other by using these observables and their correlations.

On the other hand, in view of the absence (so far) of any clear signal of new particles at the LHC, the NP scale Λ should be much higher than the electroweak (EW) scale µ EW 246 GeV.
Assuming further that there exist no undiscovered but weakly coupled light particles, any NP effect in the processes proceeding at energy scales well below Λ but above µ EW can be effectively described by a series of higher dimensional operators that are built out of the SM fields and are invariant under the SM gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y [55,56]. The resulting effective field theory (EFT) is conventionally called the Standard Model Effective Field Theory (SMEFT) [57][58][59], which has now emerged as one of the most interesting tools to probe systematically the data from the LHC and elsewhere for possible NP hints 2 . For energies 1 The advantage of considering the ratios R(D ( * ) ) instead of the branching fractions themselves lies in the fact that, apart from the significant reduction of the experimental systematic uncertainties, the CKM matrix element V cb cancels out and the sensitivity to the B → D ( * ) transition form factors becomes much weaker. 2 See, for example, Refs. [60][61][62][63] for recent reviews on the SMEFT. below Λ, the leading NP contributions in the SMEFT formalism arise from the dimension-six operators 3 , which were firstly classified in Ref. [55], but found to be redundant for some of them. The first complete and non-redundant basis of dimension-six operators was derived in Ref. [56] and is now commonly called the Warsaw basis 4 . The complete one-loop anomalous dimensions of these dimension-six operators have also been calculated in Refs. [70][71][72].
The EFT approach is also an essential ingredient for B-physics analyses both within and beyond the SM. As the typical energy scale µ b is around the bottom-quark mass m b 5 GeV, being much smaller than the EW and the NP scale, all the B-physics processes can be well described by an effective Lagrangian constructed by integrating out the SM and NP heavy degrees of freedom (for classical reviews, see for example Refs. [73,74]). The resulting EFT includes only the QCD and QED gauge interactions coupled to all the six leptons and the five lightest quarks, plus a full set of dimension-six local operators built with these matter fields as well as the gluon and photon field-strength tensors, and is conventionally called the weak effective theory (WET) [75][76][77]. In contrast to the SMEFT case, the dimension-six operators in WET are not invariant under the full SM gauge group, but only under SU (3) C ⊗ U (1) em , as this EFT is defined below the EW scale where SU (2) L ⊗ U (1) Y is already broken. A complete and non-redundant set of dimension-six operators relevant for B physics, together with the complete one-loop anomalous dimensions in QCD and QED, can be found in Refs. [75][76][77].
For a given set of SMEFT dimension-six operators with the corresponding Wilson coefficients specified at the scale Λ, to study their effects on the B-physics processes, one has to follow the following three steps [78]: perform the renormalization group evolution (RGE) of the SMEFT Wilson coefficients from the NP down to the EW scale [70][71][72]; match the given set of SMEFT operators onto the WET ones at the EW scale [76,79]; perform the RGE of the WET Wilson coefficients from the EW down to the scale µ b [75][76][77]. With the aid of these three steps, one can then bridge the gap between the SMEFT Lagrangian and the low-energy measurements in B physics. In this paper, following this procedure and motivated by the R(D ( * ) ) anomalies, we shall study the B → D ( * ) τ ν, Λ b → Λ c τ ν, B c → (J/ψ, η c )τ ν, B → X c τ ν, as well as 3 There exists only a single dimension-five operator in the SMEFT, up to Hermitian conjugation and flavour assignments [56,64]. It violates the lepton number and, after the EW symmetry breaking, gives Majorana masses for the SM neutrinos. This operator is irrelevant to this paper. 4 Apart from the Warsaw basis [56], other bases were also proposed, with the most prominent ones being the HISZ [65] and the SILH [66,67] basis. For an easy translation between these different bases, one can resort to the computer codes Rosetta [68] and WCxf [69].
B c → τ ν decays, all being mediated by the same quark-level b → cτ ν transition, in the SMEFT formalism. It is found that the most relevant operators for these processes are Q lequ , and Q (3) lequ in the Warsaw basis. The RGEs of the corresponding Wilson coefficients from the NP scale Λ down to the typical scale µ b is performed at three-loop in QCD and one-loop in EW/QED (see Refs. [80][81][82] and references therein). Confronted with the currently available data, we shall also perform a detailed phenomenological analysis of these decays.
Our paper is organized as follows. In section 2, after recapitulating the SMEFT Lagrangian, we list the most relevant dimension-six operators for b → cτ ν transitions, and then discuss the evolution and matching of these operators in both the SMEFT and WET. In section 3, all the observables considered in the paper are listed, and the corresponding inputs for the transition form factors are also mentioned. Our numerical results and discussions are presented in section 4. Finally, we make our conclusions in section 5. Explicit expressions of the helicity amplitudes for Λ b → Λ c τ ν decay are collected in the appendix.
2 Theoretical framework

SMEFT Lagrangian
Following the common practice to truncate the SMEFT Lagrangian at dimension-six level and assuming that the EW symmetry breaking is realized linearly, we can write the SMEFT Lagrangian as SM is the usual SM Lagrangian before spontaneous symmetry breaking (SSB). The dimension-six operators Q i , which are obtained by integrating out all the heavy NP particles and are invariant under the SM gauge symmetry, are given by and so on [55,56]. Here τ I are the Pauli matrices, and ε jk is the totally antisymmetric tensor with ε 12 = +1. The fields q and l correspond to the quark and lepton SU (2) L doublets, while u, d and e are the right-handed SU (2) L singlets. All the NP contributions are encoded in the Wilson coefficients C i , which are dependent on the renormalization scale. This scale dependence will, however, be canceled in a physical amplitude by that of the matrix elements of Q i .
In this paper, we focus only on the operators Q lequ and Q (3) lequ , as well as their hermitian conjugates, which contribute to the b → cτ ν transitions at tree level [79,82]. Note that the operator Q (3) lq is already self-conjugate [55,56]. We also assume that the flavour of the neutrino in these operators is pure ν τ .

Evolution and matching
To explore the NP effect on the b → cτ ν transitions, we should firstly link the SMEFT Lagrangian given at the NP scale Λ to the WET Lagrangian given at the typical energy scale µ b associated with the processes considered. This can be achieved through the following three steps, details of which could be found, for example, in Refs. [75][76][77][78][79].
Firstly, we should evolve the Wilson coefficients C i of the SMEFT Lagrangian from the initial scale Λ down to the EW scale µ EW , under the SM gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y . For simplicity, here we do not discriminate the masses of W ± , Z 0 , the top quark t, and the Higgs boson h, and set approximately all of them to be µ EW . The one-loop RGE flow of C i (µ) can be written schematically as Neglecting terms suppressed by the Yukawa couplings, which are found to be negligibly small in our case, the one-loop beta functions are given, respectively, by [70][71][72]83] β (3) lq prst Here we have introduced the abbreviations C Refs. [76,77,79,84], we can write the spontaneously broken SMEFT Lagrangian in terms of the mass-eigenstate fermion fields (f (weak) L,R = P L,R f (mass) ) except the left-handed d-type quarks, for which the usual relation between the weak and mass eigenstates reads [76] d (weak) Lm where P R,L ≡ 1±γ 5 2 are the right-and left-handed chiral projectors. As we are concerned mainly on the operators Q lequ and Q lequ , as well as their hermitian conjugates, the effective quark-mixing matrix V appearing in Eq. (2.8) coincides with the SM CKM matrix.
The second step is to perform the matching at the EW scale µ EW . After integrating out the SM heavy particles, the W ± , Z 0 , the top quark, and the Higgs boson, we can obtain the WET Lagrangian suitable for describing the b → cτ ν transitions [75,76,79] L WET = L with the WET dimension-six operators given, respectively, by 5  5 Neutrinos are assumed to be left-handed throughout this paper and, hence, we need not consider the tensor operator (cσ µν P R b)(τ σ µν P L ν), which is obtained from O T by changing the chirality of the quark current, because it is identically zero due to Fierz transformations.
Here we do not consider the Wilson coefficient C V R , because it is explicitly lepton-flavour universal in the SMEFT formalism, up to contributions of O(µ 4 EW /Λ 4 ) [76,79,[85][86][87]. We shall also neglect terms proportional to the small CKM factors V ub and V cb [88], corresponding to n = 1 and n = 2, respectively. In such a case, the b → cτ ν transitions can only be affected by the Wilson coefficients C . The last step is to evolve the WET Lagrangian L WET from µ EW down to µ b under the gauge group SU (3) C ⊗ U (1) em , with the corresponding RGEs given schematically by where , and α e = e 2 /(4π) and α s = g 2 s /(4π) are the electromagnetic and strong coupling constants, respectively. The non-zero elements of the one-loop electromagnetic anomalous dimension matrix γ em read [75, 77, 82, 89- The QCD anomalous dimension matrices γ (k) s are known to three loops, with all the non-zero entries given by [75,77,82,92,93] γ (1) As the reference energy scale in b → cτ ν transitions is at around µ b 5 GeV, the RGE from µ EW down to µ b does not involve crossing any threshold, and the effective number of quark flavours n f can be fixed at n f = 5.
There exist several ready-made packages, such as Wilson [78] and DsixTools [94], to implement the evolution using the full one-loop anomalous dimension matrices as well as the tree-level matching. In our numerical analysis, we shall work at three-loop in QCD and oneloop in EW/QED, together with the same order for the corresponding coupling constants α s , g, g and α e .

Observables in
There have been a lot of calculations for the differential decay rates of B → D ( * ) τ ν in the presence of all the operators given in Eq. (2.12). In this paper, we shall follow the analytical expressions given in Refs. [95][96][97], and consider the following observables: • q 2 -dependent and q 2 -integrated ratios where, on the theoretical side, we define • τ forward-backward asymmetry where θ is the angle between the three-momenta of the τ lepton and the B meson in the τ ν rest frame.
• τ spin polarization which can be inferred from the distinctive τ decay patterns.
• D * longitudinal and transverse polarizations which can be measured by fitting to the double differential decay distribution or from the D * decays.
In analogy to the ratios R(D ( * ) ), we can also define the following observables with the denominators involving only the light-lepton modes: • τ forward and backward fractions • τ spin 1/2 and −1/2 fractions • D * longitudinal and transverse polarization fractions while θ τ is the angle between the three-momenta of the τ lepton and the Λ c baryon in the τ ν rest frame. The helicity ampli- , with the indices λ Λ b , λ Λc and λ τ denoting respectively the helicities of the Λ b , Λ c baryons and the τ lepton, can be calculated by following the helicity method described in Refs. [101][102][103][104]; for convenience, their explicit expressions are given in the appendix.

The rest observables
In this subsection, we introduce the rest observables relevant for B c → (J/ψ, η c )τ ν, B → X c τ ν and B c → τ ν decays, which could provide additional constraints on the NP parameters.
Similar to the definitions of R(D ( * ) ), the ratios R(J/ψ) and R(η c ) for B c → (J/ψ, η c )τ ν decays are defined, respectively, by in Ref. [110], which are consistent with the preliminary lattice QCD results [124,125] at all available q 2 points, but would result in lower central values of R(J/ψ) and R(η c ) [24].
For the inclusive decay B → X c τ ν, we consider the ratio The analytic expression of the total decay width within the SM is given by [126] Γ , and S em accounts for the short-distance electromagnetic correction to the SM four-fermion operator mediating the semi-leptonic decay [89,90]. The coefficients C (0) 0 and C (1) 0 represent the partonic-level contributions with the leading-and next-to-leading-order corrections in α s , respectively; while C µ 2 π , C µ 2 G and C ρ 3 D , C ρ 3 LS account for contributions from the 1/m 2 b and 1/m 3 b corrections in the heavy-quark expansion, respectively. Explicit analytic expressions of C (0) 0 , C µ 2 π , C µ 2 G and C ρ 3 D can be found, for example, in Refs. [126][127][128], whereas can be deduced, on the other hand, from Refs. [129][130][131].
The non-perturbative parameters µ 2 π , µ 2 G and ρ 3 D , ρ 3 LS are defined in terms of the forward matrix elements of dimension-five and -six operators, respectively. To calculate the ratio R(X c ), we take [33,132]: (21) GeV and m kin c (1GeV) = 1.092 (20) GeV in the kinetic scheme [133]; the correlations between these parameters [33,132] are also considered.
To discuss the NP effects from Eq. (2.9) on the inclusive B → X c τ ν decay, we take the partonic-level approximation, and decompose the decay width as [134] Γ where the first term arises solely from the SM and is given by Eq. (3.19), while Γ NP (1) and Γ NP (2) represent respectively the interference term with the SM as well as the term that is of second order in the NP couplings, explicit expressions of which are taken from Ref. [134]. Some recent works, discussing NP effects in this inclusive mode, can be found in Refs. [126,[135][136][137][138][139][140].

B c → τ ν
The decay B c → τ ν, despite being at the moment out of the experimental reach [141], can provide a powerful constraint on NP scenarios involving scalar operators [138,139,142,143].
In terms of the WET Lagrangian given by Eq. (2.9), its branching ratio can be written as where m b and m c are the bottom-and charm-quark running masses in the MS scheme evaluated at the scale µ b . In our numerical analysis, we take as input the lifetime τ Bc = 0.507(9) ps, the mass m Bc = 6.2751(10) GeV, and the decay constant f Bc = 0.434(15) GeV [144].
An upper bound obtained from the LEP data, B(B c → τ ν) 10% [143], is stronger than the conservative constraint, B(B c → τ ν) 30% [142], by demanding that the rate does not exceed the fraction of the total width allowed by the calculation of the B c lifetime within the SM. Here we shall use the former in our numerical analysis.

Numerical results and discussions
Before presenting the numerical results, we firstly collect in Table 1 the remaining theoretical input parameters used throughout this paper. The CKM parameters are taken from Ref. [145], in which the leptonic and semi-leptonic decays involving the µ and τ leptons have been removed from the global fit to the CKM parameters, following the current experimental indications that the electronic modes are in agreement with the SM predictions.

Numerical effects of evolution and matching
In this subsection, we illustrate the numerical effects of the evolution and matching procedure, Wilson coefficients are all assumed to be real): (Λ), (4.1) (Λ), making the EW/QED evolution of the SM four-fermion operator also taken into account. In the following discussions, we shall use the abbreviations for the sake of brevity.

SM results and comparison with data
Our predictions for the observables listed in section 3 within the SM are collected in Table 2.
The values of observables for B → D ( * ) τ ν decays are always obtained by averaging over the charged and neutral modes. Although the relations H is D, D * or Λ c , and i = 5 for H is D * or Λ c ) hold, we are still presenting all of them in Table 2, because these observables involve different normalization and systematics and can, therefore, provide complementary information on the NP scenarios. This is clearly indicated by the reduced uncertainties of the observables X i (H) compared to that of R(H).
Among the observables listed in Table 2, the following ones have been measured: R(D) exp = 0.407(39)(24) and R(D * ) exp = 0.306(13)(7) with a correlation of −0.203 [14], P τ (D * ) exp =     (86) The rest observables These discrepancies will be used to constrain the SMEFT Wilson coefficients. 6 This value is obtained by using the world average for the semi-leptonic branching fractions into the light leptons, B(B → X c ν) = (10.65 ± 0.16)% [146], and an averaged constraint from LEP, B(b → Xτ ν) = (2.41 ± 0.23)% [146], which is dominated by b → X c τ ν because of |V ub | 2 /|V cb | 2 ∼ 1% and, after correcting for the b → u contribution that is about 2% due to the larger available phase space, is reduced to B(b → X c τ ν) = (2.35 ± 0.23)% [139]. It should be noted that the LEP measurement corresponds to a known admixture of initial states for the weak decay [147]. The inclusive decay rate does, however, not depend on this admixture to leading order in 1/m b . The corrections to this limit are hadron-specific and only partly known [127,135]. 10%. The dark-green, green, and light-green areas represent the 1-, 2-, and 3-σ differences between the measurements and the SM predictions for the observables, respectively.

Constraints on the SMEFT Wilson coefficients
In this subsection, we shall use ∆R(D), ∆R(D * ), ∆P τ (D * ), ∆R(J/ψ), and ∆R(X c ) to constrain the SMEFT Wilson coefficients C 1−4 (see Eq. (4.6)). Firstly, we show in Figure 1 the contributions to these observables in the presence of only a single C i . It can be seen that, after taking into account the constraint B(B c → τ ν) 10%, the scenario with a single C 3 is already ruled out by ∆R(D ( * ) ) at 3σ ( and ∆R(D * ) are the strongest, but the one from B(B c → τ ν) 10% is very complementary to them, making parts of the regions allowed by ∆R(D ( * ) ) already excluded. It can also be seen that the B(B c → τ ν) constraint in the (C 2 , C 3 ) plane is stronger than in the other five cases.
By minimizing the χ 2 (C i ) function in different scenarios, we can get the corresponding bestfit solutions, the results of which are shown in Table 3. Here the first column shows all possible cases with either a single C i or a combination of two C i s, in addition to the SM case. The second column gives the values of χ 2 min with respect to different numbers of degrees of freedom (dof), with the corresponding best-fit points as well as the 1σ ranges (∆χ 2 = χ 2 (C i )−χ 2 min ≤ 1) for the single-parameter fits shown in the third column. Only the cases satisfying the condition      (23) found that, after taking into account the combined constraints from ∆P τ (D * ), ∆R(J/ψ), and ∆R(X c ), the scenario with a single C 4 is no better than that with a single C 1 for resolving the R(D ( * ) ) anomalies.

Predictions for the observables in different NP scenarios
In order to further discriminate among the eleven most trustworthy scenarios obtained in the last subsection, we now calculate all the observables listed in section 3 within these different scenarios. Our final numerical results are collected in Tables 4 and 5. During the calculation, we use the central values of the NP Wilson coefficients obtained in scenarios S1 to S11, and take into account the uncertainties caused by the input parameters. Table 5: Predictions for the observables involved in Λ b → Λ c τ ν decay, as well as for R(J/ψ), R(η c ), R(X c ), and B(B c → τ ν) in all the NP scenarios.
Obs. S1, S2 S3 S4, S5 S6, S7 S8, S9 S10 S11  From Tables 4 and 5, we can see that the scenarios S1 and S2, S4 and S5, S6 and S7, as well as S8 and S9, all of which involve the NP Wilson coefficient C 1 that would induce only the left-handed vector current at the scale µ b (see Eq. (4.1)), cannot be distinguished from each other. There are, however, a number of observables, such as P τ (D), X 4 (D), A FB (D * ), P τ (D * ), P L (D * ), X 5 (D * ), P τ (Λ c ), P Λc , and X 5 (Λ c ), that can be used to distinguish the scenario S3 from the other ones. In addition to the scenario S3, there exist another two scenarios S10 and S11 that do not involve the Wilson coefficient C 1 . As the predicted branching fraction of B c → τ ν decay in the scenario S11 is much smaller than in the other scenarios as well as in the SM, we can use the observable B(B c → τ ν) to distinguish the scenario S11 from the other ones. On the other hand, the observables P τ (D), A FB (D * ), A FB (Λ c ), P τ (Λ c ), and B(B c → τ ν) have the potential to distinguish the scenario S10 from the other ones.
The scenarios S4, S5 and S6, S7 might be distinguished only by the observable B(B c → τ ν). While the observables X i can help to distinguish the scenarios S1, S2 from the SM, the corresponding observables normalized by the tauonic modes, such as the τ forward-backward asymmetries A FB (D), A FB (D * ), and A FB (Λ c ), fail to do, because they are all identically the same in the scenarios S1 and S2 as well as in the SM.
In order to further differentiate these different scenarios, we now consider the correlations among the observables discussed in this paper. There are totally 465 correlation plots, with a small part of them shown in Figure 3. As can be seen from the R(D)−R(D * ) correlation plot, it is interesting to note that all the NP scenarios can resolve the R(D ( * ) ) anomalies at 1σ very well and, except in S1 and S2, the predicted R(D in the scenario S3 are also found to be very different from the ones in the other scenarios.
Based on all the above observations, we can, therefore, conclude that all the eleven NP scenarios, except S1 and S2, S4 and S5, S6 and S7, as well as S8 and S9, can be distinguished from each other by the above observables as well as their correlations.

The SU (2) L -invariant implications
Due to the SU (2) L invariance of the SMEFT Lagrangian, the non-zero Wilson coefficients C 1−4 at the high-energy scale Λ enter not only in the b → cτ ν τ processes studied in this paper but also in other low-energy charged and/or neutral current processes [85,88,149]. With our prescription for the weak and mass eigenstates of fermion fields (see Eq. (2.8)), the processes c → d n τ ν, d m → d n νν and d n → d m τ + τ − , with d m being one of the d-type quarks in the mass eigenstate, will also receive the NP contributions from C 1−4 . However, compared with the b → cτ ν transitions, the NP effects on c → d n τ ν (d n = d or s) processes are suppressed by the small factor V cb V tn /V cn 1.6 × 10 −3 , and we can, therefore, neglect safely the NP impacts on the D (s) -meson decays. On the other hand, it is found that the upper bound on the branching fraction of B + → K + νν decay given by the Belle [150] and BaBar [151] collaborations will disfavour the larger parameter regions for C 1 given in Table 3. Combining the χ 2 -fit results with the constraint from the branching fraction of B + → K + νν [152][153][154][155][156][157][158][159][160][161], we also find that the SM SM S1,S2 S1 ,S2  S3  S3  S4,S5  S4,S5  S6,S7  S6,S7  S8,S9 S8,S9 S10 S10 S11 S11 C 1 contributions to the branching fractions of some b → sτ + τ − processes, such as B s → τ + τ − , B → K ( * ) τ + τ − , and B s → φτ + τ − decays, can be enhanced by about two orders of magnitude compared to the SM [88]. The NP effects on Υ(nS) → τ + τ − decays are, however, suppressed by the small factor V cb V ts /V cs compared to these b → sτ + τ − processes. Finally, it should be noted that there also exist some collider signals directly implied by the R(D ( * ) ) anomalies. For example, the partonic-level process gc → bτ ν implied by crossing symmetry from the b → cτ ν decay should also take place at the LHC [162]. Furthermore, the τ + τ − resonance searches at the LHC [163,164] should also be confronted with what have been found in this paper [165].
Detailed analyses of the SU (2) L -invariant implications will be presented in a forthcoming paper.

Conclusions
In this paper, we have discussed the B → D ( * ) τ ν, We then explored the contributions to the observables R(D), R(D * ), P τ (D * ), R(J/ψ), and R(X c ) in the presence of a single SMEFT Wilson coefficient. It is found that the scenario with a single C 1 or C 4 can be used to resolve the R(D ( * ) ) anomalies at 1σ, especially with the finding that the experimental central values can be well reproduced with a single C 4 . A single C 3 is, however, already ruled out by the measured R(D ( * ) ) and the constraint B(B c → τ ν) 10% at more than 3σ. In the case where two SMEFT Wilson coefficients are present simultaneously, on the other hand, we found that the constraints from ∆R(D) and ∆R(D * ) are the strongest, but the one from B(B c → τ ν) 10% is very complementary to them, making parts of the regions allowed by ∆R(D ( * ) ) already excluded. Under the combined constraints from the measured R(D), R(D * ), P τ (D * ), R(J/ψ), and R(X c ), we obtained the best-fit points and the allowed regions at 99.73% C.L., which are shown in Table 3 and Figure 2, respectively. Due to the extra combined constraints from P τ (D * ), R(J/ψ), and R(X c ), the scenario with a single C 4 is also found to be no better than that with a single C 1 for resolving the R(D ( * ) ) anomalies.
Through a global fit, we have identified eleven most trustworthy scenarios, each of which can provide a good explanation of the R(D ( * ) ) anomalies at 1σ. In order to further discriminate these different scenarios, we have also predicted the observables in each NP scenario and considered the correlations among them. It is found that most of the scenarios can be differentiated from each other by using these observables as well as their correlations. In particular, the predicted B(B c → τ ν) in the scenario S11 is found to be much smaller than in the other scenarios as well as in the SM. The observables P τ (D), X 4 (D), A FB (D * ), P τ (D * ), P L (D * ), Note added: After this work was finished, we are informed that there has been a preliminary Belle measurement of the D * longitudinal polarization fraction in B → D * τ ν [166]. This preliminary result P L (D * ) = 0.60 ± 0.08 ± 0.035 shall exclude the scenario S3, which predicts a very small P L (D * ) = 0.142 ± 0.001 (see Table 4). This implies that the solution to the R(D) and R(D * ) anomalies with the tensor operator is not favored.
Starting with the effective Lagrangian given by Eq. (2.9) and using the helicity-based definition of the Λ b → Λ c transition form factors in Ref. [100,167], we can obtain the hadronic helicity amplitudes as follows: • The non-zero scalar and pseudo-scalar helicity amplitudes, where m b and m c are the b-and c-quark running masses in the MS scheme and should be evaluated at the typical energy scale µ b .
• The non-zero vector and axial-vector helicity amplitudes, • The non-zero tensor helicity amplitudes,  For the leptonic helicity amplitudes, on the other hand, we obtain [100,168]: • The non-zero scalar and pseudoscalar leptonic helicity amplitudes, Integrating the two-fold angular distribution given by Eq. (3.9) over cos θ τ but without the first two summations over λ Λc and λ τ , we can obtain the following expression for the helicitydependent differential decay rate: for a polarized τ lepton.