Revisiting the new-physics interpretation of the b → cτν data

We revisit the status of the new-physics interpretations of the anomalies in semileptonic B decays in light of the new data reported by Belle on the lepton-universality ratios RD(*) using the semileptonic tag and on the longitudinal polarization of the D* in B → D*τν, FLD∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {F}_L^{D\ast } $$\end{document}. The preferred solutions involve new left-handed currents or tensor contributions. Interpretations with pure right-handed currents are disfavored by the LHC data, while pure scalar models are disfavored by the upper limits derived either from the LHC or from the Bc lifetime. The observable FLD∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {F}_L^{D\ast } $$\end{document} also gives an important constraint leading to the exclusion of large regions of parameter space. Finally, we investigate the sensitivity of different observables to the various scenarios and conclude that a measurement of the tau polarization in the decay mode B → Dτν would effectively discriminate among them.

Belle has also measured the longitudinal polarization of the τ (P D * τ ) [6] and of the D * (F D * L ) [50] in the B → D * τ ν decay,  [65] of experimental data for R D and R D * use data from BABAR [1,2], Belle [3,4,6] and LHCb [5,8,9], while the HFLAV 2019 average includes the Belle measurement of both, R D and R D * , with the semileptonic tag [51]. The LHCb measurement of R J/Ψ is reported in ref. [10] and the Belle measurements of P D * τ and F D * L in refs. [10,50]. The two experimental errors correspond to statistical and systematic uncertainties, respectively. SM predictions are obtained as specified in section 2. 3. where λ X refers to the helicity of the particle X. While P D * τ is reconstructed from the hadronic decays of the τ and is still statistically limited, the reported measurement of F D * L is rather precise and disagrees with the SM prediction with a significance of 1.7σ.
Recently, Belle announced a new combined measurement of both R D and R D * using semileptonic decays for tagging the B meson in the event [51]. This presents a significant addition to the data set because the previous combined measurements of R D ( * ) had been performed at the B factories using a hadronic tag. The new result is more consistent with the SM than the previous HFLAV average. Thus, these new data call for a reassessment of the significance of the tension of the signal with the SM and of the possible NP scenarios aiming at explaining it. The purpose of this work is to provide such an analysis using effective field theory (EFT) [52][53][54][55][56][57][58][59][60][61][62][63] and to relate it to (partial) UV completions in terms of simplified mediators. We assume that the lepton non-universal contribution affects only the couplings to the tau leptons. A comprehensive analysis of bounds on NP affecting b → c ν transitions can be found in ref. [64]. A summary of the recent data (averages) is shown in table 1, which is compared to the SM predictions which are obtained as specified in section 2.3.

Low-energy effective Lagrangian
The most general effective Lagrangian describing the contributions of heavy NP to semitauonic b → cτν processes can be written as  [66,67], τ R = R + O(v 4 /Λ 4 NP ) and the right-handed operator cannot contribute to lepton universality violation at leading order in the (v 2 /Λ 2 NP ) expansion [25,68,69]. For this reason, we do not consider the effect of τ R in our fits. Nonetheless, it is important to note that this assumption could be relaxed if there was not a mass gap between the NP and the EWSB scales, or under a nonlinear realization of the electroweak symmetry breaking [70].
The chirally-flipping scalar and tensor operators are renormalized by QCD and electroweak corrections [71][72][73][74]. The latter induce a large mixing of the tensor operator into τ S L which can have relevant implications for tensor scenarios [71]. As an illustration, and where, in a slight abuse of notation, we keep the notation for the WCs of the lowenergy EFT above the EWSB scale. Operators with vector currents do not get renormalized by QCD, whereas electromagnetic and electroweak corrections produce a correction of a few percent to the tree-level contributions [71,75]. On the other hand, all the operators in the SMEFT matching at low-energies to the Lagrangian in eq. (2.1) can give, under certain assumptions on the flavor structure of the underlying NP, large contributions to other processes such as decays of electroweak bosons, the τ lepton and the Higgs, or the anomalous magnetic moment of the muon [74,76,77]. An interesting scenario where the new physics cannot be described by the local effective Lagrangian eq. (2.1) consists of the addition of new light right-handed neutrinos [13, 15-17, 32, 78, 79]. This duplicates the operator basis given in eq. (2.1) by the replacements P L → P R in the leptonic currents (and in the hadronic current for the tensor operator) [69,78,80] and whose WCs we label with Γ →˜ Γ . None of these operators interfere with the SM and their contributions to the decay rates are, thus, quadratic and positive. This also means that the size of the NP contributions needed to explain R D ( * ) in this case are larger than with the operators in eq. (2.1) and they typically enter in conflict with bounds from other processes like the decay B c → τ ν [81,82] or from direct searches at the LHC [83]. As an illustration of the features and challenges faced by these models we consider the operator with right-handed currents, (with N R denoting the right-handed neutrino), which incarnates a popular NP interpretation of the anomaly [13, 15-17, 78, 79]. Finally, imaginary parts also contribute quadratically to the rates so we assume the WCs to be real, although we will briefly study also the impact of imaginary parts below.

Simplified models
The effective operators in eqs. (2.1), (2.3) can be mediated at tree level by a number of new particles, that we list in table 2. Possibilities with new charged colorless weak bosons can be realized with the W in either a triplet (W L ) or a singlet (W R ) representation of weak isospin. In the former case, the neutral component of the triplet, a Z with a mass close to the one of the W , produces large effects in either neutral-meson mixing or di-tau production at the LHC, so that this scenario is unavoidably in conflict with data [84]. Making the W a singlet of weak isospin, , requires introducing right-handed neutrinos to contribute to b → cτν [13,[15][16][17]; parametrizing the Lagrangian for this model, one finds the contribution to the EFT, Models based on extending the scalar sector of the SM, such as the two-Higgs doublet model (labeled by H in table 2), generate the scalar operators through charged-Higgs exchange. However, these are disfavored by experimental bounds that stem from the B c lifetime [81] and from the branching fraction of B c → τ ν derived using LEP data [82]. Strong limits from direct searches at the LHC of the corresponding charged scalars have also been obtained in the literature [85]. On the other hand, leptoquark exchanges can produce all the operators in eq. (2.1). 1 The SM interactions of the scalar leptoquark S 1 =(3, 1,+1/3) can be described by the Lagrangian,

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where ab is the antisymmetric tensor of rank two and where we are labeling the flavor of the fields in the interaction basis. This model produces left-handed, scalar-tensor and right-handed contributions [24,29,33,34], where the coefficients are defined at a scale equal to the leptoquark mass, µ = m S 1 . The tilde in the coefficients of eq. (2.7) and in the rest of this subsection indicates that the quark unitary rotations have been absorbed in the definition of the couplings. For instance, if where summation of quark flavor indices is implicit. We have also defined these couplings in the charged-lepton mass basis, ignoring neutrino masses.
The leptoquark with quantum numbers R 2 =(3, 2,+7/6) and Lagrangian, Thus, one can achieve a tensor scenario by adjusting the masses and couplings of the S 1 and R 2 leptoquarks. It is important to stress that such a solution at low energies requires some tuning due to the large electroweak mixing into scalar operators in eq. (2.2). Among the vector leptoquarks we consider the U 1 =(3, 1,+2/3), which has been extensively studied in the interpretation of the B anomalies [25, 26, 35-37, 39-42, 44, 46-48], leading to left-handed and right-handed contributions, and a scalar contribution, In particular, a combination of left-handed and right-handed couplings gives rise to a scalar operator which is instrumental to achieve a better agreement with data in some UV completions of the U 1 leptoquark [37,42,47,48]. The mediators S 3 =(3, 3,+1/3) and U 3 =(3, 3,+2/3) in table 2 provide completions of the left-handed current operator equivalent to the S 1 and U 1 ones for scalar and vector leptoquark scenarios, respectively. Finally, we have not included in the table the leptoquarks R 2 = (3, 2, +1/6) andṼ 2 = (3, 2, −1/6) because they only contribute to scalar and tensor operators with right-handed neutrinos which are not considered in this work, as argued in section 2.1.

Form factors
The hadronic matrix elements in the b → c decay amplitudes are parameterized in terms of the following form factors, where q = p−k, 0123 = 1, V and P stand for vector mesons (D * and J/ψ) and pseudoscalar mesons (B and B c ), respectively. We take the quark masses in the MS scheme, i.e, m b ≡ m b (m b ) = 4.18 GeV and m c (m c ) = 1.27 GeV [88]. Note that the c-quark mass is derived by the solution of the renormalization group equation for m c (µ) at two-loop order and α s (µ) with three-loop accuracy [89]. We follow the PDG [88] for the masses of the mesons relevant in this work. For the B → D ( * ) mode, some of the form factors are taken from Lattice QCD calculations [90,91]. The rest are parameterized using heavy-quark effective theory (HQET) [92][93][94][95][96][97][98][99] whose nuisance parameters are determined by the HFLAV global fits to theB → D ( * ) −ν data [100]. Our determination of R D and R D * differs from that of HFLAV in the choice of form factors; ours, based on ref. [101], do not include some recent refinements [102][103][104].

Statistical method
We follow a frequentist statistical approach to compare the measured values of n exp observables, O exp , to their theoretical predictions O th as functions of the Wilson coefficients , and of nuisance theoretical parameters y. The nuisance parameters parameterize the lack of knowledge (theoretical uncertainties) of the form factors. For the B → D ( * ) decays, we employ the parametrization and numerical inputs (including correlations) described in ref. [101]. For the B c → J/Ψ decays we parameterize the theoretical errors reported for the form factors in ref. [110] as uncorrelated nuisance parameters. We then define a test statisticχ where y 0 are a set of central values for the nuisance parameters, and V exp and V th denote the experimental and theoretical covariance matrices, respectively. By adding the theory term χ 2 th we have in effect (from a statistical point of view) added n th (correlated) "measurements" of the n th theory parameters to the n exp measurements of the observables.
We will consider scenarios (statistical models) with different subsets of the Wilson coefficients allowed to vary and the remaining ones set to zero, and with various subsets of the experimental observables included. In each case, we obtain best-fit values for the model parameters, including the nuisance parameters, by minimizing χ 2 . To do so, in a first step we construct a profile-χ 2 function which depends solely on the subset of Wilson coefficients allowed to take nonzero values in a particular scenario, which we again refer to as . (Note that in the case of a single measurement of an observable whose theoretical expression depends linearly on a single JHEP12(2019)065 theory nuisance parameter y, such that y −y 0 is proportional to the theoretical uncertainty, the profiling reproduces the widely employed prescription of combining theoretical and experimental errors in quadrature.) In a second step, we minimize χ 2 ( ) over ; the value(s) of at the minimum χ 2 min provide(s) the best fit (maximum likelihood fit). Next, we compute a p-value to quantify the goodness of fit, i.e. how well a given scenario can describe the data. We will assume that χ 2 ( ) follows a χ 2 -distribution with n dof = n exp − n degrees of freedom, where n is the number of parameters allowed to vary in a given fit. Note that the theory parameters do not contribute to n dof because χ 2 th contains as many "measurements" as theory parameters. In each scenario, the p-value is obtained from χ 2 min as one minus the cumulative χ 2 distribution for n dof degrees of freedom. To illustrate this, let us consider only the χ 2 exp including R D and R D * and ask how well the SM describes these data. For simplicity, let us neglect theory errors altogether (they will be included in the following section, with little impact on the result), taking the SM prediction to be the central values employed by HFLAV2019, R SM,HFLAV D = 0.299 and R SM,HFLAV D * = 0.258. In this case, there are no parameters to minimize over and χ 2 is simply a number. This is easily obtained from the HFLAV2019 averages and correlation shown in table 1, substituting the SM values for the observables, which gives ∆χ 2 SM = ∆χ 2 ( 0) as defined below, and adding a constant χ 2 min = 8.7 as stated by HFLAV. 2 Nine measurements entered the combination and we are determining zero parameters, resulting in n dof = 9. With χ 2 SM = χ 2 ( = 0) = 22.8, this gives a p-value of 6.56 × 10 −3 corresponding to 2.72σ, slightly reduced from 3.00σ obtained in an analogous manner from the HFLAV2018 combination.
Finally, for each one-parameter BSM scenario, we construct ∆χ 2 ( ) = χ 2 ( ) − χ 2 min and obtain nσ confidence intervals from the requirement ∆χ 2 ≤ n 2 . Similarly, for each 2-parameter scenario we construct the corresponding ∆χ 2 and obtain two-dimensional 1σ and 2σ regions from the conditions ∆χ 2 ≤ 2.3 and ∆χ 2 ≤ 6.18, respectively. We also determine, for each model, to quantify at what level the SM point is excluded in that model. The ∆χ 2 SM is converted to an equivalent number of standard deviations, referred to as the pull Pull SM , by employing the cumulative χ 2 -distribution with n dof set to 1 or 2, the number of jointly determined parameters, as appropriate.
Let us close this section by contrasting to the usual approach for comparing the R D and R D * measurements to the SM, as employed by HFLAV. In this approach, the true values of R D and R D * are treated as free parameters, which effectively amounts to a twoparameter BSM model. In this model, HFLAV obtain an SM pull of 3.08σ. We stress that this is a statement about how much better than the SM a BSM model can potentially describe the data. It is conceptually analogous to the pulls in our two-parameter Wilson coefficient fits. (In fact, we will find in the next section a slightly higher pull for two of our 1-parameter models. This comes about because a given ∆χ 2 value implies a lower p-value to joint determination of two parameters.) Conversely, our SM p-values are a statement how well the SM describes the data, without reference to any comparator BSM model. As we have seen, the data is marginally consistent with the SM at 3σ, little changed from 2018. As we will see in the subsequent sections, the impact of the new Belle data on the best-fit values in BSM scenarios is much stronger.

Results
In this section, we investigate the values of the WCs determined by fitting to the experimental data of R D , R D * , R J/ψ , P D * τ and F D * L given in table 1. We also discuss the constraints on scalar operators derived from the limits Br(B c → τ ν) ≤ 30%(10%) which are obtained using the B c lifetime [81] (LEP searches of the decays B (c) → τ ν [82]). Note that these limits have been critically discussed in refs. [58,119,120]. Finally, an upper bound on the values of the WCs can be derived from the tails of the monotau signature (pp → τ h X+MET) at the LHC [83,121,122] (see below). We will perform fits to two types of dataset: R D ( * ) only, as well as to the full dataset in table 1 including in addition R J/ψ and the polarization observables.

Fits to R D ( * ) only
In figure 1 we show the "trajectories" representing the correlated impact on R D and R D * of NP scenarios where only a single operator is present at a certain scale. Namely, the JHEP12(2019)065  Table 3. Best fit values, χ 2 min , p-value, pull and 1σ confidence intervals of the WCs in the fits to all the R D ( * ) data. We perform fits to one or two WCs at a time with the understanding that the others are set to 0. For the cases of two Wilson-coefficient fits, the 1σ interval of each Wilson coefficient is obtained by profiling over the other one to take into account their correlation.
"vector" curve is followed by scenarios with new pure left-handed ( τ L ) or pure right-handed (˜ R τ ) currents (which are not affected by short distance QCD corrections). "Tensor" and "scalar-tensor" interpretations involve both τ T and τ S L coupled by the radiative corrections in the SM, cf. eq. (2.2). The tensor trajectory describes a solution with only the tensor operator produced at the heavy scale (cf. produced by the combination of S 1 -and R 2leptoquark contributions described in section 2.2), that we take to be 1 TeV. The scalartensor description assumes the relation τ S L (1 TeV) = −4 τ T (1 TeV), again, at the heavy scale (cf. produced by the S 1 leptoquark). The arrows in the curves signal the direction of positive increment of the WCs. The experimental data in table 1 is represented by the different ellipses: the gray one is the 1σ contour of the 2018 HFLAV average, the blue ellipse is the 1σ region of the 2019 Belle measurement with semi-leptonic tag and, finally, the red ellipses are the 1σ, 2σ and 3σ contours of the combination of these two.
The interference of the SM with left-handed or scalar-tensor contributions can produce a simultaneous increase of R D and R D * , as illustrated in figure 1 by the positive slope of the corresponding curves at the SM point. This effect drives these solutions to agree well with the 2018 HFLAV average. In case of the tensor scenario, interference with the SM increases R D at the expense of reducing R D * or vice versa. This effect is illustrated by the negative slope of the "Tensor" curve in figure 1. Therefore, the agreement of this scenario with the older data set is due to the quadratic contributions of the tensor operator to the rates. With the new Belle measurement, R D becomes more consistent with the SM while a value of R D * larger than predicted is still favored. In this new scenario, "vector" models still agree with the data but now the interference of the tensor operator with the SM can play a role in providing a satisfactory solution.
In table 3 we show the results of fits to all the data on R D ( * ) of one or two WCs at a time, while setting the others to zero. In the two-dimensional case we only investigate the interplay between operators with left-handed neutrinos. Setting all WCs to zero, one obtains a χ 2 SM = 20.75. With 9 degrees of freedom (d.o.f.) this corresponds to a p-value of 1.38 × 10 −2 . As can be inferred from the table, the "vector" operators provide the best one-parameter fit to the data, with a p-value of 0.34 and a SM pull of 3.43σ. The difference in size of the values of the WCs between the left-and right-handed vector solutions is due to the fact that the latter corresponds to a quadratic NP effect in the rates.
The tensor operator also gives a good fit to the data, where the solution driven by the interference piece is now preferred. Scalar models do not provide good fits and require values that may be in conflict with the bounds from B c → τ ν. In figure 2 we show the functions χ 2 of the one-parameter fits for each of the WCs. We also show in dashed lines the results obtained from the fits to the 2018 HFLAV average, to emphasize the change in the structure and values of the WCs needed with the new data. Horizontal lines showing the values of the 1-to 4σ ranges have been computed taking the best model (vector operators) giving χ 2 min = 9.00 as reference. In figure 3 we show the contour plots that are obtained from each of the six twodimensional fits to the 2019 HFLAV averages of R D and R D * . In the appendix, table 7, we provide the correlation matrices for the fits to two WCs. We also show with empty red contours the results of the fits to the 2018 HFLAV averages. Black empty contours represent the 2σ upper limits that can be set by analyzing the tails of pp → τ X+MET at the LHC (solid line) and by estimating the projected sensitivity at the HL-LHC (dashed line) [83]. Adding the new Belle data in the fit results in regions which are slightly closer to the SM, although all NP scenarios still describe the data better with a significance of 3.03σ. As expected, constraints from Br(B c → τ ν) play an important role in excluding regions of the parameter space of the scalar models. For instance, in case of the pure scalar fit, with ( τ S L , τ S R ), the 1σ region is almost excluded by the softer limit based on the B c lifetime. Even the 2σ region is also excluded if the more aggressive limit of 10% on Br(B c → τ ν) is used. Constraints in the ( τ S L , τ T ) plane are interesting for UV completions involving S 1 and R 2 leptoquarks. In this scenario, data favors the parameter space in which the two WCs have the opposite sign, like the contribution of the S 1 and unlike the one of JHEP12(2019)065 the R 2 , cf. eqs. (2.7) and eq. (2.9). A fit with the scalar-tensor contribution produced by the S 1 leptoquark (evaluated at µ = 1 TeV) gives a fit with a p-value 0.15 that is considerably better than for the SM. However, this scenario performs worse than those with pure left-handed or tensor operators. Constraints in the ( τ L , τ S R ) plane are interesting for UV completions of the U 1 leptoquark involving left-and right-handed currents to the fermions [37,42,47,48].

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The LHC data also probes the parameter space of the preferred regions in the different scenarios. As already anticipated in [83], scenarios involving large quadratic contributions of the tensor operator are excluded by more than 2σ. Furthermore, the current LHC exclusion region independently covers a large portion of the 1σ ellipse in the pure scalar scenario and all the parameter space of the 2σ region will be probed by the HL-LHC. In fact, with the high-luminosity data set we should be able to probe all the interesting regions in all the scenarios, although less deeply than for the results of the fits to the 2018 HFLAV average.
A potential caveat concerning the interpretation of these LHC bounds is that their validity relies on the assumption that the NP scale is significantly larger than the partonic energies probing the effective interaction in the pp → τ ν collisions at the LHC. In ref. [83] this was studied by assessing the sensitivity to NP of the distribution in the tau transversemass, m T , of the pp → τ h X+MET analyses [121,122]. Most of the sensitivity of the LHC stems from m T 2 TeV and, for mediator masses above this mark, the EFT provides a faithful description of the NP signal. Extending the comparison to the right-handed currents, the value˜ τ R = 0.38 (6) obtained in the fit would still be challenged by the bound |˜ τ R | ≤ 0.32 at 2σ resulting from the collider analysis in the EFT. Turning to explicit UV completions in the range of masses below 2 TeV, LHC bounds are stronger than the EFT counterpart for the W but weaker for the leptoquarks [83]. 3 Finally, in our fits we have assumed real WCs, but is clear from section 2.2 that these are generally complex. The imaginary part of WCs does not interfere with the leading, SM contribution. Hence it is expected that fits with purely imaginary WCs are susceptible to bounds from, eg, the rate B c → τ ν. This is particularly the case for scalar WCs, that require large magnitude of WCs to account for R D ( * ) . For example, in the left panel of figure 4 we show the fit of the complex WC S L with best fit value located at   Table 4. Best fit values, χ 2 min , p-value, pull and 1σ confidence intervals of the WCs in the fits to all the data in R D , R D * , R J/ψ , P D * τ and F D * L . We perform fits to one or two WCs at a time with the understanding that the others are set to 0. For the cases of two WC fits, to take into account correlation between the two WCs, the 1σ interval of each WC is obtained by profiling over the other WC.
fit. An interesting example is that of the combination τ S L = 4 τ T that is the case of the R 2 leptoquark mediator, eq. (2.9); the right panel of figure 4 shows the constraints on the complex τ S L (µ = m b ) plane, with best fit point at S L = −0.08 ± 0.30 i, and having imposed the condition τ S L = 4 τ T at the matching scale µ = M R 2 = 1 TeV. 4 4 Complex coefficients for a model based on R2 were considered in ref. [43].
JHEP12(2019)065 Figure 5. Constraints in the WCs planes from the fits to all the data in R D and R D * , and to R J/ψ , P D * τ and F D * L setting two WCs to zero. The solid ellipses represent 1σ and 2σ allowed regions while the empty black solid (dashed) ellipses indicate the 2σ upper bounds from the LHC data (HL-LHC projections) on pp → τ h X+MET. Regions in gray and light gray represent 30% and 10% exclusion limits from Br(B c → τ ν), respectively.

Fits to
In this section, we perform a global fit of τ L , τ T , τ S L and τ S R to all the data including R D and R D * , R J/ψ , P D * τ and F D * L . We implement the LHC monotau constraints by demanding that the WCs are within the corresponding 2σ bounds, i.e., we take | τ L | ≤ 0.32, | τ T | ≤ 0.16, | τ S L | ≤ 0.57 and | τ S R | ≤ 0.57. In addition, we impose the constraint from the B c lifetime by requiring that Br(B c → τ ν) ≤ 30%. One obtains a χ 2 min,SM = 26.53 with 12 degrees of freedom (d.o.f) if all the WCs are set to 0, corresponding to a p-value of 9.02 × 10 −3 . The

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Best fit 1σ range 2σ range 3σ range  Table 5. Different confidence-level intervals of the WCs in the fits to all the data in R D , R D * , R J/ψ , P D * τ and F D * L , obtained from the profile χ 2 where the rest of WCs are minimized within the 2σ LHC mono-tau bound. We have also applied the 30% bound on Br(B c → τ ν).
resulting WCs from the fit are, and where χ 2 min = 12.80 for 8 d.o.f., corresponding to a p-value of 0.12 and a Pull SM = 2.64. This provides an approximation of the χ 2 min in the immediate vicinity of the minimum that is closest to the SM, although it is not appropriate to obtain realistic confidence-level regions. For instance, the 1σ intervals may seem to violate the LHC bounds described above.
In order to investigate this in more detail we perform, first, fits of two WCs to R D and R D * , R J/ψ , P D * τ and F D * L setting the others to 0. This allows one to compare to the results of the two-parameters fits to R D and R D * presented in section 3.1. The corresponding six possible combinations of two WCs fits are shown in figure 5 and the results of the fits are shown in table 4. In the appendix, table 8, we provide the correlation matrices for these fits. As compared with figure 3, one notes that although not precise, the data R J/ψ , P D * τ and F D * L is sensitive enough to exclude the same regions allowed at 2σ by the fit to R D ( * ) independently excluded by the LHC monotau signature or B c → τ ν (see also ref. [124]). However, for the favored regions of the fits closer to the SM the addition of the current data on these observables has a small impact.
Finally, in order to obtain realistic confidence-level regions with the four active WCs we obtain profile likelihoods functions depending on one or two WCs at a time. The monotau LHC constraints and the B c lifetime bound are implicitly imposed when profiling over the other "nuisance" WCs in each case. In figure 6, we show the results of the fits as constraints in the six two-WCs plots. In table 5 we show the final 1σ, 2σ and 3σ confidencelevel intervals for the WCs. The 1σ intervals are consistent with those obtained from the fit in eq. (3.1), while the 2σ and 3σ intervals differ from those obtained using the gaussian approximation of the χ 2 .

The sensitivity of observables to New Physics
As shown above, different NP scenarios currently give a good description of the data, so the natural question is which other observables, beyond R D and R D * , allow one to discriminate among them. Only total rates are sensitive to the effects from the vector operators as their effects cancel in normalized observables. On the other hand, scalar and tensor operators change the kinematic distributions of the decays and show up in observables such as tau and recoiling-hadron polarizations (if the latter carries spin), q 2 -distribution of the rate or angular analyses. In figure 7, we study the q 2 spectra of R D ( * ) and of a selection of polarization and angular observables 5 showing their sensitivity to NP. We select scenarios that can be motivated by UV completions such as those involving scalar-tensor or vector-scalar combinations of operators, and we also study the tensor scenario. The values of the WCs are fixed to the results of the fits to the R D ( * ) data, i.e, τ T = −0.03, ( τ S L , τ T ) = (0.07, −0.03), ( τ L , τ S R ) = (0.08, −0.05). In table 6 we show the results of these observables integrated over the whole kinematic region for the SM and the different NP scenarios considered. In-5 All of them have been defined in section 1, except the tauonic forward-backward asymmetry, which is independent of overall normalization [125].

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Observables SM τ T = −0.03  terestingly, none of the preferred scenarios with up to two WCs can satisfactorily describe the Belle measurement of F D * L along with the experimental enhancements reported in R D and R D * .
From the plots in figure 7 and predictions in table 6, one concludes that a clear pattern emerges in these observables for the different NP scenarios currently favored by the data, although high precision measurements will be required to discriminate among them. The most sensitive ones for this purpose turn out to be the tau polarization and forwardbackward asymmetry of the B → Dτ ν decay mode. Interestingly, with the 50 ab −1 expected to be collected by Belle II a relative statistical uncertainty better than ∼ 10% has been estimated for these observables integrated over the whole q 2 region [125].

Summary and outlook
In this work, we have studied in detail the status of the new-physics interpretations of the b → cτ ν anomalies after the addition of the Belle measurements of R D ( * ) using the semileptonic tag and F D * L to the data set. We perform two types of fits: first, we fit with one and two parameters (Wilson coefficients) to the 2019 HFLAV average of R D and R * D with particular attention to the evolution of the preferred scenarios with the new data and to the consistency with the upper bounds that can be derived from the lifetime of the B c meson and the pp → τ h X+MET signature at the LHC. The main conclusion is that NP interpretations driven by left-handed currents and tensor operators are favored by the data with a significance of ∼ 3.5σ with respect to the SM hypothesis. Solutions based on pure right-handed currents remain disfavored by the LHC data while scenarios with that only have scalar contributions are in conflict with both, the LHC and the B c -meson experimental inputs. In fact, the LHC upper bounds currently exclude large regions of JHEP12(2019)065 the parameter space allowed by the R D ( * ) data, and in the high-luminosity phase it should start probing all the interesting regions.
We also perform a second global fit of all the NP operators with (left-handed neutrinos) to the R D ( * ) data, R J/Ψ , F D * L and P D * τ . The main effect of the added observables, in particular of F D * L , is to exclude the regions involving large values of the WCs, in complementarity with the upper LHC bounds. Otherwise, the favored regions by the global fits are equivalent to the ones resulting from the fit to R D ( * ) .
A caveat to our conclusions is that the LHC bounds derived from the analysis in terms of effective operators are not applicable if the mass scale of the new mediators they correspond to is lighter than ∼ 2 TeV. Scenarios based on S 1 and U 1 leptoquarks coupled to right-handed neutrinos remain challenged by the monotau signature at the LHC except for the mass range which is being independently probed by pair-production at the LHC. A S 1 leptoquark producing a scalar-tensor scenario does not provide a solution as optimal as with the 2018 HFLAV average, whereas in combination with the R 2 leptoquark it can provide the optimal tensor scenario. The R 2 leptoquark alone can also explain the data successfully when the couplings take complex values and, interestingly, its detection should be at reach in the HL-LHC. Best solutions are incarnated by the S 1 and U 1 leptoquarks with pure left-handed couplings, possibly in combination with right-hand currents in the latter case.
Finally, we investigate the sensitivity of different observables to NP. We find that the tau polarization in the B → Dτ ν decay is sensitive to the various scenarios favored by the data. Interestingly, Belle II could achieve a precision in this observable that would provide discriminating power among them.