Constraining new physics in $b\to c\ell\nu$ transitions

$B$ decays proceeding via $b\to c\ell\nu$ transitions with $\ell=e$ or $\mu$ are tree-level processes in the Standard Model. They are used to measure the CKM element $V_{cb}$, as such forming an important ingredient in the determination of e.g. the unitarity triangle; hence the question to which extent they can be affected by new physics contributions is important, specifically given the long-standing tension between $V_{cb}$ determinations from inclusive and exclusive decays and the significant hints for lepton flavour universality violation in $b\to c\tau\nu$ and $b\to s\ell\ell$ decays. We perform a comprehensive model-independent analysis of new physics in $b\to c\ell\nu$, considering all combinations of scalar, vector and tensor interactions occuring in single-mediator scenarios. We include for the first time differential distributions of $B\to D^*\ell\nu$ angular observables for this purpose. We show that these are valuable in constraining non-standard interactions. Specifically, the zero-recoil endpoint of the $B\to D\ell\nu$ spectrum is extremely sensitive to scalar currents, while the maximum-recoil endpoint of the $B\to D^*\ell\nu$ spectrum with transversely polarized $D^*$ is extremely sensitive to tensor currents. We also quantify the room for $e$-$\mu$ universality violation in $b\to c\ell\nu$ transitions, predicted by some models suggested to solve the $b\to c\tau\nu$ anomalies, from a global fit to $B\to D\ell\nu$ and $B\to D^*\ell\nu$ for the first time. Specific new physics models, corresponding to all possible tree-level mediators, are also discussed. As a side effect, we present $V_{cb}$ determinations from exclusive $B$ decays, both with frequentist and Bayesian statistics, leading to compatible results. The entire numerical analysis is based on open source code, allowing it to be easily adapted once new data or new form factors become available.


Introduction
interactions in b → c ν. The main aim of this paper is to perform a comprehensive modelindependent analysis of all possible types of NP effects in b → c ν, making use of the wealth of experimental data. This paper is organized as follows: In section 2, the effective Hamiltonian for b → c ν is defined and relations among the operators implied by SM gauge invariance are discussed. In section 3, we discuss our treatment of B → D and B → D * form factors that are crucial ingredients in the analysis of exclusive b → c ν decays. In section 4, we perform fits to V cb from B → D ν and B → D * ν. Reproducing the values in the literature, this step is useful as a crosscheck of our numerics. We also perform the analysis with frequentist and Bayesian statistics, explicitly demonstrating their agreement. In section 5, we perform the NP analyses, starting with a discussion of NP models with tree-level mediators and their characteristic patterns of Wilson coefficients, and subsequently discussing each of the relevant operator combinations. Section 6 contains our conclusions.
An important feature of our analysis is that it is entirely based on open-source code. We have implemented all observables of interest as well as our predictions for B → D and B → D * form factors in the flavio flavour physics package [39]. This has three benefits: First, it makes our analysis transparent and reproducible. Second, it allows anyone to update the best-fit values of V cb or the allowed ranges for the Wilson coefficients when new experimental data or new lattice form factor computations become available. Third, it easily allows to study the viability of more involved NP models with multiple Wilson coefficients, that cannot be easily visualized in two-dimensional plots. Additionally, to corroborate the reliability of our results, we have obtained all numerical results with a completely independent Mathematica code.

Effective Hamiltonian
The effective Hamiltonian for b → c ν transitions can be written as 1 where the sum runs over the following operators: with in general charged-leptonand neutrino-flavour-dependent coefficients. Since we are focusing on decays with light leptons in the final state, we only consider = e or µ, but allow for = e, µ, τ , which cannot be distinguished experimentally. We have defined the coefficients C i in (2) such that they vanish in the SM. A lepton-flavour universal and diagonal NP effect in C V L can always be absorbed by a shift in V cb , since V cb is a free parameter in the SM and presently not meaningfully constrained by CKM unitarity. In the following, we will use the shorthands where appropriate.
The operators in the effective Hamiltonian arise from more funadamental interactions at or above the electroweak scale. The available high-energy data from LHC indicate the existence of another energy gap between the electroweak scale and that of NP. In such a scenario interactions beyond the SM can be cast into another effective theory, with operators symmetric under the full SM gauge group. For linearly realized electroweak symmetry breaking this effective theory is called Standard Model effectivie field theory (SMEFT), the operators of which can be ordered in terms of their mass dimension, with those at dimension six giving the dominant contributions here [41,42]. The tree-level matching of SMEFT operators onto the effective Hamiltonian (2) reads [43][44][45][46] where the definition of the SMEFT operators can be found in [42] 2 and we are employing a weak basis for SMEFT where down-type and charged-lepton mass matrices are diagonal. An important prediction of this framework is that the Wilson coefficient C V R is lepton flavour universal and diagonal [45,47]. A deviation from this prediction would hence indicate a nonlinear realization of electroweak symmetry breaking [45]. Presently such a deviation is not observed, however, and we will use C V R ≡ C V R δ in the remainder of this paper. Another implication of SMEFT is that that the Wilson coefficients C 3i ledq , that give rise to C S R , also generate neutral-current operators of the form (q L b R )(¯ R L ), where q = d, s, b. For q = s or d, these operators are constrained very strongly by the leptonic decays B d,s → , that are forbidden for = and strongly helicity suppressed for = in the SM. From existing bounds and measurements, we find that the SMEFT Wilson coefficients C 31 ledq and C 32 ledq can induce effects in C S R at most at the per mil level, which would not lead to any visible effects in b → c ν at the current level of precision. However, sizable effects in C S R cannot be excluded, since the coefficients C 33 ledq contributing to the sum in (5) only generate the flavour-conserving operators (b L b R )(¯ R L ), that are allowed to be sizable. 3 Finally, the availability of the full anomalous dimension matrix for SMEFT dimension-six operators [49][50][51] allows for the prediction of operator mixing due to electroweak renormalization effects; this will be discussed briefly at the end of section 5.1.

B → D ( * ) form factors
The hadronic form factors of the B → D ( * ) transitions are crucial both for the determination of the CKM element V cb in the SM and for constraining NP contributions to b → c ν. An important difference between the two scenarios is that in the SM V cb only changes the overall normalization of the rates, but does not modify the shapes of differential distributions. NP contributions on the other hand can modify these shapes and can also involve additional form 2 We use a normalization LSMEFT = i CiOi, i.e. dimension-6 operators have dimensions of inverse mass squared. 3 Note that these operators do not contribute to (and thus are not constrained by) leptonic decays of Υ(nS) [48].
factors, in particular tensor form factors. Since our main interest is constraining NP in b → c ν, we want to use as much information on the form factors from theory as possible, while at the same time remaining conservative enough not to introduce ficticious tensions with the precise experimental data due to too rigid parametrizations. We therefore use information from four complementary methods to determine the B → D ( * ) form factors: • LQCD. We use all available unquenched LQCD calculations of B → D ( * ) form factors.
The B → D vector and scalar form factors have been computed by the HPQCD [5] and the Fermilab/MILC collaborations [6] at several values of q 2 , constraining the shape of these form factors in addition to their normalizations. The FLAG collaboration has performed an average of these two computations, fitted to the BCL parametrization [12]. For the B → D * form factors, so far only calculations at zero hadronic recoil have been reported [7,8]; we use their average calculated in [8].
• QCD light-cone sum rules allow to compute the form factors in the region of large hadronic recoil, depending on B meson light-cone distribution amplitudes as non-perturbative input [52]. This method is complementary to LQCD, being valid in the opposite kinematic limit. We use the form factor values and ratios obtained in Ref. [52] at 4 q 2 = 0, and extract from their values given for ρ 2 D,D * two more, correlated pseudo-datapoints at w(q 2 ) = 1.3.
• Heavy Quark Symmetry and Heavy Quark Effective Theory (HQET). Treating both the bottom and charm quark as heavy compared to a typical scale of QCD interactions, QCD exhibits a symmetry among the heavy quarks [53,54]. As a consequence, all B → D ( * ) form factors either vanish or reduce to a single function -the leading Isgur-Wise function -in the infinite mass limit [55]. Perturbative QCD and power corrections to this limit are partly calculable [13,[56][57][58][59][60][61][62][63][64], to be discussed below.
• By crossing symmetry, the form factors describing the semi-leptonic transition also describe the pair production of mesons. Unitarity can then be used to impose constraints on the form factors. Taking into account contributions from other two-body channels with the right quantum numbers leads via a conservative application of HQET to the strong unitarity constraints [65,66]. We employ the updated bounds given in [9] for B → D and the simplified bounds derived in [11] for B → D * .
We proceed by using the HQET parametrization of all B → D ( * ) form factors, including corrections of O(α s , Λ QCD /m c,b ), in the notation of [13], with two differences: 1. Instead of using the CLN relation between slope and curvature of the leading Isgur-Wise function, we include both as independent parameters in the fit.
2. The treatment of higher-order corrections of O(Λ 2 QCD /m 2 c ) has recently been shown to have a significant effect on the extraction of V cb in the SM, see the discussions in [9][10][11][12][13][14][15][16]. We make these corrections explicit by including in addition to the subleading Isgur-Wise functions at order Λ QCD /m c,b corrections of order Λ 2 QCD /m 2 c in those HQET form factors that are protected from O(Λ QCD /m c ) corrections. 5 From comparison of the HQET predictions at O(α s , Λ QCD /m b,c ), using the three-point sum rule results for the subleading 4 The kinematic variable w is related to q 2 as w = (m 2 B + m 2 D ( * ) − q 2 )/(2mBm D ( * ) ). 5 Note that two of these corrections are implicitly included in [13] when the normalizations of the B → D ( * ) form factors hA 1 ,+ are decoupled from their HQET values.
Isgur-Wise functions [60][61][62] with LQCD results, we observe a shift of about −10% from these corrections in h A 1 (1), while the corrections in h + (1) are only a few per cent. For h T 1 (1) we allow for an independent correction of 10% which is not constrained by lattice data. Note that these corrections are in principle obtainable from LQCD in all form factors, however, so far only results for those appearing in the SM are available.
We then perform Bayesian and frequentist fits of this parametrization to pseudo data points corresponding to the described inputs. The result is a posterior probability distribution or profile likelihood for all the form factor parameters, respectively. These can be interpreted as theory predictions for all B → D ( * ) form factors in the entire (semi-leptonic) kinematic range. This theory prediction is then used in our numerical analysis as a prior on (in Bayesian fits) or a pseudo-measurement of (in frequentist fits) the form factor parameters.

V cb from exclusive decays in the Standard Model
Fits for the CKM element V cb from the exclusive decays B → D ν and B → D * ν measured at the B factories BaBar and Belle have already been performed in the literature (see e.g. [6,[9][10][11][12][13][14][15][16] for recent fits). Here, we repeat this exercise to define our assumptions on form factors and our experimental input. Furthermore, all of our fits are reproducible using open source code, allowing them to be adapted or modified with different theoretical or experimental inputs. We perform fits to B → D ν and B → D * ν decays, where the theoretical uncertainties are dominated by the hadronic form factors. To study the impact of different statistical treatments of these "nuisance" parameters, we consider three different fits: • A frequentist fit where the theoretical knowledge of form factors is treated as a pseudomeasurement and the individual parameters are profiled over.
• A Bayesian fit where the theoretical knowledge of form factors is treated as a prior probability distribution and the individual parameters are marginalized over.
• A "fast fit" where the theoretical uncertainty on each bin is determined by varying the form factor parameters according to the theoretical constraints and is added in quadrature with the experimental uncertainties.
The BaBar collaboration has measured the differential branching ratio of B → D ν, reconstructed with hadronic tagging [35], in ten bins, averaged over electrons and muons as well as charged and neutral B decays. Since only statistical uncertainties are given, we follow [9] and add a fully correlated systematic uncertainty of 6.7% on the rate. To avoid a bias towards lower values of V cb caused by underfluctuations in individual bins ("d'Agostini bias" [71]), we always treat relative systematic errors as relative to the SM predictions rather than the experimental central values. In addition to this differential measurement, we also include a BaBar measurement of the total branching ratio from a global fit, split by electrons and muons [67], that is statistically independent of the former; following Ref. [11], we assume the measurement of the total branching ratio to be unaffected by the form factor parametrization. We take into  Table 1: Experimental analyses considered. The analyses labeled by B → D ( * ) ν do not differentiate between the lepton species and are hence not used when analyzing scenarios with non-universal coefficients.
account the significant correlation with the D * modes extracted in the same analysis. Following HFLAV [72], we apply a rescaling of −3.7 % to the published branching ratio to account for updated D branching ratios.
We note that we cannot use the global HFLAV average of the B → D ν branching ratio because • it contains older measurements that assumed a particular form factor parametrization and would be inconsistent to use in a NP analysis, • it contains measurements from the anaylses that we include in binned form, such that including it would amount to double-counting, • it only considers the average of the electronic and muonic branching ratios, so we cannot use it for the lepton-flavour dependent NP analysis.
The Belle collaboration has measured the differential branching ratios separately for electrons and muons as well as charged and neutral B decays in ten bins each [37], specifying the full correlation matrix. This measurement does not rely on a specific form factor parametrization. The combined fit to Belle and BaBar data with the three different statistical approaches, using the form factors described in section 3, yields where the errors in the frequentist fit refer to a change by 0.5 in the profile likelihood and in the Bayesian case correspond to the highest posterior density interval with 68.3% credibility. The agreement of the central value between the frequentist and the Bayesian fit is not surprising as the minimum of the two likelihoods coincides (the prior probability for theory parameters has the same mathematical form as the "pseudo measurements" used in the frequentist fit). That We observe good compatibility with other recent extractions from this mode [9,13,16]. A direct comparison is not possible, since neither the form factor parametrizations nor the data sets used are identical. The inclusion of additional data in our case is responsible for the slightly smaller uncertainties. Note that the inclusion of the measurement [67] shifts V cb to smaller values compared to [9].
In our numerical analysis of new physics effects in section 5, when allowing for lepton flavour non-universal effects, we only use measurements where electron and muon samples are separated, since the generally different but unknown electron and muon efficiencies prohibit a consistent interpretation of the combined measurements in such scenarios. It is instructive to extract the value of V cb only from these subsets of measurements. Using the frequentist approach, we find We observe consistency among these values and with the global fit, albeit with larger uncertainties.
Since the D * is a vector meson and further decays to e.g. Dπ, a four-fold differential decay distribution in three angles and w can be measured. Belle has recently published an analysis with one-dimensional distributions in all four kinematic quantities with full error correlations, based on hadronic tagging [38]. An earlier (and statistically independent) untagged analysis by Belle [36] considers the w-differential branching ratios into longitudinally or transversely polarized D * , that can be extracted from the angular distribution. Since the error correlations are not publicly available, we simply assume the statistical uncertainties to be fully uncorrelated and the systematic ones to be fully correlated. In addition, we rescale the systematic uncertainties, that are given as relative uncertainties with respect to the measured central values, into relative uncertainties with respect to the SM prediction instead, again to avoid the D'Agostini bias mentioned before. In addition to the differential measurements, we include four measurements of the total branching ratio by BaBar, listed in table 1, which are statistically independent to a good approximation [73]. As mentioned in section 4.1, we do however take into account the significant correlation between the B → D * (e, µ)ν and B → D(e, µ)ν measurements of the BaBar "global fit". As in the case of B → D ν, we cannot use the HFLAV average for the B → D * ν branching ratio, but we apply the same rescalings as HFLAV to account e.g. for changes in D * branching ratios. The central value of the BaBar global fit is hardly modified, but the other three branching ratio measurements are reduced by 5-6% compared to the published values.
The combined fit to these data, using our form factor parametrization with the constraints discussed in Sec. 3, yields Again, we observe excellent agreement of the three different approaches and the same comments as in section 4.1 apply. We conclude that the extraction of V cb from exclusive decays does not depend in a relevant way on the statistical approach taken. The extracted values are also comparable to those in the recent literature. Again we observe good compatibility with other recent extractions from this mode [10,13,14,16], and the same comments regarding comparability apply as in B → D. The inclusion of additional data in our case is responsible for the slightly smaller uncertainty and lower central value. The latter is also related to a shift from using the experimental values from HEPData, as discussed below.
As for B → D ν at the end of section 4.1, we also repeat the extraction of V cb using only the measurements that separate the electron and muon samples, since these measurements are used in lepton flavour non-universal NP scenarios in section 5. Using the frequentist approach, we find While consistent with each other and with the global fit, we observe that the muonic data prefer a value of V cb that is lower by about one standard deviation. Finally we would like to comment on the robustness of the extracted V cb value from binned data. The value obtained from the Belle 2017 data alone depends on the precise inputs used: the difference between the binned data given on HEPData [74] and that in the publication, where the values are rounded to two significant digits, yields a shift in V cb of about one standard deviation. 6 This is problematic, since the uncertainties and correlations themselves have an uncertainty that is likely to be larger than the difference between these inputs. This should be kept in mind when analyzing the unfolded spectrum. We proceed using the inputs given on HEPData where available.

New physics
Having extracted the CKM element V cb from data assuming the SM, we next proceed to constrain NP effects. As discussed in section 2, there are five Wilson coefficients per leptonflavour combination, with relations for the right-handed vector current C V R ≡ C V R δ , i.e. up to 25 independent Wilson coefficients for b → c(e, µ)ν transitions. Since the ones for = are indistinguishable, these are effectively reduced to 17, and in the lepton-flavour diagonal case only 9 operators remain. Before analyzing them in more detail, we discuss in section 5.1 all possible simplified models that can be generated by the exchange of a single new mediator particle, implying specific combinations of these Wilson coefficients.
While we focus on exclusive modes in the SM V cb fits, for the NP fits we also compare to the inclusive decays B → X c ν. Since the full inclusive analysis involves fitting several moments of the spectrum simultaneously with V cb , quark masses, and HQET parameters, reproducing it is beyond the scope of our present analysis. Rather, we use a simplified approach where we approximate the total rate in the presence of new physics as where Γ(B → X c eν) SM is the full state-of-the-art SM prediction [3] and the rates with superscript "LO" refer to the partonic leading-order calculations. We confirm the known expressions for the LO contributions in the m → 0 limit [28,34,75,76]: and Γ 0 = G 2 F m 5 b |V cb | 2 /(192π 3 ). 1/m 2 c,b and α s corrections to the NP contributions are partly known and can be sizable [28,34,[76][77][78][79][80]. Specifically for scalar operators the α s corrections can qualitatively change the result [76]. However, since we do not include effects on the spectra in any case, we stick to the simple expressions given above; we therefore do not consider the inclusive constraints used here as on the same footing as the exclusive ones and do not combine them in a global fit.
Finally, we include the constraint from the total width of the B c meson, which can be modified significantly in scenarios with scalar couplings [81,82].

Tree-level models
Since the b → c ν transition is a tree-level process in the SM, NP models with sizable effects in these modes typically involve tree-level contributions as well. The known quantum numbers of the involved fermions allow to determine all possible mediator quantum numbers. These correspond to the following simplified models: • New vector-like quarks modifiying the W -couplings of the SM quarks, • tree-level exchange of a heavy charged vector boson (W ), • tree-level exchange of a heavy charged scalar (H ± ), • tree-level exchange of a coloured vector or scalar boson (leptoquark).
Vector-like quarks are among the simplest renormalizable extensions of the SM. They are also present, e.g., in models with partial compositeness in the form of fermionic resonances of the strongly interacting sector. An SU (2) L singlet vector-like quark with mass m Ψ , coupling to the SM quark doublet and the Higgs with strength Y , can generate a modified left-handed W coupling of order Y 2 v 2 /m 2 Ψ , while an SU (2) L doublet, coupling to the SM quark singlet and the Higgs, can generate a right-handed W coupling. The resulting contributions to the Wilson coefficients C V L or C V R , respectively, are lepton-flavour universal.
The tree-level exchange of W , H ± , or leptoquarks has been studied extensively in the literature in the context of explanations of the b → cτ ν anomalies (see e.g. [24,[83][84][85][86][87][88][89][90][91][92].) The case of leptoquarks is particularly diverse as there are six different representations -three scalars and three vectors -that can in principle contribute to b → c ν at tree level. The b → c ν Wilson coefficients generated in each of the tree-level models considered are shown in table 2. Crosses correspond to a possible tree-level effect at the matching scale, corresponding to the scale of new physics. If two crosses appear in a line, the contributions are governed by independent parameters in the model.
An interesting observation concerning table 2 is that none of the models generates the tensor operator O T on its own. At the matching scale, it is only present in the two scalar leptoquark models S 1 and R 2 in a characteristic correlation with the operator O S L .
Since table 2 is valid at the new physics scale, it is also important to consider renormalization group (RG) effects between this scale and the b quark scale that could possibly change this picture. Normalized as in Eq. (2), it can be easily seen that the vector operators are not renormalized under QCD as they correspond to conserved currents; the same is true for the combinations m b O S L,R , such that the Wilson coefficients renormalize under QCD like the quark mass. Thus, the operator O T , which is mildly QCD-renormalized, cannot mix with the other operators under QCD. Therefore the only qualitatively relevant RG mixing effects among the five operators of interest could come from electroweak renormalization in the SMEFT above the electroweak scale. Indeed, the operators O lequ , that match onto O S L and O T , respectively, mix with each other through the weak coupling constants [93]. The RG-induced values of the respective coefficients at the electroweak scale v can be written in leading logarithmic approximation as Model Table 2: Pattern of Wilson coefficients generated by tree-level models. Crosses correspond to a possible effect at the matching scale. If two crosses appear in a line, the contributions are governed by independent parameters in the model.
where the self-mixing contributions have been omitted for brevity, but are included in the numerical analysis. For Λ NP = 1 TeV, the numerical prefactors are roughly −0.1 and −0.002, respectively, so the effect is small unless the scale separation is very large, in particular given that the tree-level models already predict C S L C T at the matching scale. We thus conclude that table 2 is useful as classification of tree-level models in terms of low-energy effects and is qualitatively stable under quantum corrections. Of course, a realistic model may combine several of the discussed particles and thus lead to a more diverse pattern of effects.

Right-handed currents
C V R has been considered as a possible source for a tension between inclusive and exclusive determinations in V cb for a long time [94]. However, updated analyses based on total rates alone have already shown that the scenario is disfavoured as a solution to the present tension [33].
The main novelty of the present analysis is the inclusion of the B → D * ν angular observables in the fit. For the total rate, an effect in C V R can always be absorbed by an appropriate shift in V cb (orṼ cb ), such that constraining right-handed currents requires considering several modes. Including angular observables changes this picture, as the shape of these observables is directly sensitive to right-handed currents. As a consequence, a fit to B → D * ν data alone is able to constrain right-handed currents directly, as shown 7 in figure 2, together with the constraints from B → D ν and the inclusive decay, both constraining only combinations of V cb and C V R . In this plot we assume lepton-flavour universality as discussed above.
We observe that the tensions between the V cb determinations from B → D ν, B → D * ν, and B → X c ν, that are anyway milder than in the past, cannot be completely removed by postulating new physics in right-handed currents. What is new is that even B → D * ν alone cannot be brought into perfect agreement with B → X c ν for any value of C V R .

Lepton flavour universality violation
In view of the observed tensions with SM expectations in b → cτ ν and b → s transitions, investigating e-µ universality in b → c ν with light leptons is important. Specific new physics models suggested as solutions to the b → cτ ν anomalies actually predict such violation. Some of the experimental analyses assume LFU to hold. These analyses cannot be used in a modelindependent fit allowing for LFU violation. This is because the measurements are not simply averages of the respective electron and muon observables, but linear combinations with weights depending on the experimental efficiencies that can differ between electrons and muons even as a function of kinematical variables. Thus it is of paramount importance that experimental collaborations present their results separately for electrons and muons. In the meantime, the existing analyses that already include separate results for electrons and muons (see table 1) can be used to perform a fit with a non-universal modification of the SM operator, i.e. C e V L = C µ V L . The fit result in terms of the lepton-flavour-dependent effective CKM elementsṼ cb is shown in figure 3. Both for B → D ν and B → D * ν the fit not only shows perfect agreement with LFU, but also implies a stringent constraint on departures from the LFU limit. Given the good agrement of the constraints from B → D ν and B → D * ν, we have also performed a combined Bayesian fit of the scenario to both decay modes, marginalizing over all nuisance parameters. We find with a small correlation of −10%. Equivalently, we find which can be used as a generic constraint on e-µ universality violation in b → c ν processes. This is the first global combination of LFU constraints in b → c ν transitions and provides a significantly stricter bound on LFU than the recent measurement in Ref. [38] alone. As already discussed in section 2, a violation of LFU can also manifest itself as a contribution from a "wrong-neutrino" decay generated by C V L with = . However, this case does not have to be discussed separately as it merely leads to a rescaling of all observables that can be absorbed by definingṼ

Scalar operators
The interference of scalar contributions with the SM amplitude in exclusive and inclusive decays arises only from lepton-mass-suppressed terms. Consequently, scalar contributions always lead to an increase in the rates for a fixed value of V cb . The suppression of interference terms also implies that there is no qualitative difference between operators with = or = , so we focus on the former for simplicity. Again the differential distributions contain valuable information about possible scalar contributions. The most striking example is the endpoint of the B → D ν differential decay rate, see also, e.g., [95]. In the SM, close to while in presence of NP contributions to scalar operators, it behaves as This implies an exceptional sensitivity of the highest q 2 (or lowest w) bin in B → D ν to the sum of scalar Wilson coefficients, C S R + C S L . To illustrate this effect, in figure 4 we show on the left the predictions for the SM (using V cb from eq. (7)) and a scenario with a large NP contribution in C S L for fixed V cb , compared to the experimental data. The scenarios are chosen such that they give the same prediction for the total branching ratio. The qualitatively different endpoint behaviour can clearly distinguish between them, excluding such large scalar contributions. As a consequence, a fit to the differential B → D ν rates leads to a more stringent constraint on scalar operators than B → D * ν or the inclusive decay. While the total rates of these modes constrain only combinations of |V cb | and |C S L ± C S R | or |C S L | 2 + |C S R | 2 , these are accessible individually with differential distributions. This is demonstrated in figure 4 on the right, where the constraint resulting from the total rate alone is compared to the one resulting from all available information on that channel. Importantly, including the information from the differential rate,Ṽ cb and Re(C S R ) can be determined individually.
The constraining power of B → D ν and the complementary dependence of B → D ν and B → D * ν on the scalar Wilson coefficients C S L and C S R is also demonstrated in figures 5 and 6. In these figures, the constraints are shown separately for the modes involving electron and muons, as there is in general no reason to expect LFU to hold for scalar contributions. Furthermore, here and in the following only the measurements of electronic modes are used to constrain the electronic coefficients and the measurements of muonic modes for the muonic coefficients. We stress that more stringent constraints would be obtained by assuming the coefficient with the other lepton flavour to be free from NP and using both sets of data. In this respect, our constraints are conservative.
In figure 5, corresponding to scenarios with a U 1 or V 2 leptoquark, the electronic modes do not show any preference for non-zero C e S R , while B → D * µν prefers a sizable value of C µ S R slightly over the SM, but such values are excluded by B → Dµν in this scenario, and also disfavoured by the B c lifetime. A second interesting scenario is having NP in both C S R and C S L , as can arise in models with a charged Higgs boson (cf. table 2). The resulting constraints are shown in figure 6, again separately for electronic and muonic modes. In addition to the form factor parameters, also V cb is varied as a nuisance parameter. Consequently, the inclusive decay alone, only entering our analysis via its total rate, cannot constrain the coefficients individually and is not shown. In principle, the same qualification applies to the leptonic decays B c → ν. To give an indication, we nevertheless show the constraints that would apply for a conservatively low value of V cb = 0.035. The feature that B → D ν only constrains the sum and B → D * ν only the difference of these Wilson coefficients is clearly seen. In the muonic case, we observe a mild preference for non-zero values of both Wilson coefficients from B → D ( * ) ν. In constrast to the scenario in figure 5, the constraint from B → D ν does not exclude sizable NP effects if C S R ∼ C S L . However, the constraint from the B c lifetime excludes such a scenario, even when allowing for very large modifications of ∼ 40% and a small value of V cb = 0.035.

Tensor operator
In contrast to the scalar operators, for tensor operators B → D * ν leads to a much more stringent constraint. This is due to the existing lepton-specific data on the transverse and longitudinal decay rates (cf. table 1). In the SM the D * is fully longitudinally polarized at maximum recoil (q 2 min = m 2 ). This is no longer true in the presence of tensor operators. Indeed, in the limit of q 2 → q 2 min ≈ 0, the differential rate to transversely polarized D * in the presence of NP contributions to C V L and C T behaves as where the kinematic relation T 2 (0) = T 1 (0) was used and the D * mass was neglected for simplicity. As a consequence, this observable is exceptionally sensitive to tensor operators near the maximum recoil point. To illustrate this fact, in figure 7 on the left we show the predictions for the SM and a scenario with a sizable NP contribution in C T compared to the experimental data. The scenarios are chosen such that they give the same prediction for the total branching ratio. The different behaviour at q 2 = 0 allows to clearly distinguish them and disfavours tensor contributions of this size. On the right of figure 7 we demonstrate again this qualitative difference by comparing the constraints from the total B → D * µν rate alone and including the differential distribution. An additional observable that would be very sensitive to the tensor operator, but has not been measured yet, is the "flat term" in B → D ν. The normalized differential decay rate as a function of the angle θ between the charged lepton and the B in the lepton-neutrino mass frame can be written as In complete analogy to the B → K + − decay, the observable F H vanishes in the SM up to tiny lepton mass effects, but can be sizable in the presence of new physics in the tensor operator. Neglecting the lepton masses and allowing for NP in C T and C V L , one finds (31) Figure 8 shows the constraints on the tensor and left-handed scalar operators, which always appear together in models with a tree-level mediator, see Table 2, specifically in leptoquark models. The displayed constraints from B → D ν and B → D * ν, shown separately for electrons and muons, demonstrate clearly the strong sensitivity of B → D * ν to tensor contributions. While the individual modes B → D * eν, B → Dµν, and B → D * µν show a slight preference for non-zero NP contributions in either C S L or C T , the combination of B → D ν and B → D * ν constraints allows neither of these solutions and leads to a strong constraint on both operators.

Conclusions
Semi-leptonic charged-current transitions b → c ν with = e or µ are traditionally used to measure the CKM element V cb . In principle, this transition could be affected by new physics with vector, scalar, or tensor interactions, possibly violating lepton flavour universality. This is motivated by the long-standing tensions between inclusive and exclusive determinations of V cb , but also by hints of a violation of lepton-flavour universality in b → cτ ν and b → s transitions. We have conducted a comprehensive analysis of general new-physics effects in b → c ν transitions, considering for the first time the full operator basis and employing for the first time in a new physics analysis measurements of B → D * ν angular observables.  Our main findings can be summarized as follows.
• Extracting the absolute value of the CKM element V cb from fits to the full sets of B → D ν and B → D * ν data in Table 1 yields values consistent with the recent literature and no significant tension between determinations from B → D ν vs. B → D * ν.
• We find no dependence of the V cb extraction on the statistical approach, but find a significant dependence on the treatment of systematic uncertainties in binned observables due to the "d'Agostini bias".
• We find that NP in right-handed currents cannot improve the agreement between inclusive and exclusive determinations of V cb . Thanks to our use of differntial and angular dsitributions, this conclusion can even be drawn considering B → D * ν vs. B → X c ν alone.
• We find strong constraints on violations of e-µ universality, specifically for C V L .
• We demonstrate that the zero-recoil endpoint of the B → D ν spectrum is exceptionally sensitive to NP in scalar operators.
• We demonstrate that the maximum-recoil endpoint of the transverse B → D * ν spectrum is exceptionally sensitive to NP in tensor operators.
Our analysis could be improved in several respects. The treatment of experimental data had partly to rely on crude estimates of the systematic uncertainties or correlations, where these were not public. We urge the experimental collaborations to publish this information for future and also existing analyses. Our treatment of the inclusive decay is also approximate, as discussed at the beginning of section 5. Clearly, a full fit to the moments of the inclusive mode would be interesting, but is beyond the scope of our present analysis. Finally, our treatment of B → D * form factors had to some extent to rely on the heavy quark expansion, with only partial inclusion of 1/m 2 c,b contributions. A full calculation of the q 2 dependence from lattice QCD, ideally including tensor form factors, would make these constraints much more reliable. We emphasize again that our analysis can be easily improved once this information becomes available as all of our code is open source.

A. Numerical results for B → D ( * ) form factors
In this appendix, we give details on the parametrization of form factors discussed in section 3 and present the numerical results of our fit to lattice and LCSR calculations employing the HQET parametrization and unitarity bounds. The B → D ( * ) form factors h i in the HQET basis 8 can be found in ref. [13]. In the heavy quark limit, they vanish or reduce to a common form factor, the leading Isgur-Wise function ξ(w). It is thus convenient to write the form factors as h i (w) = ξ(w)ĥ(w). The expressions for allĥ(w) at next-to-leading order in α s and next-to-leading power in b,c =Λ/2m b,c can be found in ref. [13]. As discussed in section 3, we modify these expressions by allowing for an additional O( 2 c ) correction to the form factors that are protected from O( c ) corrections: We neglect a possible w dependence of the δ i terms. The leading order Isgur-Wise function ξ can be written to second order in the z expansion as ξ(z) = 1 − 8ρ 2 z + (64c − 16ρ 2 )z 2 .
We then perform a Bayesian fit (a Markov Chain Monte Carlo employing flavio [39] and emcee [97]) to the theory constraints described in section 3 of the ten parameters parametrizing the functionsĥ i and ξ. 9 The mean, standard deviation and correlation matrix of nine of those parameters is −0.058 ± 0.019 −0.001 ± 0.020 0.035 ± 0.019 0.358 ± 0.043 0.044 ± 0.125 1.306 ± 0.059 1.220 ± 0.109 −2.299 ± 0.394 0.485 ± 0.269 The tenth parameter, δ T 1 , is not constrained by the fit, thus its posterior is equal to the prior, which we conservatively take to be a Gaussian with mean 3 centered around 0. These form factors have been implemented and set as defaults in flavio version 0.26.