Symmetries in $B \to D^* \ell \nu$ angular observables

We apply the formalism of amplitude symmetries to the angular distribution of the decays $B \to D^* \ell \nu$ for $\ell=e,\mu,\tau$. We show that the angular observables used to describe the distribution of this class of decays are not independent in absence of New Physics contributing to tensor operators. We derive sets of relations among the angular coefficients of the decay distribution for the massless and massive lepton cases which can be used to probe in a very general way the consistency among the angular observables and the underlying New Physics at work. We use these relations to access the longitudinal polarisation fraction of the $D^*$ using different angular coefficients from the ones used by Belle experiment. This in the near future can provide an alternative strategy to measure $F_L^{D^*}$ in $B \to D^* \tau \nu$ and to understand the relatively high value measured by the Belle experiment. Using the same symmetries, we identify two observables which should exhibit a tension if the experimental value of $F_L^{D^*}$ remains high. We discuss how these relations can be exploited for binned measurements. Finally we propose an alternative path to measure the lepton-flavour universality ratio $R_{D^*}$ by connecting it with $F_L^{D^*}$ and various angular coefficients.

L in B → D * τ ν and to understand the relatively high value measured by the Belle experiment. Using the same symmetries, we identify two observables which should exhibit a tension if the experimental value of F D * L remains high. We discuss how these relations can be exploited for binned measurements. Finally we propose an alternative path to measure the lepton-flavour universality ratio R D * by connecting it with F D * L and various angular coefficients.

Motivation
Over the last six years, the hints of a tension with respect to Standard Model (SM) expectations have been growing concerning two different classes of b-quark decays, generically described as b-anomalies.
On the one hand, the interest of neutral-current b → sµµ transitions was highlighted by the measurement of B → K * µµ angular observables, and in particular the observable called P 5 [1] exhibiting discrepancies with respect to the SM at the level of 3.7σ [2][3][4][5][6]. Consistent deviations appeared in other channels such as B → Kµµ and B s → φµµ (mainly for the branching ratios), but also in a different type of observable, namely Lepton Flavour Universality Violating (LFUV) observables probing the universality of the lepton coupling in b → s comparing = e and = µ. Recent experimental updates have confirmed the presence of these deviations at the level of 2.5σ [7][8][9]. Global fits within an Effective Field Theory (EFT) approach performed on the large set of observables available have shown the remarkable consistency of the deviations observed, which can be explained through various New Physics (NP) scenarios affecting only a limited number of operators by shifting the short-distance physics encoded in Wilson coefficients. For instance, in Refs. [10,11], it was shown that adding NP to one or two Wilson coefficients is sufficient to obtain an improvement of the fit with respect to the SM (measured by the corresponding pull) by more than 5σ.
On the other hand, charged-current b → c ν transitions have also exhibited deviations in LFUV observables comparing = τ and lighter leptons. First measured as deviating significantly from the SM in 2012 [12,13], the relevant ratios R D and R D * have been updated regularly, leading to a recent decrease of the deviation with respect to the Standard Model down to 3.1σ [14][15][16][17][18]. Additional observables have been considered for B → D * τ ν concerning the polarisation of both the D * meson [19] and the τ lepton [20,21]. If the latter agrees with the SM within large uncertainties, the precise Belle measurement of the integrated F D * L yields a relatively high value compared to the SM prediction, which appears difficult to accommodate with NP scenarios, as can be seen in Refs. [22][23][24] which considered a wide set of NP benchmark points.
While neutral-current anomalies hinted at in a large set of channels and observables can be caused by small NP processes competing with SM contributions generated at the loop level, charged-current anomalies seen in two LFUV ratios should correspond to much larger NP contributions able to compete with tree-level SM processes. In this sense, the latter were much more unexpected and should be scrutinised in more detail, in order to confirm their existence.
In this note we pay close attention to the decay B → D * ν governed by the quark level transition b → c ν with = τ and = e, µ, and more specifically to its angular distribution. Depending on the NP hypotheses chosen, we will identify a set of symmetries for the massless (electron and muon) and massive (tau) distributions that will lead us to find a set of dependencies or relations among the angular coefficients of the distribution. A similar exercise was done in Refs. [25][26][27] for the case of the decay mode B → K * µµ. Here we will follow closely the detailed work in Ref. [25] to use the symmetries of the distribution in order to show that depending on the assumptions of the type of NP at work and the mass of the leptons, not all angular coefficients are independent. These relations can be used in the case of the B → D * ν decay as a way of cross-checking the consistency of the measurements of angular observables, but also to provide orientation on which kind of NP can be responsible for deviations with respect to the SM observed in these observables.
These relations among the observables, based on the symmetries of the angular distribution, lead to a new way of measuring F D * L for B → D * τ ν, relying on different coefficients of the distribution compared to the direct measurement performed by the Belle experiment. This can provide a different handle for experimentalists to cross-check the polarisation fraction and confirm or not its high value. Such an alternative extraction of the longitudinal D * polarisation can also be useful if instabilities occur when extracting the p.d.f. of angular observables due to values of F D * L beyond physical boundaries for instance 1 . We will provide general expressions for the relations among observables but we will focus mainly on a baseline case without tensor contributions (for the benchmark points analysed in Ref. [24], the presence of tensor operators decreases the value of F D * L for B → D * τ ν substantially, increasing the discrepancy with the measured value). On the other hand, we will consider the contribution of the pseudoscalar operator that can help to increase F D * L and bring it closer to the Belle measurement, as found in Ref. [24]. We will also discuss the simplified case where there are no large NP phases in the Wilson coefficients, i.e. when we assume the coefficients are real or the NP phases are small.
In Section 2 we recall the structure of the angular distribution and define the most relevant observables following Ref. [24]. In Section 3 we describe the formalism and explain how to count the number of symmetries and dependencies for each particular case and we work out the dependencies in the massless and massive cases, paying special attention to the presence of pseudoscalar operators. In Section 4 these dependencies are used to determine F D * L (or equivalently F D * T ) in terms of the other observables in various ways and we discuss the impact of binning when using these relations. In Section 5 we link the measurement of F D * L with the one of R D * and the alternative definitions of the former observable. We give our conclusions in Section 6. In App. A some details on the derivation of the exact massive dependencies are provided and illustrations of the binning effects for the relations discussed in this article are given in App. B.

Effective Hamiltonian and angular observables
The angular distribution for B → D * ν has been studied in Ref. [24]. Assuming that there are no light right-handed neutrinos, the distribution can be computed using the effective Hamiltonian: One may also use the equivalent notation of Refs. [22,23] (for instance) with the corresponding effective coefficients The resulting angular distribution is I 1c cos 2 θ D + I 1s sin 2 θ D + I 2c cos 2 θ D + I 2s sin 2 θ D cos 2θ (4) + I 6c cos 2 θ D + I 6s sin 2 θ D cos θ + I 3 cos 2χ + I 9 sin 2χ sin 2 θ sin 2 θ D + I 4 cos χ + I 8 sin χ sin 2θ sin 2θ D + I 5 cos χ + I 7 sin χ sin θ sin 2θ D , where the angular coefficients I i ≡ I i (q 2 ) are given in Ref. [24]: where N is a normalisation ) and the amplitudesH correspond to linear combinations of transversity amplitudes for various currents. We can write them in the following way to make the dependence on m explicit: where i = 0, +, − and H i correspond to vector and axial currents whereas H T,i correspond to tensor currents, andH P combines two amplitudes H t and H P : The H i amplitudes depend on form factors and on q 2 , but not on the lepton mass. In particular, the presence of 1/m inH + i means that the discussion of the limit m → 0 should be considered after expressing all the angular coefficients in terms of H i .

Observables
Contrary to B → K * [1,28], there are no specific discussions to consider concerning the possibility of optimised observables, since all B → D * form factors either vanish or yield the same Isgur-Wise function ξ in the heavy quark limit, so any ratio of angular observables is appropriate to reduce uncertainties from form factors. We thus take almost the same list as Ref. [24] for the 12 observables that form a basis: Compared to Ref. [24], we do not include the observable A λ in this list because it is related to the τ polarisation and requires one coefficient not included in the angular distribution. Instead we must introduce an additional observable (not included in Ref. [24]) so that the numbers of angular coefficients and observables match. We may choose for instance: We recall here the definition of the observables defined in Ref. [24] that will play an important role in this article: • The differential decay rate • The longitudinal and transverse D * polarisation decay rates: In order to make a more explicit contact with the integrated longitudinal polarisation we also introduceF D * L = (dΓ L /dq 2 )/Γ andF D * • The ratio R A,B describing the relative weight of the various angular coefficients in the partial differential decay rate with respect to θ , in analogy with the longitudinal polarisation fraction Eqs. (22), (23) and (24) are the "standard definitions" of dΓ/dq 2 , F D * L and F D * T respectively, and they are used to determine these observables with this particular functional dependence of the I's.
Similarly to the discussion in Ref. [29], the definition of observables integrated over a bin (or over the whole phase space) requires some care. Experimentally, the measurement yields the integrated angular coefficients I k with the definition 2 where the subscripts and 0 indicate the massive case (with m ) and the massless case respectively. We can then define the integrated longitudinal and transverse polarisations The Belle measurement is actually F D * L Belle τ = 0.60 ± 0.09.

Global fits
At this stage, a brief overview of our current understanding of the possible NP contributions is useful. Global fits to b → cτ ν favour overwhelmingly a NP contribution through a real g V L for b → cτ ν, as it allows one to modify the tauonic branching ratios involved in R D and R D * by the same amount without altering the angular observables, in agreement with the current data (apart from F D * L already discussed) [22][23][24]. For real contributions, scenarios based purely on scalar and pseudoscalar contributions exhibit some tension with the B c lifetime, depending on the relative size of the contribution allowed for B c → τ ν in the total lifetime, which requires the pseudoscalar contribution to be somewhat small. Similarly, real tensor contributions are disfavoured, as they tend to decrease the longitudinal polarisation of the D * meson compared to the SM [24], when the first measurement from the Belle experiment indicated a value higher than SM expectations [19]. If g V L is allowed as well as contributions of other operators, the former is dominant and the other operators (scalar, pseudoscalar, tensors) are subleading. Other constraints on b → cτ ν come from direct searches at LHC involving mono-τ jets [30]. The corresponding bounds are again much tighter on tensor operators than on vector or scalar operators.
Some of these scenarios allow large imaginary parts [22][23][24], with a similar hierarchy of scenarios as in the real case. However, one must take into account that such large imaginary parts are allowed due to the limited number of observables. Additional observables could bring a dramatic modification of the landscape of the allowed scenarios, restricting the possible size of imaginary parts and the applicability of scenarios currently viable severely. Indeed some of the NP scenarios favour large imaginary parts so that there are no interferences between the SM and NP contributions, which add up in quadrature only (see for instance the scenario of a purely imaginary g S L discussed in Ref. [31]). Restricting the size of these imaginary parts would enhance the interferences between SM and NP parts and would restrict the viability of the NP models where these interferences are negative. This trend is confirmed by model-dependent analyses. Most of the models with a singleparticle exchange aiming at reproducing the data in b → c ν do not generate tensor contributions, apart from the scalar SU (2) L -doublet leptoquark S 2 (see for instance Ref. [32]), which however generates much larger contributions to g S L (i.e. g S and g P ) than to g T L (i.e. g T and g T 5 ). This effect is enhanced by the running from the NP scale (1 TeV) down to the m b scale, so that scalar contributions are likely to be significantly larger than the tensor contributions if they are present at all [22].
We will thus consider as a baseline scenario that tensor contributions are subleading. We will also consider that the imaginary parts of the amplitudes can be neglected. In the SM as  Table 1: Symmetries and dependencies among the B → D * ν angular observables depending on the mass of the lepton and the contribution of tensor and pseudoscalar operators.
well as in the case of real NP, the only phase comes from the CKM matrix element, and it is actually the same for all the amplitudes. Under our baseline scenario, for instance, the angular coefficients corresponding to imaginary parts (I 7,8,9 ) are either small or vanishing, as well as any imaginary contribution. For completeness we will provide full expressions for the relations among the coefficients including these terms (see App. A for the general expressions in the massive case).

Relations among angular coefficients 3.1 Symmetries and dependencies
The decay B → D * ν has a rich angular structure, and it is interesting to investigate whether all the angular observables defined in the previous section are independent, following the same steps as in Refs. [25][26][27][28] for B → K * . We can consider the angular coefficients as being bilinears in An infinitesimal transformation will be given by For the infinitesimal transformation to leave the coefficients I unchanged, the vector δ has to be perpendicular to the hyperplane spanned by the set of gradient vectors ∇I i (with the derivatives taken with respect to the various elements of A). If the I i are all independent, the gradient vectors should span the whole space available for the coefficients, i.e. the dimension of the space for the gradient vectors should be identical to the number of angular coefficients. One can define: • The number of coefficients n c , given directly by the angular distribution • The number of dependencies n d , given by the difference between the number of angular coefficients I i and the dimension of the space given by the gradient vectors (provided by the rank of the matrix M ij = ∇ i I j ) • The number of helicity/transversity amplitudes n A , leading to 2n A real degrees of freedom • The number of continuous symmetries n s explaining the degeneracies among angular coefficients One has the following relation which we can investigate in various cases for B → D * ν summarised in Table 1.
As discussed above, the assumption of no tensor contributions seems favoured by the current global fits and we will stick to this assumption. In this case it is expected according to Table 1 the existence of 5 or 6 relations. The presence or absence of the pseudoscalar operator does not modify the outcome of the analysis and the number of dependencies in the massive case due to Eq. (19). However, we find interesting to discuss its effect separately as it was found in Ref. [24] that such a pseudoscalar contribution can help to alleviate the tension in F D * L for B → D * τ ν. We can now explore the dependence relations between angular coefficients, depending on the lepton mass, the presence of pseudoscalar and tensor operators. These relations can be used as a consistency test among the observables if all of these observables are measured in order to check the very general assumptions made to derive them. If these relations are not fulfilled, it means that there is an issue with one or more of the measurements or some of the underlying assumptions (negligible NP in tensor operator, negligible imaginary parts) are not correct. Such tests are completely independent of any assumption on the details of the NP model or the hadronic inputs.

Massless case with no pseudoscalar operator and no tensor operators
The expressions for the angular observables become in terms of the amplitudes themselves In this case, the only continuous symmetry that can be found is simply and only 5 of the 11 observables 3 are independent and 6 dependencies are found. Consequently, one can invert the system to determine the value of the real and imaginary parts of the amplitudes in terms of some of the angular coefficients, and re-express the other ones in terms of the same angular coefficients leading to the following relations: These relations can be used as a consistency test among the observables if all of these observables are measured, under the hypothesis that we have outlined (negligible lepton mass, negligible pseudoscalar and tensor operators).
Another way of exploiting these equations consists in combining the non-trivial relations Eqs. (47)-(50) under the assumption that I 7,8,9 = 0 (taking all imaginary parts to be zero). For future use under this assumption we reorganize these equations, allowing us to make contact with the massive ones later on: (51) One of the dependencies disappears once I 7,8,9 = 0 is taken.

Massless case with pseudoscalar operator but no tensor operators
The same relations between angular observables and amplitudes hold as in the previous case, apart from One can see that the two symmetries are Again, by inverting the system one can obtain the same relations as in the massless case without pseudoscalar contributions, see Eqs. (46)-(50), except for Eq. (45) which is not fulfilled.
Like in the previous case, these relations can be used as a consistency test among the observables if all of these observables are measured, under the hypothesis that we have outlined (negligible lepton mass, negligible tensor operators).

Massive case with pseudoscalar operator but no tensor operators
The symmetries in the massive case with pseudoscalar operator but no tensors are in principle a simple extension of the analogous massless case. However, obtaining the expression of the dependencies in the massive case is a rather non-trivial task. The absence of tensors implies that there is no distinction between "+" and "-" components ofH + i andH − i (see Eq. (18)) and the only surviving symmetry in this case is (56) One finds five dependencies in this case, which are identified by solving the system of non-linear equations. The first one is trivial: and the other exact four non-trivial dependencies are detailed in App. A. Like in the previous section, we will consider the simplifying case where all Wilson coefficients are real so that I 7,8,9 and all imaginary contributions can be neglected (see App. A for the general case without these assumptions). The remaining four dependencies are then simplified substantially The first three equations above are the generalisation of Eqs. (51)-(53) in the massive case while the last equation is new: it would vanish in the massless limit with no tensors. These relations can be used as a consistency test among the observables if all of these observables are measured, under the hypothesis that we have outlined (no tensor operators, imaginary contributions negligible). The last two equations can be combined to get rid of the I 2 6c term and obtain the massive counterpart of Eq. (53): Eq. (61) has obviously no counterpart in the massless case, as it vanishes then 4 .

Cases with tensor operators
In the massive case with tensors the degeneracy between theH + i andH − i is broken and two symmetries are identified. The symmetries are better described in terms of the tilde-fields: Unfortunately there are no dependencies in this case. The same is true in the massless case.

Expressions of the D * polarisation
In the previous section, we have obtained several relationships between the angular coefficients under various hypotheses, assuming that tensor contributions are negligible. We can use these relations in order to obtain alternative determinations of the longitudinal polarisation F D * L . From Sec. 4.2 to Sec. 4.4, we will provide these exact relationships in their binned form, but the corresponding unbinned versions have exactly the same form.

Massless case without pseudoscalar operator
For completeness we discuss the case with zero mass and no pseudoscalar operator, but still including all imaginary terms. Eqs (45)-(46) are trivial. Eqs. (47)-(50) can be rewritten in terms of observables providing different determinations of F D * L : We recall that A i are defined from the angular observables up to a numerical normalisation given in Ref. [24]. A similar set of expressions can be written forF D * L ,Ã i andÃ F B rather than F D * L , A i and A F B , respectively, by substituting the normalization in terms of dΓ/dq 2 by the integrated decay rate Γ. These expressions can then be binned trivially, however they are rather cumbersome to use. In the following two subsections we will restrict to the case of removing any imaginary contribution corresponding to our baseline scenario that will be relevant to the extraction of F D * L .
4 In the massive case, this relation provides access to a sum of two related observables A6s and AF B :

Massless case without imaginary contributions
Using Eqs. (46) and (51) we obtain one of the important results of this article: This expression can be used as an alternative way to determine the integrated F D * L in the massless case (without imaginary contributions but allowing for the presence of pseudoscalars) from experiment instead of the traditional determination in terms of I 1s and I 2s in Eq. (27) and Eq. (28).
This expression can be generalised to the case of smaller bins spanning only part of the whole kinematic range, leading to where i means that the integral in Eq. (26) is taken over the bin i with a narrower [q 2 i,min , q 2 i,max ] range 5 .
If we restrict further to the case without pseudoscalars (in this case I 1c = −I 2c is fulfilled), we obtain further expressions using Eqs. (52) and (53): where R A,B is positive and non-vanishing by construction.

Massive case with pseudoscalar operator but without imaginary contributions
In this case, we focus on Eqs. (57),(58) and (59) to derive new descriptions of F D * L since Eq. (60) is too involved to provide a useful alternative approach to F D * L . Eqs. (57) and (58) yield: where we define the auxiliary kinematic quantities (whose value in the massless case is two) One can write an equivalent equation to Eq. (72) for narrower q 2 bins similary to the previous section. In the case of Eq. (59) we do not substitute I 2c , leading to: Relating this equation with the massless case is not straightforward given that in the massless case I 2c was substituted (before integrating) in terms of F D * L and R A,B .

Cases with pseudoscalar operator and imaginary contributions
This corresponds to the most complete expression allowing for the presence of pseudoscalars and also imaginary parts, but no tensors. This can be achieved by using I 1s and I 2s instead of I 1c and I 2c as a starting point. The corresponding expression in the massless case is: and in the massive case Similar expressions can be written for F D * T i defined for narrower q 2 bins. These two expressions represent the most general alternative ways to determine the massless and massive polarisation fractions. Compared to the previous case, one can see that the presence of imaginary contributions comes simply from the additional I 9 term in Eqs. (75) and (76), see also Eq. (112) in App. A.
Within this more general framework, Eqs. (57) and (112) yield the following simple relation among the observables defined in Section 2.2: whereÃ i stands for the observables A i normalized to Γ rather than dΓ/dq 2 , x 1 = (m 2 − q 2 ) 2 , x 2 = 4π 2 (m 2 + 2q 2 ) 2 and x 3 = 4x 1 x 2 /(729π 2 q 4 ) (A 9 vanishes in the absence of large imaginary contributions). This relation implies that the large (small) value of F D * L (F D * T ) requires a corresponding suppression in A 2 3 + A 2 9 , in A 6s or both. For this reason it would be particularly interesting to have available predictions in specific models for this couple of observables in case that the unexpectedly large value of this polarisation fraction remains.

Binning
We have obtained alternative expressions for F D * L (or F D * T ) assuming that there are no tensors and (in some cases) no large imaginary contributions at short distances. Experimentally we have to consider binned versions of these expressions, which are nonlinear functions of the angular coefficients. Since the binned angular coefficients are the only quantities measured, we should be careful that f ( I k ) = f (I k ) when f is non-linear.
From an experimental perspective there are two ways to proceed: i) measure the coefficients I 3 and I 6s of the massless or massive distribution in very small bins in order to reconstruct a q 2 dependence of these functions, so that we can perform the integration in Eq. (68) for the massless case or in Eq. (72) in the massive case (or their counterparts including imaginary parts Eq. (75) and Eq. (76)); ii) use an unbinned measurement method (as was done for B → K * µµ [33]) to determine the q 2 dependence of the coefficients and introduce the obtained expressions inside Eq. (68) or Eq. (72) as explained above.
Both approaches are however difficult to implement when the statistics is low, and one has to choose between the extraction of the whole angular distribution and the study of the q 2 dependence of simpler observables like the decay rate. Currently, the measurements are integrated over the whole kinematic range, which constitutes a single bin for the analysis.
By comparing with our exact results, we will thus investigate the accuracy of the approximation f ( I k ) = f (I k ) , which requires the following transformation on the unbinned expressions: where w stands for any positive weight depending on m and q 2 . This leads to the following approximate formulae in the massless case, starting from Eq. (75): In the massive case, one should measure the I i and multiply each event by a numerical factor A for I 3 , I 9 and B for I 6s .
Similarly, in the absence of imaginary parts, we obtain the approximate binned expression, starting from Eq. (72): and the approximate expression starting from Eq. (74) All these expressions have a corresponding expression for F D * T i for narrower bins where is transformed into i corresponding to the integration over the narrow bin i.
In order to get an idea of the accuracy of these approximate relations, we perform the following numerical exercise. We consider a set of benchmark points corresponding to the best-fit-points of the 1D and 2D NP hypotheses in Ref. [22,23]. Among the 1D hypotheses, the most favoured one is assuming NP in g V L , followed by NP in g S R . Specifically we will take for this numerical analysis as benchmark points the best-fit-points of the following four different NP hypotheses (in each case, the remaining couplings are set to zero): where once again we run these coefficients down to µ = m b . In Ref. [24], a set of benchmark points is determined by considering the best-fit points of different scenarios with one free complex parameter. The resulting 2D benchmark points (in each case, the remaining couplings are set to zero) at the scale µ = m b are: In the following we will check the relations given in the previous sections against these benchmark scenarios. We have used the binned approximation of the relations using 6 bins of equal length as shown in Fig. 1. On the one hand, this allows us to test the quality of the binned approximation.
On the other hand, we can check the impact of the assumptions used in order to derive the various relations: for instance, checking the expressions obtained for real NP contributions in Sec. 4.3 in the case of the scenarios (C0) − (C9) with complex parameters provides an estimate of the impact of realistic NP imaginary contributions on these expressions. We need to choose a set of form factors to evaluate the hadronic contributions and to be able to test how accurate the relations remain within the binned approximation discussed above, taking into account possible unexpected NP contributions (imaginary parts, tensor contributions). Since our goal is only to check the accuracy of this approximation for the various NP benchmark points it is enough to work using a simplified setting. For this reason, we refrain from using form factors obtained by elaborate combinations of sum rules, heavy-quark effective theory and lattice simulations [22,24] and we stick to the simpler quark model in Ref. [34] without attempting to assign uncertainties to these computations.
A sample of the results is shown in Figs. 1, 2 and 3 to illustrate the accuracy of the determinations from Eqs. (80) (taking into account the contribution from imaginary parts) and (81) or (82) (neglecting this contribution). Additional scenarios are considered in App. B. Let us add that the I i are integrated with the kinematical weight A or B defined in Eq. (73) for the evaluation of the massive expressions whenever needed. We obtain the following results for the benchmark points considered: • The binned approximation works very well in all cases when testing the relations in the case of scenarios where they are expected to hold. Conversely, when one considers a NP scenario with significant tensor contributions (like (C0) or (C5)), the expressions are off by ∼ 70% in the worst cases. Only when the NP contribution to the tensor coefficients is very small (|g T | 1), the expressions work quite well, for example ∼ 5% for (R4). . We stress again that this does not apply to scenarios with tensor contributions such as (C0) and (C5). • We also tested the massless expressions in the case of NP scenarios affecting light leptons at the same level as the τ lepton. Such scenarios are ruled out by the current data, but they provide a further check of the robustness of our expressions. In these cases, the expressions that do not contain the angular coefficients containing imaginary parts of the amplitudes (I 7,8,9 ) (Sec. 4.2) are off by ∼ 20% at worst. The agreement can be restored once we generalize the corresponding expressions so that they include these angular coefficients (Sec. 4.4), where we find a perfect agreement. • In the first bin of most of the massless expressions, the relations are not completely fulfilled, with a difference up to 10% due to binning effects enhanced at the endpoint of the massless distribution.
This study shows that the expressions derived above under the assumption of no imaginary NP contributions and no tensor contributions in Secs. 4.2 and 4.3 work very well even in the binned approximation. They are very accurate even in the presence of imaginary NP contributions. Their simple generalization including imaginary parts in Sec 4.4 are as expected to be even more accurate also in the binned approximation. Finally, all relations fail in the presence of large tensor contributions.

Decision Tree
We have proposed different ways of determining F D * L (or F D * T ) which can be compared to the usual definition, based on the existing symmetries if additional assumptions are made about the nature of NP (no tensors, real contributions). One may then wonder how to interpret the situation when the determination of F D * T in a narrow bin in the case of the tau lepton yields different results from Eq. (72) and from the traditional determination. While we have provided different possible determinations we will focus on Eq. (72) because it includes pseudoscalar contributions and it is easily generalized in the presence of phases, see Eq. (76). There are three possible conclusions: 1) Our first hypothesis is the absence (or negligible size) of tensors. In the presence of tensors, there are no dependencies among the angular observables, and we cannot use Eq. (72) to determine F D * T . This first possibility seems to be in disagreement with the study in [24] that shows that tensors tend to substantially worsen the situation reducing even further the value of F D * L (or increasing F D * T ). If needed, this question can be tested by probing the relationships shown in Sec. 3 among the angular coefficients.
2) The second hypothesis is the absence of large imaginary parts. In this case one can generalize the expression Eq. (72) to the presence of imaginary parts to get Eq. (76), simply substituting: and similarly for the massless case. This simple substitution covers the presence of large phases but of course at the cost of measuring also I 9 . Alternatively one can also measure I 7,8,9 which are sensitive to large imaginary parts and determine if they differ from zero in a significant way.
3) The third option is the presence of an experimental issue in the determination of F D * L in the traditional way for B → D * τ ν. The alternative determination proposed here could help to determine the problem to be fixed and whether this second determination is also in disagreement not only with the SM but also with NP models.
5 Relation between F D * L τ and R D * The observable R D * corresponds to the ratio of branching ratios of the B → D * τ ν decay versus B → D * ν with = e, µ. This is a particularly interesting observable since its tension with the SM prediction signals a possible lepton flavour universality violation and, for this reason, it deserves a closer inspection. The most recent experimental combined analysis together with R D has reduced the discrepancy with the SM to 3.1σ while the observable R D * alone exhibits a tension of 2σ [18]. Given the importance of any input that may help to clarify if this tension is related to NP or to some experimental issue we propose a different path to determine it. For instance, constructing R D * as a combination of different observables we can test each piece separately. This may give us a handle to understand better the origin of the tension with the SM, namely, if it is global and affects many observables of the angular distribution or it is focused on a particular observable. From the standard definition of F D * L Eq.(23) (or more precisely in theF D * L convention) and taking advantage of the different normalizations for the massive and massless case it is straightforward to recast R D * in the following form where we used the relation Eq. (57). In the massive case it is necessary to multiply each event by a factor K when measuring I 1s (a similar expression can be obtained in terms of I 2s ). Using Eqs. (75) and (76) as a different determination of F D * L we obtain: whose evaluation from the binned angular observables would require a similar approximation as in Sec. 4.5. Notice that the previous expressions are only valid when integrated over the whole phase space and that there are many other possible ways to construct these combinations with different types of observables. This requires the measurement of both massless and massive F D * L as well as the corresponding I 3 and I 6s (and I 9 if large imaginary parts are considered). This measurement is more challenging experimentally, but this approach identifies different observables required to reconstruct R D * in Eq.(102) suggesting to perform an independent test of each of them separately.
This relation implies that if the tension in R D * is real necessarily one should find tension in the observables of the right-hand side of Eq. (101) or Eq. (102). On the contrary if a measurement of each of the observables on the right-hand side is done following a method independent from the one used to measure R D * and the result found for each of them is SM then this would clearly point to a problem in the measurement of R D * . We illustrate this situation in Fig. 4. The relations proposed in Eqs. (102), (101) and (80) hold properly in the scenarios without tensor contributions as expected (Fig. 4a), as can be seen from the closeness of the three symbols in each case, but they break down as we introduce tensor contributions (Fig. 4b).
Numerically, the results are in good but not perfect agreement with the results in Refs. [22,24]: this can be seen from all three types of determinations in Fig. 4a, but it is true only for the measurement in the usual way indicated with a star in Fig. 4b). Once again, we stress that our results have been obtained using the form factors from the simple quark model in Ref. [34], without assigning any uncertainty, and our main objective here is to check the consistency and accuracy of the relations obtained above, which is indeed very good in the absence of tensor contributions. Conversely, a significant difference between the three methods would be the sign of an experimental problem or of tensor contributions.
In summary, a better understanding of R D * and F D * L for B → D * τ ν can be obtained by looking at the set of observables entering Eqs. (102) and (101) and by checking the corresponding relations. This would allow one to determine if their value can be consistently explained in the SM, within a NP model with vector and/or scalar contributions or if one has to rely on tensor contributions or experimental problems.

Conclusions
The charged-current B → D * ν transition has been under scrutiny recently, as it exhibited a deviation from the SM in the LFUV ratio R D * comparing the branching ratios = τ and lighter leptons. Moreover, the polarisation of both the D * meson and the τ lepton have been measured for B → D * τ ν. If the latter agrees with the SM within large uncertainties, the Belle measurement of F D * L yields a rather high value compared to the SM prediction, which appears difficult to accommodate with NP scenarios.
We could understand better this situation by considering in more detail the angular observables that could be extracted from the differential decay rate, as described in Ref. [24]. We applied the formalism of amplitude symmetries of the angular distribution of the decays B → D * ν for = e, µ, τ . We showed that the set of angular observables used to describe the distribution of this class of decays are not independent in absence of NP contributing to tensor operators. We derived sets of relations among the angular coefficients of the decay distribution for the massless and massive lepton cases. These relations can be used to probe in a very general way the consistency among the angular observables and the underlying NP at work, and in particular whether it involves tensor operators or not.
We used these relations to access the integrated longitudinal polarisation fraction of the D * using different angular coefficients from the ones used by Belle experiment. This in the near future can provide an alternative strategy to measure F D * L for B → D * τ ν and to understand the relatively high value measured by Belle. We presented expressions in Eqs. (75) and (76) for the massless and massive case that cover the most general NP scenario including also pseudoscalars and imaginary contributions, with the only exception of tensor contributions. We also proposed an alternative path to measure LFUV ratios R D * by connecting it with F D * L , using the coefficients I 3 , I 9 and I 6s following Eq. (75) for the massless case and Eq. (76) for the massive case instead of the traditional expression in terms of I 1c and I 2c . Exploiting these relations with the current data available requires us to perform a further approximation due to the binning, leading to Eqs.  We then studied the accuracy of these expressions if only binned observables are available, or if they are used in the case of scenarios beyond the assumptions made in their derivation (imaginary contributions, tensor contributions). We used several benchmark points corresponding to bestfit points from global fits to b → cτ ν observables, relying on a simple quark model for the hadronic form factors for this exploratory study. The expressions derived under the assumption of no imaginary NP contributions and no tensor contributions work very well even in the binned approximation. They are very accurate even in the presence of imaginary NP contributions. As expected, their generalisations, derived assuming the presence of imaginary contributions, are very well behaved also in the binned approximation. All relations fail in the presence of large tensor contributions, where no dependencies can be found among the angular observables.
Besides presenting the most general expressions for F D * L in the massless and massive case, we also derived a relation among observables that identifies two potentially interesting observables from the NP point of view. Having specific model building predictions for these observables would be highly interesting.
We have explored alternative determinations of R D * and F D * L based on our symmetries. In the absence of tensor contributions, these determinations based on other angular observables are fulfilled very accurately. This provides an important cross check for the experimental measurements: if our relations are not fulfilled by the experimental measurements, this would mean either a problem on the experimental side or the presence of large tensor contributions.
These additional measurements needed for this extraction make obviously this determination more challenging experimentally, but they can help to corner the kind of NP responsible for this high value or to understand the experimental problem responsible for this unexpected value of the D * polarisation. We hope that our results will be of particular interest once the LHCb and Belle II experiments are able to analyse the B → D * ν decays in more detail and thus to provide us with a more detailed picture of the intriguing deviations currently observed in b → c ν transitions. with n = 0, 1. However, this sign ambiguity product of the twofold nature of the solution can be fixed, since physical combinations prevent interference terms that could be problematic. This set of solutions can be used to determine the square of the four amplitudes once H 0 is fixed to be real and positive through the symmetry of the angular distribution. One can also rewrite the real and imaginary parts of H t in terms of the variables in Eq. (103) and H 0 : Im With these definitions, one can find the whole set of dependencies among angular coefficients. Besides the trivial dependency Eq. (57), there are four more relations which are obtained by taking combinations of the modulus of H + , H − and Re(H t ), Im(H t ).

B Comparison of the binned expressions in benchmark NP scenarios
Following the setup of Sec. 4.5, we illustrate in Fig. 5 to Fig. 10 the errors induced on the binning by the approximation Eq.(78) on relations derived using the amplitude symmetries under various assumptions on the NP scenario in the τ lepton case. The blue (orange) curve corresponds to the left-hand side (right-hand side) of the unbinned expression. The points correspond to the binning of the curves whereF D * T and each I i is binned separately.