Complete framework for tau polarimetry in B →d (∗)τν decays

The meson decays B → D τν and B → D (cid:2) τν are sensitive probes of the b → c τν transition. In this work, we present a complete framework to obtain the maximum information on the physics of B → D ð(cid:2)Þ τν with polarized τ leptons and unpolarized D ð(cid:2)Þ mesons. Focusing on the hadronic decays τ → πν and τ → ρν , we show how to extract seven τ asymmetries from a fully differential analysis of the final-state kinematics. At Belle II with 50 ab − 1 of data, these asymmetries could potentially be measured with percent-level statistical uncertainty. This would open a new window into possible new physics contributions in b → c τν and would allow us to decipher its Lorentz and gauge structure.


I. INTRODUCTION
Leptonic and semileptonic hadron decays are important probes of the fundamental quark-lepton interactions within and beyond the Standard Model (SM). Decays of B mesons with τ leptons in the final state, in particular, provide a unique way to determine the properties of fermion interactions involving the third generation. They allow us to test the flavor structure of the SM and search for New Physics (NP) predominantly coupled to the heavier fermions. In addition, the large τ mass leads to an enhanced sensitivity to the scalar component of the weak interaction. Semitauonic B decays are therefore especially sensitive to the timelike component of the virtual W boson [1] or to the exchange of new (pseudo)scalar particles [2][3][4].
At flavor experiments, the decays B → Dτν and B → D Ã τν, both triggered by the charged-current transition b → cτν, are the most accessible semitauonic hadron decays. The branching ratios of these decays normalized to those into light leptons, R D ðÃÞ ¼ BRðB → D ðÃÞ τνÞ= BRðB → D ðÃÞ lνÞ with l ¼ e, μ, have been measured with good precision at BABAR [5,6], Belle [7][8][9], and LHCb [10,11]. Interestingly, the combination of these measurements appears to be about 20% larger than the SM prediction with a significance of 3.08σ [12]. The normalized branching fraction of B c → J=ψτν, which is based on the same b → cτν transition, has been measured by LHCb and also appears to be larger than the SM expectation [13]. Beyond total rates, in B → D Ã τν the longitudinal τ polarization P L ðτÞ [14,15] and the fraction of longitudinally polarized D Ã mesons F L ðD Ã Þ [16,17] have been measured. This shows the potential of the current flavor experiments, Belle II and LHCb, to extract the properties of the b → cτν transition by measuring the τ kinematics in the decay. Precise analyses of these transitions are important to understand the origin of the observed discrepancies with the SM and to decipher the structure of NP in case they persist.
Due to its fast decay, the production properties of the τ lepton cannot be directly measured, but have to be extracted from the decay products where part of the information on the τ momentum is carried away by at least one neutrino in the final state. Extracting the properties of the b → cτν transition from the visible τ-decay products in B → D ðÃÞ τν has evolved into a comprehensive research program [18][19][20][21][22][23][24][25][26][27][28][29][30]. One aims to construct the full differential decay rate and then integrate out all kinematic variables that are unobservable due to the presence of neutrinos [24,30].
In this paper, we develop a complete framework to extract the full set of B → D ðÃÞ τν observables (with polarized τ and unpolarized D ðÃÞ ) from the visible final state. We focus on the hadronic τ decays τ → πν and τ → ρν, which preserve more information on the τ kinematics than the leptonic decays τ → lνν [26]. In the two-body decays, the τ spin orientation is directly imprinted on the pion or rho direction of flight. The τ helicity and kinematics can thus be deduced from the energy and angular distributions of the visible final-state particles [21,26,31,32]. The main result of our paper allows us to express the differential decay rate of B → D ðÃÞ τð→ dνÞν as Here q 2 , cos θ d , and s d describe measurable kinematic quantities in the leptonic rest frame (the momentum transfer to the lepton pair, the angle between the τ daughter d and the D ðÃÞ , and the energy of the d), and nðq 2 Þ is a normalization factor. Importantly, the "leptonic functions" to be defined in the next section. This formula directly relates these asymmetry observables to the kinematic distribution of the τ daughter. By measuring the kinematics of the d particle, one can extract nearly all the physics of the b → cτν transition, including the possible presence of new physics affecting the transition. The asymmetry observables represent a useful intermediate step between the data and the underlying Wilson coefficients. Previous studies [2,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] have identified a subset of these asymmetries (A FB , P L , P ⊥ ) and shown how to extract them from differential distributions of the final state. In this work, we show that a total of nine asymmetries, together with the differential decay rate dΓ B =dq 2 , suffice to describe the full physics of B → D ðÃÞ τν with unpolarized D ðÃÞ . The remaining two not listed in Eq. (1.2), P T and Z T , are nonzero only in the presence of CP violation, and furthermore are only accessible by including additional information-e.g., from D ðÃÞ decays [30]-in the kinematic distributions. We reserve a complete study of these additional observables for a future publication [50].
We will demonstrate how one could theoretically measure the asymmetries by performing an unbinned maximum-likelihood fit to the d distribution [Eq. (1.1)]. While we do not include realistic experimental considerations such as systematic uncertainties, detector acceptance, or backgrounds (these are beyond the scope of this work), we show that at least the statistical power with 50 ab −1 of Belle II data should be enough to measure the asymmetry observables to percent-level precision.
Analytic formulas like Eq. (1.1) could prove useful in experimental studies. Besides being needed for maximumlikelihood fits, they could be adapted for Monte Carlo generators [51]. Another line in this direction has been providing efficient methods to reweight Monte Carlo event samples interpreting experimental data directly in terms of SM or NP parameters [19,25,29].
The outline of the paper is as follows: In Sec. II, we decompose the B → D ðÃÞ τν kinematics into a complete set of τ asymmetries. These asymmetries contain all information that could be obtained if the τ momentum were fully accessible. In Sec. III, we show how to extract seven of the nine τ asymmetries from the kinematics of the τ decay products. By performing a full-fledged statistical analysis in Sec. IV, we give a theoretical estimate of the expected sensitivity of Belle II to the asymmetries, assuming a given number of events and neglecting experimental effects. We also demonstrate how to decipher the structure of new physics in τ production in the framework of an effective theory and in the context of the current anomalies found in the R D ðÃÞ ratios. We conclude in Sec. V with a summary and outlook.

II. TAU ASYMMETRIES IN B → D ðÃÞ τν
In this section, we focus on the B → Mτν decay kinematics, where M ¼ D or D Ã , without considering the τ decays yet. The narrow width of the τ enables a factorization of the full decay chain into a τ-production part and a τ-decay part.
The basis for the τ asymmetries is the differential decay rate for B → Mτν with the τ spin quantized along an arbitrary directionê a , jM λ τ ;a B j 2 dΦ 3 ðp B ; p M ; p τ ; p ν Þ: ð2:1Þ Here λ τ ¼ AE is the direction of the τ spin along theê a axis, and the Lorentz-invariant phase space for a particle i decaying to n daughters is Throughout this work, we sum over the polarization states of the D Ã meson. The total differential decay rate can be calculated from the spin-dependent decay rates along any axis as is always defined along the particular axisê a . We work in the "q 2 frame," the center-of-mass frame of the lepton pair, with q 2 ¼ ðp B − p M Þ 2 being the momentum squared transferred to the leptons. Figure 1 illustrates the various momentum vectors, polarization vectors, and angles involved in the q 2 frame. Let fê 1 ;ê 2 ;ê 3 g be an orthonormal coordinate system in this frame and choosê e 3 ¼p τ ≡ê L ; ð2:5Þ wherep τ is the direction of the τ momentum. The spindependent differential decay rate (2.1) and the asymmetries (2.4) can then be expressed using τ helicity amplitudes M λ τ ;L B . From here on, the index a ¼ L will be suppressed. The resulting expressions for the asymmetries depend on how the axesê 1 andê 2 are chosen. Choosinĝ wherep M is the direction of the M momentum, results in B ;p M ;p τ ;p ν Þ: ð2:7Þ These four differential distributions capture all the information in the matrix elements M AE B in the B → D ðÃÞ τν decay (with unpolarized D Ã ).
The matrix elements do not depend on the azimuthal angle of the τ momentum with respect to the M momentum; this angle is thus integrated out. The two remaining degrees of freedom in the final state are chosen to be q 2 and cos θ τ , where θ τ is the angle between the flight direction of the τ and the negative direction of the M momentum in the q 2 frame. The decay rates and asymmetries for B → Mτν can be expanded in spherical harmonics encoding the conservation of angular momentum [1]: Re½C l ðq 2 ÞP 1 l ðcos θ τ Þ; Im½C l ðq 2 ÞP 1 l ðcos θ τ Þ; ð2:8Þ where P 0;1 l ðcos θ τ Þ are the associated Legendre functions. Together with the total differential rate dΓ B =dq 2 , the angular coefficient functions B AE 0;1;2 ðq 2 Þ ∈ R and C 1;2 ðq 2 Þ ∈ C describe the full kinematic information in B → Mτν decays with unpolarized mesons.
FIG. 1. The two decay planes of B → Mτν (black) and τ → dν (red) in the q 2 frame. The angle between the two planes is denoted ζ. The angle between the flight direction of the τ and the negative direction of M is denoted θ τ . In the decay plane of the τ, φ is the angle between the direction of the τ and the direction of the daughter particle d. Finally, the angle θ d is the angle between the direction of the daughter particle (in the τ decay plane) and the negative direction of M (in the B-decay plane).
(3) We can additionally consider double asymmetries with respect to both λ τ and cos θ τ These asymmetries have not been considered before and give access to the previously unexplored combinations of angular coefficient functions B AE 1 and C 2 . 2 (4) The angular coefficient functions B AE 2 cannot be expressed in terms of simple asymmetries like the other functions. They denote the quadrupole part of the partial wave expansion in Eq. (2.7). We combine these angular coefficient functions to define the following asymmetry observables: where the 5=2 prefactor captures the Legendre polynomial normalization. (5) Finally, given that we have extracted an overall factor of dΓ B =dq 2 in the definition (2.8) of the angular coefficient functions, they satisfy the rela-

III. TAU ASYMMETRIES FROM THE VISIBLE FINAL STATE
Since the τ decays promptly in the detector with one or two neutrinos in the final state, it is generally not possible to reconstruct its full four-momentum. 3 Therefore, the τ asymmetries in B → Mτν described in Sec. II are not directly measurable. In this section, we will show how they can be extracted from final-state observables with fully reconstructed mesons M. We focus on the two-body decays τ → dν with d ¼ π, ρ, as they preserve more information compared to the three-body decay τ → lνν. However, our formalism can be straightforwardly generalized to τ → lνν or other τ-decay modes. Figure 1 shows the two decay planes of B → Mτν and τ → dν, and the various angles and momenta involved in the decays in the q 2 frame. The angle θ d between ⃗p d and − ⃗p M is the only directly measurable angle. Meanwhile, φ (the angle between ⃗p τ and ⃗p d ), ζ (the angle between the two decay planes), and θ τ are not directly measurable. However, φ is completely determined by the kinematics of the twobody decay as Here we have introduced the notation where E d and m d denote the energy and mass of the daughter particle in the q 2 frame. It is straightforward to show that the allowed range of s d is Finally, the angles θ τ and ζ are related to φ and θ d via The matrix element for the full B → D ðÃÞ τð→dν 2 Þν 1 decay is where M B is the matrix element for B → Mτν as introduced in Sec. II, Γ τ is the total width of the τ, and M τ is the matrix element for τ → dν. In the narrow width approximation, the four-body phase space factorizes as 1 Z stands for zweifach. 2 In B → Dτν, some of the τ asymmetries are absent because the pseudoscalar nature of the D meson restricts the possible angular coefficient functions. In particular, it has been shown that B − 1 ¼ 0 for the most general effective Lagrangian with scalar, vector, and tensor operators (with left-handed neutrinos only) [27]. This implies that Z L ¼ A FB in B → Dτν and no independent information is gained from Z L . In B → D Ã τν, B − 1 is generated by the transverse polarization of the D Ã vector meson [24], so that A FB and Z L carry independent information. 3 The full τ kinematics could potentially be accessed with displaced three-prong decays [54].
with the τ set on shell. Then the full fourfold differential decay rate is where repeated λ τ indices are summed over. All terms are factorized into a B-decay part [see Eq. (2.8)] and a τ-decay part defined as Similarly to how dΓ λ τ B and dP ⊥;T B could be expanded in cos θ τ , the expressions above can be expanded in the angles φ and ζ. For a two-body τ → dν decay, where the coefficient functions for d ¼ π, ρ are given by ð3:10Þ Throughout our analysis, we neglect the π mass but not the ρ mass. To transform Eq. (3.7) into a fully observable, fully differential decay rate, we need to integrate over the two unobservable angles θ τ and ζ and replace them with the single observable angle θ d . Formally, this can be accomplished using Eq. (3.4) to obtain In Appendix A, we carry out these integrals explicitly. The result is given by One can verify using Eq. (3.10) that the first four leptonic functions in Eq. (3.13) agree precisely with those considered in Ref. [26]. We see that the fully differential final-state decay rate breaks down into a linear combination of the asymmetries, or equivalently, of the angular coefficient functions B AE 0;1;2 and Re½C 1;2 . We emphasize that Eq. (3.12) is completely general even in the presence of arbitrary heavy new physics altering the b → cτν transition. The leptonic functions are independent of the b → cτν transition and depend only on the τ-decay mode. Therefore, one could use Eq. (3.12) to directly extract the asymmetries from the data in a completely model-independent way. We will investigate the theoretical sensitivity of such an approach in the next section.

IV. SENSITIVITY TO ASYMMETRY OBSERVABLES AT BELLE II
Having derived analytic expressions for the fully differential final-state decay rate and related them to the B → Mτν asymmetry observables, we now turn to a toy study of how the asymmetry observables could be measured in practice, and what precision one could hope to achieve. We cannot comment on the systematic uncertainties associated with our proposal at different experiments. A detailed simulation of backgrounds and detector effects is also beyond the scope of this work. We will limit ourselves to calculating the achievable statistical uncertainty; this should furnish a "best-case scenario" for the sensitivity of any future measurement.
For the analysis in this section, we will need explicit numerical formulas for all the asymmetries in terms of the dimension-6 effective Hamiltonian. The dependence of the asymmetries P a ðq 2 Þ (a ¼ L; ⊥; T) and A FB ðq 2 Þ on all the relevant dimension-6 operators has already been calculated [27]. Following the notation of Refs. [27,55], we include the analytic expressions for the new asymmetries A Q ðq 2 Þ and Z a ðq 2 Þ (a ¼ L; ⊥; T; Q) in Appendix B.

A. Maximum likelihood method
The energy s d and the angle cos θ d of the daughter in τ → dν decays are directly measurable at Belle II. Using the fully differential distribution [Eq. (3.12)], we apply the unbinned maximum likelihood method in s d and cos θ d to fit for the asymmetry observables in q 2 bins and determine the covariance matrices around the best-fit values. We do not assume any templates for the q 2 dependence from the SM or otherwise; we consider a separate and independent measurement of the asymmetries in each q 2 bin. 4 Let Oðq 2 i Þ for O ¼ A FB , P L , etc., be the parameters that we want to fit for in q 2 bin i. According to Eq. (3.12), the probability distribution of events in q 2 bin i in terms of these parameters is given by We assume that the event numbers in each q 2 bin are large enough that the asymptotic form of the maximum likelihood method can be used. Then the log-likelihood statistic to be maximized is is the fraction of events in q 2 bin i with bin width Δq 2 . The elements of the inverse covariance matrix for bin i are given by In the following, we report the sensitivity to the q 2integrated asymmetries, defined by where O ¼ A FB ; P L , etc. These integrated asymmetries provide us with a sensitivity estimate in the case of limited The total covariance matrix is then In the following subsections, we will report values and make plots of the variances σ 2 a (the diagonal elements of Σ) and the correlation coefficients ρ ab ¼ σ ab =ðσ a σ b Þ (derived from the off-diagonal elements of Σ).
FIG. 3. 68% (solid) and 95% (dashed) confidence intervals for the statistical sensitivity to the τ asymmetries in the SM in B → Dτð→ πνÞν (pink) and B → Dτð→ ρνÞν (blue) decays. The central values are marked for the SM (pink star), as well as for the NP scenarios U 1 (yellow square) and S 1 − R 2 (green diamond). The correlation coefficients ρ π and ρ ρ for each pair of asymmetries are shown in a boxed insert. Assumed is a dataset of N ¼ 3000 total events in each channel.

B. Standard Model sensitivity
Using this method, we determine the theoretical sensitivity to the q 2 -integrated asymmetries, assuming the SM prediction for all the parameters-i.e.,Ôðq 2 i Þ ¼ O SM ðq 2 i Þ. It is trivial to repeat the analysis for a scenario with a different prior. Figure 2 shows the asymmetries as functions of q 2 in the SM for B → Dτν and B → D Ã τν, using hadronic form factors of Ref. [24]. In addition to the SM predictions, we also show the predictions from two benchmark NP scenarios, which are discussed in detail in Sec. IV C.
The values of the integrated asymmetries in the SM are displayed in Table I, along with the projected statistical FIG. 4. 68% (solid) and 95% (dashed) confidence intervals for the statistical sensitivity to the τ asymmetries in the SM in B → D Ã τð→ πνÞν (pink) and B → D Ã τð→ ρνÞν (blue) decays. The central values are marked for the SM (pink star), as well as for the NP scenarios U 1 (yellow square) and S 1 − R 2 (green diamond). The correlation coefficients ρ π and ρ ρ for each pair of asymmetries are shown in a boxed insert. Assumed is a dataset of N ¼ 3000 total events in each channel. sensitivities for N ¼ 3000 total events in each final-state π or ρ. 5 Figs. 3 and 4 visualize these numbers and provide the correlation between each asymmetry pair. The achievable sensitivities for these observables are at the percent level, comparable to the projected sensitivity to R D ðÃÞ [57].
Interestingly, Table I suggests that the new observables Z L , Z ⊥ , Z Q , A Q could be measured with comparable precision to the previously studied observables P L , P ⊥ , and A FB . We also find that the sensitivities to the asymmetries are comparable in both B → Dτν and B → D Ã τν decays. However, there is a stark difference between π and ρ: for all the observables, the τ → πν channel has a better sensitivity compared to the τ → ρν channel. Measuring the ρ polarizations would presumably enhance the sensitivity in the latter channel.
Of all the asymmetries in Table I, only P L for B → D Ã τν has been measured so far. The projected statistical uncertainty (obtained by rescaling the current measurement with luminosity) is AE0.06-see Table 50 in Ref. [57]-this is in the same ballpark as our projection in Table I. The difference may be attributable in part to the background effects we have neglected, as well as detector acceptance and efficiency. Nonetheless, the fact that our purely theoretical estimate of the sensitivity is within a factor of 2 of the official projection provides some confidence in the sensitivity estimates for the other observables.

C. New physics in b → cτν
Heavy new physics at scales Λ ≫ m W can modify the total rates and kinematic distributions of the τ lepton and the D ðÃÞ meson in the B → D ðÃÞ τν decays. Such modifications can be parametrized in a model-independent way in terms of an effective Hamiltonian ð4:8Þ where G F ¼ 1= ffiffi ffi 2 p v 2 and V cb is the CKM element. The various effective operators describe local scalar, vector, and tensor four-fermion interactions, defined as The Wilson coefficients C i XY in Eq. (4.8) contain information pertaining to the short-distance structure of the b → cτν transitions induced by new physics above the weak scale. In our conventions, the SM corresponds to C i XY ¼ 0. A given NP model induces specific modifications of the Wilson coefficients that can be analyzed by measuring various observables in these decays. We neglect corrections of Oðv=ΛÞ that arise from higher-dimensional operators in the effective theory.
As we discussed in the Introduction, current measurements of total rates in terms of the ratios R D ðÃÞ are in tension with the SM at a significance of about 3σ, which could be due to the presence of new physics in b → cτν transitions. Several models have been proposed that can explain this difference [55,. One class of models particularly favored by data involve a vector leptoquark U 1 TABLE I. Numerical predictions of the integrated observables in the SM, together with their theoretical uncertainties σ th and the estimated statistical uncertainties in the π and ρ channels, σ π and σ ρ . The theoretical uncertainties are obtained by scanning theoretical inputs as in Ref. [24]. The statistical uncertainties assume a dataset of N ¼ 3000 events for each final state. Both the theoretical and statistical uncertainties refer to the 68% confidence level.  [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77]. Another possibility are the scalar leptoquarks S 1 ∶ð3; 1; þ1=3) and R 2 ∶ð3; 2; þ7=6Þ, which produce a correlated effect in the scalar and tensor operators O S LL and O T LL [55,[78][79][80][81][82][83]. 6 We use these two models to demonstrate the sensitivity of our asymmetries to new physics. Our benchmarks correspond to where the Wilson coefficients are evaluated at the bottomquark mass scale. Both benchmarks are motivated by a fit to the current R D and R D Ã measurements [49]. In Fig. 2, we show the q 2 dependence of all the asymmetry observables in the U 1 (dashed) and the S 1 − R 2 (dotted) leptoquark scenarios. We also show the predictions of the q 2 -inclusive observables in these models in Figs. 3 and 4 as yellow squares and green diamonds, respectively. We have not included the expected confidence regions around the NP points, but we have checked that the statistical sensitivities are nearly indistinguishable in size and shape from the SM ellipses. Finally, in Fig. 5, we show the q 2 -integrated results of the observables, including both the theoretical and statistical uncertainties for the SM predictions [24]. 7 As can be seen from the figures, in the U 1 leptoquark scenario, most of the observables are very similar to the SM. Only P L in B → Dτν causes an appreciable deviation from the SM prediction. The reason is that the vector leptoquark primarily induces the operator O V LL . This effect merely changes the overall normalization of the decay rate in the SM and cancels out in the normalized asymmetry observables. Any observable effect of the U 1 leptoquark is due to the small scalar contribution to O S RL , which P L is especially sensitive to. On the other hand, in the scalar leptoquark scenario the deviation from the SM is quite significant for many of the observables. This scenario involves a combination of scalar and tensor operators, which significantly affect the angular distributions in the b → cτν decay.
All in all, we conclude from Fig. 5 that the most promising single observables for distinguishing between these two NP scenarios are P L ðDÞ, P ⊥ ðD Ã Þ, and Z L ðD Ã Þ. At the same time, no single observable presents a "slam dunk" case for one NP scenario or the other; differences are at 1σ-2σ at best. However, Fig. 5 and Figs. 3-4 indicate that the combination of multiple observables offers a way to achieve higher sensitivity. This emphasizes the potential of a simultaneous measurement of all of these observables to clarify the nature of the currently observed discrepancies.

V. DISCUSSION AND CONCLUSION
The study of B → D ðÃÞ τν transitions offers a unique window into couplings between quarks and leptons involving the third generation. In this work, we have shown how to extract the maximum information about the b → cτν transition from kinematic distributions of the observable particles in B → D ðÃÞ τð→ πν; ρνÞν decays. The physics of B → D ðÃÞ τν decays with polarized τ leptons and unpolarized D ðÃÞ mesons beyond total rates can be fully captured by nine coefficient functions in a partial wave expansion. Linear combinations of some of these functions correspond to widely studied τ observables, such as the longitudinal FIG. 5. Theoretical (blue bands) and statistical (black bars) uncertainties on the asymmetries at the 68% confidence level for B → Dτν (left) and B → D Ã τν (right). The statistical uncertainties correspond to the τ → πν channel, assuming 3000 events. The deviations of the central values for the NP scenarios U 1 (yellow square) and S 1 − R 2 (green diamond) from the SM central values are also shown. 6 Other models involving colorless gauge bosons W 0 [84][85][86][87][88][89], extending the Higgs sector [90][91][92][93][94][95], or adding right-handed neutrinos [96][97][98][99][100] could explain the discrepancy, but they are more in tension with other low-energy observables or collider searches [103][104][105]. 7 Figure 5 indicates that the theoretical uncertainties on the observables are always comparable to or smaller than the statistical uncertainties. This further motivates performing the measurement at Belle II, as the precision will not be theoretically limited. polarization asymmetry P L and the forward-backward asymmetry A FB . We showed that seven of the nine coefficient functions can be recovered from the kinematic distributions of the observable particles D ðÃÞ and π, ρ. The remaining two functions are sensitive to CP violation and can only be extracted by including additional information, for instance from the D ðÃÞ decay [30]. We leave such a study for future work [50].
A similar analysis has previously been performed for a subset of the asymmetries in B → Dτν [26]. In this work, we generalized this analysis to include B → D ðÃÞ τν and developed a common framework to describe both processes. Using this framework, we discovered four new asymmetries, Z L , Z ⊥ , Z Q , and A Q . These observables probe independent fundamental properties of b → cτν transitions and can also be extracted from the observable kinematic distributions, which previously had not been realized.
To assess the potential of the Belle II experiment to measure the seven asymmetries, we have performed a statistical analysis assuming the full dataset of 50 ab −1 . Our unbinned maximum-likelihood fit to the fully differential final-state distribution in B → D ðÃÞ τð→ πν; ρνÞν decays shows that almost all asymmetries could be accessed with a statistical uncertainty of a few percent. These predictions do not include realistic experimental effects such as detector acceptance/efficiency/smearing, backgrounds, or systematic uncertainties-see, e.g., Ref. [29] for further discussion. It would be interesting to further our study by taking these issues into account.
Additional sensitivity can be obtained by combining the τ → πν and τ → ρν channels with each other and with the leptonic decay modes τ → lνν. While the lepton kinematics do not contain as much information about the asymmetries as the π or ρ, the leptonic decays occur at a higher rate and should be included in a global analysis of all τ-decay modes.
These positive measurement prospects and the precise predictions of the asymmetries in the SM allow us to detect possible deviations in the presence of heavy new physics. For two new physics scenarios with scalar and vector leptoquarks, motivated by the currently observed deviations in B → D ðÃÞ τν decays, we have determined the statistical sensitivity compared to the SM expectations. In B → Dτν, the longitudinal τ polarization asymmetry P L discriminates particularly well between the two NP models; in B → D Ã τν the perpendicular polarization asymmetry P ⊥ and the double asymmetry Z L show the best individual discriminating power. Of course, the ability to discriminate between different NP models increases by combining all seven asymmetries in a global fit.
In this paper, we have endeavored to demonstrate the usefulness of the asymmetries in B → D ðÃÞ τν decays and the feasibility of measuring them at Belle II. The asymmetries furnish an important intermediate step between the raw data and the underlying Lagrangian parameters, e.g., the Wilson coefficients. The framework developed in this paper provides us with a solid interpretation scheme for τ polarimetry in B → D ðÃÞ τν decays, ready to be confronted with fresh data at Belle II.

APPENDIX A: ANGULAR INTEGRALS
In this Appendix, we give details about the integration over angles that are not observable in the final state. Using the angular expansions of Eqs. (2.8) and (3.9), the angular integrals in Eq. (3.11) are found to be of the form If f is an odd function of ζ, the integral vanishes. This is the reason why P T and Z T , which are proportional to sin ζ in the total decay rate, vanish. Changing variables from ζ to cos ζ, it is straightforward to calculate where POUYA ASADI et al. PHYS. REV. D 102, 095028 (2020)  and the Jacobian is given by After integrating over ζ, the delta function in Eq. (A2) restricts the possible range of θ τ . Solving Eq. (3.4) for cos θ τ and inserting the ζ integration limits gives The choice of sign configuration does not matter, since the other configuration can be obtained by sending φ → −φ; this angle is only defined in terms of cos φ.
Choosing φ ≥ 0 and cosðθ d þ φÞ as the lower integration limit gives The procedure above is equivalent to the change of variables in Ref. [24]. The resulting functions uðcos θ d Þ for all functions fðζÞgðcos θ τ Þ present in the full decay rate are listed in Table II.

APPENDIX B: ANALYTICAL EXPRESSIONS FOR THE NEW ASYMMETRIES
In this Appendix, we report the analytic expressions for all new asymmetries introduced in Sec. II-i.e., Z L , Z ⊥ , Z T , A Q , and Z Q . Similar formulas for the remaining asymmetries can be found in the Appendix of Ref. [27]. 8 In B → Dτν, the asymmetries are where C X YZ refer to the Wilson coefficients of the relevant dimension-6 operators; see Eq. (4.8). The hadronic functions H can be found in Refs. [27,55].
Similarly, for B → D Ã τν, the asymmetries are given by where while the quadrupole observables A Q and Z Q are The hadronic functions H are pure functions of q 2 and contain the hadronic matrix elements. All theory uncertainties are therefore contained in these functions.