Asymmetry Observables and the Origin of $R_{D^{(*)}}$ Anomalies

The $R_{D^{(*)}}$ anomalies are among the longest-standing and most statistically significant hints of physics beyond the Standard Model. Many models have been proposed to explain these anomalies, including the interesting possibility that right-handed neutrinos could be involved in the $B$ decays. In this paper, we investigate future measurements at Belle II that can be used to tell apart the various new physics scenarios. Focusing on a number of $\tau$ asymmetry observables (forward-backward asymmetry and polarization asymmetries) which can be reconstructed at Belle II, we calculate the contribution of the most general dimension 6 effective Hamiltonian (including right-handed neutrinos) to all of these asymmetries. We show that Belle II can use these asymmetries to distinguish between new-physics scenarios that use right- and left-handed neutrinos, and in most cases can likely distinguish the specific model itself.


Introduction
Among the most tantalizing hints of new physics (NP) currently are a number of flavor anomalies [1][2][3][4][5][6][7][8][9]. Of these, one of the largest and longest-standing statistical discrepancies with the Standard Model (SM) is observed in the decays B → D ( * ) τ ν. This can be seen in the ratios R D and R D * , defined as A combined analysis [17] shows a ∼ 3.8σ discrepancy [10] between the experimental results (1.2) and the SM predictions (1.3).
Theoretical models proposed to explain these anomalies rely on new heavy mediators which enhance theB → D ( * ) τ ν decay rate. These mediators can be classified by their spin (scalar or vector) and by whether they carry SU (3) color. The possibilities essentially boil down to three categories: a colorless charged scalar (possibly part of an extended Higgs sector), a heavy charged vector boson (W ), or various types of scalar and vector leptoquarks (LQs). The generic tree-level diagrams with these mediators are shown in Fig. 1.
Most models explaining these anomalies have so far relied on the left-handed (LH) SM neutrinos to provide the missing energy in theB → D ( * ) τ ν decays. However, recently there has been increased interest in the possibility that right-handed (RH) sterile neutrinos are instead present in the decays [19][20][21][22][23][24][25]. Specifically, it was shown in [24,25] that a W coupling to light RH neutrinos could explain both anomalies, while evading severe bounds from flavor physics and direct collider searches that rule out W models with LH neutrinos [26,27].
With the large number of mediators proposed to explain the R D ( * ) anomalies, it is important to understand which are phenomenologically viable, and to figure out ways to distinguish them experimentally [22,23,[28][29][30]. In particular, it is interesting to ask: what measurements can we make in order to tell the difference between models with LH and RH neutrinos? In this paper, we will explore the possibility of using various angular and polarization asymmetry observables for this purpose. Of particular interest here are the forward-backward asymmetry of the leptonic pair in bothB → Dτ ν and B → D * τ ν decays [22,[28][29][30][31][32][33][34][35][36] and the asymmetry in the polarization of the τ lepton in the decays [12,22,29,30,[35][36][37][38]. 1 We will focus on the future measurement of these asymmetry observables at Belle II. 2 An upgrade of the Belle experiment, Belle II is an e + e − collider with asymmetric beams and center of mass energy of ∼ 10 GeV, producing Υ(4S) which subsequently decay to pairs of B mesons. It is projected to collect more than forty times the data of Belle (around 50 ab −1 , a total of ∼ 55 million BB pairs) by 2025. With the total planned dataset, the uncertainty on R D (R D * ) is expected to be as low as ∼ 3% (∼ 2%) [40]. If the current global average (1.2) persists after Belle II, it will indicate an undisputed discovery of new physics. In the case of such a discovery, the asymmetry observables we study in this paper can provide information about the beyond-the-SM physics responsible for these anomalies.
In this work we calculate the contribution to these observables for all the different types of proposed mediators, and we present our results as numerical formulas for each observable in terms of the Wilson coefficients of the general dimension-6 effective Hamiltonian. Previous studies [22,23,[28][29][30]41] have considered how to separate different explanations of the R D ( * ) anomalies using angular observables. In this paper, we include additional operators involving new RH neutrinos in the comparison and specifically aim   (3), SU (2), U (1)). We further indicate the type of neutrino they require in theB meson decay to explain the anomalies.
to distinguish simplified models with different neutrino chiralities. We also take the experimental limitations into account and focus on the observables for which there are proposed measurement strategies. Besides requiring that these models simultaneously explain both the R D and R D * anomalies, we also impose other experimental constraints (in particular, modifying the branching ratios for B c → τ ν [42][43][44][45][46] and B → X s νν [47][48][49][50]). With all the experimental constraints taken into account, the list of currently viable mediators that can individually explain the anomalies can be found in Tab. 1. (We use the nomenclature from [51] for the LQs.) We show that these asymmetry observables can significantly differentiate between models with LH or RH neutrinos, even after taking into account projections of future experimental precision [36].
The questions of what mediators remain viable and how distinguishable they are from one another after Belle II depend heavily on the R D ( * ) ratios measured by the new experiment. To highlight the power of these asymmetry observables, in this work we will consider two hypothetical outcomes for the global averages after Belle II, which have substantially different implications for our study. In the first scenario, we imagine that Belle II will measure R D ( * ) equal to their current global averages. (This would correspond to a ∼ 10σ discrepancy with the SM.) In the second scenario, we posit that R D ( * ) will be reduced, but remain 5σ discrepant with the SM. As we will show, in the 10σ scenario, the neutrino chiralities can be easily distinguished from one another using the asymmetry observables, and in all but the worst case scenarios even individual models can be told apart. Meanwhile in the 5σ scenario, we show that the neutrino chiralities can almost always be distinguished from each other. In order to distinguish a small subset of models we require additional CP-odd polarization asymmetry measurements.
A measurement strategy for these CP-odd observables is yet to be delivered, but due to their discriminating power, they should be considered a high priority. Our results highlight the importance of finding experimental strategies for their measurement.
The outline of our paper is as follows. In Sec. 2, we describe those single operators and simplified models which can explain both R D ( * ) anomalies and survive other experimental constraints. We also define in detail the two different scenarios for Belle II measurements of R D ( * ) described above, and show how these measurements alone can significantly reduce the set of viable models. In Sec. 3, we define the angular observables (forwardbackward and polarization asymmetries) and calculate their dependence on the Wilson coefficients. We further discuss their experimental status and review a recent proposal [36] with higher projected sensitivity for their measurements at Belle II. Finally, in Sec. 4, we show which combinations of angular observables can be used to distinguish the viable models with LH and RH neutrinos. We show that for the bulk of possible outcomes at Belle II, we will be able to tell different types of neutrinos apart, and in many cases can distinguish individual models as well. We conclude in Sec. 5 with a brief summary and outlook.
Several appendices are included in the end. In App. A we list the leptonic matrix elements used in our calculation as well as some hadronic functions needed for the calculation involving RH neutrinos. App. B includes further details on the calculation of the asymmetries and full analytic formulas for each of them. Finally, in App. C we point out a linear relationship between different CP-even observables we study in this work and explain a numerical scan that we perform over the viable range Wilson coefficients.

Simplified Models for R D ( * )
The set of all possible dimension-6 operators modifying the b → cτ ν decay rate can be written as where the pre-factor normalizes the SM Wilson coefficient to unity, and the four-fermion effective operators are defined as for M, N = R or L. These operators can be generated by integrating out heavy new mediators; the Wilson coefficients C X M N parametrize the most general contribution. 3 Different UV models can be categorized using the operators they give rise to (typically more than one), see Sec. 2.2.
In the operator basis of (2.1), the contribution of new physics to the ratios R D ( * ) can be calculated in terms of the ten (possibly complex) Wilson coefficients: five involving a SM left-handed neutrino, and five requiring a new right-handed neutrino. The numerical contribution of all the operators from (2.1) to the ratios are [24]: Further details on deriving these numerical equations are included in App. B.

Single Operator Solutions
The range of R D ( * ) that each individual operator can generate (with general complex Wilson coefficients) is indicated in Fig. 2, along with the present-day experimental and theoretical combined uncertainty in the R D ( * ) measurements, showing the 1, 2, and 5σ contours (gray-dashed ellipses). For a review of experimental correlations in the measurements of R D ( * ) , see [4,6]. In Fig. 2, we use the current average of the correlations, ρ corr = −0.2 [10]. We see that out of all ten effective operators in ( six that can explain both anomalies simultaneously: We are not aware of any UV-complete models in the literature for these anomalies that rely solely on any of the operators O V RL , O T LL , or O V LR . Despite this lack of UVcomplete models, we will include these three single operator explanations in our analysis for the sake of completeness.

Simplified Model Solutions
We can now enumerate the full set of "simplified" models that can explain both the R D ( * ) anomalies. In this context, "simplified" means a single new mediator particle that can be integrated out to provide one or more of the effective operators which modify R D ( * ) .
An over-complete list of all the simplified models that can generate the operators in (2.1) with LH or RH neutrinos can be found in [48,52,53]. We gather these mediators Table 2: A complete list of the simplified mediator models and resulting effective operators that are possibly relevant for the R D ( * ) anomalies. The U µ 1 and S 1 LQs as well as the colorless scalars can give rise to two independent Wilson coefficients, while the rest of the mediators can generate only one. We use x = 1/8 in this work, see the text for more details. We indicate in the last column if the model is still viable (by ) and if not, what experimental constraint rules it out (see Sec. 2.3 for discussion of these constraints).

Viability Colorless Scalars
in Tab. 2. Notice that the S 1 and U 1 LQs and a heavy W can couple to either LH and RH fermions and so give rise to operators involving either type of neutrinos. In this work we consider these possibilities as separate solutions to the anomalies and will try to distinguish them from one another.
The factor of x in Tab. 2 relates the Wilson coefficients of scalar and tensor operators in some models after Fierz transformation. At the mediator scale, x = 1/4 for all the models in Tab. 2; as we run down to the GeV scale x changes to ∼ 1/8 [54][55][56], with the exact value depending on the mediator scale. For simplicity, we use the fiducial value x = 1/8 in our analysis.
In Fig. 3, we show the values of R D and R D * which can be obtained by each of the relevant mediators in Tab. 2, scanning over complex Wilson coefficient(s). In these plots the superindices L and R on S 1 and U 1 LQs refer to the neutrino chirality they couple to. Some mediators yield lines in this parameter space; these are single-coefficient models whose contribution to R D and R D * are independent of the phase of the coefficient. Other operators can cover a region of R D ( * ) as the coefficients are varied, either because the R D ( * ) values depend on both magnitude and phase of single operator, or the model results in two independent Wilson coefficients.

Additional Constraints and Final List of Viable Models
In addition to explaining R D ( * ) , a viable mediator must also avoid a number of other stringent constraints. In this subsection we will review these and then list the surviving viable solutions. A subset of the couplings which modify theB → D ( * ) τ ν decay can enhance the branching ratio B c → τ ν [42][43][44][45][46]. In terms of the Wilson coefficients in (2.1), (2.4) Given the mass ratios above, these equations imply tighter bounds on the scalar operators than the vector ones. The SM prediction is Br(B c → τ ν)| SM ∼ 2%. The B u → τ ν decay in LEP at the Z boson peak can be used to place the constraint [45] Br(B c → τ ν) 10%, (2.5) which in turn puts a constraint on the possible Wilson coefficients in (2.4). Using the theoretical calculation of the B c lifetime and its uncertainties, a looser bound of Br(B c → τ ν) 30% can be obtained as well [43]. These branching ratio constraints put particularly severe bounds on models relying on O S M N operators to explain the anomalies -to the extent that if a model relies solely on a scalar operator to explain the anomalies, it is ruled out by the constraint (2.5). This remains true even if the global average of the anomalies reduces to the magenta dot in Fig. 3 after Belle II.
The other relevant flavor constraint is the B → X s νν branching ratio [47,48]. The current bound is from the ALEPH Collaboration [57], at 90% CL. While the mediators introduced for R D ( * ) generate charged currents, the B → X s νν decay requires a neutral current beyond the SM. However, in some models that rely on leptoquarks which couple to left-handed fermions [48,49], there is an inevitable neutral current due to the SM SU (2) L symmetry. For instance, the S 3 LQ can give rise to the following terms (among others) [49] L ⊃ g ij LQ c,i where i, j are flavor indices and a is an SU (2) adjoint index. After Fierz transformation, this LQ can give rise to O V LL with where G F is the fermi constant and M S 3 is the S 3 LQ mass. Due to the SU (2) L symmetry, this term will contribute to B → X s νν as well. 4 The bound from B → X s νν translates into [49] |g 4 It is possible to generate C V LL with these leptoquarks by invoking g i =3,j L couplings as well. In this case, however, we will have a substantial CKM suppression and will need non-perturbative couplings to explain the anomalies. As a result, we discard this possibility.
Putting these together, we obtain an upper-bound on C V LL : C V LL 0.06. (2.10) A similar bound also applies to the U 3 and S 1 LQs that are coupled to LH fermions [49]. Since S 1 can generate O S LL and O T LL operators from other couplings in the Lagrangian, it can still be a viable explanation of the anomalies despite this severe bound on C V LL . S 3 and U 3 , on the other hand, can only generate O V LL and are therefore ruled out [49]. Finally, due to the SU (2) structure of the operators that it gives rise to, this bound does not apply to U 1 LQ [47,49], even though this LQ does generate O V LL . Other than these flavor constraints, there are some bounds from direct searches for these mediators. For the case of leptoquarks, the current bounds are not severe enough to rule out any further models [53,56,58]. On the other hand, the bounds on the W are fairly constraining [24,25,27,50]. In particular, if the W couples to LH fermions, the bounds on the accompanying Z effectively rule out the explanations of the anomalies [27,50].
The combination of these constraints significantly reduces the viable explanations of the R D ( * ) anomalies. In the last column of Tab. 2 we indicate which models survive. In all, there are three viable simplified models that couple to LH neutrinos, and four that couple to RH. Along with the viable single operators O V LR , O V RL , and O T LL , these will be the focus of the rest of the paper.

Benchmark Belle II Scenarios
Belle II will measure R D ( * ) with much smaller errors compared to the present, thus greatly reducing the possible range of Wilson coefficients in each model. As can be seen in Figs. 2-3, central values near the present averages would by themselves rule out at high significance many models which are presently under consideration. Meanwhile, values closer to the SM prediction (while still allowing a 5σ discovery at Belle II) would leave all the mediators and single operators we currently consider as possibilities, before constraints from the asymmetry observables are applied. Aside from having a potentially huge impact on the list of models that explain the anomalies, this can also greatly affect our ability to distinguish between these models with further measurements (such as the asymmetries).
As a result, we will consider two different outcomes of the Belle II measurement of R D ( * ) as benchmarks for our study.
1. The 10σ scenario: Belle II measures R D ( * ) with central values equal to the present average. With the projected Belle II sensitivities, this would correspond to a O (10σ) discovery. We then consider ranges of R D ( * ) within the 1σ Belle II error ellipse about this central value (the innermost red ellipse in Fig. 2-3). As we will show, in the 10σ scenario, the task of discerning different models is simplified considerably.
2. The 5σ scenario: The measured R D ( * ) values are closer to the SM expectation while still allowing a 5σ discovery; specifically, we assume the central value of the anomalies after Belle II shifts to R D = 0.34 and R D * = 0.275. This point was chosen to have 5σ significance with Belle II projected error bars, to be within ∼ 2σ of the current global average, and (crucially) to allow for all of the simplified models to continue to explain the R D ( * ) anomalies (see Fig. 3). Compared to the 10σ scenario, distinguishing between different models is much more challenging here.
These two benchmark scenarios provide a sense of the range of possibilities that we can expect from Belle II. In particular, while there are outcomes which allow a robust discovery and would make nailing down the UV model even easier than in our 10σ benchmark scenario, there are no outcomes that, while still implying a discovery, are more difficult than our 5σ scenario for distinguishing between different models.

Asymmetry Observables
The relevant models for R D ( * ) and their predictions for these ratios were reviewed in the previous section. However, one can extract more information from the decay processes than just the total decay rate and the ratios R D ( * ) . Shown in Fig. 4 is a diagram of the detailed kinematics of the decay process. Many of these angles and momenta can be measured or reconstructed, and they provide a much finer probe of the effective Hamiltonian responsible for the decay.
In particular, using the event kinematics, we can construct asymmetry observables which are sensitive to the different Wilson coefficients in (2.1). Four such observables are the forward-backward asymmetry of the τ lepton with respect to p D ( * ) in Fig. 4, denoted by A ( * ) F B , and its polarization asymmetry in all three of theê directions in Fig. 4, denoted by P ( * ) e . All of these asymmetries are defined in the leptonic center of mass frame, which we will also refer to as the "q 2 frame", where q = p B − p D ( * ) = p τ + p ν denotes the fourmomentum transferred to the leptonic system by the decaying B meson. As we will see, models with LH and RH neutrinos have a qualitatively different contribution to these asymmetry observables. Figure 4: The kinematics ofB → D ( * ) τ ν and subsequent τ → dν decay processes, in the center-ofmass frame of the leptonic system (the "q 2 frame"). The black plane indicates the original decay plane, defined by the B momentum p B (or the D ( * ) momentum p D ( * ) ) and the leptonic pair. The red plane is the decay plane of the τ , defined by the visible daughter meson d and invisible daughter neutrino ν of the τ . The three directions in which we will project the τ polarization asymmetries are indicated in green.
We will calculate the dependence of these observables on all the Wilson coefficients in (2.1) and report the result in the form of numerical formulas (like (2.3) for R D and R D * ). In particular, we carry out the calculation including the contribution of the operators with right-handed sterile neutrinos with negligible masses compared to the other energy scales in the decay. Full analytic versions are available in the appendices. Wherever possible, we have checked that parts of our calculations (results from the numerical equations, q 2 distributions, the SM predictions, etc.) are in agreement with previous studies, e.g. [22,35,39,59]. A further consistency check is that the numerical equations for the observables will manifest a symmetry between left-and right-handed neutrinos such that by applying the following transformations, (where h τ refers to the τ helicity) the observables will transform as In writing 1 + C V LL in (3.1) (and in all the up-coming numerical equations), we are explicitly separating the contribution of the SM operator. 5 These symmetries indicate 5 The complex conjugate in the way the Wilson coefficients are transformed is only relevant for the that if we flip the spin of all the external particles and the associated Wilson coefficients, we should get the same result for the decay rate in a particular q 2 and θ direction. The interference between the SM term in (3.1) and the sign flip in (3.2) are the two sources of the qualitatively different contributions from different types of neutrinos.

Forward-backward Asymmetry
The first observable of interest is the forward-backward asymmetry in the τ lepton decay with respect to the D ( * ) direction. This observable and its correlation with R D ( * ) have been studied previously [22,[28][29][30][31][32][33][34][35][36]. It is defined as where θ is the angle between the τ and D ( * ) momenta in the leptonic system rest frame, see F B that follows from this is:

Tau Polarization Asymmetries
Our second set of observables is comprised of the different polarization asymmetries of the τ lepton in the decay. Such asymmetries are defined as where ± refer to the two possible outcomes of measuring τ spin along directionê. The vectorê can be in any arbitrary direction. We consider the three directions [36], where p τ ( p D ( * ) ) is the spatial momentum of the τ (D ( * ) ) in the final state (all in the q 2 frame). P

Longitudinal polarization
The numerical expression for the contribution of all the Wilson coefficients to P ( * ) τ is:

Perpendicular polarization
Similar to the previous section we include the numerical expression for contribution of all the Wilson coefficients to P ( * ) ⊥ .

Transverse polarization
Finally, we present the numerical formulas for P ( * ) T : T observables are particularly interesting to measure as they can provide us with a way to hunt for CP-violation in B-meson decays. The SM prediction for these observables is zero. In this work we focus on the P ( * ) T observables for theB meson decay. Due to its CP-odd nature, the associated observables in the decay of B mesons can be obtained by complex conjugation of all the Wilson coefficients, i.e. an overall sign.

Overview of the Experimental Results and Proposals
So far the only asymmetry observable studied experimentally is P * τ , by Belle in a series of works [7,60,61]. The missing energy in these decays prevents us from fully reconstructing all the momenta and thus complicates the measurement of different angular observables. However, Belle was able to extract P * τ from single-prong τ decays, τ → dν with d = π, ρ, using the observation that the differential decay rate ofB → D * τ ν, τ → dν can be written as where θ hel is the angle between d and the opposite of the W * direction in the τ rest frame, see Fig. 5. The constant α d captures the sensitivity to P * τ of the particular τ decay channel under study.
Unfortunately, the τ rest frame is not reconstructible, even at the B-factories. What is reconstructible is the q 2 frame, i.e. the leptonic center of mass frame, by boosting to the frame where the (fully measurable) B and D * momenta are pointed in the same direction. Furthermore, in the q 2 frame, the angle θ τ d between the τ and its daughter meson d is given by The RHS is completely known, because the magnitude of the τ momentum is a function of q 2 in the q 2 frame As evident from Fig. 5, the angle θ τ d is related to θ hel via a boost along the τ momentum direction. Although we do not know the direction, it is enough to know the magnitude: Figure 5: A schematic showing the Lorentz boost that relates the angles θ τ d in the q 2 frame on the left and θ hel in the τ rest frame on the right. The former angle is reconstructible at the B-factories, while the latter is used to extract P ( * ) τ . Although the τ momentum vector cannot be fully reconstructed at the B factories, its magnitude is measurable, and this is sufficient to relate the two frames.
is the momentum of the daughter meson in the τ rest frame, and γ = E τ /m τ . This relation determines θ hel in terms of all measurable quantities, and allowed Belle to obtain a measurement of P * τ = −0.38 ± 0.51 +0.21 −0.16 (compared to a SM prediction of (P * τ ) SM = −0.501). Although this method works, it resulted in an enormous uncertainty, and has so far only been applied to P * τ . There are further proposals in the literature on how we can infer additional asymmetry observables from the angular distribution of the visible daughter mesons in the τ lepton decays. In particular, [36] puts forward methods for measuring P τ , P ⊥ , and A F B in B → Dτ ν decays (with τ → dν), claiming a better attainable precision than the Belle procedure described above.
In their method, q 2 , E d , and the angle θ d between d and D -all evaluated in the q 2 frame, and all directly measurable -are used to express the differential decay rates, where B d is the branching ratio of τ into the daughter meson under study, N is a normalization factor, and I 0,1,2 are functions of q 2 and E d defined in [36]. After integrating over θ d , adding together or subtracting the decay rates into the two spatial hemispheres give rise to double distributions, from which P τ , P ⊥ and A F B can be extracted. In Tab. 3, we list the projected Belle II sensitivity claimed in [36] (which we also adopt in this work), as well as our calculation for the SM predictions. Although there are currently no analogous proposals to measure the D * asymmetry observables in the literature, we believe that a similar method to the one proposed in [36] should be applicable.
At present, there is no substantiative experimental proposal for how to measure P ( * ) T at Belle II. 6 However, we have included it in our analysis, owing to the important role 6 In [35] it has been shown that the total decay rates above and their dependence on the azimuthal Table 3: Observables studied in this work, our numerical calculation for the prediction in the SM, and the projected Belle II sensitivity (assuming the 50 ab −1 full data set) where available. We use these observables to identify different explanations of the anomalies. In the upcoming sections we will assume the observables in B → D * τ ν are measured with the same uncertainty as in B → Dτ ν.
it can play in distinguishing models with LH and RH neutrinos, and in the hopes that viable proposals for how to measure it will emerge in the future.

Distinguishing Different Solutions
Having calculated these asymmetry observables, we now use them to distinguish between different simplified models for the R D ( * ) anomalies (see Sec. 2). As the range of possible Wilson coefficients depends on the value of R D and R D * after the Belle II data set is collected, we consider the two benchmark scenarios described in Sec. 2.4 and indicated in Figs. 2 and 3.

10σ Scenario
In this scenario, for the models involving the LH neutrinos, the two LQs U 1 and S 1 , as well as the single operators O T LL and O V RL , will be able to explain the anomalies while satisfying the experimental bounds mentioned above. Among the RH neutrino proposals, only U 1 and S 1 LQs will remain viable. Fig. 6 shows the ranges of CP-even asymmetry observables that are achievable in each model, projected here into 2d plots, one for each pair of observables. In each model, we have scanned over the (complex) Wilson coefficients of the model, subject to the following constraints: R D and R D * should be within the 1σ Belle II error ellipse for this scenario; Br(B c → τ ν) < 10%; and the Br (B → X s νν) bound from (2.10) (on the S 1 model coupled to LH neutrinos). Further details on how to efficiently carry out this scan are included in App. C.
angle χ between the two planes in Fig. 4 contains information about P ( * ) T . We cannot confirm the claim that this angle is experimentally accessible and are not aware of any experimental proposals for its measurement at Belle II. Fig. 6 that by measuring all these observables we can distinguish well all LH models (green) from all RH models (red). It is possible to accomplish this even without 2d plots -single observables, namely P τ , P * τ and possibly P ⊥ , are sufficient to separate LH from RH models. This conclusion would be unchanged even if we had applied the looser Br(B c → τ ν) < 30% bound.

It is obvious from
Given that the asymmetry observables can easily tell apart the LH and RH neutrino models, it makes sense to ask if we can further distinguish individual models with the same neutrino chirality. Including multiple observables in the effort (as in the 2d plots in Fig. 6) is essential for this purpose. Fig. 6 does indicate that there are indeed many possible values of the asymmetry observables for which these models can be singled out, see especially P * ⊥ -P τ , A * F B -P τ and A * F B -P ⊥ , although a lot will depend on the precision with which they can be measured.
However, we also see that there are places where the models overlap with each other or are within the projected experimental precision of the observables. Specifically: two of the green regions, which correspond to S 1 and U 1 LQs coupled to LH neutrinos, overlap with one another in this way. The same is the case for the two red regions, which refer to the same LQs coupled to RH neutrinos.
To quantify how well we can separate these models, we use a crude χ 2 measure that includes the six CP-even asymmetry observables from Fig. 6 as well as the R D ( * ) ratios themselves, for a total of 8 d.o.f.. As there are no data available at this point, the correlation between different asymmetry observables (and R D ( * ) ) is not known; we neglect these correlations in this χ 2 estimation and only use the current ρ corr = −0.2 between the R D ( * ) ratios. We calculate this χ 2 between all pairs of scanned points from the models that we want to distinguish (here the S 1 and the U 1 mediators with the same type of neutrinos). We use the relative uncertainties from Tab. 3 in this calculation; as there are no projections for the precision in measuring the quantities inB → D * τ ν, we use the same relative uncertainty as their counterparts inB → Dτ ν from Tab. 3 in our calculation.
In Tab. 4 we list two benchmark pairs of measurement outcomes in which the two mediators S 1 and U 1 , coupled to LH neutrinos, are not distinguishable; benchmark points for the case of RH fermions are included in Tab. 5. Here we take the vector operators to have real Wilson coefficients without loss of generality, since the observables are insensitive to an overall rephasing of all the Wilson coefficients. The C S and C V refer to different scalar and vector Wilson coefficients for the different models; see Tab. 2 for details.
These benchmark points illustrate that the six CP-even asymmetry observables are P τ Figure 6: Two-dimensional plots of asymmetry observables for the 10σ scenario. We scan over Wilson coefficients that result in R D ( * ) values within the 1σ Belle II error ellipse centered on the presentday world averages. We also impose the Br (B → X s νν) bound [49] (on the S L 1 LQ model) and the Br(B c → τ ν) 10% bound [45]. The projected Belle II precision for each observable, centered on the SM prediction, is indicated by the dashed gray lines, see the text. Regions which can be realized by models with LH SM neutrinos (shown in green) are from S 1 and U 1 LQs and single operators O T LL and O V RL , while those requiring new RH neutrinos (shown in red) are S 1 and U 1 LQs. It is obvious that we can clearly distinguish models with LH and RH neutrinos from one another by measuring these asymmetry observables.  Table 4: Pairs of benchmark points for the LQ models S 1 and U 1 coupled to LH fermions that are less than 1σ apart in our estimation. The approximate uncertainties using Tab. 3 are quoted in the first row as well. We need further measurements to distinguish these models in these cases.  Table 5: Pairs of benchmark points for the LQ models S 1 and U 1 coupled to RH fermions that are less than 1σ apart in our estimation. The approximate uncertainties using Tab. 3 are quoted in the first row as well. We need further measurements to distinguish these models in these cases. T observables for the points from Fig. 6 that are less than 1σ apart in our estimation. Left: the LQs only interact with the LH neutrinos. Right: the LQs only interact with the RH neutrinos. Notice the matching slopes of the lines across the two plots, which is a consequence of the symmetry outlined in (3.1)-(3.2). These figures indicate that the CP-odd asymmetries P T and P ( * ) T may be useful for further distinguishing the U 1 and S 1 leptoquark models; however, the fact that they cross at the origin also indicates that these asymmetries cannot resolve the difference in all cases.

Model
not enough to completely break the degeneracy between the S 1 and U 1 LQs when they are coupled to the same types of neutrinos. However, we still have a pair of observables at our disposal that could distinguish these models: P T and P * T . After our χ 2 estimation with all eight CP-even observables, we keep the points from the S L 1 and the U L 1 (or S R 1 and U R 1 ) models that are less than 1σ apart from each other and study their contribution to the CP-odd observables P ( * ) T . The results are depicted in Fig. 7. Clearly, a lot depends on the precision that could be achieved in a hypothetical future Belle II measurement of P ( * ) T . But the fact that the lines cross in Fig. 7 means that P ( * ) T will never be a foolproof way to break the degeneracy between the S 1 and U 1 LQs.
To recap, in this scenario, measurement of the CP-even asymmetry observables at Belle II, for which theoretical proposals exist [36], can easily discern the models of different types of neutrinos. Models with the same type of neutrinos can be distinguished in many (but not all) cases using the same CP-even measurements or with the additional measurement of CP-odd polarization asymmetries P

5σ Scenario
In our second scenario for the outcome of Belle II measurements, we study the situation in which the observed values of the R D ( * ) anomalies in the Belle II data are reduced significantly from the present average, but still significant enough to be claimed as a 5σ discovery. See Section 2.4 for details.
Similar 2D plots for this scenario as in the previous one are included in Fig. 8. We  [45]. The projected Belle II precision for each observable, centered on the SM prediction, is indicated by the dashed gray lines, see the text. All the currently viable models and single operators remain viable in this scenario. Regions which can be realized by models with LH SM neutrinos are shown in green, while those requiring new RH neutrinos are in red. In many cases, the green and the red regions are quite distinct. However, there is also significant overlap between these regions. Further measurements are required to disentangle the RH and LH models in such cases, see the text.  Table 6: Pairs of benchmark points for the LQ models S 1 coupled to either LH or RH fermions that are less than 1σ apart in our estimation. The approximate uncertainties using Tab. 3 are quoted in the first row as well. We need further measurements to distinguish these models in these cases.
see immediately that the various regions are much closer together than in Fig. 6, as expected from the reduced requirement from R D ( * ) . In contrast with the 10σ scenario, here individual observables are not sufficient to distinguish RH from LH models. Fortunately, we can achieve decent separation by going to pairs of observables, as shown in Fig. 8. The best pairs (again, highly contingent on the projected precision of the Belle II measurement) appear to be P τ -P * ⊥ and P τ -P * τ , and possibly also P τ -A * F B . There are however possible experimental outcomes that cannot be definitively attributed to RH or LH models. Using our χ 2 measure, we find that except for the two models S L 1 and S R 1 , all the LH neutrino models can be distinguished from all the RH neutrino models by at least 1σ. In Tab. 6 we list two benchmark pairs of points for these two models that are not distinguishable.
To further distinguish between the remaining overlapping models (S L 1 and S R 1 ) we can again study their contribution to the CP-odd polarization asymmetry, P ( * ) T . Shown in Fig. 9 are the values of P ( * ) T for points from these two models that are less than 1σ apart in our χ 2 measure. Notice the distinct effect that different models have on these CP-odd observables. While the CP-even ones were not able to completely break the degeneracy among the LH and RH models (more specifically S L 1 and S R 1 ), the added CP-odd ones are able to do so, given enough experimental precision. (Even with ±0.1 precision, it should be easily possible to tell the models apart.) This puts more emphasis on the importance of measuring these CP-odd polarization asymmetries, for which no Figure 9: The P T and P * T observables for the points from Fig. 8 that are less than 1σ apart according to our χ 2 constructed from all the CP-even observables. The green (red) points correspond to S 1 LQs coupled to LH (RH) neutrinos. Notice the identical slope of the lines, which is a consequence of the symmetry outlined in (3.1)-(3.2). If these CP-odd asymmetries can even be measured, and with enough experimental precision, then they will be able to distinguish between the LH and RH neutrino cases.

Conclusion
In this work we studied various τ asymmetry observables that can potentially be measured at Belle II and that could help to resolve the BSM origin of the long-standing R D ( * ) anomalies. In We also catalogued all the simplified models involving both LH and RH neutrinos that explain the R D ( * ) anomalies and are not ruled out by the severe Br (B c → τ ν) and Br (B → X s νν) constraints, see Tab. 2. We then showed that, using the CP-even asymmetry observables A ( * ) ⊥ for which proposed measurement methods exist, it is possible to tell apart solutions with different types of neutrinos (SM LH vs. RH sterile ones) from one another in the vast majority of cases, see Fig. 6 and Fig. 8. In many instances, it is even possible to tell apart different mediators with the same neutrino chirality. The most useful observables for this purpose were P τ and P In some of the most difficult cases, the CP-even asymmetries are not enough. Here we show that the information carried in the CP-odd asymmetries P ( * ) T plays a further, crucial role in distinguishing different models. As these observables do not yet have a fullydeveloped experimental strategy, our results provide a strong motivation to construct one.
Our ability to distinguish between different BSM models for the R D ( * ) anomalies depends on what Belle II actually measures for R D ( * ) . If Belle II measures the R D ( * ) ratios near the present values with much smaller error bars, then this measurement alone will greatly reduce the number of viable new physics models with either left-or right-handed neutrinos, compared to the present situation. In this case, it will be relatively straightforward to distinguish different models from one another using asymmetry observables. If instead, Belle II finds R D ( * ) ratios which are closer to the Standard Model prediction, while still constituting a 5σ discovery of new physics, then more models remain viable, and distinguishing between them becomes more difficult. However, in either scenario, we show that it is at least possible to distinguish between models with LH neutrinos and models with RH neutrinos in almost all cases, though measurements of the CP-odd P ( * ) T observables may be necessary in a small subset of cases to break degeneracies.
One caveat in our work is the lack of a study of the Belle II sensitivity to the τ asymmetries inB → D * τ ν decays. We simply assumed these observables can be measured with the same projected precision as the ones from the decay into D mesons quoted in [36]. A similar, dedicated study of how to measure τ asymmetries at Belle II in theB → D * τ ν mode is very well motivated.
Different asymmetry observables studied here are integrals over an angle and q 2 of double-differential distributions (see the derivations in App. B). While we have studied the angular dependence using the asymmetries, the q 2 dependence could in principle be used as well. Whether the q 2 distributions of all the τ asymmetries can be measured at Belle II, and whether they are useful for distinguishing between different models, are interesting questions for future study.
It is also important to think about other angular measurements and their ability to tell different models apart. In particular, the D * polarization may be able to differentiate between various new physics explanations, see for example [12,28,29,32,34]. However there is currently no published study of how feasible it is to measure the D * polarization at Belle II.
Additionally, although we have focused on measuring these observables at Belle II in this paper, one can also consider the possibilities of doing this at LHCb. There are various reasons that suggest that LHCb will have significant difficulty in measuring these quantities with reasonable precision -in particular lack of knowledge of the initial rest frame and generally higher background. However, it is conceivable that high statistics at LHCb and lower-background decay channels like τ → lνν may be leveraged to obtain comparable precision in the measurement of these angular asymmetries.
Finally, it would be interesting to consider complementary approaches to distinguishing different mediators from one another. For instance, in our study the most difficult cases generally boiled down to distinguishing various leptoquark models (e.g. S 1 and U 1 ) from each other. Even though the current collider bounds on these are not severe enough to constrain these models, future direct searches would provide a complementary strategy to distinguish them, since the S 1 and U 1 leptoquarks decay differently [56].
Overall, this is an exciting time for B physics and the R D ( * ) anomalies. With Belle II coming online in the very near future, we will soon know if these anomalies are due to new physics or not. If they are due to new physics, it will be crucial to pin down the precise BSM origin of the anomalies. The results presented here are meant to be a significant step in this direction.

A Leptonic and Hadronic Functions
In [39], the physics of the leptonic and the hadronic side of the processes in the R D ( * ) anomalies are factorized and the relevant matrix elements are calculated. We use the leptonic and the hadronic matrix elements reported therein in our work. However, as we are working with right-handed neutrinos, one needs to calculate a few more matrix elements. In this appendix we report the new leptonic matrix elements involving righthanded neutrinos, and the new hadronic matrix element with tensor current.
The leptonic matrix elements are defined as whereλ (λ τ ) denotes the polarization of the mediator (τ lepton). We use the same convention for the as [62]. Explicitly carrying out the calculation, we find the following results for different polarizations.
Lr ± λλ = 0, (A.12) where θ is again the angle between the τ lepton and the D ( * ) in the leptonic system restframe, see Fig. 4, and v = 1 − m 2 τ /q 2 . The subscript t refers to the fourth polarization of a virtual mediator.
Throughout our study we have used the hadronic functions listed in [39,48] as well. However, when working with the right-handed neutrino models for R D ( * ) , there is one missing hadronic function, namely where σ µν = i/2[γ µ , γ ν ]. To calculate this matrix element, we can simply borrow the results in [39] for the hadronic side of the operator O T LL (cσ µν (1 − γ 5 )b operator) and merely flip the sign of the axial current. The resulting hadronic functions, denoted by where the superscript s indicates that these functions are corresponding to D meson and H T functions are defined in [39,48], and for D * where again the H T functions and the form factors A 1 (w) and R 1 (w) are defined in [39,48], m M is the final meson (here D * ) mass, and

B Analytic Expressions for the Observables
In order to get an expression for the polarization asymmetries, we write the total decay rate with the spin of the final state τ lepton in the arbitrary directionŝ as [37] dΓ ( * ) (ŝ) = 1 2 dΓ where we have suppressed all other final state indices, e.g. D * polarization, and dΓ ( * ) where M ± are the corresponding matrix elements with ± τ helicity. The phase space element dΦ is given by with m M being the final meson (D ( * ) ) mass, q 2 being the four-momentum transferred to the leptonic side, and θ being the angle between the τ momentum and the final meson M in the q 2 frame. Using (B.2) in (3.5) we find an expression for the integrated asymmetries in every direction F B , to find analytic formulas for different decay rates used in Sec. 3. As indicated in the previous appendix, we use the convention and the notation in [39,48] for the hadronic functions.
We start with Γ ( * ) tot and Γ ( * ) τ . For the LH neutrinos contribution to theB → Dτ ν we have Similarly, the contribution of the RH neutrinos to these rates are The dependence of all the hadronic functions H on q 2 is implicit. These equations can be used to calculate the contribution of each type of neutrinos to R D and P τ . For the LH neutrinos contribution toB → D * τ ν we have dΓ * The corresponding decay rates with RH neutrinos instead are can be found in [39]. The expressions for H T 2 functions were reported in App. A.
The symmetry outlined in (3.1)-(3.2) between RH and LH neutrino contribution is manifested in the decay rates (B.6)-(B.9). In other words, the unpolarized decay rates have the same q 2 dependence for either types of neutrinos. The only difference between the two cases is the irreducible SM contribution. Even considering this difference, there are scenarios with different types of neutrinos that have indistinguishable q 2 distributions. Examples of this are illustrated in Fig. 10 for various LQ models interacting with different types of neutrinos. In this figure we normalize the differential rate to the SM total rate, so that the area under each plot is proportional to its R D ( * ) prediction. Each curve results in an R D ( * ) close to the current global averages (see the 10σ scenario in Sec. 2.4). These plots show that in this scenario there are benchmark points that, unlike the asymmetry observables we studied, the q 2 distribution of the decay rates will not be able to distinguish models with different types of neutrinos.
For Γ ⊥ and Γ T we have  Figure 10: The q 2 distribution for benchmark Wilson coefficients for models interacting with LH neutrinos (green curves) and those interacting with RH neutrinos (red curves). The decay rate for B → Dτ ν (B → D * τ ν) is shown on left (right). In each plot we show models whose effective operators from Tab. 2 are related through the symmetry in (3.1)-(3.2). The dashed gray line is the SM prediction. The area under each curve is proportional to its prediction for R D ( * ) . Up to a rescaling factor the plots for different types of neutrinos have indistinguishable shapes.
Equivalently, we can write the polarization asymmetries inB → D * τ ν as where Σ * is given by Let us now move on to the forward-backward asymmetries A ( * ) F B . Here we report the analytic results for a finer observable, namely the forward-backward asymmetry of τ with a specific helicity. For theB → Dτ ν decay involving the LH neutrinos we have Here the superscripts ± refer to specific τ helicities. Similarly, for the right-handed neutrinos contribution we have Equivalently, for the decays into D * we have for the LH neutrinos contribution. For the RH neutrino contribution we have We note that A ±,( * ) F B contain more information than the A ( * ) F B observables we studied in this work. The experimental proposal in [36] for measuring the forward-backward asymmetry is applicable to the total asymmetry summed over final τ helicity. It is intriguing to find a similar proposal for measurement of A  [39,48] and from App. A, and integrating over q 2 . We use the same numerical parameter values as in [24], which we list here again in Tab. 7 for completeness. 2.010 0.511 × 10 −3 0.106 1.777 Table 7: The parameter values used in the calculation of the numerical formulas in this paper.

C Scanning the Parameter Space of LQ Models
In this appendix we describe a numerical method to calculate all the CP-even asymmetry observables as a function of R D ( * ) and Br (B c → τ ν). This will allow us to efficiently carry out a scan over the parameter space of different models that generate the correct value of R D ( * ) and respect the bound (2.5) on Br (B c → τ ν).
For models that result in a single Wilson coefficient, we simply scan over the two degrees of freedom (the real and imaginary coefficient values) and find the ones respecting the conditions on R D ( * ) and B c decay rate.
For the LQ mediators that result in two independent complex coefficients (S 1 and U 1 LQs), we have a scalar current Wilson coefficient that we denote by C S , and a vector current denoted by C V . When these LQs are coupled to LH neutrinos, namely the S L 1 and the U L 1 models, we absorb the SM contribution into C V as well. Given the CP-even property of R D ( * ) observables and Br (B c → τ ν), we can write are real numbers. We can calculate these coefficients once and for all for each model and observable, and then use (C.3) to find each observable from R D ( * ) and Br (B c → τ ν). This allows us to perform a highly efficient scan over the entire viable parameter space and calculate all the CP-even observables.
For the S L 1 model, we further need to implement the bound (2.10) from Br (B → X s νν). This bound only applies to the new physics Wilson coefficient C V LL . With the parametrization we used in this appendix, for S L 1 model (2.10) translates into We can invert (C.1) to find C V from R D ( * ) and Br (B c → τ ν) and apply the bound from (C.4).