New physics and the tau polarization vector in $b\to c \tau \bar\nu_\tau$ decays

For a general $H_b\to H_c\tau\bar\nu_\tau$ decay we analyze the role of the $\tau$ polarization vector ${\cal P}^\mu$ in the context of lepton flavor universality violation studies. We use a general phenomenological approach that includes, in addition to the Standard Model (SM) contribution, vector, axial, scalar, pseudoscalar and tensor new physics (NP) terms which strength is governed by, complex in general, Wilson coefficients. We show that both in the laboratory frame, where the initial hadron is at rest, and in the center of mass of the two final leptons, a $\vec {\cal P}$ component perpendicular to the plane defined by the three-momenta of the final hadron and the $\tau$ lepton is only possible for complex Wilson coefficients, being a clear signal for physics beyond the SM as well as time reversal (or CP-symmetry) violation. We make specific evaluations of the different polarization vector components for the $\Lambda_b\to\Lambda_c$, $\bar B_c\to\eta_c,J/\psi$ and $\bar B\to D^{(*)}$ semileptonic decays, and describe NP effects in the complete two-dimensional space associated with the independent kinematic variables on which the polarization vector depends. We find that the detailed study of ${\cal P}^\mu$ has great potential to discriminate between different NP scenarios for $0^-\to 0^-$ decays, but also for $\Lambda_b \to \Lambda_c$ transitions. For this latter reaction, we pay special attention to corrections to the SM predictions derived from complex Wilson coefficients contributions.


Introduction
The tension between the Standard Model (SM) predictions and experimental data in semileptonic decays involving the third quark and lepton generations points to the possible existence of new physics (NP) affecting those decays. The strongest evidence for this lepton flavor universality violation (LFUV) is in the ratios ( " e, µ) ΓpB Ñ Dτν τ q ΓpB Ñ D ν q " 0.340˘0.027˘0.013, R D˚" ΓpB Ñ D˚τν τ q ΓpB Ñ D˚ ν q " 0.295˘0.011˘0.008, R J{ψ " ΓpB c Ñ J{ψτν τ q ΓpB c Ñ J{ψµν µ q " 0.71˘0.17˘0.18. (1.1) The R D p˚q values have been obtained by the Heavy Flavor Averaging Group (HFLAV) [1], combining different experimental data by the BaBar [2,3], Belle [4][5][6][7] and LHCb [8,9] collaborations. The corresponding SM results given in Ref. [1], R D " 0.299˘0.003 and R D˚" 0.258˘0.05, are obtained from the SM predictions in Refs. [10][11][12][13][14]. Similar results are obtained in Ref. [15] using the heavy quark effective theory parameterization of the form factors with up to Op1{m 2 c q corrections. The tension with the SM is at the level of 3.1 σ, although it will reduce to just 0.8 σ if only the latest Belle results from Ref. [7] were considered. In this respect, in Refs. [16,17] it is argued that the inclusion of the new Belle data heavily restricts the number of allowed NP solutions, claiming that a precise measurement of theB c Ñ τν τ branching ratio can distinguish among them. The important constraints, on new-physics interpretations of the anomalies observed inB Ñ D p˚q τν τ decays, derived from the lifetime of theB c meson were firstly pointed out in [18], and they have commonly be considered in all subsequent analyses.
The ratio R J{ψ has been recently measured by the LHCb Collaboration [19] and it shows a 1.8 σ discrepancy with SM results, which are in the range R SM J{ψ " 0.25´0.28 [20][21][22][23][24][25][26][27][28][29][30][31][32]. B c decays induced by the c Ñ s, d transition at the quark level are also being investigated as a possible source of information on NP [33], taking advantage of the recent results of Ref. [34]. In this latter work, the possibilities of extracting constraints on NP by using the current data on the leptonic and semileptonic decays of pseudoscalar mesons, not only driven by the b Ñ c transition, have been exhaustively discussed.
NP effects on R D p˚q and R J{ψ are studied in a phenomenological way including scalar, pseudoscalar and tensor b Ñ cτν τ effective operators, as well as NP corrections to the SM vector and axial ones. NP terms are governed by Wilson coefficients which are complex in general and should be fitted to data. As a result of this fitting procedure, different NP scenarios actually lead to an equally good reproduction of the above ratios in Eq. (1.1) (see for instance Refs. [35][36][37] 1 ). Then, other observables are needed to constrain and determine the most plausible NP extension of the SM. Typically, the τ -forward-backward (A F B ) and τ -polarization (A λτ ) asymmetries have also been considered. A greater discriminating power can be reached by analyzing the four-bodyB Ñ D˚pDπ, Dγqτν τ [39][40][41][42] and the full five-bodyB Ñ D˚pDY qτ pXν τ qν τ [43,44] angular distributions.
Another test of this non-universality can be obtained from the analog semileptonic R Λc ratio, which has been predicted within the SM in several works [45][46][47]. In Ref. [47], the result from a solid calculation including leading and sub-leading heavy quark spin symmetry (HQSS) Isgur-Wise (IW) functions, which were simultaneously fitted to LQCD results and LHCb data, was provided. The effects of different NP scenarios have been also examined in Refs. [36,[48][49][50][51][52][53][54][55][56][57]. We note that the case of a polarized decaying Λ b baryon has also been addressed in Ref. [58].
In Refs. [59][60][61], we have analyzed the relevant role that different contributions to the differential decay widths d 2 Γ{pdωd cos θ τ q and d 2 Γ{pdωdE τ q could play to the NP search, both for unpolarized and helicity-polarized final τ -lepton. Here, ω is the product of the two hadron four-velocities, θ τ is the angle made by the tau lepton and final hadron threemomenta in the center of mass of the final two-lepton pair (CM), and E τ is the final tau energy in the laboratory frame (LAB), where the initial hadron is at rest. In Refs. [59,60], we give a general description of our formalism, based on the use of general hadron tensors parameterized in terms of Lorentz scalar functions. It is an alternative to the helicity amplitude scheme, and becomes very useful to describe processes where all hadron polarizations are summed up and/or averaged. In these two works, we presented results for the Λ b Ñ Λ c τν τ decay and showed that the helicity-polarized distributions in the LAB frame provide additional information about the NP contributions, which cannot be accessed only by analyzing the CM differential decay widths, as is commonly proposed in the literature. In Ref. [61] we extended the study toB c Ñ η c τν τ ,B c Ñ J{ψτν τ as well as theB Ñ D p˚q τν τ decays. What we have found is that the discriminating power between different NP scenarios was better for 0´Ñ 0´and 1{2`Ñ 1{2`decays than for 0´Ñ 1ŕ eactions.
In this work, we present our results in a different way by looking at the τ polarization vector P µ . Furthermore, the transverse (referred to the direction of the τ ) components of P µ allows us to evaluate new observables, which do not appear in the study of LAB and CM helicity-polarized decays. The possibility of searching for NP signatures in different τ´polarization related contributions was suggested already twenty five years ago in Ref. [62] forB Ñ D p˚q -decays, in the context of SM extensions with charged Higgs bosons. The idea has been further developed in more recent works [52,[63][64][65][66][67], and in particular, a complete framework to obtain the maximum information with polarized τ leptons and unpolarized D p˚q mesons is discussed in Ref. [67], where the full decay chain down to the detectable particles stemming from the τ is considered. As mentioned above, we use here a technique different to the usual helicity-amplitude method, and we also show results for the Λ b and B c decays, for which such exhaustive analyses are not available yet.
We provide an overview of the spin density matrix formalism for semileptonic decay reactions, including NP operators, and discuss how P µ is defined in that context. As we shall show, for a given configuration of the momenta of the involved particles, the polarization vector components (projections of P µ onto some spatial-like unit four-vectors) depend on two variables pω, cos θ τ q or equivalently pω, E τ q, and they can be used as extra observables in the search for NP. To our knowledge, this is the first time that such a study has been performed in the context of LFU anomalies. For fixed ω, the dependence on cos θ τ (or E τ ) of these observables could be inferred from the general results of Ref. [60], since the polarization components turn out to be ratios of linear or quadratic functions of the product of the initial hadron and finalν τ (or τ ) four-momenta 2 . The denominators of these aforementioned ratios are determined by the unpolarized differential decay widths, which can be straightforward seen in our previous works of Refs. [60,61]. Thus, we will show here results for the coefficients of the polynomials that appear in the numerators of these ratios for Λ b Ñ Λ c τν τ andB Ñ D p˚q τν τ decays.
Here we will present results for all the semileptonic decays mentioned above, keeping in mind that a combined analysis of all them can better restrict the possible extensions of NP. We will pay special attention to the Λ b Ñ Λ c τν τ reaction, as there are good prospects that LHCb can measure it in the near future, given the large number of Λ b baryons which are produced at the LHC. Indeed, the shape of the Λ b Ñ Λ c µν µ differential decay rate was already reported by LHCb in 2017 [70]. Any measurement for the tau mode will be extremely valuable, since the evidences for SM anomalies in b Ñ c semileptonic decays are currently restricted to the meson sector, and the sensitivity of Λ b -decay observables to NP operators would likely be different.
The work is organized as follows. In Sec. 2 we introduce the general theory on the spin-density matrix and the polarization vector P µ for a H b Ñ H c τν τ decay. Analytical expressions for P µ including NP terms are then given in Sec. 3, and a detailed analysis of parity and time-reversal violation in the decay is presented in Sec. 3.1. The results are presented and discussed in Sec. 4. In Appendix A we give useful information on the kinematics in the CM and LAB frames and in Appendix B we give some angular averages of the P µ -components in the CM and LAB frames.

Spin-density matrix and polarization vector in semileptonic decays
We obtain in this section general results valid for any baryon/meson semileptonic decay for unpolarized hadrons, though we refer explicitly to those induced by the b Ñ c transition.

Spin-density operator
Let us consider a H b Ñ H c τν τ semileptonic decay of a bottomed hadron (H b ) of mass M into a charmed one (H c ) with mass M 1 . For a given momentum configuration of all the particles involved, and when the polarizations of all particles except the τ lepton are being summed up (averaged or sum over polarizations of the initial or final particles, respectively), 4 the modulus squared of the invariant amplitude for the production of a final τ´lepton in a upk 1 q state 5 can always be written as with k 1 the four-momentum of the final τ´lepton and r, r 1 hadron polarization indexes. The differential decay rate is given by [71] where G F " 1.166ˆ10´5 GeV´2 is the Fermi coupling constant and s 23 (s 13 ) is the invariant mass squared of the outgoing τν τ (H c τ ) pair. The operator O, which depends on the momenta of all particles, is determined by the physics that governs the H b Ñ H c τν τ transition and satisfies defines a trace-one hermitian operator (ρ : "ρ) in the two-dimensional Hilbert space spanned by the spin states of the τ particle 6 . A general polarization basis (covariant spin) for the τ states with four-momentum k 1 , can be constructed as follows. For the τ at rest, we take the two states u n 1 pm τ , 0 q corresponding to spin˘1{2 along the direction defined by a normalized three vector n, then apply to these states a boost of velocity k 1 {k 10 . The resulting u N 1 pk 1 q spinors are eigenstates, with corresponding eigenvalues˘1, of the γ 5 { N operator, where N µ is the transformed of the four-vector p0, nq by the boost [72]. The projectors onto the u N 1 pk 1 q states are given by P N 1 " 1 2 p1˘γ 5 { N q. Notice that N 2 "´ n 2 "´1 and that N¨k 1 " 0. Helicity is a particular case of covariant spin where n "k 1 " k 1 {| k 1 | and N µ " r s µ " p| k 1 |, k 10k1 q{m τ . For the given configuration of momenta, the spin-density operatorρ encodes all information that can be obtained on the spin of the τ leptons produced in the H b Ñ H c τν τ decay when no other particle spin state is measured. Actually, the matrix elements ofρ readρ and give the probability that in an actual measurement the τ is found in the u S 1 pk 1 q state, as follows from Eq. (2.1).

Polarization vector: definition and properties
Sinceρ is hermitian, it can be diagonalized, and there exists a polarization basis u N 1 1 pk 1 q for which the corresponding matrix elements satisfȳ where the eigenvalues,ρ 1 h , are positive real numbers, as they are just the probabilities of finding the τ in the u N 1 1 pk 1 q states. In this basis of eigenstates, the spin-density matrix can be written as where we have defined the polarization vector P µ as P µ "`ρ 1´1´ρ1`1˘N 1µ .
(2.8) 6 The formalism for antiparticles runs in parallel to the one that will be discussed below, with the obvious replacements of p { k 1`m τ q by pmτ´{ k 1 q and of Dirac u´spinors by v´spinors. Besides, O will also change.
The four-vector P µ depends on the dynamics that governs the H b Ñ H c τν τ decay, through the operator O, and it trivially satisfies Note that, for a given momentum configuration of all the particles involved, P µ depends only on three independent quantities 7 . In the present context, it seems natural to take those quantities as one of the two eigenvalues ofρ and the two angles that fix the privileged direction n 1 in the τ rest frame, which gives rise to the polarization eigenbasis u N 1 1 pk 1 q. All three are determined by the dynamics of the transition, which enters through the operator O introduced in Eq (2.1).
The information on the spin of the produced τ is solely contained in the polarization vector P µ . Thus, the probability of measuring a τ in a state u S h pk 1 q, with h "˘1, is given by where we have used thatū S h pk 1 qγ 5 { Pu S h pk 1 q "´2m τ h P¨S. 8 The same result also leads to Moreover, sinceρ 1 1 ,ρ 1 1 ě 0 and Trrρs " pρ 1 1`ρ 1 1 q " 1, we have that P 2 is then limited to the interval´1 The case P 2 " 0 implies P µ " 0 and it corresponds to the physical situation in which the emitted τ is unpolarized, i.e., the probability of measuring any polarization state is the same and equal to 1 2 . The case P 2 "´1 corresponds to a fully polarized τ , and eitherρ 1 1 " 0 orρ 1 1 " 0, and the τ is produced in the u N 1 1 pk 1 q or the u N 1 1 pk 1 q eigenstates, respectively. The case with´1 ă P 2 ă 0 corresponds to a partial polarization scenario, in which the τ 7 This follows trivially considering thatρ is a hermitian operator with trace one in a two-dimensional Hilbert space. 8 It is obtained by replacingū S h pk 1 q and u S h pk 1 q by hū S h pk 1 qγ5 { S and h γ5 { Su S h pk 1 q respectively.
is produced in an admixture of the u N 1 1 pk 1 q and u N 1 1 pk 1 q states, with probabilities given bȳ ρ 1 1 andρ 1 1 respectively. This latter interpretation is substantiated by the following result P rupk 1 qs " 1 2m τū pk 1 qρ upk 1 q "ρ 1 1ˇū pk 1 qu N 1 1 pk 1 q 2m τˇ2`ρ 1 1ˇū pk 1 qu N 1 1 pk 1 q 2m τˇ2 , (2.13) that gives the probability of finding the τ in a upk 1 q state as a sum over the probabilities that the τ is produced in the u N 1 1 pk 1 q states times the probabilities that, upon measurement, the latter are found in the upk 1 q state.
3 Tau polarization vector for H b Ñ H c τ´ν τ decays in the presence of NP We shall consider the general effective Hamiltonian that is discussed in detail for instance in Ref. [36]. The fermionic operators involve only neutrino left-handed currents, while the, complex in general, Wilson coefficients C i parameterize possible deviations from the SM, the latter given by the 4G F V cb O V L { ? 2 term. The Wilson coefficients could be lepton and flavor dependent, though normally they are assumed to be present only for the third quark and lepton generations, where anomalies have been seen.
In terms of the above effective Hamiltonian the invariant amplitude for the H b Ñ H c τν τ process is written as [60] The lepton currents are given by with k the final antineutrino four-momentum and where h "˘1 stands for the two possible τ lepton polarizations (covariant spin) along a certain four vector S µ that we choose to measure in the experiment. The dimensionless hadron currents read (here cpxq and bpxq are Dirac fields in coordinate space), with C V,A " p1`C V L˘C V R q, C S,P " pC S L˘C S R q and hadron states normalized as x p 1 , r 1 | p, ry " p2πq 3 pE{M qδ 3 p p´ p 1 qδ rr 1 , with r, r 1 polarization indexes. In addition, p and p 1 are the four-momenta of the initial and final hadrons, respectively. Summing/averaging over the final/initial hadron polarizations one can identify the O operator in Eq.
While this can be used to obtain the τ polarization vector P µ through Eq. (2.4) and the last of Eq. (2.9), in fact this work was already done in Ref. [60], where it was found that for a final τ with well defined helicity h one has 9 where q " p´p 1 " k`k 1 is the four-momentum transferred and ω is the product of the initial and final hadron four-velocities (related to the invariant mass squared of the outgoing τν τ pair via q 2 " M 2`M 12´2 M M 1 ωq, which varies from 1 to ω max " pM 2M 12´m2 τ q{p2M M 1 q. We note that the term in N H 3 pω, p¨kq was not explicitly shown in Ref. [60] since for the CM and LAB frames considered in that work one has sk 1 qp " 0 for r s µ " p| k 1 |, k 10k1 q{m τ . The N and N H 123 scalar functions are given by The ten functions, A, B, C, A H , B H , C H , D H , E H , F H and G H , above are linear combinations of the 16 Lorentz scalar structure functions (SFs) introduced in Ref. [60], and denoted as Ă W 1 s in that work. These Ă W 1 s SFs describe the hadron input to the decay, and they are constructed out of the NP complex Wilson coefficients (C 1 s) and the genuine hadronic responses (W 1 s). The latter are expressed in terms of the form-factors used to parameterize the matrix elements of the hadron operators. Symbolically, we have Ă W " CW . The functions A, B, C and A H , B H , C H , D H , E H in Eq. (3.7) are given in Appendix D of Ref. [60]. As for F H and G H they read As can be seen from the general results of Ref. [60], the Ă W SFs present in N H 3 are generated from the interference of vector-axial with scalar-pseudoscalar terms ( Ă W I1 ), scalarpseudoscalar with tensor terms ( Ă W I3 ), and vector-axial with tensor terms ( Ă W I4,I5,I6 ). Since the vector-axial terms are already present in the SM, at least one of the C S , C P , C T Wilson coefficients must be nonzero for N H 3 to be nonzero. Besides, N H 3 is proportional to the imaginary part of SFs, which requires complex Wilson coefficients, thus incorporating violation of the CP symmetry in the NP effective Hamiltonian. This feature makes the study of such contribution to the polarization vector of special relevance and it has been discussed before in the context ofB Ñ D p˚q decays [62,65]. Moreover forB Ñ D˚, some CP-odd observables, defined using angular distributions involving the kinematics of the products of the D˚decay, have been also presented [39,40,43,44]. These are known as the CP violating triple product asymmetries, which should be sensitive to the relative phases of the Wilson coefficients, as the F H and G H scalar functions are.
We note that the knowledge of the ten functions A, B, C, A H , B H , C H , D H , E H , F H and G H fully determines ř rr 1 |M| 2 , obtained after summing/averaging all spin third components of all particles except the τ lepton. These functions contain then the maximum information on NP that can be inferred by analyzing the H b Ñ H c τν τ decay. As discussed in Ref. [60], for a fixed value of ω, Apωq, Bpωq and Cpωq can be indistinctly obtained by looking at the dependence on cos θ or on E of the CM d 2 Γ{pdωd cos θ q or the LAB d 2 ΓpdωdE q unpolarized differential decay widths, respectively. To obtain all the rest of CP-conserving A H , B H , C H , D H and E H functions, it is however necessary to simultaneously use the cos θ and E dependencies of the τ -helicity polarized CM and LAB distributions, which provide complementary information. Since those two distributions do not depend on F H pωq and G H pωq, further measurements are needed to obtain these two latter CP odd quantities.

Parity and time-reversal violations in the decay width
Note that, in the most general case reflected in Eq.(3.9), P µ contains both vectors and pseudovectors and then it does not have well defined properties under parity and time reversal transformations 10 . This will give rise to parity and time-reversal violating contributions to the probability P ru S h pk 1 qs9p1`h P¨Sq or equivalently in the decay width. To see that we also need to know how hS µ transforms under parity ([hSs P µ ) and time reversal ([hSs T µ ). By using γ 5 ph { Squ S h pk 1 q " u S h pk 1 q, we find [73] ru S h pk 1 qs P " γ 0 u S h pk 1 q " γ 0 γ 5 γ µ phS µ qu S h pk 1 q " γ 5 γ µ p´hS µ qγ 0 u S h pk 1 q " γ 5 rh { Ss P ru S h pk 1 qs P , (3.10) ru S h pk 1 qs T " τ ru S h pk 1 qs˚" τ γ5 γμphS µ qru S h pk 1 qs˚" γ 5 γ µ phS µ qτ ru S h pk 1 qs" where we have ignored possible overall phases, that do not affect the transformation properties of hS µ , and we have used that γ 0 γ µ γ 0 " γ µ and τ γ5 γμτ´1 " γ 5 γ µ . Finally, we deduce We conclude that the quantity p1`h P¨Sq, and hence the polarized differential decay width, is not invariant under parity due to the presence of the p µ K and q µ K terms in P µ .
Similarly, p1`h P¨Sq is not invariant under time reversal due to the presence of the µk 1 qp contribution in P µ . This latter result is expected since, as noted above, the very existence of the µk 1 qp term in P µ relies on some of the Wilson coefficients not being real 11 .

Different components of the polarization vector
In this section we are interested in giving a decomposition of the polarization vector in the CM and LAB reference systems in which either the final pair of two leptons (CM) or the initial hadron (LAB) are at rest. For both frames, we choose as an orthogonal basis of the four-vector Minkowski space where the vectors used in their construction are understood to be measured in the corresponding frame. Note that N µ L , N µ T and N µ T T define polarization states corresponding to n L " k 1 {| k 1 |, n T " rp k 1ˆ p 1 qˆ k 1 s{|p k 1ˆ p 1 qˆ k 1 | and n T T " p k 1ˆ p 1 q{| k 1ˆ p 1 |, respectively. Since P¨k 1 " 0, we will have that in a given reference system Note that the quantity which gives the degree of polarization of the τ , is a true scalar under Lorentz transformations as can be inferred from Eq. (3.9). However, the P L and P T components are different in the two frames. This derives from the fact that N CM µ , with Λ the boost which takes four-momenta from the LAB system to the CM one. This is so because the corresponding auxiliary three vectors n LAB,CM L,T depend on the reference frame. On the other hand, P T T is the same in the two systems since it is a component perpendicular to the velocity p 1 LAB {pM´M 1 ωq defining the LAB-to-CM boost. Indeed, in this case , a " L, T, (3.16) which trivially follows from Strictly speaking, what one needs is that not all of them are relatively real. 12 I.e., p k 1 As a consequence, for a given tau kinematics determined by a pair pω, E τ q or pω, cos θ τ q, one can express the P LAB L,T pω, E τ q as linear combinations of P CM L pω, cos θ τ q and P CM T pω, cos θ τ q 13 , and thus the LAB and CM P L,T,T T components carry the same information. Note however that this equivalence is lost for the averages xP LAB, CM a ypωq that we discuss below. In any of the CM or LAB frames, P CM ,LAB L is given by where the appropriate CM or LAB four-vectors should be used in each case. As previously mentioned, N µ L corresponds to well defined helicity and, thus, P L is related to the helicity asymmetry via (see Eq. (2.11)) where here h stands for the τ helicity measured in the CM or the LAB frames. From Eq. (2.2), it is then clear that P CM ,LAB L can be obtained from the experimental asymmetries where, as already mentioned, cos θ τ is the cosine of the angle made by the CM threemomenta of the final hadron and τ lepton, and E τ is the energy of the τ lepton in the LAB frame. While the CM angle θ τ is not restricted, the LAB energy E τ is limited, for a given ω value, to the interval defined by Similarly, for the CM or LAB systems, one further has Note that both P T and P T T can also be obtained from asymmetries of the decay distributions, as in Eq. (3.21), for polarizations along N µ T and N µ T T respectively. From the discussion above, a nonzero P T T component in the LAB or CM frames is a signal for time-reversal violation that originates from the presence of non-real Wilson coefficients in the NP effective Hamiltonian. Since N µ T T does not have a zero component, 13 Note that the pΛN LAB b q¨N CM a products are fully determined by the pair of variables pω, Eτ q or equivalently by pω, cos θτ q with Eτ and cos θτ related via with Mω " M´M 1 ω. P T T comes from a non vanishing projection of the τ´polarization three-vector in the orthogonal direction to the plane defined by the outgoing hadron and τ three-momenta. Further details on the vector products appearing in the evaluation of P L , P T and P T T are given in Appendix A.
In Ref. [65], the name polarization vector components is used for what actually are averages. Here, we will denote those averages as xP a ypωq, a " L, T, T T and, within our scheme, they are given by the expressions 14 with the normalizations related by N E " pEτ´Eτ qN θ {2, and N θ explicitly given in Eq. (B.2). These averages correspond to the, easier to measure, experimental asymmetries where h CM ,LAB a "˘1 stand for positive/negative polarization along N µ a in the CM or LAB system, as appropriate. In particular, xP CM ,LAB L y is nothing but the τ polarization asymmetry A CM ,LAB λτ also used in the literature and evaluated for instance in Refs. [61,74]. In Appendix B we give expressions for the xP CM a ypωq and xP LAB a ypωq averages in terms of the scalar functions in Eq. (3.7). The equivalence between LAB and CM values present for the two-dimensional P L,T , and represented by Eq. (3.16), is now lost for the averages xP L,T y, as can be easily be inferred from the expressions in Appendix B. The reason is that the coefficients of the linear combinations that relate CM and LAB P L,T components depend on the variable (cos θ τ or E τ ) which is integrated to obtain the averages.
Therefore, if only the averages xP L,T ypωq are measured, CM and LAB values give complementary information, as we already mentioned above for the case of tau helicitypolarized differential decay distributions.
One can also define the average xP 2 ypωq. In this case, it is the same in the CM and LAB frames as a consequence of both P 2 pω, k¨pq and N pω, k¨pq being scalars. Actually, xP 2 ypωq is a Lorentz invariant and in any reference system, for a given ω, is given by where pk¨pq˘" M pM ω´Eτ pωqq and N pωq is given by We conclude the section with the trivial remark xP 2 ypωq "´xP 2 L`P 2 T`P 2 T T ypωq ‰´ÿ a"L,T,T T rxP a ypωqs 2 "´| P pωq| 2 , (3.29) with | P | defined for instance in Ref. [65] for the CM frame, and which is not even a Lorentz scalar.

Numerical Results
In this section we present the results for P L , P T and P T T , evaluated for the Λ b Ñ Λ c and B Ñ D p˚q semileptonic decays. The averages introduced in Eq. (3.25) will be presented for the above reactions as well as for theB c Ñ η c , J{ψ decays. We studied those decays in Refs. [59,60] (Λ b Ñ Λ c ) and [61] (B c Ñ η c , J{ψ andB Ñ D p˚q ), where we analyzed different observables related to the unpolarized and helicity-polarized CM d 2 Γ{dωd cos θ τ and LAB d 2 Γ{dωdE τ distributions, and their possible role in distinguishing between different NP scenarios. In this work, we shall show results for observables mentioned above, evaluated both in the CM and LAB frames, and within the SM and with the NP Wilson coefficients corresponding to Fits 6 and 7 of Ref. [36].
For the particular case of the Λ b Ñ Λ c decay, and in order to illustrate the effect of complex Wilson coefficients, we will also show results for one more NP scenario from Ref. [37]. It corresponds to a R 2 leptoquark mediator model that only gives contributions to the C S L and C T Wilson coefficients and that was first analyzed for complex values of those coefficients in Ref. [75].
For the Λ b Ñ Λ c decay we use form factors that are directly obtained (see Appendix E of Ref. [60]) from those calculated in the lattice quantum Chromodynamics (LQCD) simulations of Refs. [76] (vector and axial ones) and [51] (tensor NP form factors) using 2`1 flavors of dynamical domain-wall fermions. The NP scalar and pseudoscalar form factors are directly related to the vector and axial ones and we use Eqs. (2.12) and (2.13) of Ref. [51] to evaluate them. We use errors and the statistical correlation-matrices, provided in the LQCD papers, to Monte Carlo transport the form-factor uncertainties to the different observables shown in this work.
For the case ofB Ñ D p˚q decays, the form factors are calculated using a parameterization, based on heavy quark effective theory, that includes corrections of order α s , Λ QCD {m b,c and partly pΛ QCD {m c q 2 [14]. In this case there exist also some experimental q 2´s hape information [3,4], which is used to further constrain some matrix elements. Inputs from LQCD [77][78][79][80], light-cone [81] and QCD sum rules [82][83][84] are also available. Here, we use the set of form factors and Wilson coefficients found in [36], since in that work, not only the Wilson coefficients, but also the 1{m b,c and 1{m 2 c corrections to the form factors were simultaneously fitted to experimental data. In this way for these decays, we can also consistently estimate theoretical uncertainties, since we shall use statistical samples of Wilson coefficients and form factors, selected such that the χ 2´m erit function computed in [36] changes at most by one unit from its value at the fit minimum.
For theB c Ñ η c , J{ψ transitions, there exist no systematic LQCD calculations, except for the very recent work of the HPQCD collaboration [85] where the SM vector and axial form factors of theB c Ñ J{ψ decay have been determined. Here we use the form factors obtained within the non-relativistic quark model scheme of Ref. [22]. It has the advantage of consistency, since all the form factors needed can be evaluated within the model. These form factors are consistent with heavy quark spin symmetry and its expected pattern of breaking corrections. In addition, in Ref. [22], five different inter-quark potentials were considered allowing us to provide an estimate of the theoretical uncertainties. We expect the systematic errors present in the NRQM evaluation of the form factors should largely cancel out in ratios 15 .
We should mention that in our previous works of Refs. [60,61], we discussed in great detail, for all these decays, the helicity differential distributions obtained in the SM and NP Fits 6 and 7 of Ref. [36]. Thus, the analysis presented below for the longitudinal P L projection shows, using a different language, the same physical content, with the exception of the results related to the NP tensor R 2 leptoquark model fit of Ref. [37], which were not considered in [60,61].
However, the study of the transverse component P T T carried out here is novel, and it directly provides independent physics information (F H and G H SFs in Eqs. (3.8) and (3.9)) to that inferred from our previous works. In what respects to P T , this projection is determined by the scalar functions A, B, C, A H , B H , C H , D H and E H (see Eq. (3.9)), as it occurs with P L . As described in detail in [60], all these eight functions can be extracted from the combined study of the CM d 2 Γ{pdωd cos θ τ q and LAB d 2 Γ{pdωdE τ q helicity-polarized distributions. Therefore, though P T pω, k¨pq can be indirectly obtained from the results shown in Refs. [60,61], this polarization projection was not explicitly discussed in any of these works.

CM and LAB two-dimensional distributions
Two-dimensional (2D) distributions of the P µ projections provide observables that can also be used to distinguish between different types of NP. In this subsection, we discuss results obtained within the SM and the NP scenarios corresponding to Fits 6 and 7 of Ref. [36]. Since in this latter work, all Wilson coefficients are real, the P T T component comes out identically zero.
First in Figs. 1, 2 and 3, we show CM 2D distributions for P L , P T and P 2 and the Λ b Ñ Λ c ,B Ñ D andB Ñ D˚decays, respectively, and obtained with the central values for the Wilson coefficients and form factors. In all cases, predictions from Fit 6 are closer to the SM results, and we clearly observe, except for theB Ñ D˚decay, different 2D patterns for Fits 6 and 7, which would certainly allow to distinguish between both NP scenarios.
The transverse component P T is always negative for Λ b Ñ Λ c andB Ñ D˚decays, with a cos θ τ´d ependence that becomes flatter as ω decreases from ω max to the vicinity of zero recoil (ω " 1), where P T reaches, in modulus, its minimum value. Large negative values of P T , which can reach´0.9, are found for ω ą 1.2 and intermediate values of cos θ τ far from the˘1 limits. For these two decays, the longitudinal polarization shows a large variation, going from P L " 0.9 for angles close to π to values in the p´0.9,´0.8q range in the forward direction, where the dependence on ω is significantly more pronounced than at backward angles. Moreover, we see regions close to zero recoil, and in the forward direction, where the τ´lepton is produced largely unpolarized (P 2 "´0.2), with |P 2 | growing as both θ τ and ω increases, reaching values in the interval p´0.95,´0.9q for ω in the vicinity of ω max (see the 2D´P 2 distributions in the bottom panels). The exception is found for NP Fit 7 in the baryon decay, for which the τ is produced almost polarized, P 2 ă´0.95 at forward angles and close to ω max (right-top corner), with a large P T polarization component, around 0.9. However, in this case for backward angles, P 2 does not become so close to´1 as ω approaches ω max .
The discussion for theB Ñ D transition should take into account that for this decay P 2 "´1, implying that the τ emitted is always fully polarized. In Ref. [61], it was already pointed out that for 0´Ñ 0´transitions at zero recoil and θ τ " 0 or π, angular momentum conservation forces the τ helicity to equal that of the antineutrino which is positive, thus P L "´1 (see Eq. (3.20) or (3.21)), which implies P 2 "´P 2 L "´1 and P T " 0. Indeed, we see in the bottom panels of Fig. 2 that P 2 "´1 in the whole pω, cos θ τ q phase-space, and not only for θ τ " 0 or π at zero recoil. Therefore, longitudinal and transverse polarizations are not independent for non-CP violating physical scenarios, and in the full phase-space both components satisfy the relation P 2 L`P 2 T " 1. As in the other decays, Fit 7 predictions differ from SM ones significantly more than those obtained in the NP Fit 6, with P L exhibiting a pronounced dependence on cos θ τ , when ω departs from the zero recoil point. While P L takes negative and positive values within the SM and both Fits 6 and 7 of Ref. [36], we observe that P T is negative for SM and Fit 6, while for the NP Fit 7, this transverse component also takes positive and negative values, and even it vanishes along a pω, cos θ τ q´curve, for which P L "`1. As we will see below, for the kinematics encoded in this curve, the τ´lepton is produced in a negative-helicity state. The reason why the τ is always fully polarized for a general 0´Ñ 0´transition is the following. Since, in the massless limit, theν τ is fully polarized, we have that the invariant amplitude M, apart from momenta, only depends on the τ spin degrees of freedom. If we have Mphq, where here h "˘1 represents the τ helicity, one can always define two coefficients a˘1 "˘M ph "¯1q ř h 1 "˘1 |Mph 1 q| 2 What this result tells us is that the probability to produce a τ in the state a˚1us 1 pk 1 qà˚1 uś 1 pk 1 q is identically zero. Thus, the probability to produce a τ in the orthogonal state, a´1us 1 pk 1 q´a`1uś 1 pk 1 q, should be one. The τ is then fully polarized. Apart from irrelevant phases these two polarization states correspond to a˚1us 1 pk 1 q`a˚1uś 1 pk 1 q " u P 1 pk 1 q, a´1us 1 pk 1 q´a`1uś 1 pk 1 q " u Ṕ 1 pk 1 q, (4.3) i.e., they are the two spin-covariant eigenstates associated to the four-vector 16 N µ " P µ . For a given k 1 , these states depend on the pair of variables pω, cos θ τ q or pω, E τ q that determine all P µ components (or equivalently M) in the CM or LAB frames respectively. The above argumentation fails as soon as M depends on the spin variable of the hadrons involved in the decay. This is so since, in general, it is not possible to find a˘1 such that a h M λ phq " 0, (4.4) 16 From Eq. (2.10), the probability of measuring the τ in a state u P h pk 1 q, eigenstate of the operator γ5 { P with eigenvalue h, is given by p1´hq{2 since P 2 "´1 forB Ñ D decays. Therefore, we assign the state u Ṕ 1 pk 1 q to the produced polarized tau. This result is consistent with P µ rω " 1, cos θτ "˘1qs "´s µ rω " 1, cos θτ "˘1s, since for these two CM kinematics PL "´1. . LAB P T (first row) and P L (second row) polarization observables for the Λ b Ñ Λ c τν τ decay evaluated within the SM (left column) and with the NP Wilson coefficients from Fits 6 (middle column) and 7 (right column) of Ref. [36]. We display the 2D distributions as a function of the pω, cos θ τ q variables, and use Eq. (3.18) to compute the cos θ τ for fixed ω and a given E τ LAB energy. In all cases, central values for the form factors and Wilson coefficients have been used.  for all λ " pr, r 1 q values, where different λ values represent different hadronic spin configurations. Note however that for a fixed λ (corresponding to fixed r{r 1 polarization of the initial/final hadron) Eq. (4.4) has always a solution. Thus, for fixed λ, the τ is also fully polarized but with a polarization state that depends on λ. This is in agreement with the results obtained in Ref. [62] forB Ñ D p˚q decays.
As noted above, for the rest of the transitions, P 2 approaches´1 at maximum recoil (ω max ), with the exception of Fit 7 for the Λ b Ñ Λ c decay in the cos θ τ ă 0 region. This is better understood by looking at the polarization projections in the laboratory frame.
In Figs. 4-6, we present the LAB P L and P T 2D distributions for the same NP scenarios and decays discussed previously in Figs. 1-3. In the LAB plots, we have made use of the relation in Eq. (3.18) to represent the polarization observables as a function of pω, cos θ τ q instead of pω, E τ q. On the other hand, since P 2 is a scalar [and thus P 2 LAB pω, E τ pω, cos θ τ qq " P 2 CM pω, cos θ τ q ] we will no show it again. Though the LAB P L and P T 2D distributions shown in Figs. 4-6 can be obtained from the CM ones depicted above in Figs. 1-3, we stress that the coefficients of the linear combinations (Eq. (3.16)) depend on ω and cos θ τ . Moreover, the longitudinal or transverse character is not preserved, which also makes interesting a short discussion of the main features of the LAB polarization components. In the LAB frame, the τ 's are mainly being emitted with negative helicity (P LAB L « 1) in the high ω-region close to ω max , as can be seen in the second row of plots in Figs. 4-6. The explanation for this behavior, at least in part, is that close to maximum recoil, the τ momentum in the LAB is large and hence positive helicity is suppressed by the dominant contribution that selects negative chirality for the final charged lepton 17 . As mentioned in Ref. [61], only the O S L ,S R and O T NP terms select positive chirality. Looking at the values for the corresponding Wilson coefficients (see 17 At very large momentum helicity almost equals chirality Table 6 of Ref. [36]) one expects larger deviations from the above behavior for Fit 7. 1 1.  .7) and (3.9)) for the Λ b Ñ Λ c (bluish),B Ñ D (reddish) andB Ñ D˚(greenish) decays evaluated for the SM (solid), Fit 6 (dotted) and Fit 7 (dashed) of Ref. [36]. Error bands take into account the uncertainties associated to the Wilson coefficients and form factors, and they are calculated as explained in Refs. [60,61].
To finish this subsection, we recall here that for fixed ω, the polarization components turn out to be ratios of linear or quadratic functions of pp¨kq, as inferred from Eqs. (3.19) and (3.23). Restricting the discussion to CM observables, the denominator of these ra-tios, N pω, p¨kq, is proportional to p N pω, cos θ τ q " a 0 pωq`a 1 pωq cos θ τ`a2 pωq cos 2 θ τ , with the coefficients a i pωq appearing in the angular decomposition of the tau-unpolarized d 2 Γ{pdωd cos θ τ q differential decay width. We have already presented results for them in our previous works [60,61], and we will not make any further comment here. On the other hand, taking into account the dependence of pp¨kq, pp¨N CM L,T q and pq¨N CM L,T q on cos θ τ , we find P CM L pω, cos θ τ q " p 0 pωq`p 1 pωq cos θ τ`p2 pωq cos 2 θ τ a 0 pωq`a 1 pωq cos θ τ`a2 pωq cos 2 θ τ , P CM T pω, cos θ τ q " sin θ τ p 1 0 pωq`p 1 1 pωq cos θ τ a 0 pωq`a 1 pωq cos θ τ`a2 pωq cos 2 θ τ (4.5) with the five coefficients, p 0 , p 1 , p 2 , p 1 0 and p 1 1 , of the numerator polynomials being linear combination of the five A H pωq, B H pωq, C H pωq, D H and E H pωq functions, introduced in Eqs. (3.7) and (3.9) to generally describe the decay with polarized taus in the final state. We observe that P CM L and P CM T are not just polynomials in cos θ τ and that the simultaneous knowledge/measure of both of them, in conjunction with the unpolarized d 2 Γ{pdωd cos θ τ q distribution, provide the maximum information which can be obtained from the decay with polarized taus 18 . In addition, the longitudinal component, or equivalently the CM tauhelicity d 2 Γ{pdωd cos θ τ q double differential decay width, provides only three independent conditions (p 0 , p 1 and p 2 ) and it is not enough to determine all undetermined A H ,¨¨¨E H functions. This was already pointed out in Ref. [60], where it is also shown that all these functions can be obtained using also input from the LAB tau-helicity d 2 Γ{pdωdE τ q distribution (or equivalently P LAB L ), as expected from the discussion in Eq. (3.16) since this brings in some information of P CM T . This is another way to point out that the CM and LAB tau-helicity differential distributions provide complementary results.
Thus, we show results for A H pωq, B H pωq, C H pωq, D H and E H pωq, since this is another, more simple, way of presenting the physical information contained in the above 2D polarization observables. This is done in Fig.7 for the Λ b Ñ Λ c τν τ andB Ñ D p˚q τν τ decays. We note that these functions could also be reconstructed from the exhaustive results included in Refs. [60,61] on the tau CM angular and LAB energy dependencies of the helicity d 2 Γ{pdωd cos θ τ q and d 2 Γ{pdωdE τ q distributions, but that they have never been directly shown. In most cases we see the capability of these observables to distinguish the SM and Ref. [36] Fits 6 and 7 predictions, with the latter deviating more from the SM results. One can have direct access to these functions by measuring the τ polarization in the decay or, indirectly, through the measuring of the polarization vectors components P L,T . The latter can be obtained for instance from the analysis of the subsequent τ decay. Both methods require however to be able to measure the τ momentum (in the first case also its polarization), something that it is extremely difficult, since the decay products of the tau include an undetected neutrino.
In this sense, we should comment that the framework presented in Refs. [66,67] for B´decays, where so-called visible distributions of detectable particles from the τ -decay are analyzed, aims to determine the A H ,¨¨¨E H functions without having to measure the τ momentum. Indeed, it is integrated out in these works, and the proposed (visible) kinematical variables are referred to the initialB and outgoing D p˚q three-momenta. Further and complementary constrains, within this scheme of visible kinematics, can also be obtained 18 Non-conserving CP contributions, FH and GHpωq, related to PT T are not considered in this discussion.
from different angular asymmetries that can be constructed using the products of the final hadron decay (D˚Ñ Dπ) [44]. T y, xP CM L y and xP 2 y averages, defined in the CM system and calculated for the SM and the NP Fits 6 and 7 of Ref. [36], as a function of ω. In addition, for theB Ñ D decay, the gray curves stand for the SM (solid), Fit 6 (dotted) and Fit 7 (dashed) results obtained for´| P CM | 2 (Eq. (3.29)). Error bands take into account the uncertainties associated to the Wilson coefficients and form factors, and they are calculated as explained in Refs. [60,61]. ω Figure 9. The same as in Fig. 8, but for projections defined in the LAB system. Some of the features discussed above in the presentation of the P L,T results are easier to observe in the one dimensional plots displayed in Figs. 8 and 9. There, we now show the CM and LAB xP L ypωq, xP T ypωq and xP 2 ypωq averages, the latter given by xP 2 ypωq "´xP 2 L`P 2 T`P 2 T T ypωq, as introduced in Eqs. (3.25) and (3.27). As discussed in Subsec. 3.2, these averages are related to the CM/LAB tau polarization asymmetries obtained from dΓ{dω, whose measurement require the detection of the momentum and spin-state of the τ . Equivalently, these averages can be obtained from the analysis of the full angular distribution of the pion or rho mesons, originated in the subsequent hadron decay of the tau, measured in the τ -rest frame [65]. Following the discussion at the end of the previous subsection, these observables seem more difficult to access experimentally than those proposed in Refs. [66,67], which do not require the detection of the tau lepton and that we will study elsewhere.

One-dimensional polarization averages
In Figs. 8 and 9, only NP Fits 6 and 7 of Ref. [36] are still considered, where all Wilson coefficients are real and therefore the P T T component vanishes. Additionally, we also show results for theB c Ñ η c τν τ andB c Ñ J{ψτν τ decays, not presented for the 2D distributions and the A H ,¨¨¨E H functions discussed in the previous subsection. We include, in all cases, 68% confident-level (CL) error bands that take into account the uncertainties associated to the Wilson coefficients and form factors, as explained in Refs. [60,61].
The ω´shape patterns for theB Ñ D andB c Ñ η c or theB Ñ D˚andB c Ñ J{ψ reactions are qualitatively similar, while those obtained from the Λ b Ñ Λ c decay show some resemblances with the 0´Ñ 1´ones. A good number of the distributions depicted in Figs. 8 and 9 can be used to disentangle between SM and the two NP cases considered there. In particular, Fit 7 leads to results clearly distinctive, even taking into account theoretical uncertainties bands, while SM and Fit 6 predictions are more difficult to separate. Nevertheless, from the results of Figs. 8 and 9 one can safely conclude that, with the exception of theB c Ñ J{ψ, and to a lesser extentB Ñ D˚, the observables shown could theoretically tell apart Fit 6 from Fit 7.
We note that the averages of the LAB longitudinal and transverse projections xP LAB L,T ypωq can not be obtained as linear combinations of the xP CM L,T ypωq with known kinematical coefficients. They provide thus complementary information. This is easily seen in the expressions collected in Appendix B. In addition to A, B and C, which could be extracted from either the unpolarized CM d 2 Γ{pdωd cos θ τ q or the LAB d 2 Γ{pdωdE τ q differential decay widths [60], we observe that xP CM T ypωq depends on A H and C H while, in xP CM L ypωq, the scalar ωfunctions B H , D H and E H also appear. In turn, xP LAB Another consequence of this discussion is that for a given decay, all former five ω-functions cannot be determined only from the four averages xP CM L,T ypωq and xP LAB L,T ypωq, and it would be necessary to have additional information, as for example the two-dimensional dependencies of the different polarization components discussed in the previous subsection. Alternatively, as noted above, all these scalars (A H ,¨¨¨E H ) can also be obtained from the combined study of the CM d 2 Γ{pdωd cos θ τ q and LAB d 2 Γ{pdωdE τ q helicity-polarized distributions [60].
However, it is clear that the combined use of all averages, for the five decays, shown in Figs. 8 and 9 will greatly restrict the characteristics of possible extensions of the SM, and certainly in a more efficient way than if only one particular decay is considered.
Finally, for theB Ñ D decay, we also show (gray curves and bands) the frame dependent quantity´| P | 2 (see Eq. (3.29)), introduced in Ref. [65]. Clearly,´| P | 2 fails to convey the information on the degree of polarization of the τ . For theB Ñ D decay, and except at zero recoil, for which´| P | 2 "´xP L y 2 " xP 2 y "´1, its value is never exactly minus one. We also test that while xP 2 y is a scalar and leads to the same LAB and CM ω-distributions, | P | 2 depends on the reference system where it has been defined. We observe that in the high ω-region,´| P | 2 in LAB is closer to´1 than when it is calculated in CM, being in the first frame almost indistinguishable from´1 for the SM and Fit 6 cases. This follows from the discussion above of the LAB 2D distributions, where we pointed out that P L approaches 1 at maximum recoil, as a consequence of an approximate negative-helicity selection by the dominant operators in that ω region (large τ LAB momentum).  Figure 10. Polarization averages xP CM T ypωq, xP CM L ypωq, xP CM T T ypωq and xP 2 ypωq defined in the CM system and calculated for the SM (black) and the NP Wilson coefficients from Fit 7 (red) of Ref. [36] and the R 2 leptoquark model fit (green) of Ref. [37] (see text for details). Error bands take into account the uncertainties associated to the Wilson coefficients and form factors, and they are calculated as explained in the main text for the R 2 leptoquark model fit, and in Ref. [60] for the SM and Fit 7.

Complex Wilson coefficients
For the particular case of the Λ b Ñ Λ c transition, we also show in Fig. 10 results for the R 2 leptoquark model fit of Ref. [37]. This fit is particularly interesting since the two nonzero Wilson coefficients, C S L and C T , are complex giving rise to a nonzero xP CM T T ypωq value. For this particular model C S L and C T at the bottom mass scale, appropriate for the present calculation, are given in terms of just the value ofĈ T at the scale of 1 TeV, witĥ C S L p1 TeVq " 4Ĉ T p1 TeVq, and the corresponding evolution matrix (see Ref. [37]). The right panel of figure 4 of Ref. [37] shows the constraints on the complex C S L plane, with best fit point at C S L "´0.08˘0.30 i. As we have done in Figs. 8 and 9, the error on the observables inherited from the form-factor uncertainties is evaluated and propagated via Monte Carlo, taking into account statistical correlations between the different parameters. It is shown as an inner error band that accounts for 68% CL intervals. The uncertainty induced by the fitted Wilson coefficients is determined using different 1σ Wilson coefficients configurations provided by the authors of Ref. [37]. The two sets of errors are then added in quadrature giving rise to the larger uncertainty band that can be seen in the figure.  Table 1. Total decay widths Γ τ " Γ pΛ b Ñ Λ c τν τ q and Γ epµq " Γ`Λ b Ñ Λ c epµqν epµq˘a nd ratios R Λc " Γ pΛ b Ñ Λ c τν τ q {Γ`Λ b Ñ Λ c epµqν epµq˘o btained in the SM and in the NP scenarios corresponding to Fit 7 of Ref. [36] and the R 2 leptoquark model fit of Ref. [37] (see text for details).

SM
In Fig. 10, and for the sake of comparison, we also include the polarization observables obtained with the SM and Fit 7 of Ref. [36]. We do not show in the figure any result for Fit 6 of [36], because this latter NP fit leads to predictions close to the SM ones.
The results for xP L,T y obtained with the R 2 fit of Ref. [37] are closer to the SM ones than the ones obtained from Fit 7 of Ref. [36]. This is particularly true for xP L y where the SM result is contained in the error band of the R 2´m odel prediction. Things change for xP T T y. As mentioned above, the complex C S L and C T Wilson coefficients of the R 2 fit of Ref. [37] generate a nonzero average-polarization xP T T ypωq, which is shown in the lower-left panel of Fig. 10. The nonzero-result for xP T T y comes from the interference of SM vector-axial with the NP terms, as well as the interference between the NP terms themselves. While for the R 2 model most observables are quadratic in the imaginary part ofĈ T p1 TeVq, like xP CM L,T y here but also the R D p˚q , R Λc , R J{ψ ratios, and the τ (A λτ ) and the longitudinal D˚(F DL ) polarization asymmetries, xP T T y is indeed linear in the imaginary part ofĈ T p1 TeVq. This allows to break the degeneracy present in the other observables with respect to the sign of Im rĈ T p1 TeVqs.
As discussed above, the projection P T T is invariant under co-linear boost transformations, and as a consequence the LAB average xP T T ypωq would be identical to that shown in Fig. 10, and evaluated in the CM frame. This average can be used to determine the linear combination of the functions F H and G H given in Eq. (B.1) of the appendix. However, additional information on the CM angular dependence of the P T T projection would be required to separately extract the time-reversal odd functions F H and G H . The experimental finding of signatures of non-zero tau polarization in a direction perpendicular to the plane formed by the CM (or LAB) three momenta of the outgoing hadron and the τ would be a clear indication, not only of NP beyond the SM, but also of CP (or time reversal) violation.
The results for R Λc are collected in Table 1. The result obtained with the R 2 fit of Ref. [37] is not far from to the SM one. Part of the reason for this behavior could be in the use, in Ref. [37], of B Ñ D p˚q form factors evaluated in the heavy quark limit. The use of the improved form factors obtained in Ref. [36], which included sub-leading corrections, in the R 2 fit gives rise to a larger Im rĈ T p1 TeVqs value that results in R Λc being larger around " 0.385.

Summary
For a given configuration of the momenta of all particles involved, we have introduced the tau spin-density matrixρ and the polarization vector P µ associated to a general H b Ñ H c τν τ decay. These two quantities contain all the information on the spin state of the τ provided no other particle spin is measured. For different semileptonic decays, we have evaluated P µ in the LAB and CM frames including the effects of NP. We have seen that the independent components P L , P T and P T T provide useful information to distinguish between different NP scenarios. This is specially true for the meson 0´Ñ 0´and also for the baryon Λ b Ñ Λ c reactions analyzed in this work. For this latter reaction, we have presented results for an extension of the SM that contains complex Wilson coefficients.
The LAB and CM helicity-polarized differential decay widths do not allow access to observables related to P T T , which is the component of the polarization vector orthogonal to the plane defined by the final hadron and tau three-momenta. Moreover, P T , which is the projection of P contained in the former plane and perpendicular to the τ´momentum, can only be obtained indirectly from these helicity-distributions, provided that results from both reference systems are analyzed simultaneously. The transverse polarization P T T is of special interest, since it is only possible for complex Wilson coefficients. Measuring a non-zero P T T value in any of the two frames will be a clear indication of physics beyond the SM and of time reversal (or CP) violation. For the NP scenarios corresponding to Fits 6 and 7 of Ref. [36] the Wilson coefficients are real and thus P T T is identically zero. In such a case, P is contained in the hadron-lepton plane. The R 2 fit of Ref. [37], which contains C S L and C T complex Wilson coefficients generates, however, a nonzero P T T value.
The NP effective Hamiltonian in Eq. (3.1) contains five Wilson coefficients, in general complex, although one of them can always be taken to be real. Therefore, nine free parameters should be determined from data. Even assuming that the form factors are known, and therefore the genuinely hadronic part (W ) of the Ă W SFs, it is difficult to determine all NP parameters from a unique type of decay, since the experimental measurement of the required polarization observables is an extremely difficult task. It is therefore essential to simultaneously analyze data from various types of semileptonic decays, as we have done in this work. We have used state of the art form-factors for all reactions, and the results presented in this work nicely complement those presented in our previous works of Refs. [60,61], and all together can be efficiently employed to disentangle among different NP scenarios.
Finally, we would like to draw the attention to the hadron-tensor method used in this work, previously derived in [60], which has shown to be a particularly suited tool to study processes where all final/initial hadron polarizations have been summed up. The scheme leads to compact expressions, valid for any baryon/meson semileptonic decay for unpolarized hadrons in the presence of NP and it clearly is an alternative to the helicity amplitude framework commonly used in the literature. Subsequent decays of the produced τ , after the b Ñ cτν τ transition, H b Ñ H c τ´ν τ ë ν τ µ´ν µ , ν τ π´, ν τ ρ´¨¨¨(5.1) can be straightforwardly studied within this hadron-tensor scheme and they will be presented elsewhere. has been supported by the Spanish Ministerio de Economía y Competitividad (MINECO) and the European Regional Development Fund (ERDF) under contracts FIS2017-84038-C2-1-P and PID2019-105439G-C22, the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement no. 824093 and by Generalitat Valenciana under contract PROMETEO/2020/023.

A CM and LAB kinematics
In this appendix we collect the different vector products needed to evaluate the P L , P T and P T T polarization vector components in the CM and LAB reference frames. We have p 2 " M 2 , k 2 " 0, k 12 " m 2 τ , p¨q " M M ω , k¨k 1 " q¨k " with M ω " M´M 1 ω. In addition, the scalar products that depend explicitly on the charged lepton variables used in the differential decay widths read LAB: In this case, p µ " pM, 0 q, q µ "`M ω , M 1 ? ω 2´1q LAB˘a nd with θ LAB τ the angle made by the final hadron and τ lepton LAB three-momenta, which is fixed, once E τ and ω are known, by the relation For a given ω, the fact that | cos θ LAB τ | ď 1 limits the possible E τ energies to the interval E τ P rEτ pωq, Eτ pωqs, (A. 4) with Eτ and Eτ given in Eq. (3.22). In terms of Eτ pωq one also can write Note that from Eq. (3.26), xP CM L ypωq can be written as xP CM L ypωq " ra 0 pω, h "´1q´a 0 pω, h " 1qs`1 3 ra 2 pω, h "´1q´a 2 pω, h " 1qs a 0 pωq`1 3 a 2 pωq (B.3) where the a 0,2 pωq and a 0,2 pω, h "˘1q functions are given in Eqs. (18) and (25) of Ref. [60] in terms of the eight Apωq, Bpωq, Cpωq, A H pωq, B H pωq, C H pωq, D H pωq and C H pωq ones.