The role of right-handed neutrinos in $b\to c \tau\, (\pi \nu_\tau, \rho \nu_\tau, \mu \bar \nu_\mu \nu_\tau) \bar\nu_\tau$ from visible final-state kinematics

In the context of lepton flavor universality violation (LFUV) studies, we fully derive a general tensor formalism to investigate the role that left- and right-handed neutrino new-physics (NP) terms may have in $b\to c \tau\bar\nu_\tau$ transitions. We present, for several extensions of the Standard Model (SM), numerical results for the $\Lambda_b\to\Lambda_c\tau\bar\nu_\tau$ semileptonic decay, which is expected to be measured with precision at the LHCb. This reaction can be a new source of experimental information that can help to confirm, or maybe rule out, LFUV presently seen in $\bar B$ meson decays. The present study analyzes observables that can help in distinguishing between different NP scenarios that otherwise provide very similar results for the branching ratios, which are our currently best hints for LFUV. Since the $\tau$ lepton is very short-lived, we consider three subsequent $\tau$-decay modes, two hadronic $\pi\nu_\tau$ and $\rho\nu_\tau$ and one leptonic $\mu\bar\nu_\mu\nu_\tau$, which have been previously studied for $\bar B\to D^{(*)}$ decays. Within the tensor formalism that we have developed in previous works, we re-obtain the expressions for the differential decay width written in terms of visible (experimentally accessible) variables of the massive particle created in the $\tau$ decay. There are seven different $\tau$ angular and spin asymmetries that are defined in this way and that can be extracted from experiment. Those asymmetries provide observables that can help in constraining possible SM extensions.


Contents
3 Sequential H b Ñ H c τ pπν τ , ρν τ , µν µ ν τ qν τ decays 10 3.1 Transition matrix element and the τ´polarization vector G C n , C P L , C A F B , C Z L , C P T , C A Q , C Z Q and C Z K coefficients and their dependence on pω, ξ d q 35

Introduction
Although there is no single experiment that can still claim the discovery of new physics (NP) beyond the Standard Model (SM), there seems to be however mounting evidence that points in that direction. Lepton flavor universality (LFU), which is inherent to the SM (the exception made of lepton-Higgs couplings), is being challenged in different experiments.
the angle made by the three-momenta of the tau and final hadron in the center of mass of the final two-lepton pair (CM), and E τ is the energy of the tau lepton in the frame where the initial hadron is at rest (LAB). Our studies showed that the helicity-polarized distributions in the LAB frame provide information on NP contributions that cannot be accessed from the study of the CM differential decay width, the one that is commonly analyzed in the literature. Besides, we have found that 0´Ñ 0´and 1{2`Ñ 1{2`decays seem to better discriminate between different left-handed neutrino NP than 0´Ñ 1´reactions.
The present work is organized as follows: In section 2, together with appendices A, B, C and D, we review our tensor formalism, and extend it to include right-handed neutrino NP terms. We want to stress here that although we always refer to b Ñ c transitions, the hadron and lepton tensors, together with the expressions for the semileptonic differential distributions derived in this work, in the presence of both left-and right-handed neutrino NP terms, are valid for any q Ñ q 1 ν charged-current decay. In section 2.3 we collect some of the main theoretical expressions obtained in Ref. [54] concerning the spin density operator and the τ polarization vector, which will be needed in the next section. Besides, an extension of these results to the case of ab Ñc transition is presented (see also appendix E in this latter respect). In section 3, we make a thorough study of the H b Ñ H c τν τ reaction including the subsequent τ -decay, for which we shall consider the two hadronic decay modes τ Ñ πν τ and τ Ñ ρν τ and the leptonic one τ Ñ µν µ ν τ . Although we also provide differential decay widths with respect to variables defined in the τ rest frame 1 , we mainly concentrate in obtaining the differential decay width in terms of visible kinematic variables and we identify (section 3.4) the seven τ angular and spin asymmetries mentioned above. Some details on the evaluation of the phase-space integrals, which can be rather involved in the leptonic decay mode, are presented in appendix F, while the kinematical coefficients that multiply each of the observables are discussed in appendix G. Results for the τ asymmetries in the Λ b Ñ Λ c transition are presented in section 4. They are obtained within the SM, three different NP extensions that include right-handed neutrino fields, and a NP model constructed with left-handed neutrino operators alone. A short summary of our main findings is given in section 5.

Hadron and lepton tensors in semileptonic decays including new physics with right-handed neutrinos
In Refs. [55,56], we derived a general framework, based on the use of general hadron tensors parameterized in terms of Lorentz scalar functions, for describing any meson or baryon semileptonic decay. It is an alternative to the helicity-amplitude scheme for the description of processes where all hadron polarizations are summed up and/or averaged. In these two works, NP with left-handed neutrinos were considered, and here we extend the formalism to include also right-handed neutrino operators.

Effective Hamiltonian
We consider an extension of the SM based on the low-energy Hamiltonian comprising the full set of dimension-6 semileptonic b Ñ c operators with left-and right-handed neutrinos [61] H eff " with left-handed neutrino fermionic operators given by O V pL,RqL " pcγ µ b L,R qp¯ γ µ ν L q, O S pL,RqL " pc b L,R qp¯ ν L q, O T LL " pc σ µν b L qp¯ σ µν ν L q (2.2) and the right-handed neutrino ones and where ψ R,L " p1˘γ 5 qψ{2, G F " 1.166ˆ10´5 GeV´2 and V cb is the corresponding Cabibbo-Kobayashi-Maskawa matrix element. Note that tensor operators with different lepton and quark chiralities vanish identically 2 . The 10 Wilson coefficients C X AB (X " S, V, T and A, B " L, R) parametrize possible deviations from the SM, i.e. C X ABˇS M =0. They are lepton and flavour dependent, and complex in general.

Hadron and lepton currents
The semileptonic differential decay rate of a bottomed hadron (H b ) of mass M into a charmed one (H c ) of mass M 1 and ν , measured in its rest frame, after averaging (summing) over the initial (final) hadron polarizations, reads [66], where Mpk, k 1 , p, q, spinsq is the transition matrix element 3 , with p, k 1 , k " q´k 1 and p 1 " p´q, the decaying H b particle, outgoing charged-lepton, neutrino and final hadron four-momenta, respectively. In addition, ω is the product of the two hadron four velocities ω " pp¨p 1 q{pM M 1 q, which is related to q 2 " pk`k 1 q 2 via q 2 " M 2`M 12´2 M M 1 ω, and s 13 " pp´kq 2 . Including both left-and right-handed neutrino NP contributions, we have 2 It follows from σ µν p1`χγ5q b σµν p1`χ 1 γ5q " p1`χχ 1 qσ µν b σµν´pχ`χ 1 q i 2 µν αβ σ αβ b σµν , (2.4) where we use the convention 0123 "`1. 3 The Lorentz-invariant matrix element, T , introduced in the review on Kinematics of the PDG [66] and M used in Eq. (2.5) are related by (up to a global phase) with the polarized lepton currents given by (u and v dimensionful Dirac spinors) where h "˘1 stand for the two possible charged-lepton polarizations (covariant spin) along a certain four vector S α that we choose to measure in the experiment. This is to say, the outgoing charged-lepton is produced in the state u S pk 1 ; hq defined by the condition The four vector S α satisfies the constraints S 2 "´1 and S¨k 1 " 0, and the choice S α " p| k 1 |, k 10k1 q{m , withk 1 " k 1 {| k 1 | and m the charged lepton mass, leads to chargedlepton helicity states. For later purposes we define here the projector (2.10) In addition, h χ accounts for both neutrino chiralities, Rph χ " 1q and Lph χ "´1q.
The dimensionless hadron currents read J pαβq Hrr 1 χp"L,Rq pp, p 1 q " xH c ; p 1 , r 1 |cp0qO pαβq Hχ bp0q|H b ; p, ry, (2.11) with cpxq and bpxq Dirac fields in coordinate space, 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 spin indexes, and (we recall h χ"R "`1 and h χ"L "´1) The Wilson coefficients C S,P,V,A,T χ"L,R in the above definitions are linear combinations of those introduced in the effective Hamiltonian of Eq. (2.1) and are given in Appendix A. Neglecting the neutrino mass, m ν , there is no interference between the two neutrino chiralities, and the decay probability becomes an incoherent sum of ν L and ν R contributions, with E ν the neutrino energy. The diagonal lepton tensors needed to obtain |M| 2 are readily evaluated and they are collected in Appendix B for m ν " 0. After summing over polarizations, the hadron contributions can be expressed in terms of Lorentz scalar structure functions (SFs), which depend on q 2 , the hadron masses and the 10 NP Wilson coefficients, C X AB , introduced in the effective Hamiltonian of Eq.(2.1). Lorentz, parity and time-reversal transformations of the hadron currents (Eq. (2.12)) and states [67] limit their number, as discussed in detail in Ref. [56]. The hadron tensors are expressed as linear combination of independent Lorentz (pseudo-)tensor structures, constructed out of the vectors p µ , q µ , the metric g µν and the Levi-Civita pseudotensor µνδη . The coefficients multiplying the (pseudo-)tensors are the Ă W 1 s χ"L,R SFs. They depend on q 2 , the hadron masses, the Wilson coefficients for each neutrino chirality (C V,A,S,P,T χ"L,R ), and the genuine hadronic responses (W 1 s). The latter ones are determined by the matrix elements of the involved hadron operators, which for each particular decay are parametrized -6 -in terms of form-factors. Symbolically, we have Ă W χ " C χ W . There is a total of 16 independent SFs ( Ă W 1 s χ ) for each neutrino-chirality set of Wilson coefficients, as shown in Ref. [56]. However, the consideration of both neutrino chiralities does not modify the number of genuine hadronic responses W 1 s, and the number of Ă W SFs increases due to the greater number of Wilson coefficients. For the sake of clarity, the definition of the Ă W 1 s χ SFs are compiled here in Appendix C.
From the general structure of the lepton and hadron tensors, collected in Appendices B and C, and which are at most quadratic in k, k 1 and p, one can generally write for the decay with a polarized charged lepton. [54,56] with Sk 1 qp " αβρλ S α k 1 β q ρ p λ and the N and N H 123 scalar functions given by There are three independent functions, A, B, and C , for the non-polarized case, and seven additional ones, A H , B H , C H , D H , E H , F H and G H , to describe the process with a defined polarization (h "˘1) of the outgoing along the four vector S α . Expressions for all of them in terms of the Ă W SFs are given in Appendix D. As can be seen there, these functions receive contributions from both neutrino chiralities. For A, B, C, F H and G H , it always appears the combination pL`Rq, i.e. p Ă W iL`Ă W iR q, while for the other functions (A H , B H , C H , D H and E H ) the structure is (L´Rq : ( Ă W iL´Ă W iR q. An obvious consequence is that the NP L´and R´neutrino-chirality contributions cannot be disentangled using only the non-polarized decay, and some information is needed from charged-lepton polarized distributions. As can be seen in Appendices C and D, the Ă W SFs present in N H 3 are generated from the interference of vector-axial with scalar-pseudoscalar terms ( Ă W I1χ ), scalar-pseudoscalar with tensor terms ( Ă W I3χ ), and vector-axial with tensor terms ( Ă W I4χ,I5χ,I6χ ). Since the vector-axial terms are already present in the SM, at least one of the C S,P,T χ coefficients must be non-zero to generate a non-zero N H 3 term. 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 of special relevance.
As expected, the N pω, k¨pq and N H 123 pω, k¨pq scalar functions give also the antiquarkdriven decay Hb Ñ Hc `ν , as shown in Appendix E. Moreover, Eq. (E.3) and the results for Ă W SFs collected in this work, for NP operators involving both left-and right-handed -7 -neutrino fields, can be straightforwardly used to describe quark charged-current transitions giving rise to a final `ν lepton pair (e.g. c Ñ s `ν ). One can use all the formulae given in [56] to obtain the differential decay widths for a final τ with a well defined helicity either in the laboratory (LAB) or the center of mass (CM) frames, where the initial hadron or the outgoing p ν q-pair are at rest, respectively. Namely, the d 2 Γ{pdωdE q and d 2 Γ{pdωd cos θ q distributions for positive and negative helicities of the outgoing charged-lepton and where E is the LAB energy of the charged lepton and θ is the angle made by its three-momentum with that of the final hadron in the CM frame. Note that these distributions do not depend on the CP-symmetry breaking term N H 3 , since for both CM and LAB systems Sk 1 qp " 0, when helicity states are used, i.e. S α " p| k 1 |, k 10k1 q{m .
The CM distribution can be written as where the a 0,1,2 pω, hq coefficients are explicitly given in [56] as linear combinations of A, B, C, A H , B H , C H , D H and E H . Analogously, the detailed dependence on E for the LAB distribution is also fully addressed in [56]. The scheme is totally general and it can be applied to any charged current semileptonic decay, involving any quark flavors or initial and final hadron states. Expressions for the Ă W iχ SFs in terms of the Wilson coefficients (C X AB ) and the form-factors, used to parameterize the genuine hadronic responses (W i ), can be obtained from the Appendices of Refs. [56] and [65], for any 1{2`Ñ 1{2` ν , 0´Ñ 0´ ν or 0´Ñ 1´ ν semileptonic decay, regardless of the involved flavors (see Eq. (D.4) for details).
In Refs. [55,56], 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 by analyzing only the CM differential decay widths, as it is commonly proposed in the literature (see also the discussion of Eq. (4.5) in Ref. [54]). In Ref. [65] 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 τ ν τ decays. What we have found is that the discriminating power between different NP scenarios was better forB c Ñ η c ,B Ñ D and Λ b Ñ Λ c decays than for B c Ñ J{ψ andB Ñ D˚reactions.
In the works of Refs. [55,56,65] only NP left-handed neutrino terms were considered.

Spin density operator and charged-lepton polarization vector
The charged-lepton polarization vector P µ pω, k¨pq can be readily obtained from Eq. (2.14), -8 -with l K " rl´pl¨k 1 {m 2 τ qk 1 s, l " p, q, which guaranties k 1¨P " 0. We refer the reader to Ref. [54] for a detailed discussion on the properties of P µ and numerical calculations, within the SM and different beyond the SM (BSM) scenarios, for the Λ b Ñ Λ c τν τ ,B c Ñ η c τν τ , B c Ñ J{ψτν τ andB Ñ D p˚q τ ν τ decays. Here, we only collect some relations from Ref. [54], which will be useful to describe the sequential H b Ñ H c τ pπν τ , ρν τ , µν µ ν τ qν τ decays. The spin density operator,ρ, and the polarization vector are related by The operator O is defined by its relation with the modulus squared of the invariant amplitude for the production of a final τ´lepton in a u S pk 1 ; hq state, for a given momentum configuration of all the particles involved and when all polarizations except that of the τ lepton are being averaged or summed up, ÿ |M| 2 "ū S pk 1 ; hqOu S pk 1 , hq " From the above equation and Eqs. (2.14) and (2.18), it follows Finally, the Dirac matrix O can be expressed as where the neutrino mass has also been neglected here. The operator O SL pr, r 1 q gives the Feynman amplitude for the H b Ñ H c ν vertex, M "ū S pk 1 ; hqO SL pr, r 1 qvν pkq, (2.23) with r, r 1 hadron spin indexes. In the above equation, the antineutrino polarization is not specified, since O is obtained after summing also over this degree of freedom, resulting in { k in Eq. (2.22). From Eq. (E.4) of Appendix E, we conclude that the polarized antiquark-driven semileptonic Hb Ñ Hc `ν decay is also described by the polarization vector P µ pω, k¨pq given in Eq. (2.18). The corresponding spin-density operator (ρb Ñc ) reads in this case, The operator Ob Ñc is defined by its relation with the modulus squared of the invariant amplitude for the production of a final anti-tau lepton in a v S pk 1 ; hq state, for a given momentum configuration of all the particles involved and when all polarizations except that of the anti-tau are being averaged or summed up. One has (see Eq. (E.4)) ÿ |M| 2 "v S pk 1 ; hqOb Ñc v S pk 1 , hq " 1 2 Tr which guaranties that the unpolarized decay distributions are equal for both Hb Ñ Hcτ`ν τ and H b Ñ H c τ´ν τ reactions. Besides, with these definitions, the probability P rv S pk 1 , hqs that in an actual measurement the anti-tau is found in the v S pk 1 , hq state is given by P rv S pk 1 , hqs " 1 2m τv S pk 1 , hqρb Ñc v S pk 1 , hq " 1 2 p1´hP¨Sq " 1 2m τū S pk 1 ,´hqρ u S pk 1 ,´hq " P ru S pk 1 ,´hqs (2.27) and it is equal to the probability P ru S pk 1 ,´hqs that the τ is found in the u S pk 1 ,´hq state in the quark b Ñ c semileptonic decay.
The τ in the final state poses an experimental challenge, because it does not travel far enough for a displaced vertex and its decay involves at least one more neutrino. The maximal accessible information on the b Ñ cτν τ transition is encoded in the visible decay products of the τ lepton, for which the three dominant decay modes τ Ñ πν τ , ρν τ and ν ν τ ( " e, µ) account for more than 70% of the total τ width (Γ τ ). Hence in this section, we study subsequent decays of the produced τ , after the b Ñ cτν τ transition, in the presence of NP left-and right-handed neutrino operators. Since the lepton τ Ñ eν e ν τ distribution can be obtained from the muon-mode ones, assuming LFU in the light sector and replacing m µ Ø m e , we will only refer to the latter from now on.

Transition matrix element and the τ´polarization vector
In all cases the Lorentz-invariant amplitude 4 for the decay chain H b Ñ H c τ pd ν τ qν τ can be cast as (d " π, ρ, µν µ ) with O SL introduced in Eq. (2.23), the virtual τ four-momentum k 1 " q´k " p´p 1´k and r and r 1 spin indexes for the hadrons. In addition, M M 1 and d µ psq is a four-vector (see below), which depends on the τ decay mode, and finally s is a polarization index required to specify the state of the produced rho or muon. Now using Eqs. (2.19) and (2.22), the modulus squared of the invariant amplitude, after averaging and summing over polarizations of the initial and final particles, reads

3)
4 From now on, we follow the PDG conventions.
-10 -where we have neglected the τ´neutrino mass, and p d stands for the pion, rho or muon outgoing four-momenta. Next, we can use Eq. (2.19) to obtain R d in terms of the tau polarization vector P, with α s the ρ-meson polarization vector, a ρ " pm 2 τ´2 m 2 ρ q{pm 2 τ`2 m 2 ρ q, f π " 93 MeV and f ρ " 150 MeV. The meson decay constants and the CKM matrix element V ud determine while for the lepton mode we have (y " Note that off-shell effects have been neglected in the derivation of Eqs. (3.4)-(3.6). Actually, we will make use of the approximation which puts the τ on the mass-shell, and it is extremely accurate since Γ τ {m τ " 10´1 2 .

Integration of the phase-space of the final neutrinos
The total width for the sequential decay H b Ñ H c τ pd ν τ qν τ is given in the initial hadron rest frame by where for the muon mode, an additional phase-space integration ş d 3 pν µ 2| pν µ |p2πq 3 for the outgoing muon antineutrino is needed and also to take into account its four-momentum in the delta of conservation. The outgoingν τ pkq-ν τ pp ντ q tau antineutrino-neutrino pair, together with the muon antineutrinoν µ in the case of lepton decay mode, are very difficult to detect and hence it is convenient to integrate over their variables. This is easily done using the product of Dirac delta functions δ 4 pq´k 1´k qδ 4 pk 1´p d q of Eq. (F.2), which is obtained from the -11 -delta of conservation of total four momentum and the on-shell approximation of Eq. (3.9) for the τ´propagator. This procedure introduces an integral over the tau phase-space, and using Eq. (F.4) to perform the ν τ -ν µ neutrino integrations for the muon decay mode, we get where B d"π, ρ, µνµ are the branching fractions of τ Ñ πν τ , τ Ñ ρν τ and τ Ñ µν µ ν τ decays. In addition, d 2 Γ SL {dωds 13 is the unpolarized semileptonic H b Ñ H c τν τ differential distribution introduced in Eq. (2.5), which is re-obtained thanks to the relation of Eq. (2.21), s 13 " pp´kq 2 " pp 1`k1 q 2 , p " pM, 0 q and q " p´p 1 " pM´M 1 ω, M 1 ? ω 2´1q LAB q, witĥ q LAB a unitary vector in an arbitrary direction. Indeed, we can always take the plane OXZ as the one formed by the three momenta k 1 and p 1 of the outgoing tau and final hadron and perform two of the three d 3 k 1 integrations with the help of the Dirac delta function, with ω varying between 1 and ω max " pM 2`M 1 2´m2 τ q{p2M M 1 q and the limits of s 13 given by The scalar functions η d and χ d can only depend on masses and the scalar product pk 1¨p d q, where k 1 is rebuilt in terms of ω and s 13 . The contribution independent of the tau-polarization vector reads where Hr...s is the step function and x " 2pp µ¨k 1 q{m 2 τ (muon energy in the τ rest frame, except for a constant) in the lepton mode case. Note that and the integration on ds 13 reconstructs dΓ SL {dω. A further integration on dω will give the expected result Γ d " Γ SL B d . The term proportional to the polarization vector, which contribution vanishes when one fully integrates over d 3 p d , reads with a d"π " 1 and a d"ρ " pm 2 τ´2 m 2 ρ q{pm 2 τ`2 m 2 ρ q, as defined above.
-12 - The d 3 p d integrals in the expression for the Γ d decay width in Eq. (3.12) can be further worked out analytically thanks to the invariance of the integrand under proper Lorentz transformations. There are different choices as to what variables to integrate and in what follows we give the result for two different kinematics of the visible product after the τ -decay.
If the momentum of the τ lepton is detected, the direction of the outgoing visible particle after its decay can be referred to the plane formed by the tau and the final hadron. Taking k 1 and p k 1ˆ p 1 qˆ k 1 in the positive Z and X directions respectively, the Lorentz-scalar pp d¨P q can be evaluated in the τ rest frame (P˚µ " Λ µ ν P ν , with Λ the boost which takes the tau to rest) with θd and φd, the polar and azimuthal angles of p d in the τ´rest frame, and where the scalar functions PX ,Y,Z pω, s 13 q determine the polarization four-vector in this system, which is given by P˚µ " p0, PX , PY , PZq. Note that these Cartesian components are obtained as Lorentz scalar products (PZ "´P˚¨n L , PX "´P˚¨n T and PY "´P˚¨n T T ) of the polarization four-vector with spatial unit vectors in the positive Z, X and Y axis, which by construction coincide with the directions of k 1 , p k 1ˆ p 1 qˆ k 1 and p k 1ˆ p 1 q, respectively. These scalar products can be now evaluated in the original system with k 1 ‰ 0 by using Λ´1, boost of velocity k 1 {k 10 , and we find PZ ,X,Y "´pP˚¨n L,T,T T q "´pP¨N L,T,T T q, with which allows to identify the Cartesian components of the polarization vector in the τ rest frame with the usual longitudinal and transverse components of the polarization vector PZ " P L , PX " P T and PY " P T T in an arbitrary frame [54]. Now integrating over d| p d |, we obtain d´gd P " P T pω, s 13 q sin θd cos φd P T T pω, s 13 q sin θd sin φd`P L pω, s 13 q cos θd ‰¯, (3.20) After integration, g µνµ " 1 and g µνµ P "´p1´yq 5`1`5 y`15y 2`3 y 3˘{ r3f pyqs.
-13 -  [54]) and Wilson coefficients. Bottom: F H pωq (left) and G H pωq (right) scalar functions entering in the definition of the CP-violating N H3 term of the differential decay width (Eqs. (2.14) and (2.15)). The error inherited from the form-factor uncertainties is evaluated and propagated via Monte Carlo, taking into account statistical correlations between the different parameters, and it is depicted as an inner band that accounts for 68% confident-level intervals. The uncertainty induced by the fitted Wilson coefficients is determined using different 1σ statistical samples configurations by the authors of Ref. [39]. The two sets of errors are then added in quadrature giving rise to the larger uncertainty band. Note that uncertainties on G H pωq are largely dominated by the errors due to the form-factors.
Appropriate θd and/or φd asymmetries can be used to determine the longitudinal and the two transverse components of the tau-polarization four vector, which are twodimensional functions of the variables ω and s 13 [54]. The observable P T T is of great interest, since it is given by the CP odd term N H 3 of the P µ decomposition in Eq. (2.18), which to be different from zero requires the existence of relative complex phases between Wilson coefficients. This component, transverse to the plane formed by the outgoing hadron -14 -and tau, could be obtained integrating over cos θd and looking at the φd asymmetry The projection P T T is invariant under co-linear boost transformations, and thus it is the same in both CM and LAB systems. Measuring a non-zero P T T value in any of these sequential decays will be a clear indication of physics BSM and of time reversal (or CP) violation. One can proceed similarly to obtain P T and P L . Upon integration on s 13 , the semileptonic dΓ SL {dω differential decay width can be factorized out in Eq. (3.20) by replacing the two-dimensional polarization components P a pω, s 13 q by averages on the s 13 variable weighted by the semileptonic distribution The d 3 Γ d {pdωd cos θd dφdq distributions thus obtained coincide with the results given in Ref. [50] for the CM frame (center of mass system of the τν τ pair). Both the two-dimensional and the averaged polarization components in the CM and LAB frames were detailedly studied, in Ref. [54], in the presence of NP involving only left-handed neutrino operators. Results were obtained for the Λ b Ñ Λ c τν τ ,B c Ñ η c τν τ , B c Ñ J{ψτν τ andB Ñ D p˚q τ ν τ decays, and in the case of the baryon decay, an special attention to BSM signatures derived from complex NP contributions was paid. We complete here the analysis of section 4.2.1 of Ref. [54] for the Λ b Ñ Λ c τν τ decay by showing in Fig. 1, the CP-violating observables P T T pω, cos θ τ q´"´P¨N LAB{CM T T¯, F H pωq and G H pωq (see Eqs. (2.14) and (2.15)) obtained within the leptoquark model [39] employed in that section, and which predicts complex Wilson coefficients. The polarization xP T T ypωq displayed in the left-bottom plot of Fig. 10 of Subsec. 4.2.1 of Ref. [54] can be obtained from the average indicated in Eq. (3.22) using the two-dimensional P T T shown in the top plot of Fig. 1, and which could be measured by looking at the azimuthal asymmetry proposed in Eq. (3.21). This average xP T T ypωq will be a linear combination of the F H pωq and G H pωq scalar functions, also displayed here in Fig. 1, which encode the maximal information contained in the CP-violating N H 3 term of the Λ b Ñ Λ c τν τ differential decay width. One cannot determine F H pωq and G H pωq only from xP T T ypωq. Therefore, to extract both of them, it would be necessary to analyze the dependence of the d 3 Γ d {pdωd cos θ τ dφdq sequential decay distribution on φd, which will allow to obtain the two-dimensional P T T pω, cos θ τ q polarization component.

Visible pion/rho/muon variables in the CM frame.
When the tau momentum cannot be fully reconstructed experimentally, the previous expressions are no longer useful, since the kinematics of the decay-product is referred to the τ direction. It is therefore suitable to construct observables directly from final-state kinematics of the visible decay particle π, ρ, µ, without relying on the reconstruction of the tau momentum, which needs to be integrated out (s 13 q. We take the energy of the charged particle in the τ´decay, E d and its angle θ d with the final hadron H c , both variables defined in the CM frame ( q " 0, W boson at rest). This kinematical set up has been extensively used in the literature to analyze NP signatures inB Ñ D p˚q τ pπν τ , ρν τ , µν µ ν τ qν τ -15 -decays [48,[51][52][53]69], although these studies have not considered BSM right-handed neutrino fields. Moreover, a similar polarimetry analysis for the Λ b Ñ Λ c τν τ reaction has not be done yet, despite the good prospects that LHCb can measure it in the near future, given the large number of Λ b baryons which are produced at LHC.
Following the notation in Ref [70], we introduce with γ and β defining the boost from the τ rest frame to the CM one. In addition, the dimensionless variable ξ d is the CM ratio of the energies of the tau-decay massive product and the tau lepton. Let us call now θ CM τ d P r0, πr, the angle formed by the p τ and p d in the CM reference system. We have where we have used y " m d {m τ for all decay modes. For the hadronic channels cos θ CM τ d , obtained in that case from the condition pk 1´p d q 2 " 0, is totally fixed by ω and the energy of the tau-decay massive product, while for the lepton mode it also depends on the additional variable x " 2pp µ¨k 1 q{m 2 τ introduced above. Next, requiring pcos θ CM τ d q 2 ď 1, we obtain the allowed region for the energy of the pion or rho mesons 5 τ Ñ pπ, ρqν τ ñ 1´β 2`1`β 2 25) or bounds that the variable x should satisfy in the lepton mode, Also for this latter case, the maximum allowed value of ξ d is still ξ 2 , which corresponds to E max d " pq 2`m2 d q{p2 a q 2 q. However, in certain circumstances, ξ d can be as low as y{γ for all reachable q 2 . This is to say a kinematics where the daughter massive lepton is at rest in the CM frame, which would be compatible with the energy-momentum conservation thanks to the other two neutrinos present in the tau-decay final state. In general, one finds For the b Ñ c semileptonic decays analyzed here, we have y " m µ {m τ and a q 2 ď pMḾ 1 q ă m 2 τ {m µ , which corresponds to the range of Eq. (3.27). Thus, the outgoing muon can 5 The outgoing π or ρ hadron could exit at rest only for a single value a q 2 " m 2 τ {m d of the phase-space, which is likely not accessible, since we expect mπ,ρ ă m 2 τ {pM´M 1 q. exit at rest for any q 2 value, fixing in this way the minimum reachable value for ξ d to y{γ (E d " m µ q independently of q 2 (or ω). In a hypothetical case, for which y 2 ą p1´βq{p1`βq (Eq. (3.28)), the outgoing massive particle could not exit with zero momentum. The bounds of Eq. (3.26) should be combined with the product of step functions Hpx´2yqHp1`y 2´x q which appears in the definition of η d"µνµ and χ d"µνµ in Eqs. (3.14) and (3.16), respectively. While 2y is always smaller than the lower bound in Eq. (3.26), the combined use of p1`y 2 q and the upper bound x`is more subtle and it leads to the following available phase space (3.30) in agreement with the results of Ref [70] obtained for y " 0. Taking in this case the outgoing hadron momentum p 1 as the positive Z direction and p 1ˆ k 1 as the positive Y direction, one can write k 1µ " m τ γp1, β sin θ τ , 0, β cos θ τ q and In addition, we can express the scalar product pp d¨P pω, s 13 qq in terms of the CM-variables as: where we have used P µ "`P L N µ L`P T N µ T`P T T N µ T T˘CM , with the vectors N µ L,T,T T computed using Eq. (3.19), the CM final hadron and tau four momenta and the relations P L,T,T T "´pP¨N L,T,T T q (see Ref. [54] for further details). We recall that in Eq. -17 -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 , ones finds [54] (the P CM T T term will not contribute after integrating in φ d ) P CM L pω, cos θ τ q " δa 0 pωq`δa 1 pωq cos θ τ`δ a 2 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 , (3.34) where, for i " 1, 2, 3, one has Now, we are in conditions to address the integration of the φ d and cos θ τ (or s 13 ) 7 , • For the hadronic modes, the η d"µνµ and χ d"µνµ functions contain an energy conservation Dirac delta function, which can be used to integrate dφ d , with z 0 introduced in Eq. (3.31). Now, we use ,´a1´z 2¯ı (3.38) to carry out the φ d integration. As a result, we find that the P T T contribution vanishes and considering sin θ τ sin θ d in Eq. (3.37), we obtain a common factor Now, the integration over cos θ τ can be easily done using the analytical integrals compiled in Eq. (F.6), which appear when the factor from the above equation, the longitudinal and transverse polarization components given in Eq. • In the leptonic mode, the η d"µνµ and χ d"µνµ functions do not contain any Dirac delta. However, many of the results of the previous case can also be used here. To fix the OXZ plane, it is necessary to detect both the hadron and tau momenta, and given the expected experimental difficulties to reconstruct the τ -trajectory, we integrate the azimuthal φ d angle. It holds Here, we see again that the contribution of the CP-violating polarization component P T T cancels out, and some kind of φ d -asymmetry, similar to that of Eq. (3.21), would be needed to isolate this term. In addition, z 0 " z 0 pxq and the change of variables , (3.42) with P 2 the Legendre polynomial of order two. The contributions of the different waves for the hadronic modes read F π,ρ 0 pω, ξ d q " 1 2βp1´y 2 q˜1`a π,ρ`1`y 2´2 ξ dβ p1´y 2 q xP CM L ypωq¸, For the lepton channel, for which we remind that cos θ CM τ d depends also on the integration with the functions G 1 px, yq " xp3´2xq´y 2 p4´3xq, G L px, yq " px´2ξ d q`1`3y 2´2 x˘{β and G T px, yq "`1`3y 2´2 x˘. The integrations on the variable x are straightforward in all cases since only polynomials are involved. The actual expressions, lengthy ones in some cases, have been collected in Appendix G where we also provide, more visual, twodimensional graphic representations of their pω, ξ d q dependence. Note that G 1 and G L provide the overall normalization where the equivalent ones for the hadronic modes are trivially satisfied. Upon integration on cos θ d , and taking the massless limit y Ñ 0, we recover the results of Ref. [70] identifying 2F d 0 pω, ξ d q here with f pq 2 , ξq`P L pq 2 qgpq 2 , ξq in that reference. Note that, besides some differences in the notation, there is a sign change in the definition of the polarization terms we provide here with respect to the ones in Refs. [51,70].
In the sequential τ -decay distribution of Eq. (3.42), all information on the b Ñ cτν τ transition is encoded in the ω-dependent functions a i , δa i , p 1 1 and xP CM L,T y. As already mentioned, they can be expressed in terms of the A, B and C and A H , B H , C H , D H and E H ones introduced here in Eqs. (2.14) and (2.15). The first set of three functions (or equivalently a 0,1,2 ) determine the unpolarized H b Ñ H c τν τ semileptonic d 2 Γ SL {pdωd cos θ τ q distribution 8 . The helicity-asymmetry coefficients δa i"0,1,2 pωq involve only the second set of functions, while p 1 1 only involves C H . Finally, the angular weighted averages of the longitudinal and transverse components of the tau polarization vector are exhaustively discussed in Ref. [54], where (Appendix B) analytical expressions in terms of A H , B H , C H , D H and E H and the combination p3a 0`a2 q, can be found. Note also that xP CM L y " 3δa 0`δ a 2 3a 0`a2 and xP CM T y " 3πp 1 0 4p3a 0`a2 q . Thus for fixed ω, the combined pξ d , cos θ d ) analysis of the d 3 Γ d {pdωdξ d d cos θ d q distribution provides, in addition to a 0 , a 1 and a 2 , five independent observables δa i"0,1,2 pωq, xP CM T y and p 1 1 , which can be used to fully determine the five A H , B H , C H , D H and E H ω-functions, and that give the maximal information on NP in Table 1. For each row, the observables in the second column contain the same physical information as those compiled in the third one. The quantities in the first row determine the decay for unpolarized taus, while the ones in the second and third rows describe the decays for polarized taus. Finally, observables in the third row are zero unless the Wilson coefficients are complex.
the b Ñ cτν τ transition, without considering CP-violation. CP non-conserving contributions, encoded in the P T T component of the tau polarization vector, canceled out when we carried out the φ d integration. As noted above, the measurement of such angle would require to detect the τ -three momentum. Hence, the F H and G H functions, which are responsible for CP violation, are only accessible by including additional information. For B Ñ D˚, some CP-odd observables (triple product asymmetries), defined using angular distributions involving the kinematics of the products of the D˚decay, have also been presented [41,42,45,46]. These asymmetries are sensitive to the relative phases of the Wilson coefficients, as are the F H and G H scalar functions. We note that the expression found here for the visible distribution in Eq. (3.42) recovers the results presented in Refs. [51][52][53] forB Ñ D p˚q transitions, accounting in the leptonic mode also for effects due to the finite mass of the outgoing muon/electron from the tau decay. Thus, there is a correspondence between npq 2 q and the asymmetries A F B , P L , P K , Z L , Z K , Z Q and A Q introduced in Eq. (1.1) of Ref. [53] for the hadron modes, and a i"0,1,2 , δa i"0,1,2 , p 1 1 and xP CM T y (or A, B and C, and A H , B H , C H , D H and E H ) used here. In fact, the relationships become apparent when comparing equations (3.12) and (3.13) of [53] and the Eqs. (3.42)- (3.43) in this work, npq 2 q 9 p3a 0 pωq`a 2 pωqq , A F B pq 2 q " 3a 1 pωq{2 3a 0 pωq`a 2 pωq , A Q pq 2 q " a 2 pωq 3a 0 pωq`a 2 pωq P L pq 2 q "´xP CM L y, Z L pq 2 q "´3 δa 1 pωq{2 3a 0 pωq`a 2 pωq , Z Q pq 2 q "´δ a 2 pωq 3a 0 pωq`a 2 pωq P K pq 2 q "´xP CM T y, Z K pq 2 q "´p 1 1 pωq 3a 0 pωq`a 2 pωq . (3.46) In addition, the remaining two asymmetries P T (related to our xP T T y) and Z T mentioned in [53] should correspond to linear combinations of the F H and G H scalar functions within the tensor formalism presented in Sec. 2. As mentioned above, these CP-violating contributions cancel out on integration over the azimuthal angle φ d , which measurement would require detecting both the hadron and tau momenta. These relations are schematically shown in Table 1 where, for each row, the observables in the second and the third columns are equivalent in the sense that they contain the same physical information. In addition, we show which quantities determine the decay for unpolarized (first row) and polarized (second and third rows) final taus, as well as which of them require complex Wilson coefficients (third row  including left-and/or right-handed NP neutrino operators. Each of the observables in Eq. (3.46), which are embedded in one of the F d 0,1,2 pω, ξ d q partial waves introduced in Eq. (3.42), are affected by kinematical C n , C A F B , . . . , C Z K coefficients. Specifically, one can write Those coefficients are tau-decay mode dependent and in the case of the π and ρ hadronic ones they can be easily read out from Eq. (3.43). The corresponding expressions for the fully µν µ leptonic mode are collected in Appendix G. There, in Figs. 3 and 4, we also provide, for all three tau-decay modes considered in this work, their pω, ξ d q-graphic representations.
What we actually show are the products of each of the coefficients times the kinematical factor Kpωq " ? ω 2´1 p1´m 2 τ {q 2 q 2 that makes part of the dΓ SL {dω semileptonic decay width. The visual inspection of the different panels in Figs. 3 and 4 provides immediate information on which regions of the available pω, ξ d q phase-space might result more sensitive to (or adequate to extract from) each of the observables of Eq. (3.46). Taking into account the numerical values of the coefficients, the hadron channels, and in particular the pion mode, seem, in general, to be more convenient to determine the semileptonic quantities of Eq. (3.46). Probably, the best strategy would be to perform a multi-parametric fit of the

Results for the visible pion/rho/muon distributions in the presence of NP right-handed neutrino operators
We will consider three different extensions of the SM including right-handed neutrino fields, that correspond to the more promising ones, in terms of the pulls from the SM hypothesis, among those discussed in Ref. [61]. We will show predictions for the observables collected in Eq. (3.46), extracted from the visible distributions of the tau-decay massive products, for the baryon Λ b Ñ Λ c reaction. We will compare these NP results with those obtained in the SM, and within an extension of the SM determined by Fit 7 of Ref. [33] constructed only with left-handed neutrino operators. We focus in the baryon decay for the sake of brevity, since some of the observables of Eq. (3.46) with right-handed neutrinos were already shown in [61] for the mesonB Ñ D p˚q semileptonic decays, where the extensions considered in this work were fitted. Moreover,B c Ñ η c , J{ψ transitions, studied in our previous works, follow in general a similar pattern to that seen in the analog ones from B-meson decays. In addition, we will not include results from Fit 6 of Ref. [33], as we did in previous studies [54][55][56]65], since this NP scenario, which involves only left-handed neutrinos, provides polarized tau-distributions more similar to the SM ones than those obtained with the model Fit 7 of the same reference. The Λ b Ñ Λ c form factors used here are directly obtained (see Appendix E of Ref. [56]) from those calculated in the lattice quantum Chromodynamics (LQCD) simulations of Refs. [19] (vector and axial ones) and [20] (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. [20] to evaluate them. We use the errors and 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 model Fit 7 and the right-handed neutrino scenarios, we shall use statistical samples of Wilson coefficients selected such that the χ 2 -merit function computed in Refs. [33] and [61], respectively, changes at most by one unit from its value at the fit minimum. Both sets of errors are then added in quadrature and displayed in the predictions.
The analysis carried out in Refs. [33,61] considers only input from theB Ñ D p˚q meson transitions. Namely, the most recent world-average correlated values of R D and R D˚f rom the Heavy Flavor Averaging Group [71], the value of the q 2 -integrated lepton polarization asymmetry [P τ pD˚q " ş dq 2 pdΓ SL {dq 2 qP L pq 2 q{Γ SL s and the longitudinal Dp olarization, F DL , measured by Belle [72,73], and the q 2 distributions of the D and Dm esons [74,75], together with bounds from the leptonic decayB c Ñ τν τ .
The scenario 3 of Ref. [61] induces exclusively b Ñ cτν τ R right-handed neutrino NP interactions, and particularly the vector boson mediator only contributes to the vector Wilson coefficient C V RR . It trivially follows that for any H b Ñ H c decay, the A, B and C [A H , B H , C H , D H and E H ] functions will take the SM values scaled by a factor p1`|C V RR | 2 q [p1´|C V RR | 2 q]. Therefore, npq 2 q and consequently the total semileptonic width in the tau mode will be enhanced by p1`|C V RR | 2 q with respect to the SM result. No signatures of NP will appear in the A F B pq 2 q and A Q pq 2 q pion/rho/muon angular asymmetries, while P L pq 2 q, Z L pq 2 q, Z Q pq 2 q, P K pq 2 q and Z K pq 2 q will be scaled down by the factor p1´|C V RR | 2 q{p1`|C V RR | 2 q as compared to the SM predictions. The presence of a vector leptoquark at the high-energy scale leads to the scenario 5 of Ref. [61], where both left-and right-handed neutrino operators contribute at the m b scale. In Fit 5a, only right-handed neutrino fields are considered, which give rise to non-vanishing C V RR and C S LR Wilson coefficients, though the latter one is determined in Ref. [61] with large errors. Including also left-handed neutrino operators does not improve the χ 2 and the left-handed Wilson coefficients are compatible with zero within one sigma.
A scalar leptoquark is considered in scenario 7a of Ref. [61], where a solution dominated by C V RR , with an additional Wilson coefficient C T RR compatible with zero within one sigma, and C S RR «´8C T RR , is found. As in the previous case, adding the left-handed operators that contribute in the presence of the scalar leptoquark leads to a solution compatible with vanishing left-handed Wilson coefficients.
We note that none of these three possibilities with only right-handed neutrino fields can generate values of the longitudinal D˚polarization within its current one sigma ex--23 -perimental range. NP models, like Fit 7 of Ref. [33], with a significant contribution from C V RL reduces the tension with the F DL measurement. We should also mention that for the right-handed neutrino scenarios 3, 5a and 7a, the Wilson coefficient C V RR is found to be in the range 0.3´0.5, taking into account uncertainties, and such relatively large values are challenged by mono-tau searches at LHC [64].
In Fig. 2, we show predictions for p3a 0`a2 q, xP CM L y, A F B , Z L , xP CM T y, A Q , Z Q and Z K defined in Eq. (3.46), for the SM, the NP model Fit 7 of Ref. [33] and for scenarios 3, 5a and 7a from Ref. [61], which incorporate NP operators constructed using right-handed neutrino fields. All these quantities can be obtained from the S, P -and D-wave contributions (F d 0,1,2 pω, ξ d q) to the d 3 Γ d {pdωdξ d d cos θ d q differential distribution, associated to any of the Λ b Ñ Λ c τ pπν τ , ρν τ , µν µ ν τ qν τ sequential decays studied in the previous section. As already stressed, in the absence of CP-violation, this set of observables provides the maximal information (scalar functions A, B, C, and A H , B H , C H , D H , E H in Eqs. (2.14) and (2.15)) which can be extracted from the analysis of the semileptonic Λ b Ñ Λ c τν τ transition, considering the most general polarized state for the final tau (see Table 1).
The p3a 0`a2 q 9 dΓ SL {dω distributions displayed in the first panel of the figure lead to the results for the integrated widths compiled in Table 2, and it cannot disentangle among the three right-handed neutrino scenarios examined in this work. However, these distributions are useful to efficiently separate between the SM and any of its extensions fitted to the violations of LFU observed in B-meson decays. Moreover, for relatively large values of ω ą 1.15, neutrino left-handed and right-handed NP models predict significantly different dΓ SL {dω differential decay widths.
In the other seven panels of Fig. 2, we show tau angular and polarization asymmetries, as a function of ω. Relative errors in these observables are smaller than for p3a 0`a2 q, since they are defined as ratios for which the form-factor uncertainties largely cancel out. None of these observables are useful in distinguishing between the three scenarios with righthanded neutrinos taken from Ref. [61]. Furthermore, the angular asymmetries A F B pq 2 q and A Q pq 2 q, and to some extent the longitudinal polarization average xP CM L y, do not distinguish between SM and these latter NP models either. The predictions from Fit 7 of Ref. [33] are significantly different from those obtained within the SM and the right-handed neutrino models in all cases, except for Z L , where all the extensions of the SM give similar results. The D-wave polarization asymmetries Z Q and Z K seem quite adequate to distinguish the left-handed Fit 7 and the right-handed neutrino models, since the first type of NP extension produces an increase in the prediction of the SM, while the latter NP scenarios reduce the results of the SM.

Summary
We have given the hadron and lepton tensors and the semileptonic differential distributions in the presence of both left-and right-handed neutrino NP terms, and the most general polarization state for the final tau. The formalism is valid for any quark q Ñ q 1 ν or antiquarkq Ñq 1¯ ν charged-current decay, although we have usually referred to b Ñ c transitions. This framework is an alternative to the helicity amplitude one to describe processes where all hadron polarizations are summed up and/or averaged. The results of the first part of this work complete the scheme presented in Ref. [56], where only left-handed neutrino fields were considered.
-24 -  Figure 2. Predictions for the semileptonic observables p3a 0`a2 q, xP CM L y, A F B , Z L , xP CM T y, A Q , Z Q and Z K introduced in Eq. (3.46), as a function of ω, for the Λ b Ñ Λ c τν τ semileptonic decay. We show results obtained within the SM, the NP model Fit 7 of Ref. [33], which involves only lefthanded neutrinos, and other three ones taken from Ref. [61], where all included NP operators use right-handed neutrino fields. Error bands account for uncertainties induced by both form-factors and fitted Wilson coefficients (added in quadrature). The right-handed neutrino scenario 3 and the SM lead to the same results for the A F B pq 2 q and A Q pq 2 q angular asymmetries.
In section 3.3, we have discussed the d 3 Γ d {pdωd cos θ τ d cos θd dφdq sequential decay distribution in the τ rest frame, and how it can be used to extract the LAB or CM two dimensional P L pω, cos θ τ q, P T pω, cos θ τ q and P T T pω, cos θ τ q components of the τ -polarization -25 -vector. These observables, together with the unpolarized d 2 Γ{pdωd cos θ τ q distribution, provide the maximum information from the H b Ñ H c semileptonic decay with polarized taus [54], including the CP-violating contributions driven by the F H and G H scalar functions (Eqs. (2.14) and (2.15)). These latter functions are non zero only when some of the Wilson coefficients are complex, and are extracted from P T T pω, cos θ τ q, the polarization vector component transverse to the plane formed by the outgoing hadron and tau. We have detailed how P T T could be obtained integrating over cos θd , and looking at the φd asymmetry defined in Eq. (3.21). Results for the CP-violating contributions in the baryon Λ b Ñ Λ c τν τ reaction are shown in Fig. 1 within the R 2 leptoquark model of Ref. [39], for which the two nonzero Wilson coefficients (C S LL and C T LL ) are complex. If the tau momentum is not determined, the τ rest frame cannot be defined and the former results cannot be experimentally accessed.
Reconstructing the τ momentum in the final state poses an experimental challenge, because the τ does not travel far enough for a displaced vertex and its decay involves at least one more invisible neutrino. Direct τ polarization measurements are even more complicated to perform. Therefore, the maximal accessible information on the b Ñ cτν τ transition is encoded in the visible decay products of the τ lepton. For that reason, we have studied the sequential H b Ñ H c τ pπν τ , ρν τ , µν µ ν τ qν τ decays.
Without relying on the reconstruction of the tau momentum, we have derived the socalled visible decay particle π, ρ, µ, distributions [51][52][53], valid for any H b Ñ H c semileptonic decay. We take as visible kinematical variables the energy E d (or the variable ξ d , which is proportional to the energy) of the charged particle in the τ decay and the angle θ d made by its three-momentum with that of the final hadron H c , both variables defined in the CM frame (W boson at rest). The scheme allows to account for the full set of dimension-6 semileptonic b Ñ c operators with left-and right-handed neutrinos considered in Ref. [61].
In the absence of CP-violation, the analysis of the dependence on (ω, ξ d ) of the S, Pand D-wave contributions (F d 0,1,2 pω, ξ d q, d " πν τ , ρν τ , µν µ ν τ ) to the d 3 Γ d {pdωdξ d d cos θ d q differential distribution provides the maximal information, which can be extracted from the analysis of the semileptonic H b Ñ H c τν τ transition, considering the most general polarized state for the final tau. This exhaustive information (scalar functions A, B, C, and A H , B H , C H , D H , E H in Eqs. (2.14) and (2.15)) can be rewritten in terms of the overall unpolarized normalization distribution dΓ SL {dω, and seven angular and spin asymmetries [ see Table 1 and Eq. (3.46)] introduced in Ref. [53] for B-meson decays. We have found that, in general, the hadronic tau-decay channels, and in particular the pion mode, are more convenient to determine the H b Ñ H c τν τ semileptonic observables than the lepton τ Ñ µν µ ν τ channel. For this latter mode, we have provided, for the very first time, expressions where the muon mass is not set to zero.
We have considered three different extensions of the SM, taken from the recent study in Ref. [61], that include right-handed neutrino fields, and we have shown predictions (Fig. 2) for the semileptonic observables defined in Eq. (3.46), for the Λ b Ñ Λ c decay. We have compared these NP results with those obtained in the SM, and within an extension of the SM determined by Fit 7 of Ref. [33] constructed with left-handed neutrino operators alone.
None of the semileptonic decay asymmetries turned out to be useful in distinguishing between the three scenarios with right-handed neutrinos. The predictions from Fit 7 of Ref. [33] are, however, significantly different from those obtained within the SM and the right-handed neutrino models in all cases, except for Z L , where all the extensions of the SM give similar results. The D-wave polarization asymmetries Z Q and Z K seem quite adequate to distinguish the left-handed Fit 7 and the right-handed neutrino models.
We are aware that the measurement of these observables is rather difficult. At present, Λ b 's are only produced at the LHC, where the corresponding τ decay modes are difficult to reconstruct. However the LHCb collaboration has already published semileptonic decay results where the τ has been reconstructed through the τ Ñ µν τνµ decay mode [2,76]. It is reasonable to expect and extension of this selection strategy to Λ b semileptonic decays 9 . The other two τ decay modes, πν τ and ρν τ , analyzed in this work have a lower reconstruction efficiency and are not being exploited at the moment.
A Wilson coefficients C S,P,V,A,T χ"L,R We compile in this appendix the coefficients that enter into the definition of the hadron operators in Eq. (2.12). For left-handed neutrinos (χ " L), we have while for right-handed neutrinos (χ " R), where C X AB (X " S, V, T and A, B " L, R) appear in the BSM effective Hamiltonian of Eq. (2.1), taken from Ref. [61].

C Hadron tensors
We collect here the hadron tensors that should be contracted with the corresponding lepton ones, compiled in the previous appendix, to obtain ř |M| 2 ν χ , χ " L, R. The tensorial decompositions, for a given set C S,P,V,A,T χ of NP Wilson coefficients (see Eqs. (A.1) and (A.2)), are taken from Ref. [56].
• The spin-averaged squared of the O α Hχ operator matrix element leads to The sum is done over initial (averaged) and final hadron helicities, and the above tensor should be contracted -28 -with the lepton one L αρ pk, k 1 ; h, h χ q (Eq. (B.5)) to get the contribution to ř |M| 2 ν χ , χ " L, R. The tensor can be expressed in terms of five SFs as where all Ă W 1χ,2χ,3χ,4χ,5χ pq 2 , C V χ , C A χ q SFs are real. Following the notation in Ref. [56], • The diagonal contribution of the tensor operator O αβ Hχ gives rise to which contracted with the lepton tensor L αβρλ pk, k 1 ; h, h χ q in Eq. (B.7) provides the L or R contributions to the differential decay rate. The total tensor can be expressed in terms of four real SFs, g αρ q β q λ´gαλ q β q ρ´gβρ q α q λ`gβλ q α q ρ ih χ´ ρλαδ q β q δ´ ρλβδ q α q δ¯ı`W T 4 M 2 " g αρ pp β q λ`pλ q β q g αλ pp β q ρ`pρ q β q´g βρ pp α q λ`pλ q α q`g βλ pp α q ρ`pρ q α q ih χ´ ρλαδ pp β q δ`q β p δ q´ ρλβδ pp α q δ`q α p δ q¯ı The W T 1,2,3,4 SFs are found from pW αβρλ T T`W αβρλ pT pT q [56] and accomplish the constraint which can be used to re-write W T 1 in terms of W T 2,3,4 . In any case, the contraction of the W T 1 -part of the tensor with L αβρλ pk, k 1 ; h, h χ q is zero, and thus the contribution of W αβρλ χ to ř |M| 2 ν χ is given only in terms of W T 2 , W T 3 and W T 4 . The common factor |C T χ | 2 was absorbed in [56] real functions of q 2 [56].