New physics in the angular distribution of Bc−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}_c^{-} $$\end{document} → J/ψ(→ μ+μ−)τ−(→ π−ντ)ν¯τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_{\tau } $$\end{document} decay

In Bc−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}_c^{-} $$\end{document} → J/ψ(→ μ+μ−)τ−ν¯τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_{\tau } $$\end{document} decay, the three-momentum pτ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\boldsymbol{p}}_{\tau^{-}} $$\end{document} cannot be determined accurately due to the decay products of τ− inevitably include an undetected ντ. As a consequence, the angular distribution of this decay cannot be measured. In this work, we construct a measurable angular distribution by considering the subsequent decay τ− → π−ντ. The full cascade decay is Bc−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}_c^{-} $$\end{document} → J/ψ(→ μ+μ−)τ−(→ π−ντ)ν¯τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_{\tau } $$\end{document}, in which the three-momenta pμ+,pμ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\boldsymbol{p}}_{\mu^{+}},{\boldsymbol{p}}_{\mu^{-}} $$\end{document}, and pπ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\boldsymbol{p}}_{\pi^{-}} $$\end{document} can be measured. The five-fold differential angular distribution containing all Lorentz structures of the new physics (NP) effective operators can be written in terms of twelve angular observables ℐi(q2, Eπ). Integrating over the energy of pion Eπ, we construct twelve normalized angular observables ℐ̂i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\hat{\mathrm{\mathcal{I}}}}_i $$\end{document}(q2) and two lepton-flavor-universality ratios RPL,TJ/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R\left({P}_{L,T}^{J/\psi}\right) $$\end{document}(q2). Based on the Bc → J/ψ form factors calculated by the latest lattice QCD and sum rule, we predict the q2 distribution of all ℐ̂i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\hat{\mathrm{\mathcal{I}}}}_i $$\end{document} and RPL,TJ/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R\left({P}_{L,T}^{J/\psi}\right) $$\end{document} both within the Standard Model and in eight NP benchmark points. We find that the benchmark BP2 (corresponding to the hypothesis of tensor operator) has the greatest effect on all ℐi and RPL,TJ/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R\left({P}_{L,T}^{J/\psi}\right) $$\end{document}, except ℐ̂5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\hat{\mathrm{\mathcal{I}}}}_5 $$\end{document}. The ratios RPL,TJ/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R\left({P}_{L,T}^{J/\psi}\right) $$\end{document} are more sensitive to the NP with pseudo-scalar operators than the ℐi. Finally, we discuss the symmetries in the angular observables and present a model-independent method to determine the existence of tensor operators.

In order to distinguish between the SM and NP scenarios, and further characterise the underlying effects of NP, besides considering the total decay rate, the full angular distribution of B − c → J/ψτ −ν τ and B − c → J/ψ(→ µ + µ − )τ −ν τ decays should also be taken into account sometimes, see for instance refs. [54,60,64]. However, as pointed out in refs. [80,[99][100][101][102][103][104][105][106][107][108][109], the information of the polar and azimuthal angles of the emitted τ − cannot be determined precisely because the decay products of τ − inevitably contain an undetected ν τ . This means that the observables depending on the polar or azimuthal angle of τ − , such as the corresponding coefficients of the angular distribution and the forwardbackward asymmetry of τ − , cannot be directly measured. Therefore, in this work, we construct a measurable angular distribution by further considering the subsequent decay τ − → π − ν τ . The full cascade decay is B − c → J/ψ(→ µ + µ − )τ − (→ π − ν τ )ν τ , which includes three visible final-state particles µ + , µ − , and π − whose three-momenta can be measured. We first calculate the full five-fold differential angular distribution including all Lorentz structures of the NP effective operators, and then carefully study the NP effects in the angular distribution from many aspects.

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Our paper is organized as follows. In section 2, after defining the effective Hamiltonian, we give the analytical results of the independent transversity amplitudes and the measurable angular distribution of the five-body Definitions of the integrated observables are included in section 3. In section 4, we show the numerical results of the entire set of normalized angular observables I i (q 2 ) and the lepton-flavor-universality ratios R(P J/ψ L,T )(q 2 ) and R(J/ψ). A model-independent method for determining the existence of tensor operator is given in section 5. Our conclusions are finally made in section 6. In the appendices A and B, we present the detailed procedures related to the calculations of angular distribution and dependence relations, respectively.

Analytical results
In this section, after giving some necessary definitions, we directly list the analytical results of angular distribution. The more detailed calculations, including some useful conventions, can be found in appendix A.

Effective Hamiltonian
Assuming that the NP scale is higher than the electroweak scale, one can integrate out the possible NP particles as well as the SM heavy particles -the W ± , Z 0 , the top quark, and the Higgs boson, thus obtaining the effective Hamiltonian suitable for describing the b → cτ −ν τ transition 2 where G F is the Fermi constant, V cb is the CKM matrix element, σ µν ≡ i 2 [γ µ , γ ν ], and ν τ L = P L ν τ denotes the field of left-handed neutrino. The NP effects are encoded in the Wilson coefficients g i , which are defined at the typical energy scale µ = m b . In the SM, g V = −g A = 1 and g S = g P = g T = 0.

Transversity amplitudes
In the calculation, the hadronic matrix elements contain the nonperturbative QCD effects and can be parameterized as the Lorentz invariant form factors. The vector and axial-vector 2 Neutrinos are assumed to be left-handed in this work. The effective Hamiltonian containing righthanded neutrinos can be found in refs. [109][110][111]. It can be derived from the identity σ µν γ5 = − i 2 µναβ σ αβ that the operator (cσ µν (1 + γ5)b)(τ σµν ντL) is absent. We use the convention 0123 = − 0123 = 1.

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current matrix elements can be written as the following four form factors [27,55,112] where q = p−k, ε µ denotes the polarization vector of J/ψ meson. In our numerical analysis, we will use the vector and axial-vector form factors computed in lattice QCD [54,55].
Using the equation of motion, the scalar and pseudo-scalar matrix elements can be obtained by Based on the above four form factors V (q 2 ) and A 0,1,2 (q 2 ), one can define four independent transversity amplitudes as follows

6)
where m b and m c are the current quark masses evaluated at the scale µ = m b , and The tensor matrix element can be parameterized as [27,64,112] and J/ψ|cσ µν γ 5 b|B c = − i 2 µναβ J/ψ|cσ αβ b|B c . In the presence of the tensor operators, we find three additional independent transversity amplitudes as follows where the superscript T indicates that an amplitude appears only when one considers the tensor operators.

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In the SM, the angular observables I 7 , I 8 , and I 9 are vanishing. Therefore, in future measurements, a non-vanishing I 7 , I 8 , or I 9 would be a solid signal of NP, which induces a complex contribution to the amplitude.

E π -integrated angular observables
The differential decay rate (2.15) depends on five parameters q 2 , E π , θ J/ψ , θ π and φ π , and a complete experimental analysis may be limited by statistics. Integrating over the E π and after a proper normalization, we can get the following angular function with the twelve normalized angular observables I i (q 2 ) defined as Our choice of the normalization in eq. (3.1) results the relationship 3 I 1c q 2 − I 2c q 2 + 6 I 1s q 2 − 2 I 2s q 2 = 1. The cancellations through normalization to the decay rate lead to the observation that the observables I i (q 2 ) have less theoretical uncertainty to facilitate the discussion of the NP effects. In section 4, we will analyze numerically the entire set of observables I i (q 2 ) within the SM and in some NP benchmark points. The forward-backward asymmetry of π − meson as a function of q 2 can be defined as This asymmetry observable only exists in τ channel, and specifically for the τ − → π − ν τ decay. Obviously, it can be expressed linearly in terms of angular observables I i (q 2 ). By integrating over the lepton-side parameters E π , θ π , φ π , one can obtain the two-fold differential decay rate as follows

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are the longitudinal and transverse polarization fractions of the J/ψ meson, respectively. The differential decay rates for the longitudinally and transversely polarized intermediate state J/ψ are given, respectively, by with the factor N ≡ The polarization observables P J/ψ L,T (q 2 ) constructed above are not affected by τ decay dynamics since we have integrated over all the lepton-side kinematic parameters, so they are also applicable to light leptons µ and e. We denote P J/ψ L,T τ and P J/ψ L,T µ as extraction from B c → J/ψτ ν and B c → J/ψµν decays respectively, and define the following ratios to probe the universality of lepton flavor (3.9) The q 2 distribution of the decay rate can be obtained by adding up eqs. (3.6) and (3.7) as follows dΓ Our dΓ/dq 2 (apart from B τ B J/ψ ) is consistent with that in refs. [27,54].
Neglecting the π mass, our results are in agreement with those in refs. [104,108]. The sign difference in h ⊥ is due to the different choice of reference direction. It should be pointed out that in the absence of Z L (q 2 ), the differential forward-backward asymmetry dA π /dω π (i.e. I 1 (q 2 , ω π )) cannot be expressed in terms of A τ (q 2 ) and P ⊥ (q 2 ) as given by eq. (16) of ref. [104].

The form factors
The B c → J/ψ transition form factors are the main source of theoretical uncertainties. For the B c → J/ψ vector and axial-vector form factors, V (q 2 ) and A 0,1,2 (q 2 ), we use the latest

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high-precision lattice QCD calculation results given in ref. [55]. Since the B c → J/ψ tensor form factors T 1,2,3 (q 2 ) are not included in ref. [55], we will adopt the T 1,2,3 (q 2 ) calculated in the QCD sum rule method [64]. 3 These form factors are parameterized in a simplified z expansion to extend to the full q 2 range.

The NP benchmark points
The model-independent analyses of NP effects in B → D ( * ) τ ν decays have been completed in many previous works [17][18][19][20][21][22][23][24][25]. In order to show the influences of these NP effects on the angular distribution of we select various best-fit values as the NP benchmark points. These best-fit values are usually performed on a set of chiral base, which is equivalent to eq. (2.1) by the following relations According to the following steps, we select a total of eight NP benchmark points under seven different NP hypotheses.
Switching one coupling C i at a time, there are five NP hypotheses. The hypothesis of a single C V L can resolve the R(D ( * ) ) anomalies well, but there is no effect on the normalized observables defined in section 3, so we should not choose it. The hypothesis of a single C S L or C S R is ruled out by the decay rate of B c → τ ν decay [30,45,116]. We take a benchmark point from each of the two remaining NP hypotheses as follows [24] BP1: ( Although BP1 (BP2) and BP1 * (BP2 * ) are formally different benchmark points, they produce the same results for angular observables I 1c,1s,2c,2s,6c,6s,3,4,5 and opposite results for I 7,8,9 . Observables I 7,8,9 can distinguish between the NP benchmark point and its complex conjugate partner very well. In the following analysis, we do not consider BP1 * and BP2 * , and the same treatment is also applicable to the following BP6 * , which is the complex conjugate of the benchmark point BP6.
Considering the combinations induced by specific UV models, we choose the best-fit points in the following four different NP hypotheses as our NP benchmark points (the remaining C i are set to zero in each case) [22] BP3:

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where the Wilson coefficients are given at the NP scale 1TeV, and we should run them down to the scale m b [21]. Finally, taking into account all NP Wilson coefficients, except C V R which is explicitly lepton-flavor universal in the standard model effective field theory formalism up to contributions of O(µ 4 EW /Λ 4 ) [18], we choose a set of values labelled "Min 1b" in table 8 of ref. [20] as our NP benchmark point BP7 BP7: (C V L , C S R , C S L , C T ) = (0.09, 0.086, −0.14, 0.008) We adopt the same treatment as in many literatures (e.g. [54,78,108,117,118]), that is, only the central value of best-fit result is considered as the benchmark point to qualitatively discuss the influence of the NP effect.

Angular observables I i (q 2 )
In figure 2, we show the predictions for the entire set of angular observables I i (q 2 ) within the SM and in eight NP benchmark points. It is easy to see that the BP2 (corresponding to the red band in figure 2) has the greatest effect on all I i (q 2 ) except I 5 (q 2 ). The value of I 5 (q 2 ) in BP2 is almost the same as that in the SM. The NP corresponding to BP2 even makes the angular observables I 6c (q 2 ) negative, which is not present in the SM and in other NP benchmark points.
In the BP6 (corresponding to the blue band in figure 2), the contributions of NP to all I i (q 2 ) except I 5 (q 2 ) are in the same direction as in BP2, but the impacts are smaller than that in BP2. The BP6 can obviously decrease the value of I 5 (q 2 ). Observables sensitive to BP6 can be used to study specific UV models, such as the scalar SU(2) L doublet S 2 (also called R 2 ) leptoquark [32], which can produce the relationship C S L = 4C T at the NP scale.
As we expected, only BP1, BP2, and BP6 which can provide complex phases can produce nonzero angular observables I 7,8,9 (q 2 ). The BP1 (corresponding to the cyan band in figure 2) makes I 5,6c,6s (q 2 ) decrease slightly, and hardly contributes to I 1c,1s,2c,2s,3,4 (q 2 ). The results of all I i (q 2 ) predicted by BP4A and BP4B (corresponding to the purple and yellow bands in figure 2, respectively) coincide almost completely with each other. This indicates that I i (q 2 ) cannot be used to distinguish the two best-fit points of NP hypothesis (C S R , C S L ), which is motivated by models with extra charged Higgs. This is different from the situation in the angular observables of Λ 0 b → Λ + c (→ Λ 0 π + )τ − (→ π − ν τ )ν τ decay, which can distinguish between BP4A and BP4B very well [80]. The BP4A and BP4B make I 1c,2s,5,6c (q 2 ) decrease slightly and I 1s,2c,3,4,6s (q 2 ) increase slightly. The NP effects of BP3, BP5, and BP7 have little impact on I i (q 2 ).

Lepton-flavor-universality ratios R(P J/ψ L,T )(q 2 ) and R(J/ψ)
The q 2 distribution of lepton-flavor-universality ratios R(P J/ψ L,T ) is shown in figure 3, which includes the results within the SM and in eight NP benchmark points. All NP benchmark points except BP5 and BP1 can be distinguished by R(P

Symmetries in the angular observables without tensor operators
In the absence of tensor operators, the twelve angular observables I i (q 2 , E π ) defined in section 2.3 are not independent. These angular observables change to (5.7) 10) We can consider these angular observables as being bilinear in Generally, the experimental and theoretical degrees of freedom can be matched by the following formula [118][119][120][121] n c − n d = 2n A − n s , (5.14) where n c is the number of angular observables I i ; n d is the number of dependencies between the different observables I i , which can be obtained by the difference between the number of observables I i and the dimension of the space given by the gradient vectors ∇I i (with the derivatives taken with respect to the various elements of A); n A is the number of transversity amplitudes (each A j is complex and therefore has two degrees of freedom); n s is the number of continuous symmetries.
Without tensor operators, there are still twelve angular observables I i but only four amplitudes A t,0,⊥, . So n c = 12 and n A = 4. In this case, the only continuous symmetry that can be found is Eqs. (5.16)-(5.20) can be used as a model-independent method to determine the existence of tensor operators. The "model-independent method" here not only means that it does not depend on the NP models, but also means that it does not depend on the calculation of B c → J/ψ transition form factors. Furthermore, we can obtain the dependence relations among the normalized angular observables I i (q 2 ) by replacing the I i (q 2 , E π ) and the dimensionless factors S t,1,2,3 and R t in eqs. (5.16)-(5.22) with I i (q 2 ),S t,1,2,3 andR t , respectively. The factorsS t,1,2,3 andR t are defined, respectively, as (5.23)

Conclusions
Inspired by the R(D ( * ) ) anomalies, the angular distribution of decay has been used to explore possible NP patterns in b → cτ −ν τ transition in many previous works. However, angular observables depending on the solid angle of final-state τ − are unmeasurable theoretically, since the decay products of τ − inevitably contain an undetected ν τ and the solid angle of τ − cannot be determined precisely. Therefore, in this work, we study the measurable angular distribution of the five-body decay B − c → J/ψ(→ µ + µ − )τ − (→ π − ν τ )ν τ , which includes three visible final-state particles µ + , µ − , and π − , with their three-momenta all being measured.
The five-fold differential decay rate containing all NP effective operators can be expressed in terms of twelve angular observables I i (q 2 , E π ), which can be completely expressed by seven independent transversity amplitudes and some dimensionless factors. As

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long as one of the angular observables I 7 , I 8 and I 9 is nonzero, this will be an unquestionable sign of NP, and indicates that the NP can cause extra weak phases. Integrating the five-fold differential decay rate over the E π and normalized by dΓ/dq 2 , we can construct twelve normalized angular observables I i (q 2 ). By integrating over all lepton-side parameters, we find that there are only two angular observables P J/ψ L,T (q 2 ) whose determination can be obtained without reconstruction of the dilepton solid angle. The P J/ψ L,T (q 2 ) are not affected by the lepton dynamics, so they can be used to construct the ratios R(P J/ψ L,T ) to probe the universality of lepton flavor. Based on our five-fold differential decay rate, we show how to extract the complete set of τ asymmetries in B − c → J/ψτ −ν τ decay from the visible final-state kinematics.
Using the B c → J/ψ vector and axial-vector form factors calculated by the latest lattice QCD and the tensor form factors calculated by the QCD sum rule, we predict the q 2 distribution of the twelve normalized angular observables I i and the two lepton-flavoruniversality ratios R(P J/ψ L,T ) both within the SM and in eight NP benchmark points, which are a variety of best-fit points in seven different NP hypotheses. We find that the benchmark BP2 (corresponding to the hypothesis of tensor operator) has the greatest effect on all I i and R(P J/ψ L,T ), except I 5 . Especially for the observables I 6c and R(P J/ψ L,T ), BP2 makes their predictions very different from those in the SM and other benchmark points. The results of all I i and R(P J/ψ L,T ) predicted by the two best-fit points (i.e. BP4A and BP4B) of NP hypothesis (C S R , C S L ), which is motivated by models with extra charged Higgs, coincide almost completely with each other. This is different from the situation in the angular observables of the baryonic counterparts, which can distinguish between BP4A and BP4B very well. In addition to the benchmarks BP2, BP4A, and BP4B, the BP1, BP6, and BP7 can also have some influence on the observables. Compared with the I i , the ratios R(P J/ψ L,T ) are more sensitive to the NP with pseudo-scalar operator. All NP benchmark points can improve the value of R(J/ψ), which makes it closer to the experimental measurement.
We discuss the symmetries in the angular observables without tensor operators, and present five dependence relations. Once all twelve angular observables are measured, these five relations will be a very useful way to determine the existence of tensor operators. If these relations are not fulfilled, it means that there must be tensor operators. This method is completely independent of any assumptions on the details of the NP model and B c → J/ψ transition form factors.
ν τ provides a good prospect for measurement by the LHCb experiment, because it has excellent final-state signatures with a strongly peaking µ + µ − spectrum and a well π − identification. Additionally, the lifetime of B c meson is almost three times shorter than that of B u,d,s mesons, which can be used to improve the separation of B c decay from B u,d,s decays, thus providing an extra handle to distinguish the large background that derives from the B u,d,s mesons [16,53]. One possible background is the very rare B − → π − µ + µ − decay [122], which has about one-tenth the number of events as the signal decay. This background can also be distinguished from the signal by the kinematic properties of the visible final-state particles. Future precise measurements of the angular observables in

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especially precise measurements of the normalized ones, would be very helpful to provide a more definite answer concerning the anomalies observed in b → cτ −ν τ transition, restricting further or even deciphering the NP models.

A The calculation of the angular distribution
In the appendix of ref. [80], we have given the detailed calculation procedures for the similar five-body cascade decay of unpolarized Λ b baryon. The calculation of V * → τ − (→ π − ν τ )ν τ part is exactly the same as that in this work. Therefore, in this section, we mainly present some important definitions and conventions, and calculate the J/ψ → µ + µ − decay. At the end of this section, the dimensionless factors are listed for the sake of completeness of this paper.

A.1 Definitions and conventions
The differential decay rate of cascade decay The λ µ − and λ µ + respectively represent the helicity of final-state particles µ − and µ + , as well as the λ ( ) where µ (λ) denotes the polarization vector of the virtual vector boson V * with helicity λ. The modified leptonic helicity amplitudes are L (λ,λ ) , which can be obtained directly from the appendix of ref. [80].
In the B c rest frame, the polarization vector of J/ψ meson can be written as [123,124] The polarization vector of virtual V * can be written as [123,124]  where η t = 1 and η ±1,0 = −1.
For scalar and pseudo-scalar operators, there is only one nonzero hadronic helicity amplitude For vector and axial-vector operators, there are four nonzero hadronic helicity amplitudes listed as follows For the tensor operators, there are twelve nonzero hadronic helicity amplitudes listed as follows and one can obtain that the total decay rate is

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In the limit κ π → 0, the E π -integrated factorsS i (R i ) ≡ S i (R i )dE π are given, respectively, byS We provide the full analytical results for these factors electronically in the supplementary material.

B The detailed derivation of the dependence relations
In this section we provide the detailed derivation of the dependence relations among the angular observables I i . It is useful to re-express Im[A A * 0 ], Re[A A * 0 ], Re[A t A * ⊥ ] and Im[A t A * ] as advantage of this view is that it implies the continuous symmetry (5.15). There are only seven independent real variables due to the following three relationships Inverting the eqs. (5.2)-(5.11), one can rewrite the variables in terms of the angular observables I i as follows Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.