New physics in the angular distribution of $B_c^- \to J/\psi (\to \mu^+ \mu^-)\tau^- (\to \pi^- \nu_\tau)\bar{\nu}_\tau$ decay

In $B_c^- \to J/\psi (\to \mu^+ \mu^-)\tau^-\bar{\nu}_\tau$ decay, the three-momentum $\boldsymbol{p}_{\tau^-}$ cannot be determined accurately due to the decay products of $\tau^-$ inevitably include an undetected $\nu_{\tau}$. As a consequence, the angular distribution of this decay cannot be measured. In this work, we construct a {\it measurable} angular distribution by considering the subsequent decay $\tau^- \to \pi^- \nu_\tau$. The full cascade decay is $B_c^- \to J/\psi (\to \mu^+ \mu^-)\tau^- (\to \pi^- \nu_\tau)\bar{\nu}_\tau$, in which the three-momenta $\boldsymbol{p}_{\mu^+}$, $\boldsymbol{p}_{\mu^-}$, and $\boldsymbol{p}_{\pi^-}$ 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 $\mathcal{I}_i (q^2, E_\pi)$. Integrating over the energy of pion $E_\pi$, we construct twelve normalized angular observables $\widehat{\mathcal{I}}_i(q^2)$ and two lepton-flavor-universality ratios $R(P_{L,T}^{J/\psi})(q^2)$. Based on the $B_c \to J/\psi$ form factors calculated by the latest lattice QCD and sum rule, we predict the $q^2$ distribution of all $\widehat{\mathcal{I}}_i$ and $R(P_{L,T}^{J/\psi})$ 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 $\widehat{\mathcal{I}}_{i}$ and $R(P_{L,T}^{J/\psi})$, except $\widehat{\mathcal{I}}_{5}$. The ratios $R(P_{L,T}^{J/\psi})$ are more sensitive to the NP with pseudo-scalar operators than the $\widehat{\mathcal{I}}_{i}$. Finally, we discuss the symmetries in the angular observables and present a model-independent method to determine the existence of tensor operators.

Lepton-flavor-universality ratios R(P J/ψ L,T )(q 2 ) and R(J/ψ) 11 5 Symmetries in the angular observables without tensor operators 13 6 Conclusions 15 1 Introduction Exploring new physics (NP) beyond the Standard Model (SM) has been one of the most important tasks in high energy physics, especially since the discovery of Higgs boson [1][2][3].
In recent years, the existence of NP that breaks the universality of lepton flavour in b → cτ − ντ transition has been implied by the anomalous measurements [4][5][6][7][8][9][10][11][12] on B → D ( * ) τ − ντ decays.Moreover, the averaging results performed by the Heavy Flavor Averaging Group (HFLAV) [13] show that the measurements of R(D ( * ) ) ≡ B( B → D ( * ) τ ν)/B( B → D ( * ) ν) nian, we give the analytical results of the independent transversity amplitudes and the measurable angular distribution of the five-body B − c → J/ψ(→ µ + µ − )τ − (→ π − ν τ )ν τ decay.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 − ντ transition 2 + g S (cb)(τ ν τ L ) + g P (cγ 5 b)(τ ν τ L ) + g T (cσ µν (1 − γ 5 )b)(τ σ µν ν τ L ) + H.c., (2.1) 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 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 (2.7) with 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.

Angular distribution
The measurable angular distribution of the five-body B − c → J/ψ(→ µ + µ − )τ − (→ π − ν τ )ν τ decay can be written as where p J/ψ = √ Q + Q − /(2m Bc ) denotes the magnitude of three-momentum of the J/ψ meson in the B c rest frame.B τ ≡ B (τ − → π − ν τ ) and B J/ψ ≡ B (J/ψ → µ − µ + ) are the branching fractions of τ − → π − ν τ and J/ψ → µ − µ + decays, respectively.Here q 2 is the invariant mass squared of the τ − ντ pair; θ J/ψ denotes the polar angle of µ − in the J/ψ rest frame; E π , θ π , and φ π represent the energy, polar angle, and azimuthal angle of π − in the τ − ντ center-of-mass frame, respectively.A more intuitive definition of the angles is shown in figure 1.The function I q 2 , E π , cos θ J/ψ , cos θ π , φ π can be decomposed into a set of trigonometric functions as follows The twelve angular observables I i (q 2 , E π ) can be completely expressed in terms of the seven transversity amplitudes defined in subsection 2.2 and the dimensionless factors listed in appendix A.3.Explicitly, we have (2.28) 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 where 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 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 The q 2 distribution of the decay rate can be obtained by adding up eqs.(3.6) and (3.7) as follows Our dΓ/dq 2 (apart from B τ B J/ψ ) is consistent with that in refs.[27,54].
The eq. (3.11) can be rewritten as with Here, P 0,1,2 (cos θ π ) are the Legendre functions, the ω π and the following κ π and κ τ are defined in eq.(A.39).The τ asymmetries P L , A τ , P ⊥ , Z L , Z ⊥ , Z Q , and A Q are defined in the section 2 of ref. [108].We find that the functions f i are given, respectively, by where cos is the cosine of the π-τ opening angle θ πτ in the τ − ντ center-of-mass frame [80], and (3.17) ) 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].
4 Numerical results

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 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]. 3These 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 B − c → J/ψ(→ µ + µ − )τ − (→ π − ν τ )ν τ decay, 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: ( 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: 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
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
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 J/ψ L,T )(q 2 ), especially in the small q 2 region.The results of R(P J/ψ L,T )(q 2 ) predicted by BP4A and BP4B coincide almost completely with each other.The longitudinal polarization ratio R(P  The angular observables I i (q 2 ) as a function of q 2 , predicted both within the SM and in eight NP benchmark points.The width of each curve is derived from the theoretical uncertainties of B c → J/ψ form factors.
by benchmarks BP2 and BP6, and increased by benchmarks BP4A, BP4B, BP7, and BP3.Especially, the NP effect of BP2 makes the ratio R(P J/ψ L )(q 2 ) significantly less than 1.The transverse polarization ratio R(P J/ψ T )(q 2 ) is increased by benchmarks BP2 and BP6, and Figure 3. Lepton-flavor-universality ratios R(P ) as a function of q 2 , predicted both within the SM and in eight NP benchmark points.The width of each curve is derived from the theoretical uncertainties of B c → J/ψ form factors.

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 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] 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 (5.15) Only 7 of the 12 angular observables I i are independent and 5 dependencies are found.We present the dependence relations directly here and provide the detailed derivation in appendix B: 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 ), St,1,2,3 and Rt , respectively.The factors St,1,2,3 and Rt are defined, respectively, as St,1,2,3 ≡ S t,1,2,3 dE π , Rt ≡ R t dE π . (5.23)

Conclusions
Inspired by the R(D ( * ) ) anomalies, the angular distribution of B − c → J/ψτ − ντ or B − c → J/ψ(→ µ + µ − )τ − ντ decay has been used to explore possible NP patterns in b → cτ − ντ 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 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 − ντ 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 As the J/ψ → µ + µ − decay is dominated by electromagnetic interaction, we can write the helicity amplitude as follows where N J/ψ ≡ −8iπα EM f J/ψ / 3m J/ψ , f J/ψ is the decay constant of J/ψ meson, α EM is the fine-structure constant.There are six nonzero helicity amplitudes as follows and one can obtain that the total decay rate is

A.3 Dimensionless factors
The dimensionless factors induced in the calculation of

Figure 2 .
Figure2.The angular observables I i (q 2 ) as a function of q 2 , predicted both within the SM and in eight NP benchmark points.The width of each curve is derived from the theoretical uncertainties of B c → J/ψ form factors.