Angular analysis of $\bar{B}\to D_2^*(\to D \pi)\ell \bar{\nu}$ decay and new physics

We derive the four-fold angular distribution for the semileptonic decay $\bar{B}\to D_2^*(\to D \pi)\,\ell \bar{\nu}$ where $D_2^*$(2460) is a tensor meson. We start with the most general beyond the Standard Model (SM) dimension-six effective Hamiltonian which comprises (axial)vector, (pseudo)scalar and tensor operators for both quark and lepton currents, and it also includes the right-handed neutrinos. The decay can be described by 16 transversity amplitudes and it provides a multitude of observables which can be extracted from data. We investigate the observables in the context of the SM and the new physics scenarios which can explain the intriguing discrepancies observed in the $b \to c \tau \bar{\nu}$ data.


Introduction
In the absence of any clean signal of beyond the Standard Model (SM) particle, the effective theory analysis has become one of the primary directions to pursue. In a situation where the scale of new physics (NP) might be quite higher and thus the direct production remains awaited at the colliders, higher dimensional operators can capture its effect. It is a historical fact that several discoveries in particle physics were preceded by indirect evidence through quantum loop contributions. The decays of B mesons are the important probes for such searches. While the fully hadronic decay modes are subject to large and, in cases, not-so-well understood strong interaction corrections, the situation is much more under control for semileptonic decays.
Among such, the b → c ν modes are of special interests. In the SM, this decay proceeds through a tree level W boson exchange and thus is not suppressed as compared to flavor-changing neutral current transitions. In spite of being a charged current channel, some intriguing hints of discrepancies have been observed by several experimental collaborations. The ratios of branching fraction (BR) are particularly clean probes of physics beyond the SM, due to the cancellation of the leading uncertainties inherent in individual BR predictions, defined as only one left-handed vector current four-fermion operator. However, physics beyond the SM can contribute via operators with same and/or different Lorentz structures. Thus we start with the most general dimension-six beyond the SM effective Hamiltonian for b → c ν transition where the four-fermion operators are defined for M, N ∈ {L, R} as The Wilson coefficients C X M N = 0 in the SM and they encode the short distance physics which can be generated by heavy NP mediators. The neutrino oscillation experiments confirm the tiny mass of neutrinos which in turn implies the neutrinos are not purely left-handed and hence we also include the right-handed neutrinos in Eq. (3). It can be shown using Fierz rearrangements, that for M = N , the tensor operators vanish identically.
where µ T (h) = µν (h)q ν /m B with q µ = (p B − p D * 2 ) µ being the momentum transfer. It can be shown that theB → D * 2 matrix element for the scalar current vanishes and the pseudoscalar current reduces to Next the tensor operators are parameterized with the well-known form factors T i defined as To expressB → Dπ matrix elements in terms ofB → D * 2 form factors we assume the D * 2 decays resonantly. This allows us to use the narrow-width approximation for the D * 2 propagator as following.
We can write the hadronic matrix elements in Eqs. (9)-(12) as where A µα contains theB → D * 2 form factors. With the effective Lagrangian describing the we obtain the total decay width Γ D * 2 = Using the standard expression for the sum over polarization tensor we get the desired matrix element where With the framework defined above, we are now in a stage to compute the differential distribution for theB → D * 2 (→ Dπ) −ν decay.

Angular distribution and observables
In this section, we derive the full four-body angular distribution of the semileptonic decaȳ on the mass shell. This process is completely described by four independent kinematic variables. These kinematic variables are the lepton-pair invariant mass squared q 2 = (q 1 + q 2 ) 2 , and the three angles φ, θ and θ D . The angles θ and θ D are defined as follows: assuming that the D * 2 has a momentum along the positive z direction in the B rest frame, θ D is the angle between the D and the +z axis and θ is the angle of the − with the +z axis. The angle φ is the angle between the decay planes formed by −ν and Dπ. Squaring the matrix element, summing over spins of the final state particles and using the kinematical identities given in App. A we obtain the differential decay distribution of B → D * 2 (→ Dπ) −ν as The angular coefficients I i are functions of tranversity amplitudes given by The transversity amplitudes are the projections of the total decay amplitude into the explicit polarization basis. In the SM, the decayB → D * 2 −ν can be described by total four transversity amplitudes. Notice from Eqs. (7) and (8) that µν (±2)q ν = 0 in the B-rest frame, implying only three states of polarization contribute to the considered decay. As a result we have four transversity amplitudes corresponding to one longitudinal (A 0 ), two transverse (A ⊥, ) direction and a time-like amplitude (A t ) for the virtual vector boson decaying to lepton-antineutrino pair. However, inclusion of right-handed neutrinos distinguish the left-and right-chirality of the leptonic current and we get total eight amplitudes A L,R 0,⊥, ,t . Now in presence of the NP operators given in Eq. (3), the new (axial)vector contributions can be incorporated in the above mentioned eight transversity amplitudes modified with new Wilson coefficients, however, the (pseudo)scalar and tensor operators induce eight further (four for each chirality of leptonic current) amplitudes. These are two (pseudo)scalar amplitudes A L,R P and six for the tensor operators A L,R T 0,T ⊥,T . Thus with the most general effective Hamiltonian (Eq. (3)), the four-body decay can be described by total sixteen tranversity amplitudes. Note that we have suppressed the q 2 dependence in the angular coefficients and as well as in the transversity amplitudes and will continue to do so for simplicity.
Defining the normalization factor and by introducing the following notation for the NP Wilson coefficients we write the expressions for the transversity amplitudes arising from (axial)vector operators as The (pseudo)scalar and tensor amplitudes can be defined as The amplitudes A L,R t and A L,R P arise in a combination and hence we define The CP -conjugate mode B →D * 2 (→ Dπ) + ν can be described by replacing the angular coefficients I i in Eq. (18) withĪ i which differs by the sign flip in weak phase. Due to the change in the definition of relative angles θ l → θ l − π and φ → −φ in the amplitudes the differential distribution d 4Γ dq 2 d cos θ D d cos θ l dφ for the conjugate mode can be written with the following substitutions.
The richness of the angular distribution is such that by performing a fit to data, each of these angular coefficients for both the mode and its conjugate mode can be extracted at experiments. This however requires more statistics and next we define several observables which individually can be accessed directly from data without going into the full fit procedure.
The differential distribution w.r.t q 2 can be obtained by integrating all three angles cos θ , cos θ D and φ as dΓ and in absence of any direct CP violation, dΓ/dq 2 ≡Γ f = Γ f .
Integrating two angles at a time in Eq. (18), generates the uniangular differential distributions. The distribution in cos θ D looks as Here F L,T are the longitudinal and transverse polarization fractions for the D * 2 meson defined as respectively, which satisfy F L + F T = 1. Similarly, for the CP -conjugate mode, one can havē F L,T which are equal to F L,T , respectively, when CP violation is absent. Now the distribution in the angle φ has much simpler form given as where one can easily extract out the coefficients of cos 2φ and sin 2φ terms from data. By considering the similar distribution for the CP -conjugate mode, we define two CP -averaged asymmetry A 3 and A 9 as Integrating φ and cos θ D in Eq. (18), we get The well-known CP -averaged forward-backward asymmetry A FB is defined conventionally as, Contributions from I 4 and I 5 in Eqs. (24) and (25) are extracted by the two angular asymmetries, (55) We further define two observables A 7 and A 8 which are vanishing in the SM limit or in other words, in real amplitude limit. These asymmetries are nonzero only if NP introduces complex contribution to the amplitude. Similar holds true for the asymmetry A 9 as well. (57)

Model independent phenomenology with New Physics
In this section, we perform a phenomenological study for the observables defined in the preceding section. The form factors forB → D * 2 have been calculated in literature [14,15]. In Ref. [14] a subset of form factors are estimated within three-point QCD sum-rule approach and a recent analysis [15] extends the previous work by providing results for the full set ofB → D * 2 transition form factors (including the tensor form factors), up to twist-four accuracy of B-meson LCSR as well as incorporating the finite virtual quark mass effects. We adopt the formalism developed in Ref. [15] where the extrapolation of form factors from the calculated LCSR input points (q 2 0 GeV) to larger q 2 values is performed by a simple pole form with z-expansion where z(s) The values for the fit parameters α F 0,1 are extracted in Ref. [15] and the masses of resonances associated with the quantum numbers of the respective form factor F are taken from Ref. [18].
Only with the information about the form factors, we can easily predict the observables (discussed in the previous section) in the SM as all the Wilson coefficients C X M N are vanishing in this limit. The dilepton invariant mass square varies from m 2 We obtain the bin-averaged value for observable O obs (q 2 ) defined as in different q 2 bins and the SM predictions for each observables with ±1σ uncertainties in the form factors and quark masses are shown in Table 1 for both the muon and tau mode. We divide the kinematically allowed region into eight(five) q 2 bins, each with ∼ 1 GeV 2 range, for B → D * 2 µ(τ )ν modes, respectively.  In Fig. 1 we show the variation of these observables, in the entire q 2 range, in red bands for the SM prediction ofB → D * 2 τν mode. Note that the forward-backward asymmetry A FB has a zero crossing in the q 2 axis. It implies there exists a relation between the form factors involved in the matrix elements in the SM limit at A FB (q 2 0 ) = 0 which is given by With the estimates of the central values of the form factors we find q 2 0 = 6.0 GeV 2 . As it can be seen from Eq. (53) that the above relation originates from the cancellation between I c 6 and 2I s 6 terms, such cancellation is absent inB → D * 2 µν mode due to the low mass of muon compared to tau. Interestingly in presence of the NP operators the relation will be modified with the Wilson coefficients C X M N and the tensor form factors T i (q 2 ) and will have a bit lengthy form. However the relation becomes simpler with NP contribution (with real Wilson coefficients) to only (axial)vector operators which can be written as Hence we infer that if the zero crossing point of A FB is measured in future, it will provide important information about the NP Wilson coefficients.
Next we test the sensitivity of the observables in presence of NP contributions. The intriguing discrepancies in the b → cτν transition have predicted non-zero value(s) for one and/or several NP Wilson coefficients. We follow one of the most recent model independent analysis where global fit to the general set of Wilson coefficients of an effective low-energy Hamiltonian (with only left-handed neutrinos) is performed [19]. As a benchmark scenario, we consider the 'Min 4' from Table 6 of [19], where χ 2 minimization to R(D ( * ) ), D * longitudinal polarization fraction F D * L , binned q 2 distributions for B → D ( * ) τν data is presented for five NP coefficients. The bound from BR(B c → τν) ≤ 10% has also been imposed in the fit. We consider the central values of the fitted Wilson coefficients and using Eq. (32) the combination of coefficients entering in the tranversity amplitudes are found to be The observables with the above mentioned NP coefficients are shown in solid green bands in Fig. 1 where all other Wilson coefficients are assumed to be vanishing.
It has been mentioned in Ref. [19] that there is a degeneracy between the set of Wilson coefficients and a sign-flipped minimum exists corresponding to the same minimized χ 2 value. For such case, the values in Eq. (62) alter to In Fig. 1 we highlight the variation of observables corresponding to this mirror minima in dashed green bands. It can be seen that the difference in two NP scenarios (which are indistinguishable in terms of minimization) is quite prominent for the helicity fractions F L , F T and asymmetries A 5 and A FB . However, A 3 and A 4 show alteration only near the kinematic endpoint.
It should be noted that the angular observables A 7 , A 8 and A 9 depend on the imaginary part of the transversity amplitudes and therefore are vanishing in the SM. Only NP contributions with complex Wilson coefficients can make them finite and hence measurements of non-zero values of these observables will be clean signal of NP contributing to this decay mode.

Summary and discussion
In this paper, we have explored the semileptonic decayB → D * 2 (→ Dπ) ν, where D * 2 is a tensor meson with a mass and decay width of 2460 MeV and ∼ 48 MeV, respectively. We start with the most general beyond the SM effective Hamiltonian in dimension-six operator basis which comprises (axial)vector, (pseudo)scalar and tensor operators for quark as well as lepton currents. We also include the right-handed neutrinos in the effective Hamiltonian.
The further decay of D * 2 → Dπ states allow us to derive the full four-fold angular distribution. The entire decay can be expressed in terms of sixteen transversity amplitudes while in the SM limit the number of transversity amplitudes simply reduces to four. We find the NP contribution to (axial)vector operators can be incorporated in the SM amplitudes modified with the NP Wilson coefficients, however, the (pseudo)scalar and tensor operators induce four new transversity amplitudes for each chirality of the lepton current.
A multitude of CP -averaged observables are constructed from the full differential distribution. These are the helicity fractions of the D * 2 meson F L , F T , the forward-backward asymmetry A FB and six angular asymmetries A i , i ∈ {3, 4, 5, 7, 8, 9} which can be extracted at experiments. Among these, three asymmetries A 7,8,9 depend only on the complex part of the decay amplitude and hence are vanishing in the SM as well as real NP Wilson coefficient limit.
Next we predict the bin-averaged values of the observables for several q 2 bins forB → D * 2 (→ Dπ)µν andB → D * 2 (→ Dπ)τν channels in the SM. We also illustrate the behavior of the observables in the entire kinematical allowed range in presence of NP contributions which can explain the intriguing discrepancies observed in b → cτν transition. By using the results from the latest global fit to all the relevant observables, we find in some observables e.g., F L , F T , A 5 and A FB , the effects arising from the NP contributions are quite prominent. We show the zero crossing of the forward-backward asymmetry A FB can also provide important information about the NP coefficients.
The results derived in this work are not only restricted to the particular decay but can also be applied to other channels with tensor meson in the final state likeB s → D * 2s (→ DK) ν where D * 2s (2573) is a 2 + state. The experimental sensitivity to perform a fit to the full four-fold distribution for extracting each angular coefficients for these modes are subject to the statistics, however, we hope the observables constructed in this work can be tested at the LHC and Belle-II experiments in near future.