Footprints of New Physics in the angular distribution of $B_{c}\to D_{s}^{\ast}(\to D_{s}\gamma,(D_{s}\pi))\ell^{+}\ell^{-}$ decays

We investigate the angular decay distribution of the four-fold $B_{c}\to D^{\ast}_{s}(\to D_{s}\gamma)\mu^{+}\mu^{-}$, and $B_{c}\to D^{\ast}_{s}(\to D_{s}\pi)\mu^{+}\mu^{-}$ decays that proceed through $b\to s\mu^{+}\mu^{-}$ quark level transition. We use the model independent effective Hamiltonian with vector and axial vector new physics operators to formulate the angular observables and study the implications of different latest new physics scenarios, taken from the global fits to all the $b\to s$ data, on these observables. We also give Standard Model and new physics predictions of several observables such as differential branching ratios, forward backward asymmetry, longitudinal polarization fraction of $D_s^{\ast}$, and the unpolarized and polarized lepton flavor universality violating ratios. Future measurements of the predicted angular observables, both at current and future high energy colliders, will add to the useful complementary data required to clarify the structure of new physics in $b\to s\ell\ell$ neutral current decays.


Introduction
High Energy Physics community has put a lot of effort over the past decade in searching the new physics (NP) via exclusive decays of B meson based on flavor changing neutral (FCNC) transitions, in particular b → sℓ + ℓ − mode. These FCNC transitions occur only at loop level in the Standard Model (SM) and hence provide a fertile ground to investigate NP as well as the SM parameters. B → K * (→ Kπ)µµ, and B → Kµµ decays and their angular distributions have been studied in great detail at the LHCb experiments [1][2][3][4][5][6][7]. From the angular distributions of such decays, new set of observables have been constructed which are free from the Cabibbo-Kobayashi-Maskawa (CKM) uncertainties, and therefore furnish a complementary way to diagnose the status of NP [8]. However the main hindrance to chalk out the status of NP via angular observables are the hadronic uncertainties. There is an improvement in controlling the uncertainties in the hadronic matrix elements of local quark operators and in few cases the uncertainty is about 10%. On the other hand, the matrix element of the non-local quark operators appearing from the coupling of charmonium states, remains a daunting task to ( * ) s µ + µ − decay observables have been investigated model independently with various 1D and 2D NP scenarios whereas the authors of Ref. [49] analyzed the B c → D ( * ) s µ + µ − , and B c → D ( * ) s νν decays in a Z ′ and leptoquark models.
In this work, we use the model independent effective Hamiltonian in the presence of only vector and axial vector NP operators and perform the four-fold angular analysis of B c → D * s (→ D s γ, D s π)µµ decays using the relativistic quark model (RQM) form factors in the low energy q 2 range. For the decay channels D * s → D s γ the probability is 93%, and the probability of the channel D * s → D s π is 5%. As our NP extensions cater both new vector and axial vector couplings, therefore for the NP scenarios, we choose the best fit values of NP couplings in different 1D and 2D scenarios, from the recent global fit analysis [42]. We give the predictions of different physical observables such as differential branching fractions, forward-backward asymmetry, longitudinal helicity fraction of D * s meson, lepton flavor universality violating (LFUV) ratios, when D * s meson is longitudinally and transversely polarized and the individual angular observables within the SM and in different NP scenarios.
The organization of the paper is as follows. In section 2, we start with the general effective Hamiltonian, for b → sµµ transition, in the presence of vector and axial vector NP operators after which we express the matrix elements in terms of form factors for B c → D * s µ + µ − decay. Further, the helicity formalism is followed by the expressions of the helicity amplitudes, angular coefficients and the physical observables for the decay B c → D * s (→ D s γ, D s π)µ + µ − . In section 3, we present the phenomenological analysis of all the observables, in the SM and the NP scenarios, and section 4 concludes our discussion.

Theoretical Framework
In this section, we present the effective Hamiltonian which is used to compute the full angular distribution of B c → D * s → (D s γ, D s π)µ + µ − decays. We give the expressions of the helicity amplitudes and express all the angular coefficients in terms these helicity amplitudes. Using the full form of the four fold angular decay distribution, we can extract the q 2 dependent angular coefficients, which will be used to analyze the effects of various 1D and 2D NP scenarios.

Effective Hamiltonian and Decay Amplitude of
The most general low energy effective Hamiltonian for rare |∆B| = |∆S| = 1 transition, in the presence of new vector and axial vector operators is written as [48], where G F is the Fermi coupling constant, V ij are the CKM matrix elements. The expressions of the dipole operators O 7 (′) , and the semileptonic operators O 9 (′) ,10 (′) are given as, where e (g s ) is the electromagnetic (strong) coupling constant, and m b in O 7 (′) , is assumed to be the running b−quark mass in the MS scheme. O i ′ are the chirality flipped operators. Within the SM, contributions of O 7 ′ operator are suppressed by m s /m b , therefore we neglect them and further we do not consider NP scenarios with radiative coefficients C NP 7 (′) as they are well constrained [50]. Moreover, for the present study, we have ignored the non-factorizable contributions such as the long distance charm-loop corrections in the effective Hamiltonain, although they are expected to be significant at large recoil.
In Eq.(1), C i (µ) are the corresponding Wilson coefficients at the energy scale µ. The expressions of the C eff 7 (q 2 ) and C eff 9 (q 2 ) Wilson coefficients [51][52][53][54][55][56], that contain the factorizable contributions from current-current, QCD penguins and chromomagnetic dipole operators O 1−6,8 are explicitly given in appendix A. Using the above effective Hamiltonian, the amplitude for the B c → D * s ℓ + ℓ − decay in the framework of SM as well as NP can be written as, where where T i,D * s µ , i = (1, 2), contain the matrix elements of B c → D * s .

Matrix Elements for
The hadronic matrix element for B c → D * s µ + µ − can be parameterized in terms of form factors as follows, where P µ = p µ + k µ , q µ = p µ − k µ , and with A 3 (0) = A 0 (0). We have used ϵ 0123 = +1 convention throughout the study. The additional tensor form factors are expressed as,

Helicity Formalism of
For B c → D * s µ + µ − decay, the amplitude can be expressed in terms of helicity basis. For kinematics of the four-body decay (see Fig. 1), we closely follow Ref. [57], where detailed formalism is given. The completeness and orthogonality properties of helicity basis can be expressed as follows, with g nl = diag(+, −, −, −). From the completeness relation given in Eq. (11), the contraction of leptonic tensors L (k)αβ and hadronic tensors H ij , can be written as where the leptonic and hadronic tensors can be written in the helicity basis as follows Both leptonic and hadronic tensors shown in Eq. (13), can be evaluated in two different frame of references. The lepton tensor L (k) nl is evaluated in µ + µ − centre of mass (CM) frame, and the hadronic tensor H ij nl is evaluated in the rest frame of B c meson. For the said decay one can write the hadronic tensor as follows, where, from angular momentum conservation, r = n and s = l for n, l = ±, 0 and r, s = 0 for n, l = t. The explicit expressions of the helicity amplitudes for B c → D * s , are obtained in terms of the SM and NP Wilson coefficients as, 2.4 Four-fold Angular Distribution of B c → D * s (→ D s γ(D s π))ℓ + ℓ − Decays In an effective theory, the NP effects are due to the Wilson coefficients and new operators given in Eq. (1). For the decay modes B c → D * s (→ D s γ(D s π))ℓ + ℓ − , these effects are contained in the four-dimensional differential decay distribution that depends on the square of the momentum transfer q 2 , angles θ ℓ , θ V , and ϕ as described in Fig. 1. For the decays under consideration, the full differential angular distribution can be written as, where I γ nλ,⊥ and I nλ,∥ are the angular coefficients. The explicit expressions of I γ nλ,⊥ in terms of the helicity amplitudes are obtained as, whereas the expressions of I nλ,∥ in terms of the helicity amplitudes are written as, where with λ ≡ λ(m 2 Bc , m 2 D * s , q 2 ) and β l = 1 − 4m 2 l /q 2 .

Physical
Observables for B c → D * s (→ D s γ(D s π))ℓ + ℓ − Decays In this section, we construct the physical observables for the B c → D * s (→ D s γ(D s π))ℓ + ℓ − decays, in terms of the angular coefficients. The observables which we consider are the differential branching ratios (dB/dq 2 ), lepton forward-backward asymmetry (A FB ), longitudinal polarization fraction of D * s (f L ), unpolarized (R D * s ), and polarized (R L,T D * s ) LFUV ratios, and the angular coefficients (⟨I γ nλ,⊥ ⟩, ⟨I nλ,∥ ⟩). Other than the differential decay rates and the ratios, all observables are normalized to the corresponding differential decay rate. (i) Differential decay rates: From the full angular distribution Eq. (16), q 2 dependent differential decay rate expressions are obtained in terms of angular coefficients as follows, (ii) Lepton forward backward asymmetry: The lepton forward backward asymmetry as a function of q 2 can be expressed as, (iii) Longitudinal helicity fraction: The longitudinal helicity fraction of the decay B c → D * s (→ D s γ, (D s π))µ + µ − , when D * s meson is longitudinally polarized can be expressed as, (iv) LFUV ratios for B c → D * s ℓ + ℓ − Decay: The unpolarized and polarized LFUV for the decay B c → D * s ℓ + ℓ − can be written as, (v) Normalized angular observables: (vi) Binned normalized angular observables: 3 Phenomenological Analysis

Input Parameters
To investigate NP effects in the observables of the B c → D * s (→ D s γ, (D s π))ℓ + ℓ − decays, we use input parameters such as the transition form factors, which are calculated in the framework of the relativistic quark model (RQM) [58]. The RQM, based on the quasipotential approach, reliably determines the form factors in the whole q 2 range by incorporating relativistic effects including contributions of intermediate negative energy states and relativistic transformations of the meson wave functions. Furthermore, the form factors obtained in the RQM, satisfy all the model independent symmetry relations arising in the limits of heavy quark mass and large recoil of the final meson [59]. The form factors calculated, in the RQM, through the overlap integrals of the initial and final meson relativistic wave functions [58], can be expressed in terms of the expressions involving fitted parameters. Such as, the form factors, V (q 2 ), A 0 (q 2 ) and T 1 (q 2 ) given in Eqs. (6,7), and (9), are parameterized in the whole kinematical q 2 region as, where the form factor A 0 (q 2 ) contains a pole at q 2 = M 2 ≡ M 2 Bs and the form factors V (q 2 ), The numerical values of these pole masses are M Bs = 5.36692 GeV, and M B * s = 5.4154 GeV [60]. Moreover, the form factors A 1 (q 2 ), A 2 (q 2 ), T 2 (q 2 ), and T 3 (q 2 ), given in Eq. (7), and Eq. (10), can be parameterized as follows, For completeness, the numerical values of form factors at q 2 = 0, and fitted parameters σ 1 and σ 2 , are collected in Table 1. In order to gauge the effects of the form factor uncertainties on various observables, we allow the parameters in the fitted form factors to deviate by ±5%.
The numerical values of Wilson coefficients in the SM, evaluated at the renormalization scale Table 1: The numerical values of transition form factors for B c → D * s µ + µ − decay at q 2 = 0, and the fitted parameters σ 1 and σ 2 [58]. The reported uncertainties represent the ±5% deviations in the parameters of the fitted form factors.  [61], are presented in Table 2. The other input parameters are listed in Table 3.

NP Scenarios
In this section, we first specify our choice of the NP scenarios which are used to investigate the effects of NP on various physical observables in the angular distribution of We choose the best fit values of NP couplings in different scenarios, from the recent global fit analysis [42]. The global fit analysis performed by authors of Ref. [42], shows that with the assumption of NP present only in the muon sector, two 1D NP scenarios C NP 9µ < 0, and C NP 9µ = −C NP 10µ , continue to be the most favored scenarios, whereas the 2D scenarios (C NP 9µ , C NP 10 ′ µ ), (C NP 9µ , C NP 9 ′ µ ) and (C NP 9µ , C NP 10µ ) provide better fit to data with preference decreasing in the listed order. The best fit values of these 1D and 2D NP scenarios are listed in Table-4.   Table 4: Best-fit values of the 1D and 2D NP scenarios considering NP in muon sector only [42].

Scenario
Best-fit value S1 C

Analysis of Physical Observables in
In this section, we now analyze the NP effects via observables which are constructed from the combination of the angular coefficients, such as the differential branching ratios (dB/dq 2 ), lepton forward-backward asymmetry ( Binned averaged numerical values of the SM and NP predictions of all these observables, with errors due to the form factors, in different q 2 bins, are given in Tables 5-11, of appendix B. In Figs. 2-3, we plot differential branching ratios, forward-backward asymmetry, and the longitudinal polarization fraction of D * s , as a function of q 2 , leading to following observations. Moreover, our results regarding to LFUV ratios, in all NP scenarios, do not show sizable deviations from the SM predictions, therefore, we do not present their q 2 plots. • In Fig. 2(a) and 2(b), we have plotted the differential branching ratios for B c → D * s (→ D s γ)µ + µ − and B c → D * s (→ D s π)µ + µ − decays as a function of q 2 in the framework of the SM as well as the NP scenarios under consideration. In SM, the differential branching ratio 2. × 10 -9 3. × 10 -9 4. × 10 -9 5. × 10 -9 6. × 10 -9 Figure 2: Differential branching ratio for the decay B c → D * s µ + µ − in the SM and the NP scenarios. (a) depicts the differential branching ratio for the cascade decay B c → D * s (→ D s γ)µ + µ − , and (b) depicts the differential branching ratio for the cascade decay Fig. 2(a) and 2(b), depict that NP scenarios predictions are compatible with the SM predictions as the error bands emerging due to the uncertainties of the form factors overlap. However, the central value predictions of all the NP scenarios show trend towards the lesser values of differential branching ratios as compared to the SM expectations.
• Fig. 3, depicts the forward-backward asymmetry (A FB ), and longitudinal helicity fraction (f L ), of B c → D * s µ + µ − decay as a function of q 2 , in the SM framework as well as NP scenarios presented in Table 4. Regarding the zero position of the A F B , it is important to mention here that the uncertainty due to the form factors is small and hence the A FB provides stringent tests to see the NP effects. Fig. 3(a) shows that the zero position of A FB shifts towards right for all the NP scenarios. The zero crossing in the A FB at q 2 = 3 GeV 2 , q 2 = 2.8 GeV 2 , and q 2 = 3.4 GeV 2 in the presence of NP scenarios (S1, S5), S2, (S3, S4), respectively is quite distinct from the SM prediction at q 2 = 2.6 GeV 2 .
• Another physical observable useful to investigate the structure of NP is the longitudinal helicity fraction of the final state meson (f L ). In Fig. 3 helicity fraction f L for B c → D * s µ + µ − decay as a function of q 2 . We can recognize from Fig. 3(b), that the given NP scenarios in the longitudinal helicity fraction of D * s can be distinguished quite easily in the region q 2 = (1 − 2.5) GeV 2 , and all the NP predictions point out lesser values of f L , compare to the SM. However, for q 2 > 2.5 GeV 2 the given NP scenarios overlap with each other.
• Fig. 5, depicts the angular coefficients ⟨I γ 5,⊥ ⟩, ⟨I γ 6s,⊥ ⟩, and ⟨I γ 6c,⊥ ⟩ as a function of q 2 both in the SM and in 1D and 2D NP scenarios. For ⟨I γ 5,⊥ ⟩ (cf. Fig. 5(a)), the value of zero crossing in the SM is q 2 ≈ 2.5 GeV 2 . The deviation of the zero crossing of ⟨I γ 5,⊥ ⟩ arises in the case of scenarios S2, S3, S4, and S5, but scenario S1 is not distinguishable. For the angular coefficients ⟨I γ 6s,⊥ ⟩ and ⟨I γ 6c,⊥ ⟩, there is a shift in zero crossing compared to that of SM, with distinct zero crossing points for scenarios S2, S4, and S5. However, the scenarios S1 and S3 are not much distinguishable.

Conclusions
Study of rare semileptonic decays of B meson gives us a path to investigate physics beyond the SM. In literature various exclusive semileptonic decays mediated by the flavor changing neutral current transitions and flavor changing charged current transitions show reasonable deviations from the SM predictions. As various global fit analyses suggest the presence of NP, in different physical observables of B → (K, K * )µ + µ − decays, in terms of the fit values of the NP coupling, we analyze the implications of these NP scenarios onto the angular observables of the complementary four-fold B c → D * s (→ D s γ, (D s π))µ + µ − decays, which are governed by the same quark level transition. Using the effective Hamiltonian by incorporating the vector and axialvector NP operators (O 9 , O 9 ′ , O 10 , O 10 ′ ), we have derived the four-fold angular distributions for B c → D * s (→ D s γ)µ + µ − , and B c → D * s (→ D s π)µ + µ − decays from which the individual angular coefficients and various physical observables can be extracted. To analyze the NP effects, in these observables, we use the best fit values of the Wilson coefficients coming from the global fit analysis with the assumption of NP present only in the muon sector.
To summarize our work, we have observed sizeable difference between the NP predictions of different physical observables and the angular coefficients for B c → D * s (→ D s γ)µ + µ − and B c → D * s (→ D s π)µ + µ − decays, compared to the SM expectations. The NP results of the differential branching ratios for the considered decays indicate decreased values compared to that of the SM, however due to large error bands coming from the errors due to the form factors, NP results remain compatible with the SM estimates. Considering the forward-backward asymmetry and the longitudinal helicity fraction of D * s meson, a number of NP scenarios can be distinguished from the SM predictions as well as from each other, in some kinematical ranges. For the unpolarized and polarized LFUV ratios i.e. R D * s and R L,T D * s , our analysis shows that there is no sizeable deviations expected from the SM prediction. Furthermore, the NP analysis considering the individual angular coefficients also shows sizeable deviations from the SM predictions along with distinct predictions for different NP scenarios. Hence the precise measurement of the studied physical observables for B c → D * s (→ D s γ)µ + µ − , and B c → D * s (→ D s π)µ + µ − decays at LHCb and the future collider experiments will give useful complementary information, required to clarify the structure of new physics in b → sℓℓ decays.

B Binned Predictions of Physical Observables
In this appendix, we give the SM as well as NP predictions of physical observables in different q 2 bins.   Table 4. The listed errors arise due to the uncertainties of the form factors.  Table 6: Same as in Table 5, but for q 2 = [1.0 − 2.0] GeV 2 bin.  Table 7: Same as in Table 5, but for q 2 = [2.0 − 3.0] GeV 2 bin.  Table 8: Same as in Table 5 Table 9: Same as in Table 5, but for q 2 = [4.0 − 5.0] GeV 2 bin.   Table 5, but for q 2 = [5.0 − 6.0] GeV 2 bin.  Table 11: Same as in Table 5, but for q 2 = [1.0 − 6.0] GeV 2 bin.
GeV 2 bin, for the SM as well as the NP scenarios presented in Table-