Predictions of Angular Observables for $\bar{B}_s\to K^{\ast}\ell\ell$ and $\bar{B}\to \rho\ell\ell$ in Standard Model

Exclusive semileptonic decays based on $b\to s$ transitions have been attracting a lot of attention as some angular observables deviate significantly from the Standard Model (SM) predictions in specific $q^2$ bins. B meson decays induced by other Flavor Changing Neutral Current (FCNC), $b\to d$, can also offer a probe to new physics with an additional sensitivity to the weak phase in Cabibbo-Kobayashi-Masakawa (CKM) matrix. We provide predictions for angular observables for $b\to d$ semileptonic transitions, namely $\bar{B}_s\to K^{\ast}\ell^+\ell^-$, $\bar{B}^0\to\rho^0 \ell^+\ell^-$, and their CP-conjugated modes. For $\bar{B}^0\to\rho^0 \ell^+\ell^-$ mode, $B^0-\bar{B}^0$ mixing effects have been included and predictions are made for Belle and LHCb separately. Study of these decay modes will be useful in its own right and to understand the pattern of deviations in $b\to s$ transitions.


INTRODUCTION
In recent years, a lot of attention has been given to semileptonic decays of bottom hadrons as a result of increasing experimental evidence of new physics. Many decays have been observed involving the FCNC b → s + − and charged current b → c ν. Most reliable measurements include R K ( * ) [1,2] and R D ( * ) [3][4][5][6][7] which hint towards lepton flavor universality (LFU) violation. These measurements are important for precision tests of the standard model as well as for searches of new physics.
Albeit there exist rich data for b → s + − induced processes, the b → d counterpart of the weak decay, i.e. b → d + − , has not caught much attention perhaps because of low branching ratio. At the quark level, the lowest order contribution arises at one loop level through diagrams similar to b → s which include box diagrams and electroweak penguin diagrams. The weak phases incorporate CKM matrix elements ξ i q = V * qi V qb , where q ∈ {u, c, t} and i ∈ {s, d}. For b → s transition, ξ s c,t ∼ λ 2 and ξ s u ∼ λ 4 where λ = 0.22. Since uū contribution introduces CKM phase which is negligible for b → s , CP violating quantities are very small in SM. On the other hand, since ξ d u ∼ ξ d c ∼ ξ d t ∼ λ 4 for b → d , the B decays mediated through this transition allow for large CP violating quantities. Also, leading order contribution in this case is smaller than the leading contribution in b → s which makes this channel more sensitive to new physics. Hence, it is desirable to study processes like B → {π, ρ} + − and B s → {K,K * } + − experimentally as well as theoretically. The first transition of this variety to be measured is B → π + − by LHCb with 5.2σ significance [8] which is in good agreement the expected value in * bharti@prl.res.in † nmahajan@prl.res.in SM [9,10]. Other than this B 0 (B 0 s ) → π + π − µ + µ − has also been observed by LHCb, where the muons in final state do not originate from a resonance [11]. In this paper, we focus on two decay modes: B → ρµ + µ − and B s →K * µ + µ − . Predicted value of branching ratio for B 0 → ρ 0 + − is of the same order as B → π + − , thus making it possible to be measured with upgraded experimental facilities. Since experiments already have good measurements of B → K * + − mode, it is likely that B s →K * + − mode may get early attention. The branching ratio of this mode is expected to be a factor of two more that B 0 → ρ 0 + − owing to the factor of 1/ √ 2 in the definition of ρ 0 ∼ (uū − dd)/ √ 2 as compared to K * ∼sd. Further, neglecting SU(3) breaking effects, one expects the branching fractions to be O(λ 2 ) smaller than those in b → s + − decays. Predictions for certain observables including branching ratio, direct CP asymmetry, and forward-backward asymmetry have been given for B → ρ + − [14][15][16][17]19]. For B s →K * µ + µ − , only branching ratio has been studied based on relativistic quark model [13] and light cone sum rules (LCSR) based on heavy quark effective theory (HQET) approach [12]. However, a complete study of angular distribution is lacking. We aim to fill the gap in this paper.
Phenomenological analysis of the decays induced by this channel will provide complementary information about the nature of New Physics (NP). The most prevailing problem in the theoretical description is due to the long distance effects of cc and uū resonant states. In the q 2 region close to these resonances, only model dependent predictions are available which result in large uncertainty. To avoid these uncertainties, we restrict our study to a region which is well below J/ψ resonance region ∼ 6 GeV 2 .
The paper is organised as follows. In section II we present the effective Hamiltonian for a general semileptonic B decay mediated by b → d transition at the quark arXiv:1803.05876v3 [hep-ph] 15 Nov 2018 level, and pseudoscalar to vector transition at hadronic level. In section III, we discuss the form factors and inputs used to obtain numerical results. In section IV, we list all the observables considered in this paper. Results of these observables are given in section V followed by a summary in section VI.

II. DECAY AMPLITUDE
The theoretical description of semileptonic B decays is based on the effective Hamiltonian approach in which heavy degrees of freedom, i.e. top quark and gauge bosons, are integrated out. This approach allows the separation of short and long distance effects which are encoded in Wilson coefficients C i and effective operators O i respectively. The effective Hamiltonian for b → d + − transitions, within SM, is expressed as [20]: where, G F is the Fermi constant, ξ q = V * qd V qb are the CKM factors and, where, the dominant contribution is due to the following operators: Here, T a represents generators of SU (3) group. In new physics scenario, operators other than given in Eq. (2) can also contribute significantly. The full operator basis for the effective Hamiltonian can be found in [20]. Unitarity of CKM matrix has been utilized to obtain Eq. (2). H t eff term contains the contribution of tt and cc quark-antiquark pair in the loop, while H u eff represents contribution of cc and uū pair in the loop. C i 's are the Wilson coefficients and are calculated at scale µ = m W and expressed as a perturbative expansion in the strong coupling constant α s (µ W ): The Wilson coefficients have been worked out in [21][22][23][24][25][26] upto next-to-next-to leading order (NNLO). They are then evolved down to scale µ = m b using renormalization group equations which is again expressed as a series with α s /(4π) as expansion parameter. This step requires a calculation of anomalous dimension matrix γ(α s ) upto three-loop level to compute C i (µ b ) (2) which has been computed in [27]. To the NNLO approximation, Wilson coefficients are given as [23], π ω 9 (ŝ))A 10 (5c) whereŝ = q 2 /m 2 b is the momentum squared of the lepton pair normalized to squared mass of b quark. One major difference, as compared to b → s + − transition, is the presence of a large ξ u /ξ t term in the Wilson coefficients. In b → s transition, this term is negligible in comparison to other terms and can be conveniently neglected. This implies that the Wilson coefficients, say C eff 9 , receive a small imaginary contribution. But for b → d case, the imaginary part is quite significant.
Spectator Scattering: The intermediate quark loop (up or charm) or chromomagnetic operator (O 8g ) can emit a hard gluon which can be absorbed by the spectator quark. The T q,spec a for spectator scattering are obtained from Eqs. (20,22) of [22] and Eqs. (49,50) of [17].
Weak Annihilation: The b quark in the B meson can annihilate with the spectator quark to give the meson in the final state. This contributes to the leading order term in α s as the QCD correction to weak annihilation is highly suppressed. The non-zero contribution of weak annihialtion (T q,WA ) is given by Eqs. (46)(47)(48) of [17].
Soft-gluon correction: Quark in the intermediate loop can emit a soft gluon which contributes to nonfactorizable correction. The contribution is proportional to 1/(4m 2 c − q 2 ), and rises near the vector resonances. Hence, it can be calculated in the region [2 − 6]GeV 2 . In the region beyond that, hadronic dispersion relations are employed which systematically includes the contribution of charm resonance [55]. For B → K * the contribution of these two effects, soft-gluon emission and charmonium resonances, can be described by the parameterization defined in Eqs. (8)(9)(10)(11)(12) of [18] which is valid in the q 2 region [1 − 9]GeV 2 . Such corrections have not been explicitly computed for B → ρ or B s →K * . However, flavour SU(3) symmetry would imply that these corrections can be assumed to be roughly same as B → K * , which are  given by, ∆C 0,soft 9,c (q 2 ) = where, the mean values of parameters are given in Table  II. However, the expressions for soft-gluon emission from the up loop are still absent and need to be computed properly. Though the corresponding expressions exist for B → π mode but they can not be naively used for the present purpose. For current study, we are assuming an uncertainty of ∼ 10% in C 9 to account for this missing piece: where, |a| ∈ {0, 0.5} and θ ∈ {0, π}. The evaluation, particularly the sign, of this correction requires a complete LCSR calculation which is beyond the present work. The impact of these contributions doesn't turn out to be very significant except for one or two observables. However, to be complete and to indicate possible effect of these corrections, we include them in our numerical study.
These corrections are added systematically in transversity amplitudes which are given in Appendix A.

III. FORM FACTORS
The matrix elements corresponding to operators O 7,9,10 are expressed in terms of seven form factors which are functions of q 2 : where q µ = (p − k) µ , P µ = (p + k) µ , and c V = 1/ √ 2 in the case ofB 0 → ρ 0 ; 1 forB s → K * and B ± → ρ ± . Form factors can be calculated using the method of QCD Sum Rules on Light-Cone (LCSRs) in the low-q 2 region. For semileptonic B decays, the method involves calculation of correlation function of the weak currents involving b quark, evaluated between the vacuum and light meson in the final state. The correlation function is factorized into non-perturbative and processindependent hadron distribution amplitudes (DAs), φ, convoluted with process-dependent amplitudes T .
where, n represents twist. The contributions with increasing twist decreases by increasing powers of virtualities of the currents involved (∼ m 2 b in the low q 2 range). We follow [31] for form factors of B → ρ and B s → K * hadronic decays, which provides an improved determination of B → V form factors compared to those in [32]. In [31], updated values of hadronic parameters are used and contributions upto twist-5 in DAs have been systematically included. Further, making use of equations of motion, it is shown that the uncertainties in the ratios of form factors are reduced and so does the dependence on mass scheme. Another advantage is that the combined fits to sum rules and lattice calculations at low and high q 2 are given which provides form factors valid over the whole range. In this paper, we call the updated form factors as BSZ (Bharucha-Straub-Zwicky) form factors while those in [32] as BZ (Ball-Zwicky) form factors.
The form factors are written as a series expansion in terms of the parameter [31], Form factors are parameterized as: where m R,i is the resonance mass which is equal to 5.279 Gev for A 0 (s), 5.325 GeV for T 1 (s) and V (s), and 5.724 GeV for rest of the form factors.
Below, we provide detailed SM prediction employing BSZ form factors , computed using LCSRs, which we will refer as BSZ1 form factors in this paper. To compare the numerical impact of the improved form factors, we also provide a direct comparison with results obtained using BSZ form factors with lattice and LCSR results combined together (referred as BSZ2 form factors in this paper), and BZ form factors, in the case ofB s → K * + − . While forB 0 → ρ 0 + − , we use BSZ form factors (LCSR) only, since combined fit with lattice results are not available for this mode.

IV. OBSERVABLES
For a four body decay, B → V (→ M 1 M 2 ) + − , the decay distribution can be completely described in terms of four kinematic variables; the lepton invariant mass squared (q 2 ) and three angles θ V , θ l , and φ. The angle θ V is the angle between direction of flight of M 2 with respect to B meson in the rest frame of V , θ is the angle made by − with respect to the B meson in the dilepton rest frame and φ is the azimuthal angle between the two planes formed by dilepton and M 1 M 2 . The full angular decay distribution of B → V (→ M 1 M 2 ) + − is given by [20], where, Here, V is an intermediate vector meson which decays to M 1 and M 2 whereas + − can be any lepton pair.
The corresponding angular decay distribution (d 4Γ /(dq 2 dcosθ π dθ dφ)) for the CP-conjugated process,B →V (→M 1M2 ) + − , is obtained from Eq. (15) with the replacement, .Ī i is equal to I i with the weak phase, i.e. CKM phase in this case, conjugated. The functions I i can be written in terms of transversity amplitudes [20]. In the b → s transition, since the Wilson coefficients are effectively real, modulo a small imaginary part coming due to function h(m 2 , s) in C eff 9 ,Ī i are essentially I i and observables sensitive to imaginary part of I i are rather small within SM. This is not the case in b → d induced decays and we see this feature explicitly in the results below. Various observables are constructed from Eq. (15) by integrating over angles in various range. These observables are generally plagued with large uncertainties due to form factors. To avoid this, a lot of work has been done to construct observables which are theoretically clean in low-q 2 region [33][34][35][36][37][38][39]. Such observables are free from this dependence at the leading order and are called form factor independent (FFI) observables. Those which have a form factor dependence in the leading order are called form factor dependent (FFD) observables. We study both classes of observables in this paper, as discussed below. We shall see below, SU(3) breaking effects are clearly visible in some of the observables.
• FFD observables are (which have been experimentally studied in the context of B →K * [40]): where, dΓ dq 2 is the dilepton spectrum distribution, A F B (q 2 ) is the forward-backward asymmetry and F L (q 2 ) is the fraction of longitudinal polarization of the intermediate vector meson. Similar observables are constructed for the CP-conjugate process using the decay distribution d 4Γ /(dq 2 dcosθ π dθ dφ) discussed above.
• FFI observables or "clean observables" are independent of form factors in the leading order of 1/m b and α s thus exhibiting low hadronic uncertainties and enhanced sensitivity to new physics. Much attention has been given to the construction of such observables and some of them have been measured experimentally [41,42]. We consider following set of FFI observables here: • We also consider observables analogous to R K * for which the form factor dependence cancels exactly for B s →K * , defined as: where, numerator and denominator are integrated over Observables defined in Eqs. (17,18,19) are valid for B s →K * + − decay mode. It has been pointed out in literature that the zeroes (value of q 2 where observables is zero) are also clean observables [43,44]. Also, the relation between zeroes of different observables provide crucial tests of Standard Model. Thus, we also provide values of zeroes of different observables. Observables for the CP-conjugate decay,B s → K * + − are also defined in the same way, with the substitution I i →Ī i ≡ ζ iĨi . Results for B s (B s ) →K * (K * )µ + µ− are given in the next section which can be compared with data collected at LHCb as well as Belle.
However, for B 0 → ρ 0 + − , results corresponding to LHCb and Belle have to be computed separately. Since ρ 0 → ππ, which is not a flavor specific state, the observables are affected by B 0 −B 0 oscillations and the expressions of angular functions (I i s) defined in Eq. (15), are modified. These modified, time-dependent functions have been computed in [45] and given as, where, x = ∆m/Γ, y = ∆Γ/Γ,J i ≡ ζ i J i , and the additional functions (h i and s i ) arise because of the mixing in B 0 meson system. This leads to two types of quantities, time-dependent observables and time-integrated observables. In this work, we consider observables which include time-integrated angular functions over a range t ∈ [0, ∞) in the case of LHCb and t ∈ (−∞, ∞)(in addition to exp(−Γt) → exp(−Γ|t|)) at Belle [46,47]. After time-integration, the modified angular functions are given by, where, represents time-integrated quantity. Other difference at LHCb and Belle arises due to the fact that flavor of the meson can be tagged using flavor-specific decays at Belle. Thus, flavor of the meson decaying to the final state is known at time t = 0 and the appropriate angular function (J i orJ i ) can be used. On the other hand, there is no method to determine the flavor of meson at t = 0 at LHCb. As a result, the measured quantity at LHCb is dΓ( which is a CP-averaged quantity for i ∈ {1, 2, 3, 4, 7} and CP-violating quantity for i ∈ {5, 6, 8, 9}. Due to the difference in the method of measurement, we consider different observables to be studied at LHCb and Belle. For B 0 → ρ 0 + − at Belle, the definition of observables (say, O) in Eqs. (17,18,19)  Even though tagging power at LHCb is low, new algorithms have been suggested to improve the tagging power by 50% [48,49]. Thus, for completion we also give pre-dictions for observables which can be measured at LHCb using tagging of B mesons. The definition of these observables is again given by O Tagged . Moreover, having measurements of angular distribution with and without tagging can be of phenomenological importance [53].

V. RESULTS
In this section, we present observables as a function of q 2 and their binned values over two q 2 ranges: [0.1-1] GeV 2 and [1-6] GeV 2 and consider the di-muon pair in the final state. ForB s → K * µ + µ − , we provide the results obtained using three sets of form factors. As discussed earlier, we mainly employ the BSZ(LCSR) form factors and compare the results obtained using BZ form factors and BSZ form factors (LCSR+Lattice results). For B 0 → ρ 0 µ + µ − , we present results using the BSZ form factors only.
We first discuss the observables without the inclusion of various non-factorizable corrections. After discussing the main results, we shall return to thr discussion of the impact of these corrections. In Table IV, we give values of observables for B s →K * µ + µ − andB s → K * µ + µ − corresponding to the form factor set BSZ1, BSZ2, and BZ. For B(B) → ρµ + µ − , the values of angular observables are given in Table V for the form factor set BZ. The errors in the binned values are due to errors in form factors as given in [31,32].
As we mentioned above, due to ξ u /ξ t term in the present case, which is practically negligible in the case of b → s transition, the observable for the mode and the CP conjugated mode show clear differences and hence is a clear sign of CP violation. A precise measurement would determine whether the amount of CP violation is in conformity with the CKM picture or there are extra phases present. The observable P 6 is of particular interest in this regard as it is proportional to an imaginary part of Wilson coefficients. It can be noted that its value in low q 2 is significantly different for CP-conjugate modes, giving large value of CP asymmetry. Table IV, it is easy to note that different choices of form factors yield values for FFI observables that are reasonably close to each other while for FFD observables, like branching ratio, the impact is significant and there is a larger spread in the predictions. Compar-ingB 0 → ρ 0 µ + µ − withB s → K * µ + µ − , effects of strange quark versus up/down quark is apparent in many observables. Along with the observables discussed in the previous section, we report branching ratio for B 0 → ρ 0 µ + µ − and B s →K * µ + µ − over the full kinematically allowed range. For B 0 → ρ 0 µ + µ − , time-integrated branching ratio within SM is found out to be,   From the definition of observables given in Appendix B it is clear that the measurable quantity is actually timeintegrated branching ratio normalized by decay rate. For results given in Eq. (26), we have taken the mean value of decay rate to be Γ B 0 = 6.579 × 10 11 s −1 [50]. Thus, the actual observable dΓ/dq 2 Belle defined in section B is obtained by multiplying the results in Eq. (26)

From the
Branching ratio ofB s → K * µ + µ − in SM using BSZ form factors based on LCSR calculation is, For branching ratio in full kinematic range, form factors based on LCSR are not much reliable as they are valid in low-q 2 region only while the kinematic range extends upto ∼ 20 GeV 2 ((M Bs − M K * ) 2 ). Hence, we also give below values of branching ratio using form factors obtained from combined fits of lattice and LCSRs results.
We now study the impact of various corrections stemming from the four quark operators. The factorizable corrections are already included in the definition of C eff 9 and C eff 7 to NNLO. The non-factorizable ones i.e., weak annihilation, spectator scattering, and soft gluon emission are systematically included for predictions in the bin [1 − 6]GeV 21 . As mentioned before, the contribution of soft gluon emission from the up quark loop is not available at present. A very rough estimate leads us to include 10% uncertainty in C eff 9 due to this particular correction. The crucial issue here is not just the rough magnitude but also the sign and thus without a proper LCSR based calculation, this is the best one can do. In Tables VI, VIII, IX, we present the value of the observables with these corrections included. In these tables, the first error is due to the form factors while the second shows the spread due to soft gluon emission from the up quark loops. These are presented for the BSZ2 set of form factors. It is found that the inclusion of these corrections has significant impact on observables like P 5 , branching ratio, and A F B . This confirms the broad pattern observed in B → K * µµ. with and without these corrections is more meaningful and reliable for [1 − 6] GeV 2 bin as for q 2 < 1GeV 2 , the soft gluon contribution tend to be very large. The observables are also plotted as function of q 2 as shown in Fig. 1,2,3. The values of the zeroes are given in Table  VII,X. Since the error due to soft gluon emission from up quark is very small, we only show the error due to form factors in the value of zeroes 2 .
A potentially important missing piece is the inclusion of finite width effects, especially relevant for B → ρ modes. Since, ρ 0 → ππ width is large, it must be taken into account. In [60], an attempt is made to include these effects as a part of form factors. However, these effects are computed only for vector and axial vector form factors while no calculation exists for tensor form factors. These effects could be large and must be evaluated.
To the best of our knowledge, this is the first dedicated study in this direction. The b → d mediated modes bring along several interesting features due to intrinsic CP violating phase within SM with a non-negligible contribution. We have provided q 2 dependence as well as binned values of angular observables, which can be directly compared once the data is available. LHCb's result for B s →K * µ + µ − [52] announced recently is consistent with our result of branching ratio. Precise measurements of various angular observables will lead to complimentary information to b → s mediated decays.

Appendix A: Transversity Amplitudes
The non-factorizable corrections discussed in Section can be added to transversity amplitudes or the Wilson coefficient C eff 9 . Following [22,55], we add the corrections in the following way:  where, where P ≡B,B s and V ≡ ρ, K * . The values of input parameters used to calculate the corrections are given in Tables I and III.

Appendix B: Observables
In this section, we explicitly write the definitions of observables considered while giving predictions for B → ρµ + µ − process.

Tagged
The observables for B 0 → ρ 0 µ + µ − corresponding to tagged events which can be measured at LHCb and Belle are defined as, The observables for the CP-conjugate decayB 0 → ρµ + µ − are obtained by replacing J i byJ i (≡ ζ iĪi ) in Eqs. (B1-B6). These definitions are common for observables at LHCb and Belle, but the definitions of angular functions are different for the two cases. For Belle, the functions J i andJ i used are time integrated functions obtained from Eq. (22) and given as, while for LHCb, the angular functions are given by,

Untagged
For untagged events, the observables for B 0 → ρ 0 µ + µ − are defined as, . (B13g) In section B 1, all the J i andJ i are time integrated functions. Similarly in section B 2, the combination J i + J i are time integrated functions and the symbol is suppressed.
ACKNOWLEDGEMENT Authors thank S. Mohanty for making the computational resources of his TDP projects available for a large part of numerical analysis carried out in this paper. Authors also thank J. Virto, T. Gershon, and T. Blake for their useful comments and suggestions.