Study of semileptonic $\bar{B}^* \to P \ell \bar{\nu}_\ell$ decays

In anticipation of abundant $B^*$ data samples at high-luminosity heavy-flavor experiments in the future, the tree-dominated semileptonic $\bar{B}^*_{u,d,s} \to P \ell^- \bar{\nu}_\ell$ $(P=D\,,D_s\,,\pi\,,K)$ decays are studied within the Standard Model. After a detailed calculation of the helicity amplitudes, the theoretical predictions for branching fraction~(decay rate), lepton spin asymmetry, forward-backward asymmetry and ratio $R_D^{\ast(L)}$ are firstly presented. It is found that the CKM-favored $\bar{B}^* \to D \ell^- \bar{\nu}_\ell$ decays have relatively large branching fractions of ${\cal O}(10^{-9})$$\sim$${\cal O}(10^{-7})$, and are in the scope of running LHC and forthcoming SuperKEKB/Belle-II experiments.


Introduction
The semileptonic B meson decays induced by the tree-level b → p ν (p = u , c) transition provide an ideal ground for testing the Standard Model (SM) and probing possible hints of new physics (NP). For instance, (i) such decays offer ways of extracting the magnitudes of the CKM matrix element V cb and V ub . Moreover, the extractions from exclusive vs. inclusive semileptonic decays exhibit a long-standing ∼ 2.5σ discrepancy [1,2]; (ii) The measurements of ratios R D ( * ) ≡ B(B→D ( * ) τ −ν τ ) B(B→D ( * ) −ν ) ( = µ , e) reported by BaBar [3,4], Belle [5][6][7] and LHCb [8] collaborations exhibit significant deviations from the SM expectations at > 3σ level [9][10][11][12][13], which are the socalled "R D ( * ) puzzles". A lot of efforts have been made for possible solutions within various NP models, for instance, new four fermion operators, two-Higgs-doublet models, R-parity violating supersymmetry models, leptoquark models, Alternative Left-Right Symmetric Model and so on . In addition to B mesons, some other hadrons, such as Λ b and B * , could also decay through b → c ν transition at quark level, and therefore, these decay modes would play a similar role as semileptonic B decays mentioned above.
TheB * q meson with quantum number of n 2s+1 L J = 1 3 S 1 and J P = 1 − is the partner of B meson in the heavy-meson doublet of (bq) system [35][36][37][38]. Its decay occurs mainly through the electromagnetic processB * q →B q γ, and the weak decay modes are generally very rare. Until now, there is no available experimental information forB * q weak decays due to the limited center-of-mass energy and integrated luminosity in the previous experiments of heavy flavor physics. Fortunately, such situation is expected to be improved by the upcoming SuperKEKB/Belle-II experiment [39], which has started test operations and succeeded in circulating and storing beams in the electron and positron rings recently. For instance, using the target annual integrated luminosity 13 ab −1 /year [39], the cross section of Υ(5S) production σ(e + e − → Υ(5S)) = 0.301 nb [40] and the branching fractions of Υ(5S) decays into B * final states [41], one can find that about ∼ 4×10 9 (B * u,d +B * u,d ) and ∼ 2×10 9 (B * s +B * s ) samples could be collected per year, which implies that the B * decays with branching fractions > O(10 −9 ) are possible to be observed by Belle-II.
In addition, the running LHC may also provide some experimental information for B * decays, such as B * s → l + l − decay analyzed in Ref. [42], due to the much large beauty production cross section of pp collision compared with e + e − collision [43][44][45]. Thanks to the rapid devel-opment of heavy flavor physics experiments, the theoretical studies of B * weak decays, which could provide some useful suggestions and references for the measurements, are urgently required. Recently, a few theoretical evaluations of B * weak decays have been done, for instance, the studies of the semileptonic B * c decays within the QCD sum rules [46][47][48], the pure leptonic B * s → andB * u,c → ν decays [42], the impact ofB * s,d → µµ onB s,d → µµ decays [49], and the nonleptonicB * 0 d,s → D + d,s M − (M = π , K , ρ and K * ) decays [50,51]. In this paper, we will pay attention to the charged b → (u, c) ν transitions inducedB * u,d,s → P ν (P = D , D s , π , K) decays within the SM. Especially, theB * → D ν decays are suppressed neither by CKM factors (compared to otherB * decays) nor by loop factors, and thus expected to be observed with relatively large branching fractions.
Our paper is organized as follows. In section 2, the theoretical framework and calculations ofB * → P ν decays are presented in detail. Section 3 is devoted to the numerical results and discussion. Finally, we give our conclusions in section 4.

Effective Hamiltonian and Amplitude
Within the SM, the quark-level b → p −ν (p = u , c and = τ , µ , e) transitions occur through W -exchange and could be described by the effective low-scale O(m b ) Hamiltonian where G F is Fermi coupling constant, and V pb denotes the CKM matrix elements. With Eq. (1), the square matrix element forB * → P −ν decay can be written as in which, leptonic (L µν ) and hadronic (H µν ) tensors are built from the respective products of the lepton and hadron currents.
For convenience of evaluation, H(m, n) and L(m, n) will be calculated in the B * -meson rest frame and the virtual W * rest frame (or −ν center-of-mass frame), respectively.

Kinematics forB * → P −ν Decays
Before the further evaluation, we would like to clarify some conventions and definitions for kinematics ofB * → P −ν decays used in this paper.
In the rest frame of B * -meson with daughter P -meson moving in the positive z-direction, the momenta of particles B * , P and virtual W * could be written respectively as where q 0 = m B * −E P = (m 2 B * −m 2 P +q 2 )/2m B * and | p| = λ 1/2 (m 2 B * , m 2 P , q 2 )/2m B * with function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca) and q 2 being the momentum transfer squared bounded at m 2 ≤ q 2 ≤ (m B * − m P ) 2 . For the four polarization vectors of virtual W * ,¯ µ (λ W * = t, 0, ±), one can conveniently choose [52,53] Meanwhile, the polarization vectors of initial B * -meson could be written as Turning to the −ν center-of-mass frame, the four momenta of lepton and antineutrino are given as where E and | p | are the energy and the magnitude of the three-momentum of the charged lepton, respectively, given by E = (q 2 + m 2 )/2 q 2 and | p | = (q 2 − m 2 )/2 q 2 ; and θ is the angle between the P and three-momenta. In this reference frame, the polarization vectors of virtual W * take the form

Hadronic Helicity Amplitudes
describes the decay of three helicity states of B * meson into a pseudo-scalar P meson and the four helicity states of virtual W * . In Eq. (10), H µ (λ B * ) represents hadronic matrix elements of the vector and axial-vector currents within the SM. For B * → P transition, they are described by four QCD form factors V (q 2 ) and A 0,1,2 (q 2 ) through with the sign convention 0123 = −1.
Then, by contracting above hadronic matrix elements with the B * and W * polarization vectors given by Eqs. (6) and (7), we obtain four non-vanishing helicity amplitudes It is obvious that only the amplitudes with λ B * = λ P − λ W * = −λ W * survive 1 .

Helicity Amplitudes and Observables ofB * → P −ν Decays
Following the strategy of Refs. [9,52,56], one can expand the leptonic tensor in terms of a complete set of Wigner's d J -functions. As a result, L µν H µν is reduced to a very compact form where J and J run over 1 and 0, λ ( ) W * and λ run over their components, and λν = 1 2 . One may note that the non-diagonal interference contribution appears between the states of J = 1, λ W * = 0 and J = 0, λ W * = t, but it has no contributions to the differential decay rate d 2 Γ/dq 2 after integrating over cos θ, which can be seen from the following Eq. (22).
The h λ ,λν in Eq. (16) are the leptonic helicity amplitudes in the −ν center-of-mass frame, and given by where λ W * = λ − λν . The cases λ = −1/2 and 1/2 are referred to as the non-flip and flip transitions, respectively. Taking the exact forms of the spinors and polarization vectors, we finally obtain two nonvanishing contributions which have exactly the same expressions as the one gotten in semileptonic B and hyperon decays [9,56].
By now, the basic building blocks of amplitudes have been obtained. Then, we present the observables considered in our following evaluations. The double differential decay rate of B * → P −ν decay could be written as where the factor 1/3 is caused by averaging over the spin of initial stateB * . Further, using the standard convention for d J -function [41], we finally obtain the double differential decay rates with a given helicity state (λ = ± 1 2 ), which are Using Eqs. (21) and (22), one can get the explicit forms of various observables ofB * → P −ν decays.
Performing the integration over cos θ and summing over the lepton helicity, we obtain the singly differential decay rate from which the integrated decay rates, the branching fractions and the ratios defined by R * P (q 2 ) ≡ dΓ(B * →P τ −ν τ )/dq 2 dΓ(B * →P −ν )/dq 2 ( = µ , e) are easily to be obtained. In addition, picking out the H 2 00 and H 2 0t terms in Eq. (23), one also can get the singly differential longitudinal decay rate dΓ L /dq 2 , as well as R * L P (q 2 ), which are sensitive to the NP contributions of a charged scalar [22]. Besides the decay rate, there are also two important observables, the lepton spin asymmetry and the forward-backward asymmetry, which are defined as respectively. In Eq. (24), the polarized differential decay rates dΓ[λ = ±1/2]/dq 2 are obtained after integration over cos θ of doubly differential ones given by Eqs. (21) and (22). Explicitly, we obtain For A P θ (q 2 ), again using Eqs. (21) and (22) and summing over the lepton helicity, we arrive at the explicit expression The lepton spin asymmetry A λ is very sensitive to the NP corrections, and therefore, has been widely studied in B → D * ν decays within various NP scenarios. However, unfortunately, the lepton polarization can not be measured directly in the high energy experiments due to the lack of effective technology and method. For the case of τ lepton, its polarization could be determined in principle through analyzing the full angular distribution of τ subsequent decay, but it is not very easy. Moreover, such way is not suitable for the case of light leptons (µ and e). It is hoped that the theoretical researches on A λ could motivate the development of the experimental technology and approach.

Input Parameters
Before presenting our predictions forB * → P −ν decays, we would like to clarify the input parameters used in our numerical evaluations. For the CKM matrix elements, we use the fitted results |V cb | = 41.80 +0.33 −0.68 and |V ub | = 3.714 +0.072 −0.060 given by CKMFitter Group [2]. For the wellknown Fermi coupling constant G F and the masses of mesons and leptons, we take the averaged values given by PDG [41].
In order to evaluate the branching fractions, the total decay widths (or lifetimes) Γ tot (B * ) are essential. Due to the facts that there is no available experimental and theoretical information for Γ tot (B * ) at present and the electromagnetic processes B * → Bγ dominate the decays of B * mesons, we take the approximation Γ tot (B * ) Γ(B * → Bγ) in our evaluations of branching fraction. The theoretical predictions on Γ(B * → Bγ) have been given in many different theoretical models [57][58][59][60][61][62][63]. In this paper, we will take the most recent results [62,63] Besides, the transition form factors are also essential inputs, but no ready-made results could be used at present. In this paper, the Bauer-Stech-Wirbel (BSW) model [64,65] is employed for evaluating the form factors. Within the BSW framework, the form factors A 0,1,2 (q 2 ) and V (q 2 ) Table 1: The values of form factors A 0,1,2 (0) and V (0) within BSW model.
where p ⊥ is the transverse quark momentum. With the meson wave function ϕ( p ⊥ , x) as solution of a relativistic scalar harmonic oscillator potential [64],  Table 1.
To be conservative, in our following evaluation, we assign 15% uncertainties to these values. Moreover, with the assumption of the nearest pole dominance, the dependences of form factors on q 2 are explicitly written as [64,65] where B p (J P ) is the state of B p with quantum number of J P (J and P are the quantum numbers of total angular momenta and parity, respectively). In addition, it should be noted that, instead  Table 3: Predictions for q 2 -integrated observables A P λ ,θ ( = τ ) and R * (L) P .

Theoretical Prediction and Discussion
With the input values and the formula given above, we then present our theoretical predictions and discussion. In Table 2, we summarize the predictions of branching fractions, in which  Table 3, in which the theoretical uncertainties are caused by the form factors only. In Figs. 1 and 2, the q 2 -dependence of differential decay rates dΓ (L) /dq 2 and A P λ ,θ , R * (L) P are shown, respectively. The following are some discussions: (1) Compared withB * (s) → D (s) −ν decays,B * (s) → π(K) −ν decays are suppressed by both an additional factor λ and the relatively small form factors. Therefore, the branching fractions ofB * (s) → π(K) −ν decays are expected to be much smaller than the ones of correspondingB * (s) → D (s) −ν decays by a factor of O(10 −1 ) ∼ O(10 −2 ), which can be seen from Table 2.
Moreover,B * → D −ν decay modes are also expected to be measured by LHC experiments, which can be seen from the following rough analysis. Here, we take the possible measurement ofB * 0 → D + −ν decay at LHCb as an example. Firstly, it is expected that about 2 × 50/3 × 3.63 × 10 5 = 1.21 × 10 7B0 → D * + µ −ν µ decay events will be found after LHCb upgrade due to the facts that (i) using the data corresponding to integrated luminosities of 1.0 fb −1 and 2.0 fb −1 collected at pp center-of-mass energy √ s = 7 and 8 TeV, respectively, 3.63 × 10 5B → D * + µ −ν µ decay events have been found by LHCb collaboration [8]; (ii) After high-luminosity upgrade, a data sample of 50 fb −1 will be collected by LHCb collaboration at a much higher √ s = 14 TeV, which will results in a further enhancement of bb production by a factor about 2 [44,68]. Secondly, one can assume that the most of B mesons detected at LHC are mainly produced through B * → Bγ decay because B * mesons are often produced by about 3 times more than the B mesons, which has been confirmed by the measurements at Z 0 peak by LEP [69]. Finally, further To distinguish the possible NP hints, it will become important to control the theoretical uncertainties as well as possible. From our predictions for R * (L) D given in Table 3, as expected, one may find that the uncertainty caused by the hadronic factors is significantly reduced compared to the decay rates. Moreover, when the range of q 2 integration is the same in the numerator and the denominator of R * D , the cancellation of the nonperturbative error further improves, allowing for more precise predictions of the ratio of partial rates [70,71]. Numerically, for instance, choosing the q 2 integration range [m 2 τ , q 2 max ] for both numerator and denominator, we get (4) For the lepton spin asymmetry and the forward-backward asymmetry, our numerical results are listed in Table 3. Similar to R * (L) D , because of the cancellation of the hadronic errors between numerator and denominator, the theoretical uncertainties are significantly small compared with the branching fraction. Regarding their differential distributions, which are shown in Figs. 2 (c) and (d), a characteristic feature is the zero-crossing point, which is usually used to distinguish the NP effects from the SM, or different NP scenarios.
Numerically, we get that A P λ (q 2 ) and A P θ (q 2 ) cross the zero point respectively at q 2 = 3.4 GeV and 5.8 GeV for P = D, and q 2 = 4.0 GeV and 6.2 GeV for P = π , K.

Summary
The B * weak decays are legal within the Standard Model, although their branching ratios are tiny compared with the electromagnetic decays. In this paper, motivated by abundant B * data samples at high-luminosity heavy-flavor experiments in the future, we have studied the tree-dominated semileptonicB * u,d,s → P −ν (P = D , D s , π , K and = τ , µ , e) decays within the Standard Model. The helicity amplitudes are calculated in detail, and the predictions of observables including branching fraction (decay rate), lepton spin asymmetry, forwardbackward asymmetry and ratio R * (L) D are firstly presented in Tables 2, 3