Semileptonic $B$ and $B_s$ decays involving scalar and axial-vector mesons

We report our theoretical calculations on the branching fractions for the semileptonic $B$ and $B_s$ decays, i.e., $B (B_s) \to (P,\, V,\, S,\,A) \ell \nu_\ell$, where $P$ and $V$ denote the pseudoscalar and vector mesons, respectively, while $S$ denotes the scalar meson with mass above 1 GeV and $A$ the axial-vector meson. The branching fractions for the semileptonic $B\to P$ and $V$ modes have been measured very well in experiment and our theoretical values are in good agreement with them. The ones for $B\to S$ and $A$ modes are our theoretical predictions. There is little experimental information on the semileptonic $B_s$ decays although much theoretical effort has been done. In addition, we predict the branching fractions of $B\to D^*_0(2400) \ell \bar\nu_\ell$ and $B_s\to D^{*-}_{s0}(2317) \ell \bar\nu_\ell$ as $(2.31\pm 0.25)\times 10^{-3}$ and $(3.07\pm0.34)\times 10^{-3}$, in order, assuming them as the conventional mesons with quark-antiquark configuration. The high luminosity $e^+e^-$ collider SuperKEKB/Belle-II is running, with the data sample enhanced by a factor of 40 compared to Belle, which will provide huge opportunity for the test of the theoretical predictions and further help understand the inner structure of these scalar and axial vector mesons, e.g., the glueball content of $f_0(1710)$ and the mixing angles for the axial-vector mesons. These decay channels can also be accessed by the LHCb experiment.


I. INTRODUCTION
The CP violation is one of the necessary Sakharov conditions for the emergence of matterantimatter asymmetry [1], which is a key question in the nature. The Cabibbo-Kobayashi-Maskawa (CKM) matrix [2] has been an indispensable skeleton of the Standard Model (SM), which successfully describes the CP violation in the quark sector. In the CKM matrix, the unitarity relation |V ub | 2 + |V cb | 2 + |V tb | 2 = 1 is fulfilled, and any deviation of this unitarity constraint will be a signal of New Physics. Thus the precise determination of the CKM matrix elements has been a key task and activity in the community of flavor physics [3,4].
A recent review for the leptonic and semileptonic B decays is compiled in Ref. [5]. In our current work, we will consider B(B s ) → (P, V, S, A)ℓν ℓ decay 1 , where P and V denote the pseudoscalar and vector mesons, respectively, while S denotes the scalar meson with mass above 1 GeV and A the axial vector meson. In these processes, either |V ub | or |V cb | is involved. And notably, there is a mismatch for the extraction of |V ub | from the inclusive and exclusive decays, which is the so-called |V ub | puzzle [3,9]. The measurement of these channels can certainly help for rendering more information on the determination of CKM matrix elements, at least as a supplement to the conventional exclusive decays. Inversely, to calculate the branching fractions, we rely on their concrete values from Particle Data Group (PDG) [3]: |V cb | = (42.2 ± 0.8) × 10 −3 , and |V ub | = (3.94 ± 0.36) × 10 −3 as a combination of the determinations from inclusive and exclusive decays.
The mixing angles are not fully fixed yet, see e.g., Ref. [10]. The structure of the scalar meson is more obscure, see the review [11]. For example, f 0 states (f 0 (1370), f 0 (1500), f 0 (1710)) are interpreted as the mixed states of qq, ss and glueball (G), but which one consists mainly of G is not fully determined 2 . The observables depend on, or even are sensitive to these mixing angles. The three-body semileptonic decay is an ideal place to study the weak hadronic transition form factor as well as the underlying structure of such mesons 1 We will consider B + as an example, and B 0 decay is the same as B + case due to the isospin symmetry. This point can be verified by the branching fractions B(B + →D 0 ℓν ℓ ) = B(B 0 → D − ℓν ℓ ) = (2.20 ± 0, 10)% [3], [3] due to the additional factor of 1/ √ 2 appearing in the quark components in the neutral π and ρ. We also note that these channels could also be treated in the PQCD or light-cone models, e.g., in Refs. [6][7][8]. 2 We also note that the knowledge of the two-photon couplings to the scalars is helpful to understand their structures [12].
3 due to the absence of the final-state interactions (FSIs) between hadrons 3 . As such, the theoretical prediction of the relevant semileptonic decay channels becomes crucial for the future experimental measurement.
From the experimental point of view, Belle has accumulated huge data samples, that can be exploited to measure the branching fractions and hadronic transition form factors for the various semileptonic decay channels, in order to test or constrain the various theoretical models. There are (772±11)×10 6 BB [15] and (6.53±0.66)×10 6 B sBs pairs [16] collected at Υ(4S) and Υ(5S) resonances, respectively, by the Belle detector at the KEKB asymmetric energy electron-positron collider. The statistics will be enhanced by a factor of 40 for Belle-II, and by the mid of next decade, 50 times more data is expected comparing to the Belle experiment. Our predicted branching fractions are typically in the order of 10 −5 so that they can be, in principle, easily accessed by the Belle/Belle-II and LHCb experiments.

II. THEORETICAL FRAMEWORK
The transition form factor is a probe to the inner structure of the hadron. Among the various theoretical tools for the form factor, we will concentrate on the application of the light-front quark model (LFQM) [17,18] and its covariant extension (CLFQM) [19], see also [20][21][22]. A distinct feature of the light-front frame is that the diagrams involving quarks created out of or annihilating into the vacuum can be eliminated, i.e., only the valence quarks are considered in the meson or baryon [23]. This leads to a relativistic quark model which retains the qq structure for a meson. The relevant form factors can be extracted by choosing the plus component of the matrix elements in the LFQM. In fact, there is spurious contribution proportional to the lightlike four-vector ω = (2, 0, 0 ⊥ ) in transforming the covariant Feymann integral into the light-front form, which makes the theory non-covariant.
The covariance requires inclusion of the zero-mode effect which eliminates the undesired ω dependence. Such development is elaborated in Ref. [19]. In this way, all the form factors that are necessary to represent the Lorentz structure of a hadronic matrix element can be calculated on the same footing, which is not possible in the standard LFQM. In the framework of CLFQM, the vertex function of a meson (bound state) coupling to its constituent quarks consists of the momentum part and also the spin part, where the former 3 The proton-antiproton FSI has been elaborately examined in various decay channels [13,14]. 4 describes the momentum distribution of the constituent quarks, and the latter is constructed from the light-front helicity state involving the Melosh transformation. In the vertex wave function, there is a free parameter β, that will be fixed by the decay constant of the meson.
The fermion line is just represented by the relativistic propagator. The electroweak vertex is given by the Standard Model. Following the line of Ref. [19], Cheng, Chua and Hwang have systematically studied the decay constants and form factors for the S-and P -wave mesons in 2003 [24], while an update was done in Ref. [25] in two points: i) The experimental information of the branching fractions wherever available or the lattice results for the decay constants was used to constrain the parameters β in the wave functions; ii) the extension to the counterpart with s quark (D s and B s ) has also been considered.
All the form factors for B(B s ) → (P, V, A, S) transitions considered by us have been calculated in Refs. [24,25], and thus we omit to repeat these calculations. However, we note that to make a direct comparison between theory and experiment, we should further provide the branching fractions which are the true observables in experiment and can be directly accessed to test our theoretical predictions. This constitutes one of our main results.
Similar studies have been done for the case of charmed meson, D and D s decay [26], where the formalism corresponding to the differential decay rates and branching fractions are explicitly given. Those expressions are certainly applicable to the B and B s decay with only some replacements of the relevant masses. While Ref. [26] has aroused great interest of BES colleagues and some of our results have been confirmed, a natural question is what will happen in the beauty B and B s cases. Combining the running Belle-II, the predictions for the branching fractions of various channels are important for our experimental colleagues.
The future measurements will provide valuable information on the form factors as well as the structure of the axial-vector mesons and scalar mesons, as already mentioned in the Introduction.
Here we discuss the difference and merit of measuring B meson decays comparing to D decays. The mass of B meson is heavy enough that the methods of Perturbative QCD and Soft-Collinear Effective Theory are available, while there are little reliable theoretical tools to treat the corresponding D decays. Therefore, some B and B s decay channels considered by us in the manuscript have also been calculated in such approaches, as shown in e.g., Ref. [6].
We compared our results with them and a nice agreement is achieved. Consequently, we show the experimental values or the ones reported by PDG in the tables. On the other hand, the D meson has smaller phase space resulting in small branching fractions for decaying into f 0 (1310), f 0 (1500), f 0 (1710), e.g., B(D + → f 0 (1710)e + ν e ) ∼ 10 −9 [26] can not be measured at the BESIII factory due to limited statistics, but for the B meson case, the corresponding branching fraction is at the order of 10 −6 which is in the scope of Belle and Belle-II detectors.
Besides the larger phase space available for the B decay, the experimental situation is also much better. There are much more B data samples than the ones of D, and especially when considering the running of the Belle-II, as the upgrade of Belle detectors. This constituents one of our direct motivations to reconsider the B and B s decay. The total event numbers of D/D s and B/B s pairs are listed in Tabs. I and II, respectively. Clearly, we can see the difference of the order in magnitude. For example, the branching fractions B(D + → f 0 (1500)e + ν e ) ∼ 1 × 10 −6 and B(D + → f 0 (1710)e + ν e ) ∼ 5 × 10 −9 [26] are hard to be detected by BESIII collaboration, while Belle-II is capable of measuring those channels due  We finally make two supplemental remarks: (1) in Refs. [24,25] all the form factors correspond to the V − A current, and the "x" in Fig.1b therein [24,25] denotes the insertion of W boson, thus we will not consider the b → s transition, penguin operators etc. Or stated differently, transitions like B → K, K * , K 1 (1270), K 1 (1400) or B s → φ will not be considered in the current framework. (2) In general, for the transitions involving the (axial-) vector meson, the form factors receive the contribution of additional B functions, and for our current case, those have been confirmed to be negligibly small by numerical calculations [24]. The issues concerning self-consistency and covariance of the light-front quark models are discussed in Ref. [28].

III. RESULTS AND DISCUSSION
In this part, we report our theoretically predicted branching fractions for various semileptonic decay channels considered by us, and discuss some details and implications.
In Tab. IV, we list out the mesons that we considered in the B and B s semiletonic decays: • P denotes the pseudoscalar bosons containing π, K, η, η ′ , D 0 , D s , S contains the heavy scalar nonet a 0 (1450), f 0 (1370), f 0 (1500)/f 0 (1710), K * (1430) suggested by qq quark model [29], and the charmed mesons D * 0 (2400), D * s0 (2317), i.e., the calculation and results are based on assuming them as the conventional qq meson. In fact, except for the structures of a 0 (1450) and K * 0 (1430) which are less controversial, those of others still need to be ascertained. Especially, a common viewpoint is to interpret D * s0 (2317) as a DK molecular or a tetraquark state, see recent reviews in Refs. [30][31][32][33]. The D * 0 (2400) 4 is the excited state of D meson and can be understood from the heavy-quark spin symmetry, where the light system has j = s q + L, with s q denoting the spin of light quark and L the orbital angular momentum, and thus there are two doublets with J P as: and However, other interpretations also exist, e.g., D * 0 (2400) shows two-pole structure and the lower pole associated with D * s0 (2317) forms an SU(3) multiplet [35]. Concerning the f 0 states, it is generally argued that they are the qq meson mixed by glueball contents, but differing in which state is dominated by glueball component in literature 5 . Considering the available lattice and experimental information, the authors of Refs. [38,39] where the number in the parenthesis indicates the uncertainty for the last digit of the central value; the scalar f 0q (f 0s ) is the pure qq (ss) states with the spin-parity J P = 0 + , whose mass is 1.474 GeV (1.5 GeV), while the glueball (G) is 1.7 GeV [38,39]. Clearly, f 0 (1710) contains mainly glueball and f 0 (1500) has the flavor octet structure. To be specific, we show the corresponding B → f 0q and B s → f 0s transition form factors [25] in Tab. III. In fact, we wish to stress that the proposed measurements of the semileptonic B(B s ) decays to f 0 states will be a powerful test for their inner structure due to the absence of the final-state interaction between f 0 and the lepton pair.
Let A L be the light axial vector, and A H the heavier one.
where A q and A s denotes the corresponding components (uū+dd)/ √ 2 (note the factor of 1/2 for calculating the branching fraction) and ss in the wave functions. Following the strategy in Refs. [10,26] we will take the values α f 1 = 69.7 • , α h 1 = 86.7 • . Recently, h 1 (1380) has been confirmed by the BES-III collaboration [42] in the decay channel J/ψ → η ′ KKπ, where its mass and width, and the product branching fraction have been measured. Also, the mixing angle is determined to be 90.6 • ± 2.6 • [42] based on the mixing angle θ K 1 = 34 • and the masses of the axial-vector mesons. This is consistent with the value that is adopted by us above. Clearly, the quark contents of h 1 (1170) is dominated by h 1q , while the h 1 (1380) mainly consists of ss. Note that in the literature, e.g., Ref. [43], the mixing angle θ is often referred to the singlet-octet one, and α = θ + 54.7 • . An ideal mixing is defined as The physical mass eigenstates K 1 (1270) and K 1 (1400) are the mixture of the 1 P 1 state K 1B and 3 P 1 state K 1A [11], and we will take θ K 1 = 33 • from the analysis of Ref. [10].

9
As mentioned in Sec. II, the form factors and the formula for calculating the branching fractions can be found in Refs. [24,25] and [26], respectively. Only one point is needed to be notified: generally, the form factor is expressed by with the parameters F (0), a, b given in Ref. [25]. As discussed in [24], the form factor V 2 (q 2 ) for B(B s ) → A(1 +− ) transition approaches zero at very large −|q 2 | where the threeparameter parametrization, Eq. (7), becomes questionable. Instead, a variant has been exploited, The form factor with the expression of Eq. (8) is applied to the 1 P 1 case, i.e., b 1 , h 1 and K 1B . One may consider to replace m B by m Bs in Eqs. (7) and (8)  We thus refrain from showing the branching fractions involving f 0 (1370).
• BaBar [48] and Belle [49] have measured the four semileptonic decay modes involving the P-wave charmed mesons, cf. Eqs. (1) and (2), which of course, includes D * 0 (2400). The PDG average value of (2.5 ± 0.5) × 10 −3 means the joint branching fraction . Comparing with our theoretical value for B + → D * 0 (2400)e + ν e in Tab. V, we expect the mode of D * 0 (2400) → Dπ is the dominant one in D * 0 (2400) decays. In fact, the semileptonic decay B → D * 0 (2400), as a background contributing to one of the leading sources of the systematical uncertainty for the extraction of |V cb | from B → D * ℓν ℓ , is still poorly known, see the review in Ref. [50]. The Belle-II and LHCb detectors will provide the opportunity for the precision measurements. We also display the differential decay rate for B → D * 0 (2400) as well as B → D * s0 (2317) + lepton pairs in Fig. 1 for convenience of comparison with the future experiments.
• Due to the factor of |V cb /V ub | 2 ≈ 115, the branching fraction of b → c decay is generally enhanced by two orders compared to b → u decay, as can be seen in Tabs. V, VI and VII (Note that the additional factor of sin 2 φ or cos 2 φ appears in the processes of B + → η, η ′ to calculate the branching fractions). B(B s → D − s + X) = (93 ± 25)% [3] again shows the dominance of b → c tansition. In Tab. VII, the B s decay branching fraction is at the order of 10 −4 for b → u and 10 −2 for b → c. Unfortunately, there is scarce experimental information on the semileptonic B s decay except for the inclusive semileptonic decay B(B s → Xℓν ℓ ) = (9.6 ± 0.8)% [3]. As can be clearly seen, the sum of the branching fractions for the channels considered in Tab. VII does not exceed this limit. The theoretical predictions for B(B s → D s ℓν ℓ ) vary from 1.0% to 3.2% and for B(B s → D * s ℓν ℓ ) vary from 4.3% to 7.6% [51], see e.g., Refs. [52][53][54][55]; B(B s → D * s0 (2317)ℓν ℓ ) ∼ 0.20% − 0.57% [55][56][57]. Regarding D * s0 (2317) as a DK molecular state, the authors of Ref. [58] predict B(B s → D * s0 (2317)ℓν ℓ ) = 0.13%. The process B s → K − ℓν ℓ has been calculated in Refs. [55,59] and also examined in lattice QCD [60,61]. The B s → K * J ℓν ℓ decay is investigated in Ref. [44], and our result for B s → K * − 0 (1430)ℓν ℓ agrees very well with theirs, but not for K 1 (1270) and K 1 (1400) sector. Adopting their values of the mixing angles as input still does not remedy such discrepancy, so we will regard such discrepancy as the different predictions from the two models. Given the branching fractions of B(K 1 (1270) → Kρ) = (42 ± 6)% and B(K 1 (1400) → K * (892)π) = (94 ± 6)%, the B s → K 1 transitions could be measured with the current statistics in Belle/Belle-II and LHCb. Overall speaking, little has been known for the experimental information on the exclusive semileptonic B s decay, while plentiful theoretical predictions have been done. This situation highly calls for the true experimental measurements, and this can be realized with Belle/Belle-II and LHCb detectors.
• We wish to comment that even-parity light mesons, including the axial-vector meson, the scalar meson above 1 GeV, and the P −wave charmed meson, can be also studied via hadronic two-body B decays within the factorization scheme [62][63][64][65]. The semileptonic decay modes investigated here will provide a much cleaner environment to explore the nature of these mesons owing to the absence of the strong hadronic final-state interactions manifested in the two-body hadronic decay. At least, the investigation of such semileptonic modes could serve as a supplement to the hadronic two-body decay.
• The CKM matrix element |V ub | suffers from large uncertainty around 19%, while |V cb | has been determined better with the uncertainty of 4%. Roughly assigning 10% error induced by form factors, we have the combined uncertainty of 22% and 11% for the processes b → u and b → c, respectively. Additionally, the uncertainty induced by the mixing angle needs more care. Guided by Ref. [10] we allow the variations of point of view, it should be understood that the future measurements on these channels will be highly meaningful for a "precise" determination of the mixing angles, as also mentioned in the Introduction.
• There are several recent tests of the lepton universality via the semeileptonic decay modes, e.g., B(D → πµν µ )/B(D → πeν e ) done by the BES-III collaboration [66], and B(B → D * τ ν τ )/B(B → D * µν µ ) done by the LHCb collaboration [67]. Motivated by this, we also calculate the branching fractions of the semileptonic decays involving the τ lepton mode. Our results agree very well with the experimental values B(B + → D 0 τ ν τ ) = (7.7 ± 2.5) × 10 −3 and B + →D * 0 τ ν τ = (1.88 ± 0.20)% [3]. That is, what we compared is the direct number of the branching fraction of the τ mode, but not the ratio between the τ and electron one. The latter is related to the recently wellknown R D or R D * puzzle [68]. In fact, we do not touch this issue since we are working in the framework of Standard Model (keeping the lepton universality) and also there is the tricky estimate of uncertainties. However, as a passing comment, we want to remind the importance of the precise determination of the uncertainties. Starting from the decay B → D * ℓν ℓ , the authors of Ref. [69] also discussed the corresponding strange quark partner, B s → D s τ ν τ , which is also investigated by us. Very recently, B * → (D, D s , π, K)ℓν ℓ is also discussed for probing the New Physics effects [70]. On the experimental aspect, the electron mode is usually the easiest one to be measured, while one may encounter the large misidentification between µ and π 7 . For the τ case, the experimental error will be even larger: the two largest decay channels of τ are [3] B(τ → µ −ν µ ν τ ) = (17.39 ± 0.04)% and B(τ → e −ν e ν τ ) = (17.82 ± 0.04)%; both of them contain two neutrinos, which hinders the full construction resulting in large background, and also there is no way to use the recoiling information due to the existence of multi-neutrinos. 7 The decaying of muon to electron occurs outside the detector and thus muon can be regarded as a stable particle inside the detector.

IV. CONCLUSION
Based on the analysis of the form factors from the covariant light-front quark model [24,25], we provide the branching fractions for B → (P, V, S, A)ℓν ℓ with P, V, S, A denoting the corresponding pseudoscalar, vector, the scalar mesons with mass above 1 GeV, and the axial-vecor mesons, respectively. Those mesons are listed in Table IV. Under the framework of the lepton flavor universality, the branching fractions for the semileptonic decay involving the τ mode are also provided. The predicted branching fractions are typically in the range of 10 −6 ∼ 10 −4 . On the experimental side, (772 ± 11) × 10 6 BB and (6.53 ± 0.66) × 10 6 B sBs pairs have already been collected by the Belle detector, and Belle-II will have a larger 15 Channel (10 −5 )  Assuming D * 0 (2400) and D * s0 (2317) as the conventional quark-antiquark mesons, we predict the branching fractions of B(B → D * 0 (2400)ℓν ℓ ) = (2.31 ± 0.25) × 10 −3 and B(B s → D * − s0 (2317)ℓν ℓ ) = (3.07 ± 0.34) × 10 −3 . Confronting these values with future experimental results will provide a further scrutiny for the possible assignment of qq interpretation.