Research Article | Open Access
Qin Chang, Xiao-Lin Wang, Jie Zhu, Xiao-Nan Li, "Study of Induced Decays", Advances in High Energy Physics, vol. 2020, Article ID 3079670, 12 pages, 2020. https://doi.org/10.1155/2020/3079670
Study of Induced Decays
In this paper, we investigate the tree-dominated ( and ) decays in the Standard Model with the relevant form factors obtained in the light-front quark model. These decays involve much more helicity states relative to the corresponding and decays, and moreover, the contribution of longitudinal polarization mode ( meson) is relatively small, , compared with the corresponding meson decays. We have also computed the branching fraction, lepton spin asymmetry, forward-backward asymmetry, and ratio . Numerically, the branching fractions of decays are at the level of and are hopeful to be observed by LHC and Belle-II experiments. The ratios have relatively small theoretical uncertainties and are close to each other, , which are a bit different from the predictions in some previous works. The future measurements are expected to make tests on these predictions.
In the past years, a large amount of events have been accumulated by Babar, Belle, Tevatron and LHCb experiments, and most of -meson decays having branching fractions have been measured . Moreover, some deviations between the standard model (SM) predictions and the experimental data have been observed, for instance, the angular observable of decay with discrepancy [2–6], the differential branching fraction of decay with discrepancy [7, 8], and the well-known “ CP puzzle” [9, 10]. Besides the flavor-changing-neutral-current precesses mentioned above, the -meson semileptonic decays induced by transition also play an important role in testing the SM and probing the hints of possible new physics (NP). For instance, the well-known “ anomaly” reported by BaBar [11, 12], Belle [13–15], and LHCb [16, 17] collaborations exhibits a significant deviation between the SM prediction and experimental data [1, 18, 19]. Many studies have been done within the model-independent frameworks [20–27], as well as in some specific NP models, for instance Refs. [28–48]. One can refer to Refs. [49, 50] for recent reviews.
The spin-triplet vector meson with a quantum number of and [51–54] has the same flavor components as the spin-singlet pseudoscalar ( and ) meson and can also decay through the transition at quark-level; therefore, its -induced semileptonic decays can play a similar role as meson decays for testing the SM and probing possible hints of NP.
The meson is an unstable particle, it cannot decay via strong interaction due to that MeV ; meson decay is dominated by the radiative process , ; the weak decay modes via the bottom-changing transition (for instance, the induced semileptonic decays considered in this work) are generally very rare, and their branching fractions are expected to be very small within the SM. Until now, there is no experimental information and few theoretical works concentrating on the weak decays. Fortunately, thanks to the high luminosity and large production cross section at the running LHC and SuperKEKB/Belle-II experiments, a huge amount of the meson data samples would be accumulated. At Belle-II experiment, the and mesons are produced mainly via decays. With the target annual integrated luminosity, , and the cross section of production in collisions, , it is expected that about samples could be produced per year by Belle-II. Further considering that meson mainly decays to final states with a pair of mesons and using the branching fractions of decays given by PDG , it can be estimated that about and samples can be accumulated by Belle-II per year. Unfortunately, the meson and its decays are out of the scope of Belle-II experiment. In addition, a lot of samples can also be produced via collision and be accumulated in the future by LHC with high collision energy, high luminosity and rather large production cross section [58–60], and some weak decays are hopeful to be observed, such as the leptonic decay with branching fraction .
Encouraged by the abundant data samples at future heavy-flavor experiments, some interesting theoretical studies for the weak decays have been made within the SM, for instance, the pure leptonic and decays [61, 62], the impact of on decays , the studies of the semileptonic decays within the QCD sum rules [64–66], the semileptonic with decays within the Bethe-Salpeter (BS) method , and an approach under the assumption of heavy quark symmetry (HQS) , with  and the nonleptonic ( and ) [70, 71], , , , , and  decays. Moreover, the NP effects on the semileptonic with decays have been investigated in a model-independent scheme  and the vector leptoquark model . In this paper, we pay our attention to the CKM-favored and tree-dominated semileptonic weak decays, which are generally much more complicated than the corresponding decay modes because they involve much more allowed helicity states.
Our paper is organized as follows. In Section 2, the helicity amplitudes and observables of decays are calculated. Section 3 is devoted to the numerical results and discussions, and the transition form factors obtained within the covariant light-front quark model are used in the computation. Finally, we give our summary in Section 4.
2. Theoretical Framework and Results
2.1. Effective Lagrangian and Amplitude
In the SM, decays are induced by transition at quark level via W-exchange and can be described by the effective Lagrangian at low energy scale , where is the Fermi coupling constant and denotes the CKM matrix element. Using Eq. (1), the amplitude of decay can be written as the product of the hadronic matrix element and leptonic current. Then, in terms of leptonic () and hadronic tensors built from the respective products of the leptonic and hadronic currents, the square amplitude can be expressed as
Inserting the completeness relation of the polarization vector of virtual boson, the product of and can be rewritten as where and are Lorentz invariant and therefore can be evaluated in different reference frames. In our following evaluation, and will be calculated in the -meson rest frame and the center-of-mass frame, respectively.
In the rest frame of meson, assuming the final state -meson moving along with positive -direction, the momenta of , , and could be written as respectively, where and , with and being the momentum transfer squared, are the energy and momentum of virtual . The polarization vectors of the initial -meson and daughter -meson, and , can be written as respectively. For the four polarization vectors of virtual , , one can conveniently choose [79, 80] in which, has to be understood as and .
Turning to the center-of-mass frame, the four-momenta of lepton and antineutrino are given as where , , and is the angle between and three-momenta. In this frame, the polarization vectors have the form
2.3. Hadronic Helicity Amplitudes
For hadronic part, one has to calculate the hadronic helicity amplitudes of decay defined by which describes the decay of three helicity states of meson into the three helicity states of daughter meson and the four helicity states of virtual . For the transition, the matrix elements can be factorized in terms of ten form factors and as [81, 82] with the sign convention .
Then, by contracting these hadronic matrix elements with the polarization vector of virtual boson, we can finally obtain the nonvanishing hadronic helicity amplitudes, , given as
Obviously, only the amplitudes with survive due to the helicity conservation.
2.4. Helicity Amplitudes and Observables
For the leptonic part, the leptonic tensor could be expanded in terms of a complete set of Wigner's -functions, which have been widely used in the study of hadron semileptonic [79, 83, 84]. As a result, can be reduced to a very compact form where and run over 1 and 0, , and run over their components. For the standard expression of function, we take their value from PDG . The leptonic helicity amplitude in Eq. (15) defined as
Using the amplitudes obtained above, we can then further evaluate the observables of decays. The double differential decay rate is written as where the factor is caused by averaging over the spins of initial meson. The double differential decay rate with a given helicity state of lepton () is written as
Integrating over and summing over the lepton helicity, we can obtain the differential decay rate written as where the three nondiagonal interference terms in Eq. (20) vanish. In addition, paying attention to the polarization states of meson, one can obtain the longitudinal differential decay width by picking out , , , and terms in Eq. (21).
Using Eqs. (19) and (20) given above, we can also construct some useful observables as follows. The -dependent ratios is defined as where denotes the light leptons and (in the following calculations, we take ). The lepton spin asymmetry and forward-backward asymmetry are defined as and respectively. These observables are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent, therefore, they can be predicted with a rather high accuracy.
3. Numerical Results and Discussions
In our numerical calculation, for the well-known Fermi coupling constant and the masses of mesons and , we take their central values given by PDG . For the CKM element, we take given by CKMFitter Group . In order to evaluate the branching fractions, the total decay widths (or lifetimes), , are also essential inputs. However, there is no available experimental or theoretical information until now. While, due to the fact that the electromagnetic processes dominates decays, we can take the approximation . In the light-front quark model (LFQM), the decay width of decay is given by  where with , with being the invariant mass of bound-state, is the fine-structure constant, is the kinematically allowed energy of the outgoing photon. The radial wavefunction (WF) of bound-state is responsible for describing the momentum distribution of the constituent quarks. In this paper, we shall use the Gaussian-type WF where is the relative momentum in -direction and has the form . One can refer to Ref.  for more details. Using the constituent quark masses and the Gaussian parameter given in Table 1, we obtain the numerical results for as follows,