Research Article  Open Access
Effects of Flavor Dependence on Weak Decays of and
Abstract
We carry out a detailed analysis of effects of flavor dependence of average transverse quark momentum inside a meson on and transition form factors and twobody weak hadronic decays of and employing the factorization scheme. We predict the branching ratios of semileptonic and nonleptonic weak decays of and mesons in Cabibboangleenhanced and Cabibboanglesuppressed modes.
1. Introduction
Due to remarkable improvements of experimental techniques and instrumentation in the recent years, it is expected that more accurate measurements may now be available for rare decays also. The BES collaboration has observed some rare decays including the semileptonic as well as nonleptonic modes [1, 2]. As a result, it has revived the interest in the rare weak decays ofâ€‰â€‰â€‰â€‰into the light quarks, whose branching ratios are expected to be of the order of [3â€“11]. The future experiments [3, 12â€“14] of Beijing Electron Positron Collider (BESIII) and Large Hadron Collider (LHC) hope to accumulate data for more than events of per year, which would make it possible to measure such rare decays. From the theoretical point of view, such weak decays are particularly interesting because these are expected to explore mechanism responsible for hadronic transitions and are also important for the study of nonperturbative QCD effects. Further, decays of a vector meson involve polarization effects that may help in probing the underlying dynamics and hadron structure. Within the standard model framework, the flavor changing decays of and states are also possible, though naively these are expected to have rather lower branching ratios in comparison to their conventional hadronic and radiative decays.
Earlier, Verma, Kamal, and Czarnecki (VKC) [6] had given the first estimates of weak decay rates using the factorization scheme. VKC employs the Bauer, Stech, and Wirbel (BSW) model to estimate theâ€‰â€‰ transition form factors ignoring their dependence on their predictions. Further, Sharma and Verma [7] reanalyzedâ€‰â€‰the decays in the same model by including dependence and new values of the form factors and decay constants. The predictions in the earlier work [6, 7] are based onâ€‰â€‰wave dominance for . The analysis has also been extended to predict branching ratios of weak decays of based on heavy quark effective theory. Recently, Wang et al. [8, 9] and Shen and Wang [10] have calculated transition form factors using the QCD sum rules and employing covariant lightfront quark model, respectively, to predict the decay rates of meson.
In the present work, we employ BSW model framework [15â€“17] to investigate the effects of flavor dependence on â€‰â€‰and transitions form factors and subsequently on and decays, caused by possible variation of average transverse quark momentum inside a meson. In the light of Heavy Quark Symmetry (HQS) [18], we use the dipole dependence for the form factors , and and monopole dependence for the form factor and include contributions fromâ€‰â€‰â€‰â€‰andâ€‰â€‰waves for and decays. We also predict branching ratios of weak semileptonic and nonleptonic decays of and in Cabibboangleenhanced and Cabibboanglesuppressed modes. In support of flavor dependence of the form factors and corresponding branching ratios, we also perform an alternate QCD inspired calculation to obtain for heavy quarkonium andâ€‰â€‰â€‰â€‰states.
â€‰â€‰Order of presentation is as follows. In Sections 2 and 3, we outline the framework employed for analysing semileptonic and nonleptonic weak decays of . â€‰ Form factors and branching ratios in BSW model are presented in Section 4. Section 5 deals withâ€‰â€‰effects of flavor dependence on decays. Semileptonic and nonleptonic weak decays of are analyized in Section 6. â€‰Summary and discussion are given in the last section.
2. Semileptonic Weak Decays of
The semileptonic decay amplitude can be expressed as where is the appropriate CKM matrix element for transition and is the usual weak VA current. matrix element is given by where is the polarization vector of and and are the four momenta of and pseudoscalar meson, respectively, and . is related to and as:
The total decay width for â€‰â€‰is the sum of longitudinal and transverse decay widths given by where the longitudinal decay width is defined asâ€‰â€‰
â€‰â€‰ â€‰â€‰ and the transverse decay width is expressed as
The helicity amplitudes â€‰â€‰andâ€‰â€‰â€‰â€‰are given by where is the mass of the lepton, , and = âˆ’ is related to the threemomentum of the daughter meson in the rest frame of meson by
3. Nonleptonic Weak Decays of
3.1. Weak Hamiltonian
The QCD modified weak hamiltonian [19] generating the quark decays for Cabibboangleenhanced mode is given by
â€‰â€‰ and for Cabibboanglesuppressed mode where represents the color singlet VA current and denote standard CabibboKobayashiMaskawa (CKM) mixing matrix elements. â€™s are the undetermined coefficients assigned to the effective charge current, , and the effective neutral current, , parts of the weak Hamiltonian. These parameters are related to the QCD coefficients as follows: where , is the number of colors. Usually is treated as a free parameter to be fixed by the experiment. However, we follow the conventional limit to fix the QCD coefficients and , where [19] are obtained on the basis of decays.
3.2. Branching Ratios
In the standard factorization scheme, the decay amplitudes of are obtained by sandwiching the QCD modified weak Hamiltonian (up to the weak scale â€‰â€‰coefficient), where . Matrix elements [15â€“17] of the weak currents are defined as
â€‰â€‰
The decay rate formula for decays [20, 21] is given by where is the magnitude of the three momenta of final state meson in the rest frame of meson and denote its mass. In general, the threemomentum is defined as
Thus the decay amplitude , say for colorenhanced decay, can be expressed as
For colorsuppressed decays the QCD factor is replaced by .
The decay rate of decays [22, 23], composed of three independent Helicity amplitudes and , is expressed as
The amplitudes and are defined as follows: where
The coefficients , , and describe the , , and wave contributions, respectively.
4. Form Factors in BSW Framework
We employ the BSW [15â€“17] model for evaluating the meson form factors. In this model, the meson wave function is given by where denotes the meson mass, denotes the th quark mass, andâ€‰â€‰ is the normalization factor. is the average transverse quark momentum, ,â€‰â€‰whichâ€‰â€‰is of the order of [18].
Expressing the current in terms of the annihilation and creation operators, the form factors are given by the following integrals: where and denote masses of the nonspectator quarks participating in the quark decay process. â€‰From (21) it is clear that the form factors are sensitive to the choice of ,â€‰â€‰which is treated as a free parameter in the model. We wish to remark that in BSW [15â€“17] model the form factors are calculated by taking same value of for initial as well as final states. With the quark masses (in GeV), form factors thus calculated are presented in rows 2 and 10 of Table 1.

It has been pointed out in the BSW2 model [18] that consistency with the Heavy Quark Symmetry (HQS) requires certain form factors such as , and to have dipole dependence, whereas has monopole dependence, that is, with appropriate pole masses.
4.1. Numerical Results
In this section, we present the branching ratios calculated from the form factors (without flavor dependence) obtained in the last section for semileptonic and nonleptonic weak decays of meson.
4.1.1. Branching Ratios of Semileptonic Decays
Using these form factors, we obtain the branching ratios of semileptonic weak decays of , which are presented in column 2 of Table 2. The branching ratios ,â€‰â€‰and for various , corresponding to the Helicity amplitudes and , are given in columns 2, 3, and 4 of Table 3. We find that net branching ratios for semileptonic decays, ,â€‰â€‰, , and , are well below the experimental limits.

4.1.2. Branching Ratios of Nonleptonic Decays
(a) Decays: for and emitting decays, we take the following basis: where ; we take [24]. â€‰With the following decay constants (in ) [24â€“26], Branching ratios for the Cabibboangleenhanced decays are and (column 2 of Table 4). (b) Decays: using the following decay constants (in ) [24â€“26]: obtained branching ratios for various decays are and (column 2 of Table 5).
For the sake of comparison of the relative contributions of the Helicity amplitudes and , we have calculated the corresponding branching ratios , and for these decays, which are given in column 2 of Table 6.
