#### 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 two-body weak hadronic decays of and employing the factorization scheme. We predict the branching ratios of semileptonic and nonleptonic weak decays of and mesons in Cabibbo-angle-enhanced and Cabibbo-angle-suppressed 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 of into the light quarks, whose branching ratios are expected to be of the order of [3–11]. The future experiments [3, 12–14] of Beijing Electron Positron Collider (BES-III) 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 the transition form factors ignoring their dependence on their predictions. Further, Sharma and Verma [7] reanalyzed 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 on -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 light-front quark model, respectively, to predict the decay rates of meson.

In the present work, we employ BSW model framework [15–17] to investigate the effects of flavor dependence on 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 from - and -waves for and decays. We also predict branching ratios of weak semileptonic and nonleptonic decays of and in Cabibbo-angle-enhanced and Cabibbo-angle-suppressed 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 and states.

Order of presentation is as follows. In Sections 2 and 3, we outline the framework employed for analysing semileptonic and nonleptonic weak decays of . Form factors and branching ratios in BSW model are presented in Section 4. Section 5 deals with effects of flavor dependence on decays. Semileptonic and nonleptonic weak decays of are analyized in Section 6. 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 *V-A* 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 is the sum of longitudinal and transverse decay widths given by where the longitudinal decay width is defined as

and the transverse decay width is expressed as

The helicity amplitudes and are given by where is the mass of the lepton, , and = − is related to the three-momentum 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 Cabibbo-angle-enhanced mode is given by

and for Cabibbo-angle-suppressed mode where represents the color singlet *V-A *current and denote standard Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix elements. ’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 coefficient), where . Matrix elements [15–17] of the weak currents are defined as

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 three-momentum is defined as

Thus the decay amplitude , say for color-enhanced decay, can be expressed as

For color-suppressed 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 [15–17] 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, and is the normalization factor. is the average transverse quark momentum, , which 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. From (21) it is clear that the form factors are sensitive to the choice of , which is treated as a free parameter in the model. We wish to remark that in BSW [15–17] 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 , 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, , , , 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]. With the following decay constants (in ) [24–26], Branching ratios for the Cabibbo-angle-enhanced decays are and (column 2 of Table 4). (b) Decays: using the following decay constants (in ) [24–26]: 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.

#### 5. Flavor Dependent Effects on Decays

Since , being a dimensional quantity, may possess some flavor dependence, it may not be justified to take the same value for all the mesons. In our recent work [27, 28], we have investigated such flavor dependence through in form factors and consequently in decay widths, which may be measured in near future experiments.

##### 5.1. Flavor Dependence on Form Factors

In this section, we investigate effects of the flavor dependence on decays. Following the prescription of [27, 28], we estimate for different mesons from , that is, wave function overlap at the origin, using the following ansatz: which may be justified in light of inverse size of the mesons and dimensionality arguments. is extracted from the hyperfine splitting for the meson masses [29], where and , respectively, denote masses of the vector and pseudoscalar mesons composed of and quarks. Conventionally, meson masses fix quark masses (in GeV) to be , , , and for , and (for light flavors , , and ). However, uncertainty of , particularly for the light quark sector, may lead to some variation in the quark masses [25, 26, 30–34]. So, in our analysis, we allow the following range in the quark masses:

Calculated values of for different mesons are given in column 2 of Table 7. We use the well-measured form factor [35–37] to fix which in turn yields for other mesons as given in column 3 of Table 7. We find that all the transition form factors get significantly enhanced due to the flavor dependence of parameter , particularly, . The transition form factors, required for the weak decays under investigation, are found to be increased, which are given in rows 3 and 11 of Table 1. Uncertainties shown in the form factors arise due to variation in the quark masses discussed above.

Note that the normally uncertainties lie between 3% and 11% for all the form factors except for , which is drastically increased from 80% to 100%. These are likely to change theoretical predictions for branching of semileptonic as well as nonleptonic decays. Since the form factors , and , describe the -, -, and -wave contributions to the nonleptonic decay amplitudes, respectively, the *s-d* interference term may acquire significance when both and become large. Incidentally, being small, such contribution turned out to be very small as compared to the terms proportional to . Therefore, the variations in even of the order of 80% to 100% are not going to affect the branching ratios seriously.

##### 5.2. QCD Inspired Calculation of

The wave function for the heavy meson like, for example, meson is known to have a pronounced peak at ( being the *B* meson mass). It is thus obvious that the only region close to the peak position contributes to any degree of significance. This implies a good approximation for parameter to be of the order of . This may be easily understood since , if and has peak when . Also, the average transverse quark momentum of the heavy quark equals to that of heavy meson such that . In accord with the heavy quark effective theory, it also implies equal velocities of the -quark and of the meson up to corrections of order . In the light of these arguments, we present another method to determine , for instance by looking at the typical inverse size of the system under scrutiny, which is of order for the lighter mesons and the heavy-light ones and is of order meson mass multiplied by strong coupling constant () for the heavy quarkonium () states. It is interesting to note that following the relation
gives larger values of , that is, , ignoring the small uncertainties in for heavier quarks. A plot for thus calculated for various mesons (especially for states) has interesting pattern with respect to the physical masses of the mesons, which is shown in Figure 1.

It may also be pointed out that Sharma et al. [38] have reported that , which may be extracted from leptonic decays of vector mesons, cannot simultaneously fit with that needed for their masses. Perhaps similar analysis needs to be carried out for extracted from form factors and hyperfine splitting.

##### 5.3. Branching Ratios Including Flavor Dependent Effects

Different values of for initial and final state mesons yield transition form factors, which are given in Table 1. Using these form factors we calculate the branching ratios of semileptonic decays of as presented in column 3 of Table 2. Contributions from the Helicity amplitudes and are also obtained for the corresponding branching ratios , and for these decays and are given in column 3 of Table 3.

Further, we calculate branching ratios of various decays, which are listed in column 3 of Tables 4 and 5. In order to compare flavor dependent effects for different Helicity terms with those obtained at fixed , we present branching ratios , and in column 3 of Table 6. For the sake of comparison, we also give results of other works in Tables 2, 4, and 5. The following observations are made. (i)Branching ratios of all semileptonic as well as nonleptonic decays of meson get significantly enhanced because of inclusion of flavor dependent effects.(ii)The enhanced branching ratios for semileptonic decays are and . Presently, only experimental [1] upper limits are available, (iii)Among the Cabibbo-angle-enhanced decays, we find that branching ratios of dominant decays are = and .(iv)For Cabibbo-angle-suppressed but color-enhanced modes and , we obtain the following ratios based on the naive factorization scheme: (v)In case of decays, for the color-enhanced decay of the Cabibbo-angle-enhanced mode, we calculate , which is higher than the branching ratio of . While for color-suppressed decay we obtain . Our analysis also yields

##### 5.4. Branching Ratios Using QCD Inspired

Using QCD inspired values of for heavy quarkonium states, the obtained form factors are given in rows 4 and 12 of Table 1. It is interesting to note all the form factors except get significantly enhanced. Correspondingly, branching ratios of semileptonic and nonleptonic decays are calculated, which are presented in column 4 of Tables 2 to 6. We observe the following.(i)Branching ratios of all semileptonic decays of meson are nearly the same though marginally enhanced, = and in comparison to those obtained in our work including the flavor dependent effects. This happens due to the larger value for state, leading to increased overlap between the initial and final state wave functions. (ii)Branching ratios of all the decays, involving the form factor only, are slightly decreased in comparison to the flavor dependent branching ratios predicted in our analysis.(iii)Branching ratios of all the decays are comprable to our predictions for the flavor dependent case.

#### 6. Semileptonic and Nonleptonic Weak Decays of

##### 6.1. Form Factors and Branching Ratios

Using the framework described in Sections 4 and 5, we extend our analysis for bottom sector. We obtain the form factor for transition at both and using the flavor dependent , which are compared in rows 2 and 3 of Table 8. Note that the form factors get enhanced by several orders of magnitude for flavor dependent . This happens due to the increased overlap of and the final state wave functions for flavor dependent in comparison to that at fixed as shown in Figures 2 and 3. We wish to remark that the flavor dependent form factors come close to the expectation [7] based on HQET considerations (row 4 of Table 8). Consequently, the branching ratios of semileptonic and nonleptonic weak decays of get significantly enhanced.

We also use the QCD inspired method to determine , that is, , which match well with our value 1.80 for . Subsequently obtained form factors are given in row 4 of Table 8, which are largely the same as our results, so we exclude the results for decay rates based on the QCD inspired method for further discussion.

##### 6.2. Semileptonic Weak Decays of

Using form factor, appearing in transition, and the decay rate formula given in (5), firstly, we calculate the branching ratios for semileptonic decays of at fixed , which are given in column 2 of Table 9. The predicted branching ratios of semileptonic weak decays of using the flavor dependent effects are given in column 3 of Table 9. We find that the branching ratio of dominant semileptonic decay is that is sufficiently enhanced in comparison to obtained using fixed . Here also, we give branching ratios of longitudinal and transverse components (, and ) separately, for the semileptonic decays in Table 10.

##### 6.3. Nonleptonic Weak Decays of

In this section, the analysis is extended to decays. The effective weak Hamiltonian generating the dominant quark decays involving transition is given by for the CKM-favored mode. In our analysis we use .

Similar to decays, the factorization scheme expresses weak decay amplitudes as a product of matrix elements of the weak currents (up to the scale ) as

For instance, the decay amplitude for the color-enhanced mode of the CKM-favored decays is given by

###### 6.3.1. Decays

Following the procedure employed in Sections 3 and 4, we calculate the branching ratios for CKM-favored mode both for fixed and for flavor dependent , which are presented in columns 2 and 3, respectively, of Table 11. In addition to the decay constants given in (30) we use and . The dominant decays in this mode are found to be and , where values in the parentheses are calculated at fixed . It may be noted that our branching ratios including the flavor dependent effects compare well with the earlier results, and , obtained by Sharma and Verma [7] using HQET considerations.

###### 6.3.2. Decays

Employing the decay rate formula for such decays as discussed in Section 3 and following the similar procedure used for decays, we determine the decay amplitudes for various decays for the CKM-favored modes. We use [25, 26] and to calculate the branching ratios which are given in Table 12. We observe that the dominant mode is , following by = . Here also, the values in the parentheses are those obtained for fixed . For comparison of the contributions of the Helicity amplitudes and involved, we have calculated the corresponding branching ratios , and for various decays both for and for flavor dependent , which are presented in Table 13.

#### 7. Summary and Discussion

In this paper, we have predicted the rare semileptonic and nonleptonic weak decays of and mesons. It may be mentioned that the present work differs from the previous ones [6, 7] based on the BSW framework in three aspects. (i)Firstly, in the light of HQS based BSW2 model [12], we use the dipole dependence for the form factors , and , while the earlier work [6, 7] used the monopole dependence for these form factors. (ii)Secondly, the results given in [7] are based on -wave dominance, while our results take into account the contributions from - and -waves also. (iii)Lastly, we incorporate flavor dependent effects on and form factors in our analysis through the different values of for initial and final state mesons. Also, to support the case of flavor dependence of the form factors, we use an alternate QCD inspired approach to determine for heavy quarkonium states. The following conclusions are readily drawn out of our analysis. (A)Weak decays of .(i)Initially, we calculate transition form factors at for all the mesons: among the Cabibbo-angle-enhanced decays, branching ratios of dominant decays are calculated to be and for Cabibbo-angle-enhanced mode decays, branching ratios of dominant decays are obtained to be and . (ii)We investigate the effects of possible flavor dependence of by determining from meson masses to fix for all the mesons through an ansatz; we, then, calculated transition form factors which get significantly enhanced as compared to those at fixed ; as a result of this, branching ratios of all semileptonic as well as nonleptonic decays are enhanced. (iii)Among the Cabibbo-angle-enhanced decays, the dominant decays are predicted as = , = , = , and .(B)Weak decays of :(i)It is observed that inclusion of flavor dependent effects through significantly enhances the form factors and, consequently, the branching ratios of semileptonic and nonleptonic weak decays of .(ii)Branching ratio of the dominating semileptonic mode is predicted to be .(iii)Among the Cabibbo-angle-enhanced decay modes, the dominant ones are predicted as , , , and .(iv)Though the predicted branching ratios in the QCD inspired method of obtaining are marginally changed, the alternate calculation of for heavy quarkonium states seems to support the flavor dependent effects on rare weak decays of and mesons.(v)Our predictions also agree well with those obtained by Sharma and Verma [7] using the HQET considerations.

It is hoped that these branching ratios would lie in the detectable range and may be measured in future experiments.

#### Acknowledgment

Financial assistance from UGC, New Delhi (India) is gratefully acknowledged.