Effects of Flavor Dependence on Weak Decays of J/psi and \Upsilon

We give the detailed analysis of effects of flavor dependence of average transverse quark momentum inside a meson on J/psi -->P and \Upsilon -->Bc transition form factors and two-body weak hadronic decays of J/psi and \Upsilon employing the factorization scheme. We predict the branching ratios of semileptonic and nonleptonic weak decays of J/psi and \Upsilon mesons in Cabibbo-angle-enhanced and Cabibbo-angle-suppressed modes.


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 mode [1,2]. As a result, it has revived the interest in the rare weak decays of ) (HQS) [15], we use the dipole 2 q -dependence for the form factors The helicity amplitudes h 0 and h ± are given by 2 and 2 2 2 where m l is the mass of the lepton and

Weak Hamiltonian
The QCD modified weak hamiltonian [16] generating the − c quark decays in Cabibbo-angle-enhanced mode 1) and for Cabibbo-angle-suppressed mode 0) where c N 1 = ζ , c N is number of colors. Usually ζ is treated as a free parameter to be fixed where Matrix elements [14] of the weak currents are defined as The decay rate formula for PP J → ψ / decays [17] is given by where c p is the magnitude of the three momentum of final state meson in the rest frame of ψ / J meson and ψ / J m denote its mass. In general, the three momentum c p is defined as Combining (8) The amplitudes 0 H and 1 H ± are defined in terms of the coefficients a, b and c as follows: where ), ( and The coefficient a, b and c describe the s-, d-and p-wave contributions respectively.

Form Factors in BSW Framework
We employ the BSW [14] model for evaluating the meson form factors. In this model, the meson wave function is given by where m denotes the meson mass and i m denotes the ith quark mass, m N is the normalization factor and ω is the average transverse quark momentum, 2 ω 2 T p = . By expressing the current µ J in terms of the annihilation and creation operators, the form factors are given by the following integrals: The P J → ψ / form factors thus calculated and are presented in rows 2 and 10 of Table 1.

Numerical Results
It has been pointed out in the BSW2 model [15] with appropriate pole masses i m .

Branching ratios of semileptonic decays
Using these form factors, we obtain the branching ratios of semileptonic weak decays of ν ψ Pl J → / , and are presented in column 2 of h and 1 h ± are also calculated and are given in columns 2, 3 and 4 of Table 3. We find that

Branching ratios of nonleptonic decays
For η and η′ emitting decays, we take the following basis: We use the following values for the decay constants (in GeV ) [19,20]: is higher than the branching ratio )% 10 51

Flavor dependent effects on
Since, ω is a dimensional quantity it may show flavor dependence. Therefore, it may not be justified to take same value of ω for all the mesons. In our recent work [21], we have investigated the possible flavor dependence through ω in decay widths with improved potential measurement in near future.

Form Factors
In this section we investigate the effects of flavor dependence on / Following the prescription of [21], we estimate ω for different mesons from  Table 7. We find that all the P J → ψ / transition form factors get significantly enhanced due to the flavor dependence of parameter ω . Corresponding uncertainities in form factors due variation in the quark masses are also shown in tables. Note that the uncertainities ranges between 3% to 11% for all the form factors except for A 2 (q 2 ), where the change is roughly 80% to 100%. It may be pointed out that the form factors A 1 (q 2 ), V(q 2 ) and A 2 (q 2 ) describe the s-, p-and d-wave contributions to the final state vector meson helicity amplitudes, respectively (see eqns (22) - (24)). Thus, s-d interference is significant only if both A 1 (q 2 ) and A 2 (q 2 ) are large. At higher q 2 , amplitudes are dominated by form factor A 1 (q 2 ) (see eqn (20)), and further, if A 2 (q 2 ) is small (as it appears), contributions from the terms propotional to A 2 (q 2 ) are very samll as compared to terms proportional to A 1 (q 2 ). Therefore, the variations even of the order of 80% to 100% are not going to affect the branching ratios by large.

QCD inspired calculation of ω
In this section, we present another method to determine omega (for instance by looking at the typical inverse size of the system under scrutiny, which is of order QCD Λ for the lighter  Tables 2, 4  GeV and using flavor dependent ω , which are shown in rows 2 and 3 of Table 8. It is observed that flavor dependence significantly enhances the form factors, bringing them close to the expectation [5] based on HQET cosiderations (row 4 of Table 8). Consequently, the branching ratios of semileptonic and nonleptonic weak decays of ϒ get significantly enhanced.
We also use the QCD inspired method to determine ω i.e. =1.83 ω ϒ and subsequently obtain the form factors given in row 4 of Table 8. It may be pointed out that due to marginal change in value of ω in comparison to flavor dependent value (see Table 7), the change in form factors and branching ratios is negligible, therefore, we exclude these results for further discussion.

Semileptonic weak decays of ϒ ϒ ϒ ϒ
Using c B → ϒ form factor, appearing in b → c transition, and the decay rate formula given in (6), firstly, we calculate the branching ratios for semileptonic decays of ϒ at fixed 0.40 = ω GeV and are given in column 2 of Table 9. The predicted branching ratios of semileptonic weak decays of ϒ using flavor dependent effects are given in column 3 of Table 9. We find that the branching ratio of dominating semileptonic decay is separately, for semileptonic decays in Table 10.

Nonleptonic weak decays of ϒ ϒ ϒ ϒ
In this section, the analysis is extended to PV PP/ → ϒ decays. The effective weak Hamiltonian generating the dominant b quark decays involving c b → transition is given by For instance the decay amplitude for the color enhanced mode of the CKMfavored decays is given by Following the same procedure employed in Sections 3 and 4, we calculate the branching ratios for CKM-favored mode both for fixed 0.40 = ω GeV and for flavor dependent ω , and are presented in Table 11 as column 2 and 3 respectively. In addition to the decay constants given in (36) Table 13.

Summary and Discussions
In this paper, we have predicted the rare semileptonic and nonleptonic weak decays of ψ / J and ϒ meson. It may be mentioned that the present work differes from the previous ones [4,5] based on the BSW framework in three aspects: Firstly, in the light of HQS based BSW 2 model [12], we use the dipole 2 q dependence for the form factors • Including the flavor dependent effects through ω and this prescription, it is found that the our predictions agree well with those obtained by Sharma and Verma [5] using the HQET considerations.
• Though the predicted branching ratios in QCD inspired method of obtaining ω are marginally changed, the alternate calculation of ω for heavy quarkonium cc &bb states seems to support the flavor dependent effects on rare weak decays of ψ / J and ϒ mesons.
It is hoped that these branching ratios would lie in the detectable range and may be                Table 9: Branching Ratios (in the units of 10 10 − ) of Plν ϒ → decays