Nonfactorizable Contribution to B-Meson Decays to s-Wave Mesons

There has been a growing interest in studying the nonfactorizable terms [1-4] of weak hadronic decays of charm and bottom mesons. We study the nonfactorizable contributions to various Cabibbo–Kobayashi–Maskawa (CKM) favored decays of B-mesons. Unfortunately, it has not been possible to calculate such contributions from the first principle, as these are non-perturbative in nature. Earlier attempts involved to find how much nonfactorizable contributions are required from the empirical details for weak charm hadronic decays [5-7]. We determine these contributions in the respective isospin I =1 2 and 3/2 amplitudes for B D B D → → π ρ / and B D → π * decay modes by taking NC = 3 to calculate the factorizable terms. The ratio of the nonfactorizable amplitude in these channels also seems to follow a universal value for all the above decay modes.


Introduction
There has been a growing interest in studying the nonfactorizable terms [1][2][3][4] of weak hadronic decays of charm and bottom mesons. We study the nonfactorizable contributions to various Cabibbo-Kobayashi-Maskawa (CKM) favored decays of B-mesons. Unfortunately, it has not been possible to calculate such contributions from the first principle, as these are non-perturbative in nature. Earlier attempts involved to find how much nonfactorizable contributions are required from the empirical details for weak charm hadronic decays [5][6][7]. We determine these contributions in the respective isospin I = 1 2 and 3/2 and B D → π * decay modes by taking N C = 3 to calculate the factorizable terms. The ratio of the nonfactorizable amplitude in these channels also seems to follow a universal value for all the above decay modes.

Methodology
The effective weak Hamiltonian for Cabibbo enhanced B-mesons decays is given by where q q q q where µ = m B 2 , the values of c 1 and c 2 are taken from [5], and Fierz transforming the product of two Dirac currents of (1) in N c color-space, we get And similar term for (cu)(db) , where λ a are the Gell-Mann matrices. By using (3) and its analogue we reduced the effective Hamiltonian to describe color-favored (CF) and color-suppressed (CS) decays, respectively.

Results and Discussion
We applied the isospin formalism, and express decay amplitudes in terms of isospin reduced amplitudes Branching ratio for two body B-meson decays to pseudoscaler mesons is related to decay amplitude where τ B is the life time of B-meson, V V ud cb is the product of the CKM matrix elements [1], p is the magnitude of the 3-momentum of the final state particles in the rest frame of B-meson and A B P P → ( ) 1 2 is the decay amplitude. We have calculated isospin reduced amplitudes, A D . , using the experimental value [1], where the factorizable parts are calculated by using BSW model [3], expressed as

( ) GeV
There are many calculations for form factors and decay constants, such as light-cone sum rules [8], perturbative QCD approach, and lattice QCD [9][10][11][12][13] etc. We write nonfactorizable part in terms of isospin C. G. coefficients as scattering amplitudes for spurion + B D → π process: Repeating the same procedure used above for B D → ρ and B D → π * decays the nonfactorizable amplitudes ratio can be obtained. For the sake of comparison we have summarized all the results in Table 1 given below.

Summary and Conclusions
The motivation for the exploration of nonfactorizable term has been the failure of the large-N c limit, which was supposed to be supported by the D-meson phenomenology, especially when extended to the B-meson sector. For instance, D-decays demand a negative value for a 2 , indicating N c → ∞ limit, whereas B-meson decays clearly favor positive value for a 2 . Therefore, it has been suggested to investigate the effect of nonfactorizable terms in the heavy quark decays keeping the real value of color N c = 3 . We determine A nf This behavior is understandable, since low momentum states are likely to be affected more through the exchange of soft gluons and can acquire larger non-factorizable contributions [8]. We observe that in all the decay modes, the non-factorizable isospin amplitude A nf 1 2 / bears the same ratio, with in the experimental errors, as well as same sign, A nf 3 2 / amplitude.