Factorization versus duality in nonleptonic decays: A quark model approach.

We study in a quark model the contradiction between factorization and duality found in nonleptonic decays at next to leading order in 1/${\mathit{N}}_{\mathit{c}}$, concentrating on the quark exchange mechanism. The contradiction originates in the fact that the standard factorization assumption approximates the asymptotic final states by a nonorthogonal set of states, thus leading to an overcounting of the decay probability. We consider a system with two heavy quarks treated as classical color sources with constant velocity, and two mass-degenerate antiquarks. Exploiting permutation symmetry in an adiabatic approximation, we find that the final state interaction restores duality. Three O(1/${\mathit{N}}_{\mathit{c}}$) effects are exhibited: (i) a proper treatment of orthogonality yields a global correction 1/${\mathit{N}}_{\mathit{c}}$\ensuremath{\rightarrow}1/2${\mathit{N}}_{\mathit{c}}$ within a generalized factorization in the manner of BSW (such a factor was present in an ansatz by Shifman); (ii) the distortion of the meson wave functions at the time of the weak decay; (iii) relative phases generated by the later evolution. The latter effect becomes dominant for light antiquarks or for a small velocity of the final mesons, and may thoroughly modify the factorization picture. For exclusive decay it may interchange the role of class I and class II final channels (but it does not influence the sum I plus II), and for semi-inclusive decay it may lead to an equal sharing of the probability between the two sets of final states. In the heavy antiquark and large velocity limit, the replacement 1/${\mathit{N}}_{\mathit{c}}$\ensuremath{\rightarrow}1/2${\mathit{N}}_{\mathit{c}}$ is the dominant correction.