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 $1/N_c$, concentrating on quark exchange mechanism. The contradiction originates in the fact that the standard factorization assumption approximates the asymptotic final states by a non-orthogonal 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 final state interaction restores duality. Three $O(1/N_c)$ effects are exhibited: i) a proper treatment of orthogonality yields a global correction $1/N_c\to 1/2N_c$ within a generalized factorization \`a la BSW, (such a factor was present in an Ansatz by Shifman), ii) the distorsion 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, 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 $1/N_c\to 1/2N_c$ is the dominant correction.