Higher Order Two Step Mechanisms in Nucleon Antinucleon Annihilation and the OZI Rule

We evaluate three meson doorway mechanisms for nucleon-antinucleon annihilation at rest for the first time. Detailed results are presented for the final state phi pi0 originating from the 3S1 initial state and for the phi rho channel originating from 1S0. The results presented also include the improved contributions from two meson doorway states and from the tree diagrams. For all the channels considered a consistent explanation of large and small OZI violations emerges.