Mandelstam diagrams are not enough to study the high-energy behavior ofφ3theory

Viewed in the crossed channel, the ${\phi }^3$ Mandelstam diagrams for the Reggeon-particle cut represent repeated interaction between two of the three exchanged particles. A much larger class of Feynman diagrams is obtained by allowing all three exchanged particles to participate in the interaction, as required by Bose statistics. It is found that, in the limit of high energies, all diagrams of the same order in this larger class contribute comparably. In particular, the contributions from the Mandelstam and non-Mandelstam diagrams are of the same order of magnitude. This larger class of diagrams is summed in terms of an integral equation. It is found from this integral equation that if the momentum transfer is not too large, then the leading singularity in the angular momentum plane is a pole. As the momentum transfer increases this pole eventually disappears into a cut. These considerations are generalized, in a nontrivial way, to the case of the Reggeon-Reggeon cut. The signature factor plays a most important role, and some of the contributing diagrams show unexpected high-energy behavior. The profound effect of these additional diagrams on the program of Reggeon calculus is discussed.