Analyticity Properties of the Momentum-Space Vertex Function

Abstract

The problem of finding the maximal domain of analyticity of the momentum-space vertex function implied solely by Lorentz covariance, local commutativity, and mass spectrum with thresholds not all zero, is restated as a holomorphy envelope problem. If only one of the threshold masses is nonzero, the problem can be divided into two simpler problems. The triangle diagram suggests the holomorphy envelope of one, but only gives an upper bound on the holomorphy envelope of the other. It is shown that the boundary of the latter most probably consists of the three cuts and a single quasi-analytic hypersurface with certain specified properties. If two of the threshold masses are equal and the third vanishes, an appealing conjecture suggested by the triangle diagram is shown to contradict a generalization of Jost's example. In the general case some upper and lower bounds are obtained. The Källén-Toll representation and conjecture are briefly discussed. The relation of the first three terms in the representation of the position-space vertex function to the Mercedes diagram in perturbation theory is displayed, and it is shown that there is no analogous relation for the fourth term. In the single threshold case this fourth term must account for the singularities of the momentum-space vertex function on the quasi-analytic hypersurface which bounds the holomorphy envelope. The motivation for studying the analyticity properties of vertex diagrams is discussed, and the simplest totally symmetric ones are investigated.