Spectral Representations in Perturbation Theory. I. Vertex Function

The vertex operator is examined in lowest order perturbation theory. It is found that, as a function of the invariant momentum transfer $q^2={\mathrm{q}}^2-{q_0}^2$, it is analytic in a cut plane with the branch point on the negative real axis. A spectral representation (dispersion relation) may therefore be inferred. The threshold of the spectrum depends on the masses of all fields involved unless certain inequalities hold between the masses of the incident and outgoing particles on one hand and the particles in intermediate states on the other; in that case the threshold depends only on the intermediate masses.