Some Analytic Properties of the Vertex Function

The absorptive part of the vertex function $F[k^2, p^2, {\mathrm{}(k-p)\mathrm{}}^2]$ is an analytic function of the mass variables $k^2$ and $p^2$. On the basis of causality and the spectral conditions, the region of regularity $D\left(\sigma \right)$ of the absorptive part $A(k^2, p^2, {\sigma }^2)$ is obtained for fixed values of $\sigma >~c$. The boundary of $D\left(\sigma \right)$ is calculated explicitly for the case $k^2=p^2$, which is of interest in connection with form factors. By the use of examples based upon perturbation theory, it is shown that this boundary is characteristic for the physical assumptions that have been made. The intersection $D$ of all domains $D\left(\sigma \right)$ for $\sigma >~c$ is the region for which $F$ is an analytic function of all three variables, with ${\mathrm{}(k-p)\mathrm{}}^2$ in the cut plane and ($k^2, p^2$) in $D$. The relation of these general results to the composite structure of particles is discussed.