The Discontinuities of the Triangle Graph as a Function of an Internal Mass. II

Abstract

We consider the discontinuities of the triangle‐graph amplitude as a function of an internal mass variable. These discontinuities are important, since they form the kernel of the Aitchison‐Anisovich integral equation, which is derived from the Khuri‐Treiman three‐body final‐state‐interaction dispersion relation. We evaluate the discontinuities by explicitly performing the Feynman α integrations. We also discuss their analytic continuations. Finally we consider the applicability of the Cutkosky rules to such an internal mass variable discontinuity. It is argued that these rules must be modified in two ways. One of these is straightforward, having to do with the appearance of spacelike masses. The other is more involved and is a consequence of the results of homology theory. We apply the modified Cutkosky rules to the triangle‐graph discontinuities and obtain the same results as found by the direct method, so confirming the modifications which we have made.