The analytic properties of the scalar vertex function

Abstract

A single spectral representation of the vertex function F, for the conversion of a scalar particle binto a scalar particle d via interaction with a third scalar particle s, is obtained heuristically by using unitarity to evaluate the contribution to Im F from a state ( a c ) with the same internal quantum numbers as ( d b ) and then making a one-pole approximation to the amplitude for d b → a c and a constant approximation to the vertex function for the vertex (cas). For the case where the masses satisfy the inequality m a + m c ⩾ m b + m d, the above approximations are shown to give the same result as that obtained from the Feynman parametrisation of the vertex function arising from the triangle diagram. Two methods are then used to obtain a spectral representation for the triangle diagram vertex function for all allowed mass configurations, including cases for which the threshold is anomalous. The first is to evaluate the triangle diagram vertex function directly; the second is to continue in m b 2 and m d 2 the result for the case m a + m c ⩾ m b + m d.