More Integral Representations for Scattering Amplitudes with Complex Singularities

Abstract

Previous work is generalized in order to achieve a better understanding of the role of complex singularities in connection with integral representations. The most general conditions under which the box‐diagram contribution to a scattering amplitude satisfies a representation with real integration contours are derived. Explicit representations are derived in several special cases. It is frequently found possible to obtain representations that are essentially of the Mandelstam type, although the more general Bergman‐Oka‐Weil representation must be invoked in general. One example of a three‐dimensional representation is given, which exhibits the analytic structure in one of the external masses in addition to the kinematical variables. The significance of the ``physical sheet'' is discussed.