Spectral Forms for Four-Point Functions

Abstract

Continuing the program initiated in a recent work on three-point functions, we present a detailed study of the establishment in source theory of spectral forms for four-point functions. Both single- and double-spectral forms are treated, and the particles are all allowed different masses (although some inequalities among the masses are employed). Most of the work is carried out for the lowest-order nontrivial contributions, but some considerations are also presented for higher-order contributions. The major new element here, relative to the three-point-function work, is the matter of mass extrapolation. That is, the double-spectral form as first obtained refers to external particles with certain off-shell momenta, since it is derived from a causal realization of the amplitude in which the momenta are thus specified. So one must then extrapolate to real external particles, and that in turn causes an extrapolation of the original spectral-mass domain. A somewhat different extrapolation occurs for the single-spectral form. The paper concludes by generally reviewing the source-theoretic procedures for establishing spectral forms, with some speculations regarding further developments, and by presenting a brief comparison with the conventional, analytic approaches.