Spectral Forms for Three-Point Functions

Abstract

Single- and double-spectral forms for three-point functions are studied, in a detailed manner, within the framework of source theory. The methods developed, which are applicable beyond the present problem, are based on causal considerations and appear to provide some simplifications and advances over the conventional analytic methods. The spectral variables are any one or two of the momentum scalar products (${{p}_{\ensuremath{\alpha}}}^{2}, {{p}_{\ensuremath{\beta}}}^{2}, {{p}_{\ensuremath{\gamma}}}^{2}$) on which the three-point function depends, ${{p}_{\ensuremath{\alpha}}}^{2}$ and ${{p}_{\ensuremath{\alpha}}}^{2}\ensuremath{-}{{p}_{\ensuremath{\gamma}}}^{2}$, to be specific. After studying the lowest-order nontrivial contributions, a source-theoretic calculational scheme for contributions of arbitrary order is qualitatively developed, and it is used in establishing the spectral forms for general order. In lowest order the spectral weight functions are explicitly given, while in general order the main concern is the existence of the spectral forms. The spectral forms considered here are only ones with normal thresholds, and the methods give regions of those variables not in spectral form (${{p}_{\ensuremath{\beta}}}^{2}\ensuremath{-}{{p}_{\ensuremath{\gamma}}}^{2}$ and ${{p}_{\ensuremath{\beta}}}^{2}$) for which such spectral forms of general order occur, with all particles being allowed different masses. For the single-spectral studies the region is in agreement with that obtained conventionally. The region for the double-spectral form is all spacelike values of ${{p}_{\ensuremath{\beta}}}^{2}$; double-spectral forms for three-point functions of general order do not appear to have been investigated previously.