Complex-propagator insertions in the square diagram

Summary  The problem of inserting a complex propagator in the crossed channel of the square diagram is considered. The propagator is spectrally represented and the square diagram with full propagator insertion is expressed as a spectral integral of square diagrams over an internal mass variable. The analytic properties of such diagrams in two complex variables, one internal mass and one external energy, are investigated in detail. These properties are then used to find the analytic properties of the full diagram with propagator insertions.