Feynman diagrams in terms of on-shell propagators

TL;DR

This paper presents an alternative way to express Feynman diagrams using on-shell propagators, simplifying calculations and providing new insights into their imaginary parts and unitarity properties.

Contribution

It introduces a novel representation of Feynman diagrams with on-shell propagators, linking to Kadyshevsky's expansion and aiding in unitarity-based computations.

Findings

Equivalent expression for Feynman diagrams using on-shell propagators.

Simplified calculation of the imaginary parts of diagrams.

Application to scalar models and QED demonstrating practical advantages.

Abstract

It is shown that the usual expression for a Feynman diagram in terms of the Feynman propagator $\Delta_F(x-y)$ can be replaced by an equivalent expression involving the positive-energy on-shell propagator $\Delta^+(x-y)$, supplemented by appropriate functions associated with time-ordering. When this alternate way of expressing a Feynman diagram is Fourier transformed into momentum space, the momentum associated with each function $\Delta^+(x-y)$ is on-shell, and is only conserved at each vertex if an energy is attributed to the contributions of the time-ordering functions. The resulting expression is analogous to what Kadyshevsky had obtained by deriving an alternate expansion for the $S$--matrix. A detailed explanation of how this alternate expansion is derived is given, and it is shown how it provides a straightforward way of determining the imaginary part of a Feynman diagram, which makes it useful when using unitarity methods for computing a Feynman diagram. By considering a number of specific Feynman diagrams in self-interacting scalar models and in QED, we show how this alternate approach can be related to the old perturbation theory and can simplify direct calculations of Feynman diagrams.