Feynman diagrams in terms of on-shell propagators

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.

13 pages, 11 figures, accepted for publication in the Physical Review D

References in corpus (5)

• Direct extraction of one-loop integral coefficients
• A Tree-Loop Duality Relation at Two Loops and Beyond
• Loops and trees
• Multi-Parton Scattering Amplitudes via On-Shell Methods
• From multileg loops to trees (by-passing Feynman's Tree Theorem)