Abstract
We relate an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l-loop graph. Causality requires that the propagators of the trees have a modified iδ prescription, and we present a simple formula for the correct iδ prescription.
Highlights
Introduction.—Relating loop integrals to trees goes back to Feynman [1]
The integrand of the phase space integral corresponds to a tree, not a forest, and is obtained from the original integrand by exactly l cuts
In this Letter we present the generalization to an arbitrary loop number l
Summary
Introduction.—Relating loop integrals to trees goes back to Feynman [1]. The Feynman tree theorem allows us to relate an l-loop Feynman integral with N internal propagators to an l-fold phase space integral with a number of cuts Ncut on the original integrand, with Ncut ranging from l to N. The integrand of the phase space integral corresponds to a tree, not a forest, and is obtained from the original integrand by exactly l cuts. Notation.—Let Γ be a Feynman graph with l loops, n external lines, and N internal edges. Let us assume that the polynomial PΓ is such that all energy integrations over half-circles at infinity vanish.
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