Loops and trees

Abstract

We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to L-loop expressing the loops as integrals over the on-shell phase space of exactly L particles. We argue that the corresponding integrand for L>2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude. Beyond the planar limit, abandoning direct integral representations, we propose that loops continue to be determined implicitly by the forward limit of physical connected trees, and we formulate a precise conjecture along this line. Finally, we set up technology to compute forward amplitudes in supersymmetric theories, in which specific simplifications occur.

36 pages + appendices, 8 figures

References in corpus (8)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Direct Extraction Of One Loop Rational Terms
• On the equivalence between the Boltzmann equation and classical field theory at large occupation numbers
• From loops to trees by-passing Feynman's theorem
• A Tree-Loop Duality Relation at Two Loops and Beyond
• Particle production in field theories coupled to strong external sources. II: Generating functions
• One-Loop Gluonic Amplitudes from Single Unitarity Cuts
• Towards a Loop-Tree Duality at Two Loops and Beyond

Cited by in corpus (1)

• Notes on the scattering amplitude / Wilson loop duality