Loops and trees

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.