Loops and trees

TL;DR

The paper develops a causality-driven framework to relate loop amplitudes to forward, on-shell tree data by using retarded boundary conditions, generalizing Feynman’s tree theorem to L loops. It shows that for L>2, Minkowski-space representations are obstructed except in planar gauge theories, where forward data can yield physical amplitudes; for non-planar theories a forward-limit conjecture is proposed to implicitly determine loops. The Schwinger-Keldysh formalism is employed to define response functions and provide practical rules that reproduce known 1-loop results and guide higher-loop manipulations. Supersymmetry is leveraged to achieve improved ultraviolet behavior and explicit N=4 SYM examples demonstrate how forward amplitudes can dictate 4- and 5-point 1-loop integrands, highlighting the potential of forward-tree data to streamline loop calculations across theories, including finite-temperature contexts.

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.

Figures (8)

• Figure 1: (a) A 2-loop electron self-energy diagram. The photon energies flowing in it can be parallel or anti-parallel as in (b) and (c), each case having to be considered separately in order to respect energy positivity. In some other diagrams (d), the energies in two cut propagators cannot be compared as they do not add nor subtract in any denominator.
• Figure 2: Color structures for forward amplitudes are in one-to-one correspondence with cut planar diagrams in open-string theory.
• Figure 3: The three-step analytic continuation which takes retarded amplitudes to time-ordered ones: from Minkowski (real $q^0$) to Euclidean signature, then in Euclidean signature from the positive to the negative imaginary axis, then from Euclidean to Minkowski signature through a Wick rotation.
• Figure 4: Examples of tree and one-loop Schwinger-Keldysh diagrams for response functions.
• Figure 5: Anatomy of a 2-loop diagram containing a $\hbar$-suppressed vertex, shown as the dotted one. The retarded propagators on the figure are really strings of retarded propagators with insertions of external fields, as illustrated on the right.