New Representations of the Perturbative S-Matrix

TL;DR

This work tackles the challenge of representing perturbative scattering amplitudes without forward-limit ambiguities by introducing a $Q$-cut representation. Derived from a partial-fraction decomposition of the Feynman expansion and a multi-parameter deformation of loop momenta, each off-shell propagator coefficient is expressed as a sum over $Q$-cuts built from gauge-invariant tree amplitudes with shifted on-shell momenta. A democratic contour prescription for the $Q$-cut integrals is proposed, ensuring results reproduce the conventional Feynman contour and avoiding forward-limit divergences. The method is demonstrated at one loop with explicit bubble and all-plus gluon amplitudes and is argued to extend naturally to higher loops, including non-planar theories, opening the door to all-orders computations directly from tree data. This framework complements scattering-equation approaches and could integrate with existing reduction techniques to streamline perturbative calculations.

Abstract

We propose a new framework to represent the perturbative S-matrix which is well-defined for all quantum field theories of massless particles, constructed from tree-level amplitudes and integrable term-by-term. This representation is derived from the Feynman expansion through a series of partial fraction identities, discarding terms that vanish upon integration. Loop integrands are expressed in terms of "Q-cuts" that involve both off-shell and on-shell loop-momenta, defined with a precise contour prescription that can be evaluated by ordinary methods. This framework implies recent results found in the scattering equation formalism at one-loop, and it has a natural extension to all orders---even non-planar theories without well-defined forward limits or good ultraviolet behavior.