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 nonplanar theories without well-defined forward limits or good ultraviolet behavior.

• Received 15 September 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.061601