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.Comment: 4+1 pages, 4 figure
