A bstract We study the recently introduced curve integral formalism that defines a new family of formulas for the scattering amplitudes of the colored scalar tr ϕ 3 theory. We find that the curve integral manifests a very surprising fact about these amplitudes: the dependence on the number of particles, n , and the loop order, L , is effectively decoupled in these formulas. We derive the curve integrals at tree-level for all n . We then show that, for higher loop-order, it suffices to study the curve integrals for L -loop tadpole-like amplitudes, which have just one particle per color trace-factor. By combining these tadpole-like formulas with the tree-level results, we find formulas for the all n amplitudes at L loops. We illustrate this result by giving explicit curve integrals for all the amplitudes in the theory, including the non-planar amplitudes, through to two loops, for all n .
Arkani–Hamed et al. (Tue,) studied this question.