Dual-conformal regularization of infrared loop divergences and the chiral box expansion

Abstract

We revisit the familiar construction of one-loop scattering amplitudes via generalized unitarity in light of the recently understood properties of loop integrands prior to their integration. We show how in any four-dimensional quantum field theory, the integrand -level factorization of infrared divergences leads to twice as many constraints on integral coefficients than are visible from the integrated expressions. In the case of planar, maximally supersymmetric Yang-Mills amplitudes, we demonstrate that these constraints are both sufficient and necessary to imply the finiteness and dual-conformal invariance of the ratios of scattering amplitudes. We present a novel regularization of the scalar box integrals which makes dual-conformal invariance of finite observables manifest term by term, and describe how this procedure can be generalized to higher loop-orders. Finally, we describe how the familiar scalar boxes at one-loop can be upgraded to ‘chiral boxes’ resulting in a manifestly infrared-factorized, box-like expansion for all one-loop integrands in planar, $$ \mathcal{N}=4 $$ super Yang-Mills. Accompanying this note is a Mathematica pack-age which implements our results, and allows for the efficient numerical evaluation of any one-loop amplitude or ratio function.