Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion

TL;DR

This work addresses two key issues in one-loop amplitudes for planar $\mathcal{N}=4$ SYM: the hidden symmetry under dual-conformal invariance (DCI) prior to integration and the obfuscation of such symmetry by standard IR regulators. It introduces a DC I regularization that acts at the integrand level, making DC invariance manifest term-by-term for finite observables, and provides a complete integrand-level framework based on leading singularities and momentum-twistor techniques. The authors develop a chiral box expansion that aligns the integrand residues with the physical quad-cuts, while accounting for infrared divergences through convergent, parity-odd, triangle contributions, thereby enabling a regulator-independent, DC-invariant description of all one-loop ratio functions. They also outline higher-loop generalizations and non-planar extensions, and provide a Mathematica package to implement and numerically evaluate the proposed formalism, enhancing both analytic and computational access to one-loop amplitudes in this foundational theory.

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, N=4 super Yang-Mills. Accompanying this note is a Mathematica package which implements our results, and allows for the efficient numerical evaluation of any one-loop amplitude or ratio function.