Consistency conditions from generalized-unitarity

TL;DR

The work reframes gauge anomaly cancellation in purely on-shell terms, showing that locality constraints on rational terms in unitarity-based loop constructions reproduce anomaly-like consistency conditions. In $D=4$, locality forces the parity-odd rational term to cancel spurious residues unless the cubic Casimir vanishes, yielding $d^{abc} f^{de}{}_a = 0$. In $D=6$, the symmetric trace condition $sTr(1234)=0$ either preserves locality or, if violated, signals the Green-Schwarz two-form via a specific rational term that reproduces GS on-shell. Overall, anomalies arise as obstructions to enforcing locality on the unitary S-matrix in an on-shell framework, with the GS mechanism emerging naturally when locality demands additional degrees of freedom.

Abstract

In the modern on-shell approach, the perturbative S-matrix is constructed iteratively using on-shell building blocks with manifest unitarity. As only gauge invariant quantities enter in the intermediate steps, the notion of gauge anomaly is absent. In this letter, we rephrase the anomaly cancellation conditions in a purely on-shell language. We demonstrate that while the unitarity-methods automatically lead to a unitary S-matrix, the rational terms that are required to enforce locality, invariably give rise to inconsistent factorization channels in chiral theories. In four-dimensions, the absence of such inconsistencies implies the vanishing of the cubic Casimir of the gauge group. In six-dimensions, if the symmetric trace of four generators does not vanish, the rational term develops a factorization channel revealing a new particle in the spectrum: the two-form of the Green-Schwarz mechanism. Thus in the purely on-shell construction, the notion of gauge-anomaly is replaced by the difficulty to consistently impose locality on the unitary S-matrix.