QCD Scattering Amplitudes and Prescriptive Unitarity

TL;DR

This paper develops a systematic framework for analyzing the maximally-transcendental parts of planar QCD scattering amplitudes, enabling the first bootstrap computation of six-gluon MHV amplitudes at the symbol level and revealing new analytic structures.

Contribution

It introduces a novel approach relating maximal transcendental parts to prescriptive unitarity integrals and classifies the associated prefactors using on-shell diagrams, advancing amplitude computations in QCD.

Findings

Computed two-loop prefactors for planar MHV gluon amplitudes at arbitrary multiplicity.

Constructed a complete symbol ansatz for two-loop six-gluon MHV amplitudes.

Identified a 137-letter alphabet likely complete for two-loop six-point massless MHV scattering.

Abstract

We present a systematic framework for the maximally-transcendental part of planar QCD scattering amplitudes and perform the first bootstrap computation of six-gluon MHV amplitudes in massless QCD at the symbol level. By analyzing the maximal weight projection of amplitudes at the integrand level, we relate their maximally-transcendental parts to prescriptive unitarity integrals. This reveals a novel analytic structure: the prefactors multiplying the functions of maximal transcendentality are identified with the four-dimensional leading singularities of the theory. As a consequence, these prefactors admit a complete classification and can be computed using on-shell diagrams, a formalism originally developed in $\mathcal{N}{=}4$ super Yang-Mills theory. As a concrete application, we determine the two-loop prefactors for planar MHV gluon amplitudes at arbitrary multiplicity. Combining these prefactors with recent advances in the planar two-loop six-point function space and explicit six-point prescriptive-unitarity input, we construct a complete symbol ansatz and uniquely fix the maximally-transcendental part of the two-loop six-gluon MHV QCD amplitudes by imposing physical constraints. The resulting symbols are expressible in a reduced 137-letter alphabet, suggesting that this alphabet is complete for two-loop six-point massless MHV scattering. We also discuss the implications for multi-collinear splitting and multi-soft functions.