Bootstrapping Six-Gluon QCD Amplitudes

TL;DR

This work extends the amplitude bootstrap program to massless QCD by constructing the symbol of the planar two-loop six-gluon amplitude in the $(-,-,+,+,+,+)$ helicity sector, focusing on the maximal-weight ($ ext{weight-4}$) terms. By leveraging four-dimensional leading singularities from on-shell diagrams and enforcing physical limits, the authors uniquely determine the weight-4 symbol, revealing a reduced alphabet of 137 letters out of 167 possible. They include fermions in the planar limit and demonstrate that the fermionic piece is fixed by a subset of constraints, with the complete result satisfying extended Steinmann relations. As byproducts, they extract novel symbol-level triple-collinear and double-soft splitting functions and uncover structure hints that connect to supersymmetric theories and perhaps flag-variety geometry, suggesting broader applicability and future extensions to higher multiplicities and function-level analyses.

Abstract

We present a symbol-level bootstrap construction of the planar, two-loop six-gluon scattering amplitude for the --++++ helicity configuration in QCD, focusing on the maximal weight pieces-the "most complicated terms" in the sense of Lipatov et al. Building on recent advances in the understanding of the relevant function space, we incorporate as a crucial new ingredient the complete set of leading singularities, obtained from an explicit analysis of on-shell diagrams. The resulting expressions are manifestly conformally invariant and clarify the structure of previous five-particle results. Combining this with the symbol bootstrap, we show that constraints from physical limits are sufficient to uniquely determine the answer. We thus obtain the first concrete characterization of two-loop six-gluon amplitudes at the symbol level and at highest weight. Remarkably, we find that the effective function space involves only 137 symbol letters, significantly fewer than the full set of 167 possible letters, suggesting a yet-unexplained underlying structure akin to that seen in maximally supersymmetric Yang-Mills theory. From the novel amplitude results we extract previously unknown symbol-level results describing two-loop triple collinear and double soft limits.

Figures (1)

• Figure 1: On-shell diagrams giving rise to leading singularities $R_{i,j}$, for the $\mathord{-}\mathord{-}\mathord{+}\mathord{+}\mathord{+}\mathord{+}$ helicity configuration.