Bootstrapping Six-Gluon QCD Amplitudes

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.

💡 Research Summary

In this work the authors present the first complete symbol‑level construction of a planar, two‑loop six‑gluon scattering amplitude in massless QCD for the maximally helicity‑violating (MHV) configuration with two negative and four positive helicities (‑‑++++). The calculation is restricted to the maximal transcendental weight‑four part of the amplitude, which corresponds to the most complicated terms in the sense of Lipatov and collaborators. By focusing on this highest‑weight sector the authors can exploit a number of powerful structural constraints that are otherwise hidden in the full amplitude.

The starting point is a decomposition of the colour‑stripped amplitude into rational prefactors (R_i) multiplied by transcendental functions (f_j). The rational prefactors encode the helicity‑dependent kinematic information, while the functions (f_j) belong to a known space of iterated integrals (multiple polylogarithms) that appears at two loops in planar QCD. The crucial observation, motivated by earlier work on (\mathcal{N}=4) super‑Yang‑Mills (sYM), is that at maximal weight the rational prefactors are completely determined by four‑dimensional leading singularities obtained from on‑shell diagrams. The authors compute all such leading singularities for the six‑point MHV configuration; they are given by simple rational functions of spinor‑helicity brackets (\langle ij\rangle) (eqs. (5) and (15)) and are manifestly invariant under the dual conformal generator. This provides a compact basis ({R_{i,j}}) (seven independent objects) for the prefactors.

To isolate the finite, infrared‑subtracted part of the amplitude the authors define a hard function (H^{(2)}) (eq. (6)) by removing the known one‑ and two‑loop infrared operators (I^{(1)}) and (I^{(2)}). At symbol level only the weight‑four pieces survive, and the authors construct an ansatz for the symbol of (H^{(2)}) as a linear combination of the seven rational prefactors multiplied by unknown weight‑four symbols (G_{i,j}). The symbols (G_{i,j}) are required to belong to the planar two‑loop six‑point function space, which is built from a set of 167 possible alphabet letters (arguments of logarithms). However, by imposing a series of physical and discrete symmetry constraints the authors dramatically reduce the effective alphabet.

The constraints are threefold:

1. Discrete symmetries – cyclic invariance and a flip symmetry ((123456)\leftrightarrow(216543)) force the unknown symbols to transform in the same way as the prefactors, cutting the number of independent parameters to 2 412.

2. Absence of spurious poles – the rational prefactors contain high‑order poles in non‑adjacent spinor brackets. The symbols must vanish sufficiently fast at these loci, which translates into linear conditions on the coefficients of the ansatz.

3. Physical collinear and soft limits – in the limit where two adjacent momenta become collinear, the six‑point hard function must factorise into a five‑point hard function times a known one‑loop splitting function (eq. (10)). Similarly, triple‑collinear and double‑soft limits impose factorisation conditions (eqs. (13) and (11)). The authors enumerate all such constraints in Table I; together they provide exactly 2 412 independent linear equations, matching the number of unknowns.

Solving this linear system yields a unique solution for the weight‑four symbol of the planar two‑loop six‑gluon MHV amplitude. Remarkably, the resulting symbol alphabet collapses to only 137 letters, a substantial reduction from the full 167‑letter set. This reduced alphabet coincides with the one previously identified in studies of two‑loop six‑point Wilson loops with Lagrangian insertions, hinting at a deeper, yet‑to‑be‑understood algebraic structure reminiscent of the cluster‑algebra picture in (\mathcal{N}=4) sYM.

The authors also extend the analysis to include fermion loops, working in the planar limit with a large number of flavours (N_f) while keeping the ratio (N_f/N_c) fixed. They decompose the QCD hard function as \