Novel cluster-algebraic letters for 5- and 6-point QCD processes

Abstract

By breaking dual conformal invariance, we transform cluster-algebraic predictions for the alphabet of 9-point amplitudes in $\mathcal{N}=4$ super Yang-Mills theory to analogous predictions for 5- and 6-point processes in QCD. We start by obtaining, for the first time, candidate letters for 6-point processes with one massive external leg, and discover that they surprisingly also contain nested square roots. We confirm that our results essentially contain the alphabet of all 1-loop integrals with these kinematics, and in their massless limit also the recently computed alphabet of finite, planar 2-loop amplitudes for 6-point massless QCD processes. In the latter case, we additionally find 162 letters that may appear at higher loops. We similarly produce candidate letters for 5-point 2-mass processes, whose comparison with the literature reveals a nontrivial overlap that also includes new letters.

Figures (10)

• Figure 1: The dual conformal 4-mass box integral reduces to the Lorentz-invariant 3-mass triangle integral in the reference frame where $x_7\to \infty$.
• Figure 2: Breaking the dual conformal invariance of the DCI subalphabet of Henke:2021ity with seven points and two adjacent masses, we obtain for the first time candidate letters for 6-point 1-mass processes in QCD. Their massless limit, after cyclic symmetrisation, essentially encompasses the recent direct results of Abreu:2024feiHenn:2025xrc, and includes new predictions.
• Figure 3: There exist two inequivalent ways to break the symmetry of the DCI subalphabet with six points and three adjacent masses, and we take the union of the resulting LI letters for 5-point 2-mass hard integrals (the easy configuration is treated straightforwardly as well). Also considering all relevant permutations of the external legs, we find a highly nontrivial partial overlap with the alphabet of Abreu:2024yit.
• Figure 4: Integral topologies expected to contain the nested square root letters of eq. \ref{['eq:nested_sq_letters']}.
• Figure 5: The 5-point 2-mass LI alphabet is obtained from the 6-point 3-mass 'medium' DCI subalphabet, in complete analogy with the derivation of the 6-point 1-mass alphabet illustrated in the left half of figure \ref{['fig:7to6pt']}.