Abstract
Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster \( \mathcal{A} \)-coordinates on Gr(4, 7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle amplitudes we initiate the systematic study of the symbols of heptagon functions. We find that there is exactly one such symbol of weight six which satisfies the MHV last-entry condition and is finite in the 7 ∥ 6 collinear limit. This unique symbol is both dihedral and parity-symmetric, and remarkably its collinear limit is exactly the symbol of the three-loop six-particle MHV amplitude, although none of these properties were assumed a priori. It must therefore be the symbol of the threeloop seven-particle MHV amplitude. The simplicity of its construction suggests that the n-gon bootstrap may be surprisingly powerful for n > 6.
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Drummond, J.M., Papathanasiou, G. & Spradlin, M. A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon. J. High Energ. Phys. 2015, 72 (2015). https://doi.org/10.1007/JHEP03(2015)072
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DOI: https://doi.org/10.1007/JHEP03(2015)072