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Notes on cluster algebras and some all-loop Feynman integrals

A preprint version of the article is available at arXiv.

Abstract

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is \( {D}_2\simeq {A}_1^2 \), we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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He, S., Li, Z. & Yang, Q. Notes on cluster algebras and some all-loop Feynman integrals. J. High Energ. Phys. 2021, 119 (2021). https://doi.org/10.1007/JHEP06(2021)119

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Keywords

  • Scattering Amplitudes
  • Wilson
  • ’t Hooft and Polyakov loops