Cluster algebras for planar conformal kinematics are identified as G(4,n) subalgebras and used to bootstrap the symbol of an 8-point three-loop wheel integral via D3 and new algebraic letters.
Cluster Polylogarithms for Scattering Amplitudes
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abstract
Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define "cluster polylogarithm functions". We find that all such functions of weight 4 are made up of a single simple building block associated to the A_2 cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A_2 building blocks arrange themselves to form a unique function associated to the A_3 cluster algebra. This A_3 function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n=7 amplitude as an example.
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Explicit three-loop computation of negative geometries for F(g,z) with all-loop resummation of one-cycle diagrams and extraction of the cusp anomalous dimension via z-integration.
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Kinematics, cluster algebras and Feynman integrals
Cluster algebras for planar conformal kinematics are identified as G(4,n) subalgebras and used to bootstrap the symbol of an 8-point three-loop wheel integral via D3 and new algebraic letters.
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Multi-Loop Negative Geometries
Explicit three-loop computation of negative geometries for F(g,z) with all-loop resummation of one-cycle diagrams and extraction of the cusp anomalous dimension via z-integration.