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تسجيل مستخدم جديدNotes on cluster algebras and some all-loop Feynman integrals
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We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2simeq A_1^2$, we show that penta-box ladder has an alphabet of $D_3simeq A_3$ and provide strong evidence that the alphabet of double-penta ladder can be identified with a $D_4$ cluster algebra. We relate the symbol letters to the ${bf u}$ 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 ${rm d}log$ representation, which allows us to predict higher-loop alphabet recursively; by applying such recursions to six-dimensional hexagon integrals, we also find $D_5$ and $D_6$ cluster functions for the two-mass-easy and three-mass-easy case, respectively.
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