Abstract
We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a C_{2} cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding C_{2} inside the A_{3} cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes and for form factors recently considered in N=4 super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the G(4,8) cluster algebra.
Highlights
We initiate the study of cluster algebras in Feynman integrals in dimensional regularization
We provide evidence that four-point Feynman integrals with one off-shell leg are described by a C2 cluster algebra, and we find cluster adjacency relations that restrict the allowed function space
How general is the appearance of cluster algebras in quantum field theory? On the one hand, all known cases are related to planarity and concern finite parts of amplitudes in N 1⁄4 4 supersymmetric Yang-Mills (SYM), which have additional symmetries [18,19,20]
Summary
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany. DESY Theory Group, DESY Hamburg, Notkestrasse 85, 22607 Hamburg, Germany (Received 22 December 2020; accepted 28 January 2021; published 5 March 2021). We provide evidence that four-point Feynman integrals with one off-shell leg are described by a C2 cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. Planar six- and seven-gluon scattering amplitudes appear to be governed by the finite A3 and E6 cluster algebras, respectively This suggests that their function space is a certain set of generalized polylogarithms, which is the starting point for the bootstrap program [6,7]. Further constraints come from the absence of discontinuities in overlapping channels, the (extended) Steinmann relations [8,9,10], which are closely related to cluster adjacency properties [11,12] These findings have been instrumental for bootstrapping amplitudes to very high loop orders; see, e.g., [13,14,15,16], and the review [17]. Mutating a cluster ða; BÞ along the kth variable, with 1 ≤ k ≤ d, we obtain the new cluster ða0; B0Þ, whose exchange matrix B0 is related to the previous one by
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