Cluster algebras for Feynman integrals

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 $\mathcal{N}=4$ 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.

10 pages, 5 figures

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• Leading-Color Two-Loop Amplitudes for Four Partons and a W Boson in QCD
• A Three-Point Form Factor Through Five Loops
• Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC
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• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Antipodal Self-Duality for a Four-Particle Form Factor
• Two-loop master integrals for a planar topology contributing to $pp\to t\bar{t}j$
• Notes on cluster algebras and some all-loop Feynman integrals
• Symbol Alphabets from the Landau Singular Locus
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbol Alphabets from Tensor Diagrams
• Planar three-loop master integrals for $2\to 2$ processes with one external massive particle
• Symbology for elliptic multiple polylogarithms and the symbol prime
• First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Bootstrapping a Two-Loop Four-Point Form Factor
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• One-loop hexagon integral to higher orders in the dimensional regulator
• The recursive structure of Baikov representations I: generics and application to symbology
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering
• A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses
• Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
• Learning scattering amplitudes by heart