Truncated cluster algebras and Feynman integrals with algebraic letters

Abstract

We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) $G_{+}(4,n)/T$ for the $n$-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with $n=7,8,9$, we find finite cluster algebras $D_4$, $D_5$ and $D_6$ respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine $D_4$ cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of $G_{+}(4,8)/T$ with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of $38$ rational letters and $5$ algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight $4$, by imposing last-entry conditions inspired by the $n=8$ double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the $n=8$ double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop $d\log$ forms, and we find a remarkable pattern about the appearance of algebraic letters.

v2: typos corrected and a comment on smaller alphabet added

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Cited by in corpus (16)

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• First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production
• Schubert Problems, Positivity and Symbol Letters
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Intersection theory rules symbology
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• An Infinite Family of Elliptic Ladder Integrals
• The recursive structure of Baikov representations I: generics and application to symbology
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• The two-loop eight-point amplitude in ABJM theory
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills