Star Integrals, Convolutions and Simplices

Abstract

We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop $n$-gon integrals in $n$ dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli's formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the $6d$ hexagon and $8d$ octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.

23 pages

References in corpus (23)

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• Writing CFT correlation functions as AdS scattering amplitudes
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
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• Hexagon Wilson loop = six-gluon MHV amplitude
• Conformal Regge theory
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Uniqueness of two-loop master contours
• Notes on the scattering amplitude / Wilson loop duality
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• AdS Field Theory from Conformal Field Theory
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• The Yangian origin of the Grassmannian integral
• New differential equations for on-shell loop integrals
• Loops and trees
• On BCFW shifts of integrands and integrals
• Analytic Results for MHV Wilson Loops
• Generalised ladders and single-valued polylogarithms
• The six-point remainder function to all loop orders in the multi-Regge limit
• Some analytic results for two-loop scattering amplitudes
• Uplifting Amplitudes in Special Kinematics

Cited by in corpus (35)

• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• How to Succeed at Holographic Correlators Without Really Trying
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Elliptic Feynman integrals and pure functions
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• An analytic solution for the equal-mass banana graph
• On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$
• A Note on (Non)-Locality in Holographic Higher Spin Theories
• Holographic Four-Point Functions in the (2, 0) Theory
• Yangian Bootstrap for Conformal Feynman Integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Three Point Energy Correlators in the Collinear Limit: Symmetries, Dualities and Analytic Results
• On-Shell Methods for the Two-Loop Dilatation Operator and Finite Remainders
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• Logarithmic forms and differential equations for Feynman integrals
• The elliptic double box and symbology beyond polylogarithms
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Conformal anomaly of generalized form factors and finite loop integrals
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Building Bases of Loop Integrands
• Mellin Amplitudes for Fermionic Conformal Correlators
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• The Two-Loop Remainder Function for Eight and Nine Particles
• Four-point function in general kinematics through geometrical splitting and reduction
• Exploring Perturbative Conformal Field Theory in Mellin space
• An Infinite Family of Elliptic Ladder Integrals
• Geometrical splitting and reduction of Feynman diagrams
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory