Comments on all-loop constraints for scattering amplitudes and Feynman integrals

Abstract

We comment on the status of "Steinmann-like" constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar $\mathcal{N}=4$ super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for $n=6,7$ to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any $n$; with suitable normalization such as minimal subtraction, they hold for $n=8$ MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for $n$-point MHV amplitudes derived from $\bar{Q}$ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

26 pages, 1 figure; v2: references added, typos corrected and improved discussions on Sklyanin brackets

References in corpus (16)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Bootstrapping an NMHV amplitude through three loops
• New differential equations for on-shell loop integrals
• Cluster algebras for Feynman integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Exponential suppression with four legs and an infinity of loops
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Symbol Alphabets from Tensor Diagrams
• Bootstrapping six-gluon scattering in planar $\mathcal{N}=4$ super-Yang-Mills theory
• The Two-Loop Remainder Function for Eight and Nine Particles
• The Three-loop MHV Octagon from $\bar{Q}$ equations

Cited by in corpus (8)

• Implications of the Landau Equations for Iterated Integrals
• Antipodal Self-Duality for a Four-Particle Form Factor
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
• Hierarchies in relative Picard-Lefschetz theory