Bootstrapping two-loop Feynman integrals for planar N=4 sYM

Abstract

We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $\mathcal{N}=4$ super-Yang--Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N${}^2$MHV amplitudes even at one loop.

49 pages

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Cited by in corpus (26)

• Pentagon functions for massless planar scattering amplitudes
• Rationalizing Loop Integration
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Implications of the Landau Equations for Iterated Integrals
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Logarithmic forms and differential equations for Feynman integrals
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Schubert Problems, Positivity and Symbol Letters
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
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• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
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• The Two-Loop Remainder Function for Eight and Nine Particles
• Intersection theory rules symbology
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Aspects of High Energy Scattering
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond