Lectures on differential equations for Feynman integrals

Abstract

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the differential equations. Finally, as an application of the differential equations method we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a differential equation.

40 pages; v3: typos fixed, note at the end of section 4 extended; to appear in J.Phys.A

References in corpus (20)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• FIRE5: a C++ implementation of Feynman Integral REduction
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Feynman Diagrams and Differential Equations
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Iterative structure of finite loop integrals
• The three-loop cusp anomalous dimension in QCD
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Sharpening The Leading Singularity
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Simple loop integrals and amplitudes in N=4 SYM
• Eliminating spurious poles from gauge-theoretic amplitudes
• Master integrals for the two-loop penguin contribution in non-leptonic B-decays
• Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO
• Adequate bases of phase space master integrals for $gg\to h$ at NNLO and beyond
• An Overview of Maximal Unitarity at Two Loops
• QCD and High Energy Interactions: Moriond 2014 Theory Summary

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