Lectures on differential equations for Feynman integrals (1412.2296v3)

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.

• The paper presents advanced strategies for transforming Feynman integrals into canonical forms using differential equations and uniform transcendental weight.
• It details the use of strategic basis choices, leading singularities, and d-log representations to simplify complex multi-loop computations.
• Practical implications include improved numerical evaluations and precision in scattering amplitude calculations relevant for collider experiments.

Overview of Differential Equations for Feynman Integrals

This paper provides an in-depth exploration of the use of differential equations (DE) to compute Feynman integrals, which are crucial in quantum field theory. Feynman integrals appear in the calculation of scattering amplitudes and other quantities in perturbative quantum field theory. The document provides a comprehensive review of recent advancements in the field, illustrating how differential equations can be employed effectively to solve such integrals and highlighting new techniques and methods to simplify this process.

Simplification and Canonical Forms

The paper discusses an approach to Feynman integrals using differential equations, aiming to simplify these equations into a canonical form. The authors illustrate methods to achieve simplification through strategic basis choices, exploiting properties like leading singularities and d-log representations. Notably, a canonical form makes the singularity structure of the integrals explicit, aiding in predictive calculations and numerical evaluations.

Mathematical and Computational Techniques

The utilization of differential equations involves several mathematical strategies. The paper examines the algorithmic transformation of DEs using concepts drawn from linear algebra and complex analysis. An optimal basis is sought, where integrals satisfy uniform transcendental weight, thus simplifying the functional form of DEs. The geometric approach leverages space-time integrand properties, extending the analysis to multi-loop integrals and single-scale integrals.

Canonical Basis and Uniform Weight

A significant theme is the search for a basis of Feynman integrals that exhibits uniform transcendental weight. This choice facilitates solutions expressed in terms of iterated integrals, streamlining the computation and allowing for the easy extraction of physically relevant results. The memo also addresses the utility of the Drinfeld associator in connecting differential equations to known mathematical constructs, thereby simplifying single-scale integral computations further.

Practical and Theoretical Implications

Practically, these advancements enable more efficient and precise calculations pertinent to collider experiments like those conducted at the LHC, where understanding Feynman integrals is crucial for theoretical predictions. Theoretically, the paper opens pathways to addressing deeper mathematical problems concerning the classification and computation of Feynman integrals in different dimensional settings.

Conclusion and Future Directions

The paper suggests that future research could expand on these methods, exploring their application across various dimensional setups and more complex integral families. The techniques discussed have implications beyond computational physics, potentially influencing mathematical fields related to iterated integrals and differential systems. The discourse also speculates on further algorithmic refinement, aiming for automatization in extracting canonical forms of DEs for broader classes of integrals. Potential developments may include handling complications arising from elliptic integral classes and extending these methods to multifaceted scattering problems in diverse physical theories.