Lectures on differential equations for Feynman integrals

By J. Henn. 2014

356 citations recorded by the source corpus.

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE 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 is 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 DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.

Subfields: Applied Mathematics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7aa66000a0da4cf75d03dbe53664f43c11cd9fc9

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In Applied Mathematics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, background): “kinematic invariants), or by multiplying their numerators by polynomials which modify their dimensions, or by considering arbitrary denominator powers, respectively turning the decomposition formulas into differential equations [3–10], dimensional recurrence relations [11, 12], and finite difference”
The full four-loop cusp anomalous dimension in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{do (2019, neutral): “We find that the differential equations (for reviews, see [63, 64]) for this basis takes the form”
Feynman integrals and intersection theory (2018, methodology): “[9] J.”
Collider physics at the precision frontier (2020, methodology): “This is because the UT property [482] allows for a straightforward solution of the system of differential equations in terms of Chen iterated integrals [483, 484].”
On the maximal cut of Feynman integrals and the solution of their differential equations (2016, background): “On the other hand, in [12] it was suggested that one of the crucial criteria for selecting candidate integrals which fulfill canonical differential equations is the study of their leading singularity, which can be done by inspecting their generalized cuts [9].”
Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral (2016, methodology): “In particular, it can be simplified by the choice of a canonical basis in d ≈ 4, which can be found following the methods described in [33,34].”
Feynman integrals and hyperlogarithms (2015, methodology): “The method of differential equations [10, 98] generates a system of linear, first-order partial differential equations in terms of Feynman integrals in the family of the initial graph.”
Reducing differential equations for multiloop master integrals (2014, methodology): “I am grateful to Thomas Gehrmann, Johannes Henn, and Andrei Pomeransky for the interest to the work and useful discussions.”