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Lectures on differential equations for Feynman integrals

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arxiv 1412.2296 v3 pith:5K5FVXQ2 submitted 2014-12-06 hep-ph hep-thmath-phmath.MP

classification hep-phhep-thmath-phmath.MP
keywords differentialequationsintegralsfeynmandiscusslecturesalgorithmsallows
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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