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2019 | OriginalPaper | Buchkapitel

From Modular Forms to Differential Equations for Feynman Integrals

verfasst von : Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi

Erschienen in: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Verlag: Springer International Publishing

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Abstract

In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.

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Vorheriges Kapitel A Four-Point Function for the Planar QCD Massive Corrections to Top-Antitop Production in the Gluon-Fusion Channel

Nächstes Kapitel One-Loop StringString ScatteringScattering AmplitudesAmplitudes as Iterated EisensteinEisenstein Integrals

The notation \(j'(\tau )= j(2\tau )\) is standard in this context in the mathematics literature, though we emphasise that \(j'(\tau )\) does not correspond to the derivative of \(j(\tau )\).

There are exceptions for small values of the weight and the level.

We define the genus of a congruence subgroup \(\varGamma \) to be the genus of the modular curve \(X_{\varGamma }\).

http://www.sagemath.org/

https://math.mit.edu/~drew/ClassicalModPolys.html

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Titel: From Modular Forms to Differential Equations for Feynman Integrals
verfasst von: Johannes Broedel
Claude Duhr
Falko Dulat
Brenda Penante
Lorenzo Tancredi
Verlag: Springer International Publishing
Buch: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory
Print ISBN: 978-3-030-04479-4

Electronic ISBN: 978-3-030-04480-0

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-04480-0_6

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