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Published in:
Eta QuotientsEta quotients and Rademacher SumsRademacher sums Read chapter Read first chapter

2019 | OriginalPaper | Chapter

From Modular Forms to Differential Equations for Feynman Integrals

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

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

Publisher: 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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previous chapter A Four-Point Function for the Planar QCD Massive Corrections to Top-Antitop Production in the Gluon-Fusion Channel
next chapter One-Loop StringString ScatteringScattering AmplitudesAmplitudes as Iterated EisensteinEisenstein Integrals
Footnotes
1
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 \(j(\tau )\).
 
2
There are exceptions for small values of the weight and the level.
 
3
We define the genus of a congruence subgroup \(\varGamma \) to be the genus of the modular curve \(X_{\varGamma }\).
 
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Metadata
Title
From Modular Forms to Differential Equations for Feynman Integrals
Authors
Johannes Broedel
Claude Duhr
Falko Dulat
Brenda Penante
Lorenzo Tancredi
Copyright Year
2019
Book
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 Year: 2019

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

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