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Multiple Zeta Values and Modular Forms in Quantum Field Theory

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Computer Algebra in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

This article introduces multiple zeta values and alternating Euler sums, exposing some of the rich mathematical structure of these objects and indicating situations where they arise in quantum field theory. Then it considers massive Feynman diagrams whose evaluations yield polylogarithms of the sixth root of unity, products of elliptic integrals, and L-functions of modular forms inside their critical strips.

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Notes

  1. 1.

    http://oldweb.cecm.sfu.ca/cgi-bin/EZFace/zetaform.cgi

  2. 2.

    http://www.nikhef.nl/~form/datamine/

  3. 3.

    I am told that Källén was disappointed to find that the two-loop electron propagator involves an elliptic integral, unlike the simpler photon propagator.

  4. 4.

    http://www2.research.att.com/~njas/lattices/QQF.4.g.html

  5. 5.

    http://oeis.org/A125510

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Author information

Authors and Affiliations

  1. Department of Physical Sciences, Open University, Milton Keynes, UK

    David Broadhurst

  2. Institutes of Mathematics and Physics, Humboldt University, Berlin, Germany

    David Broadhurst

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  1. David Broadhurst
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Correspondence to David Broadhurst .

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Editors and Affiliations

  1. RISC, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. DESY, Zeuthen, Germany

    Johannes Blümlein

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Broadhurst, D. (2013). Multiple Zeta Values and Modular Forms in Quantum Field Theory. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_2

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  • DOI: https://doi.org/10.1007/978-3-7091-1616-6_2

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  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-1615-9

  • Online ISBN: 978-3-7091-1616-6

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