Magnus and Dyson series for Master Integrals

  • Mario Argeri
  • Stefano Di VitaEmail author
  • Pierpaolo Mastrolia
  • Edoardo Mirabella
  • Johannes Schlenk
  • Ulrich Schubert
  • Lorenzo Tancredi
Open Access


We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 → 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Mario Argeri
    • 1
  • Stefano Di Vita
    • 2
    Email author
  • Pierpaolo Mastrolia
    • 2
    • 3
  • Edoardo Mirabella
    • 2
  • Johannes Schlenk
    • 2
  • Ulrich Schubert
    • 2
  • Lorenzo Tancredi
    • 4
  1. 1.Dipartimento di Scienze e Innovazione TecnologicaUniversità del Piemonte OrientaleAlessandriaItaly
  2. 2.Max-Planck-Institut für PhysikMünchenGermany
  3. 3.Dipartimento di Fisica e AstronomiaUniversità di Padova, and INFN Sezione di PadovaPadovaItaly
  4. 4.Physik-Institut, Universität ZürichZürichSwitzerland

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