Journal of High Energy Physics

, 2017:90 | Cite as

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

  • Samuel Abreu
  • Ruth Britto
  • Claude Duhr
  • Einan Gardi
Open Access
Regular Article - Theoretical Physics


We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.


Scattering Amplitudes Perturbative QCD 


Open Access

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Physikalisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  2. 2.School of MathematicsTrinity CollegeDublin 2Ireland
  3. 3.Hamilton Mathematics InstituteTrinity CollegeDublin 2Ireland
  4. 4.Institut de Physique ThéoriqueUniversité Paris Saclay, CEA, CNRSGif-sur-Yvette cedexFrance
  5. 5.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  6. 6.Theoretical Physics Department, CERNGenevaSwitzerland
  7. 7.Center for Cosmology, Particle Physics and Phenomenology (CP3)Université Catholique de LouvainLouvain-La-NeuveBelgium
  8. 8.Higgs Centre for Theoretical Physics, School of Physics and AstronomyThe University of EdinburghEdinburghU.K.

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