Letters in Mathematical Physics

, Volume 103, Issue 9, pp 933–1007 | Cite as

Angles, Scales and Parametric Renormalization

  • Francis Brown
  • Dirk Kreimer


We discuss the structure of renormalized Feynman rules. Regarding them as maps from the Hopf algebra of Feynman graphs to \({\mathbb{C}}\) originating from the evaluation of graphs by Feynman rules, they are elements of a group \({G=\mathrm{Spec}_{\mathrm{Feyn}}(H)}\) . We study the kinematics of scale and angle-dependence to decompose G into subgroups \({G_{\mathrm{\makebox{1-s}}}}\) and \({G_{\mathrm{fin}}}\) . Using parametric representations of Feynman integrals, renormalizability and the renormalization group underlying the scale dependence of Feynman amplitudes are derived and proven in the context of algebraic geometry.

Mathematical Subject Classification (2000)

81T15 81T18 81Q30 14D07 


Feynman rules parametric renormalization kinematics of scattering amplitudes Hopf algebras 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergbauer C., Kreimer D.: Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology. IRMA Lect. Math. Theor. Phys. 10, 133–164 (2006)MathSciNetGoogle Scholar
  2. 2.
    Bloch S., Esnault H., Kreimer D.: On motives associated to graph polynomials. Commun. Math. Phys. 267(1), 181–225 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Bloch S., Kreimer D.: Feynman amplitudes and Landau singularities for 1-loop graphs. Commun. Number Theor. Phys. 4(4), 709–753 (2011)MathSciNetGoogle Scholar
  4. 4.
    Brown F.: The massless higher-loop two-point function. Commun. Math. Phys. 287(3), 925–958 (2009)ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Brown, F.: On the Periods of some Feynman Integrals. arXiv:0910.0114 [math.AG] (2009)Google Scholar
  6. 6.
    Brown, F., Kreimer, D.: Decomposing Feynman rules, arXiv:1212.3923 [hep-th], Proceedings of Science, 11th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory 15–20 Apr 2012. Wernigerode, Germany (to appear)Google Scholar
  7. 7.
    Bloch S., Kreimer D.: Mixed Hodge structures and renormalization in physics. Commun. Number Theor. Phys. 2, 637–718 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bloch, S.: letter to the authors (2010)Google Scholar
  9. 9.
    Bloch, S., Kreimer, D.: Feynman amplitudes and Landau singularities for 1-loop graphs. Commun. Number Theor. Phys. 4, 709 (2010) [arXiv:1007.0338 [hep-thGoogle Scholar
  10. 10.
    Kirchhoff G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie 72(12), 497–508 (1847)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Humboldt UniversityBerlinGermany

Personalised recommendations