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Letters in Mathematical Physics

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

Angles, Scales and Parametric Renormalization

  • Francis Brown
  • Dirk Kreimer
Article

Abstract

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 

Keywords

Feynman rules parametric renormalization kinematics of scattering amplitudes Hopf algebras 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

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

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