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We discuss the structure of renormalized Feynman rules. Regarding them as maps from the Hopf algebra of Feynman graphs to  originating from the evaluation of graphs by Feynman rules, they are elements of a group  . We study the kinematics of scale and angle-dependence to decompose  into subgroups  and  . 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.

Angles, Scales and Parametric Renormalization