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Angles, scales and parametric renormalization

Francis Brown, Dirk Kreimer

Abstract

We decompose renormalized Feynman rules according to the scale and angle dependence of amplitudes. We use parametric representations such that the resulting amplitudes can be studied in algebraic geometry.

Angles, scales and parametric renormalization

Abstract

We decompose renormalized Feynman rules according to the scale and angle dependence of amplitudes. We use parametric representations such that the resulting amplitudes can be studied in algebraic geometry.

Paper Structure

This paper contains 63 sections, 42 theorems, 321 equations.

Table of Contents

  1. Introduction: The decomposition of $\mathrm{Spec}_{\mathrm{Feyn}}(H)$
  2. Forests
  3. Definitions
  4. Hopf structures
  5. For trees
  6. For graphs
  7. (Co-)ideals
  8. Notation for forests
  9. Derivation of renormalized Feynman rules in parametric space
  10. Schwinger parametrization and the exponential integral
  11. $s_\Gamma=0$
  12. $s_\Gamma=1$
  13. $s_\Gamma=-1$
  14. Quadratic sub-divergences
  15. The final result
  16. ...and 48 more sections

Key Result

Lemma 2

Let $\mathrm{id}_{\mathrm{Aug}}$ be the identity map $\mathrm{Aug}\to \mathrm{Aug}$ in the augmentation ideal. We have

Theorems & Definitions (100)

  • Example 1
  • Lemma 2
  • Example 3
  • Proposition 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • Proposition 7
  • proof
  • ...and 90 more