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Motivic renormalization and singularities

Matilde Marcolli

TL;DR

The paper develops a cohesive framework linking renormalization, parametric Feynman integrals, and motive theory. It introduces a Leray coboundary-based regularization that parallels dimensional regularization and preserves Connes–Kreimer/BPHZ renormalization via Birkhoff factorization. By connecting Mellin transforms of Gelfand–Leray forms to Milnor-fibration cohomology, it ties sliced parametric integrals to mixed Hodge structures and irregular singular connections, and it offers a speculative motivic interpretation using logarithmic motives and motivic sheaves. Overall, the work proposes a path toward a motivic understanding of dimensional regularization and Renormalization that could unify multiple mathematical perspectives on Feynman integrals and graph hypersurfaces.

Abstract

We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs. We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundaries and, like dimensional regularization, replaces a divergent integral with a Laurent series in a complex parameter. The Connes--Kreimer formulation of renormalization can be applied to this regularization method. We relate the dimensional regularization of the Feynman integral to the Mellin transforms of certain Gelfand--Leray forms and we show that, upon varying the external momenta, the Feynman integrals for a given graph span a family of subspaces in the cohomological Milnor fibration. We show how to pass from regular singular Picard--Fuchs equations to irregular singular flat equisingular connections. In the last section, which is more speculative in nature, we propose a geometric model for dimensional regularization in terms of logarithmic motives and motivic sheaves.

Motivic renormalization and singularities

TL;DR

The paper develops a cohesive framework linking renormalization, parametric Feynman integrals, and motive theory. It introduces a Leray coboundary-based regularization that parallels dimensional regularization and preserves Connes–Kreimer/BPHZ renormalization via Birkhoff factorization. By connecting Mellin transforms of Gelfand–Leray forms to Milnor-fibration cohomology, it ties sliced parametric integrals to mixed Hodge structures and irregular singular connections, and it offers a speculative motivic interpretation using logarithmic motives and motivic sheaves. Overall, the work proposes a path toward a motivic understanding of dimensional regularization and Renormalization that could unify multiple mathematical perspectives on Feynman integrals and graph hypersurfaces.

Abstract

We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs. We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundaries and, like dimensional regularization, replaces a divergent integral with a Laurent series in a complex parameter. The Connes--Kreimer formulation of renormalization can be applied to this regularization method. We relate the dimensional regularization of the Feynman integral to the Mellin transforms of certain Gelfand--Leray forms and we show that, upon varying the external momenta, the Feynman integrals for a given graph span a family of subspaces in the cohomological Milnor fibration. We show how to pass from regular singular Picard--Fuchs equations to irregular singular flat equisingular connections. In the last section, which is more speculative in nature, we propose a geometric model for dimensional regularization in terms of logarithmic motives and motivic sheaves.

Paper Structure

This paper contains 28 sections, 23 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Parametric Feynman integrals
  3. Feynman parameters and algebraic varieties
  4. Dimensional Regularization
  5. Mass scale dependence
  6. Integrals on projective spaces
  7. Singularities, slicing, and Milnor fiber
  8. Non-isolated singularities
  9. Projective Radon transform
  10. The polar filtration
  11. Milnor fiber
  12. The Feynman integral: slicing
  13. Oscillatory integrals: Leray and Dimensional Regularizations
  14. Oscillatory integrals and the Gelfand--Leray forms
  15. Leray coboundary regularization and subtraction
  16. ...and 13 more sections

Key Result

Lemma 2.1

For $n-D\ell/2 >0$, the Feynman integral FeynmanU can be written, after the change of variables varchangepi, in the form up to a multiplicative constant.

Theorems & Definitions (30)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Lemma 3.1
  • ...and 20 more