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Numerators in Parametric Representations of Feynman Diagrams

Marc P. Bellon

TL;DR

This work addresses the long-standing challenge of incorporating numerators into parametric representations of Feynman diagrams. It introduces Dodgson identities and the completed graph construction to produce simpler, highly symmetric parametric expressions for massless diagrams with numerators, reducing maximal powers of the Symanzik polynomial in the denominator. A general theorem expresses the numerator-containing integrand as a sum over partitions with Gamma-function and $U$-polynomial weights, and the method is extended to fermion loops via forest polynomials, yielding explicit results for the Wess–Zumino model at three and four loops. The approach preserves supersymmetry and eliminates spurious subdivergences, offering a regulator-free route to Schwinger–Dyson equations and promising extensions to higher-loop calculations and other field theories.

Abstract

The parametric representation has been used since a long time for the evaluation of Feynman diagrams. As a dimension independent intermediate representation, it allows a clear description of singularities. Recently, it has become a choice tool for the investigation of the type of transcendent numbersappearing in the evaluation of Feynman diagrams. The inclusion of numerators has however stagnated since the ground work of Nakanishi. I here show howto greatly simplify the formulas through the use of Dodgson identities. In the massless case in particular, reduction to the completion to a vacuum graph allows for a strong reduction of the maximal power of the Symanzik polynomial in the denominator.

Numerators in Parametric Representations of Feynman Diagrams

TL;DR

This work addresses the long-standing challenge of incorporating numerators into parametric representations of Feynman diagrams. It introduces Dodgson identities and the completed graph construction to produce simpler, highly symmetric parametric expressions for massless diagrams with numerators, reducing maximal powers of the Symanzik polynomial in the denominator. A general theorem expresses the numerator-containing integrand as a sum over partitions with Gamma-function and -polynomial weights, and the method is extended to fermion loops via forest polynomials, yielding explicit results for the Wess–Zumino model at three and four loops. The approach preserves supersymmetry and eliminates spurious subdivergences, offering a regulator-free route to Schwinger–Dyson equations and promising extensions to higher-loop calculations and other field theories.

Abstract

The parametric representation has been used since a long time for the evaluation of Feynman diagrams. As a dimension independent intermediate representation, it allows a clear description of singularities. Recently, it has become a choice tool for the investigation of the type of transcendent numbersappearing in the evaluation of Feynman diagrams. The inclusion of numerators has however stagnated since the ground work of Nakanishi. I here show howto greatly simplify the formulas through the use of Dodgson identities. In the massless case in particular, reduction to the completion to a vacuum graph allows for a strong reduction of the maximal power of the Symanzik polynomial in the denominator.
Paper Structure (21 sections, 1 theorem, 33 equations, 1 table)

This paper contains 21 sections, 1 theorem, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Presentation of the main results
  3. Reducing to the completed diagram
  4. A single scalar product
  5. The general case
  6. Proof of the general result
  7. The terms to find
  8. The terms in $\boldsymbol{U^{S_0}}$
  9. Fermion loops
  10. Loop of four fermionic propagators
  11. Dimensional factors for six fermionic propagators
  12. Dodgson polynomials for the six propagator loop
  13. Forest polynomials
  14. Applications to the Wess--Zumino model
  15. Supersymmetry and completion independence
  16. ...and 6 more sections

Key Result

Theorem 2.2

The integrand of a massless vacuum diagram in space-time dimension $d$ with numerator expressed as a product of $n$ scalar products can be written as a sum over the set $\mathcal{P}_k$ of partitions of $S$ in $k$ parts, with $k$ taking all values for which this set is non empty We can observe that this rational function has homogeneity $-L D/2 - n$. The value of the residue of the diagram is then

Theorems & Definitions (2)

  • Definition 2.1
  • Theorem 2.2