Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periods of Feynman Diagrams and GKZ D-Modules

Emad Nasrollahpoursamami

TL;DR

The paper establishes a principled bridge between perturbative Feynman amplitudes and GKZ A-Hypergeometric D-modules by casting amplitudes as periods of relative motives within a toric-geometry framework. It shows that amplitudes satisfy holonomic regular differential systems derived from GKZ theory, and provides a constructive parametric form based on Symanzik polynomials that connects to matroid polytopes. A comprehensive regularization strategy is developed, identifying pole structures via semi-non-resonant faces and yielding an explicit ε-expansion in dimensional regularization through GKZ relations. The culmination proves that, under suitable regularity conditions, amplitudes are periods of relative motives, enabling explicit computation without resolution of singularities and offering a clear path to renormalization within this geometric-analytic paradigm.

Abstract

We study differential equations for Feynman amplitudes and we show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.

Periods of Feynman Diagrams and GKZ D-Modules

TL;DR

The paper establishes a principled bridge between perturbative Feynman amplitudes and GKZ A-Hypergeometric D-modules by casting amplitudes as periods of relative motives within a toric-geometry framework. It shows that amplitudes satisfy holonomic regular differential systems derived from GKZ theory, and provides a constructive parametric form based on Symanzik polynomials that connects to matroid polytopes. A comprehensive regularization strategy is developed, identifying pole structures via semi-non-resonant faces and yielding an explicit ε-expansion in dimensional regularization through GKZ relations. The culmination proves that, under suitable regularity conditions, amplitudes are periods of relative motives, enabling explicit computation without resolution of singularities and offering a clear path to renormalization within this geometric-analytic paradigm.

Abstract

We study differential equations for Feynman amplitudes and we show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.

Paper Structure

This paper contains 6 sections, 42 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Feynman Diagrams
  3. Amplitude as a function of momenta
  4. GKZ A-Hypergeometric Differemtial Equations
  5. Regularization
  6. Amplitudes and Regularization

Key Result

Theorem 1.1

Given a graph $\Gamma$ with n edges and first Symanzik polynomial $\Psi$ and second Symanzik polynomial $Q$ (including mass terms), the amplitude in dimensional regularization, up to a constant, can be computed by the following integral: The left hand side is meromorphic and poles can be described in the following way. For a 2-connected subgraph $\gamma \subset \Gamma$, let $\ell_{\gamma}$ be the

Theorems & Definitions (104)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 94 more