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Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry

Yang Zhang

TL;DR

The notes present a systematic program to tackle multi-loop scattering amplitudes by marrying integrand reduction with computational algebraic geometry. By recasting loop integrals and IBP relations as polynomial ideals and using Gröbner-basis techniques, the approach enables robust ideal membership tests, multivariate fraction reductions, and solving polynomial systems that arise in unitarity analyses. The exposition details one-loop reductions (box and triangle) and their D-dimensional generalizations, then discusses the complications at higher loops that motivate the algebraic-geometry framework. The resulting methodology promises algorithmic, scalable reduction of complex amplitudes, with practical guidance, examples, and exercises illustrating the power of algebraic geometry in high-energy physics computations.

Abstract

These notes are for the author's lectures, "Integral Reduction and Applied Algebraic Geometry Techniques" in the School and Workshop on Amplitudes in Beijing 2016. I introduce the applications of algebraic geometry methods on multi-loop scattering amplitudes, for instance, integrand reduction, residue computation in unitarity analysis and Integration-by-parts reduction. Illustrative examples and exercises are included in these notes.

Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry

TL;DR

The notes present a systematic program to tackle multi-loop scattering amplitudes by marrying integrand reduction with computational algebraic geometry. By recasting loop integrals and IBP relations as polynomial ideals and using Gröbner-basis techniques, the approach enables robust ideal membership tests, multivariate fraction reductions, and solving polynomial systems that arise in unitarity analyses. The exposition details one-loop reductions (box and triangle) and their D-dimensional generalizations, then discusses the complications at higher loops that motivate the algebraic-geometry framework. The resulting methodology promises algorithmic, scalable reduction of complex amplitudes, with practical guidance, examples, and exercises illustrating the power of algebraic geometry in high-energy physics computations.

Abstract

These notes are for the author's lectures, "Integral Reduction and Applied Algebraic Geometry Techniques" in the School and Workshop on Amplitudes in Beijing 2016. I introduce the applications of algebraic geometry methods on multi-loop scattering amplitudes, for instance, integrand reduction, residue computation in unitarity analysis and Integration-by-parts reduction. Illustrative examples and exercises are included in these notes.

Paper Structure

This paper contains 18 sections, 9 theorems, 88 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Integrand reduction and Gröbner basis
  3. Basic physical objects
  4. Integrand reduction at one loop
  5. Box diagram
  6. Triangle diagram
  7. D-dimensional one-loop integrand reduction
  8. Issues at higher loop orders
  9. Elementary computational algebraic geometry methods
  10. Basic facts of algebraic geometry in affine space I
  11. Gröbner basis
  12. One-variable case
  13. Gröbner basis
  14. Multivariate polynomial division
  15. Buchberger's Algorithm
  16. ...and 3 more sections

Key Result

Theorem 2.3

The generating set of an ideal $I$ of $R=\mathbb F[z_1,\ldots z_n]$ can always be chosen to be finite.

Figures (6)

  • Figure 1: A nonplanar diagram in color ordering, may be a planar diagram in the sense of graph theory.
  • Figure 2: Diagrams to be simplified
  • Figure 3: One-loop massless box diagram
  • Figure 4: One-loop triangle diagram
  • Figure 5: two-loop double box diagram
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: Noether
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Hilbert's weak Nullstellensatz
  • proof
  • Remark
  • Example 2.7
  • ...and 26 more