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Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods

Yang Zhang

TL;DR

Problem: higher-loop amplitude calculations in renormalizable theories are intractable with traditional Feynman/IBP approaches. Approach: an automatic integrand-level reduction algorithm based on computational algebraic geometry, using Gröbner bases to build the integrand basis and primary decomposition to classify unitarity-cut solutions, with polynomial fitting reconstructing the integrand; implemented in BasisDet and applicable to $D=4$ and $D=4-2\epsilon$. Contributions: automatic basis generation, explicit (ir)reducible scalar product identification, and a decomposition of cut solutions into irreducible components across one-, two-, and three-loop examples, including detailed counts of basis terms and solution dimensions. Significance: provides a scalable, dimension- and loop-agnostic framework for multi-loop amplitudes, enabling systematic higher-loop calculations via unitarity cuts and polynomial reconstruction.

Abstract

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D=4 and $D=4-2ε$ dimensions, and we present some two and three-loop examples of applications of this algorithm.

Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods

TL;DR

Problem: higher-loop amplitude calculations in renormalizable theories are intractable with traditional Feynman/IBP approaches. Approach: an automatic integrand-level reduction algorithm based on computational algebraic geometry, using Gröbner bases to build the integrand basis and primary decomposition to classify unitarity-cut solutions, with polynomial fitting reconstructing the integrand; implemented in BasisDet and applicable to and . Contributions: automatic basis generation, explicit (ir)reducible scalar product identification, and a decomposition of cut solutions into irreducible components across one-, two-, and three-loop examples, including detailed counts of basis terms and solution dimensions. Significance: provides a scalable, dimension- and loop-agnostic framework for multi-loop amplitudes, enabling systematic higher-loop calculations via unitarity cuts and polynomial reconstruction.

Abstract

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D=4 and dimensions, and we present some two and three-loop examples of applications of this algorithm.

Paper Structure

This paper contains 19 sections, 2 theorems, 52 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of integrand-level reduction methods
  3. One-loop integrand-level reduction
  4. Two-loop integrand-level reduction
  5. The algorithm
  6. Setup
  7. Algorithm for integrand basis determination
  8. Primary decomposition of cut solutions
  9. Examples
  10. $D=4-2\epsilon$ one-loop four-point box topology
  11. Two-loop examples
  12. $D=4$ three-loop triple-box topology
  13. Conclusions and future directions
  14. Manual for the package BasisDet 1.01
  15. Set up
  16. ...and 4 more sections

Key Result

Proposition 1

The monomials in the integrand basis must be linearly independent in the quotient ring $R/I$.

Figures (3)

  • Figure 1: One-loop box diagram
  • Figure 2: Four-point two-loop planar diagram
  • Figure 3: Four-point three-loop planar diagram

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof