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Feynman Integral Reduction using Syzygy-Constrained Symbolic Reduction Rules

Sid Smith, Mao Zeng

Abstract

We present a new algorithm for integration-by-parts (IBP) reduction of Feynman integrals with high powers of numerators or propagators, a demanding computational step in evaluating multi-loop scattering amplitudes. The algorithm starts with solving syzygy equations in individual sectors to produce IBP operators that turn seed integrals into IBP equations without artificially raised propagator powers. The IBP operators are expressed in terms of index-shift operators and number operators. We perform row reduction to systematically reshuffle the IBP operators and expose reduction rules with symbolic dependence on the powers of propagators and numerators. When this is insufficient, we produce more symbolic reduction rules by directly solving the linear system of IBP equations in which some propagator/numerator powers are kept symbolic. This linear system is kept small, as the equations are generated from a small set of seed integrals in the neighborhood of the target integral. We stress-test our algorithm against two highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. As an application, we revisit the IBP reduction in a calculation of scattering amplitudes for spinning black hole binary systems, which involves two-loop Feynman integrals with complexity greater than 20, and achieve much faster IBP reduction than that of the original calculation.

Feynman Integral Reduction using Syzygy-Constrained Symbolic Reduction Rules

Abstract

We present a new algorithm for integration-by-parts (IBP) reduction of Feynman integrals with high powers of numerators or propagators, a demanding computational step in evaluating multi-loop scattering amplitudes. The algorithm starts with solving syzygy equations in individual sectors to produce IBP operators that turn seed integrals into IBP equations without artificially raised propagator powers. The IBP operators are expressed in terms of index-shift operators and number operators. We perform row reduction to systematically reshuffle the IBP operators and expose reduction rules with symbolic dependence on the powers of propagators and numerators. When this is insufficient, we produce more symbolic reduction rules by directly solving the linear system of IBP equations in which some propagator/numerator powers are kept symbolic. This linear system is kept small, as the equations are generated from a small set of seed integrals in the neighborhood of the target integral. We stress-test our algorithm against two highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. As an application, we revisit the IBP reduction in a calculation of scattering amplitudes for spinning black hole binary systems, which involves two-loop Feynman integrals with complexity greater than 20, and achieve much faster IBP reduction than that of the original calculation.

Paper Structure

This paper contains 23 sections, 77 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Integration-by-Parts: An Overview
  3. Integration-by-Parts Identities
  4. Sectors and Spanning Cuts
  5. Building Identities from Syzygy Constraints
  6. Algorithm for Reducing Complex Integrals
  7. Building Reduction Rules
  8. Generating Ordered Identities
  9. Row Reducing Identities
  10. Solving Directly for Missing Rules
  11. Summary
  12. Applying Reduction Rules
  13. Examples
  14. Double Box with External Mass
  15. Massless Pentabox
  16. ...and 8 more sections

Figures (8)

  • Figure 1: Two Loop Equal Mass Sunrise
  • Figure 2: Tower of Sectors one must consider when the top sectors are $(2,1,1,0,0)$ and $(1,2,1,0,0)$. This drawing is schematic and the horizontal levels here do not correlate with the weighting described later. The relevant information is contained within the arrows, denoting the subsector inheritance hierarchy.
  • Figure 3: Tower of Non Zero Sectors for this Example.
  • Figure 4: Flow Chart to represent the first stage of the algorithm, namely building reduction rules.
  • Figure 5: Flow Chart to represent the second stage of the algorithm, namely applying reduction rules.
  • ...and 3 more figures