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Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections

Janko Boehm, Alessandro Georgoudis, Kasper J. Larsen, Hans Schoenemann, Yang Zhang

TL;DR

This work presents a module-intersection IBP reduction framework that trims the otherwise unwieldy IBP systems by enforcing no-doubled-propagator constraints and, when useful, unitarity cuts. By formulating IBP constraints as intersections of polynomial modules and exploiting Baikov representation, the authors compute efficient, cut-specific reductions and merge them to obtain fully analytic results. The method is demonstrated on the two-loop, five-point non-planar hexagon-box with numerators up to degree four, achieving a complete reduction to 73 master integrals and validating against existing solvers. The approach promises substantial improvements in analytic multi-loop reductions and NNLO computations, with potential integration into open-source toolchains and broader classes of amplitudes.

Abstract

We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of $73$ master integrals.

Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections

TL;DR

This work presents a module-intersection IBP reduction framework that trims the otherwise unwieldy IBP systems by enforcing no-doubled-propagator constraints and, when useful, unitarity cuts. By formulating IBP constraints as intersections of polynomial modules and exploiting Baikov representation, the authors compute efficient, cut-specific reductions and merge them to obtain fully analytic results. The method is demonstrated on the two-loop, five-point non-planar hexagon-box with numerators up to degree four, achieving a complete reduction to 73 master integrals and validating against existing solvers. The approach promises substantial improvements in analytic multi-loop reductions and NNLO computations, with potential integration into open-source toolchains and broader classes of amplitudes.

Abstract

We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of master integrals.

Paper Structure

This paper contains 15 sections, 3 theorems, 57 equations, 2 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Simplification of IBP systems by module-intersection method
  3. Module intersection method without unitarity cuts
  4. Module intersection method with cuts
  5. Algorithm for computing individual modules
  6. Determining the module intersection
  7. Sparse row reduction
  8. Selection of relevant and independent IBP identities
  9. Sparse REF and RREF strategies
  10. The non-planar hexagon-box diagram example
  11. Module intersection on cuts
  12. Reduction of IBP identities on cuts
  13. Merging of IBP reductions and final result
  14. Comparison with other IBP solvers
  15. Conclusion and outlook

Key Result

Theorem 1

With notation as above, the following conditions are equivalent:

Figures (2)

  • Figure 1: The fully massless non-planar hexagon-box diagram, along with our labelling conventions for its internal lines. The lower part shows the subset of the basis integrals with the property that their graphs cannot be obtained by adding internal lines to the graph of another basis integral. The corresponding cuts $\{1,5,7\}$, $\{2,5,7\}$, $\{2,5,8\}$, $\{2,6,7\}$, $\{3,5,8\}$, $\{3,6,7\}$, $\{3,6,8\}$, $\{4,6,8\}$, $\{1,4,5,8\}$ and $\{1,4,6,7\}$ are the cuts required for deriving complete IBP identities for the non-planar hexagon-box diagram.
  • Figure 2: (Color online.) The $75$ "pre"-master integrals found by Azurite, with the global symmetry option turned off, for the non-planar hexagon-box family of eq. \ref{['Feynman_integral']}. By turning on the global symmetry option, Azurite determines that there are $73$ master integrals. These are the illustrated "pre"-master integrals excluding $\mathcal{I}_{63}$ and $\mathcal{I}_{68}$. Our labeling convention for the propagators, corresponding to the indices recorded in eq. \ref{['eq:Azurite_basis']}, are shown in the graph of the hexagon-box itself as the blue encircled numbers.

Theorems & Definitions (8)

  • Theorem 1: Buchberger
  • Lemma 3: Intersection
  • proof
  • Definition 4
  • Lemma 5: Localization
  • proof
  • Remark 6
  • Remark 7