[论文解读] Symbolic syzygy-constrained reduction rules for Feynman integrals and the LoopIn framework

S. H. Smith|arXiv (Cornell University)|Feb 23, 2026

Particle physics theoretical and experimental studies被引用 0

一句话总结

本文提出一种新算法，用于为多循环 Feynman 积分的 IBP 约简生成符号化、受 syzygy 约束的规约规则，能够直接将目标积分约简到主积分并在复杂拓扑上演示了高效性；同时提出 LoopIn，这是一个用于自动化多循环计算的模块化框架，可与现有工具对接。

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 allows us to avoid a large intermediate system of equations and instead focus on applying direct reduction rules to the integrals. We demonstrate the application of our algorithm with some highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. We also achieve much faster IBP reduction for an example of scattering amplitudes for spinning black hole binary systems. Finally, we present LoopIn, a modular framework for automating multi-loop calculations, where the IBP techniques described here can be interfaced.

研究动机与目标

• Motivate and address the computational bottleneck of integration-by-parts reductions in multi-loop Feynman integrals.
• Develop a novel algorithm that generates symbolic reduction rules to directly reduce target integrals to lower-complexity forms.
• Combine syzygy constraints, smart seeding, and operator reshuffling to produce rules with minimal explicit propagator/numerator power dependence.
• Demonstrate the method on challenging two-loop topologies (e.g., double box with external mass, massless pentabox) and a spinning black hole binary amplitude.
• Present LoopIn as a modular framework that automates multi-loop calculations and can interface with existing reduction and numerical evaluation tools.

提出的方法

• Formulate IBP reductions at the operator level to generate symbolic reduction rules rather than solving large linear systems.
• Impose syzygy constraints to restrict identities to relevant propagator subsets, reducing variable count.
• Solve sector-wise syzygy equations with sector-specific weighting to produce initial reduction rules.
• Perform row-reduction (Gaussian elimination) on seeds and identities to derive additional reduction rules iteratively.
• Use backward substitution to apply the resulting rules and reduce target integrals to master integrals.
• If necessary, solve small neighborhood systems around targeted integrals to obtain missing symbolic rules.

Figure 1 : 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 inher

实验结果

研究问题

• RQ1 Can symbolic, syzygy-constrained reduction rules reduce arbitrary target integrals to lower-complexity forms without constructing large intermediate systems?
• RQ2 How well do sector-specific syzygy constraints and operator reshuffling perform on high-rank, multi-scale two-loop integrals?
• RQ3 What gains in rule count and computational effort are achieved for complex topologies (e.g., double box with external mass, massless pentabox) and higher-rank targets?
• RQ4 Can the approach be integrated into a modular framework (LoopIn) to automate and streamline multi-loop calculations from amplitude generation to numerical evaluation?
• RQ5 What are the practical limitations and optimal choices (e.g., monomial ordering, seed selection) for generating and applying these symbolic reduction rules?

主要发现

• Symbolic reduction rules can reduce high-rank, multi-scale integrals directly to master integrals, reducing reliance on large linear IBP systems.
• Applications to challenging topologies (double box with external mass, massless pentabox) demonstrate successful reductions up to rank 20 with multiple dots.
• For spinning black hole binary amplitudes, the method achieves substantial speedups: example reduction time improves from days to hours when using the symbolic approach.
• An iterative scheme combining syzygy-constrained identities and seed perturbations yields multiple efficient rule sets with relatively small numbers of master integrals.
• LoopIn is introduced as a modular framework that automates multi-loop calculations and interfaces existing tools (Kira, LiteRed, FiniteFlow, AMFlow) for end-to-end processing.

Figure 2 : A flow chart describing the algorithm for generating reduction rules