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IBP reduction coefficients made simple

Janko Boehm, Marcel Wittmann, Zihao Wu, Yingxuan Xu, Yang Zhang

TL;DR

The paper tackles the problem of oversized analytic IBP reduction coefficients in multi-loop Feynman integrals by introducing an improved Leinartas' multivariate partial fraction algorithm implemented in Singular. It leverages UT-basis properties to ensure denominators are symbol letters or polynomials in $D$, enabling substantial size reductions, often by orders of magnitude, when present. The approach is demonstrated across simple and frontier two-loop/topologies, including elliptic cases, with reported compression factors up to around 100x and consistent applicability even without a UT basis. A publicly available Singular library accompanies the work, highlighting practical impact for fast, storage-efficient, and scalable multi-loop amplitude computations. The findings suggest broad utility for analytical simplification and potential integration with finite-field and reconstruction techniques in high-precision collider physics.

Abstract

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas' multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, We observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension $D$. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as $\sim 100$. We observe that our algorithm also works well for settings without a UT basis.

IBP reduction coefficients made simple

TL;DR

The paper tackles the problem of oversized analytic IBP reduction coefficients in multi-loop Feynman integrals by introducing an improved Leinartas' multivariate partial fraction algorithm implemented in Singular. It leverages UT-basis properties to ensure denominators are symbol letters or polynomials in , enabling substantial size reductions, often by orders of magnitude, when present. The approach is demonstrated across simple and frontier two-loop/topologies, including elliptic cases, with reported compression factors up to around 100x and consistent applicability even without a UT basis. A publicly available Singular library accompanies the work, highlighting practical impact for fast, storage-efficient, and scalable multi-loop amplitude computations. The findings suggest broad utility for analytical simplification and potential integration with finite-field and reconstruction techniques in high-precision collider physics.

Abstract

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas' multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, We observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension . With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as . We observe that our algorithm also works well for settings without a UT basis.

Paper Structure

This paper contains 14 sections, 7 theorems, 54 equations, 3 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. IBP and Master Integrals
  3. Integration-by-Parts Identities and master integrals
  4. Differential equation and UT basis
  5. Symbol of a transcendental function
  6. Improved Leinartas' Algorithm and Modern Implementation
  7. Examples
  8. A baby example: Nonplanar two-loop four-point with an external massive leg
  9. A cutting-edge example: Nonplanar two-loop five-point
  10. An elliptic example: two-loop four-point with a top quark loop and a pair of external massive legs
  11. Performance of the algorithm
  12. Summary and Discussion
  13. Manual of the Partial Fractioning Singular Library
  14. An Explicit Example of the Size Reduction

Key Result

Theorem 3.5

Let $f,g\in K[\mathbf x]$ and let $g=\prod_{i=1}^mq_i^{e_i}$ be the factorization of $g$ into irreducible factors $(e_i\in\mathbb N)$. Then there exists a decomposition where all nonzero summands satisfy the following conditions

Figures (3)

  • Figure 1: One-massive crossed box.
  • Figure 2: 2-loop 5-point nonplanar double pentagon
  • Figure 3: elliptic double box

Theorems & Definitions (23)

  • Definition 3.1
  • Definition 3.2: monomial ordering
  • Definition 3.3: Gröbner basis
  • Definition 3.4: division with remainder
  • Theorem 3.5: partial fraction decomposition
  • Lemma 3.6: Nullstellensatz certificate
  • proof
  • Lemma 3.7: Jacobian criterion
  • Corollary 3.8
  • proof
  • ...and 13 more