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Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Janko Boehm, Alessandro Georgoudis, Kasper J. Larsen, Mathias Schulze, Yang Zhang

TL;DR

<p>The paper addresses the problem of generating integration-by-parts (IBP) identities for multi-loop Feynman integrals without introducing unwanted dimension shifts. It develops a method based on the Baikov representation and the Laplace expansion of the Gram determinant to produce a complete, degree-one generating set of syzygies that realize dimension-preserving IBP relations; completeness is proven using ideals of submaximal minors and classical resolutions. The authors also extend the construction to unitarity cuts, showing how cut syzygies can be obtained via module intersection, and illustrate the approach with explicit two-loop examples, including massless planar and non-planar topologies. The results provide a practical, scalable framework for purely $D$-dimensional IBP reductions, potentially simplifying NNLO computations and enabling systematic imposition of constraints such as avoiding squared propagators. Overall, the work unifies algebraic geometry techniques with loop-integral reductions to yield explicit, low-degree generators applicable to arbitrary loop counts and external kinematics.

Abstract

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

TL;DR

<p>The paper addresses the problem of generating integration-by-parts (IBP) identities for multi-loop Feynman integrals without introducing unwanted dimension shifts. It develops a method based on the Baikov representation and the Laplace expansion of the Gram determinant to produce a complete, degree-one generating set of syzygies that realize dimension-preserving IBP relations; completeness is proven using ideals of submaximal minors and classical resolutions. The authors also extend the construction to unitarity cuts, showing how cut syzygies can be obtained via module intersection, and illustrate the approach with explicit two-loop examples, including massless planar and non-planar topologies. The results provide a practical, scalable framework for purely -dimensional IBP reductions, potentially simplifying NNLO computations and enabling systematic imposition of constraints such as avoiding squared propagators. Overall, the work unifies algebraic geometry techniques with loop-integral reductions to yield explicit, low-degree generators applicable to arbitrary loop counts and external kinematics.

Abstract

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

Paper Structure

This paper contains 12 sections, 3 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Baikov representation of loop integrals
  3. Integration-by-parts identities on unitarity cuts
  4. Syzygy generators from Laplace expansion
  5. Off-shell case
  6. On-shell case
  7. Proof of completeness of syzygies
  8. Examples
  9. Fully massless planar double box
  10. Planar double box with internal mass
  11. Fully massless non-planar double pentagon
  12. Conclusions

Key Result

Proposition 1

There are surjective maps \xymatrix@R=0em{ \mathop{\mathrm{Mat}}\nolimits_n(\mathsc{O}_X)^{\oplus2}\ar@{->>}[r]^-\pi & \mathop{\mathrm{Der}}\nolimits(-\log(\det)) \,,\\ (A,B)\ar@{|->}[r] & \sum_{i,j}(MA-BM)_{i,j}\frac{\partial}{\partial x_{i,j}} \,,\\ \mathop{\mathrm{Mat}}\nolimits_n(\mathsc{O}_{X'}

Figures (3)

  • Figure 1: The fully massless planar double-box diagram. All external momenta are taken to be outgoing.
  • Figure 2: Planar double-box diagram. The bold lines represent massive propagators with the same mass $M$. All external momenta are taken to be outgoing.
  • Figure 3: The fully massless non-planar double-pentagon diagram. All external momenta are taken to be outgoing.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • proof