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Transforming differential equations of multi-loop Feynman integrals into canonical form

Christoph Meyer

TL;DR

This work tackles transforming differential equations for multi-loop master integrals into the canonical $\epsilon$-form by constructing a rational transformation from a given IBP-reduced basis. It develops a general algorithm that leverages transformation properties, a finite $\epsilon$-expansion, recursion over subsectors, and Leinartas decomposition to express the required rational functions, solving finitely many differential equations order-by-order. The method accommodates multiple scales and rational $\epsilon$-dependence, and is validated on non-trivial two-loop double-box topologies for both single-top-quark production and vector-boson pair production, yielding explicit canonical bases and alphabets. The results enable automatic, systematic computation of higher-loop integrals by reducing them to integration of a canonical system, significantly simplifying (and potentially automating) multi-loop QCD calculations with multiple scales.

Abstract

The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

Transforming differential equations of multi-loop Feynman integrals into canonical form

TL;DR

This work tackles transforming differential equations for multi-loop master integrals into the canonical -form by constructing a rational transformation from a given IBP-reduced basis. It develops a general algorithm that leverages transformation properties, a finite -expansion, recursion over subsectors, and Leinartas decomposition to express the required rational functions, solving finitely many differential equations order-by-order. The method accommodates multiple scales and rational -dependence, and is validated on non-trivial two-loop double-box topologies for both single-top-quark production and vector-boson pair production, yielding explicit canonical bases and alphabets. The results enable automatic, systematic computation of higher-loop integrals by reducing them to integration of a canonical system, significantly simplifying (and potentially automating) multi-loop QCD calculations with multiple scales.

Abstract

The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

Paper Structure

This paper contains 13 sections, 10 theorems, 170 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithm
  4. General properties of the transformation
  5. Expanding the transformation
  6. Recursion over subsectors
  7. Leinartas decomposition
  8. Solving for a rational transformation
  9. Applications
  10. Two loop single top-quark production
  11. Vector boson pair production
  12. Conclusion
  13. Polynomial rings

Key Result

Theorem 1

Let $I\subset K[X,Y_1,\dots,Y_m]$ be an ideal and $G$ be a Gröbner basis of $I$ with respect to lexicographic order with $X>Y_1>\dots >Y_m$. Then is a Gröbner basis of the ideal $I\cap K[Y_1,\dots,Y_m]$.

Figures (4)

  • Figure 1: Two loop graph of the planar topology 1.
  • Figure 2: Two loop graph of the non-planar topology 2
  • Figure 3: Two loop graph of topology 1.
  • Figure 4: Two loop graph of topology 2.

Theorems & Definitions (19)

  • Definition 1: Algebraic Independence
  • Theorem 1: Elimination Theorem
  • Lemma 1
  • Lemma 2
  • Corollary 1: Nullstellensatz certificate
  • Theorem 2: Leinartas
  • Theorem 3: Division Algorithm
  • Definition 2: Ideal
  • Definition 3: Ideal generated by a set of polynomials
  • Definition 4: Irreducible polynomial
  • ...and 9 more