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Feynman parameter integration through differential equations

Martijn Hidding, Johann Usovitsch

TL;DR

This paper introduces an automated numerical framework for multi-loop Feynman integrals that iteratively applies Feynman's trick to merge propagators, generating simplified topologies whose master integrals are solved via differential equations in a Feynman parameter. Solutions are expressed as piecewise generalized power series, which are integrated term-by-term to obtain the original integral, avoiding manual boundary-condition specification. Regularization is handled through quasi-finite bases or epsilon-sampling with pole-resolving techniques, and threshold singularities are managed with carefully chosen iδ prescriptions. The approach demonstrates substantial reductions in the number of master integrals and computational effort, showcased by a non-planar double pentagon example and validated against AMFlow, pointing toward an efficient, automatic route for complex Feynman-integral computations.

Abstract

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, and segment by segment. This approach leads to a fully algorithmic method for computing Feynman integrals from differential equations, which does not require the manual determination of boundary conditions. Furthermore, the most complicated topology that appears in the method often has less master integrals than the original one. We illustrate the strength of our method with a five-point two-loop integral family.

Feynman parameter integration through differential equations

TL;DR

This paper introduces an automated numerical framework for multi-loop Feynman integrals that iteratively applies Feynman's trick to merge propagators, generating simplified topologies whose master integrals are solved via differential equations in a Feynman parameter. Solutions are expressed as piecewise generalized power series, which are integrated term-by-term to obtain the original integral, avoiding manual boundary-condition specification. Regularization is handled through quasi-finite bases or epsilon-sampling with pole-resolving techniques, and threshold singularities are managed with carefully chosen iδ prescriptions. The approach demonstrates substantial reductions in the number of master integrals and computational effort, showcased by a non-planar double pentagon example and validated against AMFlow, pointing toward an efficient, automatic route for complex Feynman-integral computations.

Abstract

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, and segment by segment. This approach leads to a fully algorithmic method for computing Feynman integrals from differential equations, which does not require the manual determination of boundary conditions. Furthermore, the most complicated topology that appears in the method often has less master integrals than the original one. We illustrate the strength of our method with a five-point two-loop integral family.
Paper Structure (12 sections, 28 equations, 7 figures, 2 tables)

This paper contains 12 sections, 28 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Feynman parametrization
  4. Iterating Feynman's trick
  5. Direct integration through differential equations
  6. Main method
  7. Regularization
  8. Threshold singularities
  9. Pedagogical examples
  10. Non-planar double pentagon family
  11. Computational complexity
  12. Conclusion

Figures (7)

  • Figure 1: A double box topology named topo7 which in this paper is dressed with one massive external leg and one massive propagator.
  • Figure 2: A double pentagon topology named 5p which in this paper is dressed with one massive external leg.
  • Figure 3: Step $\kappa = 1$: Combining propagators for the first time, reducing the complexity of the initial topology 5p.
  • Figure 4: Step $\kappa = 2$: Combining propagators for the second time, reducing the complexity further.
  • Figure 5: Step $\kappa = 3$: Combining propagators for the third time, reducing the complexity further.
  • ...and 2 more figures