Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions

Martijn Hidding

TL;DR

DiffExp provides a public Mathematica implementation of one-dimensional series-expansion methods for solving Feynman integral families via differential equations, enabling high-precision results at arbitrary phase-space points. The approach combines integration-sequence derivation, Frobenius-based homogeneous solutions, and generalized variation techniques, with analytic continuation and precision accelerators such as Möbius transforms and Padé approximants. It introduces dynamic and predivision line-segmentation strategies, robust handling of degenerate lines, and a comprehensive boundary-condition workflow, validated on equal- and unequal-mass three-loop banana graphs. The work offers a practical tool for multi-loop QCD computations and sets the stage for extending to elliptic and more intricate geometries in future work.

Abstract

DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its associated package, is to provide a public implementation of these series expansion methods, which works for any family of integrals for which the user provides a set of differential equations and boundary conditions (and for which the program is not computationally constrained.) The main functions of the DiffExp package are discussed, and its use is illustrated by applying it to the three loop equal-mass and unequal-mass banana graph families.

DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions

TL;DR

DiffExp provides a public Mathematica implementation of one-dimensional series-expansion methods for solving Feynman integral families via differential equations, enabling high-precision results at arbitrary phase-space points. The approach combines integration-sequence derivation, Frobenius-based homogeneous solutions, and generalized variation techniques, with analytic continuation and precision accelerators such as Möbius transforms and Padé approximants. It introduces dynamic and predivision line-segmentation strategies, robust handling of degenerate lines, and a comprehensive boundary-condition workflow, validated on equal- and unequal-mass three-loop banana graphs. The work offers a practical tool for multi-loop QCD computations and sets the stage for extending to elliptic and more intricate geometries in future work.

Abstract

DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its associated package, is to provide a public implementation of these series expansion methods, which works for any family of integrals for which the user provides a set of differential equations and boundary conditions (and for which the program is not computationally constrained.) The main functions of the DiffExp package are discussed, and its use is illustrated by applying it to the three loop equal-mass and unequal-mass banana graph families.

Paper Structure

This paper contains 39 sections, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of some aspects of Feynman integrals
  3. Basic definitions
  4. Feynman parametrization
  5. Remarks on analytic continuation
  6. Expansions around asymptotic limits
  7. The differential equations method
  8. Basic definitions
  9. Canonical basis
  10. Series expansion methods
  11. Differential equations order-by-order in $\epsilon$
  12. Deriving an integration sequence
  13. Homogeneous solutions and the Frobenius method
  14. General solutions
  15. Solutions along degenerate lines
  16. ...and 24 more sections

Figures (3)

  • Figure 1: The three-loop unequal mass banana diagram.
  • Figure 2: Plot of the master integral $B_3$ in the region $p^2/m^2 = 0 \ldots 32$. The solid lines are the real parts of the integrals, and the dotted lines the imaginary parts.
  • Figure 3: Plots of some of the precanonical basis integrals of the unequal-mass three-loop banana family. Note that $B_1 = \epsilon(1+3\epsilon)I_{1122}^\text{banana}$ and that $B_{11} = \epsilon(1+3\epsilon)(1+4\epsilon)I_{1111}^\text{banana}$.