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Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form

O. Gituliar, V. Magerya

arXiv:1701.04269v1hep-phmath-phmath.CAmath.MP

TL;DR

Feynman master integrals can be calculated from rational differential systems, but obtaining their solutions as epsilon series requires an epsilon-form transformation. Fuchsia implements Lee's algorithm to find such transformations, and the authors report reducing a 74 × 74 matrix in about an hour; some normalization cases still require manual changes of variables.

  • Problem

    Rational differential systems for Feynman master integrals are readily solved as epsilon series when their epsilon form is known, making a tool to find that transformation useful for evaluating the integrals.

    Source: PROGRAM SUMMARY · p. 2

  • Method

    Fuchsia implements Lee's algorithm to transform rational differential systems into epsilon form through fuchsification, normalization, and factorization.

    Source: §2.2 · p. 6, §4 · p. 17

  • Results

    About an hour is needed to reduce a 74 × 74 matrix with 20 rational and complex singular points and at most 3 × 3 coupled blocks on a laptop with Intel i5 CPU.

    Source: §4 · p. 18

  • Takeaways & Limitations

    Once the epsilon form is found, the system can be solved as a Laurent series in epsilon and solutions for the initial system restored by solving a linear system.

    Source: §2.2 · p. 6

    When residue eigenvalues do not fit n + m epsilon after fuchsification, suitable nonlinear changes of variables may need to be found manually because Fuchsia does not automate such cases.

    Source: §2.2.2 · p. 8

Abstract

from arXiv · show

We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $\partial_x\,\mathbf{f}(x,\epsilon) = \mathbb{A}(x,\epsilon)\,\mathbf{f}(x,\epsilon)$ finds a basis transformation $\mathbb{T}(x,\epsilon)$, i.e., $\mathbf{f}(x,\epsilon) = \mathbb{T}(x,\epsilon)\,\mathbf{g}(x,\epsilon)$, such that the system turns into the epsilon form: $\partial_x\, \mathbf{g}(x,\epsilon) = \epsilon\,\mathbb{S}(x)\,\mathbf{g}(x,\epsilon)$, where $\mathbb{S}(x)$ is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator $\epsilon$. That makes the construction of the transformation $\mathbb{T}(x,\epsilon)$ crucial for obtaining solutions of the initial equations. In principle, $\text{Fuchsia}$ can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals.

PROGRAM SUMMARY

Fuchsia provides methods for finding epsilon-form transformations for differential systems used to evaluate Feynman master integrals.

  • Finding epsilon-form transformations can make rational differential systems for Feynman master integrals solvable as an epsilon series.Source: PROGRAM SUMMARY · p. 2
  • The program uses methods of Moser and Lee, and applies only when the system is reducible to Fuchsian form with residue eigenvalues n + m epsilon, where n is integer.Source: PROGRAM SUMMARY · p. 2
  • About an hour is needed to reduce a 74 × 74 matrix with 20 singular points on a 1.7GHz Intel Core i5 PC.Running time depends on the input data, its size, and complexity.Source: PROGRAM SUMMARY · p. 2

1 Introduction

The introduction motivates epsilon-form differential equations as a route to epsilon-series solutions and presents Fuchsia as a public implementation of Lee's algorithm. The method targets Feynman integrals and can be paired with integration-by-parts reduction tools.

  • Epsilon form makes a differential-equation system readily solvable as an epsilon series, supporting the calculation of Feynman integrals.Source: §1 · p. 3
  • Lee's algorithm first finds Fuchsian form using improved Moser reduction [Mos59], then normalizes the eigenvalues of the Fuchsian system at every singular point.Source: §1 · pp. 3–4
  • After those steps, a transformation to epsilon form may be found if they succeed; the method also applies to other problems meeting the initial-system requirements.Source: §1 · p. 4
  • Until recently, no public implementation of Lee's method was available; Fuchsia was presented as the first public implementation, followed by Epsilon [Pra17].Source: §1 · p. 4
  • Fuchsia and integration-by-parts reduction algorithms form a tandem for evaluating Feynman integrals.Source: §1 · p. 4

2 Overview of the Lee method

The Lee method reduces a rational one-variable ODE system to epsilon form through fuchsification, normalization, and factorization. For block-triangular systems, reducing diagonal blocks first can streamline the process.

2.1 Notation and definitions

  • The method treats a rational matrix-valued ODE system, with unknown functions collected in a vector and singular behavior assessed at finite points and infinity.The basis transformations are restricted to be rational in both x and ǫ.Source: §2.1 · pp. 4–5

2.2 Reduction to epsilon form

  • The reduction seeks a rational transformation to a Fuchsian epsilon form, which can then be solved as a Laurent series in ǫ to recover solutions of the initial system.The method comprises fuchsification, normalization, and factorization.Source: §2.2 · p. 6

2.2.1 Fuchsification

  • Fuchsification lowers Moser order at irregular points, but a rank-lowering transformation can increase the Poincaré rank elsewhere and introduce an apparent singularity.A system is fuchsifiable only if it has no irregular singularities; sometimes the rank increase occurs at a previously regular point.Source: §2.2.1 · p. 7

2.2.2 Normalization

  • Normalization shifts integer parts of residue eigenvalues until they vanish, preserving Fuchsian form and Poincaré ranks.It applies when the residue eigenvalues have the form n + m ǫ; if they do not fit that form after fuchsification, a nonlinear variable change may be needed, which the tool does not automate.Source: §2.2.2 · p. 8

2.2.3 Factorization

  • After normalization, an x-independent transformation can factor the matrix as ǫ S(x), with the transformation determined from linear equations involving the residues.An arbitrary reference value can be used to reconstruct an invertible transformation.Source: §2.2.3 · pp. 8–9

2.2.4 Block-triangular form

  • For block-triangular systems, the method reduces diagonal blocks first, then fuchsifies off-diagonal blocks; exploiting sparse block structure can speed reduction.The diagonal blocks can be treated independently, after which the off-diagonal blocks are reduced in sequence.Source: §2.2.4 · pp. 9–10

3 Using Fuchsia

This section explains how to use Fuchsia, covering installation prerequisites, its command-line and SageMath interfaces, a worked reduction, and the function reference. It also documents matrix input, runtime options, and the exception raised when a requested reduction cannot be performed.

3.1 Installation

  • Installation requires SageMath version 7.0 or higher and fuchsia.py; the authors caution against some distribution builds because of Maxima bugs affecting Fuchsia.Source: §3.1 · p. 10

3.2 Usage from the command line

  • The command line runs through sage -python fuchsia.py with actions for epsilon reduction, individual reduction stages, block sorting, custom transformations, and variable changes.Source: §3.2 · p. 11

3.3 Usage from SageMath or Python

  • In SageMath, a sample matrix is reduced through fuchsification, normalization, simplification, and factorization; epsilon_form automates the reduction, and results can vary with step order.The worked example checks the returned transformation against the transformed matrix.Source: §3.3 · p. 14

3.3.1 Function reference

  • The form API triangularizes M, reduces diagonal blocks, then handles off-diagonal blocks and epsilon factorization; it raises FuchsiaError if the system is irreducible.Source: §3.3.1 · p. 15
  • file(filename) reads symbolic matrices in Mathematica or MatrixMarket array format, with the format detected automatically.Source: §3.3.1 · p. 17
  • Fuchsia’s options set logging verbosity from errors-only to verbose and can enable Maple to speed some calculations, especially for large matrices or complex singularities.Source: §3.3.1 · p. 17
  • FuchsiaError indicates that a Fuchsia routine cannot perform the requested reduction.Source: §3.3.1 · p. 17

4 Summary

Fuchsia implements the Lee algorithm for reducing Feynman-integral differential equations to epsilon form, with optimizations for block-triangular or sparse matrices. A 74 × 74 example with at most 3 × 3 coupled blocks took about an hour, while computations involving algebraic-number coefficients can limit performance.

4 Summary

  • Heavy-duty computations may require Maple, but algebraic-number coefficients can still cause difficulty and sharply reduce performance.Maple is more powerful for some tasks, but cannot process some huge intermediate expressions involving algebraic numbers in their coefficients.Source: §4 · p. 17
  • Fuchsia's optimization for block-triangular or sparse matrices supports good performance in many cases.The optimization speeds reduction of relatively large matrices.Source: §4 · pp. 17–18
  • On a laptop with an Intel i5 CPU, reducing a 74 × 74 matrix with 20 rational and complex singular points and at most 3 × 3 coupled blocks takes about an hour.Source: §4 · p. 18

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