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

Martijn Hidding

arXiv:2006.05510v2hep-phhep-th

TL;DR

Feynman-integral calculations require methods that remain usable when analytic representations become difficult for complicated geometries. This paper presents DiffExp, a public Mathematica implementation of differential-equation integration through truncated one-dimensional series expansions, with improvements for sequencing, coupled systems, segmentation, and applications to three-loop banana families. The package is intended for families supplied with differential equations and boundary conditions, subject to computational constraints.

  • Problem

    Complicated multiloop and multiscale Feynman integrals can involve geometries and coupled structures for which polylogarithmic or elliptic analytic representations are difficult or unavailable.

  • Method

    DiffExp integrates differential-equation systems by joining truncated one-dimensional series expansions along phase-space line segments, with automated sequencing, coupled-system strategies, and improved segmentation.

  • Results

    DiffExp is presented as the first publicly available Mathematica package implementing these series-expansion methods, and it is applied to equal-mass and unequal-mass three-loop banana families.

  • Takeaways & Limitations

    The package makes the described series-expansion strategy available as a general-purpose implementation for user-supplied differential equations and boundary conditions.

  • Takeaways & Limitations

    Its error estimates are convenience estimates and should not be relied upon for sensitive results; comparing evaluations along different contours is recommended instead.

Abstract

from arXiv · show

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.

1 Introduction

The paper presents DiffExp as a public Mathematica implementation for integrating Feynman integrals through differential equations and one-dimensional series expansions. It targets analytic-evaluation challenges arising from increasingly complicated integral geometries and explains the package’s scope and organization.

  • Contribution: DiffExp computes Feynman integrals from differential equations using one-dimensional series expansions joined along connected phase-space line segments.The method is intended to provide numerical results across phase space, including settings where conventional analytic representations are difficult.
  • Motivation: Feynman-integral evaluation is a major computational bottleneck alongside IBP reduction, especially beyond leading order in QCD and the Standard Model.The introduction contrasts relatively tractable leading-order calculations with increasingly difficult higher-order computations.
  • Related methods: Analytic methods can be faster than Monte Carlo and sector-decomposition approaches, but analytic representations are not always known or available.The paper reviews hypergeometric, iterated-integral, and differential-equation approaches as alternatives.
  • Motivation: Higher-loop and multiscale integrals can involve hyperelliptic curves, Calabi–Yau geometries, and coupled square roots beyond established polylogarithmic frameworks.The paper identifies three-loop banana graphs as an example associated with Calabi–Yau geometries.
  • Paper scope: The package provides a general-purpose implementation of series-expansion methods, with sections covering differential equations, continuation, precision, segmentation, package functions, and banana-graph applications.The stated applications include equal-mass and unequal-mass three-loop banana families.

2 Review of some aspects of Feynman integrals

This section introduces scalar Feynman-integral families, their parametrization, regularization, scaling, and analytic continuation. It also explains why asymptotic boundary conditions may require expansion by regions.

  • Basic definitions: A scalar Feynman-integral family is specified by a diagram’s loops, propagators, numerator structures, and integer propagator exponents.Numerator terms span the relevant loop- and external-momentum dot products.
  • Basic definitions: IBP identities reduce integrals within a family to a finite basis of linearly independent master integrals, while dimensional regularization treats the dimension as d0 − 2ϵ.The regulator controls ultraviolet and infrared divergences.
  • Feynman parametrization: The Feynman parametrization uses projective parameters and Symanzik polynomials U and F, with Cheng–Wu allowing the integration domain to be pulled back to a simplex.U is expressed through spanning trees, while F uses spanning two-forests and kinematic invariants.
  • Analytic continuation: The Feynman prescription supplies the infinitesimal imaginary part needed to select a physical branch, while the Euclidean region avoids interior threshold singularities.Outside the Euclidean region, threshold singularities are crossed by analytic continuation from a suitable starting region.
  • Boundary conditions: Boundary conditions for differential equations are often sought at simplifying limits, but naive massless limits can turn kinematic singularities into dimensionally regulated singularities.Expansion by regions provides a way to obtain asymptotic boundary terms without first solving the generic integral.

3 The differential equations method

Feynman integrals form solutions of linear differential systems in kinematic invariants and masses. The section develops integrability, scaling, basis changes, canonical forms, and path-ordered solutions in the dimensional regulator.

  • Differential systems: Derivatives of master integrals reduce back to the same family, yielding a linear system of ordinary differential equations in kinematic invariants and internal masses.The master integrals are packaged into a vector f and differentiated with respect to the variables in S.
  • Consistency conditions: The partial-derivative matrices satisfy an integrability condition that follows from the vanishing of the second total differential.This condition provides a consistency check for the differential equations.
  • Consistency conditions: Scaling relations encode the mass dimensions of basis integrals and offer another cross-check on the derived differential equations.The scaling matrix is diagonal with entries determined by the dimensions of the basis integrals.
  • Basis choices: A basis transformation can simplify the differential equations, and canonical forms isolate the dimensional-regulator dependence from kinematic differential forms.For multiple-polylogarithmic integrals, the resulting letters form an alphabet of kinematic functions.
  • Solutions: The general solution is represented by a path-ordered exponential along a path in the phase space of invariants and internal masses, then expanded order-by-order in ϵ.The ϵ expansion can be arranged to begin at finite order by multiplying the basis by an overall power of ϵ.

4 Series expansion methods

DiffExp solves differential-equation systems through connected one-dimensional series expansions. It manages finite convergence radii, singularities, analytic continuation, and several algorithmic improvements over earlier strategies.

  • Series construction: The method follows connected line segments from a boundary point to the desired phase-space point, using multiple local series expansions.Multiple segments are needed because each expansion has a finite convergence radius.
  • Analytic continuation: Branch points and singularities can be crossed by centering a segment at the singular point and analytically continuing with a line-parameter prescription ±iδ.The resulting series solutions may contain square roots and logarithms.
  • Improvements: The package improves earlier methods by deriving integration sequences from differential equations and optimizing homogeneous and coupled-system solutions.The integration sequence uses graph structure, while the coupled-system strategy is designed for general solutions.
  • Improvements: DiffExp also improves line segmentation by deriving explicit center points for neighboring segments.The package discusses segmentation strategies alongside series acceleration and continuation procedures.

4.1 Differential equations order-by-order in ϵ

DiffExp expands coupled master-integral differential equations order-by-order in the dimensional regulator along a line parameter. The formulation assumes finite basis integrals and controls regulator-dependent prefactors through basis rescalings.

  • The master-integral vector satisfies differential equations along a line parameter, with kinematic invariants and masses specified by the path γ(x).
  • The coefficient matrix is expanded in ϵ, and the resulting equations are solved by collecting terms order-by-order in the regulator.
  • The basis is assumed finite after normalization by an overall power of ϵ, while rescaling can remove poles or factors 1/P(ϵ) when possible.
  • The leading matrix A(0)_x governs the homogeneous component and determines which integrals are coupled at leading order in ϵ.

4.2 Deriving an integration sequence

DiffExp derives an integration order from the dependency structure of the leading-order differential equations. Strongly connected components identify integrals requiring simultaneous solution, while topological sorting orders the remaining components.

  • Integration proceeds from leading to higher orders in ϵ, with subsectors integrated before sectors that depend on them.
  • A directed graph encodes leading-order dependencies between basis integrals using the nonzero entries of A(0)_x.
  • Strongly connected components group integrals whose derivatives eventually involve one another, so their differential equations must be solved simultaneously.
  • Topologically sorting the condensation graph produces a compatible integration sequence, which must be re-derived for each path because coupling can depend on the transport direction.

4.3 Homogeneous solutions and the Frobenius method

DiffExp constructs homogeneous series solutions for coupled differential equations by reducing the system to scalar equations and applying Frobenius expansions. Wronskian-based transformations then assemble independent solutions and support term-by-term integration.

  • The homogeneous equations for coupled integrals are solved as series expansions around the line-segment origin.
  • Series expansion of the matrices requires selecting the correct analytic branch when square roots occur.
  • The homogeneous construction assumes the matrix ˜M is invertible for generic kinematic and mass configurations, with separate treatment needed on degenerate lines.
  • A scalar p-th order equation is obtained from the coupled system through a left-null-space construction, then normalized so its highest-derivative coefficient equals one.
  • The Frobenius method uses a series ansatz around the regular point x = 0; the indicial equation determines the leading exponent and recursive coefficient solution.
  • Reduction of order recursively supplies the remaining p − 1 independent series solutions from an initial Frobenius solution.
  • The resulting solutions form a Wronskian-based matrix representation of the homogeneous system, while powers and logarithms are integrated using repeated integration by parts and replacement rules.

4.4 General solutions

DiffExp combines homogeneous solutions with boundary-condition constants to obtain general solutions of the coupled inhomogeneous equations. It reuses matrix solutions along each line segment and includes alternatives for computing difficult inverses.

  • The general solution is built from a homogeneous solution matrix F and an auxiliary matrix H, with a constant diagonal matrix fixed by boundary conditions.
  • The solution can be expressed through columns of G, whose constants are determined from the supplied boundary data.
  • When direct inversion of the Wronskian is difficult because of logarithmic series, DiffExp solves a differential equation for its inverse and reconstructs it using a constant matrix.
  • DiffExp computes F and F−1 once for each coupled-integral set on a line segment, then reuses them at each order in ϵ with the appropriate B-matrix.
  • The method is equivalent in essence to variation of parameters, but the authors report that this implementation is more efficient in their examples.

4.5 Solutions along degenerate lines

For degenerate lines where the coefficient matrix is not invertible, DiffExp supplements its generic integration strategy with variation of parameters and algebraic relations among coupled integrals.

  • Variation of parameters provides an alternative integration method that remains applicable when the matrix ˜M is not invertible.It is typically slower than the generic strategy but more straightforward to generalize to degenerate cases.
  • DiffExp derives a higher-order scalar differential equation for each integral by finding vectors in the left null-space of an augmented coefficient matrix.The selected null-space vector has the most trailing zeros, yielding the lowest-order equation for that integral.
  • The method fixes constants from boundary conditions and uses Wronskian determinants to construct the inhomogeneous solution.The determinant calculations can be computationally heavy for large systems, and the strategy is enabled through the IntegrationStrategy option set to "VOP".
  • For an invertible reduced matrix, the scalar equation determines the remaining basis-integral solutions after computing the corresponding solution for one integral.The reduced matrix is formed by removing the last row of the augmented matrix.
  • For non-invertible systems, variation of parameters computes the coupled integrals, while relations among them remove redundant integration constants.The constants are fixed by substituting the solutions into the original differential relations and reducing the resulting linear system.

4.6 Analytic continuation

DiffExp analytically continues series solutions across singularities by tracking logarithmic and square-root branches through user-specified iδ prescriptions and replacement rules.

  • DiffExp can analytically continue series containing logarithms and square roots by assigning the line parameter an infinitesimal imaginary part consistent with the Feynman prescription.Replacement rules implement the continuation internally, while IntegrateSystem[...] exposes explicit θp and θm factors in returned results.
  • Users specify branch choices through DeltaPrescriptions, which assigns ±iδ terms to polynomials whose zeros define threshold singularities or square-root branch loci.This generalizes continuation beyond assigning signs independently to Mandelstam variables.
  • A square-root example shows that solving the differential equation first gives f(x)=c1x^1/2, after which the prescription updates the branch and boundary conditions fix c1=1.The resulting prescribed solution is reported as the correct answer.
  • For physical thresholds, square-root branches must agree with the Feynman prescription, whereas non-threshold roots may use the principal branch with +iδ.Unspecified square roots are automatically assigned +iδ.
  • DiffExp requires irreducible square-root arguments and differential-equation coefficients containing only rational functions and square roots, because it analytically continues only square roots and logarithms.Reducible arguments can prevent simultaneous satisfaction of the Feynman prescription and principal-branch choice.

4.7 Precision and numerics

DiffExp improves series precision by controlling the distance to singularities, remapping line parameters with Möbius transformations, and optionally accelerating evaluations with Padé approximants.

  • DiffExp is designed for differential equations whose coefficients contain rational functions and square roots of rational functions, with singularities determining series convergence.The relevant singularities are poles of rational functions and zeros of square-root arguments in the complex line-parameter plane.
  • 4.7.2 Improving the precision: Möbius transformations: Rescaling the line parameter and applying a Möbius transformation can move nearby singularities away from the expansion origin and equalize their distances.The transformed series converges within the interval mapped from the chosen boundaries, improving coefficient behavior.
  • 4.7.2 Improving the precision: Möbius transformations: At order 15, the original expansion gives S15(f)(x) ≈−6.65 · 10^6, whereas the transformed expansion gives S15(f)(y) ≈1.51 for f(x=1/4)=32/21≈1.52.The x-series does not converge at the target point, while the y-series does.
  • 4.7.3 Improving the precision: Padé approximants: Padé approximants evaluate boundary conditions for subsequent line segments by converting each series into a rational or fractional-power approximation.DiffExp computes diagonal Padé approximants and supports Laurent, fractional-power, and logarithmic series through decomposition.
  • 4.7.3 Improving the precision: Padé approximants: Padé approximants often reduce computation time at a target precision, but they can lower coefficient accuracy, require higher working precision, and be expensive for complicated series.They are disabled by default and enabled with UsePade→True.

4.8 Line segmentation strategies

DiffExp transports boundary conditions along contours by subdividing them into series-valid segments, using either dynamic error control or precomputed distance-based segmentation.

  • Dynamic segmentation: The dynamic strategy subdivides a line while keeping the differential-equation series error within a prescribed bound.Bounding the coefficient expansions also bounds the resulting series solutions, though not necessarily by exactly the same amount.
  • Predivision segmentation: The predivision strategy typically needs fewer line segments than dynamic integration to reach a given precision.Its segment boundaries are determined algebraically from the nearest singularities and the chosen division parameter.
  • Both strategies expand the differential equations at a segment center, impose boundary conditions there, and evaluate or transport the resulting series toward the endpoint.The process iterates by using the evaluated point as the next boundary condition until the endpoint is reached.
  • Möbius-transformed segments use projected singularities and rescaling to improve numerical behavior and converge within the interval (−1,1).Complex singularities are handled through selected projections onto the real axis.
  • Predivision segmentation: Predivision constructs all line segments before expansion and evaluates each series only within a fixed fraction of the distance to the nearest differential-equation singularity.This strategy is enabled by default with DivisionOrder set to 2.

5 The DiffExp package

DiffExp is a Mathematica package for transporting boundary conditions and solving Feynman-integral differential equations through one-dimensional series expansions. It provides configurable segmentation, precision, numerical, and continuation controls for computations along phase-space lines.

  • DiffExp is the paper’s main contribution: a public Mathematica package implementing the discussed series-expansion methods.
  • Configuration: MatrixDirectory is mandatory and should contain the partial-derivative matrices; DeltaPrescriptions is also generally required for analytic continuation.The package accepts kinematic invariants, masses, and the requested dimensional-regulator order as inputs or configuration data.
  • Configuration: SegmentationStrategy supports Dynamic and Predivision modes, while the solving strategy can use the default method or variation of parameters.Variation of parameters is generally slower for coupled integrals but works along degenerate lines.
  • Precision and numerics: AccuracyGoal targets boundary-condition transport at absolute precision 10^-δ, but highly coupled sectors may not achieve the same solution accuracy.With Predivision, differential equations may be expanded repeatedly until the requested precision is reached, which can bottleneck computation.
  • Main functions: TransportTo transports prepared boundary conditions to arbitrary real-valued phase-space points along specified lines.It can optionally return results, errors, and data for the individual line segments.
  • Precision and numerics: Error estimates are convenience diagnostics and should not be relied upon for sensitive results; comparing evaluations along two contours is recommended instead.The estimates compare reduced-order and original series at segment matching points and the final point, accumulating differences across segments.

6 Examples

The examples apply DiffExp to equal- and unequal-mass three-loop banana families, using differential equations and asymptotic boundary conditions to transport series expansions. The unequal-mass case demonstrates high-precision results, cross-checks, and substantial sensitivity to computational settings and path choice.

  • 6.1 Equal-mass three-loop banana family: Boundary conditions for the equal-mass family are obtained at the infinite momentum limit by expanding the Feynman parametrization into fifteen regions and summing their contributions.The regions are generated in the asymptotic limit and integrated at leading order in the line parameter.
  • 6.1 Equal-mass three-loop banana family: About 1 minute reaches p2/m2 = 32 from p2/m2 = −∞ with estimated error 10−25 using DivisionOrder 3 and ExpansionOrder 50.The resulting real and imaginary parts are plotted over the region p2/m2 = 0 ... 32.
  • 6.2 Unequal-mass three-loop banana family: The unequal-mass family uses 15 precanonical master integrals, with differential equations of about 8 megabytes and eleven coupled integrals in the top sector.The IBP reductions for this family were obtained using Kira, and the coupling makes the computation significantly more difficult than the equal-mass case.
  • 6.2 Unequal-mass three-loop banana family: At (50, 2, 3/2, 4/3, 1), DiffExp reports error 10−22 and independent-contour differences of order 10−24.A higher-precision run reports error 10−58 and contour differences of order 10−61; evaluating along the computed line is then nearly instantaneous.
  • 6.2 Unequal-mass three-loop banana family: A path through (50, 1, 1, 1, 1) reaches the unequal-mass target in around 23 minutes at estimated precision 10−70, versus nearly four hours along γ(x).At lower expansion order, the same route takes 6 minutes with estimated precision 10−34, and results agree with pySecDec within its reported errors.

7 Conclusions and outlook

The paper presents DiffExp as the first publicly available Mathematica implementation of one-dimensional series-expansion methods for solving Feynman-integral families through differential equations. It adds automation, optimization, improved segmentation and acceleration, and demonstrates the strategy on higher-degree coupled banana graphs, while leaving extensions beyond rational and square-root prefactors for future work.

  • DiffExp is the first publicly available Mathematica package implementing truncated one-dimensional series expansions for Feynman-integral families through systems of differential equations.
  • The package adds automatic integration-sequence derivation, optimized treatment of coupled integrals, improved neighbouring-segment matching, and series acceleration using Padé approximants and Möbius transformations.
  • The strategy is applied to three-loop equal-mass and unequal-mass banana families whose top sectors are coupled at orders 3 and 11, respectively.
  • Future work includes extending DiffExp to integral bases with prefactors beyond rational functions and square roots, including elliptic integrals.
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