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epsilon: A tool to find a canonical basis of master integrals

Mario Prausa

arXiv:1701.00725v3hep-ph

TL;DR

Finding a canonical basis for master integrals is challenging, though it makes their ϵ-series solution trivial when boundary conditions are known. epsilon implements Lee’s algorithm using Fermat, and the authors demonstrate its applicability on a three-loop example and the possibility of reducing systems with complex singular points.

  • Problem

    Finding a canonical basis for certain master integrals is challenging, although it makes their ϵ-series solution trivial when boundary conditions are known.

    Source: Program Summary · p. 1, §1 · p. 2

  • Method

    epsilon implements Lee’s algorithm with Fermat, exploiting explicit kinematic-variable dependence to reduce intermediate variables and support complex singularities.

    Source: §1 · p. 2

  • Results

    The authors demonstrated epsilon’s applicability on a three-loop example and the possibility of reducing systems with complex singular points.

    Source: §5 · p. 26

  • Takeaways & Limitations

    When boundary conditions are known, a canonical basis makes the ϵ-series solution of the master-integral differential equations trivial.

    Source: §1 · p. 2

    Normalization fails unless residue eigenvalues are a+bϵ with integer a,b, and multi-scale problems are unsupported.

    Source: Program Summary · p. 1

Abstract

from arXiv · show

In 2013, Henn proposed a special basis for a certain class of master integrals, which are expressible in terms of iterated integrals. In this basis, the master integrals obey a differential equation, where the right hand side is proportional to $\epsilon$ in $d=4-2\epsilon$ space-time dimensions. An algorithmic approach to find such a basis was found by Lee. We present the tool epsilon, an efficient implementation of Lee's algorithm based on the Fermat computer algebra system as computational backend.

Program Summary

epsilon implements Lee’s algorithm to find canonical bases for certain master integrals, where the differential equation’s right-hand side is proportional to ϵ. In that basis, solving the integrals in an ϵ-expansion becomes trivial.

Program Summary

  • In the canonical basis, the master-integral differential equation has a right-hand side proportional to ϵ, making its ϵ-expansion solution trivial.The described application is a certain class of master integrals.Source: Program Summary · p. 1
  • epsilon implements Lee’s algorithm to find such a canonical basis.The tool is implemented in C++ and distributed under GPLv3.Source: Program Summary · p. 1
  • Normalization fails unless residue eigenvalues are a+bϵ with integer a,b, and multi-scale problems are unsupported.These are stated restrictions of the tool.Source: Program Summary · p. 1

1 Introduction

Integration-by-parts identities reduce large collections of Feynman integrals to master integrals, whose solutions can be simplified by a canonical basis. The paper presents epsilon, an implementation of Lee’s algorithm that reduces intermediate variables and supports systems with complex singular points.

1 Introduction

  • The introduction places the method in the context of multi-loop Feynman-integral evaluation in d=4−2ϵ dimensions.The denominators are usually formed from squared momenta minus masses.Source: §1 · p. 2
  • Integration-by-parts identities reduce many Feynman integrals to a relatively small set of master integrals.After tensor reduction, diagrams can be expressed as a larger number of scalar Feynman integrals.Source: §1 · p. 2
  • A canonical basis makes the ϵ-series solution of master-integral differential equations trivial when boundary conditions are known.The basis applies to master integrals expressible in iterated integrals.Source: §1 · p. 2
  • Lee proposed an algorithm to automate finding a canonical basis, following earlier implementations of the algorithm.The earlier implementations are cited as [9] [10].Source: §1 · p. 2
  • epsilon uses Lee’s algorithm, exploiting transformations’ explicit kinematic-variable dependence to reduce intermediate variables and Fermat polymod to support complex singularities.The paper describes epsilon as a further implementation of Lee’s algorithm.Source: §1 · p. 2

2 Implementation details

The implementation applies Lee’s algorithm to differential systems with explicit kinematic-variable dependence, exploiting block-triangular structure to reduce active and off-diagonal blocks. It normalizes eigenvalues through balances between singularities and uses an x-independent transformation to factor out ε, reducing off-diagonal blocks to Fuchsian form when diagonal blocks are already in ε-form.

2.1 Definitions

  • The system is kept in an explicit-singularity form, and the active block is chosen within the block-triangular system, preferably with small dimension to reduce computational complexity.The stated restriction forbids singularities that depend on ε.Source: §2.1 · pp. 3–4

2.2 Utilizing the explicit x-dependence

  • Balances and off-diagonal reductions have explicit x-dependence, while an x-independent transformation factors out ε; coefficient-matrix operations avoid partial fractioning afterward.The coefficient-matrix form reduces the rational functions handled by the computer algebra system to one fewer variable.Source: §2.2 · pp. 4, 5, 7

2.3 Overview of Lee’s algorithm

  • Lee’s algorithm transforms the active block to Fuchsian form, normalizes residue eigenvectors, and factors out ε, then makes the off-diagonal block B Fuchsian.The first three steps are applied to the active block; the fourth addresses B.Source: §2.3 · p. 7

Fuchsification

  • Balances with a suitable projector can lower a singularity’s generalized Poincaré rank; if no suitable projector exists, using a regular point creates an apparent singularity later removed by normalization.The rank-lowering procedure is described for finite singularities, and the chosen projector must satisfy an invariant-space condition at another singular point when one is used.Source: Fuchsification · pp. 8–10

Normalization

  • For Fuchsian systems, residue eigenvalues of the form a+bε with integer a are required for normalization; balances can shift eigenvalues at two singularities by one.A normalized eigenvalue is proportional to ε; if the stated eigenvalue condition fails, redefining the kinematic variable might help.Source: Normalization · p. 10
  • Normalization balances negative eigenvalues upward against positive ones; a fallback singularity handles unmatched eigenvalues, and an apparent singularity can enable a restart at the fallback.Balances with the fallback normalize eigenvalues at every other singularity; an unnormalized eigenvalue at the fallback can be paired with one at a newly created apparent singularity.Source: Normalization · p. 12
  • After normalization, an x-independent transformation factors out ε without changing system eigenvalues, and its defining linear system can be solved by Gaussian elimination.The transformation constants must make T(ε, µ0) invertible for an arbitrary µ0.Source: ϵ-Factorization · pp. 12–13

Fuchsification of off-diagonal blocks

  • With diagonal blocks A and C already in ε-form, transformations choose bG to make the off-diagonal block’s highest-rank term vanish, reducing its singularities to Fuchsian form.The linear equations for bG can be solved by Gaussian elimination.Source: Fuchsification of off-diagonal blocks · p. 13

3 Usage

Section 3 explains how to install epsilon and its Fermat backend, prepare and represent input systems, run the reduction jobs, handle complex singularities, and use Mathematica utilities.

3.1 Installation guide on Linux systems

  • Install libFermat before epsilon; the programs and library can be placed under a chosen prefix, with environment paths adjusted for non-standard locations.The installation instructions also describe building the programs individually and installing the optional Mathematica package.Source: §3.1 · pp. 14–15

3.2 Input/Output format

  • Epsilon input files encode each coefficient matrix by its singularity and pole order, with ϵ represented by the fixed symbol ep.The format uses lists of matrix rows, and the paper points to a section 4 example.Source: §3.2 · p. 15

3.3 Usage of epsilon-prepare

  • epsilon-prepare converts Mathematica matrices of rational functions in ϵ and x into epsilon format by partial fractioning over complex x.It factors denominator polynomials over the integers and finds roots factor by factor; this root-finding step fails for irreducible factors above degree two or roots depending on ϵ.Source: §3.3 · p. 16

3.4 Usage of epsilon

  • Epsilon processes command-line jobs in order to transform active blocks toward Fuchsian and ϵ-form systems, with options for normalization and factoring out ϵ.Transformation queues can save and replay work; the tool can also generate a Dyson operator for an active block already in ϵ-form.Source: §3.4 · p. 17, Jobs · pp. 18–19

3.5 Using field extensions

  • Fermat polymods encode algebraic extensions by imposing a polynomial relation, allowing epsilon to handle complex numbers in a system.The authors report solving systems with up to three complex numbers by assigning a polymod to the most frequent variable and reloading when changing the active one.Source: §3.5 · pp. 19–20

3.6 Usage of EpsilonTools.m

  • EpsilonTools.m is optional Mathematica support for preparing and reading epsilon-format systems, checking their form, and identifying block boundaries.Its functions can also map epsilon-prepare symbols to Mathematica expressions.Source: §3.6 · pp. 20–21

4 A physical example

The example applies epsilon to a three-loop family of master integrals and reduces its differential system to ϵ-form. It illustrates complex singular points and reports a reduction time of about three minutes.

4 A physical example

  • The example defines a family of nine three-loop master integrals in d = 4−2ϵ dimensions, with internal lines of mass one or zero.Figure 4.1 depicts the integrals except I3; I3 shares I2's topology but has an additional numerator.Source: §4 · p. 21, Figure 4.1 · p. 22
  • The system has singularities at r3, −1, 1−r3, 1, 0, and ∞, with respective Poincaré ranks 0, 2, 0, 4, 1, and 1.The symbol r3 is introduced as a root of a polynomial in the example.Source: §4 · p. 23
  • The input system has a block-triangular structure, and the paper identifies candidate blocks {1, 1}, {2, 3}, {4, 4}, and {5, 9} for reduction.The blocks can be read from the matrix or identified with EpsilonBlocks.Source: §4 · pp. 22–23
  • The reduction process takes about three minutes on an Intel Core i5-3320M CPU with 2.60 GHz, and the resulting system is in ϵ-form (2.4).For block {5, 9}, µ = −1 would lead to a singular system, so µ = 1 is chosen instead.Source: §4 · p. 24

5 Summary

The paper presents epsilon as an efficient implementation of Lee's algorithm for reducing rational-coefficient differential systems to a canonical form with an ϵ-proportional right-hand side. A three-loop example demonstrates its applicability, including systems with complex singular points.

5 Summary

  • epsilon efficiently implements Lee's algorithm to reduce rational-coefficient ordinary differential systems to canonical form with a right-hand side proportional to ϵ.The paper demonstrates the tool on a three-loop example and systems with complex singular points.Source: §5 · pp. 25–26
  • In dimensional regularization, ϵ is usually the small regulator parameter in d = 4−2ϵ dimensions.Source: §5 · p. 25
  • The three-loop example demonstrates epsilon's applicability to systems with complex singular points.Source: §5 · p. 26

A.1 Balances

The appendix describes balances acting on a block-triangular system: they transform the active block and can alter off-diagonal blocks, while leaving other blocks unchanged.

A.1 Balances

  • Balances act on the active block C to reduce generalized Poincaré ranks and normalize eigenvalues at Fuchsian singularities.The system is assumed to be in block-triangular form.Source: §A.1 · p. 26
  • The projector P is determined by its action on the active block, and the balance also modifies blocks B and E.Source: §A.1 · pp. 26, 28, 29
  • The appendix's coefficient-matrix transformation leaves all other blocks unaffected.Source: §A.1 · pp. 26–28

A.2 Fuchsification of off-diagonal blocks

The appendix transforms the off-diagonal block B to Fuchsian form, assuming diagonal blocks A and C are already in ϵ-form. The stated transformation uses T = L(x1, k, G) and its inverse.

A.2 Fuchsification of off-diagonal blocks

  • The transformation targets B while assuming diagonal blocks A and C are already in ϵ-form.Source: §A.2 · p. 30
  • The stated transformation is T = L(x1, k, G), with inverse T−1 = L(x1, k, −G); the displayed relation is labeled (A.2).The definitions (2.8) are used.Source: §A.2 · pp. 30–31
  • Besides B, the transformation affects D; A and C remain unaffected.Source: §A.2 · p. 31

A.3 ϵ-Factorization

For ϵ-factorization, the required transformation is x-independent, so each coefficient matrix transforms identically by a similarity transformation. In the block-triangular structure, blocks B and E are affected alongside C.

A.3 ϵ-Factorization

  • Because the ϵ-factorization transformation is x-independent, every coefficient matrix transforms identically by a similarity transformation.Source: §A.3 · p. 32
  • The transformation is applied using the block-triangular structure of the coefficient matrices.Source: §A.3 · p. 32
  • Besides block C, blocks B and E are influenced by the transformation.Source: §A.3 · p. 32
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