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

Mario Prausa

TL;DR

The paper presents epsilon, a Fermat-backed implementation of Lee's algorithm to find a canonical basis of master integrals whose differential equations have an $\epsilon$-proportional right-hand side ($\epsilon$-form). By leveraging explicit $x$-dependence and a block-triangular system, epsilon performs Fuchsification, eigenvalue normalization, and $\epsilon$-factorization, enabling straightforward $\epsilon$-expansions. It provides detailed installation/setup guidance, a Mathematica bridge via epsilon-prepare and EpsilonTools.m, and demonstrates the approach on a non-trivial three-loop example with complex singularities. The tool thus automates a previously challenging step in multi-loop calculations, streamlining the derivation of $\epsilon$-form differential equations for Feynman integrals.

Abstract

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 $ε$ in $d=4-2ε$ 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.

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

TL;DR

The paper presents epsilon, a Fermat-backed implementation of Lee's algorithm to find a canonical basis of master integrals whose differential equations have an -proportional right-hand side (-form). By leveraging explicit -dependence and a block-triangular system, epsilon performs Fuchsification, eigenvalue normalization, and -factorization, enabling straightforward -expansions. It provides detailed installation/setup guidance, a Mathematica bridge via epsilon-prepare and EpsilonTools.m, and demonstrates the approach on a non-trivial three-loop example with complex singularities. The tool thus automates a previously challenging step in multi-loop calculations, streamlining the derivation of -form differential equations for Feynman integrals.

Abstract

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 in 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.

Paper Structure

This paper contains 18 sections, 74 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Implementation details
  3. Definitions
  4. Utilizing the explicit $x$-dependence
  5. Overview of Lee's algorithm
  6. Usage
  7. Installation guide on Linux systems
  8. Input/Output format
  9. Usage of epsilon-prepare
  10. Usage of epsilon
  11. Using field extensions
  12. Usage of EpsilonTools.m
  13. A physical example
  14. Summary
  15. Transformations
  16. ...and 3 more sections

Figures (1)

  • Figure 4.1: Three-loop master integrals to be solved with epsilon. The not pictured integral $I_3$ has the same topology as $I_2$ but with an additional numerator. The thick (thin) lines are massive (massless). The thin external lines carry the momenta $q_1$ and $q_2$, while the double line carries the momentum $q_1+q_2$.