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AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow

Xiao Liu, Yan-Qing Ma

TL;DR

AMFlow delivers a public Mathematica package for high-precision numerical evaluation of dimensionally regularized Feynman integrals via the auxiliary mass flow method. By solving differential equations in an auxiliary mass parameter and applying a numerical-fit strategy, it achieves controlled boundary conditions and efficient epsilon-expansions, with automatic and manual computation modes. The framework supports iterative reduction to manage master integrals, and integrates with established IBP reducers to enable automated calculation of complex multiloop integrals. This approach promises significant gains in speed and precision for phenomenological studies, and aims to pair AMFlow with public reduction tools to tackle even more challenging integrals.

Abstract

AMFlow is a Mathematica package to numerically compute dimensionally regularized Feynman integrals via the recently proposed auxiliary mass flow method. In this framework, integrals are treated as functions of an auxiliary mass parameter and their results can be obtained by constructing and solving differential systems with respect to this parameter, in an automatic way. The usage of this package is described in detail through an explicit example of double-box family involved in two-loop $t\bar{t}$ hadroproduction.

AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow

TL;DR

AMFlow delivers a public Mathematica package for high-precision numerical evaluation of dimensionally regularized Feynman integrals via the auxiliary mass flow method. By solving differential equations in an auxiliary mass parameter and applying a numerical-fit strategy, it achieves controlled boundary conditions and efficient epsilon-expansions, with automatic and manual computation modes. The framework supports iterative reduction to manage master integrals, and integrates with established IBP reducers to enable automated calculation of complex multiloop integrals. This approach promises significant gains in speed and precision for phenomenological studies, and aims to pair AMFlow with public reduction tools to tackle even more challenging integrals.

Abstract

AMFlow is a Mathematica package to numerically compute dimensionally regularized Feynman integrals via the recently proposed auxiliary mass flow method. In this framework, integrals are treated as functions of an auxiliary mass parameter and their results can be obtained by constructing and solving differential systems with respect to this parameter, in an automatic way. The usage of this package is described in detail through an explicit example of double-box family involved in two-loop hadroproduction.
Paper Structure (12 sections, 48 equations, 4 figures)

This paper contains 12 sections, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Auxiliary mass flow
  3. The plain method
  4. Iterative strategy
  5. Numerical fit
  6. Using AMFlow
  7. Input
  8. Automatic computation
  9. Manual computation
  10. Other functions
  11. Summary of options
  12. Summary and outlook

Figures (4)

  • Figure 1: Singularities and analytic continuations. Singularities are labeled as crosses. Solid dots are points where to perform series expansions.
  • Figure 2: A two-loop five-point massless double-pentagon topology.
  • Figure 3: Figure from Ref. Liu:2021wks. The all-small region iteration of the double-pentagon topology. For each Feynman diagram, the solid line (if exists) represents the propagator where we introduce $\eta$. The number of master integrals of the original topology and the one with $\eta$ introduced (in parentheses) are also listed below each diagram except the last scaleless topology.
  • Figure 4: A two-loop planar integral family involved in NNLO QCD corrections to $t\bar{t}$ hadroproduction.