Source-linked AI summary
AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow
Xiao Liu, Yan-Qing Ma
TL;DR
Feynman-integral computation requires reliable master-integral evaluation for precision particle-physics studies. AMFlow automates auxiliary mass flow with differential equations, iterative reduction, and numerical fitting, while reporting high-performance solving and an extreme-case limitation.
Problem
Feynman integrals are crucial for testing the Standard Model and probing new physics, while complicated problems can make auxiliary-mass differential systems costly because η may increase the number of master integrals.
Method
AMFlow implements auxiliary mass flow using differential-equation recurrence relations, an iterative strategy, and numerical fitting of high-precision series solutions.
Results
The package provides a high-performance differential-equations solver suitable for high-precision computations.
Takeaways & Limitations
AMFlow offers a public implementation intended to make auxiliary mass flow more widely usable in phenomenological studies.
Takeaways & Limitations
SolveIntegrals may fail to produce satisfactory results in some extreme cases, requiring a more involved manual computation.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper motivates AMFlow as a public implementation of auxiliary mass flow for high-precision Feynman-integral computations, addressing the need for usable methods in phenomenological studies.
- Feynman integrals are crucial for testing the Standard Model and probing new physics.
- Existing master-integral methods include sector decomposition, Mellin–Barnes representations, difference equations, and traditional differential equations.
- Auxiliary mass flow computes Feynman integrals by solving differential equations with respect to an auxiliary mass parameter η.
- Its advantages include systematic boundary conditions at η →∞, efficient high-precision ordinary differential-equation solving, and applicability to linear propagators and phase-space integrations.
- The paper implements the fully iterative strategy and a high-performance numerical solver to broaden auxiliary mass flow's use in phenomenological studies.
2. Auxiliary mass flow
Auxiliary mass flow introduces η into Feynman integrals, derives differential equations for master integrals, and analytically continues series solutions from η = ∞ to the physical limit η → i0−.
- 2. Auxiliary mass flow: The auxiliary family inserts η into propagators, after which physical results are recovered by taking η → i0−.
- 2. Auxiliary mass flow: At η →∞, region analysis reduces auxiliary integrals to combinations of equal-mass vacuum integrals that provide boundary information.
- 2. Auxiliary mass flow: Integral reduction expresses auxiliary integrals through master integrals and supplies differential equations that numerically flow from η = ∞ to η = i0−.
- 2.1. The plain method: The continuation path uses regular points for successive series expansions while avoiding singularities and the branch cut.
- 2.1. The plain method: The three stages expand at infinity, propagate between regular points, and match a formal η = 0 expansion before taking the physical limit.
- 2.1. The plain method: Substituting expansions into the differential equations yields recurrence relations that reduce coefficients to boundary data and determine the physical solution.
- 2.1. The plain method: The one-loop example's physical-limit result agrees with the analytic result.
- 2.2. Iterative strategy: Introducing η can greatly increase master-integral counts, motivating an iterative strategy that introduces η to fewer propagators.
3. Numerical fit
Numerical fit estimates expansion coefficients by evaluating a function at sample points and solving linear equations, with sample placement and precision chosen to control accuracy and cost. Applied to Feynman integrals, the method avoids truncated epsilon-series cancellations and enables independent parallel computations.
- Numerical-fit construction: Numerical fit estimates expansion coefficients by evaluating f(x) at N+1 sample points and solving a linear system.The samples lie near x=0, with N≥k for an expansion through order k.
- Numerical-fit construction: Sample points can have comparable magnitudes r much smaller than the convergence radius R or be uniformly distributed on a circle of radius r<R.These are the two useful sample-point choices identified in practice.
- Accuracy and cost: The relative accuracy of fitted coefficients decreases as the expansion order n increases, so the target order determines the required sample precision.The paper recommends setting the estimated error at the target order comparable to the desired error.
- Accuracy and cost: Sample precision must be sufficiently high for correct results, while evaluation time depends on the problem and numerical algorithm.In power-series differential-equation methods, the dominant sample cost is typically polynomial-like in the number of correct digits.
- Application to Feynman integrals: For Feynman integrals, numerical values at selected epsilon points are fitted to epsilon expansions, with real epsilon samples helping avoid potential complexities.The reference sampling formulas may require practical adjustments for satisfactory results.
- Application to Feynman integrals: Numerical fit treats sampled integrals as pure numbers, resolves epsilon-order cancellations, permits more epsilon orders through higher precision, and supports massive parallelization.The sample-point computations are independent of one another.
- Extensions: The same fitting strategy can produce asymptotic expansions at a phase-space point or auxiliary-mass value while ignoring removable singularities inside a sampling circle.This can be useful when differential equations contain many removable singularities.
4. Using AMFlow
AMFlow is used by defining an integral family and numerical kinematics, selecting an IBP reducer, and calling SolveIntegrals for automatic reduction and auxiliary-mass-flow evaluation. When automatic computation is unsatisfactory, users can generate numerical parameters, evaluate epsilon samples, and fit the resulting expansions manually.
- Package configuration: Users can select among FiniteFlow+LiteRed, FIRE+LiteRed, and Kira as IBP reducers through SetReductionOptions.Other reducers can be used after their interfaces with AMFlow are built properly.
- Example setup: The example concerns a two-loop planar integral family for NNLO QCD corrections to t¯t hadroproduction with kinematic variables s and t.The external momenta obey momentum conservation and specified on-shell conditions.
- Automatic computation: In the example, four top-sector integrals are targeted with goal=20 and epsorder=4 using LiteRed notation for the family indices.The notation is j[tt,ν1,...,ν9] ↔ I(ν1,...,ν9).
- Automatic computation: SolveIntegrals[target, goal, epsorder] reduces target integrals to master integrals and computes them with auxiliary mass flow, returning replacement rules for their values.The epsilon expansion runs from epsilon^-2L through epsilon^-2L+order, where L is the loop count.
- Automatic computation: The automatic output contains Laurent expansions in the dimensional regulator epsilon, including complex coefficients for the target integrals.An example begins with a 1/epsilon^4 term and lists coefficients through the finite part.
- Manual computation: When SolveIntegrals gives unsatisfactory results in extreme cases, GenerateNumericalConfig supplies epsilon samples, working precision, and truncation order for a manual route.Using the suggested parameters should reproduce SolveIntegrals results in principle, while users may override them after failure.
- Manual computation: BlackBoxAMFlow evaluates target integrals at the selected epsilon samples, and FitEps reconstructs their epsilon expansions from the numerical values.The leading epsilon-pole power is set explicitly, to -4 in the example.
5. Summary and outlook
AMFlow presents a Mathematica implementation of auxiliary mass flow, emphasizing automated computation and a numerical fit strategy for solving differential equations. Its high-performance solver supports high-precision calculations, while future reduction-method integration is intended to broaden automation and reduce computation time.
- AMFlow is presented as a Mathematica package implementing the auxiliary mass flow method for automated Feynman-integral computation.
- The package highlights a numerical fit strategy designed to overcome difficulties in numerically solving differential equations.
- AMFlow provides a high-performance differential-equations solver suitable for high-precision computations.
- Auxiliary mass flow requires integral reduction as the only input for calculating Feynman integrals.
- A planned implementation of a reduction method is expected to typically reduce time consumption by 2 orders of magnitude compared with other market methods.