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Automatic Loop Calculations with FeynArts, FormCalc, and LoopTools

T. Hahn

TL;DR

This work presents an integrated workflow for automatic one-loop calculations using three Mathematica-based tools: FeynArts generates and visualizes diagrams, FormCalc algebraically simplifies amplitudes and translates them for numerical evaluation, and LoopTools provides the actual one-loop integrals. The approach emphasizes speed (via FORM), modularity, and openness, with built-in support for supersymmetric models and Majorana fermions through a flipping-rule, and by offering Fortran, C++, and Mathematica interfaces. Key contributions include a seamless three-part pipeline, a robust treatment of SUSY-specific issues, and practical guidance for code generation and integration with LoopTools. The system enables efficient, reproducible one-loop computations suitable for complex models like the MSSM and beyond. It is distributed under GPL and designed to operate on common Unix platforms with minimal user intervention beyond input parameters.

Abstract

This article describes three Mathematica packages for the automatic calculation of one-loop Feynman diagrams: the diagrams are generated with FeynArts, algebraically simplified with FormCalc, and finally evaluated numerically using the LoopTools package. The calculations are performed analytically as far as possible, with results given in a form well suited for numerical evaluation. The latter is straightforward with the utility programs provided by FormCalc (e.g. for translation into Fortran code) and the implementations of the one-loop integrals in LoopTools. The programs are also equipped for calculations in supersymmetric models.

Automatic Loop Calculations with FeynArts, FormCalc, and LoopTools

TL;DR

This work presents an integrated workflow for automatic one-loop calculations using three Mathematica-based tools: FeynArts generates and visualizes diagrams, FormCalc algebraically simplifies amplitudes and translates them for numerical evaluation, and LoopTools provides the actual one-loop integrals. The approach emphasizes speed (via FORM), modularity, and openness, with built-in support for supersymmetric models and Majorana fermions through a flipping-rule, and by offering Fortran, C++, and Mathematica interfaces. Key contributions include a seamless three-part pipeline, a robust treatment of SUSY-specific issues, and practical guidance for code generation and integration with LoopTools. The system enables efficient, reproducible one-loop computations suitable for complex models like the MSSM and beyond. It is distributed under GPL and designed to operate on common Unix platforms with minimal user intervention beyond input parameters.

Abstract

This article describes three Mathematica packages for the automatic calculation of one-loop Feynman diagrams: the diagrams are generated with FeynArts, algebraically simplified with FormCalc, and finally evaluated numerically using the LoopTools package. The calculations are performed analytically as far as possible, with results given in a form well suited for numerical evaluation. The latter is straightforward with the utility programs provided by FormCalc (e.g. for translation into Fortran code) and the implementations of the one-loop integrals in LoopTools. The programs are also equipped for calculations in supersymmetric models.

Paper Structure

This paper contains 8 sections, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. FeynArts
  3. FormCalc
  4. Algebraic simplification
  5. Translation to Fortran code
  6. LoopTools
  7. Calculations in Supersymmetric Models
  8. Requirements and Availability

Figures (3)

  • Figure 1: Steps in a one-loop calculation and the distribution of tasks among the programs FeynArts, FormCalc, and LoopTools.
  • Figure 2: Flowchart for the generation of Feynman amplitudes with FeynArts.
  • Figure 3: The interplay between Mathematica and FORM in FormCalc.