LoopTools

LoopTools
Name	LoopTools
Developer	Thomas Hahn
Released	1992
Programming language	Fortran (programming language), C (programming language), C++
Operating system	Unix-like, Microsoft Windows, macOS
Genre	Computer algebra system, Mathematical software
License	GNU Lesser General Public License

LoopTools LoopTools is a numerical library for evaluating one-loop integrals that arise in perturbative calculations in Quantum Field Theory, particularly within Particle physics. It provides implementations of scalar and tensor integrals used in automated computations by frameworks such as FeynArts, FormCalc, and MadGraph. Widely used in phenomenological studies associated with experiments at facilities like the Large Hadron Collider, LoopTools interconnects with software developed in communities around CERN, DESY, and various university groups.

Introduction

LoopTools was designed to supply reliable, efficient numerical routines for one-loop integrals encountered in renormalization and radiative corrections in models including the Standard Model (particle physics), Supersymmetry, and effective field theories studied at institutions like SLAC National Accelerator Laboratory and Fermilab. It complements symbolic programs such as FORM, Mathematica, and Maple by providing numeric evaluation for expressions derived from diagrammatic generators like FeynArts and amplitude manipulators like FormCalc. The library targets users engaged in precision predictions for processes relevant to collaborations such as ATLAS, CMS, LHCb, and experiments at LEP.

Features and Functionality

LoopTools implements a comprehensive set of one-loop scalar integrals (A0, B0, C0, D0) and associated tensor coefficients used in Passarino–Veltman reduction, enabling connections to works by G. 't Hooft and M. Veltman. The package offers multi-precision support influenced by strategies from GNU Multiple Precision Arithmetic Library projects and includes routines for infrared and ultraviolet regularization consistent with dimensional regularization approaches introduced by Gerard 't Hooft and codified in literature by A. Denner and collaborators. Interoperability features allow linkage with programs developed at Max Planck Society groups and codebases originating from University of Oxford and Rutgers University.

Mathematical Foundations and Algorithms

The mathematical core relies on analytic continuations and branch-cut prescriptions rooted in contour integration techniques championed by Paul Dirac and elaborated in textbooks by Peskin and Schroeder and Itzykson and Zuber. Tensor integral decomposition follows the Passarino–Veltman algorithm, connecting to work by G. Passarino and M. Veltman, while scalar integral analytic expressions trace to classical treatments from Feynman parameterization and Schwinger representations related to contributions by Julian Schwinger. Numerical stability enhancements borrow ideas from algorithmic research at Numerical Algorithms Group and matrix condition number analyses common in studies at Courant Institute.

Software Architecture and Implementation

The implementation combines Fortran (programming language) backends with C (programming language) and C++ interface wrappers to facilitate integration into toolchains developed at CERN and university collaborations such as University of Durham groups. Build systems often use GNU Make and CMake conventions prevalent in projects from Kitware and incorporate continuous integration approaches inspired by infrastructure at Travis CI and GitHub Actions. Precision management and memory handling reflect practices from high-energy software projects produced by teams at SLAC and Brookhaven National Laboratory.

Usage and Interfaces

LoopTools exposes callable routines usable from standalone codes in Fortran (programming language), C (programming language), and bindings to scripting environments such as Python (programming language) via adapters inspired by interfaces used in NumPy-based stacks. It is commonly invoked by diagram-generation pipelines including FeynArts, amplitude calculators like FormCalc, and event generators like MadGraph and Sherpa employed in phenomenology studies feeding into collaborations such as ATLAS and CMS. Users integrate LoopTools into workflows that involve symbolic simplification in FORM or Mathematica and subsequent numerical evaluation for comparison with data from facilities like CERN and DESY.

Validation and Benchmarks

Validation of LoopTools routines has been conducted through comparisons with analytic results from literature by authors such as A. Denner and cross-checked against independent numerical libraries developed at institutions like Los Alamos National Laboratory and groups contributing to HEPTools repositories. Benchmark suites compare performance on matrix element evaluations similar to tests published by Les Houches workshops and precision studies aligned with analyses from LHC Physics Center teams. Stability tests address edge cases discussed in reports from collaborations including ILC study groups and computational reviews at KEK.

Development History and Licensing

The project emerged in the 1990s amid efforts to automate radiative correction calculations by researchers associated with Paul Zerwas-affiliated groups and has been maintained via contributions from scientists linked to MPI for Physics and numerous university groups across Europe and the United States. Licensing follows the GNU Lesser General Public License permitting linkage with proprietary front ends while preserving open-source contributions, a model also used by projects at CERN and many academic software initiatives. Ongoing development continues in community repositories influenced by collaboration practices at GitHub and code review norms from Open Source Initiative-aligned projects.

Category:Mathematical software