Hiren H. Patel
Calculating Feynman loop integrals commonly encountered in nuclear physics, particle physics and cosmology is a time consuming task that takes weeks and sometimes months to complete by hand. Extensively tested and adopted by researchers around the world, Package‑X is the definitive Mathematica program to instantly solve one loop Feynman integrals in full generality.
- Computes dimensionally regulated one-loop integrals with up to four distinct propagators of arbitrarily high rank.
- Calculates traces of Dirac matrices in d dimensions for closed fermion loops, or carries out Dirac algebra for open fermion lines
- Generates analytic results for any kinematic configuration (e.g. at zero external momentum, at physical threshold, etc.) for real masses and external invariants.
- Provides analytic expressions for UV-divergent, IR-divergent and finite parts either separately or all together.
- Computes discontinuities across cuts of one-loop integrals.
- Constructs Taylor series expansions for one-loop integrals.
- Evaluates with either machine precision or arbitrary precision arithmetic, and is always consistent with the +iε prescription.
User interface features
- Complete documentation available online and within the Wolfram Documentation Center which includes
- over 350 basic usage examples with detailed information for every symbol defined by Package‑X,
- 4 project-scale tutorials,
- and instructions on linking Package‑X to FeynCalc, LoopTools, and the COLLIER library.
- Readable and intuitive input
- Output easily manipulatable and readily numerically evaluatable
- Command-line friendly
Reference and Source
Package‑X was developed after several long years of tedious work. Please include the following citation if Package‑X was useful for your publication.
Hiren H. Patel, Comput. Phys. Commun. 197, 276 (2015),
Esentially all algorithms and formulae used by Package‑X can be viewed in handwritten form at hhpatel.net/notes (Vol II, §1 and §5, and Vol V).
Bug report and feature request
Please report any bugs to hpatel6ucsc.edu.
Any comments and suggestions for improvements, including feature requests, are welcome.
Latest Program Files, License, and Release Notes
Package‑X 2.1 is closed source and distributed under the CC by 4.0 license agreement.
v2.1.1: (November 18, 2017) Download | Primer | Documentation
Series can be applied to
Inactive form of
- Documentation pages have been pre-indexed for the Wolfram Documentation Center in Mathematica 11.2.
- Fixed a harmless bug affecting only Mathematica 10.0 which caused
LoopIntegrate to print a spurious cell during evaluation.
Any computer running Mathematica 8.0.4 or higher.
Recommended: Mathematica 10.1.0 or higher.
ATTENTION: In the near future, Package-X will not be supported on Mathematica earlier than version 10.1.
Note A bug was introduced in Mathematica 10.4.0 (fixed in 11.0.0) causing the numerical evaluation of Package‑X special functions compiled to the Wolfram Virtual Machine (e.g.
ScalarD0, ...) to take an excessive amount of time.
Uncompress downloaded file, and copy the Package‑X folder (/X) with all its contents to the folder $UserBaseDirectory/Applications/. See the primer for details.
||Interface to the COLLIER library for rapid numerical evaluation of Passarino-Veltman functions
- CollierLink now uses version 1.2 of the COLLIER library.
- Improved library dependency issues on macOS and Linux.
- Fixed minor bugs caused by certain C compilers.
||Download | Documentation
||Compute vector derivatives (gradients) of loop integrals
||CC by 4.0
||Reduce Passarino-Veltman functions to scalar functions without limiting to 4 dimensions, and without inserting explicit formulae
||CC by 4.0
To install an expansion pack, uncompress the downloaded file, and copy the directory with all its contents to the folder $UserBaseDirectory/Applications/ (but not inside the Package‑X directory itself).