Package-X: A Mathematica package for the analytic calculation of one-loop integrals
Hiren H. Patel
TL;DR
Package-X provides a comprehensive Mathematica-based framework for analytic, dimensionally regulated ($d=4-2\epsilon$) one-loop tensor integrals with up to three denominators. It combines a covariant tensor decomposition (LoopIntegrate) with an analytic refinement stage (LoopRefine) to yield compact expressions for UV/IR poles and finite parts, leveraging Denner–Dittmaier reduction and a detailed catalog of $C_0$ cases. The package also offers efficient Dirac-trace (Spur) and fermion-form-factor projection (Projector) tools, as well as an optimized treatment of the $+i\varepsilon$ prescription and logarithm simplification, with substantial speed-ups from compilation. Crosschecks against Standard Model quantities (e.g., $H\to gg$, $H\to\gamma\gamma$, $g-2$, neutrino moments) demonstrate accuracy, while the authors acknowledge limitations (three denominators, naive $\gamma_5$, no open-fermion-chain input) and outline avenues for future development.
Abstract
Package-X, a Mathematica package for the analytic computation of one-loop integrals dimensionally regulated near 4 spacetime dimensions is described. Package-X computes arbitrarily high rank tensor integrals with up to three propagators, and gives compact expressions of UV divergent, IR divergent, and finite parts for any kinematic configuration involving real-valued external invariants and internal masses. Output expressions can be readily evaluated numerically and manipulated symbolically with built-in Mathematica functions. Emphasis is on evaluation speed, on readability of results, and especially on user-friendliness. Also included is a routine to compute traces of products of Dirac matrices, and a collection of projectors to facilitate the computation of fermion form factors at one-loop. The package is intended to be used both as a research tool and as an educational tool.