Automatized One-Loop Calculations in 4 and D dimensions

TL;DR

The paper addresses the need for automated, flexible one-loop calculations in gauge theories by implementing two packages, FormCalc (symbolic/analytic) and LoopTools (numerical), that operate in either $D$-dimensional dimensional regularization or in constrained differential renormalization (CDR), which is equivalent to dimensional reduction at one loop. It provides a detailed account of the regularization methods, their equivalence, and a two-tier workflow: symbolic preparation of amplitudes and numerical evaluation of loop integrals, with interfaces to Fortran, C++, and Mathematica. The authors demonstrate the approach with a Standard Model application ($ZZ \to ZZ$), outlining practical steps, performance, and reproducibility through demo codes, and discuss software requirements and accessibility. Overall, the work delivers a practical, cross-compatible toolchain for reliable one-loop computations, combining Mathematica, FORM, and language-specific numerical libraries to facilitate precision calculations in high-energy physics.

Abstract

Two program packages are presented for evaluating one-loop amplitudes. They can work either in dimensional regularization or in constrained differential renormalization. The latter method is found at the one-loop level to be equivalent to regularization by dimensional reduction.