$Apart: A generalized Mathematica Apart function Original Research Article

We have generalized the Mathematica function Apart from 1 to image dimensions: the generalized function $Apart can decompose any linear dependent elements in image to irreducible ones. The elements in image can be viewed as the corresponding propagators which involve loop momenta, and the decomposition will be useful when one tries to perform loop calculations using packages such as Fire and Reduze, which have implemented the integration by parts (IBP) identities and Lorentz invariance (LI) identities. A description on how to use this package, combined with Fire, FeynArts, and FeynCalc packages, to do one-loop calculations in double quarkonium production in image colliders is given, and the full source code for a specific process (image) is also available.