Apart: A Generalized Mathematica Apart Function

We have generalized the \textsc{Mathematica} function \texttt{Apart} from 1 to $N$ dimension, the generalized function \texttt{\$Apart} can decompose any linear dependent elements in $\mathcal{V}_{x}^*$ to irreducible ones. The elements in $\mathcal{V}_{x}^*$ can be viewed as the corresponding propagators which involve loop momenta, and the decomposition will be useful when one tries to perform the loop calculations using the packages such as \textsc{Fire} and \textsc{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 \textsc{Fire}, \textsc{FeynArts} and \textsc{FeynCalc} packages, to do the one-loop calculations in double quarkonium production in $e^+e^-$ colliders is given, and the full source code for a specific process $(e^+e^-\to J/\psi + \eta_c)$ is also available.