$Apart: A Generalized Mathematica Apart Function

We have generalized the $\text{textsc}Mathematica$ function $\mathtt{\text{Apart}}$ from 1 to $N$ dimension, the generalized function $\mathtt{\text{\$Apart}}$ can decompose any linear dependent elements in ${\mathcal{V}}_x^{\ast }$ to irreducible ones. The elements in ${\mathcal{V}}_x^{\ast }$ 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 $\text{textsc}Fire$ and $\text{textsc}Reduze$, which have implemented the integration by parts (IBP) identities and Lorentz invariance (LI) identities. A description on how to use this package, combined with $\text{textsc}Fire$, $\text{textsc}FeynArts$ and $\text{textsc}FeynCalc$ 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 $(e^{+}{e}^{-}\to J\mathrm{/}\psi +{\eta }_c)$ is also available.