FeynRules implementation of the minimal Stueckelberg extension of the SM

We implement the Stueckelberg minimal extension of the standard model for the $Z^{\prime }$ boson within a FeynRules model file. With the FeynRules package, we use the fields belonging to the representation of the Lorentz and minimally extended gauge symmetries and the BRST symmetry to construct the entire Lagrangian. The package permitted us to reduce the parameter number of the model and the computation of the mass spectrum. We obtained Feynman's rules for all vertices, followed by the decay widths of the massive particles. For the validation procedure, we focused first on the induction of the $Z^{\prime }$ mass range from its decay widths at the tree level for several values of the model parameters. After exporting the model to the FeynArts and FeynCalc packages for semi-automatic computation, the second validation concerned the study of the scattering processes $e+{\nu }_{\mu }\to e+{\nu }_{\mu }$, where the total cross-section is obtained and discussed. In the last validation test, we evaluated the amplitude of one triangular loop diagram with three identical legs. We studied the process $Z^{\prime }\to Z^{\prime }+Z^{\prime }$ and $Z\to Z+Z$ as samples, and we established their amplitude cancellation conditions.