Probing the $Z'$ sector of the minimal $B-L$ model at future Linear Colliders in the $e^+e^-\to \mu^+\mu^-$ process

We study the capabilities of future electron-positron Linear Colliders, with centre-of-mass energy at the TeV scale, in accessing the parameter space of a $Z'$ boson within the minimal $B-L$ model. In such a model, wherein the Standard Model gauge group is augmented by a broken $U(1)_{B-L}$ symmetry -- with $B(L)$ being the baryon(lepton) number -- the emerging $Z'$ mass is expected to be in the above energy range. We carry out a detailed comparison between the discovery regions mapped over a two-dimensional configuration space ($Z'$ mass and coupling) at the Large Hadron Collider and possible future Linear Colliders for the case of di-muon production. As known in the literature for other $Z'$ models, we confirm that leptonic machines, as compared to the CERN hadronic accelerator, display an additional potential in discovering a $B-L$ $Z'$ boson as well as in allowing one to study its properties at a level of precision well beyond that of any of the existing colliders.