Aspects of a Five-Dimensional $U(1)_{L_\mu - L_\tau}$ Model at Future Muon-Based Colliders

We study a five-dimensional (5D) framework based on the $U(1)_{L_\mu-L_\tau}$ gauge symmetry, where the associated gauge field $V$ propagates in the bulk, giving rise to an infinite tower of Kaluza--Klein (KK) excitations $V^{(n)}$ that couple selectively to the second- and third-generation leptons. Originally motivated by its potential to address the muon $g-2$ anomaly, this framework remains of interest as a minimal, anomaly-free, phenomenologically well-motivated extension of the Standard Model (SM) of particle physics. We focus on high-energy muon-based colliders, which could directly probe the gauge structure without relying on the kinetic mixing between the SM hypercharge gauge boson and the 5D gauge boson $V$. We explore a set of complementary processes: the elastic $\mu^+\mu^+ \to \mu^+\mu^+$ scattering via off-shell exchange of KK (gauge) excitations $V^{(n)}$; the bremsstrahlung production of $V^{(n)}$ followed by their decays into neutrinos and into $\mu^-\mu^+$ at a future $\mu$TRISTAN collider. Further, we study the $\mu^-\mu^+ \to \mu^-\mu^+$ scattering via resonant KK excitation(s) at a future muon collider. Our results show that these future muon-based colliders could offer sensitive and complementary probes into regions in the parameter space of the scenario that are beyond the reach of low-energy experiments. In particular, such experiments would be able to probe both heavier such KK gauge bosons with TeV-scale masses for relatively large gauge couplings, as well as the much lighter ones with masses in the MeV-scale for couplings as weak as $g_D \sim \mathcal{O}(10^{-5})$, thereby offering a promising $2\sigma$ exclusion reach for such KK excitations, over an extensive range of masses, at these facilities.