Revisiting for maximal flavor violating $Z^{'}_{e\mu}$ and its phenomenology constraints

Lepton flavor violation (LFV), observed conclusively in neutrino oscillations, remains a pivotal area of investigation due to its absence in the Standard Model (SM). Beyond the Standard Model (BSM) physics explores charged lepton flavor violation (CLFV), particularly through new particle candidates such as the $Z'$. This article focuses on maximal LFV interactions facilitated by the $Z'$ boson, specifically targeting its off-diagonal interactions with the first and second generations of charged and neutral leptons. In our ultraviolet (UV) model for the origin of the $Z'$, inspired by the work of [R.Foot \textit{et al.,}, Phys.Rev. D50 (1994) 4571-4580], we utilize the discrete $Z_2$ symmetry to investigate the maximal LFV mediated by the $Z'$ between the muon ($\mu$) and electron ($e$) arising from the additional scalars. This symmetry prohibits flavor-conserving interactions between $Z'$ and $\mu^+\mu^-,\, e^+e^-$. In conjunction with collider, $(g-2)_{\mu}, (g-2)_e$, inverse $\mu$ decay, Muonium-to-Antimuonium conversion and LFV decay constraints, we provide forecasts for anticipated limits derived from processes such as $\nu_\mu N \to \nu_e \mu^+ e^- N$ in neutrino trident experiments like the DUNE search at the first time. These limits highlight the prospective scope and significance of LFV investigations within these experimental frameworks. Within the mass range of 0.01 GeV to 10 GeV, the most stringent limit arises from $\it{B} (\mu \to e + X + \gamma)$ when $M_{Z'} < m_\mu$, while $\Delta a_e$ provides effective constraints as $M_{Z'}$ approaches 10 GeV. Looking ahead, the proposed Muonium-to-Antimuonium Conversion Experiment (MACE) is expected to impose the most stringent constraints on Muonium-to-Antimuonium oscillation, improving sensitivity by about one order of magnitude against $\Delta a_e$.