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Three decades of FCNC studies in 3-3-1 model with right-handed neutrinos: from $Z^\prime$-dominance to the alignment limit

Flavor-changing neutral current (FCNC) processes play a prominent role in the search for physics beyond the Standard Model (SM) due to their sensitivity to new physics at the TeV scale. Meson-antimeson transitions and rare meson decays provide stringent constraints on new physics through precision measurements of observables such as mass differences, CP asymmetries, and branching ratios. Extensions of the SM based on the $\text{SU}(3)_C \times \text{SU}(3)_L \times \text{U}(1)_N$ gauge group offer a compelling framework for flavor physics, as FCNC processes emerge inexorably at tree level due to the non-universal transformations of the quark families. Among its various realizations, the version incorporating right-handed neutrinos (331RHN) is the most phenomenologically viable. This review synthesizes three decades of theoretical developments in FCNC phenomenology within the 331RHN model, from early $Z^\prime$-dominated studies to the recent recognition of the decisive role played by the SM-like Higgs boson and the identification of the alignment limit. We demonstrate that viable parameter space spans orders of magnitude, from $m_{Z^\prime} \sim$ a few hundred GeV to $\sim 100$ TeV, depending critically on quark mixing parametrizations and scalar alignment configurations, with significant implications for experimental searches at current and future colliders.