Meson Mixing Bounds on $Z^{\prime}$ Mass in the Alignment Limit: Establishing the Phenomenological Viability of the 331 Model

We perform a systematic study of flavor-changing neutral currents (FCNCs) in the 331 model with right-handed neutrinos (331RHNs), analyzing constraints on the $Z^\prime$ boson mass from $K$-, $D$-, $B_d$-, and $B_s$-meson oscillations. By explicitly incorporating scalar sector dynamics and quark rotation ambiguities ($V_L^{u,d}$), we demonstrate that $Z^\prime$ mass limits depend critically on the parametrization of Cabibbo-Kobayashi-Maskawa (CKM) matrix factors. Three scenarios are explored: (i) $V_L^u = V_\text{CKM}^\dagger$ (FCNCs restricted to $D$-mesons), (ii) $V_L^d = V_\text{CKM}$ (dominant $B_s$ constraints), and (iii) a hybrid mixing pattern. Strikingly, scenario (i) reduces the $Z^\prime$ mass bound to $M_{Z^\prime} \gtrsim 600\;\text{GeV}$-two orders of magnitude below literature values-by leveraging large experimental uncertainties in $D$-$\bar{D}$ oscillations. Conversely, scenario (ii) requires $M_{Z^\prime} \gtrsim 165\;\text{TeV}$ due to stringent $B_s$ data. We further establish the alignment limit $\cos(\phi+\varphi) = 0$ for the SM-like Higgs, showing its viability depends on $V_L^{u,d}$ configurations, with $B_s$ systems enforcing $|\cos(\phi+\varphi)| < 0.01$ in down-sector FCNC scenarios. Our analysis reveals that strategic choices of quark mixing matrices can suppress FCNC visibility, reconciling the 331 framework with flavor data without ultra-heavy $Z^\prime$ bosons. This work provides the first unified treatment of SM-like Higgs- and $Z^\prime$-mediated FCNCs in 331 models, identifying viable parameter spaces for collider phenomenology.