  This thesis covers a few successes of the SM and its limitations. The
limitations implement that the SM must be expanded to explain the observables
which can not be addressed by the SM such as mass of neutrinos, a few of
anomalies, DM, DE, etc. In order to extend the SM, we take the model-dependent
approach and a minimal extension to the SM. For the minimal extension, we make
use of vector-like family, SM-like scalar (plus a singlet flavon), and
$U(1)^\prime$ symmetry. A first BSM model in my first work is to explain both
the electron and muon anomalous magnetic moments at the same time, while
keeping the constraints $\mu \rightarrow e \gamma$ decay and neutrino trident
production. Especially, two mass sources, chirality flip and vector-like mass,
appear in our analytic prediction for the anomalies and the chirality flip mass
between $0$ and $200 \operatorname{GeV}$ is searched and then we show no any
value between them can satisfy the anomalies at the same time. A second BSM
model in my second work shares the same motivation, to explain both anomalies,
however it goes one step further by considering the strong hierarchical
structure of the SM into the second BSM model. In this BSM model, the SM
appears as an effective theory spontaneously broken from the $U(1)^\prime$
symmetry. Both anomalies are studied in the neutrino sector first and it
reveals that the neutrino contributions are too small to explain their
experimental bounds, so leading to a conclusion that the neutrinos can not
explain both anomalies simultaneously. Next we study the scalar contribution
and show that the scalar contributions can explain both anomalies
simultaneously. My third work investigate the diverse FCNCs within the same
second BSM model to constrain mass range of vector-like fermions.
