Lepton number as a fourth color is an intriguing theoretical idea which is combined with a possible left-right symmetry within the famous Pati-Salam (PS) model. In the conventional PS model, a spontaneous breaking of the PS gauge group down to the SM, one can only take place at very high scales (above the PeV scale) due to the stringent bounds from $K_L\to \mu e$ and $K\to \pi \mu e$ induced by the resulting vector leptoquarks. In this paper, we show that these constraints can be avoided once additional vectorlike fermions are introduced and, thus, a breaking at the TeV scale is possible. We consider the flavor phenomenology of this model in the context of the intriguing hints for new physics in semileptonic $B$ decays. The necessary violation of lepton flavor universality is induced by mixing SM and vectorlike fermions. Concerning $R(D)$ and $R(D^{*})$, we find that sizable effects are possible while respecting the bounds from other flavor observables but predicting a large enhancement of $B_s\to {\tau }^{+}{\tau }^{-}$. Furthermore, also in $b\to s{\ell }^{+}{\ell }^{-}$ transitions, the observed deviations from the SM predictions [including $R(K)$ and $R(K^{*})$] can be explained with natural values for the free parameters of the model without any fine-tuning, predicting sizable decay rates for $b\to s\tau \mu$. Finally, the anomaly in the anomalous magnetic moment of the muon can be accounted for by a loop contribution involving the vector leptoquark and vectorlike leptons.

We consider a simple extension of the Standard Model (SM) by a complex scalar doublet and a singlet along with three sterile neutrinos. The sterile neutrinos mix with the SM neutrinos to produce three light neutrino states consistent with the oscillation data and three heavy sterile states. The lightest sterile neutrino has lifetime longer than the age of the Universe and can provide correct dark matter relic abundance. Utilizing tree-level flavor changing interactions of a light scalar with mass $\sim \mathcal{O}(100)\mathrm{MeV}$ along with sterile neutrinos, we can explain the anomalous magnetic moments of both muon and electron, KOTO anomalous events and the MiniBooNE excess simultaneously.

The most recent measurement of the fine structure constant leads to a $2.4\sigma$ deviation in the electron anomalous magnetic moment $g_e-2$, while the muon anomalous magnetic moment $g_{\mu }-2$ has a long-standing $3.7\sigma$ deviation in the opposite direction. We show that these deviations can be explained in a three-loop neutrino mass model based on an $E_6$ Grand Unified Theory. We also study the impact such a model can have on the anomalous events observed by the ANITA experiment and find an insufficient enhancement of the event rate.

The formation of primordial black hole (PBH) dark matter and the generation of scalar induced secondary gravitational waves (SIGWs) have been studied in the generic no-scale supergravity inflationary models. By adding an exponential term to the Kähler potential, the inflaton experiences a period of ultraslow-roll and the amplitude of primordial power spectrum at small scales is enhanced to $\mathcal{O}({10}^{-2})$. The enhanced power spectra of primordial curvature perturbations can have both sharp and broad peaks. A wide mass range of PBHs can be produced in our model, and the frequencies of the accompanied SIGWs are ranged form nanohertz to kilohertz. We show four benchmark points where the generated PBH masses are around $\mathcal{O}({10}^{-16}{M}_{\odot })$, $\mathcal{O}({10}^{-12}{M}_{\odot })$, $\mathcal{O}({10}^{-2}{M}_{\odot })$ and $\mathcal{O}({10}^2{M}_{\odot })$. The PBHs with masses around $\mathcal{O}({10}^{-16}{M}_{\odot })$ and $\mathcal{O}({10}^{-12}{M}_{\odot })$ can make up almost all the dark matter, and the accompanied SIGWs can be probed by the upcoming space-based gravitational wave observatory. Also, the SIGWs accompanied with the formation of stellar mass PBHs can be used to interpret the stochastic GW background in the nanohertz band, detected by the North American Nanohertz Observatory for Gravitational Waves, and can be tested by future interferometric gravitational wave observatory.