Neutral Exotica at FASER$\nu$ and SND@LHC

The $(g-2)_\mu$ anomaly indicates that the second generation of leptons should have new interactions beyond the standard model. The high flux of $\nu_\mu$ and $\bar{\nu}_\mu$ at the forward experiments such as FASER$\nu$ and SND@LHC makes them suitable setups to search for new interactions of the second generation leptons. In this paper, we build a model in which the second generation left-handed leptons couple to a new right-handed neutrino, $N$ and a new Higgs doublet which also couples to the quarks. The scattering of high energy $\nu_\mu$ off nuclei can produce $N$. We investigate how forward experiments can test this model by looking for the $N$ production vertex followed by the displaced vertex of the $N$ decay. Discovering even a single such event can be a harbinger to look for the spectacular signals of the new Higgs doublet production at the LHC. We discuss the possibility of explaining the $(g-2)_\mu$ anomaly by adding more generations of $N$ which will lead to chain decays of $N$ and multiple leptons with distinct signals both at forward experiments and at the CMS and ATLAS detectors. Finally, we show that by adding a new light singlet scalar mixed with the neutral component of the new Higgs doublet ({\it i.e.,} 2HDM+$S$ model), the statistics of the data sample can be dramatically increased.