We study the impact of coupling neutrinos with a new light neutral gauge boson, $Z^\prime$, mass less than 500 {\rm MeV} in FASER$\nu$ experiment. Scenarios which boson is coupled to are motivated within numerous contexts designed explain various anomalies particle physics and cosmology. This interaction leads decay mode for charged mesons lepton plus neutrino $Z^{\prime}$, ($\pi^+(K^+)\to e^+ ...