Sensitivities to secret neutrino interaction at FASER$\nu$

We study the impact of the coupling of neutrinos with a new light neutral gauge boson, $Z^\prime$, with a mass of less than 500 {\rm MeV} in FASER$\nu$ experiment. Scenarios in which a light gauge boson is coupled to neutrinos are motivated within numerous contexts which are designed to explain various anomalies in particle physics and cosmology. This interaction leads to a new decay mode for charged mesons to a light lepton plus neutrino and $Z^{\prime}$, ($\pi^+(K^+)\to e^+ \nu Z^\prime$) followed by the subsequent decay of $Z^\prime$ into the pair of neutrino and anti-neutrino, ($Z^\prime \to \nu\bar{\nu}$). FASER$\nu$, the Forward Search Experiment at the LHC, has the potential to detect collider neutrinos for the first time. In particular, the FASER$\nu$ emulsion detector will provide the opportunity to detect $\tau$-neutrinos and to measure their energies. Using this ability of FASER$\nu$ emulsion detector, we investigate the potential of FASER$\nu$ experiment and the proposed upgraded version of this experiment, FASER2$\nu$, to constrain the coupling of a neutrino with the light gauge boson.