Constraining neutrinophilic mediators at FASER$\nu$, FLArE and FASER$\nu$2

High energy collider neutrinos have been observed for the first time by the FASER$\nu$ experiment. The detected spectrum of collider neutrinos scattering off nucleons can be used to probe neutrinophilic mediators with GeV-scale masses. We find that constraints on the pseudoscalar (axial vector) neutrinophilic mediator are close to the scalar (vector) case since they have similar cross section in the neutrino massless limit. We perform an analysis on the measured muon spectra at FASER$\nu$, and find that the bounds on the vector mediator from the current FASER$\nu$ data are comparable to the existing bounds at $m_{Z^\prime}\approx 0.2$ GeV. We also study the sensitivities to a neutrinophilic mediator at future Forward Physics Facilities including FLArE and FASER$\nu$2 by using both the missing transverse momentum and the charge identification information. We find that FLArE and FASER$\nu$2 can impose stronger bounds on both the scalar and vector neutrinophilic mediators than the existing bounds. The constraints on the scalar mediator can reach 0.08 (0.1) for $m_\phi\lesssim1$ GeV with (without) muon charge identification at FASER$\nu$2.