$Z^\prime$ induced forward dominant processes in $\mu$TRISTAN experiment

General $U(1)$ extension of the Standard Model (SM) is a well motivated beyond the Standard Model(BSM) scenario where three generations of right handed neutrinos (RHNs) are introduced to cancel gauge and mixed gauge-gravity anomalies. After the $U(1)_X$ is broken, RHNs participate in the seesaw mechanism to generate light neutrino masses satisfying neutrino oscillation data. In addition to that, a neutral gauge boson $Z^\prime$ is evolved which interacts with the left and right handed fermions differently manifesting chiral nature of the model which could be probed in future collider experiments. As a result, if we consider $\mu^+ e^-$ and $\mu^+ \mu^+$ collisions in $\mu$TRISTAN experiment $Z^\prime$ mediated $2\to2$ scattering will appear in $t-$ and $u-$channels depending on the initial and final states being accompanied by the photon and $Z$ mediated interactions. This will result well motivated resulting forward dominant scenarios giving rise to sizable left-right asymmetry. Estimating constrains on general $U(1)$ coupling from LEP-II and LHC for different $U(1)_X$ charges, we calculate differential and integrated scattering cross section and left-right asymmetry for $\mu^+ e^- \to \mu^+ e^-$ and $\mu^+ \mu^+ \to \mu^+ \mu^+$ processes which could be probed at $\mu$TRISTAN experiment further enlightening the interaction between $Z^\prime$ and charged leptons and the $U(1)_X$ breaking scale.