Search prospect for electrophilic bosons at the $\mu$TRISTAN

Mediators coupled predominantly to the electron remain weakly constrained above the kinematic reach of the $B$~factories, a region that is difficult to access at hadron colliders and only partially covered by existing $e^{+}{e}^{-}$ data. We point out that the asymmetric ${\mu }^{+}{e}^{-}$ mode of the proposed $\mu$TRISTAN collider, operating at $\sqrt{s}=346$~GeV, is well suited to this window, and quantify its reach for two minimal realisations: an axion-like particle and a $Z^{\mathrm{\prime }}$ gauge boson, treated as effective interactions with the electron current. Both are radiated off the electron line and decay to an electron-positron pair, giving a common four-lepton signature whose Standard Model counterpart is under good control. A resonance search in the electron-positron invariant mass, performed for mediator masses between $5.5$ and $300$~GeV, yields projected $95\mathrm{\%}$~CL sensitivities to couplings of order ${10}^{-5}$ at the light end of this range, improving on existing bounds by more than four orders of magnitude for the pseudoscalar and by up to two orders of magnitude for the vector, and providing the only constraint on the heaviest masses considered.