Boosting long lived particles searches at $\mu$TRISTAN

We study the prospects of the proposed $\mu$TRISTAN experiment, running in the energy asymmetric $\mu^+ e^-$ mode, in probing long lived particles (LLPs) arising from the decay of the Standard Model Higgs boson. We focus on the proposed runs with $\{E_{\mu^+}, E_{e^-}\} = \{1\,{\rm TeV},\,30\,{\rm GeV}\}$ and $\{E_{\mu^+}, E_{e^-}\} = \{3\,{\rm TeV},\,50\,{\rm GeV}\}$ and we show that, owing to the boosted nature of the produced events, a far detector placed along the beam line can collect a large fraction of the LLP flux. This allows one to set bounds on the exotic Higgs branching ratio which, for specific $\phi$ decay modes, can surpass those expected at the end of the High Luminosity LHC in the regime of large LLPs proper decay lengths. On the other hand, we find that the proposed strategy will not be able to further extend the limits that might be set by proposed LHC far detectors such as CODEX-b, ANUBIS and MATHUSLA.