  This study investigates the potential of a multi-TeV Muon Collider (MuC) for
probing the Inert Triplet Model (ITM), which introduces a triplet scalar field
with hypercharge $Y=0$ to the Standard Model. The ITM stands out as a
compelling Beyond the Standard Model scenario, featuring a neutral triplet
$T^0$ and charged triplets $T^\pm$. Notably, $T^0$ is posited as a dark matter
(DM) candidate, being odd under a $Z_2$ symmetry. Rigorous evaluations against
theoretical, collider, and DM experimental constraints corner the triplet
scalar mass to a narrow TeV-scale region, within which three benchmark points
are identified, with $T^\pm$ masses of 1.21 TeV, 1.68 TeV, and 3.86 TeV, for
the collider study. The ITM's unique $TTVV$ four-point vertex, differing from
fermionic DM models, facilitates efficient pair production through Vector Boson
Fusion (VBF). This characteristic positions the MuC as an ideal platform for
exploring the ITM, particularly due to the enhanced VBF cross-sections at high
collision energies. To address the challenge of the soft decay products of
$T^\pm$ resulting from the narrow mass gap between $T^\pm$ and $T^0$, we
propose using Disappearing Charged Tracks (DCTs) from $T^\pm$ and Forward muons
as key signatures. We provide event counts for these signatures at MuC energies
of 6 TeV and 10 TeV, with respective luminosities of 4 ab$^{-1}$ and 10
ab$^{-1}$. Despite the challenge of beam-induced backgrounds contaminating the
signal, we demonstrate that our proposed final states enable the MuC to achieve
a $5\sigma$ discovery for the identified benchmark points, particularly
highlighting the effectiveness of the final state with one DCT and one Forward
muon.
