aainstitutetext: Indian Institute of Technology Hyderabad, Kandi,
Sangareddy-502284, Telengana, Indiabbinstitutetext: Korea Institute for
Advanced Study, Seoul, 02455, Republic of Koreaccinstitutetext: Department of
Physics, Konkuk University, Seoul 05029, Republic of Korea

# Probing Inert Triplet Model at a multi-TeV muon collider via vector boson
fusion with forward muon tagging

Priyotosh Bandyopadhyay a Snehashis Parashar a Chandrima Sen c and Jeonghyeon
Song<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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 1.2 TeV, 1.85
TeV, and 2.3 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.

††preprint:  IITH-PH-0002/23, KIAS-P23063

##  1 Introduction

The landmark discovery of a Higgs boson with a mass of $125{\;{\rm GeV}}$ at
the LHC ATLAS:2012yve ; CMS:2012qbp  has reinforced the foundations of the
Standard Model (SM). The significance of this Higgs boson is particularly
profound due to its connections to several unresolved fundamental questions,
including the nature of Dark Matter (DM) Navarro:1995iw ; Bertone:2004pz ,
neutrino masses, the metastability of the SM vacuum Degrassi:2012ry , and the
naturalness problem Dimopoulos:1995mi ; Chan:1997bi ; Craig:2015pha . Given
that these questions call for a new particle physics theory Beyond the SM
(BSM), the concept of an extended Higgs sector incorporating a DM candidate is
both rational and intriguing. However, despite intensive efforts to unveil
potential extensions in the Higgs sector, no clues of new signals have emerged
as of yet: the properties of the observed Higgs boson align seamlessly with SM
expectations, and endeavors to detect additional scalar bosons have yielded no
significant findings. This situation strongly motivates the pursuit of a BSM
theory that has eluded experimental detection up to this point.

In this context, the Inert Triplet Model (ITM) emerges as a compelling
theoretical extension. The ITM introduces an additional $SU(2)_{L}$ triplet
scalar field with hypercharge $Y=0$, accommodating a neutral triplet $T^{0}$
and a pair of charged triplets $T^{\pm}$. To incorporate a DM candidate
$T^{0}$, the model adopts a discrete $Z_{2}$ symmetry, under which the triplet
scalars flip sign but the SM fields remain intact. The theoretical appeal of
the ITM is significant, offering potential solutions to a suite of core issues
such as the DM problem Araki:2010nak ; Araki:2011hm ; YaserAyazi:2014jby ;
Jangid:2020qgo , neutrino mass Kajiyama:2013zla ; Dong:2013ioa ; Okada:2015vwh
; Lu:2016dbc ; Lu:2016ucn , leptogenesis Arina:2011cu ; Lu:2016dbc ;
Lu:2016ucn , strong first-order electroweak phase transition Kazemi:2021bzj ,
the metastability of the SM Higgs vacuum Jangid:2020qgo , and the muon $g-2$
anomaly DeJesus:2020yqx . However, the detection of the new triplets in
current high-energy collider experiments is challenging due to several
factors: the absence of tree-level couplings to fermions, the requirement for
TeV-scale triplet masses to align with DM experimental data, and an extremely
compressed mass spectrum.

A crucial question arises: how can we detect the signals of this theoretically
robust yet experimentally elusive model at high-energy colliders? One
promising answer lies in the Disappearing Charged Track (DCT) phenomenon
Belyaev:2016lok ; Curtin:2017bxr ; Mahbubani:2017gjh ; Fukuda:2017jmk ; Lopez-
Honorez:2017ora ; Calibbi:2018fqf ; Curtin:2018mvb ; Han:2018wus ;
Alimena:2019zri ; Saito:2019rtg ; Bharucha:2018pfu ; Belanger:2018sti ;
Filimonova:2018qdc ; Jana:2019tdm ; Jana:2020qzn ; Chiang:2020rcv ;
Belyaev:2020wok ; Calibbi:2021fld . This phenomenon is primarily due to the
extremely small mass difference, $M_{T^{\pm}}-M_{T^{0}}\simeq 166{\;{\rm
MeV}}$ Cirelli:2005uq , which results in a proper decay length of
approximately $5.7$ cm for $T^{\pm}$. As $T^{\pm}$ is electrically charged,
its trajectory can be tracked within the inner detector layers. However, it
vanishes before reaching the calorimeters, as its decay products remain
elusive: in the dominant decay mode $T^{\pm}\to\pi^{\pm}T^{0}$, the pions are
too soft to reconstruct, and $T^{0}$ leaves no trace in any detector
component.

Building on this understanding, the HL-LHC has been explored for its potential
in probing the ITM using the DCT phenomena in the Drell-Yan (DY) production
accompanied by an additional photon Chiang:2020rcv . However, there is a
limitation: the detectable mass range is restricted to
$M_{T^{0}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}520{\;{\rm GeV}}$, even when considering
an integrated luminosity of $\mathcal{L}_{\rm tot}=3{\,{\rm ab}^{-1}}$
ATLAS:2017oal . This range is less than ideal, as it fails to cover even 10%
of the observed DM relic density Planck:2018vyg . Looking towards the future,
projections for a 100 TeV proton-proton collider with a luminosity of
$30{\,{\rm ab}^{-1}}$ indicate a potential for detecting triplet scalar masses
up to $3{\;{\rm TeV}}$, if pileup effects are efficiently mitigated
Chiang:2020rcv . However, reaching such a high luminosity remains a long-term
goal, underscoring the need for alternative research avenues.

A promising alternative is a multi-TeV muon collider (MuC) Palmer:1996gs ;
Ankenbrandt:1999cta . The MuC’s unique ability to achieve multi-TeV center-of-
mass (c.m.) energies in a compact setup, combined with the cleaner collision
environment of muon interactions, positions it as a powerful tool for BSM
searches. Unlike hadron colliders like the LHC, where the hard scattering
process depends on parton-parton collisions within protons, the MuC utilizes
collisions between two fundamental particles, thus fully harnessing the beam
energy. This feature significantly enhances the efficiency of the MuC in
exploring heavy BSM particles Choi:1999kn ; Han:2020uak ; Han:2021udl ;
Jueid:2021avn ; Han:2022edd ; Black:2022qlg . Consequently, the feasibility
and potential of the MuC have been increasingly recognized, especially since
the European strategy endorsed MuC research, culminating in the establishment
of the Muon Collider Collaboration in 2020 Schulte:2021eyr . Furthermore,
recent advancements in addressing critical challenges, such as cooling muon
beams Antonelli:2013mmk ; Antonelli:2015nla  and reducing beam-induced
backgrounds (BIB) Collamati:2021sbv ; Ally:2022rgk , have significantly
enhanced the prospects of the MuC program.

There exist extensive studies on DM signatures employing DCTs at the MuC,
focusing on the scalar/fermion $SU(2)_{L}$ multiplet, Higgsino, and Wino DM
candidates Bottaro:2021snn ; Bottaro:2022one ; Han:2020uak ; Han:2022ubw ;
Capdevilla:2021fmj . These explorations have been centered around DY pair
production, using various triggers such as mono-photon, mono-$V$
($V=Z,W^{\pm}$), $VV^{\prime}$, or mono-muon, within a pseudorapidity coverage
of $\left|\eta\right|\leq 2.5$ MuonCollider:2022ded . However, a dedicated
study of ITM signatures at the MuC remains uncharted, which we aim to address
in this paper. Our initial step involves updating the allowed parameter space
for the ITM, incorporating the latest DM experimental constraints from LUX-
ZEPLIN (LZ) LZ:2022ufs  and the H.E.S.S. Collaboration HESS:2022ygk . We will
show that the masses of triplet scalar bosons are confined to a narrow TeV-
scale range.

In our investigation of the phenomenological signatures of ITM at the MuC, we
propose two innovative approaches. First, we will focus on Vector Boson Fusion
(VBF) processes, which are distinct from DY processes in that their cross-
sections increase with c.m. energy. This characteristic makes VBF processes
exceptionally well-suited for exploiting the high collision energy of the MuC
Costantini:2020stv . Second, we utilize Forward ($\left|\eta\right|>2.5$)
muons Ruhdorfer:2023uea ; Forslund:2023reu  as innovative triggers in neutral-
current VBF processes, moving beyond the conventional pseudorapidity coverage
limit of $|\eta|<2.5$. This typical limit, primarily imposed due to tungsten
nozzles designed to shield the detector by absorbing the majority of soft BIB
particles Collamati:2021sbv ; Ally:2022rgk , is not an actual barrier for
high-energy Forward muons, as they can effectively penetrate the nozzles.
Recent initiatives within the MuC community to integrate Forward muon
detectors into the collider’s design further reinforce this approach
Accettura:2023ked ; Ruhdorfer:2023uea ; Forslund:2023reu .

Our new strategy—employing VBF pair production of triplet scalars with Forward
muon triggering—is particularly robust in probing ITM, owing to the presence
of $TTVV$-type vertices (where $V\equiv W^{\pm}/Z$ and $T=T^{0},T^{\pm}$).
These vertices, absent for fermionic DM like the Higgsino and Wino, notably
enhance the VBF cross-sections, thereby enabling a more effective analysis of
the DCT signals. Utilizing the comprehensive DCT reconstruction efficiencies
in Ref. Capdevilla:2021fmj , we will demonstrate that final states featuring a
DCT and a Forward muon at a 10 TeV MuC can effectively probe the viable
parameter space of the ITM. Our findings will highlight the MuC as an ideal
platform for investigating the ITM, marking a significant contribution to the
field.

The structure of this paper unfolds in the following manner: We begin with a
concise review of the ITM in Sec. 2, where we also discuss the constraints
from theoretical requirements and collider experiments. Following this, Sec. 3
details the identification of the parameter space which is consistent with
various DM measurements. This section also introduces three benchmark points
for further study at the MuC. In Sec. 4, our focus shifts to the VBF pair
production of the triplet scalars at the MuC. Here, we identify four distinct
final states that involve DCTs and Forward muons. Section 5 delves deeper into
the signal characteristics. Particular attention is given to the long-lived
$T^{\pm}$, along with the reconstruction efficiencies of DCTs and the
detection feasibility of Forward muons. Our results are presented in Sec. 6,
where we report the number of events for each final state at two distinct MuC
energies, 6 TeV and 10 TeV. Furthermore, in Sec. 7, we discuss the luminosity
projections required for $5\sigma$ discoveries. The paper concludes with Sec.
8, where we wrap up our study with key conclusions and insights.

##  2 Inert Triplet Model

###  2.1 Brief review of ITM

In the ITM, the SM Higgs sector is extended by introducing an $SU(2)_{L}$
triplet scalar field $\mathcal{T}$, which carries a hypercharge of $Y=0$ in
the $Q=T_{3}+Y$ convention.111The ITM variant with $Y=2$ is significantly
constrained by DM direct detection experiments due to the notably enhanced
spin-independent cross-section via $Z$-mediated scattering processes
Araki:2011hm . Consequently, we have excluded this scenario from our analysis.
The SM scalar doublet $\Phi$ (with a hypercharge of $Y=1$) and the new scalar
field $\mathcal{T}$ are described as:

$\Phi=\begin{pmatrix}\phi^{+}\\\
\phi^{0}\end{pmatrix},\quad\mathcal{T}=\frac{1}{2}\begin{pmatrix}T^{0}&\sqrt{2}T^{+}\\\
\sqrt{2}T^{-}&-T^{0}\end{pmatrix}.$ (1)

Thus, this model introduces three new scalar bosons: a neutral $CP$-even
scalar $T^{0}$ and charged scalars $T^{\pm}$. To incorporate a DM candidate,
we adopt a discrete $Z_{2}$ symmetry, under which all SM particles are
designated as even while the newly introduced triplet field is classified as
odd. This classification ensures that the new scalar bosons do not directly
couple to SM fermions, thereby earning the mode its name as the _Inert_
Triplet Model. In addition, the neutral component of $\mathcal{T}$ does not
acquire a non-zero vacuum expectation value. Within this ITM, therefore, it is
solely the Higgs doublet $\Phi$ that is responsible for electroweak symmetry
breaking, as $\Phi$ acquires a non-zero VEV given by
$\phi^{0}=\left(v+h^{0}\right)/\sqrt{2}$ (where $v\approx 246{\;{\rm GeV}}$).

In our proposed model, the Lagrangian for the scalar fields is specified as:

$\mathcal{L}=\left|D_{\mu}\Phi\right|^{2}+\text{Tr}\left[\left(D_{\mu}\mathcal{T}\right)^{\dagger}\left(D^{\mu}\mathcal{T}\right)\right]-V(\Phi,\mathcal{T}),$
(2)

where the covariant derivative of the triplet scalar field is defined by

$D_{\mu}\mathcal{T}=\partial_{\mu}\mathcal{T}-i\frac{g_{2}}{2}\left[\tau^{a}W^{a}_{\mu},\mathcal{T}\right].$
(3)

Then, the gauge interaction Lagrangian manifests as:

$\displaystyle\mathcal{L}_{\text{gauge}}$
$\displaystyle={ig_{2}}\left(\sin\theta_{W}A_{\mu}+\cos\theta_{W}Z_{\mu}\right)(T^{-}\overset{\leftrightarrow}{\partial^{\mu}}T^{+})$
(4) $\displaystyle\quad-
ig_{2}W^{-}_{\mu}(T^{0}\overset{\leftrightarrow}{\partial^{\mu}}T^{+})+ig_{2}W^{+}_{\mu}(T^{0}\overset{\leftrightarrow}{\partial^{\mu}}T^{-})$
$\displaystyle\quad-
g_{2}^{2}T^{0}\left(\sin\theta_{W}A_{\mu}+\cos\theta_{W}Z_{\mu}\right)(T^{-}W^{+\mu}+T^{+}W^{-\mu})$
$\displaystyle\quad+g_{2}^{2}T^{+}T^{-}\left(\sin\theta_{W}A_{\mu}+\cos\theta_{W}Z_{\mu}\right)^{2}+g_{2}^{2}W^{+}_{\mu}W^{-\mu}(T^{0}T^{0}+T^{+}T^{-})$
$\displaystyle\quad-\frac{g_{2}^{2}}{2}\left(W^{+}_{\mu}W^{+\mu}T^{-}T^{-}+W^{-}_{\mu}W^{-\mu}T^{+}T^{+}\right),$

where the derivative is represented by
$(f\overset{\leftrightarrow}{\partial^{\mu}}f^{\prime})=f\partial^{\mu}f^{\prime}-(\partial^{\mu}f)f^{\prime}$,
$g_{2}$ is the $SU(2)_{L}$ gauge coupling, and $\theta_{W}$ is the electroweak
mixing angle.

The scalar potential $V(\Phi,\mathcal{T})$, conforming to the $Z_{2}$
symmetry, is given by

$\displaystyle V(\Phi,\mathcal{T})\ $
$\displaystyle=\mu_{h}^{2}\,\Phi^{\dagger}\Phi+\mu_{T}^{2}\,{\rm
Tr}\left(\mathcal{T}^{\dagger}\mathcal{T}\right)+\lambda_{h}\left|\Phi^{\dagger}\Phi\right|^{2}+\lambda_{t}\left({\rm
Tr}\left|\mathcal{T}^{\dagger}\mathcal{T}\right|\right)^{2}+\lambda_{ht}\left(\Phi^{\dagger}\Phi\right){\rm
Tr}\left(\mathcal{T}^{\dagger}\mathcal{T}\right).$ (5)

Consequently, the ITM is characterized by three parameters:

$\left\\{\mu_{T},\;\lambda_{t},\;\lambda_{ht}\right\\}.$ (6)

For the DM and collider phenomenologies of the ITM, the parameters $\mu_{T}$
and $\lambda_{ht}$ are of primary focus. While $\lambda_{t}$ does not directly
impact the phenomenologies, it plays a significant role in modulating the
behavior of $\lambda_{ht}$ under renormalization group equations. The
theoretical stability of the model up to the Planck scale requires
$\lambda_{t}$ to be within the range of [0.01, 0.8] Jangid:2020qgo . In this
study, we have chosen to set $\lambda_{t}$ at a value of 0.2. This decision
aligns with both perturbativity constraints and the theoretical stability
requirements outlined in the literature.

The scalar potential in Equation 5 determines the tree-level mass spectrum:

$\displaystyle m_{h}^{2}$ $\displaystyle=2\lambda_{h}v^{2},$ (7)
$\displaystyle M_{T^{0}}^{2}$
$\displaystyle=M_{T^{\pm}}^{2}=\frac{1}{2}\lambda_{ht}v^{2}+\mu^{2}_{T}.$

At the tree level, $T^{\pm}$ and $T^{0}$ exhibit mass degeneracy. However, a
small mass deviation emerges due to loop corrections through the interactions
with SM gauge bosons. At the one-loop level, the mass difference is given by
Cirelli:2005uq :

$\displaystyle\Delta M$ $\displaystyle\equiv M_{T^{\pm}}-M_{T^{0}}$ (8)
$\displaystyle=\frac{\alpha_{2}}{4\pi}M_{T^{0}}\Bigg{[}\mathcal{F}\left(\frac{m_{W}}{M_{T^{0}}}\right)+(s_{W}^{2}-1)\mathcal{F}\left(\frac{m_{Z}}{M_{T^{0}}}\right)\Bigg{]},$

where the form factor $\mathcal{F}(x)$ is defined as

$\displaystyle\mathcal{F}(x)=-\frac{x}{4}\left[2x^{3}\ln
x+(x^{2}-4)^{\frac{3}{2}}\ln\frac{x^{2}-2-x\sqrt{x^{2}-4}}{2}\right].$ (9)

Figure 1: The mass difference between $T^{\pm}$ and $T^{0}$ as a function of
$M_{T^{0}}$. This mass difference is computed at the one-loop level, induced
by the interactions with SM gauge bosons.

In Figure 1, we present the mass splitting $\Delta M$ as a function of
$M_{T^{0}}$. It is evident that for a higher mass scale
($M_{T^{0}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}800$
GeV), $\Delta M$ stabilizes around 166 MeV. As $M_{T^{0}}$ decreases, the mass
difference experiences a slight increase but consistently remains below 200
MeV. Notably, even with the inclusion of two-loop corrections, the mass
difference undergoes only minor changes, in the order of a few MeV Ibe:2012sx
. As $T^{0}$ is the lightest stable particle with an odd $Z_{2}$ parity, it
emerges as a viable DM candidate.

Let us now examine the decay modes of the charged triplet $T^{\pm}$. Given
that the mass difference exceeds the mass of pion, $T^{\pm}$ predominantly
decays into $T^{0}\pi^{\pm}$, $T^{0}e^{\pm}\nu_{e}$, or
$T^{0}\mu^{\pm}\nu_{\mu}$. The partial decay widths and the corresponding
branching ratios are as follows Cirelli:2005uq ; Franceschini:2008pz :

$\displaystyle\Gamma(T^{\pm}\to T^{0}\pi^{\pm})$
$\displaystyle=\frac{2G_{F}^{2}\,V_{ud}^{2}\,\Delta
M^{3}\,f_{\pi}^{2}}{\pi}\sqrt{1-\frac{m_{\pi}^{2}}{\Delta M^{2}}},$
$\displaystyle($ $\displaystyle{\rm{Br}}\simeq 97.7\%),$ (10)
$\displaystyle\Gamma(T^{\pm}\to T^{0}e^{\pm}\nu_{e})$
$\displaystyle=\frac{2G_{F}^{2}\,\Delta M^{5}}{15\pi^{3}},$ $\displaystyle($
$\displaystyle{\rm{Br}}\simeq 2\%),$ $\displaystyle\Gamma(T^{\pm}\to
T^{0}\mu^{\pm}\nu_{\mu})$ $\displaystyle=0.12\,\,\Gamma(T^{\pm}\to
T^{0}e^{\pm}\nu_{e}),$ $\displaystyle($ $\displaystyle{\rm{Br}}\simeq
0.25\%),$

where $f_{\pi}\simeq 130.4{\;{\rm MeV}}$ is the decay constant of the pion
Colangelo:2000zw . The overwhelmingly dominant decay channel for $T^{\pm}$ is
into $T^{0}\pi^{\pm}$.

Another critical observation from Equation 10 is that the partial decay rate
of the dominant $T^{\pm}\to T^{0}\pi^{\pm}$ is directly proportional to
$\Delta M^{3}$. Consequently, the highly compressed mass spectrum leads to a
notably small total decay width for $T^{\pm}$, resulting in a substantial
proper lifetime of $0.19\,$ns. This corresponds to a decay length for
$T^{\pm}$ in its rest frame of approximately 5.7 cm Han:2022ubw . Such a decay
length is sufficiently long to be explored by high-energy collider detectors,
emphasizing its crucial significance in experimental observations.

###  2.2 Viable parameter space from the theoretical and collider constraints

Prior to investigating the implications from diverse DM observations, it is
imperative to identify the viable parameter points that are consistent with
theoretical requirements and collider constraints. To this end, we consider
the following aspects:222We do not consider the constraints from the
electroweak oblique parameters, $S$, $T$, and $U$ ParticleDataGroup:2022pth ,
since the extremely small $\Delta M$ and the TeV-scale triplet masses result
in negligible contributions from the triplet scalar bosons. Note that in the
ITM, $S=0$, $T\simeq(\Delta M)^{2}/(24\pi s_{W}^{2}m_{W}^{2})$, and
$U\simeq\Delta M/(3\pi M_{T^{\pm}})$ Chakrabarty:2021kmr .

Theoretical requirements

  * –

All quartic couplings, $\lambda_{h}$, $\lambda_{t}$, and $\lambda_{ht}$, must
be perturbative, i.e., $|\lambda_{h,ht,t}|\leq 4\pi$.

  * –

The unitarity condition imposes an additional requirement:
$|\lambda_{t}|\leq\pi$.

  * –

The scalar potential should be bounded-from-below, necessitating

$\displaystyle\lambda_{h}>0,\quad\quad\lambda_{t}>0,\quad\quad
2\sqrt{\lambda_{h}\lambda_{t}}>|\lambda_{ht}|.$ (11)

Given that $\lambda_{h}=0.129$, all three conditions collectively lead to

$\displaystyle-1.27<\lambda_{ht}<4\pi.$ (12)

Higgs precision data at the LHC

The charged triplets can significantly affect di-photon decay rate of the
Higgs boson via loop corrections, resulting in the partial decay width given
by

$\Gamma(h\to\gamma\gamma)=\frac{G_{F}\alpha^{2}m_{h}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}Q_{f}^{2}\mathcal{R}_{1/2}(\xi_{f})+\mathcal{R}_{1}(\xi_{W})+\frac{\lambda_{ht}v^{2}}{2M_{T^{\pm}}^{2}}\mathcal{R}_{0}(\xi_{T^{\pm}})\right|^{2},$
(13)

where $N_{c}$ and $Q_{f}$ are the color factor and the charge of the fermions,
respectively, and $\xi_{a}={4M_{a}^{2}}/{m_{h}^{2}}$. The form factors for
spin-0, spin-1/2, and spin-1 are

$\displaystyle\mathcal{R}_{0}(\xi)$
$\displaystyle=-\xi\left[1-\xi\,\mathcal{P}(\xi)\right],$ (14)
$\displaystyle\mathcal{R}_{1/2}(\xi)$
$\displaystyle=2\xi\left[1+(1-\xi)\mathcal{P}(\xi)\right],$
$\displaystyle\mathcal{R}_{1}(\xi)$
$\displaystyle=-\left[2+3\xi+3\xi(2-\xi)\mathcal{P}(\xi)\right],$

where

$\mathcal{P}(\xi)=\begin{cases}\arcsin[2](\xi^{-\frac{1}{2}})&\text{if
}\xi\leq 1,\\\\[5.0pt]
-\frac{1}{4}\left[\ln(\frac{1+\sqrt{1-\xi}}{1-\sqrt{1-\xi}}-i\pi)\right]^{2}&\text{if
}\xi>1.\end{cases}$ (15)

The di-photon signal modifier
$\mu_{\gamma\gamma}=\Gamma(h\to\gamma\gamma)/\Gamma(h\to\gamma\gamma)|_{\text{SM}}$
is constrained to be $\mu_{\gamma\gamma}=1.04_{-0.09}^{+0.10}$ at the
$1\sigma$ limit by the recent ATLAS measurement ATLAS:2022tnm , which limits
$\lambda_{ht}$ through Equation 13.

Figure 2: Excluded regions in the $(M_{T^{\pm}},\lambda_{ht})$ parameter
space, indicated by the perturbativity constraint (blue), vacuum stability and
unitarity requirements (pink), and the $1\sigma$ constraints from the Higgs
boson decay into two photons (green).

In Figure 2, we illustrate the excluded regions within the parameter space
$(M_{T^{\pm}},\lambda_{ht})$, based on the constraints discussed earlier. The
combined criteria for vacuum stability and unitarity exclude the region shown
in pink, effectively ruling out a substantial portion of the negative
$\lambda_{ht}$ space. The perturbativity condition disqualifies the region
where $\lambda_{ht}>4\pi$, shown in blue. Additionally, constraints from the
decay rate of $h\rightarrow\gamma\gamma$ eliminate a large area of the
parameter space, especially for lighter $M_{T^{0}}$ values. When
$M_{T^{0}}\lesssim 200$ GeV, the magnitude of $\lambda_{ht}$ is constrained to
be less than one. Conversely, for heavier $M_{T^{0}}$ values, such as those
above 1 TeV, the parameter space with sizable $\lambda_{ht}$ remains viable.
In summary, the white region in Figure 2 represents the parameter space that
meets all theoretical and collider-imposed requirements.

##  3 Dark Matter constraints

In this section, we reassess the constraints on the ITM parameter space in
light of the latest DM observations, focusing on the viable parameter space
delineated in the previous section. This reassessment is conducted via an
extensive random parameter scan within the following ranges:

$\displaystyle\mu_{T}\in[80.0,3000.0]{\;{\rm
GeV}},\quad\lambda_{ht}\in[0.1,1.0],\quad\lambda_{t}=0.2.$ (16)

We reiterate that $\lambda_{t}$ does not exert a significant influence on the
DM sector. The upper bound on $\lambda_{ht}$ is set with consideration of
stringent constraints from DM direct detection experiments, since the spin-
independent cross-section for DM-nucleon scattering is proportional to
$\lambda_{ht}^{2}$ Jangid:2020qgo .

Our analysis incorporates constraints from three key domains: relic density,
direct detection, and indirect detection. For the DM calculations, we
initially generated the CalcHEP model files Belyaev:2012qa  by incorporating
the new Lagrangian into SARAH Staub:2013tta . Subsequently, we utilized
micrOMEGAs version 5.3.41 Belanger:2001fz . In the subsequent subsections, we
will provide detailed accounts of these constraints.

###  3.1 DM relic abundance

The triplet DM particles, initially in thermal equilibrium with the SM
particles in the early universe, froze out when their annihilation rate fell
below the Hubble expansion rate. The latest result for the DM relic density,
as reported by the Planck Collaboration based on data from the Planck 2018
release Planck:2018vyg , is

$\Omega_{\rm DM}^{\rm Planck}h^{2}=0.1198\pm 0.0012,$ (17)

where $\Omega_{\rm DM}$ denotes the ratio of the DM relic density to the
critical density $\rho_{c}=3H^{2}/(8\pi G_{N})$. Here, $H$ represents the
Hubble constant, expressed as $H=100h$ km/s/Mpc. Roughly, $\Omega_{\rm
DM}h^{2}$ is inversely proportional to $\langle\sigma_{\rm ann}v\rangle$, the
thermal average of the DM annihilation cross-section multiplied by their
relative velocity.

We now turn our attention to DM annihilation within the ITM framework. The DM
candidate $T^{0}$ has four primary annihilation channels: $T^{0}T^{0}\to
W^{+}W^{-},\,ZZ,\,hh,\,f\bar{f}$, where $f$ represents the charged SM
fermions. The most dominant annihilation mode is $T^{0}T^{0}\to W^{+}W^{-}$,
due to $T^{0}$ being charged under $SU(2)_{L}$. This channel proceeds through
three mechanisms: the four-point vertex $T^{0}$-$T^{0}$-$W^{+}$-$W^{-}$, the
$t$-channel exchange of $T^{\pm}$, and the $s$-channel exchange of the Higgs
boson. The other annihilation channels into $ZZ,\,hh,\,f\bar{f}$ contribute
significantly less to the relic density, as they predominantly occur through
$s$-channel Higgs mediation.

Figure 3: Relic abundance plotted against the Dark Matter (DM) mass
$M_{T^{0}}$, color-coded by $\lambda_{ht}$. The horizontal dark green band
represents the observed DM relic abundance by the Planck Collaboration
Planck:2018vyg , specifically $\Omega_{\rm DM}h^{2}=0.1198\pm 0.0012$.

In the ITM, the additional contributions from coannihilation processes to the
DM relic density are sizable due to the small mass difference of $\Delta
M\simeq 166{\;{\rm MeV}}$ Griest:1990kh ; Mizuta:1992qp . Coannihilation
occurs through the scatterings of $T^{0}T^{\pm}$ and $T^{+}T^{-}$. The former
includes $T^{0}T^{\pm}\to
W^{\pm}Z,\,W^{\pm}\gamma,\,W^{\pm}h,\,f\bar{f}^{\prime}$, while the latter
comprises $T^{+}T^{-}\to W^{+}W^{-}$, $ZZ$, $Z\gamma$, $\gamma\gamma$, $hZ$,
$hh$, $h\gamma$, $f\bar{f}$, and $\nu\nu$. The most dominant coannihilation
process is $T^{0}T^{\pm}\to ZW^{\pm}$, which occurs via the $t$-channel
exchange of $T^{\pm}$, the $s$-channel exchange of $W^{\pm}$, and a contact
term involving $T^{0}$-$T^{\pm}$-$Z$-$W^{\pm}$.

In Figure 3, we display the relic abundance of DM in the ITM with respect to
the mass $M_{T^{0}}$ while varying the model parameters as in Equation 16. The
color code denotes $\lambda_{ht}$. The dark green horizontal band represents
the DM relic abundance determined by Planck measurements, as shown in Equation
17. The results clearly demonstrate that the observed DM relic density can be
accounted for within the ITM framework.

A closer examination reveals notable correlations between the model parameters
in accounting for the relic density. For a fixed $\lambda_{ht}$, a higher
$M_{T^{0}}$ results in a larger relic density. This is attributed to smaller
annihilation cross-sections for heavier triplet scalars, leading to a larger
$\Omega_{\rm DM}$. For a fixed $M_{T^{0}}$, the relic density decreases with
increasing $\lambda_{ht}$. This is because a higher $\lambda_{ht}$ increases
the Higgs boson couplings to $T^{0}T^{0}$ and $T^{+}T^{-}$, leading to larger
annihilation cross-sections and thus a smaller relic density.

Let us examine the allowed parameter space by the relic density measurement of
the Planck experiment. The possibility that the DM in the universe comprises
multiple components allows scenarios of under-abundance. Consequently, a
parameter point can be considered viable if it aligns with under-abundance
scenarios. In such cases, parameter points with $M_{T^{0}}\lesssim 2.5{\;{\rm
TeV}}$ remain viable. However, a BSM model that accounts for only a negligible
fraction of the observed relic density might be considered less favorable from
a theoretical perspective.

An ideal model should encompass a substantial part of the relic density. By
setting the threshold at $\Omega_{\rm DM}>0.5\,\Omega_{\rm DM}^{\rm Planck}$
($\Omega_{DM}>0.1\,\Omega_{\rm DM}^{\rm Planck}$), we find that $M_{T^{0}}$
should be greater than approximately 1.2 TeV (550 GeV). This establishes a
significant lower bound on $M_{T^{0}}$. Distinctly, the parameter points
yielding $\Omega_{\rm DM}h^{2}=0.1198\pm 0.0012$ are regarded as satisfying
the Planck relic density measurement. However, an over-abundance of DM is
untenable, leading us to exclude high DM masses of $M_{T^{0}}\gtrsim
2.5{\;{\rm TeV}}$. This upper limit is pivotal to ensure the testability of
the model at a multi-TeV MuC.

###  3.2 DM direct detection bounds

Several DM direct detection experiments are currently in operation, aiming to
detect signals resulting from the elastic scattering of DM with atomic nuclei.
Most of these experiments are specifically designed to investigate the spin-
independent interactions between DM particles and the target nuclei with a
high atomic number and mass.

In the ITM, the spin-independent scattering occurs through $t$-channel Higgs
boson exchange. The cross-section $\sigma_{\rm SI}$ at the leading order is
described by Chao:2018xwz :

$\sigma_{\rm SI}\simeq\frac{\lambda_{ht}^{2}f_{N}^{2}}{4\pi
m_{h}^{4}}\frac{M_{N}^{4}}{(M_{N}+M_{T^{0}})^{2}},$ (18)

where $f_{N}\approx 0.287\,(0.084)$ for $N=p\,(n)$ represents the nucleon form
factor Belanger:2013oya , $m_{h}$ is the Higgs boson mass, and $M_{N}$ is the
mass of the nucleon.

When comparing the model’s prediction for $\sigma_{\rm SI}$ with the observed
upper bounds on $\sigma_{\rm SI}$, it is essential to consider a scaling
factor, ${\Omega_{\rm DM}}/{\Omega_{\rm DM}^{\rm Planck}}$. If the model’s
predicted relic density is lower than the observed value $\Omega_{\rm DM}^{\rm
Planck}$, there is a corresponding reduction in the actual DM-nucleon
scattering cross-section. Thus, we should account for this reduction factor.
Consequently, we introduce a scaled spin-independent cross-section,
$\sigma_{\rm SI}^{\rm scaled}$, defined as:

$\displaystyle\sigma_{\rm SI}^{\rm scaled}\equiv\frac{\Omega_{\rm
DM}}{\Omega_{\rm DM}^{\rm Planck}}\times\sigma_{\rm SI}.$ (19)

Figure 4: Scaled spin-independent DM-nucleon scattering cross-section of
$T^{0}$ as a function of $M_{T^{0}}$, over the parameter points avoiding the
over-closure. We include upper bounds on $\sigma_{\rm SI}$ from PandaX-4T
PandaX-4T:2021bab  (purple dotted line) and LUX-ZEPLIN LZ:2022ufs  (black
dashed line) experiments. The green data points represent parameter
combinations that comply with the observed relic density $\Omega_{\rm DM}^{\rm
Planck}h^{2}=0.1198\pm 0.00212$.

In Figure 4, we depict $\sigma_{\rm SI}^{\rm scaled}$ as a function of
$M_{T^{0}}$, based on our random parameter scan described in Equation 16. The
color code corresponds to $\lambda_{ht}$, a key factor in the spin-independent
cross-section. Within the plot, we specifically showcase data points that
satisfy $\Omega_{\rm DM}\leq\Omega_{\rm DM}^{\rm Planck}$: the green data
points correspond to the correct relic density. Notably, the observed bounds
on $\sigma_{\rm SI}^{\rm scaled}$ from the PandaX-4T in 2021 PandaX-4T:2021bab
and LZ in 2022 LZ:2022ufs  experiments are represented by the purple dotted
and black dashed lines, respectively.333We have omitted the most recent
results from the XENONnT experiment XENON:2023sxq  as they constrain DM mass
up to 500 GeV. The most stringent constraint emerges from the 2022 LZ
experiment, which rules out values of $\lambda_{ht}$ exceeding 1.0.
Importantly, the combined constraints derived from relic density and direct
detection measurements introduce a significant limitation on $\lambda_{ht}$
for lighter $M_{T^{0}}$: for $M_{T^{0}}=550{\;{\rm GeV}}$ (the minimum to
accommodate $\Omega_{\rm DM}>0.1\,\Omega_{\rm DM}^{\rm Planck}$),
$\lambda_{ht}$ should be less than about 0.4. The upper bound on $M_{T^{0}}$
is slightly reduced into $\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}2.35{\;{\rm TeV}}$.

###  3.3 DM indirect detection bounds

In the previous subsections, we have shown that the combination of DM relic
density and direct detection measurements pinpoints the preferred mass range
for $M_{T^{0}}$ within $[0.55,2.35]{\;{\rm TeV}}$. The relatively high mass of
the real triplet DM particle opens up the possibility of DM annihilation into
energetic SM particles. This is especially relevant in high DM density
regions, such as dwarf spheroidal galaxies and galactic centers. These
annihilations can generate gamma rays as a byproduct through various
mechanisms, including hadronization, electromagnetic radiation via
bremsstrahlung, and electroweak radiation.

Dedicated experiments such as the Fermi-LAT MAGIC:2016xys  and H.E.S.S.
HESS:2022ygk  have jointly contributed to the detection of gamma rays.
However, up to the present time, none of these experiments have observed
significant excesses beyond the expected backgrounds. The absence of gamma ray
excesses imposes stringent constraints on the parameter space of the ITM,
which is the primary focus of this subsection. For our analysis, we
exclusively rely on the results from H.E.S.S. in 2022, as they provide the
most robust constraints for the mass range under investigation. It is
noteworthy that the Fermi-LAT results are effective in probing lower mass
ranges.

The gamma-ray flux from the annihilation of DM particles into the SM particles
is expressed as:

$\frac{d\phi_{\gamma}}{dE}=\frac{\langle\sigma_{\rm ann}v\rangle}{8\pi
M_{T^{0}}^{2}}\sum_{\rm ch}\,{\rm{Br}}_{\rm ch}\,\frac{dN_{\gamma}^{\rm
ch}}{dE}J(\Delta\Omega),$ (20)

where $J(\Delta\Omega)$ is

$\displaystyle J(\Delta\Omega)=\int_{\Delta\Omega}\int_{\rm LOS}\rho_{\rm
DM}^{2}[l(r,\theta)]d\Omega\,dl\,.$ (21)

Here, ${dN_{\gamma}^{\rm ch}}/{dE}$ is the energy spectrum of gamma rays
produced per DM annihilation in a given channel “ch”, ${\rm{Br}}_{\rm ch}$ is
the branching fraction of the channel, and $\rho_{\rm DM}[l(r,\theta)]$ is the
DM density profile along the line of sight (LOS) over the region of interest
with the solid angle $\Delta\Omega$. The gamma flux observations impose
constraints on $\langle\sigma_{\rm ann}v\rangle$ through Equation 20, thus
influencing the model parameters. These constraints also depend on the DM halo
profiles, for which three widely considered profile functions are used in this
work, namely, Einasto Springel:2008by , Navarro-Frenk-White (NFW)
Navarro:1996gj , and Einasto 2 Cirelli:2010xx . We utilize micrOMEGAs 5.3.41
to calculate the contribution of different DM annihilation channels to the
$\langle\sigma_{\rm ann}v\rangle$.

Figure 5: Thermally averaged DM annihilation cross-section of the dominant
channel into ${W^{+}W^{-}}$, $\langle\sigma_{\rm ann}v\rangle_{WW}$, plotted
against $M_{T^{0}}$ for parameter points that fulfill both the DM relic
density and direct detection measurements. The overlay includes bounds from
the 2022 H.E.S.S. indirect detection experiment, taking into account three
different DM halo profiles. Green points represent regions where the relic
density is satisfied, while pink points indicate regions of under-abundance.

Now let us investigate the allowed parameter space by the H.E.S.S. results.
The H.E.S.S. experiment primarily targets the photon flux resulting from
$b\bar{b}$, $t\bar{t}$, ${W^{+}W^{-}}$, $\tau^{+}\tau^{-}$, and
$\mu^{+}\mu^{-}$ pairs. Given that the annihilation channel into $W^{+}W^{-}$
is dominant in the ITM, contributing more than approximately 70%, we consider
only the ${W^{+}W^{-}}$ channel. For simplicity, we denote its contribution to
the thermally averaged annihilation cross-section as $\langle\sigma_{\rm
ann}v\rangle_{WW}$ in what follows.

In Figure 4, we present $\langle\sigma_{\rm ann}v\rangle_{WW}$ about
$M_{T^{0}}$, over the parameter points permitted by the 2022 LZ experiment and
the relic density measurement of the Planck experiment. The parameter points
that yield the correct relic density are highlighted in green. We superimpose
the 2022 H.E.S.S. experiment constraints for the ${W^{+}W^{-}}$ channel,
considering the three previously discussed halo profiles. Remarkably, all
parameter points that account for the correct relic density lie below the
bounds set by the three DM halo profiles.

However, the three halo profiles yield different mass restrictions on the
triplet DM particles. The boundary defined by the Einasto-2 profile (depicted
as a brown dotted line) proves to be the most tolerant, allowing DM masses
above approximately $600$ GeV, except for the narrow mass range around
$200{\;{\rm GeV}}$. Conversely, the most restrictive Einasto profile
(represented by a black dashed line) imposes the lower limits on masses of
approximately $1.2{\;{\rm TeV}}$. The constraints derived from the NFW profile
(represented by a blue solid line) lie in between these extremes. Therefore,
we opt for the region sanctioned by the NFW profile as the reference, which
confines the mass of the triplet DM to $M_{T^{0}}\in[850-2350]{\;{\rm GeV}}$.

Figure 6: Coupling $\lambda_{ht}$ versus the DM mass $M_{T^{0}}$ over the
allowed region obtained from the DM relic density and direct/indirect
detection. The color coding represents the ratio of the predicted DM relic
density in the ITM to the observed relic density, $\Omega_{\rm DM}/\Omega_{\rm
DM}^{\rm Planck}$. Three blue stars correspond to the selected benchmark
points for the collider study.

In conclusion of the DM study, we delineate in Figure 6 the final parameter
range of $(M_{T^{0}},\lambda_{ht})$ permitted by the combined constraints from
DM relic density and direct/indirect detection. The color code represents
$\Omega_{\rm DM}/\Omega_{\rm DM}^{\rm Planck}$. Based on these findings, we
propose the following three benchmark points (marked by blue stars) for the
MuC phenomenology study in the subsequent sections:

BP1: $\displaystyle M_{T^{0}}=1200.0{\;{\rm GeV}},\quad\lambda_{ht}=0.318;$
(22) BP2: $\displaystyle M_{T^{0}}=1852.5{\;{\rm
GeV}},\quad\lambda_{ht}=0.183;$ BP3: $\displaystyle M_{T^{0}}=2301.5{\;{\rm
GeV}},\quad\lambda_{ht}=0.837.$

Note that BP2 and BP3 align with the observed DM relic density, whereas BP1
results in an under-abundance, with $\Omega_{\rm DM}/\Omega_{\rm DM}^{\rm
Planck}\approx 0.5$.

##  4 Golden channels to probe the triplet scalars at a muon collider

In the previous section, we successfully constrained the masses of the ITM
triplet scalars to three benchmark points with
$M_{T^{0}}=1.2,\,1.85,\,2.3{\;{\rm TeV}}$. This outcome was achieved by
incorporating the theoretical requirements, collider experiment constraints,
and various DM experiment results. The exploration of such a high-mass domain
provides strong motivation for the utilization of a multi-TeV MuC
AlAli:2021let ; Accettura:2023ked ; Aime:2022flm .

Regarding the choice of the c.m. energy for the MuC, several options have been
under consideration in the literature. The current focal point of the
international MuC collaboration centers around a 3 TeV MuC, while ongoing
discussions and activities also encompass higher energies up to 30 TeV
Han:2021udl ; AlAli:2021let . For our analysis, which focuses on the pair
production of the ITM triplet scalars, we will assess the feasibility of both
a 6 TeV and 10 TeV MuC setup, each designed with projected integrated
luminosities of $4{\,{\rm ab}^{-1}}$ and $10{\,{\rm ab}^{-1}}$, respectively
AlAli:2021let .

###  4.1 Production modes of triplet scalars at a muon collider

Figure 7: Feynman diagrams illustrating the Drell-Yan production of
$T^{+}T^{-}$ through $\mu^{+}\mu^{-}$ collisions.

To identify promising channels for the ITM at a MuC, we explore various
avenues for the pair production of triplet scalars. We begin with the DY pair
production of charged triplet scalars, ${\mu^{+}\mu^{-}}\to T^{+}T^{-}$. The
corresponding Feynman diagrams are shown in Figure 7: those mediated by gauge
bosons of $\gamma$ and $Z$ in Figure 7(a) and the one mediated by the Higgs
boson in Figure 7(b).444Notably, it is worth mentioning that due to the
relatively small values of $\lambda_{ht}$ and the muon Yukawa coupling to the
SM Higgs boson, the contribution from Higgs mediation proves negligible.
However, these DY pair production modes do not exhibit high efficiency,
primarily due to the fact that the cross-section of an $s$-channel diagram
decreases as $\sqrt{s}$ increases. This reduction prevents us from fully
harnessing the high collision energy of the MuC, which was the initial
motivation for probing the heavy triplet scalars.

More efficient processes are available at the MuC, the VBF processes. A
standout advantage of VBF is its dramatically higher cross-section at high
c.m. energy, compared to the DY cross-section Costantini:2020stv . The VBF
cross-section exhibits growth according to $\log^{2}({s}/{m_{V}^{2}})$
Han:2021udl , rendering VBF processes exceptionally well-suited for a high-
energy MuC. Moreover, there are multiple VBF production channels for two
triplet scalars, as follows:

$\displaystyle\mu^{+}\mu^{-}\;$ $\displaystyle\to\;T^{+}T^{-}\nu\bar{\nu},$
(23) $\displaystyle\mu^{+}\mu^{-}\;$
$\displaystyle\to\;T^{0}\;T^{0}\,\nu\bar{\nu},$
$\displaystyle\mu^{+}\mu^{-}\;$
$\displaystyle\to\;T^{\pm}T^{0}\,\mu^{\mp}\nu,$
$\displaystyle\mu^{+}\mu^{-}\;$
$\displaystyle\to\;T^{+}T^{-}\,\mu^{+}\mu^{-}.$

These diverse production channels significantly enhance the potential for
probing the model.

Figure 8: Representative Feynman diagrams illustrating the dominant
contributions to the pair production of $T^{\pm}$ and $T^{0}$ through VBF.

The VBF processes in Equation 23 occur through several Feynman diagrams: the
Higgs boson-mediated $s$-channel diagrams, the triplet-mediated $t$-channel
diagrams, and the contact diagrams generated from the quartic couplings
associated with the $V$-$V^{(\prime)}$-$T$-$T^{(\prime)}$ vertex (where
$V^{(\prime)}=\gamma,Z,W^{\pm}$ and $T^{(\prime)}=T^{0},T^{\pm}$). Notably,
the major contributions are from the quartic vertices, of which the Feynman
diagrams are presented in Figure 8. It is worth emphasizing that the $TTVV$
vertices are distinctive features of the ITM, setting it apart from other BSM
models with fermion DM, such as Wino and Higgsino Han:2020uak ;
Capdevilla:2021fmj . The presence of these quartic vertices allows the triplet
scalars to exhibit a high VBF production rate without encountering suppression
from the triplet-mediated propagators. Although similar vertices are also
found in Two-Higgs Doublet Models Han:2021udl , the coupling strength of
triplets significantly surpasses that of doublet scalars, resulting in
substantially larger cross-sections at high collision energies.

Figure 9:  Various cross-sections of triplet scalars as a function of the MuC
c.m. energy $\sqrt{s}$ in the left panel (a), and as a function of $M_{T^{0}}$
in the right panel (b). For the left panel (a), we consider BP2 with
$M_{T^{0}}=1.85{\;{\rm TeV}}$. In the right panel (b), the c.m. energy is
fixed at $\sqrt{s}=10{\;{\rm TeV}}$. The colored region in the right panel
highlights the range of $M_{T^{0}}$ allowed by DM relic density, direct
detection, and indirect detection experiments.

To determine the most efficient production channel for triplet scalars, we
calculate the parton-level cross-sections of the DY and four VBF processes.
These calculations are performed using MadGraph5_aMCNLO with the .ufo files
generated by SARAH. Figure 9 displays these parton-level production cross-
sections. To observe the dependence of the cross-sections on the c.m. energy,
we show the cross-sections as functions of $\sqrt{s}$ for BP2 with
$M_{T^{0}}=1.85{\;{\rm TeV}}$ in the left panel (a). We observe that the
cross-section for the DY process, marked by a red dotted line, peaks near
$\sqrt{s}=5{\;{\rm TeV}}$ before rapidly decreasing. In contrast, the VBF
process cross-sections demonstrate a consistent increase with rising
$\sqrt{s}$, showcasing the $\log^{2}({s}/{M_{V}^{2}})$ pattern.

Among the four VBF channels detailed in Equation 23, the
$T^{+}T^{-}\nu\bar{\nu}$ final state yields the highest cross-section,
exceeding the DY cross-section at approximately $\sqrt{s}\approx 4.3{\;{\rm
TeV}}$. The process $\mu^{+}\mu^{-}\to T^{0}T^{0}\nu\bar{\nu}$, involving two
charged currents from the initial muon beams, has the second highest cross-
section. The third significant process, $\mu^{+}\mu^{-}\to
T^{\pm}T^{0}\mu^{\mp}\nu$, is mediated by a charged current from one muon beam
and a neutral current from the other. The VBF channel $\mu^{+}\mu^{-}\to
T^{+}T^{-}\mu^{+}\mu^{-}$, which involves two neutral currents, has the lowest
cross-section, attributed to $\cos^{2}\theta_{W}$ vertex suppressions.

In Figure 9(b), we present the cross-sections as a function of the DM mass
$M_{T^{0}}$ at $\sqrt{s}=10{\;{\rm TeV}}$. The pale green band emphasizes the
region in alignment with constraints from DM experimental results. As the mass
increases, there is a noticeable decrease in the cross-sections for both the
DY and VBF processes, attributed to phase space suppression effects. Despite
this, the hierarchy in production cross-sections, as observed in Figure 9(a),
remains intact. Consequently, the VBF channels emerge as robust options for
producing heavy triplets, especially those channels involving one or two
charged currents. Notably, even the $\mu^{+}\mu^{-}\to
T^{+}T^{-}\mu^{+}\mu^{-}$ process, which has the smallest cross-section among
the four VBF processes, still outperforms the DY process at
$\sqrt{s}=10{\;{\rm TeV}}$

Cases | Cross-section [fb] for $\mu^{+}\mu^{-}\to$ final states
---|---
|  $M_{T^{0}}$ [GeV] |  $\sqrt{s}$ [TeV] |  $T^{+}T^{-}$ | $T^{0}T^{\pm}\mu^{\mp}\bar{\nu}_{\mu}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | $T^{0}T^{0}\nu_{\mu}\bar{\nu}_{\mu}$ | $T^{+}T^{-}\nu_{\mu}\bar{\nu}_{\mu}$
BP1 | 1200.0 | 6 | 0.32 | 31.8 | 3.2 | 39.5 | 79.4
10 | 0.13 | 244.8 | 24.6 | 304.4 | 608.3
BP2 | 1852.5 | 6 | 0.20 | 4.3 | 0.4 | 5.5 | 10.9
10 | 0.12 | 110.1 | 11.1 | 137.0 | 273.0
BP3 | 2301.5 | 6 | 0.11 | 0.5 | 0.05 | 0.6 | 1.2
10 | 0.10 | 56.2 | 5.7 | 70.0 | 139.5
Table 1: Parton-level cross-sections of pair and associate production of
$T^{\pm}$ and $T^{0}$ at a MuC with two different $\sqrt{s}$ values.

In Table 1, we provide a detailed overview of the parton-level cross-sections
for the three benchmark points at a high-energy MuC, specifically focusing on
two c.m. energies: $\sqrt{s}=6{\;{\rm TeV}}$ and $10{\;{\rm TeV}}$. The
hierarchy in the VBF production cross-sections remains consistent, resulting
in the following order: $T^{+}T^{-}\nu\bar{\nu}$, $T^{0}T^{0}\nu\bar{\nu}$,
$T^{\pm}T^{0}\mu^{\mp}\nu$, and $T^{+}T^{-}\mu^{+}\mu^{-}$. The comparison
between the DY and VBF production cross-sections varies depending on the
benchmark points and the specific $\sqrt{s}$ values. At $\sqrt{s}=10{\;{\rm
TeV}}$, the DY cross-sections are significantly lower than the VBF cross-
sections across all benchmarks. However, at $\sqrt{s}=6{\;{\rm TeV}}$, BP3 is
an exception, where $\sigma_{\rm DY}$ exceeds VBF cross-sections for
$\mu^{+}\mu^{-}\to T^{+}T^{-}\mu^{+}\mu^{-}$. Another notable observation is
that the VBF cross-sections increase more significantly with rising $\sqrt{s}$
for heavier $M_{T^{0}}$ masses. The enhancement factors are approximately 7.7
for BP1, 25 for BP2, and a remarkable 110 for BP3, highlighting the benefits
of operating at higher c.m. energies for larger triplet masses.

###  4.2 Signatures of the ITM at a muon collider

Let us examine the signatures of $T^{0}$ and $T^{\pm}$ at the MuC. The DM
candidate $T^{0}$, being stable and neutral, does not interact with the
detector, creating a missing energy signature. Detecting the charged triplet
scalars, which decay within the detector, also presents a challenge due to the
small mass splitting of $M_{T^{\pm}}-M_{T^{0}}\simeq 166{\;{\rm MeV}}$. The
pion or charged lepton produced in the decay of $T^{\pm}$ lacks the necessary
energy for effective tracking or calorimeter detection, rendering $T^{\pm}$
elusive. As both $T^{0}$ and $T^{\pm}$ contribute to missing energy
signatures, we encounter significant SM backgrounds from $Z\to\nu\bar{\nu}$.

Fortunately, the intrinsic electric charge of $T^{\pm}$ enables its detection
within the innermost layers of the detector. A DCT appears as the proper decay
length of about 5.7 cm for $T^{\pm}$ yields no corresponding signals in the
external layers. However, DCT signatures alone are insufficient for effective
signal triggering, necessitating an additional mechanism. One option is a hard
mono-photon trigger, but it significantly reduces event rates.

In the VBF processes of $T^{\pm}T^{0}\mu^{\mp}\nu$ and
$T^{+}T^{-}\mu^{+}\mu^{-}$, a notable feature is the presence of one or two
high-energy, forward spectator muons, $\mu_{\rm f}$. The current MuC detector
design, focusing primarily on the central region with $|\eta|<2.5$ to reduce
BIB through the use of tungsten nozzles, mostly fails to detect these forward
spectator muons. This limitation has led to growing support for integrating a
dedicated Forward muon detector into the MuC. The feasibility of this addition
is supported by the fact that Forward muons in VBF processes typically exhibit
momenta exceeding $p_{\mu_{\rm f}}\geq 300{\;{\rm GeV}}$, enabling them to
penetrate the tungsten nozzles Ruhdorfer:2023uea ; Accettura:2023ked ;
Forslund:2023reu . This capability distinctly sets them apart from the softer
BIB particles, including muons, which generally possess less than 100 GeV of
kinetic energy Collamati:2021sbv . Consequently, we propose the use of one or
two Forward muons as triggers in our analysis of the DCT signals from the VBF
processes: $T^{\pm}T^{0}\mu^{\mp}\nu$ and $T^{+}T^{-}\mu^{+}\mu^{-}$.
Furthermore, we underscore the importance of ${\rm MET}$ as an instrumental
tool in probing DM signals, particularly given that DCTs are mainly detectable
in the central barrel region of the detector.

Building upon these observations, we suggest four distinct final states (FS)
as follows:

FS1: $\displaystyle\quad 1\,{\rm{DCT}}+1\,\mu_{\rm f}+{\rm MET}\,;$ (24) FS2:
$\displaystyle\quad 2\,{\rm{DCT}}+2\,\mu_{\rm f}+{\rm MET}\,;$ FS3:
$\displaystyle\quad 1\,{\rm{DCT}}+2\,\mu_{\rm f}+{\rm MET}\,;$ FS4:
$\displaystyle\quad 2\,{\rm{DCT}}+1\,\mu_{\rm f}+{\rm MET}.$

It is noteworthy that FS1 primarily results from the
$T^{\pm}T^{0}\mu^{\mp}\nu$ process. However, it may also emerge from the
$T^{+}T^{-}\mu^{+}\mu^{-}$ process when one Forward muon falls outside the
rapidity coverage of the proposed Forward muon detector and one DCT signal
goes undetected. On the other hand, FS2, FS3, and FS4 are exclusively
associated with the $T^{+}T^{-}\mu^{+}\mu^{-}$ process.

##  5 Characteristics of the triplet signals at a multi-TeV muon colliders

In order to refine our final selections, we delve into the characteristics of
triplet signals at the MuC, focusing on three pivotal aspects: (i) the large
decay radius of $T^{\pm}$; (ii) the considerable efficiency of DCTs; and (iii)
the high efficiency of Forward muons. We will elucidate these features using
detailed detector simulations, employing the following methodology. Event
generation was accomplished using MadGraph5_aMC@NLO Alwall:2011uj  version
3.4.2, followed by showering with Pythia8 Bierlich:2022pfr  version 8.3.07.
Subsequently, Delphes version 3.5.0 deFavereau:2013fsa  facilitated a rapid
detector simulation, utilizing the card of
delphes_card_MuonColliderDet.tcl555The MuC detector Delphes card can be found
at the GitHub repository: https://github.com/
delphes/delphes/blob/master/cards/delphes_card_MuonColliderDet.tcl. It is
noteworthy that the detection of Forward muons has been incorporated into this
Delphes card.

###  5.1 Brief review of the MuC tracker and the decay radius of $T^{\pm}$

Figure 10: (a) Schematic of the MuC tracker; (b) detailed view of the radial
plane focusing on the Vertex Detector (VXD) layers. These figures are adapted
for reader convenience from Refs. Capdevilla:2021fmj ; MuonCollider:2022ded .

Disappearing charged tracks from $T^{\pm}$ are pivotal for identifying ITM
signatures. Analyzing these DCT signals demands a thorough understanding of
the tracker configuration in the proposed MuC detector as outlined in Ref.
MuonCollider:2022ded . Figure 10 presents a schematic representation of the
MuC tracker. As illustrated in Figure 10(a), the tracker comprises three
primary components: the Vertex Detector (VXD) depicted in green, the Inner
Tracker (IT) in black, and the Outer Tracker (OT) in red. These components are
segmented into barrel (represented by solid lines) and endcap (indicated by
dashed lines) sections. The VXD barrel contains four double-layers positioned
at specific radial distances from the interaction point, as shown in Figure
10(b). Radially, the IT barrel ranges from 12.7 cm to 55.4 cm, and the OT
spans from 81.9 cm to 148.1 cm. The endcap areas, comprising tracker disks
oriented perpendicular to the beam axis Collamati:2021sbv ;
MuonCollider:2022ded ; Ally:2022rgk , predominantly encounter a majority of
the BIB hits.

Our investigation concentrates on the barrel region of the tracker, where the
incidence of BIB hits is markedly lower compared to the endcap region
Capdevilla:2021fmj . Figure 10(b) zooms into the radial plane, focusing on the
double-layers of the VTD and the initial single layer of the IT. For clear
reference in subsequent discussions, the double layers in the VXD are
designated as VXD0-1, VXD2-3, VXD4-5, and VXD6-7, while the IT layers are
sequentially labeled as IT0, IT1, and so on.

Given the detailed design of the MuC tracker, we are able to determine the DCT
reconstruction efficiency, which depends on the polar angle and decay length
of $T^{\pm}$. The decay length is expressed as $c\tau\beta\gamma$, where
$\beta$ is the velocity component $\left|\vec{p}\right|/{E}$ and $\gamma$ is
the relativistic factor ${1}/{\sqrt{1-\beta^{2}}}$. However, for our analysis,
we focus on the transverse decay length $c\tau\beta_{T}\gamma$ (with
$\beta_{T}={p_{T}}/{E}$), commonly referred to as the decay radius, since our
DCT analysis is concentrated on the central barrel region of the detector.

Figure 11: Event count distribution by the decay radius of $T^{\pm}$ in the
process of $\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ at a 10 TeV muon
collider with $\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$. Benchmark points
are indicated as follows: BP1 ($M_{T^{\pm}}=1.2{\;{\rm TeV}}$) in turquoise,
BP2 ($M_{T^{\pm}}=1.85{\;{\rm TeV}}$) in orange, and BP3
($M_{T^{\pm}}=2.3{\;{\rm TeV}}$) in purple. The figure also delineates the
radial extents of the barrel region trackers: Vertex Detector (VXD) in green,
Inner Tracker (IT) in grey, and Outer Tracker (OT) in red.

Figure 11 illustrates the event count distributions relative to the decay
radius of $T^{\pm}$ in the process $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$ at the 10 TeV MuC with an integrated luminosity of
$\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$. Our analysis includes three
benchmark points: BP1 with $M_{T^{\pm}}=1.2{\;{\rm TeV}}$ (turquoise), BP2
with $M_{T^{\pm}}=1.85{\;{\rm TeV}}$ (orange), and BP3 with
$M_{T^{\pm}}=2.3{\;{\rm TeV}}$ (purple). The radial extents of the three
tracker barrel regions are also indicated: the VXD in green, the IT in grey,
and the OT in red.

A consistent trend observed across all benchmark points is the rapid decrease
in signal events as the decay radius of $T^{\pm}$ increases. This pattern is
attributed to reduced boost effects associated with the heavy $M_{T^{\pm}}$.
Fortunately, a considerable proportion of events exhibit $T^{\pm}$ decaying
within the VXD and IT regions. Such occurrences are vital for generating DCTs
that exhibit high levels of reconstruction efficiency.

###  5.2 Disappearing charged tracks

This subsection is dedicated to the reconstruction efficiency of DCT signals
at the MuC. The current Delphes card for the MuC does not have the capability
to track heavy unstable charged BSM particles, although it can reconstruct the
tracks of SM particles using the EFlowTrack module. To bridge this gap, Ref.
Capdevilla:2021fmj  has calculated model-independent DCT reconstruction
efficiencies using a Geant4 simulation. Our methodology integrates these
results in two steps: first, by establishing criteria for selecting DCT
candidates, and second, by applying the efficiency as a weight to each signal
event.

For selecting DCT candidates, two conditions for $T^{\pm}$ are set: a decay
radius $c\tau\beta_{T}\gamma$ in the range of $5.1{\,{\rm
cm}}<c\tau\beta_{T}\gamma<148.1{\,{\rm cm}}$, and a polar angle $\theta$
within $0.7<\theta<2.44$. The lower $c\tau\beta_{T}\gamma$ threshold of 5.1 cm
ensures that $T^{\pm}$ tracks hit at least the first two double-layers of the
VXD, i.e., up to VXD2-3. The upper limit for $c\tau\beta_{T}\gamma$ is set to
guarantee that $T^{\pm}$ tracks end before reaching the final layer of the OT
at 148.1 cm. This is crucial for reducing SM backgrounds by excluding
calorimeter hits, akin to DCT search strategies at the LHC CMS:2020atg ;
Capdevilla:2021fmj . The $\theta$ restriction aims to minimize BIB hits in
endcap regions beyond $|\eta|>2.44$. Further refinement to a more central
region, $\theta\in[0.7,2.44]$ (or equivalently $|\eta|\leq 1$), is applied, as
the DCT reconstruction efficiency significantly diminishes outside this range.

The second step involves applying the DCT efficiency $\mathcal{P}_{\rm DCT}$
in Fig.11 of Ref. Capdevilla:2021fmj  as a weighting factor to each signal
event count. This process requires an understanding of its dependency on both
the decay radius and polar angle of $T^{\pm}$. For $c\tau\beta_{T}\gamma$, the
efficiency is high when the charged track extends into the VXD’s outer layers
without crossing IT0, peaking in the VXD6-7. Between IT0 and the OT’s final
layer at 148.1 cm, the efficiency, though lower, remains non-zero,
particularly within $\theta\in[0.7,1.1]$. We conservatively assume that the
DCT reconstruction efficiency in this region is comparable to that of tracks
reaching IT0. This assumption is supported by supported by consistent BIB hit
counts across all three tracker layers MuonCollider:2022ded ; Ally:2022rgk ;
Collamati:2021sbv . Additionally, the efficiency is notably sensitive to
$\theta$, with the highest efficiency observed within $\theta\in[1.05,1.22]$.
This relationship, depicted as $\mathcal{P}_{\rm DCT}(\theta,\rho_{\rm d})$,
will be incorporated into the calculation of the expected number of signal
events exhibiting a DCT signal.

Figure 12: Two-dimensional histograms of normalized event counts in
$(\theta,c\tau\beta_{T}\gamma)$ plane at a 10 TeV muon collider with
$\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$. The figure showcases three
benchmark $M_{T^{\pm}}$ values: 1.20 TeV in the upper right, 1.85 TeV in the
lower left, and 2.3 TeV in the lower right panel. For comparison, the upper-
left panel presents a reference scenario with a lower $T^{\pm}$ mass of 400
GeV.

To assess the reconstructability of $T^{\pm}$ as a DCT signal, we present in
Figure 12 the two-dimensional distribution of event counts in the
$(\theta,c\tau\beta_{T}\gamma)$ plane. This analysis is relevant to the signal
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}\nu$ at a 10 TeV MuC with an
integrated luminosity of $\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$. In
addition to three benchmark scenarios (BP1, BP2, BP3), we also include results
for a reference case with $M_{T^{\pm}}=400{\;{\rm GeV}}$, providing insights
into the impact of $M_{T^{\pm}}$ on $\mathcal{P}_{\rm DCT}$. Each panel
notably features three red dashed lines, marking the starting radii for VXD,
IT, and OT. A critical observation from Figure 12 is that a significant
portion of our signal events lies within the range favoring DCT
reconstruction. Additionally, we observe that signals from heavier $T^{\pm}$
particles tend to be more centrally concentrated, a trend consistent with the
findings in Figure 11. These results suggest that the ITM with heavy triplet
scalars exhibits a high potential for discovery at the MuC.

###  5.3 Forward muons

A key characteristic of the final states in Equation 24 is the presence of one
or two spectator muons. These muons are expected to appear predominantly in
the high pseudorapidity region. Although they can occasionally be detected in
the main detector through statistical fluctuations, the probability of this
occurring is notably low. Therefore, dedicated Forward muon detection
facilities are essential for the effective triggering of our VBF processes, an
option proactively considered by the Muon Collider R&D group Accettura:2023ked
.

The Delphes MuC card includes a Forward muon module, although it is primarily
optimized for the main detector setup, covering a pseudorapidity range of
$\left|\eta\right|\leq 2.44$.666The current nozzle configuration is tailored
for BIB at a 3 TeV MuC MuonCollider:2022glg . At higher c.m. energies, BIB
hits are expected to occur more forward, justifying the use of
$\left|\eta\right|\leq 2.44$ for our analysis at $\sqrt{s}=6,10{\;{\rm TeV}}$
as a conservative assumption. This module spans the region
$2.5\leq\eta_{\mu}\leq 8.0$, achieving an optimistic efficiency of 90%. In our
simulations, we adopt a more conservative pseudorapidity range of
$2.5\leq\eta_{\mu}\leq 7.0$ for the Forward muons Ruhdorfer:2023uea .

Figure 13:  (a) The $\eta$ distribution of muons produced in the process
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ at a 10 TeV MuC with
$\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$. Forward muons are highlighted in
orange, while muons detected by the main detector are shown in blue; (b) The
three-momentum distribution of the same Forward muons at the 10 TeV MuC.

To demonstrate the crucial role of Forward muon facilities in detecting the
spectator muons of our signal, we display in Figure 13(a) the pseudorapidity
distribution for muons produced in the $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$ process at a 10 TeV MuC with $\mathcal{L}_{\rm
tot}=10{\,{\rm ab}^{-1}}$. In this figure, Forward muons are represented in
orange, contrasting with those detected by the main detector, shown in blue.
The distribution indicates that the majority of muons in our signal
predominantly occupy the $\left|\eta\right|>2.5$ region, with only a minor
fraction being detected in the main detector.

An essential requirement for detecting the spectator muons by the Forward muon
detector is their ability to penetrate the tungsten nozzle, which necessitates
a sufficiently high energy of the Forward muons. Figure 13(b) presents the
three-momentum distribution of these spectator muons, confirming that they are
indeed highly energetic, predominantly peaking around 3 TeV. This high energy
profile significantly differentiates them from BIB particles, including muons,
which generally possess much lower kinetic energy, typically less than 100
GeV. To accurately identify our signal, therefore, we apply a threshold of
$p_{\mu_{\rm f}}\geq 300$ GeV, along with a pseudorapidity range of
$2.5\leq\left|\eta_{\mu}\right|\leq 7.0$ for the Forward muons.

###  5.4 Infeasibility of the charged pion signal

Before delving into the our analysis results, it is important to demonstrate
the infeasibility of detecting charged pions originating from the decay
$T^{\pm}\to\pi^{\pm}T^{0}$. While the long-lived nature of $T^{\pm}$ might
suggest that the displaced charged pions could be detectable against SM
backgrounds, in practice, these pions are not observable. The small mass
splitting between $T^{\pm}$ and $T^{0}$, approximately 166 MeV, results in
charged pions that lack the necessary momentum for reconstruction. Detecting
these soft pions is particularly challenging in the MuC environment,
overwhelmed by soft BIB pions.

Figure 14: Distributions of three-momentum (orange) and $p_{T}$ (blue) of the
displaced charged pion for BP2, from the process $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$ at a 10 TeV MuC with $\mathcal{L}_{\rm
tot}=10{\,{\rm ab}^{-1}}$.

The idea that high collision energy at a 10 TeV MuC could provide enough boost
to these displaced charged pions for detection is another aspect to consider.
This possibility remains largely unexplored in the literature. We present in
Figure 14 the momentum distribution of displaced charged pions from
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$, followed by
$T^{\pm}\to\pi^{\pm}T^{0}$, at a 10 TeV MuC with $\mathcal{L}_{\rm
tot}=10{\,{\rm ab}^{-1}}$. Our analysis reveals that, even at this collision
energy level, the displaced pions do not attain enough boost to meet the track
reconstruction threshold of $p_{T}\geq 0.5$ GeV. Consequently, they remain
unreconstructed, indistinguishable among the multitude of soft BIB pions.

Looking forward, a 30 TeV MuC with extremely forward BIBs might provide a
chance to detect some energetic displaced pion tracks. However, moving to a 30
TeV MuC involves a substantial jump in c.m. energy and necessitates a longer
timescale for development and analysis. Therefore, we defer the exploration of
these scenarios to future studies.

##  6 Results on the signal event counts

In the previous section, we established criteria for DCT signals and Forward
muons, summarized as:

$\displaystyle\text{DCT}:$ $\displaystyle\quad
0.7<\theta_{\text{tr}}<2.44,\quad 5.1{\,{\rm cm}}<\rho_{\rm tr}<148.1{\,{\rm
cm}},\quad p_{T}^{\text{tr}}\geq 300{\;{\rm GeV}},$ (25)
$\displaystyle\mu_{\rm f}:$ $\displaystyle\quad p_{T}^{\mu_{\rm f}}\geq
300{\;{\rm GeV}},\leavevmode\nobreak\ \leavevmode\nobreak\ \quad
2.5\leq\left|\eta_{\mu_{\rm f}}\right|\leq 7.0,$

where $\rho_{\rm tr}$ is the transverse length of the charged track.

Based on these criteria, our focus shifts to the signal-to-background analysis
for ${\mu^{+}\mu^{-}}\to T^{\pm}T^{0}\mu^{\mp}\nu$ and ${\mu^{+}\mu^{-}}\to
T^{+}T^{-}\mu^{+}\mu^{-}$ through the four final states in Equation 24. We
first generated events using MadGraph5_aMC@NLO, followed by showering in
Pythia8 and detector simulation with Delphes using
delphes_card_MuonColliderDet.tcl. For jet clustering, the inclusive Valencia
algorithm was employed, with a jet radius parameter of $R=0.5$ and a minimum
momentum cut of $p_{T}^{j}\geq 20$ GeV. Charged leptons within the calorimeter
coverage region were tagged with a cut of $p_{T}^{\ell}\geq 20$ GeV.

Final state |  VBF of ${\mu^{+}\mu^{-}}\to$ | Selections
---|---|---
FS1 |  $T^{0}T^{\pm}\mu^{\mp}\nu$, |  $N_{\text{DCT}}=1$ |  $N_{\mu_{\rm f}}=1$ |  ${\rm MET}<10{\;{\rm GeV}}$, [5pt] $N_{\ell^{\pm}}=N_{j}=0$
$T^{+}T^{-}\mu^{+}\mu^{-}$
FS2 | $T^{+}T^{-}\mu^{+}\mu^{-}$ | $N_{\rm DCT}=2$ | $N_{\mu_{\rm f}}=2$
FS3 | $T^{+}T^{-}\mu^{+}\mu^{-}$ | $N_{\rm DCT}=1$ | $N_{\mu_{\rm f}}=2$
FS4 | $T^{+}T^{-}\mu^{+}\mu^{-}$ | $N_{\rm DCT}=2$ | $N_{\mu_{\rm f}}=1$
Table 2: Summary of the selection cuts for the four final states of two signal
processes, ${\mu^{+}\mu^{-}}\to T^{\pm}T^{0}\mu^{\mp}\nu$ and
${\mu^{+}\mu^{-}}\to T^{+}T^{-}\mu^{+}\mu^{-}$. Here, $N_{i}$ represents the
number of the object $i$, and ${\rm MET}$ stands for the missing transverse
energy.

To effectively suppress SM backgrounds, we employ four key discriminators: (i)
presence of one or two Forward muons with $p_{T}^{\mu_{\rm f}}>300{\;{\rm
GeV}}$; (ii) a veto on events with any jets or leptons; (iii) the requirement
of exactly one or two DCTs with $p_{T}^{\rm tr}>300{\;{\rm GeV}}$; (iv) soft
${\rm MET}$ below 10 GeV. These discriminators were meticulously tailored to
each final state, as summarized in Table 2. In this table, we present the
number of Forward muons and DCT signals along with common requirements on
${\rm MET}$ and lepton/jet multiplicity.

The proposed discriminators are highly efficient, nearly eliminating all SM
backgrounds. Detailed information is provided in Appendix A. Consequently, the
number of expected signal events, $N_{\rm sig}$, becomes a critical factor in
assessing the potential of the MuC to probe ITM signals. We calculate $N_{\rm
sig}$ by integrating the DCT reconstruction efficiency $\mathcal{P}_{\rm DCT}$
as follows:

$N_{\rm sig}=\mathcal{P}_{\rm DCT}\times\frac{n_{\rm cut}}{n_{\rm
gen}}\sigma_{\rm tot}\mathcal{L}_{\rm tot},$ (26)

where $n_{\rm gen}$ is the total number of events generated at the MadGraph
level, $n_{\rm cut}$ is the count of events meeting the criteria in Table 2,
and $\sigma_{\rm tot}$ is the total cross-section for the respective signal
process. Two MuC configurations are considered: 6 TeV MuC with
$\mathcal{L}_{\rm tot}=4{\,{\rm ab}^{-1}}$ and 10 TeV MuC with
$\mathcal{L}_{\rm tot}=10{\,{\rm ab}^{-1}}$.

We now present our results for FS1, a final state characterized by exactly one
DCT and one Forward muon. FS1 emerges from two distinct signal processes:
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ and $\mu^{+}\mu^{-}\to
T^{+}T^{-}\mu^{+}\mu^{-}$. The notable promise of this final state is largely
attributed to the high production cross-section of $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$, positioning it as a key focus in our analysis.

$\sqrt{s}$ | $\mathcal{L}_{\rm tot}$ |  VBF process $\mu^{+}\mu^{-}\to$ products | Number of events for FS1
---|---|---|---
BP1 |  BP2 |  BP3
$6{\;{\rm TeV}}$ | $4{\,{\rm ab}^{-1}}$ | $T^{0}T^{\pm}\mu^{\mp}{\nu}$ | 12598.78 | 766.75 | 27.91
$T^{+}T^{-}\mu^{+}\mu^{-}$ | 153.86 | 11.50 | 0.49
Total | 12752.64 | 778.25 | 28.4
$10{\;{\rm TeV}}$ | $10{\,{\rm ab}^{-1}}$ | $T^{0}T^{\pm}\mu^{\mp}{\nu}$ | 402391.71 | 128795.32 | 51453.43
$T^{+}T^{-}\mu^{+}\mu^{-}$ | 4231.26 | 1575.11 | 682.30
Total | 406622.97 | 130370.43 | 52135.73
Table 3: Number of signal events for the FS1 at a 6 TeV and a 10 TeV MuC, with
an integrated luminosity of 4 ab-1 and 10 ab-1, respectively.

Table 3 shows the expected number of signal events for FS1 at both 6 TeV and
10 TeV MuC, accounting for contributions from both signal processes across
three benchmark points. Although the process $\mu^{+}\mu^{-}\to
T^{+}T^{-}\mu^{+}\mu^{-}$ also contributes a substantial number of events,
these values are smaller by almost two orders of magnitude compared to
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$. This discrepancy arises from
its lower cross-section and a small probability of missing one Forward muon.

At the 6 TeV MuC, a significant number of signal events are predicted for BP1
and BP2; however, only about 28 events are anticipated for the heavier
$T^{\pm}$ in BP3. The lower event count for BP3 is attributed to the reduced
cross-section (as shown in Table 1) and the insufficient boost for the charged
tracks to reach the detector region. In contrast, at the 10 TeV MuC, there is
a noticeable increase in event counts, particularly for $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}\nu$, due to the higher cross-section at increased
collision energy.

$\sqrt{s}$ | $\mathcal{L}_{\rm tot}$ |  VBF process $\mu^{+}\mu^{-}\to$ products |  Number of events for FS1′ $N_{\rm DCT}=N_{\mu_{\rm c}}=1,\leavevmode\nobreak\ {\rm MET}<10{\;{\rm GeV}},\leavevmode\nobreak\ N_{j,\ell}=0$
---|---|---|---
BP1 |  BP2 |  BP3
6 TeV | $4{\,{\rm ab}^{-1}}$ | $T^{0}T^{\pm}\mu^{\mp}{\nu}$ | 115.82 | 9.11 | 0.49
10 TeV | $10{\,{\rm ab}^{-1}}$ | $T^{0}T^{\pm}\mu^{\mp}{\nu}$ | 987.53 | 500.35 | 236.81
Table 4: The number of signal events for FS1′ featuring a central muon but
without tagging a Forward muon at both 6 TeV and 10 TeV MuC, with integrated
luminosities of 4 ab-1 and 10 ab-1, respectively. Here, $N_{\mu_{\rm c}}$
represents the number of hard central muons.

Prior to discussing the results for FS2, it is important to highlight the
significant role of tagging Forward muons in probing the ITM signals. To
illustrate this, we consider an alternative final state to FS1, named FS1′.
The primary distinction between FS1′ and FS1 lies in the type of muon
required: FS1′ necessitates a central muon, characterized by
$\left|\eta_{\mu_{\rm c}}\right|\leq 2.44$, while FS1 demands a Forward muon.
The other selection conditions remain the same. The number of signal events
for FS1′, originating from the main signal process $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$, is presented in Table 4.

It is apparent that the event count for FS1′ is significantly lower than that
for FS1, by almost three orders of magnitude. Notably, at a 6 TeV MuC, BP3
yields no events, and BP2 only about 10 events. While the event numbers at a
10 TeV MuC are still relatively high, FS1′ may encounter substantial non-VBF
SM backgrounds Han:2020uak , leading to a much lower discovery potential. In
conclusion, the use of Forward muons is crucial for effectively probing the
heavy triplet scalars.

$\sqrt{s}$ | $\mathcal{L}_{\rm tot}$ |  VBF process $\mu^{+}\mu^{-}\to$ products | Number of events for FS2
---|---|---|---
BP1 |  BP2 |  BP3
6 TeV | $4{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 206.29 | 6.22 | 0.06
10 TeV | $10{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 9979.87 | 2509.74 | 814.78
Table 5: Number of signal events for the FS2 at a 6 TeV and a 10 TeV muon
collider, with an integrated luminosity of 4 ab-1 and 10 ab-1, respectively.

We now turn our attention to FS2, characterized by two DCTs and two Forward
muons. This final state exclusively arises from the $\mu^{+}\mu^{-}\to
T^{+}T^{-}\mu^{+}\mu^{-}$ process. The expected number of signal events for
FS2 at both a 6 TeV and a 10 TeV MuC, with integrated luminosities of 4 ab-1
and 10 ab-1 respectively, are detailed in Table 5.

When comparing the signal event counts for FS2 with those for FS1 from the
same process, $\mu^{+}\mu^{-}\to T^{+}T^{-}\mu^{+}\mu^{-}$, some interesting
insights emerge. Intuitively, one might expect the $T^{+}T^{-}\mu^{+}\mu^{-}$
process to yield more events for FS2 than for FS1. However, this is not the
case at the 6 TeV MuC. Specifically, for BP2 and BP3, the event counts for FS2
are lower than those for FS1. Even though BP1 shows a higher event count for
FS2, the increment is only marginal. This outcome results from the requirement
of two reconstructed DCTs, leading to a compounded suppression from two
factors of $\mathcal{P}_{\rm DCT}$. The limited boosting at the 6 TeV MuC
leads to decreased reconstruction efficiency, thereby diminishing the overall
event count.

Conversely, at a 10 TeV MuC, FS2 records more events than FS1 for the
$T^{+}T^{-}\mu^{+}\mu^{-}$ process. The number of signal events is
substantial, even for the higher $T^{\pm}$ mass of 2.3 TeV. This increase is
attributed to the enhanced boost available at the higher energy level,
allowing more disappearing tracks to reach regions with higher efficiency.

$\sqrt{s}$ | $\mathcal{L}_{\rm tot}$ |  VBF process $\mu^{+}\mu^{-}\to$ products | Number of events for FS3
---|---|---|---
BP1 |  BP2 |  BP3
6 TeV | $4{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 1330.21 | 85.60 | 2.33
10 TeV | $10{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 37556.77 | 14045.45 | 6108.81
Table 6: Number of signal events for the FS3 at a 6 TeV and a 10 TeV muon
collider, with an integrated luminosity of 4 ab-1 and 10 ab-1, respectively.

We now examine FS3, characterized by one DCT and two Forward muons. This final
state arises exclusively from the $\mu^{+}\mu^{-}\to T^{+}T^{-}\mu^{+}\mu^{-}$
process, with one of the two $T^{\pm}$ particles not being reconstructed as a
DCT signal at the detector level. The expected numbers of signal events for
FS3 at both a 6 TeV and a 10 TeV MuC, with integrated luminosities of 4 ab-1
and 10 ab-1 respectively, are shown in Table 6.

Comparing FS3 with FS2, it becomes evident that FS3 consistently shows a
higher number of events. This difference is attributable to the requirement of
only a single DCT for FS3, which allows for the preservation of more signal
events due to the application of $\mathcal{P}_{\rm DCT}$ only once.

When comparing the results at the 6 TeV MuC with those at the 10 TeV MuC, a
clear pattern emerges: the 10 TeV MuC consistently yields a significantly
higher number of signal events compared to the 6 TeV MuC. Remarkably, the
enhancement factor in the number of signal events exceeds that seen at the
cross-section level in Table 1, especially for BP3. This discrepancy is
primarily due to the boost required for DCT reconstruction. While the 6 TeV
MuC yields limited boosting, hindering effective DCT reconstruction, the 10
TeV MuC achieves the necessary boost, resulting in more effective DCT
detection and a subsequent increase in signal events.

$\sqrt{s}$ | $\mathcal{L}_{\rm tot}$ |  VBF process $\mu^{+}\mu^{-}\to$ products | Number of events for FS4
---|---|---|---
BP1 |  BP2 |  BP3
6 TeV | $4{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 24.46 | 1.00 | 0.02
10 TeV | $10{\,{\rm ab}^{-1}}$ | $T^{+}T^{-}\mu^{+}\mu^{-}$ | 1084.74 | 274.14 | 83.31
Table 7: Number of signal events for the FS4 at a 6 TeV and a 10 TeV muon
collider, with an integrated luminosity of 4 ab-1 and 10 ab-1, respectively.

Finally, we analyze FS4, which consists of two DCTs and one Forward muon.
Originating from the $\mu^{+}\mu^{-}\to T^{+}T^{-}\mu^{+}\mu^{-}$ process, FS4
represents a scenario where one of the two Forward muons is not detected by
the Forward detector, while both DCTs are successfully reconstructed. The
expected number of events for FS4 at a 6 TeV and 10 TeV MuC, with integrated
luminosities of 4 ab-1 and 10 ab-1 respectively, is presented in Table 7.

FS4 records the lowest number of events among the four final states. This is
primarily due to the relatively small cross-section of the
$T^{+}T^{-}\mu^{+}\mu^{-}$ process and the low probability of missing a hard
Forward muon, given its expected detection efficiency of about 90%. At the 6
TeV MuC, fewer than 25 events are observed for BP1, while the event counts for
BP2 and BP3 are negligible. In contrast, at the 10 TeV MuC, a substantial
number of events are observed, notably around 84 events for BP3.

##  7 Discovery projections

In this section, we conduct an in-depth analysis of the discovery potential of
our model at two different MuC energies, 6 TeV and 10 TeV, taking into account
the target luminosity. A critical factor in determining this potential is the
signal significance, which depends heavily on the number of SM background
events. As detailed in Appendix A, the final states consisting of DCTs and
hard Forward muons are not contaminated by the direct SM background processes.
However, the sheer volume of BIB hits can result in a significant number of
events with fake tracks that persist even after applying all selection
criteria for the suggested final states. According to Ref. Capdevilla:2021fmj
, at a 10 TeV MuC with a total luminosity of $10{\,{\rm ab}^{-1}}$,
approximately 200 BIB events could be misinterpreted as containing one
disappearing track in the VXD barrel region. Considering the distribution of
BIB hits in the IT and OT barrel regions MuonCollider:2022ded , it is possible
that around 600 BIB events could potentially mimic DCT signals.

For each final state and collider energy, therefore, we explore two background
scenarios: one with 200 background events and the other with 600 background
events. We then conduct a likelihood-based analysis using the pyhf package
lukas_heinrich_2021_5009162 ; pyhf_joss  to perform hypothesis testing and
calculate $p$-values. We calculate the signal significance from these
$p$-values, incorporating a conservative systematic uncertainty of 20%. A
discovery is defined as a signal significance reaching $5\sigma$. For the MuC
c.m. energy of 6 TeV, we scale the event numbers to match the luminosity in
both the 200 and 600 background event scenarios.

Figure 15: Luminosity (ab-1) vs significance ($\sigma$) projections at a 6 TeV
muon collider for the final states: (a) FS1, (b) FS2, (c) FS3, and (d) FS4. A
20% background uncertainty is factored into these projections. Dashed lines
represent scenarios with 200 background events, while solid lines indicate
scenarios with 600 background events. The pink vertical line marks the 4 ab-1
luminosity goal for the 6 TeV MuC, and the red horizontal line denotes the
5$\sigma$ discovery threshold.

Utilizing our analytical frameworks, we have generated luminosity versus
significance plots for each final state at both the 6 TeV and 10 TeV MuC.
Figure 15 presents these plots for a 6 TeV MuC, with subfigures (a) to (d)
corresponding to FS1 through FS4. In each graph, dashed curves represent the
scenario with 200 background events, while solid curves depict the situation
with 600 background events. The results for BP1, BP2, and BP3 are illustrated
in blue, orange, and purple, respectively.

From Figure 15, it is evident that FS1 is the most advantageous final state,
requiring the least luminosity to achieve a 5$\sigma$ significance. Within the
target luminosity of $4{\,{\rm ab}^{-1}}$, however, discovery significance is
only attainable for BP1 and BP2. For BP1, a discovery is possible with about
3.2 (3.5) fb-1 for the 200 (600) background event scenarios. BP2 requires
around 120 (230) fb-1 under the same scenarios. For BP3, the luminosity needed
exceeds 20 ab-1 in both background scenarios, surpassing the feasible range
for a 6 TeV MuC.

FS3, depicted in Figure 15(c), emerges as the next most favorable final state.
It could lead to the discovery of BP1 with approximately 60 (100) fb-1 of
luminosity for the 200 (600) background event scenarios. For BP2, discovery
would necessitate about 6 (16) ab-1, beyond the target luminosity for a 6 TeV
MuC. With the planned 4 ab-1 luminosity, BP2 might only achieve around
4$\sigma$ (2.2$\sigma$) significance under the same background scenarios. For
BP3, the significance remains negligible for luminosities under 20 ab-1.

In contrast, FS2 and FS4, as shown in Figure 15(b) and (d) respectively,
exhibit very limited significances. Both final states result in negligible
significance with a total luminosity of 4 ab-1. The heavy masses of the
triplet scalars, required to explain the observed relic density, demand
luminosities well above 20 ab-1, thereby excluding them from feasible
discovery prospects. FS2 could potentially enable discovery of BP1 with about
1 (3) ab-1 under the 200 (600) background event scenarios. However, FS4 does
not present any significant discovery potential, even for BP1, at a 6 TeV MuC.

Figure 16: Luminosity versus significance projections, similar to those in
Figure 15, but for a 10 TeV MuC.

Shifting our focus to the 10 TeV MuC, Figure 16 presents the same luminosity
versus significance plots for this higher-energy collider. The enhanced cross-
section and improved tracking efficiency at the 10 TeV MuC enable all three
benchmark points to achieve a $5\sigma$ discovery in the final states FS1,
FS2, and FS3.

Specifically, in FS1, the high cross-section of the VBF process
$\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ plays a crucial role. Coupled
with the higher c.m. energy and an increased number of events meeting the DCT
reconstruction criteria, FS1 allows for achieving a $5\sigma$ significance
with relatively low luminosity. The luminosities required for discovery are
approximately 0.3 fb-1, 1 fb-1, and 3 fb-1 for BP1, BP2, and BP3,
respectively. The difference between the 200 and 600 background event
scenarios is minimal.

However, FS4, which records the lowest number of events, still encounters
challenges. As illustrated in Figure 16(d), BP3 does not reach a $5\sigma$
significance, even with the 10 ab-1 luminosity target of the 10 TeV MuC.
Nonetheless, BP1 and BP2 show potential for discovery, requiring luminosities
of approximately 0.2 (0.35) ab-1 and 1.5 (4.0) ab-1, respectively, for the 200
(600) background event scenarios.

Figure 17: Luminosity (ab-1) versus significance $\sigma$ projections for the
alternate final state FS1′, which includes one central muon but does not
involve Forward muons. We consider three BPs at MuC energies of (a) 6 TeV and
(b) 10 TeV. For each BP, dashed lines correspond to scenarios with 200
background events, and solid lines indicate scenarios with 600 background
events. Pink vertical lines represent the targeted luminosity of 4 ab-1 for
the 6 TeV MuC and 10 ab-1 for the 10 TeV MuC. Red horizontal lines signify the
5$\sigma$ discovery threshold.

Lastly, we discuss FS1′, an alternative to the FS1 scenario. FS1′ includes one
DCT and a central muon, but distinctively omits Forward muons. The same
luminosity versus significance plots for FS1′ are shown in Figure 17. This
alternative significantly differs from the original FS1, leading to a notable
decrease in detection sensitivity. At the 6 TeV MuC, discovering BP1 through
FS1′ is considerably more challenging. With 200 fake track events, around 3
ab-1 of luminosity is required for discovery, increasing to about 10 ab-1 for
600 background events.

However, the prospects at the 10 TeV MuC are more favorable. Here, achieving a
$5\sigma$ significance in the FS1′ scenario is feasible with luminosities of
approximately 0.2 (0.4) ab-1, 0.6 (1.5) ab-1, and 2.0 (6.0) ab-1 for BP1, BP2,
and BP3, respectively, under the assumption of 200 (600) background events.
Yet, the luminosities required for discovery through FS1′ are significantly
greater than those needed for the original FS1. This finding underscores the
high value of integrating Forward muon facilities into the experimental setup,
particularly at the 6 TeV MuC.

##  8 Conclusions

In this study, we have investigated the potential of a multi-TeV Muon Collider
(MuC) in probing the Inert Triplet Model (ITM), which introduces a scalar
triplet field with zero hypercharge ($Y=0$). The ITM, featuring a neutral
triplet $T^{0}$ and charged triplets $T^{\pm}$, emerges as an intriguing
Beyond the Standard Model scenario, offering a viable Dark Matter (DM)
candidate $T^{0}$, thanks to an incorporated $Z_{2}$ symmetry.

Our evaluation against a range of constraints, including theoretical,
collider, and DM experimental considerations, has identified three benchmark
points with heavy triplet scalar masses of 1.2 TeV, 1.85 TeV, and 2.3 TeV. A
notable characteristic of the ITM, distinguishing it from fermionic DM models,
is the existence of a $TTVV$ four-point vertex. This feature enables efficient
pair production via Vector Boson Fusion (VBF) processes, positioning the MuC
as an ideal venue for ITM exploration, especially given the amplified VBF
cross-sections at high collision energies. However, the extremely narrow mass
gap between $T^{\pm}$ and $T^{0}$ complicates the detection of $T^{\pm}$. The
dominant decay mode $T^{\pm}\to T^{0}\pi^{\pm}$ produces pions that are too
soft for detection amidst the beam-induced background (BIB) at the MuC. To
address this, we have advocated the use of Disappearing Charged Tracks (DCTs)
from $T^{\pm}$. Additionally, we propose leveraging Forward muons,
characterized by their high momenta in high $\eta$ regions, as innovative
triggers. By identifying four final states, comprising combinations of one or
two DCTs and one or two Forward muons, we have calculated event counts for the
three benchmark points at MuC center-of-mass energies of 6 TeV and 10 TeV,
with integrated luminosities of 4 ab-1 and 10 ab-1, respectively.

While Standard Model backgrounds can be effectively suppressed, BIBs at the
MuC pose challenges, potentially leading to fake tracks that contaminate the
signal. Addressing this, we consider scenarios with 200 and 600 background
events and present integrated luminosity projections for a $5\sigma$ discovery
at both the 6 TeV and 10 TeV MuC energies. The most promising outcome is
observed in the final state with one DCT and one Forward muon. Remarkably, the
4 ab-1 luminosity goal of the 6 TeV MuC is sufficient for a $5\sigma$
discovery of the benchmark points with triplet scalar masses of 1.2 TeV and
1.85 TeV. At the 10 TeV MuC, the heavier scalar with a mass of 2.3 TeV can
also be probed with less than 10 fb-1 of luminosity.

In conclusion, our study highlights the significant potential of a multi-TeV
MuC in exploring heavy triplet scalars in the ITM via VBF pair production
channels. The MuC emerges not just as a powerful tool for probing intricate
aspects of particle physics, but as a crucial instrument in unveiling
previously unknown facets of BSM.

###### Acknowledgements.

PB wants to thank SERB’s MTR/2020/000668 grant for support during project. SP
acknowledges the Council of Scientific and Industrial Research (CSIR), India
for funding his research (File no: 09/1001(0082)/2020-EMR-I). SP also thanks
Rodolfo Capdevilla, Nathaniel Craig, Simone Pagan Griso, Lorenzo Sestini, and
Massimo Casarsa for useful inputs, facilitated by the Muon4Future workshop. CS
would like to thank the MoE, Government of India for supporting her research
via SRF. CS also acknowledges Barbara Mele for valuable discussions during her
visit at the INFN Sezione di Roma. The work of JS is supported by the National
Research Foundation of Korea, Grant No. NRF-2022R1A2C1007583. PB and JS are
also supported by the Visiting Professorship at Korea Institute for Advanced
Study.

##  Appendix A SM Background rejection

In this appendix, we demonstrate the effectiveness of four discriminators in
significantly reducing the backgrounds from direct SM processes. We focus our
discussion on the signal process ${\mu^{+}\mu^{-}}\to
T^{\pm}T^{0}\mu^{\mp}\nu$ for BP2. The discriminators are as follows: (i) the
presence of one hard Forward muon with $p_{T}^{\mu_{\rm f}}>300{\;{\rm GeV}}$;
(ii) a veto on any jets or leptons in the event; (iii) the requirement of
exactly one disappearing charged track (DCT) with $p_{T}^{\rm tr}>300{\;{\rm
GeV}}$; (iv) soft missing transverse energy (MET) below 10 GeV.

The first discriminator, focusing on a hard Forward muon, is essential in
distinguishing our signals from the majority of non-VBF or neutral-current-VBF
SM background events. An example of such backgrounds is the invisible process
$\mu^{+}\mu^{-}\to\nu_{\mu}\bar{\nu_{\mu}}$, which could mimic nearly-
invisible disappearing track signals Capdevilla:2021fmj . Implementing the
hard Forward muon criterion effectively nullifies this background.

The second discriminator, vetoing out any calorimeter hits (jets and charged
leptons), leverages the unique nature of our signal, which is the absence of
hadronic or leptonic activities in the main detector region. This method
efficiently eliminates most non-VBF SM processes, such as $\mu^{+}\mu^{-}\to
VV$. However, certain VBF SM processes, such as $\mu^{+}\mu^{-}\to\mu^{\pm}\nu
W^{\mp}Z$ and $\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu_{\mu}}$, still
possess large cross-sections and could potentially contaminate our signal.

Figure 18: Panel (a) presents the distribution of charged track multiplicity
and panel (b) showcases the $p_{T}$ distribution of the track. This figure
features the signal process $\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ for
BP2, in comparison with two main SM backgrounds
($\mu^{+}\mu^{-}\to\mu^{\pm}\nu W^{\mp}Z$ and
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu_{\mu}}$) at the 10 TeV MuC.

The third discriminator, which requires one DCT with high $p_{T}$, plays a
vital role in distinguishing the signal from background events. In Figure
18(a), we illustrate the DCT multiplicity distributions for both the signal
and SM backgrounds at a 10 TeV MuC. The backgrounds include
$\mu^{+}\mu^{-}\to\mu^{\pm}\nu W^{\mp}Z$ and
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\nu_{\mu}\bar{\nu_{\mu}}$. The signal events
primarily consist of just one track, corresponding to a single $T^{\pm}$.
However, a notable number of background events also display a single DCT
signal, though they represent only a minor fraction of the total backgrounds.

In addressing this, the transverse momentum of the single track becomes an
essential discriminator. Figure 18(b) presents the $p_{T}$ distributions of
DCTs for both signal and SM backgrounds, each with precisely one track. The
greater mass of $T^{\pm}$ leads to a notably higher $p_{T}$ for its track, in
contrast to the generally softer SM background tracks, which mostly remain
below $150\ \text{GeV}$. Therefore, applying a cut of $p_{T}^{\text{tr}}\geq
300\ \text{GeV}$ removes nearly all SM backgrounds with a single track.

Figure 19: Distribution of the missing transverse energy (MET) of the BP2
signal process $\mu^{+}\mu^{-}\to T^{0}T^{\pm}\mu^{\mp}{\nu}$ against two
sample SM backgrounds at the 10 TeV MuC.

The final discriminator, requiring ${\rm MET}$ below 10 GeV, arises from the
observation that in our signal process, the two heavy triplet scalars are
almost back-to-back, resulting in relatively low MET. Figure 19 shows the
${\rm MET}$ distributions for the BP2 signal process $\mu^{+}\mu^{-}\to
T^{0}T^{\pm}\mu^{\mp}{\nu}$, alongside two representative SM backgrounds at
the 10 TeV MuC. Despite the signal involving two heavy particles that escape
detection, a significant cancellation effect results in most events having MET
below 10 GeV. This contrasts with the SM backgrounds, which typically exhibit
much higher ${\rm MET}$ values. Therefore, we implement a stringent cut of
${\rm MET}<10{\;{\rm GeV}}$ for all our signal processes.

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