††institutetext: Theoretical Physics Division, Physical Research Laboratory,
Shree Pannalal Patel Marg, Ahmedabad, 380009, Gujarat, India

# Unveiling desert region in inert doublet model assisted by Peccei-Quinn
symmetry

Anupam Ghosh  and Partha Konar<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Inert Higgs Doublet model (IDM), assisted by Peccei-Quinn (PQ) symmetry,
offers a simple but natural framework of a dark sector that accommodates
Weakly Interacting Massive Particle (WIMP) and axion as dark matter
components. Spontaneous breaking of $U(1)_{PQ}$ symmetry, which was originally
proposed as an elegant solution to the strong charge-parity (CP) problem, also
ensures the stability of WIMP through a residual $\mathbb{Z}_{2}$ symmetry.
Interestingly, additional fields necessitated by PQ symmetry further enrich
the dark sector. These include a scalar field proprietor for axion DM and a
vector-like quark (VLQ) that acts as a portal for the dark sector through
Yukawa interactions. Moreover, this combination of the axion and WIMP
components satisfies the observed DM relic density and reopens the
phenomenologically exciting region of the IDM parameter space where the WIMP
mass falls between 100 - 550 GeV. We investigate the model-independent pair
production of VLQs exploring this region at the Large Hadron Collider (LHC),
incorporating the effects of next-to-leading order (NLO) QCD corrections.
After production, each VLQ decays into a top or bottom quark accompanied by an
inert scalar, a consequence of the residual $\mathbb{Z}_{2}$ symmetry.
Utilising relevant observables with a leptonic search channel and employing
multivariate analysis, we demonstrate the ability of this analysis to exclude
a significant portion of the parameter space with an integrated luminosity of
300 $\text{fb}^{-1}$.

###### Keywords:

Dark matter, Large Hadron Collider, Vector-like quark

##  1 Introduction

The Standard Model (SM) of particle physics has established itself as a
cornerstone of modern physics, remarkably successful in describing a vast
array of experimental data during the last several decades. However, its
explanatory reach remains limited in the face of several fundamental enigmas,
such as the strong charge-parity (CP) problem Peccei:1977hh ; Peccei:1977ur ,
the existence of a substantial component of non-baryonic dark matter (DM) in
the Universe Sofue:2000jx ; Clowe:2006eq , the tiny yet non-zero masses of
neutrinos Super-Kamiokande:1998kpq ; SNO:2002tuh ; K2K:2002icj , and the
observed matter-antimatter asymmetry Riotto:1999yt ; Dine:2003ax , among
others. These persistent shortcomings motivated the theoretical and
experimental communities to explore new physics theories beyond the standard
model (BSM), extending the SM’s explanatory power. This work delves into a
simple yet elegant extension of the SM that tackles two of these outstanding
challenges: the Strong CP problem and the existence of dark matter. We propose
a detailed search strategy at the Large Hadron Collider (LHC) to probe the
associated dark sector particles.

Inert Higgs doublet model (IDM) Barbieri:2006dq ; Cirelli:2005uq  provides one
of the simplest BSM frameworks with a viable dark matter candidate. The
standard model is extended by a single $SU(2)_{L}$ scalar doublet carrying an
odd charge under $\mathbb{Z}_{2}$ symmetry, resulting in three new states:
charged scalar ($H^{\pm}$), a neutral scalar ($H^{0}$), and a neutral
pseudoscalar ($A^{0}$). Notably, the lightest of these neutral components
become stable and a dark matter candidate since an ad hoc $\mathbb{Z}_{2}$
symmetry is implemented to prevent the state from decaying. Various
experimental searches for dark matter and theoretical considerations strongly
constrain this extended DM model. Direct detection experiments, LHC searches,
and other astronomical and cosmological observations constrain the model’s
parameter space Ilnicka:2015jba ; Belyaev:2016lok ; Arhrib:2013ela ;
Dercks:2018wch . Theoretical constraints also play a crucial role. Oblique
electroweak parameters ($S,T,U$) Peskin:1991sw  quantify contributions from
BSM physics to electroweak radiative corrections. Notably, to satisfy the
$T$-parameter constraint, the masses of two out of the three new IDM particles
often need to be nearly degenerate Belyaev:2016lok . The IDM can explain the
measured dark matter relic density within two characteristically distinct
parameter regions, such as the hierarchical mass region and the degenerate
region. The hierarchical mass scenario achieves this through resonant Higgs
portal annihilation with the dark matter mass roughly half that of the SM
Higgs boson, while the other two BSM scalars ($H^{\pm},A^{0}$) are heavier and
almost degenerate. In the degenerate region, where all three new particles
have nearly equal masses, the dark matter typically has a mass exceeding 550
GeV. Consequently, the relic density remains underabundant due to strong gauge
interactions for a significant parameter space where dark matter mass lies
between 100-550 GeV. The prospect of searching for IDM particles at the LHC is
particularly interesting. For example, hierarchical regions can be probed
through the signature of jets or boosted fatjets plus missing transverse
momentum (MET) (Ghosh:2021noq, ; Belyaev:2018ext, ; Poulose:2016lvz, ).
Displaces vertex search CMS:2014gxa  can offer a potential avenue for
exploring the degenerate mass region. However, one expects a more challenging
pursuit due to the heavy final state particles and their ultra-soft decay
products.

Non-barionic cold dark matter, constituting roughly $27\%$ of the universe’s
energy budget, remains a mystery. While a simple, single-candidate model holds
appeal, no such supportive evidence exists so far. On the other hand, the
universe’s visible matter and radiation components, comprising only a tiny
fraction ($5\%$) of energy budget, are composed of a spectrum of particles
from fermionic families, gauge bosons and Higgs. Such design insinuates the
possibility of a much richer structure within the dark sector, including a
multicomponent dark matter scenario. The WIMP-axion scenario presents an
exciting potential, not only because the combined contributions with axion can
yield the observed dark matter relic density. Additionally, spontaneous
breaking of Peccei-Quinn $U(1)_{PQ}$ symmetry provides an elegant solution to
the strong CP problem and also ensures the stability of WIMP through a
residual $\mathbb{Z}_{2}$ symmetry for a natural realisation of this dark
matter picture.

Spontaneous PQ symmetry breaking necessitates a complex scalar field, with its
phase component being the axion. This $U(1)_{PQ}$ symmetry is expected to
break at an energy scale considerably higher than the electroweak scale.
Spontaneous breaking of this symmetry results in the emergence of a pseudo-
Goldstone boson, commonly known as the QCD axion. This axion possesses faint
interactions with all other particles Kim:1979if ; Shifman:1979if ;
Dine:1981rt ; Zhitnitsky:1980tq  and can have a lifetime far exceeding the
universe’s age, making it a suitable dark matter candidate. The PQ symmetry
breaking scale and the misalignment angle determine the axion relic density.
To implement the Peccei-Quinn symmetry, a vector-like quark (VLQ) is necessary
Peccei:1977hh ; Peccei:1977ur . The vector-like quark facilitates a
straightforward realisation of the $U(1)_{PQ}$ symmetry, as seen in the KSVZ
axion model Kim:1979if ; Shifman:1979if  and serves as a portal between the
dark sector and the SM fields. This connection is evident by the spontaneous
breaking of the $U(1)_{PQ}$ symmetry that generates a residual
$\mathbb{Z}_{2}$ symmetry inherently stabilising the WIMP dark matter
candidate 111Without any loss of generality, we can further continue our
analysis considering $H^{0}$ as the lightest mode and the WIMP DM candidate.,
thus eliminating the need for the introduction of an additional ad-hoc
discrete symmetry in the model. Furthermore, the two-component IDM framework
reopens the intriguing 100-550 GeV mass window for WIMPs and presents a
compelling avenue for further exploration of dark matter.

Building upon the WIMP-axion scenario, the residual $\mathbb{Z}_{2}$ symmetry
in the inert Higgs doublet model dictates that the vector-like quark ($\Psi$)
decays exclusively into BSM scalars ($H^{\pm},A^{0},H^{0}$) with SM quark.
Recent study Alves:2016bib  has explored the IDM with PQ symmetry in detail,
focusing on dark matter phenomenology. Their analysis considered a scenario
where the VLQ interacts only with the first-generation quark family of the SM.
Additionally, they reinterpreted LHC mono-jet and multi-jet and missing-energy
searches, demonstrating the feasibility of a VLQ-SM quark coupling strength
ranging from 0 to 1 across various parameter spaces. In contrast, the present
work focuses on VLQs primarily interacting with the third-generation quark
family (top and bottom quarks) of the SM, as the Yukawa interactions in the
visible sector. Interactions with the first and second generations are
considered negligible. This approach leads to a significant advantage: the VLQ
pair production cross-section becomes independent of any BSM couplings,
relying solely on the strong coupling constant. This independence renders our
searches model-independent. Furthermore, this study identifies an exciting
signal topology at the LHC for exploring the degenerate region of the IDM
parameter space. This signal consists of a single, isolated electron or muon
accompanied by two jets (one of which is identified as a b-tagged jet) with
significant missing transverse momentum (MET).

Hence, the salient features of this work are as follows: (i) The present work
explores the capabilities of the LHC to probe the WIMP-axion multicomponent
dark matter scenario within the Inert Higgs doublet framework, where the
residual discrete symmetry inherent to the model naturally stabilises the WIMP
dark matter candidate. (ii) Our study focuses on the phenomenologically
interesting degenerate region ($100\leq M_{H^{0}}\leq 700$) of the IDM
parameter space, considered as a desert region in a pure IDM scenario. (iii)
Traditionally, this heavier DM region in IDM (degenerate region) is
challenging to probe at the LHC because the heavy particles ($H^{\pm},A^{0}$)
decay into the dark matter particle ($H^{0}$) along with low-momentum Standard
Model particles. However, the present scenario opened up a novel search
strategy that leverages the presence of vector-like quarks within the model.
This strategy targets an alternative signal topology consisting of an
isolated, energetic lepton (electron or muon) accompanied by two jets (one
identified as a b-jet) and significant missing transverse momentum. (iv) We
demonstrate that a sizeable portion of the degenerate parameter space can be
explored using this approach at the 14 TeV LHC already with 300 fb-1 of
luminosity. (v) To enhance the accuracy of our predictions, we incorporate
Next-to-Leading-Order (NLO) QCD corrections for the VLQ pair production
process. (vi) Additionally, we utilise a sophisticated multivariate analysis
technique employing a Boosted Decision Tree (BDT) algorithm to refine the
collider analysis further.

The rest of the manuscript is organised as follows: Section 2 discusses the
underlying theoretical framework of the model. We provide a concise overview
of various constraints arising from theoretical considerations, astrophysical
observations, and existing collider data. This section also explores the dark
matter phenomenology associated with the two-component dark matter scenario.
In the next Section 3, we delve into the collider phenomenology, describing
the targeted signal topologies and establishing suitable benchmark points for
our analysis. We detail the primary event selection process based on simulated
signal and background events, followed by constructing relevant high-level
variables. Subsequently, we employ multivariate analysis techniques on these
input variables to calculate the corresponding signal efficiency. Finally,
this section presents the final statistical significance for different
benchmark points and establishes exclusion regions within the parameter space.
Finally, the concluding Section 4 summarises our key findings and reiterates
the potential of our proposed search strategy for exploring the degenerate
region of the inert doublet model assisted by Peccei-Quinn symmetry at the
LHC.

##  2 Theoretical Framework

###  2.1 The model

The model comprises an inert Higgs doublet model augmented by a KSVZ-type
axion model Kim:1979if ; Shifman:1979if . This axion model includes a singlet
scalar field, with its phase part being the axion field. The vacuum-
expectation value (vev) of this field breaks the $PQ$ symmetry at a scale much
larger than the electroweak (EW) scale. A residual $\mathbb{Z}_{2}$ symmetry
will remain intact after $PQ$ symmetry breaking, stabilising the WIMP dark
matter, the lightest neutral component of the IDM. The scalar field is as
described below.

$\eta=\frac{1}{\sqrt{2}}\Bigl{(}f_{a}+\sigma(x)\Bigr{)}~{}e^{\frac{ia(x)}{f_{a}}}.$
(1)

The $PQ$ breaking scale is denoted as $f_{a}$ and corresponds to vev of
$\eta$, while $\sigma(x)$ and $a(x)$ represent the radial mode and axion
field. The axion is invariably associated with $\frac{a(x)}{f_{a}}$; its
interactions with matter and gauge bosons are diminished by $f_{a}$. Since
$f_{a}$ is much larger than the EW scale, axions have faint interactions with
other particles. The effects of nonperturbative QCD result in a potential that
gives a mass to the axion GrillidiCortona:2015jxo .

$m_{a}=5.70~{}\Bigl{(}\frac{10^{12}~{}\text{GeV}}{f_{a}}\Bigr{)}~{}\mu eV~{}.$
(2)

In addition to the Standard Model Higgs doublet, $\Phi_{1}\sim(1,2,1/2)$, IDM
contains a new $SU(2)_{L}$ doublet, $\Phi_{2}\sim(1,2,1/2)$. The numbers
inside the parenthesis represent the charges under the SM gauge groups
$SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}$. Peccei-Quinn charge of
$\Phi_{2}$ is $PQ(\Phi_{2})=-1$, while all the SM fields have zero $PQ$
charges. Fields with even or zero $PQ$ charge remain even under residual
$\mathbb{Z}_{2}$ symmetry after $PQ$ symmetry breaking, while fields with odd
$PQ$ charge become odd under $\mathbb{Z}_{2}$. The two doublets can be written
as

$\Phi_{1}=\begin{pmatrix}G^{+}\\\
\dfrac{1}{\sqrt{2}}(v+h+iG^{0})\end{pmatrix},\quad\Phi_{2}=\begin{pmatrix}H^{+}\\\
\dfrac{H^{0}+i\hskip 2.84526ptA^{0}}{\sqrt{2}}\end{pmatrix},$ (3)

where $G^{0}$ and $G^{+}$ are the Goldstone bosons, $h$ is the SM Higgs boson
and $v=246\hskip 2.84526pt\text{GeV}$ is the EW vev. $H^{\pm}$, $H^{0}$, and
$A^{0}$ are the inert scalars, with $H^{0}$ and $A^{0}$ being CP even and CP
odd neutral scalars, respectively. The Higgs potential is as follows,

$\begin{split}V=&\,\mu_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}+\mu_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}+\dfrac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\dfrac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}(\Phi_{2}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})\\\
&+\dfrac{\lambda_{5}}{2}\hskip
2.84526pt[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}]\,.\end{split}$
(4)

After EW symmetry breaking, the masses for inert scalars are as follows.

$\begin{split}&M^{2}_{H^{\pm}}=\mu_{2}^{2}+\dfrac{1}{2}\lambda_{3}v^{2},\\\
&M^{2}_{A^{0}}=\mu_{2}^{2}+\dfrac{1}{2}\lambda_{c}v^{2},\\\
&M^{2}_{H^{0}}=\mu_{2}^{2}+\dfrac{1}{2}\lambda_{L}v^{2}.\end{split}$ (5)

where $\lambda_{L/c}=(\lambda_{3}+\lambda_{4}\pm\lambda_{5})$, $\lambda_{L}$
is also known as a Higgs portal coupling and can be either positive or
negative. The parameters $\lambda_{1}$ and $\mu_{1}$ can be expressed in terms
of the mass of the SM Higgs boson ($M_{h}=125$ GeV) and the EW vev ($v$).
$\lambda_{2}$ represents the self-coupling strength between inert scalars, is
a free parameter, and has no effect on scalar masses and their phenomenology.
In our study, we consider $H^{0}$ to be the dark matter candidate.
Alternatively, one can choose $A^{0}$ as the dark matter candidate without
altering any phenomenology simply by reversing the sign of $\lambda_{5}$.

Addressing the strong CP problem, the KSVZ axion model requires the inclusion
of at least one heavy vector-like quark ($\Psi$) in addition to the SM
fermions. The VLQ has the following interaction with the PQ breaking scalar
($\eta$)

$f_{\Psi}~{}\eta^{*}~{}\overline{\Psi}_{L}\Psi_{R}+h.c.~{},$ (6)

where $PQ(\Psi_{L})=-1$ and $PQ(\Psi_{R})=+1$ and $f_{\Psi}$ is the coupling
strength. The above equation facilitates the interaction of axions with a pair
of color triplet VLQs, enabling the axions to interact indirectly with gluons
via a triangle-loop diagram with VLQ in the loop. This axion-gluon field
strength interaction is necessary to solve the strong CP problem through the
$PQ$ mechanism. The charge assignments of all BSM fields are provided in table
1.

| $\eta$ | $\Psi_{L}$ | $\Psi_{R}$ | $\Phi_{2}$
---|---|---|---|---
$SU(3)_{C}$ | 1 | 3 | 3 | 1
$SU(2)_{L}$ | 1 | 1 | 1 | 2
$U(1)_{PQ}$ | 2 | -1 | 1 | -1
$\mathbb{Z}_{2}$ | + | - | - | -
Table 1: Particle contents beyond the SM and their quantum charges.

VLQ bridges dark sector particles and SM fields, featuring the following
Yukawa interaction with the SM quark doublet $q_{L}\sim(3,2,1/6)$ and the IDM
doublet $\Phi_{2}$.

$\mathcal{L}\supset f~{}\bar{q}_{L}\Phi_{2}\Psi_{R}+h.c.$ (7)

The $U(1)_{Y}$ hypercharge of VLQ is $-1/3$ and remains odd under residual
$\mathbb{Z}_{2}$ symmetry after the breaking of $PQ$ symmetry. As a result,
VLQ does not mix with the SM quarks. VLQ can interact with all three SM quark
generations. However, interactions primarily with the first two generations
are heavily constrained by flavor-violating effects, such as
$D^{0}-\bar{D}^{0}$ oscillation Garny:2014waa ; Gedalia:2009kh . To be
consistent with flavour oscillation data, interactions of the VLQ must
predominantly involve a single quark family. Another exciting possibility is
that the TeV scale VLQ predominantly interacts with the third generation of SM
quarks and minimally (or not at all) with the first two generations, similar
to the SM Higgs boson coupling with the top quark being much larger compared
to the up and charm quark couplings. In this case, the interaction vertices
from equation 7 are $\Psi\bar{t}H^{+}$, $\Psi\bar{b}H^{0}$, and
$\Psi\bar{b}A^{0}$. The free parameters of the model are the following.

$\bigl{\\{}M_{H^{0}},M_{A^{0}},M_{H^{\pm}},M_{\Psi},\lambda_{L},f,f_{a}\bigr{\\}}$
(8)

The mass of VLQ is $M_{\Psi}=f_{\Psi}f_{a}/\sqrt{2}$. The coupling $f_{\Psi}$
becomes the dependent parameter since we consider $M_{\Psi}$ and the axion
decay constant ($f_{a}$) independent parameters. All the inert scalars have
nearly degenerate mass in the degenerate region, $\Delta
M=M_{H^{\pm}}-M_{H^{0}}=M_{A^{0}}-M_{H^{0}}=\mathcal{O}(1)$ GeV. Direct
detection searches of inelastic DM require a mass splitting of more than 100
keV between inert neutral CP-even ($H^{0}$) and odd scalar ($A^{0}$)
Barbieri:2006dq ; Hambye:2009pw . Pair production of vector-like quarks in our
analysis does not depend on the $\lambda_{L}$ coupling. However, the same does
play a role in the IDM dark matter phenomenology of $H^{0}$. Various
theoretical, astrophysical, and collider data can constrain this model, and we
will discuss them briefly in the next.

###  2.2 Constraints

For a realistic model to be considered valid, the potential must be bounded
from below, and the vacuum must be charge-neutral, which leads to the
following constraints Deshpande:1977rw ; Ivanov:2006yq :

$\lambda_{1}>0,\hskip 2.84526pt\lambda_{2}>0,\hskip
2.84526pt\lambda_{3}+2\sqrt{\lambda_{1}\lambda_{2}}>0,\hskip
2.84526pt\lambda_{3}+\lambda_{4}+\lambda_{5}+2\sqrt{\lambda_{1}\lambda_{2}}>0.$
(9)

The condition $\lambda_{4}+\lambda_{5}<0$ implies that the inert vacuum is
charge neutral. The requirement of the inert vacuum to be the global minima
leads to Ginzburg:2010wa ; Swiezewska:2012ej ,

$\dfrac{\mu_{1}^{2}}{\sqrt{\lambda_{1}}}-\dfrac{\mu_{2}^{2}}{\sqrt{\lambda_{2}}}>0~{}.$
(10)

The unitarity in $2\rightarrow 2$ scalar-scalar scattering processes demands
that all $|\Lambda_{i}|\leq 8\pi$, where $|\Lambda_{i}|$ are given below
Arhrib:2012ia .

$\begin{split}&\Lambda_{1,2}=\lambda_{3}\pm\lambda_{4},\hskip
2.84526pt\Lambda_{3,4}=\lambda_{3}\pm\lambda_{5},\hskip
2.84526pt\Lambda_{5,6}=\lambda_{3}+2\lambda_{4}\pm 3\lambda_{5},\\\
&\Lambda_{7,8}=-\lambda_{1}-\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+\lambda_{4}^{2}},\hskip
2.84526pt\\\
&\Lambda_{9,10}=-3\lambda_{1}-3\lambda_{2}\pm\sqrt{9(\lambda_{1}-\lambda_{2})^{2}+(2\lambda_{3}+\lambda_{4})^{2}},\\\
&\Lambda_{11,12}=-\lambda_{1}-\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+\lambda_{5}^{2}}.\end{split}$
(11)

The SM weak gauge bosons can not decay into dark scalars, constraining the
inert scalars’ masses.

$M_{Z}<M_{H^{0}}+M_{A^{0}},~{}M_{W^{\pm}}<M_{H^{0}}+M_{H^{\pm}},~{}M_{W^{\pm}}<M_{A^{0}}+M_{H^{\pm}}~{}.$
(12)

In the region where $M_{H^{0}}<\dfrac{M_{h}}{2}$, the SM Higgs boson can decay
into a pair of WIMPs ($H^{0}$), contributing to the Higgs invisible decay
branching ratio and imposing stringent limits on the Higgs portal coupling.
For instance, if $M_{H^{0}}=60~{}\text{GeV}~{}(10~{}\text{GeV})$, then
$|\lambda_{L}|\lesssim 0.012~{}(0.007)$ Belanger:2013xza .

#### LEP bound:

Supersymmetric neutralino search at LEP Lundstrom:2008ai ; Belanger:2015kga
excludes the following region of the parameter spaces
$M_{H^{0}}<80~{}\text{GeV},~{}M_{A^{0}}<100~{}\text{GeV}$ for
$M_{A^{0}}-M_{H^{0}}>8~{}\text{GeV}$. The supersymmetric chargino search
Pierce:2007ut  at LEP further excludes $M_{H^{\pm}}<70~{}\text{GeV}$. Thus, we
have chosen,

$M_{A^{0}},M_{H^{\pm}}>100~{}\text{GeV},~{}\text{and}~{}M_{H^{0}}>90~{}\text{GeV}~{}.$
(13)

#### LHC bound:

The authors Belanger:2015kga  analyzed the dilepton plus missing energy
signature of LHC Run I data and excluded $H^{0}$ masses below 35 GeV at a
$95\%$ confidence level (CL). This constraint was based on the production
channel $q\bar{q}\rightarrow Z\rightarrow
A^{0}H^{0}\rightarrow(Z^{*}H^{0})H^{0}\rightarrow l^{+}l^{-}H^{0}H^{0}$. Our
region of interest, equation 13, remains unaffected by this search.

In a recent study conducted by the ATLAS collaboration Marjanovic:2014eca ,
researchers searched for supersymmetric particles with a final state of
multiple jets plus missing energy. Since vector-like quarks can interact with
the first two generations of SM quarks, they can produce the same final state.
The green shaded area in figure 8 represents the excluded parameter space at a
$95\%$ CL based on the reinterpretation Giacchino:2015hvk  of the ATLAS
results Marjanovic:2014eca  at the 8 TeV LHC, with an integrated luminosity of
20.3 $\text{fb}^{-1}$.

###  2.3 Direct detection

WIMPs may interact with nuclei, causing detectable energy deposits in
underground instruments such as LUX LUX:2015abn , XENON1T XENON100:2012itz ,
and CDMS SuperCDMS:2014cds . Each instrument uses different target nuclei,
such as xenon, argon, and others, and unique detection methods. These
underground detectors in direct detection experiments enforce strict
constraints on the WIMP-nucleon scattering cross-section versus the WIMP mass.

In the context of the IDM model, evading the direct detection thresholds set
by LUX can be achieved by ensuring that the mass gap between $A^{0}$ and
$H^{0}$ exceeds 100 KeV. Additionally, reducing the $\lambda_{L}$ coupling
without compromising our analysis, particularly when the values of
$|\lambda_{L}|\lesssim 0.01$, remains undetectable by the ongoing LUX
experiment Blinov:2015qva  and the projected reach of XENON1T Aprile:2012zx ;
XENON:2015gkh .

The presence of an exotic vector-like quark ($\Psi$) can mediate the
interaction of WIMP with nuclei through $\Psi$-mediated scattering
$qH^{0}\rightarrow qH^{0}$. The WIMP-nucleon scattering cross-section will be
negligibly small if the $\Psi$ predominantly couples with the third-generation
quark family. However, if the $\Psi$ couples with the first two generations of
SM quarks, the cross-section can be significant and depends on the Yukawa
coupling ($f$) and VLQ mass ($M_{\Psi}$). The most stringent constraint will
occur when $M_{\Psi}\sim M_{H^{0}}$. If $f\lesssim 0.5$, the model will be
consistent with the LUX constraint on the spin-independent scattering cross-
section. If the mass of $\Psi$ is not degenerate to the WIMP ($H^{0}$), the
bound will be much more relaxed. This holds true if $H^{0}$ contributes to all
of the dark matter relic density. However, this is not the case in our two-
component dark matter scenario. The WIMP-nucleon cross-section needs to be
rescaled by the ratio of WIMP local density to the total relic density. As a
result, the constraints will be further relaxed in cases where the axion
contribution is significant.

###  2.4 Indirect detection

The indirect detection concept involves observing particles produced when dark
matter annihilates or co-annihilates, generating final state particles like
electrons, positrons, photons, and neutrinos. In many BSM scenarios, dark
matter particles annihilate to produce SM particles. Photons and neutrinos,
being electromagnetically neutral, are likely to reach detectors without much
deviation. Experiments such as PAMELA Kohri:2009yn , MAGIC MAGIC:2016xys  and
Fermi-LAT Eiteneuer:2017hoh , among others, are searching for these indirect
detection signals. By measuring the flux of these stable particles,
constraints can be placed on the parameters of various dark matter models. For
a two-component dark matter scenario, the indirect detection cross-section
needs to be rescaled by the ratio of the local density of $H^{0}$ to the total
dark matter relic density squared. As a result, the mass of interest
$100~{}\text{GeV}<M_{H^{0}}<550~{}\text{GeV}$ is consistent with the searches
by Fermi-LAT Eiteneuer:2017hoh  and H.E.S.S. HESS:2011zpk  limits.

###  2.5 Dark matter relic density

Figure 1: The plot shows the variation of WIMP relic density for different DM
masses when inert scalars are near mass degenerate, but VLQ is not. The red
line represents the Higgs portal coupling with $\lambda_{L}=0.01$ and $\Delta
M=1$ GeV. The solid (dashed) blue line is for $\lambda_{L}=1.0$ and $\Delta
M=1$ GeV (5 GeV). Here, $\Delta M=M_{H^{\pm}}-M_{H^{0}}=M_{A^{0}}-M_{H^{0}}$.
The dashed black line is the observed DM relic density by the Planck
collaboration Aghanim:2018eyx .

In figure 1, we display the WIMP ($H^{0}$) abundance, considering VLQ is not
degenerate with the inert scalars. The relic abundance is obtained by solving
the Boltzmann equation using micrOMEGAs -v5 Belanger:2018ccd . As mentioned
earlier, in IDM, there are two possible regions: (1) Hierarchical, where
$A^{0}$ and $H^{\pm}$ have nearly identical masses significantly higher than
$H^{0}$ mass, and (2) Degenerate, where all inert scalar masses are nearly
identical, allowing for coannihilation. In the former case, no coannihilation
is present because of the larger mass gap between heavy inert scalars and
$H^{0}$. In that case, the correct relic is obtained through annihilation of
$H^{0}$ into the SM Higgs boson in the region when the mass of $H^{0}$ is
nearly half of the SM Higgs boson mass. The exploration of the hierarchical
region at the LHC has been extensively documented in the literature
Ghosh:2021noq ; Poulose:2016lvz ; Datta:2016nfz . The variation of the relic
density of $H^{0}$ in figure 1 is shown for different values of the Higgs
portal coupling $\lambda_{L}=0.01~{}\text{and}~{}1.0$, keeping $\Delta M=1$
GeV by solid red and solid blue, respectively. The blue dashed line is for
$\lambda_{L}=1.0$ and $\Delta M=5$ GeV. The orange dashed line shows the
variation for a larger mass gap between the inert scalars.

$M_{H^{0}}<100$ GeV: We first focus on the region $M_{H^{0}}<100$ GeV, where
other heavy inert scalars fall within a narrow mass range. The first dip is
around $M_{H^{0}}\sim 38-44$ GeV because of the resonance production of the
$Z~{}(H^{0}A^{0}\rightarrow Z)$ and $W^{\pm}~{}(H^{0}H^{\pm}\rightarrow
W^{\pm})$ boson. The region $M_{H^{0}}\leq 45$ GeV is ruled out from the
electroweak bound as given in equation 12. The second dip is a result of the
resonance production of the SM Higgs boson through the Higgs portal
annihilation ($H^{0}H^{0}\rightarrow h$). Furthermore, some additional shallow
dips are caused by the opening of $H^{0}H^{0}\rightarrow W^{+}W^{-}$ and
$H^{0}H^{0}\rightarrow ZZ$ annihilation modes. This region has been studied at
the LHC with a mono-jet signal, as discussed in reference Belyaev:2018ext .

$M_{H^{0}}>100$ GeV: The blue curves in figure 1 show a sharp dip around
$M_{H^{0}}\sim 125$ GeV due to the opening up of $H^{0}H^{0}\rightarrow hh$
annihilation mode ($h$ is the SM Higgs boson), which enhances the annihilation
cross-section and reduces the relic density. The $H^{0}H^{0}hh$ quartic
coupling is directly proportional to the $\lambda_{L}$, so a small
$\lambda_{L}$ value does not produce the observed dip in the red line. The
horizontal dashed black line represents the observed total DM relic density.
For $\lambda_{L}=0.01$ (1.0, solid blue), we observe that $H^{0}$ can provide
the correct relic density when $M_{H^{0}}\sim 543~{}(1000)$ GeV; below these
masses, the relic is underabundant. WIMP is more abundant for a small
$\lambda_{L}$ value since a larger $\lambda_{L}$ increases the annihilation
cross-section, decreasing relic density, as observed in the solid red and blue
lines. For $|\lambda_{L}|\geq 0.001$, the region
$100~{}\text{GeV}<M_{H^{0}}<500~{}\text{GeV}$ generally remains underabundant,
whereas the region $M_{H^{0}}>500$ GeV may or may not be underabundant,
depending on the $\lambda_{L}$ value 222If VLQ becomes nearly degenerate with
the inert scalars, then the coannihilation of VLQ with other inert scalars and
VLQ annihilation will also contribute to the relic density of $H^{0}$
Alves:2016bib . This will result in an increase in the annihilation cross-
section, and consequently, the relic density of $H^{0}$ will be smaller than
that in figure 1..

#### Axion relic density:

In our two-component dark matter scenario, the underabundant WIMP combines
with the axion to produce the correct relic density, as observed by the Planck
collaboration Aghanim:2018eyx

$\Omega_{H^{0}}h^{2}+\Omega_{a}h^{2}=\Omega_{\text{total}}h^{2}=0.120\pm
0.001$ (14)

The relic density of axions depends on whether the Peccei-Quinn symmetry is
broken before or after the period of inflation. In scenarios where the
symmetry is broken before inflation ends, axion production primarily occurs
through coherent oscillations resulting from vacuum realignment. In this case,
the axion relic density is given by Sikivie:2006ni ; Bae:2008ue ,

$\Omega_{a}h^{2}\simeq 0.18\hskip 2.84526pt\theta^{2}\hskip
2.84526pt\bigg{(}\frac{f_{a}}{10^{12}~{}\text{GeV}}\bigg{)}^{1.19}\,$ (15)

where $\theta$ represents the initial misalignment angle of the axion, and it
is of the order of unity. For $\theta=1$, the axion decay constant required to
match the correct DM relic density in our WIMP-axion model is

$f_{a}=\Bigl{(}\dfrac{\Omega_{\text{total}}h^{2}-\Omega_{H^{0}}h^{2}}{0.18}\Bigr{)}^{1/1.19}~{}10^{12}~{}\text{GeV}~{}.$
(16)

Equation 15 suggests the axion can give the total relic density when
$f_{a}\sim 10^{12}~{}\text{GeV}$. Therefore, the overproduction of axions
imposes a constrain $f_{a}\leq 10^{12}~{}\text{GeV}$.

##  3 Collider analysis

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 2: Feynman diagrams of $\Psi\bar{\Psi}$ pair production in the proton-
proton collision at leading order partonic processes (a)-(d). Diagram (e) is
the production of $\Psi$ associated with WIMP. Diagrams (f) and (g) are the
possible decay modes of VLQ when it predominantly couples with the third-
generation quark family of the SM.

The presence of VLQ opens up interesting collider searches. VLQ can be pair-
produced or produced in association with the inert scalars. The Yukawa
interaction of VLQ with the SM quarks and the inert scalars is given in
equation 7. Considering VLQ predominantly couples with third-generation of the
SM quarks, the pair production at the LHC, $pp\rightarrow\Psi\bar{\Psi}$,
representative diagrams are shown in figure 2 in parts 2a-2d. Quark-initiated
diagram 2d strongly depends on the value of the Yukawa coupling ($f$). For
protons with four or five massless quark flavours, diagram 2d is either absent
or contributes negligibly small, even with $f\sim 1$, because of the low
parton distribution function (PDF) of $b$-quarks. As a result, the first three
diagrams, parts 2a-2c, are only the production channels of the VLQ pair at the
proton-proton collision. These processes are governed entirely by the strong
interaction of the SM and are independent of any BSM couplings, though they do
depend on the mass of the VLQ 333In contrary, if VLQ significantly interacts
with the first or both the first and second families of the SM quarks, the
quark-initiated pair production processes can contribute significantly. The
cross-section of these processes depends on the Yukawa coupling values and the
masses of the inert scalars, making the partonic cross-section fully dependent
on the model specifics.. The production of $\Psi$ with WIMP is shown in
diagram 2e. It also has a negligibly small cross-section for $f\leq 1.0$.
Therefore, to investigate this model at the LHC, the focus is solely on the
pair production of VLQs. Upon their production, each VLQ decays into either a
top or bottom quark, accompanied by an inert scalar, as illustrated in
diagrams 2f and 2g. Depending on the decay modes of VLQ, the possible signal
topologies are detailed below.

$\displaystyle
pp\rightarrow\Psi\bar{\Psi}\rightarrow(t~{}H^{-})(\bar{t}H^{+})=t\bar{t}+H^{+}H^{-}$
(17) $\displaystyle
pp\rightarrow\Psi\bar{\Psi}\rightarrow(t~{}H^{-})(\bar{b}H^{0}/\bar{b}A^{0})=t\bar{b}(\text{or}~{}\bar{t}b)+H^{\pm}H^{0}/H^{\pm}A^{0}$
(18) $\displaystyle
pp\rightarrow\Psi\bar{\Psi}\rightarrow(bH^{0}/bA^{0})(\bar{b}H^{0}/\bar{b}A^{0})=b\bar{b}+H^{0}H^{0}/H^{0}A^{0}/A^{0}A^{0}$
(19)

The signal we are looking for contains at least one top (or anti-top) quark
that decays leptonically, so both topologies 17 and 18 will play a role. In
topology 17, one top quark undergoes a leptonic decay while the other decays
hadronically. Conversely, topology 18 involves one b-jet with the top quark
decaying leptonically. In both cases, we end up with a charged lepton and two
b-quarks from either the top quark’s decay or VLQ’s. Therefore, our final
state includes one isolated energetic lepton (electron or muon) and inclusive
two jet events, with one of the leading two jets expected to be b-tagged. In
addition, our final state has a large missing transverse momentum (MET or
$\not{E}_{T}$), which comes from the neutrinos from the top decay and the
nearly degenerate inert scalars. Topology 19 does not contribute to our
desired final state, as no lepton is present. This final state is highly
promising for dark matter searches at the LHC and provides a complementary
search to the multijet + MET studies. It marks an intriguing direction for
exploring the degenerate spectrum of IDM and VLQ at the LHC. Additionally, we
include the next-to-leading order (NLO) QCD correction of VLQ pair production
processes for more accurate predictions. We then match these NLO fixed-order
results with the Pythia8 parton shower, ensuring our findings are applicable
over entire phase-space regions.

Benchmark | $M_{H^{0}}$ | $M_{\Psi}$ | $\Delta M_{\Psi DM}$ | BR($\Psi\rightarrow tH^{-}$) | LO, $~{}~{}\mathcal{O}(\alpha_{S}^{2})$ | NLO, $~{}~{}\mathcal{O}(\alpha_{S}^{3})$ | K-fac
---|---|---|---|---|---|---|---
points | (GeV) | (GeV) | (GeV) |  | (fb) | (fb) |
BP1 | 645 | 850 | 205 | 0.3528 | $89.21^{+28.6\%}_{-20.8\%}$ | $133.2^{+9.6\%}_{-11.0\%}$ | 1.49
BP2 | 445 | 750 | 305 | 0.40 | $191.7^{+28.6\%}_{-20.8\%}$ | $287.2^{+9.1\%}_{-10.7\%}$ | 1.50
BP3 | 550 | 1000 | 450 | 0.4867 | $31.76^{+28.7\%}_{-20.9\%}$ | $47.33^{+10.5\%}_{-11.5\%}$ | 1.49
BP4 | 500 | 1100 | 600 | 0.494 | $16.67^{+28.8\%}_{-21.0\%}$ | $24.45^{+10.2\%}_{-11.4\%}$ | 1.47
BP5 | 700 | 1500 | 800 | 0.4964 | $1.79^{+29.2\%}_{-21.3\%}$ | $2.47^{+11.6\%}_{-12.2\%}$ | 1.38
Table 2: Few representative benchmark points (BPs) satisfy all the constraints
discussed in section 2. $M_{H^{0}}$ and $M_{\Psi}$ are the mass of the WIMP DM
and VLQ, respectively, with $\Delta M_{\Psi DM}=M_{\Psi}-M_{H^{0}}$. Here, we
fix mass difference within inert scalars
$M_{A^{0}}-M_{H^{0}}=M_{H^{\pm}}-M_{H^{0}}=5.0$ GeV. BR($\Psi\rightarrow
tH^{-}$) is the branching fraction of VLQ decaying into a top quark and
$H^{-}$, and BR($\Psi\rightarrow bH^{0}$) + BR($\Psi\rightarrow bA^{0}$) =
1-BR($\Psi\rightarrow tH^{-}$). The sixth column contains the partonic leading
order cross-section of VLQ pair production, which has order
$\mathcal{O}(\alpha_{S}^{2})$. The following columns contain next-to-leading
order cross-section ($\mathcal{O}(\alpha_{S}^{3})$), and K-factor. The
superscript and subscript denote the scale uncertainties (in percentage) of
the total cross-section.

Table 2 lists several benchmark points that meet all the constraints outlined
in section 2. The leading order (LO) partonic cross-section,
$\sigma(pp\rightarrow\Psi\bar{\Psi})$, which encompasses diagrams 2a-2c,
involves a coupling order $\mathcal{O}(\alpha_{S}^{2})$, where $\alpha_{S}$
represents the strong coupling constant. The next-to-leading order (NLO)
cross-section is of order $\mathcal{O}(\alpha_{S}^{3})$. The LO and NLO cross-
sections are presented in the table for the central choice of factorization
($\mu_{F}$) and renormalization ($\mu_{R}$) scales. The central choice is
$\mu_{F}=\mu_{R}=\text{partonic centre-of-mass energy}~{}(\sqrt{\hat{s}})$.
$\mu_{F}$ and $\mu_{R}$ are two spurious scales resulting from collinear
factorization and renormalization processes. The hadronic cross-section is
theoretically independent of these scales. Nonetheless, when the perturbative
series is truncated at a finite order, like LO, NLO, or so on, this invariance
is lost, leading to scale uncertainties in the cross-section. Such
uncertainties are more prominent at lower orders. These scales are then varied
as $\mu_{F}=\zeta_{1}\sqrt{\hat{s}}$ and $\mu_{R}=\zeta_{2}\sqrt{\hat{s}}$,
where $\zeta_{1}$ and $\zeta_{2}$ can take values from the set
$\\{1/2,1,2\\}$, creating a total of nine data sets. The superscripts and
subscripts in the cross-section denote the envelope of those nine scale
choices. As expected, the theoretical scale uncertainties for NLO-QCD events
are significantly smaller than those for LO events. The K-factor, defined as
the ratio of the NLO to LO cross-section, is approximately 1.5 for a VLO mass
of up to 1 TeV, but it decreases with an increase in mass. Thus,
$\mathcal{O}(\alpha_{S})$ corrections applied to the pair production processes
increase the production rate and lessen the scale uncertainty.

###  3.1 Simulation details

Our analysis incorporates all background processes that mimic the signal,
which is detailed in the following subsection. We generate each background
process with an addition of two to four QCD jets using an MLM matching scheme
Mangano:2006rw ; Hoeche:2005vzu . Our model is implemented in FeynRules
Alloul:2013bka , and we generate the UFO model file. The signal and background
events are generated using MadGraph5_aMC@NLO Alwall:2014hca  and pass through
Pythia8 Sjostrand:2001yu ; Sjostrand:2014zea  for showering, fragmentation and
hadronization. These showered events undergo detector simulation via Delphes3
deFavereau:2013fsa . The _NNPDF3.0_ NNPDF:2014otw  LO and NLO PDF datasets are
used to generate LO and NLO events. Anti-kT Cacciari:2008gp  jets with a
radius parameter 0.4 are formed by clustering particle-flow towers and tracks.
Finally, for the multivariate analysis (MVA), we use the adaptive Boosted
Decision Tree (BDT) method Roe:2004na ; FREUND1995256 ; Freund:1997xna  within
the TMVA Hocker:2007ht  framework.

###  3.2 Backgrounds

Our signal consists of a single energetic isolated lepton (electron or muon),
multijet, and a large amount of missing transverse energy. One of the two
leading jets must be b-tagged, and any event with no charged lepton or more
than one lepton is rejected. The background includes $t\bar{t}+\text{jets}$,
$W+\text{jets}$, and single top (including $tW+\text{jets}$), among others.

  * I.

$t\bar{t}+\text{jets}$ $t\bar{t}$ events can be categorized into two classes
that will significantly contribute to our final state. In semileptonic
$t\bar{t}$ events, one top-quark decay leptonically and the other decays
hadronically, resulting in one prompt lepton from the decay of the $W$ boson.
The other category is dileptonic $t\bar{t}$ events, where both tops decay
leptonically, producing two prompt leptons.

####

The semileptonic $t\bar{t}$ events contain only one charged lepton and a
single neutrino from $W$ decay, which is the source of MET. This process has a
lower MET due to the presence of only one neutrino. On the other hand,
dileptonic $t\bar{t}$ events mimic the signal when one of the charged leptons
is lost, sometimes referred to as the lost-lepton background. In dileptonic
$t\bar{t}$ events, two neutrinos from $W$ decay and one lost lepton
collectively give the MET. The $t\bar{t}$ background is generated with an
additional two jets with MLM matching and is normalized with the higher-order
QCD corrected production cross-section at the 14 TeV LHC Muselli:2015kba .

  * II.

Single top-quark Single top quark production at the LHC with the $W$ boson
contributes considerably as background, which is generated with two extra
parton MLM matching. We consider both cases where either the top quark or $W$
boson decays leptonically or both of them decay leptonically. Therefore, in
the former case, the single neutrino or, in the latter case, two neutrinos
with one mistag charged lepton give the MET.

####

Another small contribution comes from the single top quark production
associated with the $b$-quark ($t\bar{b}~{}(\bar{t}b)+\text{jets}$). This
background is also generated with two extra QCD radiation with MLM matching.
In this case, the top decays leptonically, giving one prompt lepton and MET.
The $tW$ and $tb$ backgrounds are normalized using higher-order QCD cross-
section Kant:2014oha ; Kidonakis:2015nna .

  * III.

$W+\text{jets}$ The $W+\text{jets}$ can mimic our signal when the $W$ decays
leptonically. This background contains only one neutrino, which gives the MET
and a single prompt-charged lepton. The background is generated with up to
four QCD jets using MLM matching, resulting in a final state with multijets.
This background is also scaled by a higher-order QCD cross-section
Balossini:2009sa .

  * IV.

di-boson+jets Among diboson productions, $WZ+\text{jets}$ and $WW+\text{jets}$
will mimic our final state. In the former case, $W$ decays leptonically to
give rise to a lepton, and $Z$ decays invisibly ($Z\rightarrow\nu\bar{\nu}$).
Those three neutrinos, two coming from $Z$ decay and one from $W$ decay,
collectively produce MET. We observed that if $Z$ boson decays hadronically,
it contributes much less than $Z$ decays invisibly. This is because in the
hadronic decay of $Z$, the $WZ$ background yields only one neutrino with a
small MET. Therefore, this background becomes significantly smaller when a
large MET cut is applied. In the $WW+\text{jets}$ background, one of the $W$
decays leptonically and the other hadronically. Both backgrounds are generated
with two extra jets. We scaled the di-boson productions with the higher-order
QCD cross-section Campbell:2011bn .

  * V.

$t\bar{t}V$ In our analysis, $t\bar{t}$ production with a vector boson
$V~{}(=W,Z)$ has a minimal contribution due to its small cross-section, but we
include it. Compared to $t\bar{t}W$, $t\bar{t}Z$ has a larger contribution
when one of the top quarks decays leptonically, and the $Z$ boson decays
invisibly.

###  3.3 Event reconstruction

The events in our analysis can be selected using triggers with a large MET and
the presence of an isolated electron or muon. MET is defined as the negative
vector sum of the transverse momentum of all particle flow candidates
CMS:2017yfk . We chose MET to be greater than 80 GeV. The lepton must have
transverse momenta $P_{T}(l)>30$ GeV and must also fall within the detector’s
acceptance, having a pseudo-rapidity $|\eta|$ less than 2.4. Additionally, if
electrons and positrons fall within the range $1.44<|\eta|<1.57$, we reject
the event since this range is the transitional area between the barrel and
endcap sections of the CMS electromagnetic calorimeter. The lepton should have
passed the isolation criteria.

#### Lepton Isolation:

Isolation of leptons $(l=\mu,e)$ is implemented via the following variable,

$I_{l}=\sum_{i(\neq
l)\subset\text{tracks}}^{P_{T}(i)>P_{T}^{\text{min}}}P_{T}(i)~{}.$ (20)

The sum runs over all tracks above transverse momentum
$P_{T}^{\text{min}}=0.5$ GeV, excluding the lepton of interest, lying within a
cone of radius $\Delta R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}=0.3$ centred
on the lepton. Muon isolation requires $I_{\mu}\leq 0.1\times P_{T}(\mu)$, and
electron isolation requires $I_{e}<5$ GeV.

#### Lepton veto:

In case the event contains more than one lepton, we use the lepton veto. We
discard such events if any additional lepton aside from the leading lepton has
transverse momentum $P_{T}(l)>5$ GeV and $|\eta|<2.4$.

Di-leptonic $t\bar{t}+\text{jets}$ background events can contribute to our
desired final state when one lepton passes all the criteria defined above, but
the second lepton passes the loose selection (lepton veto) or the lepton
isolation criteria. Furthermore, the final state contains at least two jets.
Jets are formed by clustering particle flow candidates with a radius parameter
0.4, utilizing the Anti-kT algorithm. Each leading and subleading jets are
required $P_{T}(j_{i})~{}(i=1,2)>30$ GeV and $|\eta|<2.4$. The events in which
the leading two jets or the isolated lepton overlap with the missing
transverse momentum within a cone of radius $\Delta R<0.4$ are discarded. All
those cuts mentioned above will be referred to as a preselection cut.

###  3.4 Preselection and high-level variables

(a)

(b)

Figure 3: The normalized stacked histograms of the signal (BP2) and all
background processes are shown. On the left, the distribution of the
transverse mass of the lepton is presented, while on the right, the
distribution of the total missing transverse momentum after preselection cuts
is displayed. The signals sig-$t\bar{t}$ and sig-$tb$ correspond to topologies
given in equations 17 and 18, respectively. The decay branching ratio of
$\Psi$ is detailed in table 2.

After preselection cuts, the distribution of the transverse mass of the lepton
is shown in the left panel of figure 3. In this stacked histogram, the
contribution of individual background processes is filled with different
colors. The signal is presented for BP2. The signals sig-$t\bar{t}$ and
sig-$tb$ correspond to topologies in equations 17 and 18, respectively. The
solid red line represents the total signal distribution. The contribution of
sig-$tb$ lies between the blue and red lines, while the solid blue line
represents the contribution of sig-$t\bar{t}$. The transverse mass of the
lepton is defined as Cheng:2008hk ,

$M_{T}(l,\text{MET})=\sqrt{2P_{T}(l)\not{E}_{T}(1-\cos\Delta\Phi)}$ (21)

where $P_{T}(l)$ denotes the lepton’s transverse momentum, and $\Delta\Phi$
represents the azimuthal separation between the lepton’s direction and the
missing transverse momentum (MET or $\not{E}_{T}$). After preselection cuts,
the main backgrounds that mimic our signal are the semileptonically decaying
$t\bar{t}+$ jets (filled blue) and the leptonically decaying $W+$ jets (filled
olive). In contrast, the dileptonic $t\bar{t}+$ jets (filled orange)
background is smaller. The vertical black dashed line corresponds to a
lepton’s transverse mass of 120 GeV. Applying this cut significantly reduces
the semileptonic $t\bar{t}$ and $W+$ jets backgrounds, making the dileptonic
$t\bar{t}+$ jets background the major one, followed by the semileptonic
$t\bar{t}$.

The right panel of figure 3 shows the stacked histogram of the missing
transverse momentum for the signal (BP2) and background. The vertical black
dashed line in this plot indicates a MET of 150 GeV. All the background has a
smaller MET, while the signal has a larger MET. Hence, using a higher MET cut
increases the signal-to-background ratio. As a result, we apply
$\text{MET}>150$ GeV and $M_{T}(l,\text{MET})>120$ GeV along with the
preselection cuts for the actual Multivariate analysis (MVA) in the next
section. The signal and background distribution of different kinematic
variables after applying those cuts are presented in figure 4.

Semileptonic $t\bar{t}$ and $W+$ jets are the reducible backgrounds, and by
applying cuts on lepton’s transverse mass and MET, we reduce these two
backgrounds significantly. However, dileptonic $t\bar{t}$ remains an
irreducible background with signal-like distribution. In signal, degenerate
inert scalars and neutrino (from the top quark decay) give a large MET. In the
dileptonic $t\bar{t}$ background, two neutrinos and a mistagged lepton
contribute to a large MET. Furthermore, multiple sources of missing particles
do not lead to any peak of lepton’s transverse mass around the $W$ boson mass.
As a result, we find a significant dileptonic $t\bar{t}$ background left even
after a large MET and transverse mass cut. We have constructed the following
variable Lester:1999tx ; Barr:2011xt ; Konar:2009wn .

$M_{T2}(\vec{P}_{T}^{~{}1},\vec{P}_{T}^{~{}2},\vec{\not{E}}_{T})=\min_{\vec{q}_{T}^{~{}1}+\vec{q}_{T}^{~{}2}=\vec{\not{E}}_{T}}[~{}\max\\{~{}M_{T}(\vec{P}_{T}^{~{}1},\vec{q}_{T}^{~{}1}),~{}M_{T}(\vec{P}_{T}^{~{}2},\vec{q}_{T}^{~{}2})~{}\\}]$
(22)

where $\vec{q}_{T}^{~{}i}$ is the transverse momentum of an invisible
particle, assumed to be massless, contributing to the missing transverse
momentum $\vec{\not{E}}_{T}$; $\vec{P}_{T}^{~{}i}$ is the transverse momentum
of a visible particle. Transverse mass is defined below,

$M_{T}(\vec{P}_{T}^{~{}i},\vec{\not{E}}_{T})=\sqrt{2|\vec{P}_{T}^{~{}i}|~{}|\vec{\not{E}}_{T}|~{}[1-\cos\Delta\Phi(\vec{P}_{T}^{~{}i},~{}\vec{\not{E}}_{T})]}~{}.$
(23)

In order to reduce the leading dileptonic $t\bar{t}$ background, we propose
the variable $M_{T2}^{Combo}$, which is defined below.

$M_{T2}^{Combo}=\text{min}[~{}M_{T2}(j_{1}+l,~{}j_{2},~{}\vec{\not{E}}_{T}),~{}M_{T2}(j_{1},~{}j_{2}+l,~{}\vec{\not{E}}_{T})]~{}.$
(24)

The two leading jets are the $j_{1}$ and $j_{2}$, with one being $b$-tagged.
The reconstructed charged lepton is denoted by $l$. The endpoint of the
variable $M_{T2}^{Combo}$ is expected to be around the top quark mass
$(m_{t})$ for the dileptonic $t\bar{t}$ events. As a result, the backgrounds
have a smaller value of $M_{T2}^{Combo}$, whereas the signal has a larger
value because of the presence of many undetected particles. Therefore, this
variable plays a crucial role in distinguishing the signal from the background
in multivariate analysis. The distribution of the $M_{T2}^{Combo}$ variable is
depicted in figure 4e, where there is a clear separation between the signal
and background; most background events have a smaller $M_{T2}^{Combo}$ value,
while the signal shows a significantly larger value.

Transverse mass distributions of the leading two jets and the lepton are shown
in figures 4a, 4b, and 4c, respectively. We can see that the shapes of those
normalized distributions of signal and background are nearly the same, but the
jets in the signal are slightly harder than those in the background. The MET
distribution is presented in figure 4d, where we see that at larger MET, the
background falls sharply. In contrast, because of many missing particles in
the signal, MET is large and remains nearly flat up to a certain point.
Therefore, MET is a good discriminator for distinguishing between the signal
and background. Another inclusive variable, $\sqrt{\hat{S}_{min}}$, is plotted
in figure 4f, defined as
$\sqrt{\hat{S}_{min}}=\sqrt{(\sum_{j}E_{j})^{2}-(\sum_{j}P_{z,j})^{2}}+\not{E}_{T}$
Konar:2008ei . Here, the sum includes all visible objects, such as jets and
leptons; $E_{j}$ and $P_{z,j}$ represent the energy and $Z$-component of the
particle’s momentum.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 4: The normalized stacked histograms for the signal (BP2) and
background processes are shown after applying preselection cuts along with the
lepton transverse momentum $>120$ GeV and missing transverse momentum $>150$
GeV.

The separation of the lepton from the MET and the $b$-jet in the $\eta-\phi$
plane is illustrated in figures 4i and 4j. The azimuthal separation of the
leading jet from the MET and the jet multiplicity are depicted in figures 4k
and 4l, respectively. The distribution of the transverse mass of $b$-jet and
the lepton are shown in figures 4g and 4h. The signal includes two $b$-quarks;
the one with higher transverse momentum is tagged as a $b$-jet accompanied by
several missing particles, making the azimuthal separation between the $b$-jet
and MET arbitrary. Additionally, MET in the signal is large. This results in a
large $b$-jet’s transverse mass (equation 23) in the signal compared to the
background, establishing it as a good discriminator.

###  3.5 Multivariate analysis

We select several properties of the reconstructed lepton ($l=e,\mu$), jets,
and missing transverse momentum (MET or $\not{E}_{T}$) as input variables to
the BDT. Specifically, we consider their transverse momentum ($P_{T}(i)$ for
$i\in[j_{1},j_{2},l]$), azimuthal separation and distances in the $\eta-\phi$
plane ($\Delta\Phi(i,\not{E}_{T})$ and $\Delta R(i,j)$ with $i\neq
j\in[j_{1},j_{2},l,\text{b-jet}]$), where b-jet is either the leading or
subleading jet. Additionally, we include the invariant mass of the lepton and
b-jet pair $m(l,b)$, the transverse mass of the lepton
($M_{T}(l,\not{E}_{T})$) and $b$-jet ($M_{T}(b,\not{E}_{T})$), and the number
of jets ($N_{\text{jets}}$). Moreover, we include the variable
$M_{T2}^{Combo}$ as defined in equation 24, and a global inclusive variable
$\sqrt{\hat{s}_{\text{min}}}$ that determines the mass scale of new physics.

The full set of BDT input variables is thus:

$\displaystyle\Biggl{\\{}$ $\displaystyle
N_{\text{jets}},~{}\not{E}_{T},~{}P_{T}(j_{1}),~{}P_{T}(j_{2}),~{}P_{T}(l),~{}\Delta\Phi(j_{1},\not{E}_{T}),~{}\Delta\Phi(j_{2},\not{E}_{T}),~{}\Delta\Phi(l,\not{E}_{T}),~{}\Delta
R(j_{1},\not{E}_{T}),$ (25) $\displaystyle\Delta
R(j_{2},\not{E}_{T}),~{}\Delta R(l,\not{E}_{T}),~{}\Delta
R(b,l),~{}m(l,b),~{}M_{T}(l,\not{E}_{T}),~{}M_{T}(b,\not{E}_{T}),~{}\sqrt{\hat{s}_{\text{min}}},~{}M_{T2}^{Combo}\Biggr{\\}}$

The distribution of some of those variables after applying preselection cuts
along with a transverse mass of lepton $>120$ GeV and MET $>150$ GeV are shown
in figure 4. Note that we do not use stringent preselection criteria for the
input data sets in multivariate analysis. Thus, MVA has enough scope to find
an optimal nonlinear boundary by applying cuts to appropriate variables and
providing better statistical significance for the signal over the background.

Multiple processes contribute to the signal and background; therefore, we take
the weighted sum of those processes and make a combined signal and background
class. We use the adaptive Boosted Decision Tree (BDT) algorithm in our
analysis, and each signal and background event sample is split randomly for
training and testing purposes. The variables mentioned in equation 25 are not
all linearly independent, and some are particularly important for
distinguishing between signal and background classes. Therefore, we select
variables that are not highly correlated (or highly anti-correlated) in both
signal and background classes and have significant importance in
discriminating the signal from the background. By definition, the invariant
mass of $b$-jet and lepton $m(l,b)$ depends on the azimuthal separation
between $b$-jet and lepton, $\Delta\Phi(b,l)$. This relationship reveals a
significant correlation between $m(l,b)$ and $\Delta R(b,l)$ in both signal
and background classes. In our final selection of variables, $\Delta R(b,l)$
is preferred over $m(l,b)$ due to its greater relative importance in signal-
background separation. The relative importance is obtained from the TMVA
package. Another example is that $P_{T}(l)$ is highly correlated with
$M_{T2}^{Combo}$ in both signal and background classes, and we choose to keep
$M_{T2}^{Combo}$ because of its greater relative importance compared to
$P_{T}(l)$.

The final variables are used in our multivariate analysis, and their linear
correlation among themselves in signal (left panel) and background (right
panel) are shown in figure 5. Positive coefficients indicate correlation,
while negative coefficients indicate anti-correlation. If a coefficient is not
mentioned, the correlation or anti-correlation between the variables is less
than $1\%$. $M_{T2}^{Combo}$ and transverse mass of $b$-jet have a large
correlation with the MET for the signal class but not in the background class;
therefore, we keep all of them in our analysis.

(a)

(b)

Figure 5: Linear correlation coefficients (in percentages) between various kinematic variables for the signal (left, for BP2) and corresponding background (right) are displayed. Positive coefficients indicate correlation between variables, while negative coefficients indicate anti-correlation. | $\not{E}_{T}$ | $M_{T2}^{Combo}$ | $M_{T}(b,\not{E}_{T})$ | $\Delta R(l,\not{E}_{T})$ | $\Delta R(j_{1},\not{E}_{T})$ | $\Delta\Phi(j_{1},\not{E}_{T})$ | $\sqrt{\hat{s}_{\text{min}}}$ | $\Delta R(b,l)$ | $M_{T}(l,\not{E}_{T})$ | $\Delta R(j_{2},\not{E}_{T})$ | $N_{\text{jets}}$
---|---|---|---|---|---|---|---|---|---|---|---
BP1 |  11.32 |  13.32 | 5.761 | 8.544 | 5.486 | 4.766 | 6.994 | 5.814 | 3.931 | 3.982 | 3.06
BP2 |  26.86 |  25.21 | 16.40 | 6.412 | 5.897 | 5.427 | 5.134 | 5.104 | 3.854 | 3.666 | 2.811
BP3 |  48.62 |  39.76 | 32.97 | 9.47 | 9.092 | 5.538 | 5.419 | 3.915 | 11.43 | 6.062 | 1.838
BP4 |  59.62 |  46.84 | 42.64 | 10.49 | 10.48 | 5.321 | 8.801 | 3.665 | 18.48 | 6.595 | 1.409
BP5 |  70.16 |  54.57 | 52.99 | 13.99 | 14.20 | 5.429 | 15.87 | 3.379 | 28.32 | 8.949 | 1.508
Table 3: Variable ranking by relative importance, independent of the method
used, among the observables.

(a)

(b)

(c)

Figure 6: The normalized histograms of the combined signal for BP1-BP5 and
combined background (solid black) are shown: (a) missing transverse momentum,
(b) variable $M_{T2}^{Combo}$ as defined in equation 24, and (c) transverse
mass of $b$-jet. Combined indicates the weighted sum of all the processes.

The relative importance of the variables is summarized in table 3. In our
analysis, we separately optimize for all benchmark points. For BP1, all the
variables are almost equally important during MVA. As the mass difference
between the vector-like quark ($\Psi$) and DM ($\Delta M_{\Psi DM}$) increases
towards BP5, the importance of missing transverse momentum, $M_{T2}^{Combo}$,
and the transverse mass of the $b$-jet and lepton increases significantly
compared to the other variables in the same row. Figure 6 depicts the
variation of MET, $M_{T2}^{Combo}$, and transverse mass of the $b$-jet for the
signal (all BPs) and background. With a larger mass gap, the transverse
momentum of missing particles, visible lepton and jets increases as they are
generated from the cascade decay of $\Psi$. A larger mass gap raises MET,
flattening its range and enhancing signal-background separation. A larger mass
gap also increases the transverse momentum of the $b$-jet and lepton, along
with the missing transverse energy, resulting in an increased transverse mass
for the $b$-jet and lepton. This increased transverse mass increases
$M_{T2}^{Combo}$, resulting in a more distinct separation between signal and
background.

(a)

(b)

Figure 7: On the left, we present the normalized distribution of the BDT response for both signal (BP2) and background classes for training and testing samples. On the right, we display the signal (blue) and background (red) efficiencies alongside the statistical significance of the signal (green), plotted against the cut applied to the BDT output. | $N_{S}^{bc}$ |  BD$T_{opt}$ | $N_{S}$ | $N_{B}$ |  $\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}$ ($\frac{N_{S}}{\sqrt{N_{B}}})$, 300 $\text{fb}^{-1}$ | $\frac{N_{S}}{N_{B}}$
---|---|---|---|---|---|---
BP1 | 688 | 0.2696 | 209 | 4490 | 3.05 (3.12) | 0.0465
BP2 | 2653 | 0.4883 | 876 | 2481 | 15.1 (17.6) | 0.353
BP3 | 590 | 0.4559 | 168 | 367 | 7.26 (8.77) | 0.4578
BP4 | 342 | 0.4327 | 126 | 243 | 6.56 (8.08) | 0.5185
BP5 | 35 | 0.6796 | 14 | 75 | 1.48 (1.62) | 0.1867
$N_{SM}$ | 237628 |
Table 4: The table displays the effectiveness of the current search in terms
of statistical significance for signal over background. $N_{S}^{bc}$ and
$N_{SM}$ represent the total number of events for the combined signal and
background before any BDT cuts at a luminosity of 300 $\text{fb}^{-1}$. After
optimal selection on the BDT response (BD$T_{opt}$), the surviving signal and
background events ($N_{S}$ and $N_{B}$) for the 14 TeV LHC are listed. The
statistical significance and the signal-to-background ratio are provided in
the last two columns.

Figure 7a shows the normalized distribution of the BDT response for the signal
(BP2) and background classes, including both training and testing samples. The
distributions for training and testing match well, and there is a clear
separation between the signal and background. In the right panel of figure 7,
we present the variation in signal (blue) and background (red) efficiencies as
well as the statistical significance (green) of the signal compared to the cut
applied on the BDT response.

In table 4, the numbers of signal and background events after applying cuts,
as discussed below in equation 25, are represented by $N_{S}^{bc}$ and
$N_{SM}$, respectively, for an integrated luminosity of 300 $\text{fb}^{-1}$.
$N_{S}$, $N_{B}$ represent the numbers of surviving signal and background
events after applying an optimal cut on the BDT output, BD$T_{opt}$. The
optimal cut maximizing significance. The table also contains the statistical
significance of the signal for different BPs and the signal-to-background
ratio. Importantly, for BP1-BP4, we achieve a statistical significance greater
than 3.0 for an integrated luminosity of 300 $\text{fb}^{-1}$. For example, in
BP4, where the mass of the VLQ is 1.1 TeV, and the mass of the WIMP is 500
GeV, we obtain statistical significance,
$\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}=6.56$.

###  3.6 Exclusion contour

Figure 8: Excluded regions at a $95\%$ CL in the plane of WIMP mass
($M_{\text{DM}}$) and $\frac{\Delta M_{\Psi\text{DM}}}{M_{\text{DM}}}$, where
$\Delta M_{\Psi\text{DM}}=M_{\Psi}-M_{\text{DM}}$, for $\Psi\bar{\Psi}$
production, assuming BR($\Psi\rightarrow tH^{-}$)+BR($\Psi\rightarrow
bH^{0}/A^{0})=1$, and the mass of $\Psi$ is greater than the top-quark mass
($m_{t}$). Models within the blue region are excluded. The y-axis is plotted
on a logarithmic scale. The red dashed line corresponding to $\Delta
M_{\Psi\text{DM}}=m_{t}$ and below this line top is produced in off-shell. The
dashed orange and magenta lines represent how a fixed value of
$M_{\Psi}=700~{}\text{and}~{}600$ GeV looks in this plane. On the other hand,
if $\Psi$ predominantly interacts with the first two generations of the SM
quarks, then the exclusion contour at a $95\%$ CL is shown by the filled green
region from the reinterpreted analysis Giacchino:2015hvk  of the ATLAS result
Marjanovic:2014eca .

We did a detailed collider analysis of this model, and a few representative
benchmark points are outlined in the last section. The exclusion contour with
a $95\%$ confidence level of our search in the final state of an energetic
lepton with multijet where one of the leading two jets is b-tagged and a large
missing transverse energy is shown in figure 8. The exclusion region of our
search is shown by a filled blue region in a plane of the dark matter mass,
$M_{\text{DM}}$ versus $\frac{\Delta M_{\Psi\text{DM}}}{M_{\text{DM}}}$, where
$\Delta M_{\Psi\text{DM}}=M_{\Psi}-M_{\text{DM}}$ is the mass difference
between the vector-like quark ($\Psi$) and DM. The contour is drawn assuming
that $\Psi$ primarily interacts with the third generation of the SM quark,
with BR($\Psi\rightarrow tH^{-}$)+BR($\Psi\rightarrow bH^{0}/A^{0})=1$. Also,
the mass of $\Psi$ is greater than the top-quark mass, ensuring that the top-
quark resulting from the decay of $\Psi$ is on-shell. This contour is plotted
for an integrated luminosity of 300 $\text{fb}^{-1}$ at the 14 TeV LHC. Our
search strategy is model-independent since we consider $\Psi\bar{\Psi}$
production, where the production cross-section entirely depends on the QCD
coupling strength and is independent of any BSM coupling. The y-axis is
plotted on a logarithmic scale. We find that a vast parameter space can be
probed at the LHC with the current luminosity. From this plot, for instance,
if the WIMP mass is 515 GeV (780 GeV), the VLQ mass less than 1.39 TeV (1.19
TeV) can be excluded at $95\%$ CL.

In the same figure, we present the exclusion contour at a $95\%$ CL (filled
green region) if VLQ predominantly interacts with the first two generations of
the SM quarks from the reinterpreted analysis Giacchino:2015hvk  of the ATLAS
result Marjanovic:2014eca .

##  4 Summary and conclusion

The Inert Higgs Doublet Model (IDM) offers a straightforward extension for
Weakly Interacting Massive Particle (WIMP) dark matter, typically requiring an
ad-hoc discrete symmetry to ensure stability. On the other side, Peccei-Quinn
(PQ) symmetry, originally introduced as an elegant solution to the strong CP
problem, also provides another compelling dark matter candidate in the form of
the axion. In this work, we consider a well-motivated, multicomponent WIMP-
axion dark matter model, where the spontaneous breaking of PQ symmetry
naturally stabilises the WIMP dark matter through a residual $Z_{2}$ symmetry.
Their mutual contribution ensures the realisation of the observed DM relic
density and revives the phenomenologically exciting region of the IDM
parameter space where the WIMP mass falls between 100 and 550 GeV. Moreover,
additional fields necessitated by PQ symmetry further enrich the dark sector.
These include a scalar field proprietor for axion DM and a vector-like quark
(VLQ) that acts as a portal between the dark sector and Standard Model
particles via Yukawa interactions.

Traditional searches for this degenerate mass region in the IDM rely on
displaced vertex searches. The WIMP-axion dark matter model facilitates a more
effective, model-independent approach to the Large Hadron Collider (LHC) by
exploring the pair production of VLQs. To enhance accuracy, we incorporate
Next-to-Leading-Order (NLO) QCD corrections. Following production, the
residual $Z_{2}$ symmetry dictates the decay of each VLQ into a top or bottom
quark along with an inert scalar. Employing a leptonic search channel with
relevant observables and employing multivariate analysis techniques, we
demonstrate the ability to exclude a significant portion of the parameter
space using an integrated luminosity of 300 fb-1 at the 14 TeV LHC.

In conclusion, this work highlights the potential of the IDM assisted by PQ
symmetry as a framework for exploring the WIMP-axion dark matter scenario at
the LHC. The proposed search strategy offers a promising avenue for probing
previously unexplored regions of the parameter space.

## Acknowledgements

The computational works are performed using the Param Vikram-1000 High-
Performance Computing Cluster and the TDP project resources of the Physical
Research Laboratory (PRL).

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