aainstitutetext: Physical Research Laboratory, Ahmedabad, 380009, Gujarat,
Indiabbinstitutetext: Indian Institute of Technology, Gandhinagar, 382424,
Gujarat, Indiaccinstitutetext: Department of Physics, Kyungpook National
University, Daegu, 41566, Korea

# Top-philic Dark Matter in a Hybrid KSVZ axion framework

Anupam Ghosh a Partha Konar c Rishav Roshan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore a two-component dark matter scenario in an extended Kim-Shifman-
Vainshtein-Zakharov (KSVZ) axion framework. This hybrid setup incorporates an
extra $SU(2)_{L}$ complex singlet scalar whose lightest component plays the
role of one of the dark matter, while the QCD axion of the KSVZ model acts as
a second dark matter candidate. In this work, we focus on accentuating the
role of vector-like quark that naturally emerges in the KSVZ extension on the
dark matter and collider phenomenology. Here, we demonstrate that the presence
of this colored particle can significantly affect the allowed dark matter
parameter space of the scalar dark matter by opening up additional co-
annihilation as well as the direct detection channels. Moreover, the
interaction between the color particle with the top quark and scalar dark
matter provides a unique topology to generate a boosted-top pair with
considerable missing transverse momentum at the LHC. Using jet substructure
variables and multivariate analysis, here we show that one can already exclude
a vast region of parameter space with 139 $\text{fb}^{-1}$ integrated
luminosity at 14 TeV LHC.

###### Keywords:

Dark Matter, Large Hadron Collider, KSVZ axion

##  1 Introduction

The Standard Model (SM) of fundamental particles is one of the outstanding
achievements of modern-day physics, which has been experimentally verified at
many frontiers. Ever since the discovery of the Higgs boson CMS:2012qbp ;
ATLAS:2012yve , the last missing piece of SM, the particle physics community
is eagerly scrutinizing collider data to witness if nature can offer a glimmer
of hope in the ongoing hunt for new physics. Physics beyond the SM (BSM) is
envisaged as inevitable since SM still fails to account for various issues to
give us a fully coherent description of nature. Some of the principal concerns
are related to the Strong CP problem Peccei:1977hh ; Peccei:1977ur , the
existence of the dark matter (DM) Sofue:2000jx ; Clowe:2006eq  in the
Universe, non-zero but minuscule neutrino masses Super-Kamiokande:1998kpq ;
SNO:2002tuh ; K2K:2002icj , matter-antimatter asymmetry of the Universe
Riotto:1999yt ; Dine:2003ax  etc. The incapability of the SM in explaining
these issues motivates us to look at possible extensions accommodating these
aspects.

Different celestial observational evidence Sofue:2000jx ; Clowe:2006eq  at
diverse length scales suggests the existence of a non-baryonic, non-luminous,
gravitationally interacting form of matter in the Universe, which is popularly
known as the _dark matter_. Precision measurement of anisotropies in cosmic
microwave background radiation (CMBR) in Wilkinson Microwave Anisotropy Probe
(WMAP) Hinshaw:2012aka  and PLANCK Akrami:2018odb  played a crucial role in
providing us with a valuable estimate of the present abundance of DM relic
density. Despite these shreds of evidence in different formats, the particle
nature of the DM is still obscured from us. Different DM paradigms exist in
the literature attempting to explain the DM’s particle nature. Some of the
most popular ones are weakly interacting massive particles (WIMP)
Jungman:1995df ; Bertone:2004pz ; Konar:2009ae ; Feng:2010gw ; Arcadi:2017kky
; PhysRevD.105.115038 ; Konar:2020wvl ; Konar:2020vuu , feebly interacting
massive particle (FIMP) Hall:2009bx ; Bernal:2017kxu ; Barman:2020plp ;
Barman:2021tgt ; Datta:2021elq ; Barman:2021qds ; Konar:2021oye ;
Chakrabarty:2022bcn , asymmetric dark matter models DuttaBanik:2020vfr ;
Barman:2021ost , models with axion or axion-like particle (ALP) DM
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Marsh:2015xka  etc. Although
these models can explain the DM relic density of the Universe, their parameter
spaces are restricted from different experimental searches. For instance, WIMP
dark matter is severely constrained from null detection of the DM in the
collider searches Kahlhoefer:2017dnp ; CMS:2017dcx  as well as the direct
Akerib:2016vxi ; Tan:2016zwf ; Cui:2017nnn ; Aprile:2018dbl  and indirect
Kohri:2009yn ; Ahnen:2016qkx ; Eiteneuer:2017hoh  search experiments.

Interestingly such a null outcome in experimental searches can instead be an
indication of a much richer dark sector, and the dark sector can possibly be
composed of more than one component of DM. In recent times, there have been
several proposals for a multicomponent dark matter Borah:2019aeq ;
Bhattacharya:2019fgs ; Bhattacharya:2019tqq ; DuttaBanik:2020jrj ;
Chakrabarty:2021kmr ; DuttaBanik:2016jzv ; Borah:2019epq ;
Bhattacharya:2021rwh ; Abdallah:2019svm ; Pandey:2020hoq ; Dasgupta:2013cwa ;
Chatterjee:2018mac ; Lazarides:2022spe ; Konar:2009qr  where the dark sector
is made up of different DM components. For example, one can think of a
scenario with WIMP-WIMP Borah:2019aeq ; Bhattacharya:2019fgs ;
Bhattacharya:2019tqq ; DuttaBanik:2020jrj ; Chakrabarty:2021kmr  dark matter
candidates, WIMP-FIMP DuttaBanik:2016jzv ; Borah:2019epq ;
Bhattacharya:2021rwh  dark matter candidates, FIMP-FIMP Abdallah:2019svm ;
Pandey:2020hoq  dark matter candidates, WIMP-axion Dasgupta:2013cwa ;
Chatterjee:2018mac  dark matter candidates, etc. In the present work, we aim
to explore one such scenario where the dark sector constitutes a WIMP-type
dark matter, and the QCD axion plays the role of a second dark matter.

The minimal extension of the SM that can accommodate a WIMP dark matter is its
extension by a scalar singlet field with a Higgs portal interaction
Silveira:1985rk ; McDonald:1993ex ; Burgess:2000yq . Here, the DM is assumed
to have an odd charge under a discrete unbroken $Z_{2}$ symmetry that
guarantees its stability. This particular DM scenario is in tension with
direct search experiments. The recent direct detection data from the XENON1T
experiment Aprile:2018dbl  has nearly ruled out the DM with mass below 1 TeV
Bhattacharya:2019fgs ; DuttaBanik:2020jrj ; Borah:2020nsz  apart from the near
Higgs resonance region. Reviving this sub-TeV parameter space of the scalar
singlet DM could be an exciting undertaking since it can open the possibility
of its testing at different frontiers like direct, indirect, and collider
searches. One possible way to revive this sub-TeV parameter space is to embed
this scalar DM in a two-component framework Bhattacharya:2019fgs ;
DuttaBanik:2020jrj . With various extensions of two-component DM with a scalar
singlet already in place, an interesting exercise could explore the insertion
of a scalar singlet in a QCD axion DM framework, as it can enlarge the scalar
singlet DM parameter space and can also provide rich collider phenomenology.

Extension of the SM with a global Peccei-Quinn (PQ) symmetry Peccei:1977hh ;
Peccei:1977ur  provides solutions for two of the critical issues discussed
above in one go, $i.e.$, the _Strong CP problem_ and the existence of _dark
matter_. This global symmetry is expected to be broken at a scale much larger
than the Electroweak (EW) scale. The breaking of $U(1)_{PQ}$ predicts a
pseudo-Goldstone particle, popularly known as the _QCD axion_ , that is not
absolutely stable but can have a lifetime much greater than the age of the
Universe Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Marsh:2015xka  to
play the role of DM. There are primarily three different QCD axion models that
can simultaneously explain the presence of the DM in the Universe and solve
the Strong CP problem. The (i) Peccei-Quinn-Weinberg-Wilczek (PQWW)
Peccei:1977hh ; Wilczek:1977pj ; Weinberg:1977ma  model introduces an
additional singlet scalar that also obtains a non-zero vacuum expectation
value (vev) at the time of EW phase transition. This setup is already ruled
out from the experiments. The (ii) Kim-Shifman-Vainshtein-Zakharov (KSVZ)
Kim:1979if ; Shifman:1979if  model introduces an extra colored particle
together with a complex scalar that breaks the PQ symmetry. Anomaly-free
condition is ensured by the introduction of vector-like quarks (VLQ). Finally,
the (iii) Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) Dine:1981rt ;
Zhitnitsky:1980tq  model incorporates an additional Higgs field apart from the
PQ breaking scalar. It is also interesting to point out that the breaking of
PQ symmetry in these models also leaves a remnant $Z_{2}$ symmetry that
remains intact. If such a setup is extended with an extra particle that also
carries a non-trivial $Z_{2}$, then this unbroken symmetry can naturally
ensure its stability. This motivates us to study two-component DM scenarios in
these models.

In the present work, we aim to explore a hybrid KSVZ scenario, where an extra
complex scalar singlet extends the particle spectrum of the KSVZ setup in
addition to the usual complex scalar that breaks the PQ symmetry. As mentioned
above, the breaking of $U(1)_{PQ}$ symmetry in the KSVZ construction leaves
remnant $Z_{2}$ symmetry which remains unbroken throughout, and the VLQ
present in the setup carries an odd charge under it. If the model is extended
by a complex scalar that also holds a non-trivial charge under the same
$Z_{2}$, the lightest component of the scalar can play the role of the second
dark matter, making the dark sector two-component.

Note that our setup is similar to the one considered in Dasgupta:2013cwa ;
Chatterjee:2018mac  but with some crucial differences. For example, the
present framework considers an up-type VLQ rather than a down-type VLQ, and it
opens up a non-trivial possibility in collider analysis. Current construction
also aims to explore the possibility of DM effective annihilations (including
the DM co-annihilation with the vector-like quark as well as the annihilation
of the vector-like quark to the SM particles) to the SM particles in obtaining
the correct relic density of the scalar (WIMP) DM. Next, both Dasgupta:2013cwa
; Chatterjee:2018mac  did not consider the VLQ’s contribution to the DM-
nucleon spin-independent scattering, which can play a crucial role in the DM
phenomenology of WIMP dark matter. This feature of DM effective annihilation
to the SM particles, as well as the role of the VLQ in the tree level spin-
independent DM-nucleon scattering, was also discussed in Borah:2020nsz .
Ref.Borah:2020nsz  featured a VLQ doublet and VLQ singlet fermions, unlike
ours with a singlet VLQ which naturally occurs in a KSVZ scenario. We also
want to emphasize that, although Dasgupta:2013cwa ; Chatterjee:2018mac ;
Borah:2020nsz  featured some discussions on the collider aspect of this kind
of setup, none of them explored it in detail. In the present work, we also
perform an exhaustive colliders analysis and aim to find out the relevant
parameter space, which would not only validate the dark matter phenomenology
but also be consistent with the collider searches.

The paper is organized as follows. We introduce our model in section 2 where
the particle spectrum together with their charges under different symmetry
groups have been discussed. Various theoretical and experimental constraints
in our model are presented in section 3. In section 4 we discuss the dark
matter phenomenology of the model. The collider analysis and the result based
on multivariate analysis are presented in section 5. Finally, we summarize our
findings in section 6.

##  2 The Model

As stated in the introduction, the present work aims to study dark matter and
collider phenomenology in a hybrid KSVZ framework of QCD axion. As is well
known, the vanilla KSVZ model requires a complex scalar singlet $\eta$ that
breaks a global symmetry, popularly known as $U(1)_{\text{PQ}}$. In addition,
this model also demands a $SU(2)_{L}$ singlet colored fermion $\Psi$ with a
$+1$ unit of $U(1)_{\text{PQ}}$ charge. This extra quark is vector-like and
hence does not introduce any chiral anomaly. In addition, the hybrid KSVZ
model also introduces an additional complex singlet scalar $S$ charged under
the $U(1)_{\text{PQ}}$. The BSM fermion and scalar content of the model and
their respective charges are listed in Tab. 1. The most general renormalizable
and gauge-invariant Lagrangian for the present setup can be written as,

$\displaystyle-\mathcal{L}^{\text{VLQ}}$ $\displaystyle=$ $\displaystyle
f_{i}S\overline{\Psi}_{L}{u_{i}}_{R}+f_{\Psi}\eta\overline{\Psi}_{L}\Psi_{R}+h.c.,$
(1)

where, $u_{R}$ represents right-handed up-type quarks in the SM with
$i=u,c,t$. Here, $L$ and $R$ denote left- and right-handed projections. Note
that the hypercharge of the newly introduced VLQ depends on its interaction
with SM quarks. The relevance of introducing an up-type VLQ in this setup will
be clear once we discuss the DM and collider phenomenologies in sections 4 and
5 respectively.

| $\eta$ | $S$ | $\Psi$
---|---|---|---
$SU(3)_{C}$ | 1 | 1 | 3
$SU(2)_{L}$ | 1 | 1 | 1
$U(1)_{Y}$ | 0 | 0 | 2/3
$U(1)_{PQ}$ | 2 | 1 | 1
Table 1: Particle contents and their respective charge assignments under
different symmetry groups.

Moving on to the scalar part of the Lagrangian, the most general
renormalizable scalar potential of our model, $V(H,\eta,S)$ can be written as,

$\displaystyle V(H,\eta,S)$ $\displaystyle=$
$\displaystyle\lambda_{H}(|H|^{2}-v_{H}^{2}/2)^{2}+\lambda_{\eta}(|\eta|^{2}-F_{a}^{2}/2)^{2}+\lambda_{\eta
H}(|H|^{2}-v_{H}^{2}/2)(|\eta|^{2}-F_{a}^{2}/2)$ (2) $\displaystyle+$
$\displaystyle\mu_{S}^{2}|S|^{2}+\lambda_{S}|S|^{4}+\lambda_{SH}|H|^{2}|S|^{2}+\lambda_{S\eta}|\eta|^{2}|S|^{2}+[\epsilon_{S}\eta^{*}S^{2}+h.c].$

After the breaking of both $U(1)_{\text{PQ}}$ and the SM gauge symmetry, the
different scalars involved in the present setup take the following form,

$\displaystyle H=\left(\begin{array}[]{c}0\\\
\frac{1}{\sqrt{2}}(v_{H}+h_{0})\end{array}\right)\,,\,\,\,\eta=e^{\frac{ia}{F_{a}}}\frac{(F_{a}+\sigma_{0})}{\sqrt{2}}\,,\,\,\,S=\frac{S_{1}+iS_{2}}{\sqrt{2}}\,,$
(5)

where $v_{H}$ denotes the vacuum expectation value (vev) of $H$ obtained after
the electroweak symmetry breaking (EWSB) and $F_{a}$ represents the
$U(1)_{\text{PQ}}$ breaking scale. It is to be noted that, after the breaking
of both symmetries, a non-zero $h_{0}-\sigma_{0}$ mixing leads to the
following mass terms:

$M^{2}\equiv\begin{pmatrix}2v_{H}^{2}\lambda_{H}&F_{a}v_{H}\lambda_{\eta H}\\\
F_{a}v_{H}\lambda_{\eta H}&2F_{a}^{2}\lambda_{\eta}\end{pmatrix}.$ (6)

The mass matrix can be diagonalised using

$\begin{pmatrix}h_{0}\\\
\sigma_{0}\end{pmatrix}=\begin{pmatrix}\cos\theta_{m}&\sin\theta_{m}\\\
-\sin\theta_{m}&\cos\theta_{m}\end{pmatrix}\begin{pmatrix}h\\\
\sigma\end{pmatrix}$ (7)

where the mixing angle is given by,

$\tan(2\theta_{m})=\dfrac{F_{a}v\lambda_{\eta
H}}{F_{a}^{2}\lambda_{\eta}-v^{2}\lambda_{H}}\,\,.$ (8)

Finally, after diagonalization, the physical masses of the $h$ and $\sigma$
are given as,

$\displaystyle
M_{h,\sigma}^{2}=(\lambda_{H}v^{2}+\lambda_{\eta}F_{a}^{2})\pm\sqrt{(\lambda_{H}v^{2}-\lambda_{\eta}F_{a}^{2})^{2}+F_{a}^{2}v^{2}\lambda_{\eta
H}^{2}}\,\,.$ (9)

Next, as an artifact of two different symmetry breakings, the masses of the
different components of the $S$ can be expressed as,

$\displaystyle M_{S_{1,2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2\mu_{S}^{2}+v_{H}^{2}\lambda_{SH}+F_{a}^{2}\lambda_{S\eta}\mp
2\sqrt{2}\epsilon_{s}F_{a}).$ (10)

Notice that the presence of the term proportional to $\epsilon_{S}$ in Eq. 2
plays a crucial role in generating the mass splitting among the components of
$S$. Subsequently, the mass of the VLQ is given as,

$\displaystyle M_{\Psi}$ $\displaystyle=$ $\displaystyle
f_{\Psi}\frac{F_{a}}{\sqrt{2}}.$ (11)

At this stage, it is interesting to point out that, even after the breaking of
both the symmetries, there still exists a remnant $Z_{2}$ symmetry under which
both the Lagrangian as well as the scalar potential remains invariant. This
remnant $Z_{2}$ can remain intact if $S$ does not acquire a non-zero vev.
Under such a scenario, the lightest neutral component of $S$ can provide a
vital DM candidate.

Finally, the setup also contains a pseudo-Nambu Goldstone boson $a$,
associated with scalar $\eta$, popularly known as _axion_. The axion obtains a
mass as a result of non-perturbative QCD effects given as Preskill:1982cy ;
Dine:1982ah ,

$\displaystyle m_{a}\simeq
0.6~{}\text{meV}\times\bigg{(}\frac{10^{10}~{}\text{GeV}}{F_{a}}\bigg{)}.$
(12)

Note that a suitable choice of decay constant $F_{a}$ can adjust the fraction
of which QCD axion can contribute toward the relic density of the dark matter.
That makes the preset setup a tunable two-component dark matter scenario. The
role of QCD axion as a DM candidate and its constraints are elaborated in
section 4. Now with the knowledge of all the particles and their interactions
in this hybrid KSVZ setup, we are in a position to list the set of independent
parameters important for the dark matter and collider phenomenology:

$\\{M_{\Psi},M_{S_{1}},M_{S_{2}},M_{\sigma},F_{a},\lambda_{SH},\lambda_{S\eta},f_{i}\\}.$

##  3 Experimental and Theoretical Constraints

The extended KSVZ model under consideration is subjected to various
theoretical as well as experimental constraints. In this section, we summarize
all the relevant ones.

  * •

Stability and Perturbativity: The scalar sector is extended over the vanilla
model. Hence, different scalars in the present setup can help stabilize the
electroweak vacuum. The stability of the electroweak vacuum also demands that
the scalar potential should be bounded from below in all the field directions
of the field space. On the other hand, a perturbative theory demands that the
model parameters should obey:

$\displaystyle|\lambda_{i}|<4\pi~{}{\rm{and}}~{}|g_{i}|,|y|,|f_{i}|<\sqrt{4\pi}.$
(13)

where ${g}_{i}$ and ${y}$ are the SM gauge and Yukawa couplings, whereas
$f_{i}$ are Yukawa couplings involving different BSM fields, respectively.

  * •

Relic density, Direct and Indirect detection of DM: For any dark matter model,
it is essential to satisfy the observed abundance of DM relics from the
precision measurement in the Planck experiment Aghanim:2018eyx ,

$\displaystyle\Omega_{\text{DM}}h^{2}$ $\displaystyle=$ $\displaystyle
0.120\pm 0.001.$ (14)

Apart from DM relic density, the DM-nucleon scattering cross-section is also
constrained by various direct search experiments like LUX Akerib:2016vxi ,
PandaX-II Tan:2016zwf ; Cui:2017nnn , and XEXON1T Aprile:2018dbl . Finally,
the DM annihilation to the SM particles are also subjected to the constraints
coming from the indirect search experiments like PAMELA Kohri:2009yn , Fermi-
LAT Eiteneuer:2017hoh , MAGIC Ahnen:2016qkx  etc. Nonetheless, in all these
cases, one also needs to take care of the multi-component nature of DM in our
extended scenario, which is further discussed in section 4.

  * •

Flavor constraints: The Yukawa interactions of the complex singlet scalar $S$
with VLQ and the SM right-handed quarks like $u$ and $c$ in the present setup
can contribute towards the $D^{0}-\bar{D}^{0}$ mixing Garny:2014waa . The
measured value of the $D-$meson mass splitting significantly constrained this
mixing. The Feynmann diagrams that contribute to this mixing are shown in Fig.
15; each diagram has four possible configurations with a total of sixteen
diagrams. Effective operator contributing to this mixing in the present setup
can be expressed as

$\displaystyle\mathcal{L_{\text{eff}}}=\frac{\tilde{z}}{M_{\Psi}^{2}}\bar{u}_{R}^{\alpha}\gamma^{\mu}c_{R}^{\alpha}\bar{u}_{R}^{\beta}\gamma_{\mu}c_{R}^{\beta}.$
(15)

where

$\displaystyle\tilde{z}=-\frac{f_{u}^{2}f_{c}^{2}}{96\pi^{2}}[g_{\psi}(M_{S_{1}}^{2}/M_{\Psi}^{2})+g_{\psi}(M_{S_{2}}^{2}/M_{\Psi}^{2})-2g_{\psi}(M_{S_{1}}M_{S_{2}}/M_{\Psi}^{2})].$
(16)

Here $g_{\Psi}(x)=24xf_{6}(x)+12\tilde{f_{6}}(x)$ where the expressions of
$f_{6}$ and $\tilde{f_{6}}$ can be found in Gedalia:2009kh . The measurement
of the $D-$meson mass splitting demands $|\tilde{z}|\lesssim 5.7\times
10^{-7}(M_{\Psi}/\text{TeV})^{2}$ Gedalia:2009kh ; Garny:2014waa

  * •

LHC diphoton searches: As a result of mixing between $h$ and $\sigma$, all the
tree level interactions with the SM Higgs get modified. In such a case, the
signal strength in the di-photon channel takes a form:

$\displaystyle\mu_{\gamma\gamma}=c^{2}_{\theta}\frac{BR_{h\to\gamma\gamma}}{BR_{h\to\gamma\gamma}^{\text{SM}}}\simeq
c^{2}_{\theta}\frac{\Gamma_{h\to\gamma\gamma}}{\Gamma_{h\to\gamma\gamma}^{\text{SM}}}.$
(17)

LHC sets a limit on this new mixing angle as $|\sin{\theta}|\leq 0.36$
Robens:2016xkb .

  * •

Invisible Higgs decay: Involvement of the new interactions of SM Higgs with
various BSM particles in the present setup can lead to its new decay modes if
kinematically allowed. These extra decays of Higgs can contribute toward
invisible Higgs decay. In such a situation, we need to employ the bound on the
invisible Higgs decay width as ATLAS-CONF-2020-052 :

$\displaystyle
Br(h\rightarrow\rm{Invisible})\equiv\frac{\Gamma(h\rightarrow\rm{Invisible})}{\Gamma(h\rightarrow
SM)+\Gamma(h\rightarrow\rm{Invisible})}<0.11.$ (18a)

In the case of light DM, the Higgs can decay to a pair of it when
kinematically allowed. However, in our present analysis, we primarily focus on
the parameter space where $m_{i}>~{}\frac{m_{h}}{2}$ so the above constraint
is not applicable.

  * •

Direct collider constraints: Due to the presence of colored vector-like
quarks, the present model is subjected to various collider constraints. Being
non-trivially charged under the $U(1)_{PQ}$ allows the VLQ to couple with the
complex scalar and the SM up type quarks. If kinematically allowed, the
heavier states can always decay into the DM and an SM quark. Therefore a
generic collider signature of this model contains a considerable amount of
missing (transverse) energy from the escape of final DM particles from
detection at the detector.

Vector-like fermion can be pair produced through electroweak interaction
performed at CERN’s Large Electron Positron Collider (LEP):

$e^{+}e^{-}\rightarrow\gamma^{*},Z\rightarrow\Psi\bar{\Psi}$ (19)

The interaction between vector-like fermion, light SM quarks, and the DM can
lead to the decay of $\Psi$ to a light quark associated with DM at LEP if
kinematically allowed. The reinterpreted LEPII results of squark search
Giacchino:2015hvk ; OPAL:2002bdl  exclude the mass of $\Psi$ up to 100 GeV.
Such constraint is incorporated in our final exclusion plots. Please follow
the brown region in Fig. 10. Similar searches were also carried out at the
LHC. In a recent ATLAS search, the vector-like mediator is searched while it
decays into an invisible particle and light quark up (charm) when the mass
difference between the mediator and DM is less than the top quark mass. The
green region in Fig. 10 is excluded from the reinterpreted result
Giacchino:2015hvk  of the ATLAS search Marjanovic:2014eca  for multijet (2-6
jets) plus missing transverse momentum at center-of-mass (CM) energy
$\sqrt{s}=8$ TeV with an integrated luminosity of 20.3 $\text{fb}^{-1}$.
Exploring a larger mass difference between the mediator and DM candidate, top-
antitop plus missing transverse momentum signal has been extensively studied
by both CMS and ATLAS collaborations, particularly superpartners searches of
the top quark ATLAS:2017eoo ; ATLAS:2017drc ; ATLAS:2017avc ; ATLAS:2017www ;
ATLAS:2017tmw ; CMS:2017okm ; CMS:2017mbm ; CMS:2017gbz ; CMS:2017jrd  and
some dedicated dark matter searches CMS:2017dcx . The vector-like mediator can
be pair produced at the LHC mainly through strong interaction and then decay
into an up-type quark and invisible particle. So, the search for a top pair
along with the missing transverse momentum signature by ATLAS and CMS can be
reinterpreted to exclude some of the parameter spaces of this model. The CMS
analysis CMS:2017jrd  is reinterpreted in Ref Colucci:2018vxz  at 13 TeV LHC
for an integrated luminosity of 35.9 $\text{fb}^{-1}$, assuming vector-like
mediator decay with $100\%$ branching fraction into the top and invisible
particle. In their analysis, the signal consists of two oppositely charged
isolated leptons from leptonic decays of both top and anti-top. The signal
also consists of at least two hard jets; one of them is b-tagged and a large
missing transverse momentum. The olive region in Fig. 10 is the exclusion
region ($2\sigma$) obtained from this analysis.

The existing LHC search relies on finding the top pair based on two hard
leptons and a b-tagged jet. It is evident that the sensitivity of such
detection deteriorates when these tops are boosted, especially while decaying
from a heavy mother particle. We propose an alternative search strategy in
this work by recognizing these boosted double top jets with a large missing
transverse energy signature using jet substructure variables and multivariate
analysis. We are examining the spectrum where the mass difference between
vector-like mediator and DM is larger than the mass of the top quark such that
on-shell decay into the top is possible. Our search strategy helps to explore
the significant parameter space of this model that gives observed relic
density of DM and also allowed from the direct-detection experiment with the
current luminosity of the LHC.

##  4 Dark Matter Phenomenology

In this section, we aim to elaborate on the DM phenomenology of the model
under consideration. As discussed earlier, the setup is a hybrid of the KSVZ
model that includes an extra complex scalar ($S$) whose lightest component
($S_{1}$) plays the role of one of the DM while the part of the second DM is
played by the QCD axion of the KSVZ setup. The involvement of the two DMs in
this extended KSVZ scenario makes the layout a two-component DM system.
Besides the QCD axion, the KSVZ setup naturally demands a presence of an extra
colored fermionic $SU(2)_{L}$ singlet. This fermion plays a non-trivial role
in the DM phenomenology and the collider searches of the DM as it talks
directly to it through the Yukawa interaction given in Eq. 1. Next, we discuss
the DM phenomenology of both the DM candidates of the present model.

###  4.1 Relic density and DM detection

Apart from providing a solution to the _strong CP problem_ , another
interesting consequence of introducing a PQ symmetry is the emergence of the
Nambu Goldstone boson, popularly known as _axion_. If the breaking scale
($F_{a}$) of the PQ symmetry is chosen appropriately, the resulting axion can
be light as well as stable. This QCD axion can be an excellent DM candidate in
such a scenario. Axions can be produced non-thermally as a result of the
misalignment mechanism. Here, the axion field begins to coherently oscillate
around the minimum of the PQ vacuum when its mass becomes comparable to the
Hubble parameter. This coherent oscillation of the axion field behaves like a
cold matter in the Universe. The relic density of the axion in such a case is
approximately given by Dasgupta:2013cwa ; Chatterjee:2018mac ; Bae:2008ue ,

$\Omega_{a}h^{2}\simeq 0.18\hskip 2.84526pt\theta_{a}^{2}\hskip
2.84526pt\bigg{(}\frac{F_{a}}{10^{12}\text{GeV}}\bigg{)}^{1.19}\,.$ (20)

Here, $\theta_{a}$ represents the initial misalignment angle of the axion.

For the case of the scalar DM, we consider the mass hierarchy
$M_{S_{2}}>M_{S_{1}}$ such that the lightest scalar component represents the
second DM candidate. Its interactions with the SM Higgs and the VLQ keep it in
equilibrium with the thermal bath in the early Universe. As the temperature of
the Universe drops below the DM mass, its production from the thermal bath
stops while its annihilation of the SM particle continues. Once the Universe’s
expansion rate becomes larger than the interaction rate of the DM, its
annihilation to the SM bath also stops, and its abundance freezes out. DM can
annihilate to the SM particles through: (a) its contact interactions, (b)
Higgs-mediated channels 111From Eq. 9, it is evident that until and unless
$\lambda_{\eta}$ is very small, $M_{\sigma}$ will remain much heavier than
$M_{h}$ and hence the $\sigma$ mediated annihilation channels be very much
suppressed. and (c) VLQ mediated channels (as a result of Yukawa interaction
given in Eq. 1). The presence of the Yukawa interaction also allows the DM to
co-annihilate if the mass-splitting between the DM and newly introduced quark
is sufficiently small. Note that as the VLQ and $S_{2}$ share the same $Z_{2}$
charge similar to the DM, their annihilations would also be important for
evaluating the effective annihilation cross-section. In appendix A, we present
all the important annihilation and co-annihilation channels of the DM that are
crucial in determining its final relic abundance 222Just for completeness we
have also shown the DM annihilation to the axion final states. These
annihilations are highly suppressed and do not contribute towards the relic
density of scalar dark matter. This is because most of the vertices involved
in these annihilation cross-sections are either proportional to $1/F_{a}$ or
$\frac{\sin{\theta}}{F_{a}}$ or $\frac{\epsilon_{S}}{F_{a}}$ or
$\epsilon_{S}$. . Once all the important annihilation and co-annihilation
channels are identified, one can use them to determine the final relic density
of the DM, which can be expressed as McDonald:1993ex ,

$\displaystyle\Omega_{S_{1}}h^{2}$ $\displaystyle=$
$\displaystyle\frac{1.09\times
10^{9}~{}\rm{GeV^{-1}}}{g_{*}^{1/2}~{}M_{Pl}}\frac{1}{J(x_{f})},$ (21)

where $J(x_{f})$ is given by,

$\displaystyle J(x_{f})$ $\displaystyle=$
$\displaystyle\int_{x_{f}}^{\infty}\frac{\langle\sigma|v|\rangle_{\rm
eff}}{x^{2}}~{}\rm{dx}.$ (22)

$\langle\sigma|v|\rangle_{\rm eff}$ in Eq. (22) is the effective thermal
average DM annihilation cross-sections including contributions from the co-
annihilations and is given by,

$\displaystyle\langle\sigma|v|\rangle_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{g_{s_{1}}^{2}}{g_{\rm
eff}^{2}}\sigma(\overline{S_{1}}S_{1})+2\frac{g_{s_{1}}g_{s_{2}}}{g_{\rm
eff}^{2}}\sigma(\overline{S_{1}}S_{2})(1+\Delta_{12})^{3/2}~{}\text{exp}[-x\Delta_{12}]+2\frac{g_{s_{1}}g_{\Psi}}{g_{\rm
eff}^{2}}\sigma(\overline{S_{1}}\Psi)(1+\Delta_{\Psi})^{3/2}$ (23)
$\displaystyle~{}\text{exp}[-x\Delta_{\Psi}]+\frac{g_{S_{2}}^{2}}{g_{\rm
eff}^{2}}\sigma(\overline{S_{2}}S_{2})(1+\Delta_{12})^{3}~{}\text{exp}[-2x\Delta_{12}]+\frac{g_{\Psi}^{2}}{g_{\rm
eff}^{2}}\sigma(\overline{\Psi}\Psi)(1+\Delta_{\Psi})^{3}$
$\displaystyle~{}\text{exp}[-2x\Delta_{\Psi}].$

In the equation above, $g_{s_{1}}$, $g_{s_{2}}$ and $g_{\Psi}$ are the spin
degrees of freedom for $S_{1}$, $S_{2}$ and $\Psi$. Here,
$x=\frac{M_{S_{1}}}{T}$ representing dimensionless parameter with inverse of
temperature, while $\Delta_{\Psi}$ and $\Delta_{12}$ are two dimensionless
parameters qualifying mass splittings from dark scalar candidate:

$\displaystyle\Delta_{\Psi}=\frac{M_{\Psi}-M_{S_{1}}}{M_{S_{1}}};\;\Delta_{12}=\frac{M_{S_{2}}-M_{S_{1}}}{M_{S_{1}}}.$
(24)

The effective degrees of freedom in Eq. (23) is given by,

$\displaystyle g_{\text{eff}}$ $\displaystyle=$ $\displaystyle
g_{s_{1}}+g_{s_{2}}(1+\Delta_{12})^{3/2}~{}\text{exp}[-x\Delta_{12}]+g_{\Psi}(1+\Delta_{\Psi})^{3/2}~{}\text{exp}[-x\Delta_{\Psi}].$
(25)

In the following analysis, we first generate the model using FeynRules
Alloul:2013bka  and then implement it in micrOMEGAs -v5 Belanger:2018ccd  to
find the region of parameter space that corresponds to correct relic abundance
for our scalar DM candidate in accordance with the relation,

$\displaystyle\Omega_{\text{T}}h^{2}$ $\displaystyle=$
$\displaystyle\Omega_{a}h^{2}+\Omega_{S_{1}}h^{2},$ (26)

where $\Omega_{\text{T}}h^{2}$ corresponds to the total relic density of the
DM satisfying PLANCK constraints Aghanim:2018eyx .

Next, the present model is subjected to the constraints coming from the direct
search experiments for the dark matter. Experiments like LUX Akerib:2016vxi ,
PandaX-II Tan:2016zwf ; Cui:2017nnn  looks for the DM recoil in the DM-nucleon
scattering and subsequently provides a bound on the DM-nucleon scattering
cross-section. Being a two-component DM system, the direct detection cross-
section of the scalar DM should be rescaled as,

$\displaystyle\sigma_{S_{1,\text{eff}}}^{\text{SI}}=\frac{\Omega_{S_{1}}}{\Omega_{\text{T}}}\sigma_{S_{1}}^{\text{SI}}$
(27)

As mentioned earlier, due to the direct Yukawa interaction of the scalar DM
with the up-quark, two other scattering processes contribute to the direct
detection cross-section of the scalar apart from the usual SM Higgs-mediated
scattering. In appendix A we listed all the scattering processes of the DM
$S_{1}$ with the detector nucleon.

Finally, the model is also subjected to the constraints coming from the
indirect search experiments. Indirect search experiments looking for an excess
of gamma rays can help in probing the WIMP dark matter. DM particles can
annihilate and produce SM particles, out of which photons (and also
neutrinos), being electromagnetically neutral, have better chances of reaching
the detector from the source without getting deflected. Experiments like
PAMELA Kohri:2009yn , Fermi-LAT Eiteneuer:2017hoh , MAGIC Ahnen:2016qkx  etc.
look for such excess in order to confirm the particle nature of the DM. The
present set up being a two-component DM scenario, the indirect detection
cross-section of the scalar DM should be rescaled as well,

$\displaystyle\sigma_{S_{1,\text{eff}}}^{\text{ID}}=\bigg{(}\frac{\Omega_{S_{1}}}{\Omega_{\text{T}}}\bigg{)}^{2}\sigma_{S_{1}}^{\text{ID}}.$
(28)

At this stage, it is worth commenting on some of the detection possibilities
of the axion as a DM candidate. Several ongoing and proposed experiments rely
on axion being a DM. All these experiments lean on different detection
techniques. For example, ADMX ADMX:2009iij  searches for DM-photon conversion
in the presence of the magnetic field. CASPEr Budker:2013hfa  uses nuclear
magnetic resonance to hunt for the axion DM; it is known that if the axion
exists, it will modify Maxwell’s equation. ABRACADABRA Kahn:2016aff  utilizes
this by using a toroidal magnet to source an effective electric current, and
finally, MADMAX Caldwell:2016dcw  is a proposed experiment that uses
dielectrics haloscopes.

###  4.2 Parameter Space of Hybrid KSVZ Axion Framework

Figure 1: Variation of QCD axion relic density with the decay constant $F_{a}$
for three different values of misalignment angles: $\theta_{a}=1.0$ (solid),
$\theta_{a}=0.1$ (dashed), and $\theta_{a}=0.01$ (dotted). Black thick dashed
line corresponds to observed relic $\Omega_{\text{DM}}h^{2}=0.12$. The cyan
region is disallowed from the Supernova cooling data. The light pink region
corresponds to the parameter space where the DM relic density remains
overabundant.

It is well known that the KSVZ model provides a DM in the form of QCD axion.
For this axion to play the role of the DM or contributes sufficiently towards
the relic density of the DM, the decay constant $F_{a}$ should lie in the
range,

$\displaystyle 10^{10}~{}\text{GeV}\leq F_{a}\leq 10^{12}~{}\text{GeV}.$ (29)

The lower bound on $F_{a}$ comes from the supernova cooling data
Raffelt:1987yt  whereas the upper bound results from the overproduction of the
axion or, in other words, the relic density of the axion become overabundant.
To understand this, in Fig. 1 we study the variation of the axion relic
density ($\Omega_{a}h^{2}$) with the decay constant for three different values
of the misalignment angles $i.e.$ $\theta_{a}=1.0$ (solid), $\theta_{a}=0.1$
(dashed), and $\theta_{a}=0.01$ (dotted). The region in cyan is ruled out from
the supernova cooling data, whereas the light pink region corresponds to the
overproduced DM relic density. As can be seen from Fig. 1, for
$\theta_{a}=1.0$ and $F_{a}\simeq 10^{12}~{}\text{GeV}$, the axion alone can
contribute $100\%$ towards the relic density of the dark matter. Finally, the
white region corresponds to the parameter space where QCD axion as a DM
remains underabundant.

The present setup is an extended version of the KSVZ scenario, which contains
an additional DM candidate as a singlet scalar. The presence of this extra DM
here demands us to choose a parameter space for axion from Fig. 1 where the
relic density of the axion remains underabundant so that the relic density of
the axion together with the scalar can satisfy the Planck limit. For a
demonstrative purpose, we fix $F_{a}=10^{11}$ GeV and choose the misalignment
angle as $\theta_{a}=1$ for the rest of the analysis. This choice of $F_{a}$
and $\theta_{a}$ corresponds to $\Omega_{a}h^{2}=0.012$. Without losing
generality in our analysis, we set a heavier $M_{\sigma}$ at 50 TeV (as the
setup requires it to be quite heavy). At this stage, we would like to point
out that the DM matter couples to $\sigma$ through $S_{1}-S_{1}-\sigma$
interaction. This interaction can also help DM to annihilate into the SM
particles through scalar mixing. Although these annihilations will have
suppression coming from the mass of $\sigma$, they might still not be that
small as these annihilations are also proportional to the
$F_{a}\lambda_{S\eta}$. With $F_{a}=10^{11}~{}\text{GeV}$ and not so small
value of $\lambda_{S\eta}$, the DM can still have significant annihilation
cross-sections and such cross-sections might violate perturbative unitarity
Profumo:2019ujg . This demands $\lambda_{S\eta}$ to be extremely tiny. For
simplicity, we set $\lambda_{S\eta}=0$ throughout our analysis. Next, for the
analysis purpose, we also define a mass-splitting, $\Delta
M=M_{S_{2}}-M_{S_{1}}$ and consider it to be a free parameter rather than
$M_{S_{2}}$. It is interesting to point out that once $\Delta M$ and $F_{a}$
are fixed, the parameter $\epsilon_{S}$ automatically gets fixed, as can be
seen from Eq. 10. Before diving into the detailed analysis of the second DM
candidate, we will like to mention the set of parameters that are relevant for
the analysis of the DM phenomenology of the second DM candidate:

$\\{M_{\Psi},M_{S_{1}},\Delta M,F_{a},\lambda_{SH},f_{i}\\}.$

Figure 2: Variation of $\Omega_{S_{1}}h^{2}$ (left panel) and
$\sigma_{S_{1,\text{eff}}}^{\text{SI}}$ (right panel) versus dark matter mass
$M_{S_{1}}$. In all the plots we fix
$F_{a}=10^{11}~{}\text{GeV},~{}\lambda_{SH}=0.01$. The Black dashed line in
all the left plots corresponds to $0.120-\Omega_{a}h^{2}$.

Figure 3: Variation of effective indirect detection cross-section versus DM
mass. Variation for different values of $f_{t}$, the mass of VLQ, $f_{u,c}$,
and $\Delta M$ are shown in top left, top right, bottom left and bottom right
panels respectively. In all the plots we fix
$F_{a}=10^{11}~{}\text{GeV},~{}\lambda_{SH}=0.01$. The solid black line shows
the experimental upper bound in $t\bar{t}$ final state.

To demonstrate the above discussions, we display the variation of the relic
density of $S_{1}$ with its mass in all the left panel plots of Fig. 2. In the
right panel, we also exhibit the variation of the effective direct detection
cross-section with $M_{S_{1}}$ for different choices of parameters. In the top
left panel of Fig. 2, we project the importance of the Yukawa coupling $f_{t}$
while choosing fixed values of $\Delta M=100$ GeV, $M_{\Psi}=500$ GeV and
$\lambda_{SH}=0.01$. It is interesting to point out that for
$\lambda_{SH}=0.01$, the DM does not satisfy the correct relic density in a
pure scalar singlet DM scenario. In these plots, we also set both Yukawa
couplings $f_{u}=f_{c}=0.01$ to highlight the importance of the top Yukawa
coupling $f_{t}$ for three values of $f_{t}$: 0.1 (blue), 0.5 (red) and 1.0
(green). Notice that for $f_{t}=0.1$, with the increase in the DM mass, we
first observe a resonance dip at $M_{S_{1}}=M_{h}/2$ 333In a lower DM mass
regime, DM annihilating to the three body final states $q\bar{q}g$ can also
contribute significantly towards the relic density, where chirality
suppression in the lower order process is lifted by final state radiation. We
do not consider this three-body final state in our analysis as the entire low
mass regime of the DM is already ruled out from the DD searches, as can be
seen from the right panels of Fig. 2. , next, a fall is observed at
$M_{S_{1}}=80$ GeV where the annihilation of the DM to the $W^{\pm}$ boson
opens up after which the relic density increases with the increase in the DM
mass ($\langle\sigma v\rangle\propto 1/M_{S_{1}}$) and again drops at
$M_{S_{1}}=125$ GeV when the DM starts annihilating into the Higgs boson 444We
would also like to point out that for DM mass $M_{S_{1}}<M_{t}$, DM
annihilations to $gg$ (via a box diagram) Deppisch:2018eth ; Baek:2016lnv  or
three-body final states like $tWb$ Deppisch:2018eth  can also contribute
towards the relic density of the DM for a significantly large Yukawa coupling
$f_{t}$. We do not consider these processes in our analysis as we found that
these processes remain suppressed for the choice of $f_{t}$ we are interested
in for our analysis. . Finally, at a larger value of $M_{S_{1}}$
($M_{S_{1}}=345$ GeV), when the mass difference between $M_{\Psi}$ and
$M_{S_{1}}$ becomes relatively small, and the effect of DM co-annihilation
with the VLQ comes into the picture, and a sharp drop in DM relic density is
observed. In this region, although the DM co-annihilates with $M_{\Psi}$, the
annihilation of $\Psi$ with $\bar{\Psi}$ to the gluons dominates the effective
annihilation cross-section (see Eq. 23) of the DM. Due to these co-
annihilations, the relic density dark matter finally satisfies the condition
$0.120-\Omega_{a}h^{2}$ (denoted by the black dashed line) at $M_{S_{1}}=444$
GeV. Further increasing $f_{t}$ to the higher values like 0.5 and 1.0, one
notices that the DM annihilations to top quarks mediated by $M_{\Psi}$ (see
appendix A) start to dominate once the threshold of $M_{S_{1}}=M_{top}$ is
crossed and the relic density can also satisfy the condition
$0.120-\Omega_{a}h^{2}$ near about $M_{S_{1}}\simeq 200$ GeV for $f_{t}=1.0$.
This figure illustrates the importance of VLQ in the DM phenomenology of the
present setup as it makes huge parameter space allowed from the relic density,
which was originally disallowed in the pure scalar singlet DM scenario.

In the top-right panel, we plot the spin-independent effective direct
detection cross-section of $S_{1}$ with the DM mass and compare it with the
experimental results. Notice that only the coupling $f_{u}$ enters the direct
detection cross-section of DM apart from the Higgs-portal coupling
$\lambda_{SH}$. Note that the reduced values in the effective direct detection
cross-section are the result of the rescaling in the two-component DM system
(see Eq. 28). Additionally, Higgs resonance dip at $M_{S_{1}}=M_{h}/2$ and
rise at $M_{S_{1}}\simeq 500$ GeV because of the interference among the direct
detection diagrams (see appendix A). Here, one notices that the near resonance
region remains discarded from the experimental bounds. Still, the other
regions where relic density can be satisfied remain allowed from the direct
detection searches.

In the left panel of the second row in Fig. 2 we fix $f_{t}=1.0$ and then
study the effect of varying $M_{\Psi}$ in $\Omega_{S_{1}}h^{2}-M_{S_{1}}$
plane. As expected, the final fall in the relic density pattern happens at
three different positions corresponding to the three different values of
$M_{\Psi}$. With the heavier propagator mass $i.e.~{}M_{\Psi}=600$ GeV, the
effective annihilation cross-section of the DM to the top quark remains
smaller in comparison to what is observed for $M_{\Psi}=400$ GeV and hence
relatively larger relic density is observed for $M_{\Psi}=600$ GeV than for
$M_{\Psi}=400$ GeV. Similarly, in the left panel of the third row, we depict
the effect of varying $f_{u}~{}\text{and}~{}f_{c}$. For simplicity, here we
also assume $f_{u}~{}=~{}f_{c}$. As expected, for a large value of $f_{u,c}$
the annihilation of DM to top and up (charm) quark final state also becomes
dominant the moment the threshold $2M_{S_{1}}=M_{top}+M_{u(c)}$ is achieved.
This leads to an increase in the DM annihilation cross-section, and
consequently, a decrease in the relic density is observed. Next, in the left
panel of the fourth row we show the effect of varying $\Delta M$ on the relic
density of the DM. As expected, a smaller $\Delta M$ results in a larger
effective DM annihilation cross-section and hence a smaller relic, so in order
to satisfy the correct relic density a heavier DM mass is required. On the
other hand, a larger $\Delta M$ requires a smaller DM mass to satisfy the
observed relic density and hence the plot shifts towards the lower DM mass.
Finally, the middle and bottom right panel of Fig. 2 can be followed from the
one observed in the top right panel.

As can be seen from Fig. 2, the correct relic density is mostly satisfied in
the parameter space where $M_{s_{1}}>M_{t}$. Hence, one needs to check the
prospects of indirect detection of DM in our model specifically focusing on
$t\bar{t}$ final states from the DM annihilations. We display our findings in
Fig. 3 where we plot the effective indirect detection cross-section with DM
mass and show the experimental upper bound (black solid line) of DM
annihilating to $t\bar{t}$ final states that can be obtained from antiproton
cosmic ray data Colucci:2018vxz . In the top left panel of Fig. 3, we find
that the DM mass for which the observed relic density satisfied (shown by
green $\star$) in the top left panel of Fig. 2 is also allowed from the
constraints coming from the indirect search bound. A similar situation is also
observed in the top right and bottom left panels. On the other hand, in the
bottom right panel where the variation with $\Delta M$ is studied, it found
that a larger $\Delta M\sim 100$ GeV is prefered if one also considers the
constraints coming from indirect search experiments. For this reason, we fix
$\Delta M=100$ GeV throughout our analysis.

Figure 4: Parameter space satisfying observed DM abundance and also allowed by
the direct search experiments in the bi-dimensional plane of $\frac{\Delta
M_{\Psi S_{1}}}{M_{S_{1}}}-M_{S_{1}}$, where the colour coding is done with
respect to: Left Panel: the Yukawa couplings $f_{u}=f_{c}$ and Right Panel:
the Yukawa coupling $f_{t}$. In both the plots we fix
$F_{a}=10^{11}~{}\text{GeV},~{}\lambda_{SH}=0.01,~{}\Delta M=100$ GeV while we
vary $f_{t}$ in the range 0 - 1.5 and $f_{u}=f_{c}$ in the range 0 - 1.5.

In Fig. 4, we show the parameter space that remains consistent with the DM
constraints and is also allowed by the constraints that come from the flavor
observable like $D^{0}-\bar{D}^{0}$ mixings in the bi-dimensional plane of
$\Delta M_{\Psi S_{1}}/M_{S_{1}}~{}Vs~{}M_{S_{1}}$, where $\Delta M_{\Psi
S_{1}}$ is the mass difference between VLQ and DM, $M_{\Psi}-M_{S_{1}}$. Here,
the dependence upon different Yukawa couplings ($f_{u}=f_{c}$ in the left
panel and $f_{t}$ in the right panel of the figure) is spotted with a
continuous color map. Two discrete narrow slices at the top-left corner due to
Higgs resonance. We are primarily interested in the non-resonant continuous
region extended over vast parameter space. At the lower $M_{S_{1}}$ end, this
continuous region opens up when the DM pair annihilate into a top quark and up
(charm) quark (see appendix A). Eventually, for a choice of heavier mass, the
DM pair starts annihilating into the top pair.

This allowed region can be categorized into two distinct parts as upper and
lower regions separated by a line where the mass difference between VLQ and DM
equates to the top mass. Hence the upper region can be probed at the collider
with on-shell production of top quark from VLQ decay, while the lower region
is sensitive to a probe with light quark search. We will further demonstrate
in the next section how top quark searches from the boosted top jet can
improve the search strategy in this region.

As a consequence of the narrow mass gap between scalar DM and VLQ, co-
annihilation takes a leading role in most parts of the lower region. Precisely
because of the same reason, this region is also susceptible to the direct
detection probe. Variations of color contours for different $f_{u}$ values are
evident in the lower region of the left plot in Fig. 4. This reflects the
gradually larger parameter space excluded due to direct detection constrain
for a choice of larger $f_{u}$ values. The lower value of mediator mass
increases the direct detection cross-section. In order to keep this cross-
section below the current direct detection bounds, a smaller $f_{u}$ is
required. On the contrary, the direct search experiments allow the upper
region irrespective of the choice of $f_{u}$, and hence a uniform distribution
of the colors is observed. This is because even for a large $f_{u}$, the DM-
nucleon scattering cross-section still remains small due to the presence of a
heavier mediator $i.e~{}M_{\Psi}$.

At the right part of the same plot, one finds that with an increasing DM mass,
a relatively smaller $\Delta M_{\Psi S_{1}}/M_{S_{1}}$ is required in order to
satisfy the correct relic density, while the interplay between the DM mass,
mediator’s mass and the $f_{u}$ makes these points allowed from the direct
detection constraints. In this region, the DM dominantly annihilates into the
top-quark pair and sub-dominantly into the top quark and up (charm) quark
final states. Next, in the right panel of Fig. 4, we show the color coding
with respect to $f_{t}$ in order to highlight its significance. One observes
the correct relic density in the top-left region of the plot due to the
involvement of a large $f_{t}$ as is also evident from the top left panel of
Fig. 2. As expected for a lighter mediator mass, a relatively smaller $f_{t}$
is required to satisfy the correct relic density, as is also observed while
moving downward in the plot. Finally, the role of $f_{t}$ becomes more
prominent once the $M_{S_{1}}=M_{top}$ threshold is opened, as can also be
seen from the right side of the plot.

##  5 Collider Analysis and Results

The involvement of VLQ ($\Psi$) in the present setup opens up interesting
collider prospects as they can be produced either in pair or associated with a
scalar in the proton-proton collision at LHC. Among these production channels,
$pp\rightarrow\Psi\bar{\Psi}$ and
$pp\rightarrow\overset{{$(\mkern-1.0mu-\mkern-1.0mu)$}}{\Psi}S_{1,2}$, the
cross-section of the second process strongly depends on the Yukawa coupling
$f_{u,c}$, while the pair of VLQs is produced primarily by the strong
interaction and hence model-independent. Once produced, the VLQ can decay
preferably into scalar DM candidate or its heavier pair along with one of the
up-type quarks as allowed kinematically. Hence primary LHC searches rely on
identifying such quark jets along with missing transverse energy (MET or
$\not{E}_{T}$) from DM production, as discussed in section 3.

It is noteworthy that a substantial parameter space exists in this model where
the mass difference between VLQ and DM is significantly larger than the top
quark mass while providing correct dark matter relic density and also allowed
from the direct detection experiments. Here produced top quarks are expected
to be fairly boosted by production from the decay of heavy mother (VLQ)
particles. Such a prospect motivates us to look at this signal possessing a
unique topology where hadronic decay of the top retains its collimated
structure as a boosted fatjet 555We encountered a similar feature in the
succession of different BSM scenarios Das:2017gke ; Bhardwaj:2018lma ;
Bhardwaj:2019mts ; Bhardwaj:2020llc , where boosted fatjet is probed in
association with MET. Fatjets, in these searches, still harbor the intrinsic
footprint of their root and manifest such features inside the jet
substructure. Exploring this can provide additional tools to deal with a
significant background involving QCD jets. and is identified as a top-like-
fatjet ($J_{t}$).

Figure 5: Representative Feynman diagrams for leading order partonic processes
contributing VLQ pair production $pp\rightarrow\Psi\bar{\Psi}$ at LHC.

To probe these regions at the LHC, we consider pair production of VLQs and
each of those further decay into the top quark associated with the scalar (DM
or $S_{2}$). Here we adopt a significantly smaller Yukawas $f_{u}=f_{c}$
($=0.01$) so that the primary branching fraction of the decay of $\Psi$ into
the top quark is close to $100\%$. The signal topology is below, where we
identify two final top fatjets associated with significant missing transverse
momentum from dark matter.

$pp\rightarrow\Psi\bar{\Psi}\rightarrow(t,S_{1,2}),(\bar{t},S_{1,2})\equiv
2J_{t}+\not{E}_{T}$ (30)

Note, $S_{2}$ can decay through two-body $(S_{2}\rightarrow S_{1}a)$, three-
body $(S_{2}\rightarrow S_{1}\hskip 1.42262ptj\hskip 1.42262ptj)$, and four-
body $(S_{2}\rightarrow S_{1}\hskip 1.42262ptj\hskip 1.42262ptb\hskip
1.42262ptW)$ decay modes, where suppressed multibody decay occurs through off-
shell VLQ. Partonic level Feynman diagrams of the production of VLQ pair are
shown in Fig. 5. Although the main contribution comes from the strong
interaction, we keep all the diagrams for completeness. Few representative
benchmark points (BPs) are listed in Tab. 2; those provide observed relic
density of DM and allowed from the direct and indirect detection experiments
along the constraints coming from the theoretical and LHC data as listed in
section 3. Also, the production cross-section of the partonic process
$pp\rightarrow\Psi\bar{\Psi}$ at LO for different benchmark points before
decaying into SM quark and scalar at 14 TeV LHC is shown at the last column
Tab. 2. For our analysis, we have used an NLO QCD $K-$factor of 1.33 for the
$pp\rightarrow\Psi\bar{\Psi}$ production 666 We estimate an approximate NLO
(QCD) $K-$factor for the process $pp\rightarrow\Psi\bar{\Psi}$ by replacing
$\Psi$ with the top quark of mass $m_{\Psi}$ at the MadGraph5_aMC@NLO and took
the most conservative value over this mass range..

Benchmark | $M_{S_{1}}$ | $\Delta M_{\Psi S_{1}}$ | $\Delta M$ | $f_{t}$ | $\Omega_{S_{1}}h^{2}$ | $\sigma^{SI}_{S_{1},eff}$ | $\sigma(pp\rightarrow\Psi\bar{\Psi})$
---|---|---|---|---|---|---|---
points | (GeV) | (GeV) | (GeV) |  |  | (pb) | (fb)
BP1 | 301 | 305 | 100 | 0.8 | 0.108 | $9.24\times 10^{-12}$ | 966
BP2 | 302 | 475 | 100 | 1 | 0.104 | $9.96\times 10^{-12}$ | 223
BP3 | 403 | 405 | 100 | 1 | 0.109 | $5.77\times 10^{-12}$ | 175
BP4 | 358 | 448 | 100 | 1 | 0.109 | $6.48\times 10^{-12}$ | 177
BP5 | 433 | 364 | 100 | 1 | 0.107 | $4.6\times 10^{-12}$ | 188
BP6 | 459 | 326 | 100 | 1 | 0.109 | $4.2\times 10^{-12}$ | 208
BP7 | 494 | 273 | 100 | 1 | 0.107 | $3.79\times 10^{-12}$ | 239
BP8 | 510 | 238 | 100 | 1 | 0.099 | $3.69\times 10^{-12}$ | 278
BP9 | 527 | 224 | 200 | 1 | 0.103 | $3.56\times 10^{-12}$ | 272
BP10 | 542 | 188 | 100 | 0.98 | 0.106 | $3.57\times 10^{-12}$ | 321
BP11 | 678 | 349 | 100 | 1.3 | 0.109 | $2.14\times 10^{-12}$ | 37
Table 2: Different benchmark points satisfy the observed relic density of DM,
direct and indirect detection (not shown in the table) bounds, along with the
constraints coming from the theoretical and LHC data, as listed in the text.
$M_{S_{1}}$ is the mass of the DM, $S_{1}$. $f_{t}$ is the coupling strength
of the interaction between top quark, VLQ, and the scalar $S$ (see: Eq. 1).
$\Delta M_{\Psi S_{1}}=M_{\Psi}-M_{S_{1}}$ and $\Delta M=M_{S_{2}}-M_{S_{1}}$.
$\Omega_{S_{1}}h^{2}$ (see: Eq. 26) and $\sigma^{SI}_{S_{1},eff}$ (see: Eq.
28) are the relic density of DM, $S_{1}$ and effective direct detection cross-
section, respectively. Other parameters are
$F_{a}=10^{11}~{}\text{GeV},~{}\lambda_{SH}=0.01,~{}\text{and}~{}f_{u}=f_{c}=0.01$.
The production cross-section of the partonic process
$pp\rightarrow\Psi\bar{\Psi}$ at LO for different benchmark points before
decaying into SM quark and scalar at 14 TeV LHC is given at the last column.

###  5.1 Simulation Details with Signal and Backgrounds

In preparation for our investigation of this Hybrid KSVZ framework through VLQ
pair production at the LHC, we require a realistic setup to simulate both the
signal processes as well as a careful selection of background processes that
can mimic the signal.

We implement this Hybrid KSVZ framework in FeynRules Alloul:2013bka  to
generate the UFO model file required for matrix element generation for Monte-
Carlo event generator. Parton level events are generated in the
MadGraph5_aMC@NLO environment Alwall:2014hca  and further pass through Pythia8
Sjostrand:2001yu ; Sjostrand:2014zea  for showering, fragmentation and
hadronization. Background events are generated along with two to four
additional jets MLM matching Mangano:2006rw ; Hoeche:2005vzu  with virtually-
ordered Pythia showers to avoid any double counting. We include higher-order
corrections for different processes by multiplying the appropriate $K$ factor.
An in-built NN23LO1 pdf set is adopted for the parton distribution functions
(PDF), and a default dynamical factorization scale is used for events
generation. The showered events are further passed through Delphes3
deFavereau:2013fsa  to include detector effects with the default CMS card.
Jets ($j$) of radius parameter 0.5 are constructed with the
$\mbox{anti-k}_{T}$ Cacciari:2008gp  clustering algorithm, where we used the
particle-flow towers and particle-flow tracks as input. We implement the
Cambridge-Achen (CA) Dokshitzer:1997in  algorithm to construct large radius
fatjets - $J$. Fastjet 3.2.2 Cacciari:2011ma  is used for clustering fatjets
of radius parameter $R=1.5$. A boosted top gives a fatjet whose radius
parameter is approximately govorned by $R\sim 2m_{t}/P_{T}$, where $m_{t}$
($P_{T}$) is the mass (transverse momentum) of the top quark. Hence, the
minimum transverse momentum required by each top to form such a fatjet is
$P_{T}\gtrsim 200$ GeV. Finaly, we implement the adaptive Boosted Decision
Tree (BDT) algorithm to perform the multivariate analysis (MVA) in the TMVA
Hocker:2007ht  framework.

Our analysis considers all the backgrounds that significantly contribute to
the two boosted top fatjets with large missing transverse momentum, as listed
below.

$t\bar{t}+$ jets: Top pair production with the semi-leptonic top decays is the
most dominant background for our signal process. Although pure hadronic decay
of tops can offer two boosted top jets, the requirement of a considerable
amount of missing energy reduces this background by a significant factor of
$~{}100$, where mismeasurement of hadronic activities acts as a source of MET.
In the semileptonic decay, one top decay hadronically and is reconstructed as
boosted top jet, and the other top decay leptonically gives a significant
amount of missing energy when the lepton escapes detection. Other boosted jet
comes from the QCD radiation. This background is matched with the MLM matching
scheme up to two extra jets.

QCD background: QCD background is enormous at the LHC but can be reduced to a
negligible contribution (see, for example, Ref.Plehn:2010st ). Even after the
requirements of two boosted fatjets, we are left with a remarkably large
number of events from this background. We further require at least one b tag
within the leading or sub-leading fatjet. Contribute negligibly after
additional suppression of 100 comes from fake MET from hardons and another 50
from the requirement of $b$ tag fatjet. We do not include this background in
our analysis.

$tW+$ jets: Single top production associated with W boson significantly
contributes to the SM background. The top is reconstructed as the boosted top
where the b quark is tagged within it, and the W boson decays leptonically to
give rise to missing transverse momentum. In contrast, another boosted fatjet
arises from QCD jets. MLM matching up to two extra jets is done for this
process.

$V+$ jets: (Simi-)invisible decay of W/Z vector boson in addition to QCD
radiation that emulates the fatjet can contribute sizably even with a
requirement of sizeable reconstructed mass of the fatjet. We do MLM matching
up to four extra jets for both processes. A generation level cut
$\not{E}_{T}>100$ GeV is applied for both processes to obtain statistically
significant background events.

di-boson + jets: Minor contribution can come from Di-boson + jets. We retain
all the three di-boson background processes ($pp\rightarrow WZ,\hskip
2.84526ptWW,\hskip 2.84526ptZZ$) in our analysis. Among the three, WZ + jets
contribute the most. All three processes are matched up to two extra jets with
an MLM matching scheme. In all the cases, one of the vector boson decay
invisibly $(Z\rightarrow\nu\nu)$ or leptonically $(W\rightarrow l\nu)$ to give
$\not{E}_{T}$. One of the boosted fatjet comes from QCD jets, and another
fatjet comes from either hadronically decaying vector boson or the QCD jets.

$t\bar{t}V$: Such processes have three body phase spaces and have less cross-
section than other background processes mentioned above. Both the tops can be
reconstructed as boosted fatjets, while $\not{E}_{T}$ comes from the invisible
or leptonically decay of the Z and W boson, respectively. Among these two,
$t\bar{t}Z$ contributes the most because of the larger cross-section and more
significant efficiency when applying $\not{E}_{T}$.

We consider all contributions generating those events at leading order and
normalize with the NLO (QCD) cross-section. Higher-order QCD corrected
production cross-section at the 14 TeV LHC for different background processes
accounted in this analysis are listed in Tab. 3.

Background |  $\sigma$ (pb)
---|---
top pair Muselli:2015kba  |  $t\bar{t}+$ jets | 988.57 [$N^{3}$LO]
single top Kidonakis:2015nna  | $tW$ | 83.1 [$N^{2}$LO]
mono-V boson Catani:2009sm ; Balossini:2009sa  |  $Z+$ jets |  $6.33\times 10^{4}$ [$N^{2}$LO]
$W+$ jets |  $1.95\times 10^{5}$ [NLO]
di boson Campbell:2011bn  |  $ZZ+$ jets | 17.72 [NLO]
$WW+$ jets | 124.31 [NLO]
$WZ+$ jets | 51.82 [NLO]
mono-V + $t\bar{t}$ | $t\bar{t}Z$ | 0.911 [NLO]
| $t\bar{t}W^{\pm}$ | 0.636 [NLO]
Table 3: Higher-order QCD corrected cross-section at the 14 TeV LHC of
different background processes considered in our study. The order of QCD
correction is given in brackets. For the final process, higher-order QCD
corrected cross-section in five massless quark flavors at 14 TeV LHC obtained
from MG5_aMC@NLO. Default factorization and renormalization scales and an in-
built NN23NLO pdf set are used.

###  5.2 Construction of High-Level Variables and Cut-Based Analysis

Once we have generated our signal and background processes after the realistic
detector-level simulation, the next task is constructing high-level event
variables sensitive to kinematic configuration signal and background
processes. For example, the unique point of this collider study counts on the
fatjet characteristic and its different properties related to the mass-energy
distribution within these fatjets. We categorize some of the useful variables
for our analysis in the following bulleted points:

N-subjettiness ratio: In the case of a highly boosted top quark, one can
capture all three hadronically decayed constituents of the top quark within a
single large-radius jet (fatjet). The whole energy of a reconstructed top-
fatjet is distributed around three subjet axes. Assuming N number of subjets
belong to the fatjet, N-subjettiness is defined by the angular distance in the
transverse plane of constituents of the fatjet from the nearest subjet axis
and weighted by the transverse momentum of the constituents as below
(Thaler:2010tr, ; Thaler:2011gf, ):

$\tau_{N}=\dfrac{1}{\mathcal{N}_{0}}\,\sum_{i}\,P_{T,i}\,\text{min}\\{\Delta
R_{i,1},\Delta R_{i,2},...,\Delta R_{i,N}\\}.$ (31)

Here, the summation goes over all the particles inside the jet. The
denominator is $\mathcal{N}_{0}=\sum_{i}\,P_{T,i}\,R$, where $P_{T,i}$ and $R$
are the transverse momentum of the i-th constituent and radius of the jet,
respectively. Since N-subjettiness determines the jet shape, the
N-subjettiness ratios, such as $\tau_{31}$ and $\tau_{32}$ are good
observables in signal background analysis. $\tau_{32}$ effectively
distinguishes the top signal from two-prong fatjets arising from the boosted W
or Z boson in the background. In contrast, $\tau_{31}$ is also effective for
separating the top signal from the one-prong QCD fatjets that contribute
significantly to the background.

Pruned jet mass: Jet-mass is a good variable for classifying a boosted top-
fatjet from the two-prong fatjets from the boosted W/Z boson or one-prong QCD
fatjets. The jet mass, $M_{J}=(\sum_{i\in J}P_{i})^{2}$, where four-momentum
of i-th energy hit in the calorimeter is $P_{i}$. Since large radius jets pick
additional soft contributions from underlying QCD radiations, we must remove
these soft and wide-angle radiations for more realistic predictions. Different
jet grooming techniques, pruning, filtering, and trimming Krohn:2009th ;
Butterworth:2008iy ; Ellis:2009su ; Ellis:2009me  are available to remove
those softer and wider angle radiations while we consider pruning in our
analysis. In the first step of pruning, we define fatjet using the CA
algorithm, and in the second step, we pruned its constituents in each
recombination step.

$Z=\text{min}(P_{Ti},P_{Tj})/P_{T(i+j)}<Z_{\text{cut}}\hskip
25.60747pt\text{and}\hskip 25.60747pt\Delta R_{ij}>R_{\text{fact}}\hskip
2.84526pt.$ (32)

The merging $i,j\rightarrow J$ is vetoed when both the conditions are
satisfied. Pruning is parametrized by two parameters, the softness parameter,
$Z$, and the angular distance of the constituents, $\Delta R_{ij}$. We chose
$Z_{cut}=0.1$ Ellis:2009su  and $R_{\text{fact}}=0.86\hskip 2.84526pt(\sim
m_{t}/P_{T,\text{top}})$ Ellis:2009me  in our analysis.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 6: Distributions of different kinematical variables for the signal
(BP3) and all the backgrounds contributing to the fatjets $+\not{E}_{T}$ final
state after imposing the b tag within leading or subleading fatjet,
$\not{E}_{T}>150$ GeV, and the primary event selection criteria (described in
the text) for 14 TeV LHC. The normalized distribution for the signal is given
by the solid red line. The events of each background process have been
weighted by their cross-section and the cut efficiency after applying the
previously mentioned cuts. Each background process is then normalized to the
sum of individual cross-section times cut efficiency. Colors show the
contribution of the individual background process.

Primary event selection criteria : Based on our previous discussion and
construction of high-level variables, we identify two large-radius jets,
leptons, and missing transverse energy as per the following event selection
criteria both for the signal and background events alike:

  1. 1.

Each event should contain at least two fatjets constructed by CA algorithm
with radius parameter $R=1.5$, and each of them has transverse momentum,
$P_{T}(J_{0}),P_{T}(J_{1})>200$ GeV. Here, $J_{0}$ and $J_{1}$ represent the
leading and subleading fatjet.

  2. 2.

Each event is selected with a minimum missing transverse energy
$\not{E}_{T}>100$ GeV.

  3. 3.

Since our signal does not contain lepton, we veto any event if it contains any
lepton with transverse momentum, $P_{T}(l)>10$ GeV within pseudorapidity
$|\eta(l)|<2.4$.

  4. 4.

To minimize jet mismeasurement contribution to $\not{E}_{T}$, we keep an
azimuthal separation between each fatjet and $\not{E}_{T}$,
$|\Delta\Phi(J_{0,1},\not{E}_{T})|>0.2$.

The normalized distribution of different observables of a sample signal
benchmark point, BP3, and bin-wise stacked histogram of all the backgrounds
are shown in Fig. 6. These plots are shown after demanding at least one the b
tag within leading or subleading fatjet, enhanced $\not{E}_{T}>150$ GeV, over
the preselection cuts already described for 14 TeV LHC.

The prime background $t\bar{t}+$ jets, where one of the top decay hadronically
and the other decays leptonically, is shown by the top most blue shade, while
the solid red line indicates the sample signal, BP3. The distributions of the
pruned jet mass of the leading ($M_{J_{0}}$) and subleading ($M_{J_{1}}$)
fatjets are given in Fig. 6a and Fig. 6b, respectively. At LO, $\Psi$ and
$\bar{\Psi}$ produce back to back, each one followed by decay into an
(anti)top quark and $S_{1,2}$. In most events in this benchmark point, these
tops are boosted as they are produced from the decay of heavy particles. When
the top is sufficiently boosted, all three constituents of the top quark fall
within a single large-radius jet, giving a three-prong jet substructure and
pruned jet mass very close to the top quark mass. For the signal, we get a
sharp peak around the top quark mass for both the leading and subleading
fatjet. These large radius jets sometimes misses some of the constituent sub-
jets, especially when the boost of the top quark is relatively low, causing a
secondary peak near the W/Z boson mass for both the fatjets of the signal. For
semi-leptonic $t\bar{t}+$ jets background, the top which decays hadronically
gives the leading fatjet for a significant number of events and causes a sharp
peak near top mass in the leading fatjet mass distribution. From the demand
for a very high missing transverse momentum, $t\bar{t}+$ background
contributes to a phase space region where the b-jet from the leptonically
decaying top quark generates the subleading fatjet predominantly.
Consequently, subleading fatjet mass generates its peak near 20 GeV from QCD
radiation.

| BP3 |  $t\bar{t}$+jets |  $tW$+jets | $ttZ$ | $ttW$ |  $Z$+jets |  $W$+jets |  $WZ$+j |  $ZZ$+j |  $WW$+j
---|---|---|---|---|---|---|---|---|---|---
C1 | 5969 | $9.6\times 10^{4}$ | $5.1\times 10^{4}$ | 1048 | 111 | $3.5\times 10^{5}$ | $1.9\times 10^{5}$ | $1.3\times 10^{4}$ | $1.6\times 10^{3}$ | $3.6\times 10^{3}$
[$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$] |  [$100\%$]
C2 | 5296 | $4.2\times 10^{4}$ | $2.12\times 10^{4}$ | 793 | 64 | $2.28\times 10^{5}$ | $1.06\times 10^{5}$ | $8.11\times 10^{3}$ | 969.2 | $1.6\times 10^{3}$
[$88.73\%$] |  [$43.89\%$] |  [$41.96\%$] |  [$75.71\%$] |  [$57.53\%$] |  [$65.06\%$] |  [$53.78\%$] |  [$64.34\%$] |  [$61.73\%$] |  [$43.97\%$]
C3 | 4424 | $3.21\times 10^{4}$ | $1.59\times 10^{4}$ | 656 | 54.1 | $3.36\times 10^{4}$ | $1.64\times 10^{4}$ | $1.5\times 10^{3}$ | 267 | 341.1
[$74.11\%$] |  [$33.60\%$] |  [$31.42\%$] |  [$62.63\%$] |  [$48.73\%$] |  [$9.57\%$] |  [$8.32\%$] |  [$11.89\%$] |  [$17.0\%$] |  [$9.37\%$]
C4 | 1005 | $4.02\times 10^{3}$ | $1.72\times 10^{3}$ | 185 | 16.7 | $1.54\times 10^{3}$ | 926 | 72 | 10.4 | 26
[$16.85\%$] |  [$4.20\%$] |  [$3.39\%$] |  [$17.66\%$] |  [$15.07\%$] |  [$0.44\%$] |  [$0.47\%$] |  [$0.57\%$] |  [$0.66\%$] |  [$0.71\%$]
C5 | 666 | $2.46\times 10^{3}$ | $1.07\times 10^{3}$ | 132.5 | 12 | 842 | 493 | 42.5 | 7.1 | 15.7
[$11.16\%$] |  [$2.57\%$] |  [$2.12\%$] |  [$12.64\%$] |  [$10.84\%$] |  [$0.24\%$] |  [$0.25\%$] |  [$0.337\%$] |  [$0.45\%$] |  [$0.43\%$]
C6 | 432 | 411 | 197 | 54 | 3.1 | 260 | 132 | 17.5 | 4.3 | 1.7
[$7.24\%$] |  [$0.43\%$] |  [$0.39\%$] |  [$5.12\%$] |  [$2.78\%$] |  [$0.074\%$] |  [$0.067\%$] |  [$0.139\%$] |  [$0.272\%$] |  [$0.047\%$]
Table 4: The cut efficiency and expected number of events after the
corresponding cuts for the signal and all the backgrounds contribute to the
fatjets $+\not{E}_{T}$ final state at the 14 TeV LHC and 139 $\text{fb}^{-1}$
integrated luminosity. The effectiveness of different selection cuts can be
followed in the form a cut flow from top to bottom after applying (C1)
Preselection cuts, (C2) $\not{E}_{T}>150$ GeV, (C3) requiring at least one
b-tag within $J_{0}$ or $J_{1}$, (C4) 120 GeV$<M_{J_{0}},M_{J_{1}}<230$ GeV,
(C5) $\tau_{31}(J_{0}),\tau_{31}(J_{1})<0.4$ and finally, (C6) $M_{T_{2}}>320$
GeV. A sample benchmark point, BP3, is presented in this table.

The total missing transverse energy distribution is shown as another
interesting variable in Fig. 6j. In the case of signal, we have two missing DM
particles coming from the decay of $\Psi$ pair, where they primarily produce
back to back, so the $\not{E}_{T}$ has uniform distribution as two missing
particles can avail entire phase space. In contrast, the background drops
sharply for large $\not{E}_{T}$. Distributions of the azimuthal separation of
the leading and subleading fatjets from the $\not{E}_{T}$ are presented in
Fig. 6d and Fig. 6e, respectively. As stated earlier, two missing particles
can avail the entire phase space for the signal, so both
$\Delta\Phi(MET,J_{0,1})$ have a uniform distribution. For a significant
amount of events of the $t\bar{t}+$ jets background, the b-jet from the
leptonically decaying top quark behaves as a subleading fatjet ($J_{1}$), and
the neutrino gives the $\not{E}_{T}$, where we select the events that have
large $\not{E}_{T}$. Hence, the azimuthal separation of $J_{1}$ from
$\not{E}_{T}$ gets a maximum at a lower value. In contrast, the azimuthal
separation of the leading fatjet ($J_{0}$) from $\not{E}_{T}$ peaks near
$\sim\pi\,\text{rad}$. The distribution of $\Delta R(J_{0},J_{1})$, angular
distance between $J_{0}$ and $J_{1}$ in the transverse plane is given in Fig.
6c.

The distribution of the kinematic variable $\tau_{31}$ for both leading and
subleading fatjet are shown in Fig. 6f, Fig. 6g, respectively. In both
distributions, as expected, the signal has a peak for a smaller value of
$\tau_{31}$ representing that signal fatjets have a three-prong structure.
Similarly, the distribution of the kinematic variable $\tau_{32}$, which
separates the three-prong fatjet from the two-prong fatjet, are presented in
Fig. 6h, Fig. 6i. $\tau_{32}(J_{0})$ has a peak near 0.6 and 0.75 for signal
and background, respectively. Note that we do not apply any mass window in
generating these distributions, but if we do, the peaks of $\tau_{32}$ move
towards a lower value. So, in the final event selection in the cut-based
analysis, we apply a mass window to discriminate the signal from the
background better.

The distribution of kinematic stransverse mass variable $M_{T2}$ Lester:1999tx
; Barr:2011xt ; Konar:2009wn  is given in Fig. 6k. Assuming DM mass is unknown
to us, we construct $M_{T2}$ after setting trial DM mass as zero in this
construction. SM particles have a smaller mass compared to the mass of $\Psi$,
so the $M_{T2}$ distribution of signal and background are well separated.
Since we do not want to find the correct mass of the mother particle ($\Psi$),
this variable is used to discriminate the signal from the background
efficiently. The distribution of $M_{\text{eff}}$ is given in Fig. 6l.
Effective mass is defined as

$M_{\text{eff}}=\not{E}_{T}+H_{T},$ (33)

where $H_{T}\equiv\sum\limits_{i=1}^{N_{J}}P_{iT}$ ($N_{J}$ is the number of
visible jets) is the scalar sum of the transverse momentum of the jets. The
above distributions show that all the variables are very good at
distinguishing the signal from the background.

We apply the following selection cuts to demonstrate a cut-based analysis
(CBA) over the preselection cuts (described before) to increase the signal-to-
background ratio. Note that our final results are based on sophisticated
multivariate analysis with improved statistics. So, the next part is for
demonstration purposes without putting much effort into optimizing all the
selection criteria. Here, we offer a cut-flow in cut-based analysis to better
understand the signal and background differential distributions.

BP | BP1 | BP2 | BP3 | BP4 | BP5 | BP6 | BP7 | BP8 | BP9 | BP10 | BP11
---|---|---|---|---|---|---|---|---|---|---|---
$\sigma$ | 11.4 | 12.8 | 11.1 | 9.9 | 6.9 | 5.3 | 4.0 | 3.6 | 2.5 | 1.2 | 2.8
$\frac{S}{B}$ | 0.41 | 0.47 | 0.40 | 0.35 | 0.23 | 0.17 | 0.13 | 0.12 | 0.08 | 0.04 | 0.09
Table 5: Statistical significance ($\sigma$) and the signal-to-background
ratio ($\frac{S}{B}$) are shown for the signal corresponding to different
benchmark points contributing to the fatjets $+\not{E}_{T}$ final state at 14
TeV LHC and 139 $\text{fb}^{-1}$ integrated luminosity.

Final selection cuts:

  1. 5.

We increase $\not{E}_{T}$ from 100 GeV to 150 GeV since it reduces the
background sharply than the signal.

  2. 6.

Demand an additional b-tag withing either leading or subleading fatjet is
applied. The b-tag efficiency for the signal within leading or subleading
fatjet is $84\%$. This requirement reduces $Z+$ jets and $W+$ jets backgrounds
substantially below $t\bar{t}+$ jets background.

  3. 7.

We select the events for which the pruned mass of the leading and subleading
fatjets falls within 120 GeV$<M(J_{0}),M(J_{1})<230$ GeV 777Note that, in the
MVA next section, we retain only the lower mass threshold and let the
framework select the non-linear cuts to get the optimal signal-to-background
ratio.. The lower threshold helps us reduce different backgrounds where one or
both the fatjets originated from QCD radiation or $W/Z$ boson.

  4. 8.

To discriminate further the fatjets from QCD jets, we use N-subjettines and
collect the events that satisfy $\tau_{31}(J_{0})\hskip
2.84526pt\text{and}\hskip 2.84526pt\tau_{31}(J_{1})<0.4$ 888One may use the
N-subjettiness variables $\tau_{32}(J_{0})\hskip 2.84526pt\text{and}\hskip
2.84526pt\tau_{32}(J_{1})$ to discriminate the fatjets from two-prong fatjets
originated from boosted $W/Z$ bosons. Since we analyze the same signal using
MVA in the next section, we do not check $\tau_{32}$ variables in CBA..

  5. 9.

We impose $M_{T_{2}}>320$ GeV. This requirement increases the signal-to-
background ratio ($\frac{S}{B}$). For example, in the case of BP3,
$\frac{S}{B}$ changes from 0.13 to 0.4 (Tab. 4).

The expected number of signal (for a sample benchmark point, BP3) and
background events and cut efficiency after imposing the preselection cuts and
final selection cuts at 14 TeV LHC for 139 $\text{fb}^{-1}$ integrated
luminosity are shown in Tab. 4. Statistical significance and the signal-to-
background ratio for different benchmark points are shown in Tab. 5.
$\sigma=\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}$ defines the statistical
significance, where $N_{S}$ and $N_{B}$ are the expected signal and background
events after the cuts, respectively. The statistical significance for the
signal of different benchmark points is above the discovery potential for an
integrated luminosity of 139 $\text{fb}^{-1}$. We also have good statistics
indicating that extracting the VLQ pair from the Standard Model background is
not tough.

###  5.3 Analysis based on the Multivariate Gradient Boosting Technique

Signal |  BP1 |  BP2 |  BP3 |  BP4 |  BP5 |  BP6 |  BP7 |  BP8 |  BP9 |  BP10 |  BP11
---|---|---|---|---|---|---|---|---|---|---|---
6625 |  3525 |  2341 |  2711 |  2176 |  1924 |  1424 |  1081 |  915 |  552 |  385

SM BG |  tt+jets |  tW+jets |  ttZ |  ttW |  Z+jets |  W+jets |  WZ+jets |  ZZ+jets |  WW+jets
---|---|---|---|---|---|---|---|---|---
8928.06 |  3815.42 |  294.35 |  25.93 |  3527.96 |  2408.37 |  172.35 |  27.54 |  46.49

Table 6: The expected number of signal and SM background events after applying
$M_{J_{0}}>120$ GeV, $M_{J_{1}}>120$ GeV and b-tag (within leading or
subleading fatjet), $\not{E}_{T}>150$ GeV in addition to preselection cuts at
14 TeV LHC for 139 $\text{fb}^{-1}$ integrated luminosity.

In the previous section, we constructed high-level variables and demonstrated
their potential in a CBA. This section extends that idea to perform a more
sophisticated MVA. In these analyses, MVA generally gives better sensitivity
than CBA if appropriate kinematic variables are utilized, where we may get
significance above the discovery limit for the benchmark points that is unable
through CBA. The $M_{J_{0}}$ and $M_{J_{1}}$ distribution (Fig. 6a, Fig. 6b)
have the largest peak around the top mass, and the signal is much harder than
the background for $M_{J_{0,1}}>120$ GeV. Instead of both lower and upper mass
thresholds to set an allowed window, we retain only a lower mass threshold of
120 GeV for both the fatjets for event selection in MVA for a higher number of
events. We expect the MVA framework to select nonlinear variable space to get
the optimal signal-to-background ratio. The 120 GeV cut on both fatjets
reduces the backgrounds drastically compared to the signal for which fatjets
arrises from the QCD jet (one-prong) or boosted W/Z boson (two-prong). We also
demand at least one b tag within the leading or subleading fatjet for event
selection in MVA, reducing the background much more than the signal. From the
missing energy distribution (Fig. 6j), we see most background events exist in
low missing energy, so after demanding large missing energy, we reduce the
background significantly compared to the signal. So we apply $\not{E}_{T}>150$
GeV for the event selection in MVA.

Figure 7:  Relative importance (Method unspecific) of the different kinematic
variables used in MVA. We get those numbers for BP3 from the TMVA package.
Those numbers can change a little bit if one chooses a different algorithm.

(a)

(b)

Figure 8: Linear correlation coefficients (in percentage) between different
kinematical variables for the signal (BP3, left panel) and background (right
panel). Missing entries correspond to a negligible correlation smaller than
one. Positive and negative coefficients indicate that two variables are
correlated or anti-correlated, respectively.

(a)

(b)

Figure 9: Left panel: normalized distribution of the BDT response for both
signal (blue, BP3) and background (red) classes (both training and testing
samples of both classes). Right panel: signal (blue) and background (red)
efficiencies and the statistical significance of the signal (green) as a
function of cut applied on BDT output.

With these selection criteria, we keep all other variables unrestrained,
giving enough scope to the multivariate analysis to find an optimal nonlinear
cut based on the suitable variables. The expected number of signal and
background events after applying MVA selection cuts at 14 TeV LHC for 139
$\text{fb}^{-1}$ integrated luminosity is given in Tab. 6. We apply the
adaptive Boosted Decision Tree (BDT) algorithm in our analysis and construct
statistically independent signal and background event samples. Each event
sample is split randomly for training and testing purposes. Since multiple
processes contribute to the total background, we generate them with two to
four extra jets MLM matching separately and combine them in proportion to
their proper weight to get a combined background sample. For multivariate
analysis, a final set of kinematic variables are accepted from a larger set,
where we retain only those variables that are less (anti) correlated in both
signal and background and have larger relative importance. Even before
implementing any model, a variable can have more relative importance than
another when it has larger discriminating power separating the signal class
from the background class. We find $P_{T}(J_{0})$, $P_{T}(J_{1})$, and
$\sqrt{\hat{S}_{min}}$ Konar:2008ei ; Konar:2010ma  are highly correlated with
$M_{\text{eff}}$ in both signal and background. However, we keep
$M_{\text{eff}}$ as it has larger relative importance than other variables.
$\sqrt{\hat{S}_{min}}$ is defined as

$\sqrt{\hat{S}_{min}}=\sqrt{(\sum_{j}E_{j})^{2}-(\sum_{j}P_{z,j})^{2}}+\not{E}_{T}$
(34)

where summation runs over all the visible jets. From Eq. 33 and Eq. 34, the
above correlations are expected. We also observe that
$\Delta\Phi(J_{0},\not{E}_{T})$ and $\Delta\Phi(J_{1},\not{E}_{T})$ are
moderately anti-correlated in signal but highly anti-correlated in background.
The moderate anti-correlation of the signal is because of the total
availability of phase space of the two missing particles. In the case of
background, for example, the principle $t\bar{t}+$ jets background, the only
allowed phase space is when both top and anti-top are highly boosted and move
almost in the opposite direction, where one of the reconstructed tops gives
the leading fatjet, and another one gives both subleading fatjet and large
missing transverse momentum. As a result, these two variables are highly anti-
correlated in the background. We keep $\Delta\Phi(J_{1},\not{E}_{T})$ as it
has larger relative importance than $\Delta\Phi(J_{0},\not{E}_{T})$. We notice
that $M_{T2}$ and $\not{E}_{T}$ are also highly correlated in signal and
moderately in background. Since $\not{E}_{T}$ has the largest relative
importance, we choose $\not{E}_{T}$ over $M_{T2}$ for MVA analysis. The
relative importance of the different kinematic variables used in MVA is
presented in Fig. 7 for sample benchmark point BP3. From the normalized
distributions in the previous section, we notice that all variables used in
MVA are outstanding in distinguishing the signal from the background. However,
the $\not{E}_{T}$, $\Delta R(J_{0},J_{1})$, and
$\Delta\Phi(J_{1},\not{E}_{T})$ are the finest among all these useful
variables. The linear correlation coefficients (in percentage) between
different kinematical variables for the signal (BP3, left panel) and
background (right panel) is presented in Fig. 8. BDT algorithm may lead to
overtraining for wrong choices of different (BDT specific) parameters during
training. Such overtraining can be avoided if one checks the Kolmogorov-
Smirnov probability during training. We train the algorithm separately for
every benchmark point and confirm that no overtraining exists in our analysis.

BP | $N_{S}^{bc}$ | BD$T_{opt}$ | $N_{S}$ | $N_{B}$ | $\sigma$ | $\frac{N_{S}}{N_{B}}$ | $\frac{N_{S}}{N_{B}}$
---|---|---|---|---|---|---|---
(MVA) | (CBA)
BP1 | 6625 | -0.0259 | 5643 | 8584 | 47.3 | 0.66 | 0.41
BP2 | 3525 | 0.1584 | 2325 | 2047 | 35.1 | 1.14 | 0.47
BP3 | 2341 | 0.2553 | 1222 | 1048 | 25.6 | 1.17 | 0.40
BP4 | 2711 | 0.1975 | 1556 | 1277 | 29.2 | 1.22 | 0.35
BP5 | 2176 | 0.1446 | 1366 | 2502 | 21.9 | 0.55 | 0.23
BP6 | 1924 | 0.1325 | 1170 | 2727 | 18.7 | 0.43 | 0.17
BP7 | 1424 | 0.1398 | 821 | 2845 | 13.6 | 0.29 | 0.13
BP8 | 1081 | 0.0942 | 624 | 2951 | 10.4 | 0.21 | 0.12
BP9 | 915 | 0.0785 | 627 | 4820 | 8.5 | 0.13 | 0.08
BP10 | 552 | 0.0506 | 311 | 3469 | 5.0 | 0.09 | 0.04
BP11 | 385 | 0.3875 | 134 | 346 | 6.1 | 0.39 | 0.09
$N_{SM}$ | 19246 |

$\Delta M$ | $N_{S}^{bc}$ | BD$T_{opt}$ | $N_{S}$ | $N_{B}$ | $\sigma=$
---|---|---|---|---|---
(GeV) | $\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}$
75.1 | 2315 | 0.2645 | 1252 | 1240 | 25.1
100 (BP3) | 2341 | 0.2553 | 1222 | 1048 | 25.6
153.6 | 2258 | 0.2129 | 1246 | 1176 | 25.3
200.4 | 2035 | 0.1899 | 1284 | 1384 | 24.9
244.5 | 2019 | 0.2963 | 1159 | 948 | 25.2
302.7 | 2036 | 0.208 | 1254 | 1240 | 25.1
351 | 2034 | 0.2271 | 1261 | 1199 | 25.4
402.7 | 2023 | 0.2778 | 1133 | 882 | 25.2
$N_{SM}$ | 19246 |

Table 7:  The upper table demonstrates the effectiveness of the current search
in terms of statistical significance ($\sigma$) for different benchmark points
conceived for this study. The lower table illustrates the variation of this
potential for one benchmark point, changing the mass of the heavy scalar,
$S_{2}$, and shows that this mass does not have much impact in exploring the
parameter space. $N_{S}^{bc}$ and $N_{SM}$ are the total number of events for
different signal benchmark points and the combined background before applying
any cut on BDT output (as shown in Tab. 6). After using an optimal selection
on the BDT response (BD$T_{opt}$) surviving number of signal and background
events are given by $N_{S}$ and $N_{B}$, respectively for 14 TeV LHC for an
integrated luminosity 139 $\text{fb}^{-1}$. Corresponding statistical
significance and the signal-to-background ratio are also presented for ready
reference. To better compare the sensitivities between the different analysis
methods, we add the $\frac{N_{S}}{N_{B}}$ ratio of CBA from Tab. 5 in the last
column of the upper table.

The normalized distribution of the BDT response of the signal (BP3) and the
background classes (both training and testing samples of both classes) is
shown in Fig. 9. We notice both the classes are well separated. We present the
variation of the signal and background efficiencies and the statistical
significance of the signal (BP3) with the cut applied on the BDT response in
the right panel of Fig. 9. Statistical significance is defined as
$\sigma=\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}$. The number of events that survive
after applying the $\text{BDT}_{\text{res}}>\text{BDT}_{opt}$ cut for signal
and background is $N_{S}$ and $N_{B}$, respectively. $\text{BDT}_{opt}$ is the
optimal cut for which the significance is maximum. In Tab. 7 (upper) $N_{S}$,
$N_{B}$, $\sigma$, and $\dfrac{N_{S}}{N_{B}}$ are presented for different
benchmark points at 14 TeV LHC with integrated luminosity 139
$\text{fb}^{-1}$. We find that for a few of the chosen BPs, the number of
signal events is larger than the background events after the
$\text{BDT}_{\text{res}}>\text{BDT}_{opt}$ cut, and for all eleven benchmark
points, we reach the discovery potential with integrated luminosity 139
$\text{fb}^{-1}$.

Our next interest would be to verify how significance varies with the mass of
the scalar $S_{2}$. For that purpose, we generated the event samples
separately for different masses of the $S_{2}$ with the same DM mass
$M_{S_{1}}=403$ GeV, VLQ mass $M_{\Psi}=808$ GeV, and the coupling constant
$f_{t}=1$. We train the algorithm separately for different samples of
different $S_{2}$ masses and confirm that no overtraining exists in our
analysis and perform the MVA.

There are two possible hierarchies possible: $M_{S_{2}}>M_{\Psi}>M_{S_{1}}$
and $M_{\Psi}>M_{S_{2}}>M_{S_{1}}$. We consider two boosted tops associated
with missing transverse momentum as our signal. It is interesting to note that
the significance of the former hierarchy is always greater or equal to the
second. In the case of the former, $S_{2}$ can decay into $\Psi(\rightarrow
S_{1}j)j$ or $S_{1}jj$ (through off-shell $\Psi$), where $j$ is the up-type SM
quark. If at least one of these jets is the top quark, then the signal
efficiency increases and hence the significance. So the hierarchy
$M_{\Psi}>M_{S_{2}}>M_{S_{1}}$ gives a lower statistical significance, and we
consider this scenario throughout our result for a conservative estimation.
The total number of events coming from the signal topology for different
masses of $S_{2}$ and background events after applying an optimal cut
(BD$T_{opt}$) is given in Tab. 7 (lower) for the hierarchy
$M_{\Psi}>M_{S_{2}}>M_{S_{1}}$. The statistical significance variation with
the $S_{2}$ mass is also shown here for a given mass of $\Psi$ and couplings.
The mass of $S_{2}$ has no effect on the statistical significance of the
boosted top fatjets plus a large missing momentum signature. However, if one
of the decay products of $S_{2}$ ($S_{2}\rightarrow S_{1}jj$) is at least a
top quark, then it can increase the significance.

Figure 10: The solid red line is the $5\sigma$ discovery contour, and any
point inside the red line has a statistical significance $>5\sigma$ at 14 TeV
LHC for an integrated luminosity 139 $\text{fb}^{-1}$. The dashed red line
corresponds to $\Delta M_{\Psi S_{1}}=M_{\Psi}-M_{S_{1}}=M_{\text{top}}$.
Below the dashed red line, we can not probe with the boosted tops plus missing
energy signal, as we can not get any on-shell top from the decay of $\Psi$.
The solid black and blue dashed lines are the exclusion contour ($2\sigma$) of
our analysis for an integrated luminosity of 139 $\text{fb}^{-1}$ and 35.9
$\text{fb}^{-1}$, respectively. The exclusion region ($2\sigma$) from the
existing LEP, ATLAS (multijet $+$ MET), and CMS ($t\bar{t}$ $+$ MET) analysis
are shown by brown, green, and olive color, respectively.

Finally, we present the discovery ($5\sigma$) and exclusion ($2\sigma$)
contours from our analysis at 14 TeV LHC for an integrated luminosity 139
$\text{fb}^{-1}$ in the bi-dimensional plane of $\dfrac{\Delta M_{\Psi
S_{1}}}{M_{S_{1}}}-M_{S_{1}}$ in Fig. 10 by solid red and solid black lines,
respectively. Our analysis is effective when the on-shell top is produced from
$\Psi$ decay. Hence the region below the dashed red line can not be probed in
the present channel. Considering the $100\%$ branching fraction of the decay
of $\Psi$ into the top quark associated with the scalar, the existing search
Colucci:2018vxz  can exclude vector-like quark masses up to 1 TeV. We find the
masses of the vector-like quark ranging up to 1.41 TeV can be excluded, while
the masses extent to 1.28 TeV can be discovered at 14 TeV LHC with an
integrated luminosity of 139 $\text{fb}^{-1}$. In the region below the dashed
red line, the mass difference between a vector-like quark and the scalar DM is
less than the top quark’s mass, and the vector-like quark fully decays into a
light quark associated with a scalar when kinematically allowed. So, one can
probe those regions using multi jets plus missing transverse momentum
signature, which is beyond the scope of our present analysis.

##  6 Summary and Conclusion

In this work, we analyze a hybrid KSVZ setup, where the model is extended by
an extra complex scalar singlet whose lightest component plays the role of
dark matter. We highlight the fact that the presence of a colored vector-like
quark that occurs naturally in the KSVZ model plays a crucial role both in the
dark sector and collider phenomenology of the setup. Being charged under
$U(1)_{PQ}$ allows the VLQ to couple with all up-type quarks and the DM
through the Yukawa interactions. When appropriately tuned, this coupling can
enhance the DM parameter space in comparison to what is observed in a pure
scalar singlet DM scenario. In this work, we demonstrate that the Yukawa
couplings play a non-trivial role in obtaining the observed relic density.
Moreover, the same couplings also allow the parameter space from the direct
search bounds by entering into extra Feynman diagrams that contribute toward
the direct detection cross-section of the dark matter.

A search of vector-like quarks in events with two boosted top fatjets with
large missing transverse momentum is presented. The analysis is done for 139
$\text{fb}^{-1}$ integrated luminosity at 14 TeV LHC. We discuss all the
significant backgrounds that can potentially mimic the signal. Jet
substructure variables and various other variables are used in our analysis.
Sophisticated multivariate analysis is performed to increase the sensitivity
over cut-based one. Different jet substructure variables, $\Delta
R(J_{0},J_{1})$, N-subjettiness ratios, and $M_{eff}$ are outstanding in
distinguishing the signal from the background and take a central role in
getting very high significance. However, the missing transverse momentum
distribution and the azimuthal separation between the subleading fatjet and
the missing transverse energy direction have the uppermost importance in
separating the signal from the background. With a conservative estimation, we
give discovery and exclusion contours in Fig. 10 in the region where the mass
difference between the vector-like quark and the scalar dark matter is larger
than the top quark mass.

###### Acknowledgements.

This work is supported by the Physical Research Laboratory (PRL), Department
of Space, Government of India. Computational work was performed using the HPC
resources (Vikram-100 HPC) and TDP project at PRL. R.R. also acknowledges the
National Research Foundation of Korea (NRF) grant funded by the Korean
government (NRF-2020R1C1C1012452).

##  Appendix A Feynman Diagrams

  * •

Annihilation channels of scalar dark matter $S_{1}$ are shown in Fig. 11.

  * •

Co-annihilation channels of scalar dark matter $S_{1}$ are shown in Fig. 12.

  * •

Annihilation channels of vector-like quark $\Psi$ are shown in Fig. 13.

  * •

Spin independent elastic scattering between dark matter ($S_{1}$) and nucleon
channels are shown in Fig. 14.

  * •

Diagrams contributing to the $D^{0}-\bar{D}^{0}$ mixing are shown in Fig. 15.

Figure 11: Annihilation channels of scalar dark matter $S_{1}$. U denotes the
SM up-type quark ($U\equiv u,c,t,\bar{u},\bar{c},\bar{t}$)

Figure 12: Co-annihilation channels of scalar dark matter $S_{1}$. U and D
denote the SM up-type and down-type quark, respectively; $U\equiv
u,c,t,\bar{u},\bar{c},\bar{t}$, $D\equiv d,s,b,\bar{d},\bar{s},\bar{b}$

Figure 13: Annihilation channels of vectorlike quark $\Psi$. U denotes the SM
up-type quark ($U\equiv u,c,t,\bar{u},\bar{c},\bar{t}$)

Figure 14: Spin independent elastic scattering between dark matter ($S_{1}$)
and nucleon

Figure 15: Diagrams contributing to the $D^{0}-\bar{D}^{0}$ mixing.

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