#  h
Asymmetries in invisible Dark Matter mediator production associated with
$t\bar{t}$ final states

E. Chalbaud ${}^{1\,}$111E-mail<EMAIL_ADDRESS>Rui M. Silva1 2222E-mail:
<EMAIL_ADDRESS>António Onofre${}^{2\,}$333E-mail:
<EMAIL_ADDRESS>
Ricardo Gonçalo ${}^{1\,}$444E-mail<EMAIL_ADDRESS>, Miguel C. N.
Fiolhais${}^{1\,3\,4\,}$555E-mail<EMAIL_ADDRESS>
1Laboratório de Instrumentação e Física Experimental de Partículas,
Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
2 Centro de Física da Universidade do Minho e Universidade do Porto (CF-UM-
UP),
Universidade do Minho, 4710-057 Braga, Portugal
3 Science Department, Borough of Manhattan Community College,
The City University of New York, 199 Chambers St, New York, NY 10007, USA
4The Graduate School and University Center, The City University of New York,
365 Fifth Avenue, New York, New York 10016, USA

###### Abstract

In this paper, we propose two sets of different CP-sensitive observables
inspired by the Higgs production in association with the top quark. We employ
a Dark Matter simplified model that couples a scalar particle with three
generation fermions with a mass equal to that of the top quark. The
reconstruction of the kinematic variables is presented at NLO accuracy for
events associated with this massive scalar particle, which is assumed to be
vanishing to invisible decays in a detector such as ATLAS. We build these
observables by taking advantage of the similarity between the scalar coupling
with the top quark and the factorization theorem in the total scattering
amplitude, in order to represent the basis in which the phase space is
parameterized. A twofold approach employs the direct implementation of the
four-momentum phase space measure in building CP sensitive observables such as
$b_{2}$ for the Higgs, and the spin polarization of the top-quark decays in
the narrow width approximation for the employed model. We studied the
asymmetries of these distributions to test for any improvement in increasing
the exclusion region for the $g_{u_{33}}^{S}-g_{u_{33}}^{P}$ parameters
associated with this vanishing scalar particle. We have found no significant
effect in the exclusion limits by using the forward-backward asymmetry
distributions and the full shaped ones. The best limits obtained for
$g_{u_{33}}^{S}$ at NLO accuracy were obtained: $b_{2}:[-0.0425,0.0475]$
$n_{4}:[-0.0450,0.0450]$ for an invisible scalar with a mass of 10-2 GeV for a
luminosity of 300 fb-1 expected for the end of Run 3 of the LHC.

##  1 Introduction

A plethora of phenomena in astrophysics, such as gravitational lensing,
galaxies’ rotational curves, collisions of intergalactic clusters, and the
evolution of the large-scale structure of the Universe [1, 2, 3, 4, 5, 6],
offer persuasive indirect proof for the existence of dark matter (DM).
However, the particle nature of DM is still not fully understood. Theoretical
enlargements to the Standard Model suggest a ”dark sector” comprising
particles that interact minimally with known standard model particles, leading
to the proposition of weakly interacting massive particles (WIMPs) as a
potential DM candidate. For example, the intriguing coincidence that a stable
WIMP of a specific mass could naturally explain the observed relic density of
DM observed by Planck measurements [7], is a fundamental element in the search
for the identification of DM particles. Although the Large Hadron Collider
(LHC) offers an unmatched opportunity to generate and analyze DM particles,
conventional detection techniques encounter considerable difficulties due to
the expected weak interactions of DM with regular matter [8].

Addressing the challenges linked to the weakly interactive character of DM
particles, the Large Hadron Collider (LHC) has implemented novel methods for
their identification. DM detection is not based on direct observation, but
rather on missing transverse momentum in proton-proton collisions [9, 10, 11],
suggesting the creation of undetectable particles. This approach has led to an
emphasis on ”mono-X” event topologies, with ”X” representing an observable
particle like a jet, Z/W boson, or photon, which can create the necessary
contrast to the invisible DM particle[12, 13]. To make these studies more
manageable, simplified DM models have gained popularity over the effective
field theory approach [14]. By introducing a limited number of new particles
and couplings, these simplified models allow for the examination of specific
attributes and interactions of DM [15]. This focused method enables a more
straightforward link between the experimental findings and the characteristics
of the candidates for DM, thus optimizing the search and improving our
understanding of where these elusive particles may fit within the cosmic
framework [7, 16].

Numerous simplified models have been proposed, indicating that DM could be an
isolated heavy entity that only weakly interacts with the recognized particles
of the Standard Model (SM). These interactions are theorized to be enabled by
a novel particle mediator. At lower energy scales, where the mediator mass is
greater than the energy reach of the experiment, its interactions appear as
point-like, thus permitting the use of effective field theory (EFT) to detail
the interaction through a sequence of higher-dimensional operators[17, 18].
This method has been popular and validated by experiments such as ATLAS, CMS,
and LHCb especially in the context of supersymmetry searches [19, 20]. On the
other hand, at the LHC where the energy scales can match or exceed the
mediator’s mass, the tangible effects of the mediator require a detailed
quantum field theory that explicitly includes the mediator in its particle
scope. The ability of the LHC to investigate a wide range of DM and mediator
masses, in addition to various intensities and spin configurations of their
couplings, is crucial.

The primary objectives of this study are to rigorously investigate CP-
violating observables in the context of dileptonic decays of the top quarks,
drawn from established literature on the Higgs sector [21, 15, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58], in order to improve
the signal-to-background ratio in the context of DM detection. Secondly, a
thorough analysis is presented covering the mass spectrum from 10-2 GeV to 125
GeV (the Higgs mass scale), delineating a strategic detection window for DM
particles. Finally, we determine the sensitivity of these observables with
respect to the future luminosity scenarios expected at the LHC, both during
Run 3 and at the projected end of its service. This offers a clearer
understanding of the potential of LHC to probe the frontiers of DM in the
years to come.

The composition of this paper is as follows. We outline the simplified DM
model and define the relevant parameters and angular observables in Section 2.
The methodology for event generation and simulation is detailed in Section 3.
Section 4 shows the definitions for the CP sensitive observables. Section 5
explores the asymmetries of such observables. Exclusion limits are shown in
Section 6 where we have included the results for the total cross-section as a
function of the luminosity. We draw our central conclusions in Section 7.

##  2 Simplified Dark Matter model

Since the production of DM in colliders is expected to appear in the
s-channel, cross-sections derived from Lagrangians should be accessible in the
context of heavy fermion’s production, especially within the expected LHC
luminosity from Run 3 of the LHC. However, depending on how the particle
mediator couples with the SM particles, the discovery of such a sector can be
challenging to separate from different backgrounds due to the small value of
the cross-section. In general, the Lagrangian can be written in several forms,
for example in the context of a DM simplified model [59], a vertex-like
approach it is expected to capture much of the phenomenology in the
interaction. In the case of scalar-like mediators, the coupling with fermionic
DM and the DM mediator follows a Yukawa coupling of the form:

$\mathcal{L}=\bar{X}_{D}(g^{s}+ig^{p}\gamma^{5})X_{D}Y_{0},$ (2.1)

where $g^{s}$ corresponds to the scalar coupling of the mediator, $g^{p}$ is
the pseudoscalar coupling constant with the DM fermion $X_{D}$. Additionally,
the coupling with the SM, as is explored in the model proposed in [60], the DM
mediators can be coupled with quarks in one or several generations as follows:

$\mathcal{L}=\sum_{i,j}\bigg{[}\bar{d}_{i}\frac{y^{d}_{i,j}}{\sqrt{2}}(g^{s}_{d_{ij}}+ig^{p}_{d_{ij}}\gamma^{5})d_{j}+\bar{u}_{i}\frac{y^{u}_{i,j}}{\sqrt{2}}(g^{s}_{u_{ij}}+ig^{p}_{u_{ij}}\gamma^{5})u_{j}\bigg{]}Y_{0},$
(2.2)

where $y^{u/d}_{i,j}$ denotes the quark Yukawa-like coupling matrix with the
scalar mediator, $g^{s}_{d_{ij}}$ and $g^{p}_{d_{ij}}$. Notice that we can
have models depending on the specification of the constants by including off-
diagonal terms in the coupling matrix. The computation of the LO cross-section
depends on the nature of this matrix where third generation of quark
interactions can be imposed. Although scalar mediator models that employ 4 and
5 flavor schemes have been known in the literature, NLO computations of the
cross-section do not change significantly irrespectively from the scheme
employed. Even though the renormalizability of the coupling with the b-quark
is expected to be challenging, due to the higher order uncertainties when
switching schemes, regularization appears to be formally complete [60, 18,
61]. Given this limitation, and from the limits of the CP-violating phase due
to the smallness of expected contributions coming from this sector, we can
assume no doublet couplings with heavy generation fermions keeping the top
quark coupling of the form:

$\mathcal{L}=\frac{y_{3,3}}{\sqrt{2}}\bar{t}(g^{s}+ig^{p}\gamma^{5})tY_{0},$
(2.3)

where $y_{3,3}$ is the corresponding coupling constant with the top quark, and
the $g^{s}$ denotes the CP-even (scalar) coupling with the mediator and
$g^{p}$ the corresponding CP-odd (pseudoscalar) interaction. Additionally,
lepton couplings for the mediator are left behind at this point due to the
unitarity-violating effects in the leptonic sector being difficult to make
compatible with these searches, which are out of scope of this paper. We
introduce the intrinsic assumption of the simplified model from where no
beyond the DM particle X and mediator Y at the weak scale is needed. In this
work, we employ an implementation of the Lagrangian in equation 2.3, including
LO and NLO corrections in the total cross-sections, by using FeynRules package
[62]. The decaying fixation for the DM mediator particle is performed, in such
a way that the particle can escape detectors, i.e., fix $\Gamma=0$. This
construction is analogous to the freely available DM simplified model DMsimp
for scalar mediators in the s-channel interaction, but including the 5 flavor
scheme. The main topologies obtained in the model for interaction of the top
quark with the particle mediator are given in Figure 1, where corresponding
NLO diagrams are the same as those obtained in [60] for the DMsimp package.

$g$$g$$g$$g$$g$$g$$g$$Y_{0}$$Y_{0}$$Y_{0}$$Y_{0}$$u,d$$u,d$$\overline{t}$$\overline{t}$$\overline{t}$$\overline{t}$$t$$t$$t$$t$$t$$t$$t$$t$$t$$t$$g$
Figure 1: Representative Feynman digrams for the spin 0 mediator production,
with the top quarks in the final state employed in the UFO model.

Due to the similar nature of the scalar coupling for the Higgs boson, and by
assuming the narrow width approximation for the top and antitop quarks, we can
approximate and define the unpolarized differential cross-section for the
process $gg\rightarrow t\left(\rightarrow
b\ell^{+}\nu_{\ell}\right)\bar{t}\left(\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}\right)Y$
in a ”factorized” form:

$d\sigma=\sum_{\begin{subarray}{c}b\ell^{+}\nu_{n}\\\ \text{ spins spins }\\\
\ell^{-}\bar{\nu}_{\ell}\end{subarray}}\left(\frac{2}{\Gamma_{t}}\right)^{2}d\sigma\left(gg\rightarrow
t\left(n_{t}\right)\bar{t}\left(n_{\bar{t}}\right)Y\right)d\Gamma\left(t\rightarrow
b\ell^{+}\nu_{\ell}\right)d\Gamma\left(\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}\right),$
(2.4)

where $d\sigma\left(gg\rightarrow
t\left(n_{t}\right)\bar{t}\left(n_{\bar{t}}\right)Y\right)$ is the
differential cross-section for the production of a top and antitop quarks,
where the spin vectors $n_{t}$ and $n_{\bar{t}}$ are introduced for enhancing
the relative sign sensitivity for the pseudoscalar component of the model in
addition of the quadratic enhancement proposed in [63].

The factorized terms of the form: $d\Gamma\left(t\rightarrow
b\ell^{+}\nu_{\ell}\right)$ and
$d\Gamma\left(\bar{t}\rightarrow\bar{b}\ell^{-}\bar{\nu}_{\ell}\right)$ are
the partial differential decay widths for an unpolarized top and anti-top
quark and such factorized form is valid within the narrow-width approximation.
The spin basis is defined in such a way that $t,\bar{t},\ell^{+}$and
$\ell^{-}$, normalize the 4-vectors:

$\displaystyle n_{t}=-\frac{p_{t}}{m_{t}}+\frac{m_{t}}{\left(p_{t}\cdot
p_{\ell^{+}}\right)}p_{\ell^{+}}\quad\quad
n_{\bar{t}}=\frac{p_{\bar{t}}}{m_{t}}-\frac{m_{t}}{\left(p_{\bar{t}}\cdot
p_{\ell^{-}}\right)}p_{\ell^{-}},$ (2.5)

where $p_{\ell^{\pm}}$ correspond to the momentum of the outgoing lepton
coming from the top quark decay. The spatial components of these variables are
used to define equations 4.2. Given that the representation of the total
differential cross-section is analogue to the CP-sensitive one, we can in
principle construct analogue operators as in the Higgs case.

##  3 Simulation

In this section, it is shown the results of the event generation from an
independent implementation of the DMsimp DM simplified model [60]. We have
produced 1 million parton-level events using MadGraph5_aMC@NLO [64] for both
signal and background events at $\sqrt{s}=14$ TeV for $pp\rightarrow
t\bar{t}Y$ process. We have explored DM mediator masses set to
$m_{Y_{0}}=0,1,10$ and $125\mathrm{GeV}$ and 1 TeV. The top quark mass is set
to $m_{t}=172.5\mathrm{GeV}$ considering only the dileptonic decays of the
$t\bar{t}$ system, i.e. $\left(t\bar{t}\rightarrow
bW^{+}\bar{b}W^{-}\rightarrow
b\ell^{+}\nu_{\ell}\bar{b}\ell^{-}\bar{\nu}_{\ell}\right)$. The decay of the
DM particle is fixed to zero to reproduce the situation of the particle
escaping the detector. The analysis of signal and background events is
performed within the MadAnalysis5 framework [65]. Standard Model (SM)
backgrounds were also generated with NLO accuracy, and event hadronization was
conducted using PYTHIA6Q [66], followed by detector simulation with Delphes
[67]. For the kinematic fit associated to the reconstruction of the kinematic
variables, we have followed the same approach in [46], but including NLO
corrections in both signal and background events.

The event selection was based on jet and lepton identification with specific
constraints, focusing on events with a pair of jets and isolated leptons.
Reconstruction of the system $(t\bar{t})$ involved assigning jets to their
corresponding b-quarks and utilizing multivariate analysis for improved
accuracy. This analysis included various likelihood discrimination estimators
as those used in [52, 68], where the Boosted Decision Tree with Gradient boost
(BDTG) showed superiority in classifying the events. In Figure 2, it is shown
the reconstruction of the $t\bar{t}Y$ events, presenting the impact in the
reconstruction when a mediator particle with different masses is considered in
the analysis. By using kinematic equations from [68], it is shown that heavy
mediator particles with masses of the order of magnitude of 1 TeV contribute
more to the total energy missing of the reconstruction, normally associated to
the neutrinos, translating in an increasing bias in the reconstruction of the
kinematic variables.

---
Figure 2: Two-dimensional distributions of $p_{T}$ in $t\bar{t}Y_{0}^{+}$
events. The impact of the mediator mass on the kinematic reconstruction of the
$t\bar{t}$ system for mediator masses of $10^{-2}$ GeV (upper-left), $10$ GeV
(upper-right), $125$ GeV (lower-left), and $10^{3}$ GeV (lower-right).

##  4 CP sensitive obervables

In the study, the search of sensitive CP-observables starts by considering the
total cross-section coming from a Yukawa interaction term in the DM simplified
model. For scalar particles, it can be shown that a process of the form
$gg\rightarrow Q\bar{Q}Y$ and $q\bar{q}\rightarrow Q\bar{Q}Y$ does not contain
any mixed Yukawa couplings at the tree-level. This is considered for CP-even
particles in [68], where in order to maximize the CP sensitivity for the Higgs
CP couplings it is considered the following normalized $\alpha$ constant:

$\alpha\left[\mathcal{O}_{CP}\right]\equiv\int\left[\mathcal{O}_{CP}\right]\\{d\sigma(pp\rightarrow
t\bar{t}Y)/dPS\\}dPS\ .$ (4.1)

The integral over the phase space $PS$, for a given differential observable
$\mathcal{O}_{CP}$ is such that in the case of the Higgs boson leads to the
definition of the $b_{2}$ observable that can be written in terms of the
projections of the 4-momenta for the outgoing top quarks in a given event:

$\displaystyle
b_{2}=\frac{\left(\vec{p}_{t}\times\widehat{z}\right)\cdot\left(\vec{p}_{t}\times\widehat{z}\right)}{\left|\vec{p}_{{t}}\right|\left|\vec{p}_{\bar{t}}\right|}\quad\tilde{b_{2}}=\frac{\left(\vec{p}_{t}\times\widehat{y}\right)\cdot\left(\vec{p}_{t}\times\widehat{y}\right)}{\left|\vec{p}_{{t}}\right|\left|\vec{p}_{\bar{t}}\right|}\quad\tilde{b_{2}}=\frac{(\vec{p}_{t}\times\widehat{d})\cdot(\vec{p}_{t}\times\widehat{d})}{\left|\vec{p}_{{t}}\right|\left|\vec{p}_{\bar{t}}\right|}$
(4.2) $\displaystyle
n_{2}^{t\bar{t}Y}=\frac{(\vec{n}_{t}\times\hat{k}_{z})\cdot(\vec{n}_{\bar{t}}\times\hat{k}_{z})}{\left(\left|\vec{n}_{t}\right|\cdot\left|\vec{n}_{\bar{t}}\right|\right)}\quad
n_{4}^{t\bar{t}Y}=\frac{\left(n_{t}^{z}\cdot
n_{\bar{t}}^{z}\right)}{\left(\left|\vec{n}_{t}\right|\cdot\left|\vec{n}_{\bar{t}}\right|\right)}\quad
d=(1,1,0)/\sqrt{2}$

where $\hat{n}$ corresponds to the projection direction of $\vec{p}$ following
the same definitions in [38]. This observable has been studied in the context
of minimal pseudoscalar extensions of the Higgs boson, where it appears to
have competitive sensitivity following this construction. However, given the
nature of the DM simplified coupling with the scalar mediator, we can exploit
this definition and its rotational symmetry to build two new angular
observables. More specifically, a new approach involving orthogonal
projections with respect to different axis. To test the hypothesis of these
projections improving sensitivity for CP-odd components in the model, we
extend the projections of the $b_{2}$ variable from the $\hat{z}$ axis, to the
normalized $(0,1,0)$ and
$(0,\nicefrac{{1}}{{\sqrt{2}}},\nicefrac{{1}}{{\sqrt{2}}})$ directions of
projection.

In Figures 3 and 4, it is shown all the corresponding parton-level
distributions after event selection and kinematic reconstruction, for a
reference luminosity of $100\mathrm{fb}^{-1}$ for all the considered
observables. All SM backgrounds: $t\bar{t}$ $(t\bar{t}c\bar{c}$ and
$t\bar{t}+$ light jets), $t\bar{t}b\bar{b},t\bar{t}V,t\bar{t}H$, single top
quark production ( $t$-, $s$ \- and $Wt$-channels), $W/Z+$ jets, and diboson
$(WW,ZZ,WZ)$ events are represented. The $t\bar{t}Y_{0}$ scalar and
pseudoscalar signals, with $m_{Y_{0}}=0\ \mathrm{GeV}$, are shown as well,
scaled by factors of 2 and 500, respectively, for convenience. As expected,
the main SM background contribution is the $t\bar{t}$ due to its similarity
with the signal final state topology. All other backgrounds are essentially
residual to the overall SM contribution.

Differences in shapes of the background distributions can also be noticed when
compared with the signals. In addition, selection cuts heavily impact the
relative bin value of the distribution but preserve the shape of the parton-
level distributions. In Figure 4, the $b_{4}$ distribution for the scalar
signal (in blue) is more populated in positive values rather than in the
negative ones. This behavior is inverted for the pseudoscalar case (in
orange).

---
Figure 3: Angular distributions at parton level before selection cuts (left)
and after final selection cuts and full kinematic reconstruction, considering
all the dominant backgrounds in $t\bar{t}$ production (right) of: (top)
$x_{\alpha[\mathcal{O}_{CP}]}=b_{2}$, (middle)
$x_{\alpha[\mathcal{O}_{CP}]}=\tilde{b}_{2}^{(0,\nicefrac{{1}}{{\sqrt{2}}},\nicefrac{{1}}{{\sqrt{2}}})}$
and (bottom) $x_{\alpha[\mathcal{O}_{CP}]}=\tilde{b}_{2}^{(0,1,0)}$. Mass
reference is $10^{-2}$ GeV for a luminosity of 100 fb-1.
---
Figure 4: Angular distributions at parton-level before selection cuts (left)
and after final selection cuts and full kinematic reconstruction, considering
all the dominant backgrounds in $t\bar{t}$ production (right) of: (top)
$x_{\alpha[\mathcal{O}_{CP}]}=b_{4}$, (middle)
$x_{\alpha[\mathcal{O}_{CP}]}=n_{2}$ and (bottom)
$x_{\alpha[\mathcal{O}_{CP}]}=n_{4}$. Mass reference is $10^{-2}$ GeV for a
luminosity of 100 fb-1.

##  5 Asymmetries

As it is explored in Higgs production studies [52], an asymmetry in the
distribution of the top and antitop quark production can provide an additional
source of information for the study of the properties of DM mediator
particles. In particular, the measurement of the forward-backward asymmetry,
defined as the difference between the number of top-antitop pairs produced in
the forward and backward directions in the top-quark rest frame, can provide
important constraints on the couplings of the top quark to other particles.
Furthermore, the study of the spin correlation in the top-antitop production,
which is sensitive to the relative orientation of the spins of the top and
antitop quarks, can provide additional insights into the production and decay
dynamics of these particles.

Detailed studies of these asymmetries can provide valuable tests of the
standard-model predictions and potentially reveal evidence of new physics and
the reduced impact that systematic effects have in such measurements. As an
example, the Higgs boson production in association with the top quark has been
reported in the literature [69] showing interesting results in constraining
the minimal extensions of the Higgs sector. In this work, the same forward-
backward asymmetries approach is employed for testing the impact on the
exclusions limits. The asymmetry in the distribution is defined as:

$\displaystyle
A^{\alpha[\mathcal{O}_{CP}]}_{FB}=\frac{\sigma(x_{\alpha[\mathcal{O}_{CP}]}>0)-\sigma(x_{\alpha[\mathcal{O}_{CP}]}<0)}{\sigma(x_{\alpha[\mathcal{O}_{CP}]}>0)+\sigma(x_{\alpha[\mathcal{O}_{CP}]}<0)},$
(5.1)

where the selection cut is established at the center of the distribution. The
application of this cut is presented in Tables 1 and 2 at both parton-level
distributions and after selection cuts are imposed during the reconstruction.
It is noticeable that the application of selection cuts clearly distorts the
asymmetry observed in the differential distributions at parton-level in
comparison with the ones obtained after imposing the selection cuts in the
events. Remarkably, these distortions exhibit minimal variation with the mass
of the mediator particle, indicating a consistent impact across different mass
scales. However, the variation observed between the different observables
presented in the Tables (1) and (2) is substantial. This significant disparity
among the observables could have critical implications for the exclusion
limits, a topic that will be explored in detail in the forthcoming section.

Observable | 10-2GeV | 1 GeV | 10 GeV | 100 GeV | 125 GeV
---|---|---|---|---|---
$t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$
$A_{FB}^{b_{2}(\hat{z})}$ | -0.839/-0.579 | -0.834/-0.579 | -0.819/-0.568 | -0.703/-0.409 | -0.674/-0.377
$A_{FB}^{b_{2}(\hat{y})}$ | +0.222/-0.042 | +0.219/-0.041 | +0.217/-0.049 | +0.211/-0.156 | +0.199/-0.180
$A_{FB}^{b_{2}(\hat{d})}$ | +0.098/-0.110 | +0.092/-0.109 | +0.086/-0.116 | +0.061/-0.185 | +0.046/-0.199
$A_{FB}^{n_{4}}$ | -0.243/-0.061 | -0.245/-0.062 | -0.246/-0.056 | -0.250/+0.0298 | -0.244/+0.050
$A_{FB}^{n_{2}}$ | +0.278/+0.278 | +0.242/+0.221 | +0.234/+0.149 | +0.262/0.117 | +0.212/+0.178
Table 1:  Parton level asymmetry values computed for the 5 benchmark masses employed in the model. Here we have defined the direction: $\hat{d}=(1,1,0)/\sqrt{2}$ for the observable $b_{2}$. Observable | 10-2GeV | 1 GeV | 10 GeV | 100 GeV | 125 GeV
---|---|---|---|---|---
$t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$ | $t\bar{t}Y^{+}/t\bar{t}Y^{-}$
$A_{FB}^{b_{2}(\hat{z})}$ | -0.896/-0.697 | -0.888/-0.697 | -0.874/-0.690 | -0.753/-0.512 | -0.724/-0.454
$A_{FB}^{b_{2}(\hat{y})}$ | -0.066/-0.281 | -0.073/-0.276 | -0.078/-0.293 | -0.079/-0.363 | -0.051/-0.369
$A_{FB}^{b_{2}(\hat{d})}$ | -0.191/-0.360 | -0.205/-0.349 | -0.212/-0.370 | -0.203/-0.386 | -0.180/-0.383
$A_{FB}^{n_{4}}$ | +0.132/-0.008 | +0.131/-0.031 | +0.140/-0.036 | +0.129/-0.104 | +0.122/-0.125
$A_{FB}^{n_{2}}$ | -0.285/-0.286 | -0.269/-0.283 | -0.292/-0.270 | -0.332/-0.228 | -0.323/-0.222
Table 2:  Asymmetry values after applying the selection criteria and kinematic
reconstruction computed for the same benchmark masses.

##  6 Exclusion limits

Confidence levels (CLs) for excluding the scalar and pseudoscalar nature of
the top quark couplings to the DM mediator are computed in two distinct
scenarios:

  * •

Scenario 1 : Exclusion of the SM plus the addition of a new CP-mixed DM
mediator, assuming the SM. In this instance, $H_{0}$ is the SM-only
hypothesis, while $H_{1}$ is the SM plus a new CP-mixed DM mediator;

  * •

Scenario 2 : Exclusion of the SM plus the addition of a new CP-mixed DM
mediator, assuming the SM plus a new pure CP-even DM mediator has already been
discovered. Here, $H_{0}$ is the SM plus a new CP-even DM mediator signal
hypothesis, while $H_{1}$ is the SM plus a new CP-mixed DM mediator signal.

For each scenario, these confidence levels (CLs) are computed for two
different luminosities: one roughly corresponding to the expected integrated
luminosity of the LHC Run 3, and the other to the High-Luminosity phase of the
LHC (HL-LHC). In each analyzed scenario, one hundred thousand pseudo-
experiments were generated by applying bin-by-bin Poisson fluctuations around
a mean value. This mean value is set to the number of events in each
individual bin of the angular distributions [70]. Evaluating whether a null
hypothesis ($H_{0}$) or an alternative hypothesis ($H_{1}$) can effectively
explain the pseudo-experiment involves assessing the probabilities associated
with each hypothesis. The likelihood ratio of the probabilities linked to
$H_{1}$ and $H_{0}$ serves as the test statistic for computing the confidence
level at which hypothesis $H_{1}$ can be excluded, assuming that $H_{0}$ is
true.

In Table 3, it is shown the impact of applying the asymmetry cuts in the
differential distributions. The data presented provides insightful comparisons
between full distribution shapes and post asymmetry cut distributions for
various observables. It is remarkable to observe the consistency in exclusion
limits across most observables, implying the robustness of these measurements
despite computing such values using full shape distribution and their
corresponding asymmetric 2-binned distribution. Notably, the minimal
variations in the exclusion limits, particularly for the
$\tilde{b}_{2}(\hat{y})$ and $n_{2}$ observables, reinforce the stability of
these parameters against the applied cuts. However, the observable $n_{4}$
stands out, showing a slight divergence in the scalar coupling exclusion
limits post asymmetry cut.

The computation of the exclusion limits in the $g_{u_{33}}^{S}-g_{u_{33}}^{P}$
plane are obtained using the distributions after the forward-backward cut. In
Figure 5, it is presented the exclusion limits for the $b_{2}$ observable, an
interesting trend is observed. Specifically, in Scenario 1, there is a notable
improvement in the constraint of the pseudoscalar component as the mass of the
mediator increases. This contrasts sharply with the results for Scenario 2,
where no such improvement in the exclusion limits for the pseudoscalar
component is evident. Similarly, Figure 6, focusing on the $n_{4}$ observable,
mirrors this pattern, with Scenario 1 showing improved constraints on the
pseudoscalar component with increasing mediator mass, while Scenario 2 does
not exhibit this enhancement. This discrepancy between the scenarios can be
attributed to the minimal contribution of the pseudoscalar component’s cross-
section in Scenario 2, leading to results that are predominantly influenced by
the scalar contributions.

Observable ($\alpha[\mathcal{O}_{CP}]$) | Full-shape ($\alpha[\mathcal{O}_{CP}]$) | Asymmetry $(A^{\alpha[\mathcal{O}_{CP}]}_{FB})$
---|---|---
$g_{u_{33}}^{S}$ | $g_{u_{33}}^{P}$ | $g_{u_{33}}^{S}$ | $g_{u_{33}}^{P}$
$b_{2}$ | $[-0.0475,0.0425]$ | $[-0.87,0.87]$ | $[-0.0425,0.0475]$ | $[-0.83,0.83]$
$\tilde{b}_{2}(\hat{y})$ | $[-0.0425,0.0425]$ | $[-0.87,0.87]$ | $[-0.0425,0.0425]$ | $[-0.87,0.87]$
$\tilde{b}_{2}(\hat{d})$ | $[-0.0475,0.0425]$ | $[-0.89,0.87]$ | $[-0.0425,0.0475]$ | $[-0.89,0.87]$
$n_{2}$ | $[-0.0425,0.0475]$ | $[-0.89,0.89]$ | $[-0.0425,0.0475]$ | $[-0.89,0.89]$
$n_{4}$ | $[-0.0425,0.0425]$ | $[-0.87,0.89]$ | $[-0.0450,0.0450]$ | $[-0.89,0.89]$
Table 3:  Exclusion limits at 68% for two different distribution shape
scenarios studied in this paper: 9-bin distributions (Full-shape) compared
with the 2-binned one post cut application in the distributions, namely
symmetrically applied cut (Forward-Backwards). The values refer to a particle
mediator mass of 10-2 GeV.

In Table 4, which presents exclusion limits for the coupling constant $g_{33}$
from $b_{2}$ asymmetries in forward-backward (FB) scenarios, it is observed a
consistent pattern across different mediator masses $m_{Y_{0}}$ and
luminosities. For both the scalar $S$ and pseudoscalar $P$ components of
$g_{u_{33}}$, the exclusion limits tighten as the luminosity increases from
300 fb-1 to 3000 fb-1, as expected. However, what stands out is the relative
stability of these limits across different mediator masses. For instance, at a
95% confidence level (CL), the exclusion limits for $g_{u_{33}}^{S}$ at 0.01
GeV and 125 GeV are quite similar, especially at the higher luminosity of 3000
fb-1. This consistency suggests that the exclusion limits are not
significantly sensitive to the mediator mass in this range, implying a
robustness in the predictive power of the model under varying mass conditions.
It also indicates that the physical phenomena driving these limits are not
overly dependent on the mass of the mediator particle, which is an interesting
observation from a theoretical standpoint.

---
Figure 5: Exclusion limits for the $\mathrm{SM}$ with a massive DM mediator in
the low mass region, $Y_{0}\left(m_{Y_{0}}=1\times 10^{-2},1\right.$ and the
Higgs benchmark case of $10\mathrm{GeV}$ in the top, middle, and bottom rows,
respectively). We have included the mixed scalar and pseudoscalar couplings
against the SM as null hypothesis (left), for the spin observable $b_{2}$ and
their corresponding exclusion limit obtained from the SM plus a mixed DM
mediator against the SM plus a pure scalar mediator as a null hypothesis
(right). Limits are shown for the expected luminosity of the Run 3 of the LHC
of $\left(L=300\mathrm{fb}^{-1}\right)$.
---
Figure 6: Exclusion limits for the $\mathrm{SM}$ with a massive DM mediator in
the low mass region, $Y_{0}\left(m_{Y_{0}}=1\times 10^{-2},10\right.$ and the
Higgs benchmark case of $125\mathrm{GeV}$ in the top, middle, and bottom rows,
respectively). We have included the mixed scalar and pseudoscalar couplings,
against the SM as null hypothesis (left), for the spin observable $n_{4}$ and
their corresponding exclusion limit obtained from the SM plus a mixed DM
mediator against the SM plus a pure scalar mediator as a nul hypothesis
(right). Limits are shown for a luminosity of the Run 3 of the LHC of
$\left(L=300\mathrm{fb}^{-1}\right)$.

Exclusion Limits from $b_{2}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0475]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.83,0.83]$ | $[-1.57,1.57]$ | $[-0.4725,0.4575]$ | $[-0.8775,0.8925]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2625]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1475]$
$g_{u_{33}}^{P}\in$ | $[-0.85,0.85]$ | $[-1.61,1.61]$ | $[-0.4725,0.4725]$ | $[-0.8925,0.8925]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.01,1.015]$ | $[-1.885,1.89]$ | $[-0.5625,0.5625]$ | $[-1.0575,1.0575]$
$g_{u_{33}}^{P}\in$ | $[-1.29,1.27]$ | $[-2.41,2.43]$ | $[-0.725,0.725]$ | $[-1.35,1.375]$
Table 4: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $b_{2}$
observable.

Additionally, Table 5 details the exclusion limits from $n_{4}$ asymmetries
FB, tells a similar story. Despite being derived from a different physical
basis than $b_{2}$, the results display a similar trend of minimal variation
in exclusion limits with changes in mediator mass. The scalar and pseudoscalar
components of $g_{u_{33}}$ again show a tightening of limits with increased
luminosity, but the impact of the mediator mass remains marginal. For example,
at 3000 fb-1 and a 95% CL, the exclusion limits for $g_{u_{33}}^{P}$ at
mediator masses of 1$\times 10^{-2}$ GeV and $125$ GeV are quite similar. This
similarity in the behavior of the exclusion limits across different
observables and mediator masses reinforces the notion that the underlying
physics of the model is less sensitive to the mass of the mediator particle.
The consistency across different observables and mass scales is significant,
as it underscores the model’s broad applicability and robustness, providing
confidence in its use for a range of scenarios in particle physics research.

Exclusion Limits from $n_{4}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0450,0.0450]$ | $[-0.0875,0.0875]$ | $-0.0225,0.0225$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.8900,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.5025,0.5025]$ | $[-0.9375,0.9375]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.135,0.135]$ | $[-0.255,0.255]$ | $[-0.075,0.075]$ | $[-0.1475,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.9100,0.8900]$ | $[-1.7100,1.6900]$ | $[-0.5025,0.5025]$ | $[-0.9525,0.9525]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.05,1.05]$ | $[-1.965,1.965]$ | $[-0.593,0.593]$ | $[-1.118,1.118]$
$g_{u_{33}}^{P}\in$ | $[-1.53,1.55]$ | $[-2.89,2.87]$ | $[-0.85,0.85]$ | $[-1.63,1.63]$
Table 5: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $n_{4}$
observable.

In summary, the analysis of the exclusion limits for the scenario 1 when
fixing the coupling constant $g_{33}$ from both $b_{2}$ and $n_{4}$
observables shows that: Across different mediator masses $m_{Y_{0}}$ and
luminosities, it is observed a consistent pattern of exclusion limits
tightening by increasing luminosity, yet showing minimal sensitivity to
changes in the mediator mass. This consistency is particularly notable as it
persists across different physical bases of the $b_{2}$ and $n_{4}$
observables. The relative stability of exclusion limits across varying
mediator masses suggests a robustness in the predictive power of the model for
this specific scenario, indicating that the underlying physics driving these
limits is not overly dependent on the mediator particle’s mass.

However, for the scenario 2, it is shown that the $n_{4}$ observable has a
dependence in the exclusion limit for the pseudoscalar component that varies
with the mediator mass. This is particularly visible in the case of the
forward-backward asymmetry distributions. This dependence with the mediator
mass scale can be observed in Figure 6, where an improvement in the exclusion
limit for the pseudoscalar is visible as the mediator mass approaches the
Higgs mass scale.

###  6.1 Exclusion limits in the total cross-section

In addition to the CLs for the different benchmark cases, the sensitivity of
the coupling constants with the luminosity is investigated. As a first
approach, this sensitivity is studied by changing the luminosity in the null
hypothesis, i.e. by fixing the pseudoscalar coupling constant to a vanishing
value. The focus is on determining exclusion limits in a scenario where a pure
scalar mediator is produced. As we have discussed before, all the constructed
CP observables should agree with their dependence on the luminosity because of
their same assigned parity. Figure 7, shows the dependence of the observed
95$\%$ and 68$\%$ confidence regions as a function of the luminosity for a
mediator mass of 1$\times 10^{-2}$ GeV.

It is noticeable that, as the luminosity increases, the ability to constrain
these constants improves by reducing the confidence interval by a factor of
two for the expected luminosity in the HLLHC. In addition, a flattening is
observed in the behavior of the exclusion limit with the luminosity, due to
the enhanced background production with luminosity, which leads to a reduction
of the exclusion limits for the same value of the cross-section. The same
observation is appointed to the observables coming from the spin basis
observables, i.e. $n_{4}$, where a similar behavior is obtained for the
exclusion limits with the luminosity as the $b_{2}$ (see Figure 7.)

Figure 7: Expected exclusion limits at 95% and 68% CL for the $g_{u_{33}}^{S}$
scalar coupling assuming vanishing pseudoscalar component $g_{u_{33}}^{P}=0$
as a function of the luminosity in fb-1 assuming the SM as the null hypothesis
for the most sensitive observables in this work $b_{2}$ (left) and $n_{4}$
(right).

The $b_{2}$ observable shows the small impact of the mass in the exclusion
limits for the scalar mediator production cross-section as a function of the
luminosity. In each case we have computed the cross-section, including NLO
corrections, for the range of mass values. Despite increasing the luminosity
beyond 2000 fb-1, the enhancement in the exclusion limit of the total cross-
section is not as noticeable as the one observed in the interval from 300 fb-1
to 600 fb-1. This finding suggests that a pure increase in luminosity is not
sufficient to improve these limits. Additionally, it is included the impact of
the same calculation by employing a different observable, i.e. $n_{4}$, for
the production of a pure-scalar mediator particle. Figure 8 shows the
exclusion limit for the $b_{2}$ and $n_{4}$ observables, as a function of the
luminosity, for the same benchmark mass values employed in this work.

Figure 8: Expected exclusion limits at 68% CL for the total cross-section
$\sigma_{ttY}$ for production of a pure scalar mediator particle $Y_{0}$ as a
function of the luminosity in fb-1 assuming the SM as the null hypothesis for
the most sensitive observables in this work $b_{2}$ (left) and $n_{4}$
(right).

The most sensitive spin basis observable shows an identical result as the
$b_{2}$ observable. As expected, the profile of the distribution is irrelevant
in the improvement of the total cross-section exclusion limit, as it has
already been shown from the forward backwards asymmetry study. The observed
consistency across different mediator masses reinforces the robustness of the
model and suggests that the understanding of the underlying physics is on
solid ground. Nevertheless, it also implies that exploring a wider range of
mediator masses might not yield new insights under the current experimental
conditions.

##  7 Conclusions

This study has conducted an in-depth examination of the 5 different DM
mediator particle $Y$ benchmark masses, i.e. 1$\times 10^{-2}$, 1, 10, 100 and
125 GeV, investigating its CP characteristics through scalar and pseudoscalar
interactions with the top quark. The analysis presented in this paper spanned
expected luminosities for Run 3 ($\sim 300\mathrm{fb}^{-1}$) up to the HL-LHC
era ($3000\mathrm{fb}^{-1}$), and it was centered on $t\bar{t}Y$ production in
leptonic decay channels for the top-quark, in the presence of a mediator
particle coupling with fermionic DM. From the reconstruction employed for DM
particle mediator $Y$, as a front-end result we have doubled the exclusion
limits in the scalar coupling of the model while going from the Run 3
luminosity to the 2000 fb-1. In addition, the approximation employed in the
reconstruction showed robustness at all levels in the analysis, at least for
the employed kinematic phase space with the imposed selection cuts on the
events. This validates its applicability to any interaction Lagrangian with a
DM mediator $Y$.

By introducing a suite of novel observables—$b_{2}$,$\tilde{b}_{2}$ with
projections along $\widehat{z}$, $\widehat{y}$, and $\widehat{d}$, as well as
$n_{2}^{t\bar{t}Y}$ and $n_{4}^{t\bar{t}Y}$ the exclusion limits within the
$g_{u_{33}}^{S}-g_{u_{33}}^{P}$ framework was refined. It is shown that all
the proposed observables in this work were sensitive to the CP-even and CP-odd
configurations. However, after imposing selection cuts and performing the
exclusion limits computation for the relevant couplings in the model, no
noticeable enhancement in the sensitivity of the $g_{u_{33}}^{P}$ coupling
through the application this set of observables has been found. Despite this
comprehensive approach, no significant sensitivity in the pseudoscalar
coupling $g_{u_{33}}^{P}$ was found by using spin polarization basis
observables, in contrast with previous studies involving the Higgs coupling to
the top quark [38].

Remarkably, the analysis demonstrated that asymmetries derived from angular CP
observables encapsulate the statistical significance of the dataset, obviating
the need for full-shape distribution analysis at least for type I scenarios.
This insight suggests that asymmetries alone are sufficient to encapsulate the
CP-even and CP-odd statistical information coming from the DM mediator $Y$ for
this type of exclusion limits analysis. However, this conclusion does not
apply for the spin observable $n_{4}$ from where a dependence with the mass
has been observed. This dependence with the mass for this observable implies
that for the FB analysis proposed in this work, the shape of the distribution
might be relevant for the exclusion of the CP nature of the mediator particle,
as it is drawn in the type 2 scenario.

## Acknowledgments

E.C. was supported by FCT, under the contract PRT/BD/154189/2022 and CERN/FIS-
PAR/0037/2021. R.M.S. was partially supported by FCT Grant No CERN/FIS-
PAR/0037/2021. The work of M.C.N.F. was supported by the PSC-CUNY Awards
65071-00 53 and 66073-00 54. A.O. was partially supported by FCT, under the
contract CERN/FIS-PAR/0037/2021. R.G was partially supported by FCT, under the
Contract CERN/FIS-PAR/0026/2021.

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##  Appendix A Exclusion Limits

Figures 9 and 10 illustrate the exclusion limits for the $g_{33}$ coupling
constant, graphically representing data from the detailed tables. These plots
provide a visual interpretation of the exclusion limits for different
exclusion limits configurations.

---
Figure 9: Exclusion of the SM with a massless scalar DM mediator , $Y_{0}$,
including NLO effects with mixed scalar and pseudoscalar Yukawa-like coupling
with the top quarks, against the SM as null hypothesis, for derived $b_{2}$
observable (top left), $\tilde{b}_{2}$ in the (0, 1, 1) direction of the
momentum plane (top right), $\tilde{b}_{2}$ in the (0, 1, 0) direction of the
momentum plane (bottom). Limits are shown for a luminosity of
$L=300\mathrm{fb}^{-1}$.
---
Figure 10: Exclusion of the SM with a massless scalar DM mediator, i.e.
$Y_{0}$, including NLO effects with mixed scalar and pseudoscalar Yukawa-like
coupling with the top quarks, against the SM as null hypothesis, for both spin
observables: $n_{4}$ (left), and $n_{2}$ (right). Limits are shown for a
luminosity of $L=300\mathrm{fb}^{-1}$.

##  Appendix B Full observables constraints

The subsequent tables provide a more detailed examination of the $g_{33}$
coupling constant’s exclusion limits, covering a broader range of mediator
masses and luminosities for the different observables proposed in this paper.

Exclusion Limits from $b_{2}$
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0475,0.0425]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.87]$ | $[-1.63,1.63]$ | $[-0.485,0.475]$ | $[-0.92,0.91]$
$1$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0725,0.0725]$ | $[-0.1425,0.1425]$ | $[-0.0425,0.0425]$ | $[-0.0775,0.0775]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.87]$ | $[-1.65,1.63]$ | $[-0.485,0.49]$ | $[-0.92,0.925]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2625]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1475]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.89]$ | $[-1.65,1.67]$ | $[-0.50,0.49]$ | $[-0.935,0.94]$
$100$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.76,0.74]$ | $[-1.41,1.415]$ | $[-0.4125,0.4275]$ | $[-0.7875,0.7875]$
$g_{u_{33}}^{P}\in$ | $[-1.25,1.27]$ | $[-2.37,2.35]$ | $[-0.70,0.70]$ | $[-1.325,1.325]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.01,1.015]$ | $[-1.91,1.915]$ | $[-0.5775,0.5775]$ | $[-1.0725,1.0725]$
$g_{u_{33}}^{P}\in$ | $[-1.37,1.35]$ | $[-2.57,2.59]$ | $[-0.775,0.775]$ | $[-1.45,1.45]$
Table 6: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $b_{2}$
observable. Exclusion Limits from $\tilde{b}_{2}^{(0,1,0)}$
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0425]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.87]$ | $[-1.65,1.65]$ | $[-0.485,0.49]$ | $[-0.935,0.925]$
$1$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0725,0.0725]$ | $[-0.1425,0.1425]$ | $[-0.0425,0.0425]$ | $[-0.0775,0.0775]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.89]$ | $[-1.67,1.67]$ | $[-0.50,0.49]$ | $[-0.935,0.94]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2625]$ | $[-0.0775,0.0775]$ | $[-0.1475,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.89]$ | $[-1.69,1.69]$ | $[-0.50,0.505]$ | $[-0.95,0.955]$
$100$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.76,0.765]$ | $[-1.435,1.44]$ | $[-0.4275,0.4275]$ | $[-0.8025,0.8025]$
$g_{u_{33}}^{P}\in$ | $[-1.37,1.39]$ | $[-2.57,2.59]$ | $[-0.775,0.775]$ | $[-1.45,1.45]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.06,1.04]$ | $[-1.985,1.965]$ | $[-0.5925,0.5925]$ | $[-1.1025,1.1025]$
$g_{u_{33}}^{P}\in$ | $[-1.57,1.55]$ | $[-2.93,2.91]$ | $[-0.875,0.875]$ | $[-1.625,1.625]$
Table 7: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the Scenario 1, are depicted for various $Y_{0}$ masses under fixed
luminosities of 300 fb-1 and 3000 fb-1. The limits are presented at 68% and
95% confidence levels for the $\tilde{b}_{2}^{(0,1,0)}$ observable. Exclusion
Limits from
$\tilde{b}_{2}^{(0,\nicefrac{{1}}{{\sqrt{2}}},\nicefrac{{1}}{{\sqrt{2}}})}$
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0475,0.0425]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.87]$ | $[-1.65,1.65]$ | $[-0.485,0.49]$ | $[-0.935,0.925]$
$1$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0725,0.0725]$ | $[-0.1425,0.1425]$ | $[-0.0425,0.0425]$ | $[-0.0775,0.0775]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.91]$ | $[-1.67,1.67]$ | $[-0.50,0.49]$ | $[-0.935,0.94]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2625,0.2575]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1475]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.91]$ | $[-1.69,1.69]$ | $[-0.50,0.505]$ | $[-0.95,0.955]$
$100$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.76,0.765]$ | $[-1.435,1.44]$ | $[-0.4275,0.4275]$ | $[-0.8025,0.8025]$
$g_{u_{33}}^{P}\in$ | $[-1.37,1.39]$ | $[-2.61,2.59]$ | $[-0.775,0.775]$ | $[-1.45,1.45]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.035,1.04]$ | $[-1.96,1.965]$ | $[-0.5925,0.5925]$ | $[-1.1025,1.1025]$
$g_{u_{33}}^{P}\in$ | $[-1.53,1.55]$ | $[-2.89,2.91]$ | $[-0.875,0.875]$ | $[-1.625,1.625]$
Table 8: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the Scenario 1, are depicted for various $Y_{0}$ masses under fixed
luminosities of 300 fb-1 and 3000 fb-1. The limits are presented at 68% and
95% confidence levels for the
$\tilde{b}_{2}^{(0,\nicefrac{{1}}{{\sqrt{2}}},\nicefrac{{1}}{{\sqrt{2}}})}$
observable. Exclusion Limits from $n_{2}$
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0475]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.8900,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.4875,0.4875]$ | $[-0.9375,0.9375]$
$1$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0725,0.0725]$ | $[-0.1425,0.1425]$ | $[-0.0425,0.0425]$ | $[-0.0775,0.0775]$
$g_{u_{33}}^{P}\in$ | $[-0.8900,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.5025,0.4875]$ | $[-0.9375,0.9375]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2575]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.9100,0.9100]$ | $[-1.7100,1.7100]$ | $[-0.5025,0.5175]$ | $[-0.9525,0.9675]$
$100$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.7600,0.7650]$ | $[-1.4350,1.4400]$ | $[-0.4275,0.4275]$ | $[-0.8025,0.8025]$
$g_{u_{33}}^{P}\in$ | $[-1.4100,1.3900]$ | $[-1.4100,1.3900]$ | $[-0.7750,0.8000]$ | $[-1.4750,1.4750]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.0350,1.0400]$ | $[-1.9850,1.990]$ | $[-0.5925,0.5925]$ | $[-1.1025,1.1175]$
$g_{u_{33}}^{P}\in$ | $[-1.5700,1.5900]$ | $[-2.9700,2.9500]$ | $[-0.8750,0.8750]$ | $[-1.6750,1.6750]$
Table 9: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $n_{2}$
observable. Exclusion Limits from $n_{4}$
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0425]$ | $[-0.0875,0.0875]$ | $-0.0225,0.0225$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.8700,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.5025,0.4875]$ | $[-0.9375,0.9375]$
$1$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0725,0.0725]$ | $[-0.1425,0.1425]$ | $[-0.0425,0.0425]$ | $[-0.0775,0.0775]$
$g_{u_{33}}^{P}\in$ | $[-0.8700,0.8900]$ | $[-1.6500,1.6500]$ | $[-0.4875,0.4875]$ | $[-0.9225,0.9225]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2575]$ | $[-0.0775,0.0775]$ | $[-0.1475,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.9100,0.8900]$ | $[-1.7100,1.6900]$ | $[-0.5025,0.5025]$ | $[-0.9525,0.9525]$
$100$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.7600,0.7650]$ | $[-1.4350,1.4400]$ | $[-0.4275,0.4275]$ | $[-0.8025,0.8025]$
$g_{u_{33}}^{P}\in$ | $[-1.3700,1.3500]$ | $[-2.5700,2.5900]$ | $[-0.7750,0.7750]$ | $[-1.4500,1.4500]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.0600,1.0400]$ | $[-1.9850,1.965]$ | $[-0.5925,0.5925]$ | $[-1.1175,1.1175]$
$g_{u_{33}}^{P}\in$ | $[-1.5300,1.5500]$ | $[-2.8900,2.8700]$ | $[-0.8500,0.8500]$ | $[-1.6250,1.6250]$
Table 10: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $n_{4}$
observable.

##  Appendix C FB asymmetries cut

The subsequent Tables provide a more detailed examination of the $g_{33}$
coupling constant’s exclusion limits computed using the foward-backward
symmetry cut in the distributions.

Exclusion Limits from $b_{2}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0475]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.83,0.83]$ | $[-1.57,1.57]$ | $[-0.4725,0.4575]$ | $[-0.8775,0.8925]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2625]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1475]$
$g_{u_{33}}^{P}\in$ | $[-0.85,0.85]$ | $[-1.61,1.61]$ | $[-0.4725,0.4725]$ | $[-0.8925,0.8925]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.01,1.015]$ | $[-1.885,1.89]$ | $[-0.5625,0.5625]$ | $[-1.0575,1.0575]$
$g_{u_{33}}^{P}\in$ | $[-1.29,1.27]$ | $[-2.41,2.43]$ | $[-0.725,0.725]$ | $[-1.35,1.375]$
Table 11: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $b_{2}$
observable. Exclusion Limits from $\tilde{b}_{2}^{(0,1,0)}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0425]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.87]$ | $[-1.65,1.67]$ | $[-0.4875,0.4875]$ | $[-0.9375,0.9225]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2625]$ | $[-0.0775,0.0775]$ | $[-0.1475,0.1475]$
$g_{u_{33}}^{P}\in$ | $[-0.89,0.91]$ | $[-1.71,1.69]$ | $[-0.5025,0.5025]$ | $[-0.9525,0.9525]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.06,1.065]$ | $[-1.985,1.99]$ | $[-0.5925,0.5925]$ | $[-1.1175,1.1175]$
$g_{u_{33}}^{P}\in$ | $[-1.57,1.55]$ | $[-2.93,2.91]$ | $[-0.875,0.875]$ | $[-1.65,1.65]$
Table 12: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $b_{2}$
observable. Exclusion Limits from
$\tilde{b}_{2}^{(0,\nicefrac{{1}}{{\sqrt{2}}},\nicefrac{{1}}{{\sqrt{2}}})}$
Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0422,0.0474]$ | $[-0.0876,0.0876]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.87,0.87]$ | $[-1.67,1.67]$ | $[-0.4875,0.4875]$ | $[-0.9375,0.9375]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.138,0.138]$ | $[-0.263,0.263]$ | $[-0.078,0.078]$ | $[-0.148,0.148]$
$g_{u_{33}}^{P}\in$ | $[-0.92,0.92]$ | $[-1.71,1.71]$ | $[-0.5025,0.5025]$ | $[-0.9525,0.9525]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.06,1.065]$ | $[-1.985,1.99]$ | $[-0.5925,0.5925]$ | $[-1.1175,1.1175]$
$g_{u_{33}}^{P}\in$ | $[-1.57,1.55]$ | $[-2.93,2.95]$ | $[-0.875,0.875]$ | $[-1.65,1.65]$
Table 13: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $b_{2}$
observable. Exclusion Limits from ${n}_{2}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0425,0.0475]$ | $[-0.0875,0.0875]$ | $[-0.0225,0.0225]$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.8900,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.4875,0.4875]$ | $[-0.9375,0.9375]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.1375,0.1375]$ | $[-0.2575,0.2575]$ | $[-0.0775,0.0775]$ | $[-0.1425,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.9100,0.9100]$ | $[-1.7100,1.7100]$ | $[-0.5025,0.5175]$ | $[-0.9525,0.9675]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.0350,1.0400]$ | $[-1.9850,1.990]$ | $[-0.5925,0.5925]$ | $[-1.1025,1.1175]$
$g_{u_{33}}^{P}\in$ | $[-1.5700,1.5900]$ | $[-2.9700,2.9500]$ | $[-0.8750,0.8750]$ | $[-1.6750,1.6750]$
Table 14: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $n_{2}$
observable. Exclusion Limits from $n_{4}$ Asymmetries FB
---
|  300 fb-1 |  3000 fb-1
$m_{Y_{0}}$ |  | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$0.01$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.0450,0.0450]$ | $[-0.0875,0.0875]$ | $-0.0225,0.0225$ | $[-0.0475,0.0475]$
$g_{u_{33}}^{P}\in$ | $[-0.8900,0.8900]$ | $[-1.6700,1.6700]$ | $[-0.5025,0.5025]$ | $[-0.9375,0.9375]$
$10$ GeV | $g_{u_{33}}^{S}\in$ | $[-0.135,0.135]$ | $[-0.255,0.255]$ | $[-0.075,0.075]$ | $[-0.1475,0.1425]$
$g_{u_{33}}^{P}\in$ | $[-0.9100,0.8900]$ | $[-1.7100,1.6900]$ | $[-0.5025,0.5025]$ | $[-0.9525,0.9525]$
$125$ GeV | $g_{u_{33}}^{S}\in$ | $[-1.05,1.05]$ | $[-1.965,1.965]$ | $[-0.593,0.593]$ | $[-1.118,1.118]$
$g_{u_{33}}^{P}\in$ | $[-1.53,1.55]$ | $[-2.89,2.87]$ | $[-0.85,0.85]$ | $[-1.63,1.63]$
Table 15: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings, considering
the CP-even component against irreducible backgrounds, are depicted for
various $Y_{0}$ masses under fixed luminosities of 300 fb-1 and 3000 fb-1. The
limits are presented at 68% and 95% confidence levels for the $n_{4}$
observable.