Boosted four-top production at the LHC : a window to Randall-Sundrum or
extended color symmetry

Debajyoti Choudhury<EMAIL_ADDRESS>Kuldeep
<EMAIL_ADDRESS>kuldeep.deka@nyu.edua,b, Lalit Kumar
Saini333sainikrlalit@gmail.coma,c,d,e

a Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India.

b New York University Abu Dhabi, Saadiyat Island, 129188, United Arab
Emirates.

c SGTB Khalsa College, University of Delhi, Delhi 110007, India.

d Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India.

e Homi Bhabha National Institute, BARC Training School Complex, Anushakti
Nagar, Mumbai 400094, India.

###### Abstract

Scenarios seeking to address the issue of electroweak symmetry breaking often
have heavy colored gauge bosons coupling preferentially to the top quark.
Considering the bulk Randall-Sundrum as a typical example, we consider the
prospects of the first Kaluza-Klein mode ($G^{(1)}$) of the gluon being
produced at the LHC in association with a $t\overline{t}$ pair. The enhanced
coupling not only dictates that the dominant decay mode would be to a
$t\overline{t}$ pair, but also to a very large $G^{(1)}$ width, necessitating
the use of a renormalised $G^{(1)}$ propagator. This, alongwith the presence
of large backgrounds (specially $t\overline{t}jj$), renders a conventional
cut-based analysis ineffective, yielding only marginal significances of only
around 2$\sigma$. The use of Machine Learning (ML) techniques alleviates this
problem to a great extent. In particular, the use of Artificial Neural
Networks helps us identify the most discriminating observables, thereby
allowing a significance in excess of 4$\sigma$ for $G^{(1)}$ masses of $\sim$
4 TeV.

###### Contents

  1. 1 Introduction
  2. 2 Model
  3. 3 Renormalised KK gluon propagator
  4. 4 Collider signatures
    1. 4.1 4-top Signal and 4-top SM background
    2. 4.2 Other major Backgrounds
  5. 5 Signal and Background simulations
    1. 5.1 Details of Simulation
    2. 5.2 Signal-Background profiles
      1. 5.2.1 Dominant Discriminating Observables
      2. 5.2.2 Subdominant discriminating Observables
    3. 5.3 Analysis
  6. 6 Neural Network Analysis
    1. 6.1 Artificial Neural Network(ANN) Model
    2. 6.2 ML architecture
    3. 6.3 NN Training
    4. 6.4 ANN results
  7. 7 Summary and Conclusion

##  1 Introduction

Despite its unprecedented success in describing experimental results, the
Standard Model (SM) is expected to be only an effective theory on account of
its failure to address a host of issues. Hence, despite the lack of any direct
evidence of new physics at the LHC, the search goes on. Of particular interest
is unravelling the nature of electroweak symmetry breaking (EWSB), which
continues to be be little-understood despite the discovery of a Higgs scalar.
With the top quark being the heaviest particle within the SM, with a mass
close to the EWSB scale, it is expected to provide a sensitive probe for not
only the symmetry breaking sector [1, 2], but also to a variety of possible
scenarios going beyond the SM. This is understandably true for models wherein
the new physics sector has enhanced couplings to the top quark, examples being
topcolor [3, 2], topcolor assisted technicolor [4] or certain models defined
on a warped 5-dimensional space. More interestingly, even in the absence of
such an enhancement (such as for generic coloron models [5, 6] or axigluons
[7, 8]), the unique signal profiles that, say, a $t\overline{t}$ final state
provides, as also the smaller (as compared to a dijet final state) SM
background that this channel is associated with, render (multi-)top-quark
production very attractive arenae to probe such theories.

The models mentioned above typically incorporate an extended color symmetry,
typically of the form $SU(3)_{I}\otimes SU(3)_{II}\to SU(3)_{c}$ with the
breaking occurring at a high enough scale. They differ from each other in both
the details of how $SU(3)_{c}$ is embedded in the larger group (including
whether the said group encompasses a chiral symmetry) as also in the
representations (and, hence, the couplings) of the SM fermions. Quite
understandably, the symmetry can be enhanced almost arbitrarily with extra
factors of $SU(3)$’s associated with different couplings. Furthermore,
differing symmetry breaking chains and/or embeddings of fermions would result
in different relations between the gauge couplings. In other words, these
models could be envisaged as different low-energy realizations (or
deconstructions) of a deeper theory.

The warped extra dimension framework due to Randall and Sundrum (RS) [9],
defined on a modest-sized (in the fifth direction) slice of AdS5, presents one
such scenario. Originally motivated as a solution to the Planck scale–weak
scale hierarchy problem within the SM, this was constructed by localizing the
usual (four-dimensional and massless) graviton near an end-of-the-world (UV)
brane attributed with a Planckian fundamental scale, whereas the SM sector
(including the Higgs) was localized on the opposite (IR) brane whence the
scale is protected by the warping-down of the fundamental scale. However, it
was soon realized that on considering the $5D$ effective field theory, higher-
dimensional operators are suppressed only by the warped-down scale ($\sim$ a
few TeVs), leading to untenably large contributions to flavour-changing
neutral current (FCNC) processes as well as electroweak precision observables
and/or proton decay.

Going beyond the original motivation and extending [10, 11, 12] the gauge and
fermion fields into the bulk (of the warped extra dimension) as well removes
some of these constraints. Compactifying the fifth dimension now results in
towers of fermions and gauge bosons, with the last-mentioned being the
analogues of the heavy gauge bosons in the extended color symmetry models. The
profiles (wavefunctions) of these Kaluza-Klein excitations (bosonic as well as
fermionic) would depend on the details of the warping (such as whether it is
engendered by a bulk cosmological constant alone or if, say, a nontrivial
dilaton field is involved as well) as well as the bulk masses of the fields
(other than the gauge fields). Localizing the first two generation fermions
near the Planck brane automatically suppresses the FCNC’s as well as
contributions to electroweak precision test observables [12, 13] on account of
the effective cutoff being nearly Planckian in the vicinity. Furthermore, with
the overlap of the wavefunctions playing a crucial role in the determination
of the effective Yukawa couplings, a solution to the SM flavour problem is now
obtainable with only a very moderate hierarchy in the bulk fermion masses.
Such welcome features has spawned an abiding interest in such Bulk RS models
[10, 14, 15, 16, 17, 18, 19, 20, 21].

In the majority of such models, the heavy colored gauge bosons have enhanced
couplings with at least some of the SM fermions, most often the top-quark.
While, in the generic coloron or axigluon models, this comes about in the
quest of a dynamical origin to EWSB, for the RS gluons (depicted by $G^{(1)}$)
this results from the aforementioned nontrivial profiles in the bulk.
Consequently, $t\overline{t}$ production (whether at the Tevatron or the LHC)
has long been a favoured channel to probe such models [22, 23, 24, 25].
However, such analyses need to be approached with care. Owing to the
aforementioned coupling enhancements, such a gluon has a naturally large width
and the narrow width approximation is often not valid [25, 26, 27]. And, away
from the peak, the signal often tends to get submerged in the extremely large
QCD rates. The channel also has a subdominant behaviour because of being
initiated by a quark-antiquark fusion (dominated by light quarks), where the
relatively small coupling of $G^{(1)}$ with the light quarks result in very
small production cross-section compared to the $t\overline{t}$ SM background.
Consequently, alternative channels need to be probed and, in this paper, we
consider the four-top ($t\overline{t}t\overline{t}$) signal. While this
channel has been considered in the past [28, 29] for various beyond-the-SM
scenarios, the large width presents its own challenges (both theoretical and
experimental), and we include the effects in self-consistently. The cross-
section for this channel is also very small, but we can leverage on the fact
that the two top quarks enamating from the heavy $G^{(1)}$ will be highly
boosted, giving us a way to discriminate over the towering SM backgrounds.
This, however, is not a simple task. As we shall see, simple cut-based
techniques cannot meaningfully discriminate the signal, with the main culprit
being the $t\overline{t}jj$ ($j$ being light jets) background. This compels us
to look at machine learning methods, specifically Artificial Neural Networks
(ANNs) for better discriminating power, and this we would discuss in great
detail.

The rest of this article is constructed as follows. In the next section, we
begin by outlining the class of scenarios that incorporate gluon resonances,
culminating with a specific choice that we would concentrate on as a template
model. Typically, such resonances tend to be rather broad, necessitating a
proper treatment of the propagator, and this we review in Section 3. In the
subsequent section, we delineate the signal characteristics at the Large
Hadron Collider, and discuss the various sources of backgrounds, reducible and
irreducible. In Section 5, we discuss in detail the signal and the (much
larger) background profiles and attempt a cut-based analysis, demonstraing how
the significance of a signal-excess could be established. As a complementary
approach, we examine, in Section 6, the efficacy of employing machine learning
techniques in this context. We establish that the use of Artifical Neural
Networks can significantly enhance the reach of the LHC. Finally, we conclude
in Section 7.

##  2 Model

The model of our interest is defined on a slice of AdS5 described by the
metric

$ds^{2}=e^{-2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$

with $\eta_{\mu\nu}={\rm diag}(1,-1,-1,-1)$ and the fifth dimension being a
segment of real line, namely $y\in[0,\pi r_{c}]$. The background gravity is
assumed to be defined by the Einstein-Hilbert action in five dimensions (with
a characteristic scale $M_{5}$ not too different from $M_{\rm Pl}$) augmented
by a substantial negative cosmological constant ($-\Lambda$), that is not so
large as to invalidate a semiclassical treatment. This leads to

$A(y)=-k|y|\ ,\qquad\quad k=\sqrt{\frac{\Lambda}{24M_{5}^{3}}}\ ,$

and for $kr_{c}\sim 12$ (easily achieved if $r_{c}$ is somewhat larger than
$M_{5}^{-1}$ as is also required for a semiclassical treatment to be valid),
the warping between the scales induced on the two end-of-the-world (i.e., at
$y=0$ and $y=\pi r_{c}$) 3-branes mirrors that between $M_{\rm Pl}$ and the
electroweak scale [9]. For the system to be consistent (i.e., for the Israel
junction conditions to be satisfied), the two branes need to be associated
with equal and opposite brane tensions related, in turn, to $\Lambda$.

The resolution of the hierarchy obviously requires a specific value for the
distance modulus $r_{c}$. This can be addressed by invoking a bulk-propagating
scalar field $\phi$ ascribed with a potential $V(\phi)$. A self-consistent
solution of such a graviton-radion system automatically generates a stabilized
solution for $r_{c}$. Both the graviton and the radion can be expanded in
terms of the respective Kaluza-Klein modes. The lowest (massless) graviton
mode has a profile peaked near the (UV) brane located at $y=0$, and naturally
has only a highly suppressed coupling with the SM fields which are localized
on the IR brane (at $y=r_{c}$). And while the higher modes have relatively
unsuppressed couplings (of the order of the electroweak gauge couplings), the
masses of these resonances (in the TeVs) render their effects small.

While the aforementioned (RS) model provides a natural solution for the
electroweak-Planck scale hierarchy, it does not address another vexing issue
within the SM, namely the flavour problem. Not only is the hierarchy between
the fermion masses unexplained, the low effective scale ($\sim$ TeV on account
of the warping-down) of new physics implies that higher-dimensional operators
do not suffer large suppressions and, thus, unwanted features such as flavour-
changing neutral current (FCNC) processes or proton-decay stand to have
uncomfortably large amplitudes.

An attractive solution to this problem appears if we allow the SM fields to
propagate in the bulk [10, 14, 11, 12]. The SM fermion is now identified with
the chiral zero-mode of the $5D$ fields, with the profile (localization) along
the extra dimension being determined by its $5D$ mass parameter. Very moderate
hierarchies in these parameters change the profiles considerably.
Understandably, localizing the fermion fields towards the UV (IR) branes
implies light (heavy) masses for the low-lying modes as the Higgs is
resolutely localized on (or close to) the IR brane. This automatically ensures
that the FCNCs from higher-dimensional operators are suppressed by effective
cut-off scales (typically, $\gg$ TeV) at the location of these fermions [12,
13]. Quite analogously, contributions to the electroweak precision observables
are suppressed too.

The propagation of the fermions into the bulk, of course, necessitates that
the gauge bosons must do so too. Once again, the lowest-lying mode is massless
and has a trivial profile. The higher KK-modes, though, are massive and have
nontrivial profiles. It is the last mentioned property that, coupled with the
non-identical profiles of the fermions, leads to unequal couplings of the SM
fermions (i.e., the respective lowest-lying modes) to the higher KK-gauge
bosons and, thereby results in FCNCs. However, the very structure of the
theory ensures that the non-universal part of these couplings are proportional
to the SM Yukawa couplings [12, 13]. With the new degrees of freedom being
heavy and their couplings to fermions of the first two generations being small
(a consequence of the respective localizations), there exists an approximate
symmetry structure that suppresses the FCNCs (at least amongst the lowest-
lying modes) adequately [30]. And while additional contributions to the
electroweak precision variables (primarily, the $S$ and $T$ parameters) emerge
from the gauge KK modes, the ensuing constraints can be satisfied for a KK
mass scale as low as $\sim 3$ TeV once a custodial isospin symmetry is imposed
[31].

We are now in a position to consider the interactions of the gluons
$G_{\mu}^{(n)a}$ of the $n$-th KK level with the SM fermions. The interaction
Lagrangian of interest can be expressed as

$\displaystyle\mathcal{L}^{(n)}_{RS}\,$ $\displaystyle=$
$\displaystyle\,-\,\frac{1}{4}G^{(n)a}_{\mu\nu}\,G^{(n)a\,\mu\nu}-m_{n}^{2}\,G^{(n)a}_{\mu}\,G^{(n)a\,\mu}+\sum_{q}\sum_{j=L,R}g^{(n)}_{q_{j}}\overline{q_{j}}\,\gamma^{\mu}\,T^{a}\,q_{j}\,\,G^{(n)a}_{\mu}$
(1)

where $q$ denotes the flavour and $g^{(n)}_{q_{j}}$ the corresponding coupling
to a quark of flavour $q$ and chirality444Note that, even for the same
flavour, the left– and right-chirality fields descend from different 5D fields
and, hence, can be localised differently. $j$. In particular, we would be
interested in the top/bottom sector. Clearly, if we seek to localize both
$t_{L,R}$ near the UV-brane, the ensuing top Yukawa coupling would be too
small. On the other hand, localizing555Being part of a gauge multiplet, these
two would, perforce, have to be localized identically as the 5D Lagrangian
must be manifestly gauge invariant. $(t,b)_{L}$ close to the IR-brane leads to
it having a relatively large coupling to the $Z_{\mu}^{(n)}$ (for $n>0$). The
mixing of the latter with the ordinary $Z$ (i.e., $Z_{\mu}^{(0)}$), in turn,
results in a non-universal shift in the latter’s coupling to the $(t,b)_{L}$
[31]. In particular,

$\delta g^{(Z)}_{b_{L}}\sim\sqrt{\log\frac{M_{\rm Pl}}{1\mbox{ TeV
}}}g^{Z^{(n)}}_{b_{L}}\frac{m^{2}_{Z}}{M_{n}^{2}}$

where $g^{Z^{(n)}}_{b_{L}}$ is the (non-universal) coupling of the $Z^{(n)}$.

With experimental constraints (emanating largely from electroweak precision
tests) stipulating that $\delta
g^{(Z)}_{b_{L}}/g^{(Z)}\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}1.4\%$[32], it is evident that if
the mass of the first KK-resonance $\simeq$ a few TeV, there is a seeming
tension between this constraint and the need to have a large top mass. This
can, however, be relaxed, even for dimensionless $5D$ Yukawa couplings
consistent with perturbativity. An example is afforded by quasi-localizing the
$(t,b)_{L}$ doublet near the IR-brane (so that the shift $\delta
g^{(Z)}_{b_{L}}$ remains consistent with the data) and simultaneously
localizing $t_{R}$ very close to the IR-brane so as to enable a large top
quark mass. Note that the resulting coupling of the $b_{L}$ to the gauge KK
modes (including gluon) is comparable to the SM couplings and, in being
dictated by the large top mass, is, thus, still larger than what is expected
on the basis of $m_{b}$ alone. Consequently, even with such choices, the KK
scale is required to be rather high, namely, $\sim 5$ TeV.

As can be imagined, the constraints do depend on the symmetries imposed and
the assignments, e.g. the representation of the heavy quarks under the
custodial isospin symmetry [33]. Indeed, for certain choices of profiles for
$t_{R}$ and $(t,b)_{L}$, the KK scale can be as low as $\sim 3$ TeV [34]. In
the rest of this paper, we consider the assignments of Ref. [23], wherein the
couplings of $G^{(1)}$ to a light quark (including the $b_{R}$) pair are
suppressed by a constant666Typically, $\xi\sim\sqrt{\log(M_{\rm Pl}/1\mbox{
TeV})}\sim 6$. $\xi$ with respect to the standard QCD coupling. As for its
coupling to a gluon pair, it is even further suppressed (vanishing at the
leading order) owing to the orthogonality of profiles. In other words, the
$G^{(1)}$ is “proton-phobic”. The coupling to $(t_{L},b_{L})$ is largely
unchanged. As for the $t_{R}$, which is localized very close to the IR brane
(or, in the dual picture, is a composite), its coupling to the $G^{(1)}$ is
enhanced by $\xi$. In other words, and using an obvious notation777It should
be realized that these outcomes are not special to the $G^{(1)}$, but are also
applicable to the other gauge KK-modes. However, those do not concern us.,

$\begin{array}[]{rclcrcl}g_{qqG^{(1)}}&\simeq&\xi^{-1}g_{s}&&g_{ggG^{(1)}}&\approx&{\Large
0}\\\\[8.61108pt]
g_{Q_{3}Q_{3}G^{(1)}}&\approx&g_{s}&&g_{t_{R}t_{R}G^{(1)}}&\approx&\xi g_{s}\
,\end{array}$ (2)

where $g_{s}$ is the usual strong coupling, $Q_{3}\equiv(t,b)_{L}$ and $q$
comprises all of $u_{L,R},d_{L,R},s_{L,R},c_{L,R}$ as well as $b_{R}$. For
most of our numerical work, we would use $\xi=5$ (as advocated in Ref.[23]) as
our benchmark value.

##  3 Renormalised KK gluon propagator

While it is commonplace to consider the Breit-Wigner form as the propagator
for a massive and unstable particle, this is straightforward only in the case
of a very narrow width. Else, care needs to be taken to preserve gauge
invariance. In the present case, the enhanced coupling of the KK gluon to the
right-handed top quarks leads to a significantly large decay width of the KK
gluon as given by

$\begin{array}[]{rcl}\Gamma&=&\sum_{q}\Gamma(G^{(1)}\to q\overline{q})\,\ =\,\
\dfrac{M_{G^{(1)}}}{48\pi}\sum_{q}\sqrt{1-\dfrac{4m_{q}^{2}}{M_{G^{(1)}}}}\left[(g_{q_{L}}^{2}+g_{q_{R}}^{2})\left(1-\dfrac{m_{q}^{2}}{M_{G^{(1)}}}\right)+6g_{q_{L}}g_{q_{R}}\dfrac{m_{q}^{2}}{M_{G^{(1)}}}\right]\
.\end{array}$ (3)

where the couplings $g_{q_{L}}$ and $g_{q_{R}}$ are as in Eq. 2. For a
sufficiently heavy $G^{(1)}$ and a large enough value of $\xi$ (as we are
interested in), we have $\Gamma\approx\xi^{2}\alpha_{s}M_{G^{(1)}}/12$, which
translates to several hundreds of GeVs. Such a broad resonance has to be dealt
with carefully, and gauge invariance ensured [27].

Reminding ourselves that the Breit-Wigner form originated from the imaginary
part of the self-energy correction (courtesy the optical theorem and the
Cutkosky rules), we begin by listing the same for the $G^{(1)}$. To the one-
loop order, this is straightforward and, for a $G^{(1)}$ of momentum $k$, is
given by888Since we would be primarily interested in the absorptive part of
the self-energy correction, we present here only the contributions of the SM-
quark loops.

$i\Pi^{ab}_{\mu\nu}(k^{2})=i\delta^{ab}\left[g_{\mu\nu}k^{2}\Pi(k^{2})-k_{\mu}k_{\nu}\Delta_{\Pi}(k^{2})\right]\
,$ (4)

with $\Pi(k^{2})=\overline{\Pi}(k^{2})+\delta M^{2}/k^{2}$ and

$\begin{array}[]{rcl}\overline{\Pi}(k^{2})&=&\Delta_{\Pi}(k^{2})+\dfrac{1}{16\pi^{2}}\sum_{q}(g_{q_{L}}-g_{q_{R}})^{2}\dfrac{m_{q}^{2}}{k^{2}}\left[2+\mathcal{F}_{q}(k^{2})\right]\
,\\\\[12.91663pt]
\Delta_{\Pi}(k^{2})&=&\dfrac{-1}{48\pi^{2}}\sum_{q}(g_{q_{L}}^{2}+g_{q_{R}}^{2})\left[\dfrac{2}{\varepsilon}-\gamma_{E}+\ln\dfrac{4\pi\mu^{2}}{m_{q}^{2}}+\dfrac{5}{3}\left(1+\dfrac{12m_{q}^{2}}{5k^{2}}\right)+\left(1+\dfrac{2m_{q}^{2}}{k^{2}}\right)\mathcal{F}_{q}(k^{2})\right]\
,\\\\[12.91663pt] \end{array}$ (5)

Here $\varepsilon$ is the regulator of the ultraviolet divergence,
$\gamma_{E}$ is the Euler-Mascheroni constant, $\mu$ is the arbitrary scale
which appears in dimensional regularization, and $m_{q}$ is the quark mass.
The loop integrals $\mathcal{F}_{q}(k^{2})$ are functions of the familiar
kinematic variable $\beta_{q,k}\equiv(1-4m_{q}^{2}/k^{2})$, namely

$\mathcal{F}_{q}(k^{2})=\left\\{\begin{array}[]{ll}\sqrt{\beta_{q,k}}\left(\ln{\dfrac{1-\sqrt{\beta_{q,k}}}{1+\sqrt{\beta_{q,k}}}}+i\pi\right)&\quad\mbox{for}\quad
k^{2}\geq 4m_{q}^{2}\\\\[12.91663pt]
-2\sqrt{-\beta_{q,k}}\;\arctan{\dfrac{1}{\sqrt{-\beta_{q,k}}}}&\quad\mbox{for}\quad
k^{2}<4m_{q}^{2}\ .\end{array}\right.$ (6)

Defining the subtraction scheme so that the mass is given by

$\delta
M^{2}=\dfrac{1}{16\pi^{2}}\sum_{q}(g_{q_{L}}-g_{q_{R}})^{2}m_{q}^{2}\left(\dfrac{2}{\varepsilon}-\gamma_{E}+\ln\dfrac{4\pi\mu^{2}}{m_{q}^{2}}\right)\
,$ (7)

the corrected KK-gluon propagator, to one loop, after summing all the one-
particle-reducible diagrams is

$\begin{array}[]{rcl}iG_{\mu\nu}^{ab}&=&iG_{\mu\nu}^{(0)ab}+iG_{\mu\alpha}^{(0)ac}i\Pi^{\alpha\beta}_{cd}(q^{2})iG_{\beta\nu}^{(0)db}+\cdots\\\\[8.61108pt]
&=&i\delta^{ab}\left[\dfrac{-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{k^{2}}}{k^{2}-M_{0}^{2}}\left(1+\dfrac{k^{2}\Pi(k^{2})}{k^{2}-M_{0}^{2}}\right)+\dfrac{k_{\mu}k_{\nu}}{k^{2}}\dfrac{1}{M_{0}^{2}}\
\mbox{terms}+\cdots\right]\\\\[17.22217pt]
&=&i\delta^{ab}\left(\dfrac{-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{k^{2}}}{k^{2}-M_{0}^{2}-k^{2}\Pi(k^{2})}+\dfrac{k_{\mu}k_{\nu}}{k^{2}}\dfrac{1}{M_{0}^{2}}\
\mbox{terms}\right)\ ,\end{array}$ (8)

where
$iG_{\mu\nu}^{(0)ab}=i\delta^{ab}\frac{-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{M_{0}^{2}}}{k^{2}-M_{0}^{2}}$
is the propagator at zeroth order in the unitary gauge and $M_{0}$ the KK-
gluon bare mass. Contributions from the $k_{\mu}k_{\nu}$ terms are
proportional to the masses of the external particles
($m_{q}^{2}/M_{G^{(1)}}^{2}$) and, hence, are small indeed (even for the top-
current). This, then, provides the proper expression for the propagator to
this order.

##  4 Collider signatures

With the $gG^{(1)}G^{(1)}$ coupling being unsuppressed (a consequence of gauge
invariance), the naive expectation would be that QCD-driven pair production
would constitute the dominant mode at the LHC. However, the very large mass of
the $G^{(1)}$ leads to a kinematic suppression. Similarly the rather
suppressed $ggG^{(1)}$ and $q\overline{q}G^{(1)}$ (for light quarks) couplings
come in the way of resonant single production cross section being substantial.
Although the last mode has been examined in the literature [22, 23, 24, 25],
we eschew it altogether.

###  4.1 4-top Signal and 4-top SM background

Instead, the process of interest to us is $G^{(1)}$ production in association
with a $t\overline{t}$ pair, where we take advantage of the rather enhanced
$t\overline{t}G^{(1)}$ coupling. The very same fact (of enhanced coupling)
also stipulates that any $G^{(1)}$ thus produced must decay primarily into a
$t\overline{t}$ pair. It must be borne in mind that the large mass of the
$G^{(1)}$ and the large $t\overline{t}G^{(1)}$ coupling, together, leads to a
large width for the KK-gluon (as discussed in the preceding section). These
twin facts (large mass and large width) together imply that a very substantial
part of the signal would emanate from an off-shell $G^{(1)}$. In other words,
the processes of interest are

$gg,q\overline{q}\to t\overline{t}G^{(1)(*)}\to t\overline{t}t\overline{t}$
(9)

where the gluon-initiated subprocess, naturally, dominates. Clearly the
process, as seen in terms of the intial and final states, also occurs purely
within the SM, and given the fact that the signal events are not associated
with a narrow resonance, the interference between the SM and NP amplitudes is
not negligible999This is not unique to this process and similar effects have
been seen to be important in other contexts[35, 26, 27] as well. and may
significantly affect the event profile.

The 4-top final state within the SM has been well-studied and, at the next-to-
leading order [36], has a cross-section of about 15.7 fb. The signal cross
section presents a more complicated picture, though. Had it been dominated by
the production (and subsequent decay) of a narrow $G^{(1)}$ resonance, clearly
the cross-section would have scaled largely as
$\left[g_{t_{R}}^{(1)}\right]^{2}$. Indeed, even accounting for the large
width of the KK-gluon, the total amplitude of the $G^{(1)}$-mediated diagram
would still be nearly proportional to $g_{t_{R}}^{(1)}$. However, the
considerable interference between the NP and the SM amplitudes means that the
size of the signal, defined as

$\sigma_{\rm Signal}\equiv\sigma_{SM+NP}(4t)-\sigma_{SM}(4t)$ (10)

is no longer a simple function of $g_{t_{R}}^{(1)}$. This is exhibited in the
behaviour of the total cross sections as displayed in Fig.1.

Figure 1: The total signal cross section (as defined in Eq. 10) as a function
of the the KK-gluon mass for various values of $\xi$.

Furthermore, the considerable similarity between the signal and the
irreducible background (defined as $\sigma_{SM}(4t)$) implies that the
separation between the two in the data would not be straightforward. This is
particularly worrisome as, for allowed values of the couplings and masses,
$\sigma_{\rm Signal}$ tends to be smaller than $\sigma_{\rm SM}(4t)$ and
similar signal profiles would render difficult the task of separating the
signal from the background. On the other hand, this very similarity allows us
to use the NLO $K$-factor, as derived [36] within the SM, to be used for the
entire sample, without risking large inaccuracies. It might be argued here
that the large mass of the $G^{(1)}$ would imply that the $t\overline{t}$ pair
coming off it would be highly boosted and, hence, render the topologies
markedly different. However, while the first part of the argument is certainly
true (and would be used to our advantage), note that the large width of the
KK-gluon, convoluted with the fact that the parton-fluxes fall off rapidly at
large subprocess center-of-mass energies, results in a considerable similarity
between the events so as to render this approximation (of similar $K$-factors)
to be quite admissible.

To understand the possible event profiles, it is necessary to consider the
decays of the particles produced in the primary process. As is well-known, a
top quark decays overwhelmingly into a $b$-quark and a $W$-boson with the
latter, in turn, decaying promptly into a pair of light quarks or to a lepton-
neutrino pair. In other words, the four-top state would almost always have
four $b$-jets; this apart, it can be classified into five categories [37],
namely a completely hadronic (jets) state, or states with hard and isolated
charged leptons (the count varying from one to four) accompanied by missing
transverse momenta. (In listing this, we are neglecting the further decay of
the $b$-hadrons.) In the context of the signal-like events, the large mass of
the $G^{(1)}$ means that the $t\overline{t}$ pair from the $G^{(1)}$ leg would
tend to be highly boosted. Consequently, their decay products would tend to be
collimated and lead to “fatjets”. In anticipation of this, we demand that
there be two fatjets in the event, with each having a jet mass around $m_{t}$.
We still need to decide how the other two tops decay. As we would see in the
subsequent sections, opting for the hadronic decay of one top and leptonic
decay of the other would give us the largest signal significance.

###  4.2 Other major Backgrounds

Several other disparate processes may potentially contribute to the
background, depending on how the signal profile is defined. For example, if
all the tops were to decay hadronically, then, QCD multijet production, with a
cross section that is several orders of magnitude larger, could, presumably,
overwhelm the signal. It is here that the fact that the signal events would,
typically, contain two fatjets, each reconstructing to the top (even if with a
non-negligible experimental tolerance) plays a crucial role. This requirement,
immediately, ensures that only those SM events that contain at least two tops
would contribute majorly to the background. Apart from the SM four-top events,
these include:

  * •

$t\overline{t}jj$ : With $t\overline{t}$ production cross section being very
large indeed, one would assume that this particular cross section would also
be very large, if only on account of initial-state and final-state radiation.
The size of the cross section, of course, depends on the demands made on the
transverse momenta of the jets as well as their separation from the beam
directions and each other. Canonical basic requirements are that the putative
jets be central (with rapidity obeying $|\eta|\leq 3$), have $p_{T}(j)\geq 20$
GeV after ensuring that they are reconstructible as jets and are angular-
separated101010Here, $(\Delta R)^{2}\equiv(\Delta\eta)^{2}+(\Delta\phi)^{2}$,
with $\Delta\eta$ and $\Delta\phi$ being the separation between the jets in
terms of their rapidities and azimuthal angles. by $\Delta R_{jj}>0.4$. Ref.
[38] estimates a cross section ($\sim\,$ 106 pb) for 14 TeV at NLO with the
corresponding K-factor 0.89.

Although only a small fraction of the tops in such a sample would be boosted,
yet a few of those undergoing three-pronged hadronic decays would still be
reconstructed as a fatjet, especially if a large radius $R$ is used for the
jet recontruction algorithm. Consequently, the much larger production cross
section renders this a major background. One way to reduce this background is
to demand for two independent $b$-tagged jets accompanying the top fatjets.
For a genuine 4-top event, such $b$’s would emerge from the decays of the less
energetic tops, and the signal would not be impacted by this cut. The
$t\overline{t}jj$ background would, however, reduce drastically. Additional
backgrounds accrue from events wherein a charm quark ($c$) is sometimes
mistagged as a $b$ jet. Such mistagging, though, has only a $10\%$
probability, whereas the probability of other quarks and gluons being
mistagged as a $b$ jet is less than $0.1\%$. An useful tool to further
suppress all such background is to demand one or more isolated hard leptons
(which, for the signal events, can emanate from the leptonic decay of the
$W$).

  * •

$t\overline{t}b\overline{b}$ : Compared to the preceding case
($t\overline{t}jj$), it has a much smaller cross-section ($\approx$ 2.64 pb)
for 14 TeV at NLO with the corresponding K-factor 1.77 [39, 40]). Once again,
demanding two top fatjets would reduce this background substantially. On the
other hand, it would not be further suppressed on demanding two $b$-tagged
jets. However, demanding one or more leptons as argued for in the previous
case would make this background almost negligible.

  * •

$t\overline{t}W$ : This has a comparatively smaller cross-section of $0.577$
pb for 13 TeV at NLO with the corresponding K-factor 1.58 [36], and is
expected to reduce drastically once we demand two top fatjets and two
$b$-tagged jets.

##  5 Signal and Background simulations

As the discussion in the preceding section shows, the case of both of the two
softer top quarks decaying fully hadronically would be associated with a large
SM background (primarily from $t\overline{t}jj$ production). In view of this,
we stipulate that the signal events must have at least one of the tops
decaying semileptonically (in particular, to a state containing either an
electron or a muon). The somewhat smaller branching fraction ($\approx 8/27$
as opposed to $\approx 4/9$ for the fully hadronic mode) is more than
compensated for by the background reducing manifold. Thus, we would demand
that the final state contains two fatjets (with masses close to $m_{t}$),
additional (at least four more) jets, a hard isolated light lepton ($e^{\pm}$
or $\mu^{\pm}$) and some missing tranverse momentum.

###  5.1 Details of Simulation

Implementing the model in FeynRules [41, 42], we generate signal and
background events using MadGraph5_amc@nlo [43] augmented by the implementation
of the renormalised RS gluon propagator as discussed in Sec 3. To this end, we
use the default dynamic scale ($\mu$) choice in MadGraph5_amc@nlo (which
equals the central transverse mass ($m_{T}$) given by $m_{T}$ =
$\sqrt{m^{2}+p^{2}_{T}}$ , $m$ being the mass of the top, and $p_{T}$ the
transverse momentum after the $k_{T}$ clustering on LHE-level final states
111111https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/FAQ-General-13) and
the NNPDF parton distributions [44]. Interfacing with Pythia8 [45] for parton
showering and fragmentation, the events are then passed through Delphes-3
[46], in order to implement detector effects and apply reconstruction
algorithms. The multiple jets in the event (including the two boosted jets
from the $G^{(1)}$ leg) are reconstructed and identified with the FastJet
module using the anti-$k_{T}$ algorithm. To start with, we only demand
$p_{T}\geq 20$ GeV and $|\eta|\leq 2.4$ for all jets in the event. As for the
radius parameter $R$ used in reconstructing a fatjet, we investigated the
performance for different values and found $R\approx 0.5$ to be an optimum
choice. Too small a value of $R$ would result in very few events being
reconstructed as containing fatjets, while too large a value would result in
virtually all events being reconstructed as having a fatjet. Given the mass of
the $G^{(1)}$ being explored, it can actually be estimated that $R=0.5$ would
be optimal in capturing both the (boosted) tops from the $G^{(1)}$ being
reconstructed as a fatjet, while simultaneously rejecting the less energetic
background tops owing to their failure to reconstruct within a similar cone.
For other parameters of the jet-substructure observables, we use the default
values of the Delphes card from Delphes-3 [46]. For lepton isolation too, we
use the default values, which requires that there be no other charged
particles with $p_{T}>0.5$ GeV within a cone of $\Delta R<0.5$.

In the rest of this section, as also in the next, we include all the major
backgrounds mentioned in the preceding section as also the subdominant ones
(without showing the latter in the plots). As for the signal, this is defined
as the $\overline{t}t\overline{t}t$ sample, where the SM contribution has been
exactly subtracted out (see Eq.10). In other words, it is defined as the sum
of the contributions of the pure $G^{(1)}$ piece as well as that due to the
interference of the $G^{(1)}$-mediated amplitude with the SM one.

###  5.2 Signal-Background profiles

####  5.2.1 Dominant Discriminating Observables

We begin by examining the distributions in various kinematic variables as this
would allow us to determine the optimal cuts for effecting the signal-
background discrimination. With four tops in the final state, there are many
possible independent variables, but only some of them will stand out as good
discriminating variables and we start with these. To aid this, we begin by
ordering the jets in terms of their transverse momenta ($p_{T}$). As the
$G^{(1)}$ is expected to be heavy, we would expect the two tops coming off it
to be highly boosted and their decay products collimated. In other words, the
$p_{T}$s of the two leading jets are expected to be large for the signal
events. Similarly, the masses for these individual jets are also expected to
reflect their origin. Furthermore, if these two jets were the putative top-
fatjets, their invariant mass would tend to be large. Finally, one would
expect $H_{T}$, the scalar sum of the transverse momenta of all the jets in
the event to be large too.

Figure 2: The signal (for $M_{G^{(1)}}$ = 4 TeV) and background distributions
in $p_{T}$ of the leading and next-to-leading jets (left top and left bottom),
the corresponding jet masses (middle top and middle bottom), the invariant
mass of the two jets (right top) and $H_{T}$ (right bottom).

In Fig. 2, we display the distributions in the aforementioned variables for
the leading backgrounds as well as the signal (for a typical value of the
$G^{(1)}$-mass, namely 4 TeV). As expected, for the signal events, the $p_{T}$
of the two leading jets have a much broader distribution as compared to the
backgrounds. As Fig. 2 suggests, a strong cut on the $p_{T}$s of the two
leading jets would serve to suppress a much larger fraction of the two major
backgrounds ($t\overline{t}jj$ and $t\overline{t}b\overline{b}$) as also the
subleading ones than it would affect the signal. Understandably, the
$p_{T}(j_{1})$ distribution of the SM 4-top background is the closest to the
signal, a consequence of the somewhat similar event topologies. On the other
hand, the corresponding $p_{T}(j_{2})$ distribution is much softer than that
for the signal. This can be understood from the fact that while the “signal-
like” events would largely correspond to the $G^{(1)}$ (on-shell or off-shell)
decaying to two tops with relatively large and comparable $p_{T}$s (as
indicated by the left-panels of Fig. 2). But for the SM-like 4-top sample, the
absence of any heavy resonance makes the $p_{T}$ distribution of the next-to-
leading jet much softer compared to the signal.

The very hardness of the two leading top-jets facilitates their reconstruction
within a cone of radius $R=0.5$ (for a three-prong fatjet, the optimal choice
being given by $\Delta R={3m_{j}}/{2p_{T}}$). For the signal events, the jet
mass distribution for the two leading jets shows (see Fig. 2) three distinct
peaks, where the one at low masses corresponds to $b$-jets or other secondary
QCD radiation. The other two nontrivial peaks correspond to the $W$ boson mass
and the $top$ itself when they are reconstructed within a radius of $R=0.5$.
The SM 4-top background also shows a similar behavior for the leading jet,
albeit with a less prominent peak around the top mass whereas the sub-leading
jet for this background does not carry any evidence of a top-quark at all.
This difference is but a consequence of the $p_{T}$ distribution for the
second jet as discussed above. The other backgrounds have a prominent peak at
low masses corresponding to the QCD radiation and drop rapidly as we move
towards larger masses, with the only exception being $t\overline{t}W$ which
shows a prominent peak at $W$ mass for the leading jet and a slightly obscured
peak for the sub-leading jet.

Given the fact that the $p_{T}$ distributions are much harder (especially for
the second jet) for the signal events, it is conceivable that this might be
true as well for the other jets as well. Indeed, as Fig.2 shows, the variable
$H_{T}$ (for the signal) has a much harder distribution as well. And although
it is related, in a fashion, to the two aforementioned $p_{T}$ distributions,
this variable obviously carries extra information and can, hence, be used as
an independent discriminator.

Finally, for the signal, we also expect the invariant mass of the two most
energetic top quarks to be centred around the $G^{(1)}$ mass, with the width
of the distribution comparable to the decay width. This is illustrated by the
plot for the invariant mass for the two leading jets, where the distribution
shows a flat trajectory till large values of $M_{j_{1}j_{2}}$ (and falling off
sharply once $M_{G^{(1)}}$ is crossed). This is expected to be accentuated
further once the two leading jets can be associated with top quarks with
greater certainty. On the other hand, the distributions for the other
backgrounds drop rapidly as we go towards higher $M_{j_{1}j_{2}}$.

####  5.2.2 Subdominant discriminating Observables

Figure 3: The signal (for $M_{G^{(1)}}$ = 4 TeV) and background distributions
in $p_{T}$ of the third and fourth leading jets (left top and left bottom),
the corresponding jet masses (middle top and middle bottom), the $p_{T}$ of
the leading lepton (right top) and the missing transverse energy (right
bottom).

In Fig 3, we display the distributions for a few observables which have
relatively less discriminating power, namely the $p_{T}$ and jet mass of the
third and fourth leading jet, $p_{T}$ of the leading lepton and transverse
missing energy (MET). The last mentioned is expected to arise from the
neutrinos accompanying the leptons, and, of course, is constructed from the
imbalance in the measured $p_{T}$s of all the visible objects.

As seen from Fig. 3, the $W$ peak and/or the top peak in the jet mass are just
about visible for the third jet, and cannot be discerned for the fourth. This
is not unexpected, for the full event would have to be very energetic indeed
for the third (or fourth) jet to carry sufficient energy to be sufficiently
collimated and identified as a fatjet, given our choice of the jet radius.
Indeed, the events where the third (or fourth) jet does have a high mass are
largely those where the reconstruction process has clubbed together objects
originating from disparate sources. Of course, there is also a small fraction
wherein the tops emanating from the $G^{(1)}$ carry a lower $p_{T}$ (as
compared to the other top/jets) and are not ranked as being the hardest (given
our choice of using $p_{T}$ for ranking). In particular, for approximately
20$\%$ of all the events, a fatjet top is the third leading jet in $p_{T}$.
And although the signal events boast wider distributions in both $p_{T}$ and
jet-mass (understandably, the SM 4-top background is the closest to the signal
on both these counts), it turns out that the discriminatory power is low
(especially once the variables in the preceding subsection have been used.)

Much the same holds for the $p_{T}$ of the leading lepton as well as for the
missing transverse momentum. In either case, the distribution for the signal
is much wider (with the SM 4-top sample being somewhat similar) than the
$t\overline{t}jj$ or $t\overline{t}b\overline{b}$ backgrounds. This can,
again, be understood by realizing that, for the 4-top sample (whether signal
or background), the leading lepton $p_{T}$ accrues in three steps, the $p_{T}$
of the decaying top (also see the preceding discussion about the third leading
jet), hence the $p_{T}$ of the $W$ and, finally, the $p_{T}$ imparted by the
decay of the $W$. For the $t\overline{t}W$ events, the first contribution does
not exist. For the $t\overline{t}b\overline{b}$ and the $t\overline{t}jj$
samples, the leptons largely arise from the fragmentation (once the
requirement of two fatjets is imposed) and are relatively softer. The same
argument holds for the neutrino $p_{T}$ (the irreducible source of the MET).
There is, of course, an additional contribution arising from jet $p_{T}$
measurements, but this, understandably, does not harden the spectrum
substantially.

With the demand for two top-fatjets in the final state, it is tempting to
consider observables characterising subjets within the fatjets. Observables
such as N-subjettiness [47] and Energy Correlation functions [48] etc,
however, do not improve the results because of the similar event topology for
both the signal and the 4-top SM background (with the other backrounds being
reduced substantially by the imposition of other cuts); rather, the imposition
of cuts based on these only serve in worsening the discovery significance by
reducing the number of signal events. Hence for the cut-based studies, we
would not be using such observables.

Instead, we constrain the masses of the two fatjets to be around $m_{t}$. To
be specific, we impose $100<m_{j_{1}/j_{2}}<250$ GeV, in order to maximise the
retention of the signal. Furthermore, the lower cut not only removes the very
large QCD radiation, but also the potentially significant $W$-fatjet
background.

Figure 4: $M_{j_{1}j_{2}}$ and $H_{T}$ distribution after putting a cut of
$100<m_{j1},m_{j2}<250$ GeV.

Once it has, thus, been ensured that the event contains two top-like fatjets,
the distribution in the invariant masses of the two leading jets (see Fig. 4)
is significantly different for the signal as compared to the backgrounds. For
the former, the peak should ideally lie just at $M_{j_{1}j_{2}}=M_{G^{(1)}}$.
However, there exists a slight degradation on account of the jet
reconstruction algorithm losing a few entities (either on account of some
neutrinos in hadronic decays or due to some hits not registering as part of
the jet). The second peak at a lower $M_{j_{1}j_{2}}$ value might seem
intriguing. This, however, is but an artefact of the cuts. Noting that, for a
three-prong top-jet, the radius scales like $R=3~{}m_{j}/2p_{T}$, our twin
choices of $R\leq 0.5$ and $m_{j}>100$ GeV essentially translates to
$p_{T}>300$ GeV. With the invariant mass of the two leading jets being
expressible as

$M_{j_{1}j_{2}}^{2}=2\cosh(\Delta\eta)\sqrt{p_{T_{j_{1}}}^{2}+m_{j_{1}}^{2}}\sqrt{p_{T_{j_{2}}}^{2}+m_{j_{2}}^{2}}-2\cos(\Delta\phi)p_{T_{j_{1}}}p_{T_{j_{2}}}+m_{j_{1}}^{2}+m_{j_{2}}^{2}\
,$ (11)

this implies that the $M_{j_{1}j_{2}}$ distributions start only at around
$500$ GeV. This explains the initial rise as seen in Fig. 4. However, with the
$p_{T}$ distributions falling for higher $p_{T}$s (as seen in Fig. 2), a
kinematic peak must result at relatively low $M_{j_{1}j_{2}}$. A cut of
$M_{j_{1}j_{2}}>2000$ GeV would, thus, significantly enhance the signal to
background ratio. Expectedly, the $H_{T}$ plot too shows clear distinction
with the background showing a sharp peak around $1700$ GeV (for the same
kinematic reason as $M_{j_{1}j_{2}}$), whereas the signal boasts a much
flatter distribution extending till $5000$ GeV. In this case too, a cut of
around $H_{T}>2000$ GeV would considerably improve the signal-to-noise ratio.

###  5.3 Analysis

Having gained an understanding of signal and background behaviour, we first
proceed with a cut-based analysis. As discussed earlier, the dominant decay
mode, viz. the fully hadronic decay of two of the tops (the other being
required to result in fatjets) suffers from very large backgrounds (mainly
from $t\overline{t}jj$ and, to a lesser extent, from
$t\overline{t}b\overline{b}$). Concentrating on semileptonic decays addresses
this problem by greatly reducing the $t\overline{t}jj$ background, with only a
relatively small decrease (nominally, about by only a factor of about two-
thirds, discounting efficiencies) in the signal strength. Hence, we
concentrate on this mode. Below, we outline the detailed sequence of cuts we
intend to apply.

  * •

$\bf{Baseline~{}selection~{}cuts}$ : In order to reduce the $t\overline{t}jj$
and $t\overline{t}b\overline{b}$ background, we are looking for at least one
isolated and hard (with a minimum $p_{T}$ of 20 GeV for the leading candidate)
light lepton ($e^{\pm},\mu^{\pm}$). accompanied by a missing transverse
momentum (MET) of at least 20 GeV. Furthermore, we demand that there be at
least 6 jets in the event, with a minimum $p_{T}$ of 20 GeV each.

  * •

$\bf{N_{fatjets}}$ : The two boosted tops because of their large transverse
momenta can be confined within a jet radius of $R=0.5$, thereby constituting a
fatjet. We demand that there be at least two fatjets with a jet mass
satisfying $m_{j}\in[100,250]$ GeV.

  * •

$\bf{N_{btags}}$: Once the two leading fatjets are isolated, we demand that a
minimum of two of the other jets are b-tagged.

  * •

$\bf{H_{T}}$: We demand that $H_{T}$ (the scalar sum of all hadronic $p_{T}$s)
be greater than 2 TeV.

  * •

$\bf{M_{fj_{1};fj_{2}}}$: Since the invariant mass of the two leading fatjets
should roughly correspond to the RS gluon mass, we demand
$M_{fj_{1};fj_{2}}>2$ TeV.

Cuts | Signal |  ${\bf t\overline{t}t\overline{t}}$ (SM) | ${\bf t\overline{t}b\overline{b}}$ | ${\bf t\overline{t}jj}$ | ${\bf t\overline{t}W}$
---|---|---|---|---|---
${\bf Generation~{}level}$ | 2.21 | 15.7 | $2.4\times 10^{4}$ | $4.65\times 10^{5}$ | 602
${\bf Baseline~{}selection}$ | 0.672 | 4.96 | $2.24\times 10^{3}$ | $6.31\times 10^{4}$ | 61.3
${\bf N_{fatjets}\geq 2}$ | 0.032 | 0.029 | 0.156 | 12.6 | 0.025
${\bf N_{btags}\geq 2}$ | 0.014 | 0.014 | 0.0288 | 2.09 | 0.0033
${\bf H_{T}\geq 2~{}TeV}$ | 0.008 | 0.0018 | 0.0072 | 0.163 | 0.0004
${\bf M_{fj_{1};fj_{2}}\geq 2~{}TeV}$ | 0.006 | 0.0002 | 0.0029 | 0.023 | 0.0002
Table 1:  Cut flow table for the cross section (in fb) for the one-lepton
final state at $\sqrt{s}$ = 14 TeV. Signal corresponds to $M_{G^{(1)}}=4$ TeV
and $\xi=5$.

As discussed earlier and as seen from Table 1, the largest reducible
backgrounds are $t\overline{t}jj$, $t\overline{t}b\overline{b}$ and
$t\overline{t}W$, in that order. The baseline selection cut reduces the signal
by about 67%; on the other hand, the $t\overline{t}jj$ background is reduced
by about $85~{}\%$ while the $t\overline{t}$$b\overline{b}$ and
$t\overline{t}W$ backgrounds are reduced by nearly 90%. Demanding that the
event contains at least two fatjets and atleast two b-tagged jets does reduce
the signal by a factor of nearly 20, but the dominant backgrounds are
suppressed by several orders of magnitude. At this stage, the (irreducible) SM
$t\overline{t}t\overline{t}$ background is nearly the same as the signal size
(for our benchmark point). Finally, once we demand a cut of 2 TeV on $H_{T}$
and the invariant mass of the two leading fatjets, the two largest backgrounds
($t\overline{t}jj$ and $t\overline{t}$$b\overline{b}$) comes down to
manageable levels, whereas only a small amount of $t\overline{t}W$ survives.
As far as the SM $t\overline{t}t\overline{t}$ is concerned, it is reduced by
almost eight times compared to the surviving signal cross section. At a
luminosity of $3000~{}fb^{-1}$, we get a total of 19 signal events compared to
a total of 77 events for the background. This results in a significance
($\sigma={N_{s}}/\sqrt{N_{s+b}}$) of 1.84$\sigma$. As we move towards higher
center-of-mass (c.o.m) energies in future LHC runs, we can probe more and more
massive RS gluons.

##  6 Neural Network Analysis

###  6.1 Artificial Neural Network(ANN) Model

The artificial neural network (ANN) [49], is a recent machine learning
technique inspired by the network and functioning of interconnected neurons in
the human brain. It has been employed in tasks such as discriminating between
signal and background events and identifying physics objects like b-tagging
using collider experiment observables [50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64].

The process commences with the selection of features (typically, kinematic
variables) from the collider experiment data, including both signal and
background events, and training the ANN to detect patterns that would help it
to differentiate between the two classes. ANNs consist of interconnected
nodes, also called neurons, organized, typically, into three kinds of layers,
viz. the Input layer, Hidden layers and the Output layer. A representative
architecture of a feed-forward ANN with a single hidden layer is shown in
Figure 5.

Figure 5: Representative architecture of a feed-forward artificial neural
network for single hidden layer.

The selected features (collider observables in our case) are fed to the model
through the input layer. These observables ($x_{i}$) are initially assigned
random weights ($w_{ji}$) and then a bias ($b_{j}$) is added before passing
the inputs through an activation function as in Eq. 12.

$\displaystyle\sigma_{j}$ $\displaystyle=\sum_{n}w_{ji}x_{i}+b_{j}$ (12)
$\displaystyle Y_{j}$ $\displaystyle=\mathcal{F}(\sigma_{j}).$

The activation functions ($Y_{j}$) introduce non-linearities121212Without
these, the different layers of a neural network can be combined to a single
layer thereby losing on complexity. The $Y_{j}$s allow the network to go
beyond simple linear regression and develop complex representations and
functions. into the network, and some common activation functions include the
sigmoid, the hyperbolic tangent (tanh), and the rectified linear unit (ReLU).
This processed output from the input layer is fed to the hidden layer, and the
processed output from the latter, in turn, goes to the output layer. In case
of a multiple hidden layered structure, the input of any hidden layer is the
processed output from the previous hidden layer. This sequence of processing
is known as Forward Propagation.

At this step, a cost function (loss function) is defined with the specific
choice depending on the problem at hand. This calculates the difference
between the predicted output and actual output, and minimizing this leads to
the optimization of the network. Examples of the cost function include Mean
Squared Error (MSE), Cross-Entropy Loss (Log Loss), Binary Cross-Entropy Loss
(Logistic Loss), Total Variation Loss, etc

To minimize the loss function, the weights are updated through a process
called backward propagation, typically achieved using the stochastic gradient
descent algorithm. This consists of calculating the gradient of the loss
function with respect to each weight, iteratively and layer by layer, and
propagating backward from the last layer to avoid redundant computations of
intermediate terms in the chain rule. The weights for the $(n+1)$-th epoch are
updated using the weights $w^{n}_{ji}$ from the $n$-th epoch and the gradient
of the cost function $f(\left\\{w^{n}_{kl}\right\\})$, namely

$w^{n+1}_{ji}=w^{n}_{ji}-\alpha\,\frac{\partial}{\partial
w^{n}_{ji}}\,f(\left\\{w^{n}_{kl}\right\\}),$ (13)

where the learning rate $\alpha$ is a small positive constant that determines
the step size of each update.

The observables are passed again through the model with these updated weights
and the loss function values recalculated. One such complete single iteration
through the entire training dataset during the training phase is called an
epoch. This process continues until the model weights have been so updated as
to minimize the loss function. Once the weight values reach such an optimal
set, the trained model is ready to make predictions.

The architecture of neural networks, such as the number of hidden layers and
neurons in each layer, along with the number of epochs, depends on factors
like number of inputs, desired output, data complexity, error function
intricacy, network design, and training algorithm. Insufficient epochs result
in underfitting, missing data patterns, while excessive epochs lead to
overfitting, causing specialization and poor generalization. Achieving the
right balance is critical for optimal learning. Techniques like early stopping
and regularization help find this balance.

###  6.2 ML architecture

In our scenario, the final state involves a minimum of 6 jets, which
corresponds to at least 24 hadronic observables (comprising 3 momentum
components and 1 invariant mass for each jet), along with 4 observables for
the lepton as well as the missing transverse momentum (MET)131313Because of
the large number of jets in the final state, the MET receives additional
contributions from both jet energy measurements as well as the components
missed by the detector. While it might be argued that the MET does not
represent an independent input variable for the ANN, it should be realised
that its very composition (especially the contributions from the missed
components) ensures that this is not so. Furthermore, the inclusion of a
dependent variable would only result in a slight slowing down of the ANN’s
execution, and show up in the form of strong correlations.. Since such
fundamental variables, in isolation, may or may not possess significant
discriminatory power, it is a standard approach to work with a broader array
of observables that exhibit greater discriminatory potential. Such strategic
combinations of these fundamental observables could yield valuable insights,
especially when applied within a neural network, whose primary role is to
differentiate between the signal and background by assigning weights based on
the discriminating capabilities of the observables (or combinations thereof).
In this, we will use our understanding of kinematic distributions which we
explored in the cut-based analysis (Section 5.1).

To produce the data (both signal and background) for our analysis, we begin by
applying some initial criteria141414Note that we use jet algorithms and
isolation criteria identical to those in Sec.5., which are outlined as
follows:

$\displaystyle n_{jets}\geq 4\hskip 14.22636pt{\rm and}\hskip
14.22636ptp_{T}^{jet}\,>\,20\,{\rm GeV}$ (14) $\displaystyle n_{lep}>0\hskip
14.22636pt{\rm and}\hskip 14.22636ptp_{T}^{lep}\,>\,20\,{\rm GeV}$
$\displaystyle{\rm MET}\,\ >\,20\,{\rm GeV}.$

In this analysis, our focus is on 34 observables, which are further
categorized into low-level observables, as shown in Table 2, and derived
observables, as detailed in Table 3.

transverse momentum of the 4 leading jets |  $p_{T}^{1}$, $p_{T}^{2}$, $p_{T}^{3}$, $p_{T}^{4}$
---|---
individual masses of the 4 leading jets |  $m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$
transverse momentum of the leading lepton | $p_{T}^{l}$
missing transverse energy | $E_{T}\\!\\!\\!\\!\\!\\!\,\,\\!\\!\\!/$
rapidity of the 2 leading jets |  $\eta_{1}$ and $\eta_{2}$
azimuthal angle of the 2 leading jets |  $\phi_{1}$ and $\phi_{2}$
number of jets | $n_{jets}$
number of b-jets | $n_{b}$
Total | 16 observables
Table 2: Low level observables for ANN training. separation between two jets |  $\Delta R_{j_{1}j_{2}}$, $\Delta R_{j_{1}j_{3}}$, $\Delta R_{j_{1}j_{4}}$, $\Delta R_{j_{2}j_{3}}$, $\Delta R_{j_{2}j_{4}}$, $\Delta R_{j_{3}j_{4}}$
---|---
invariant masses for the 3 leading jet pairs |  $M_{j_{1}j_{2}},M_{j_{1}j_{3}}$, $M_{j_{2}j_{3}}$
scalar sum of transverse momentum of the jets | $H_{T}$
$\tau_{32}$ for the 4 leading jets |  $\tau_{32}^{1}$, $\tau_{32}^{2}$, $\tau_{32}^{3}$, $\tau_{32}^{4}$
$\tau_{21}$ for the 4 leading jets |  $\tau_{21}^{1}$, $\tau_{21}^{2}$, $\tau_{21}^{3}$, $\tau_{21}^{4}$
Total | 18 observables
Table 3: Derived observables for ANN training.

Of the second set, the $\tau_{32}^{i}$ ($\tau_{21}^{i}$) observable for the
$i$-th jet is given by $\tau_{3}/\tau_{2}$ ($\tau_{2}/\tau_{1}$), where the
N-subjettiness $\tau_{N}$ is defined as

$\displaystyle\tau_{N}=\frac{1}{d_{0}}\,\sum_{k}\,p_{T,k}\,{\rm
min}\Big{(}\Delta R_{j_{1}j_{k}},\Delta R_{j_{1}j_{k}},....,\Delta
R_{j_{N}j_{k}}\Big{)}.$ (15)

Here $N$ is the number of candidate subjets of the jet to be reconstructed,
$k$ runs over the constituent particles in a given jet with $p_{T,k}$ being
their transverse momenta and $\Delta R_{j_{i}j_{k}}$, the angular separation
between a candidate subjet $i$ and a constituent particle $k$. Furthermore,

$\displaystyle d_{0}=\sum_{k}\,p_{T,k}\,R_{0},$ (16)

where $R_{0}$($=0.5$) is the characteristic jet-radius. Physically, $\tau_{N}$
provides a dimensionless measure of whether a jet can be regarded to be
composed of $N$-subjets. In particular, the ratios $\tau_{N}/\tau_{N-1}$ are
powerful discriminants between jets predicted to have $N$ internal energy
clusters and those with fewer clusters. For example, jets coming from the
hadronic decays of the $Z/W/H$ tend to have lower values for the ratio
$\tau_{21}\equiv\tau_{2}/\tau_{1}$ as compared to QCD or top-jets. Jets coming
from the three-pronged hadronic decay of the tops have a low value of
$\tau_{32}\equiv\tau_{3}/\tau_{2}$.

In order to reduce the $t\overline{t}jj$ cross-sections to manageable levels,
which otherwise is disproportionately large, we demand that the (individual)
masses of the two leading jets be greater than 80 GeV in addition to the
baseline cuts used in Eq. 14. Its impact is shown in Table 16. After the cuts,
it has to be ensured that there are enough events for signal as well as
background to be fed to the ANN. We have taken sufficiently large and equal
sample sizes for both, so that the network is able to predict the
characteristics more accurately.

Cuts | Signal | ${\bf t\overline{t}t\overline{t}}$ | ${\bf t\overline{t}b\overline{b}}$ | ${\bf t\overline{t}jj}$ | ${\bf t\overline{t}w}$
---|---|---|---|---|---
Generation level | 2.21 | 15.7 | $2.4\times 10^{4}$ | $4.65\times 10^{5}$ | 602
Baseline selection (Eq. 14) | 0.839 | 5.19 | $1.82\times 10^{3}$ | $6.29\times 10^{4}$ | 56.4
${\bf m_{1,2}>}$ 80 GeV |  0.0942 |  0.0917 |  0.45 |  41.463 |  0.117
Table 4:  Cut flow table of signal and background cross sections (in fb)
pertaining to the preselections for the ANN at $\sqrt{s}$ = 14 TeV. Signal
corresponds to $M_{G^{(1)}}=4$ TeV and $\xi=5$.161616After the implementation
of the above simple cuts, the ratio of signal versus $\overline{t}tjj$ events
reduces to 176 from 170317\. This ratio had finally reduced to 3.17 in the
case of the cut-based analysis in Sec. 5.3.

Unlike traditional cut-based collider analyses that rely heavily on multitude
of cuts to reduce background events, we employ only a set of basic cuts,
mentioned in Eq. 14 and Table 16, to eliminate obviously background-like
events while ensuring fairness and compatibility with the ANN methodology.
These cuts are essential as they filter out events that contribute
significantly to the total cross-section for the respective channels but
provide little discriminating ability. In this instance, demanding a higher
invariant mass for the two leading jets helps eliminate the softer QCD
emissions, allowing the neural network to focus solely on the relevant events
for signal and background discrimination. This approach prevents the network
from learning unnecessary features and allows it to assign appropriate weights
to relevant events, thereby ensuring faster and better convergence.

These input observables have varying ranges. To ensure that the gradient
descent method progresses smoothly towards the minima, and that the steps for
it are updated uniformly across all features, we normalize the data before
inputting it to the model. This, typically, involves scaling the data between
0 and 1, and is often referred to as Min-Max scaling. Such a normalization
allows for a uniform application of the gradients in the hidden layer as well
as recombination of observables into newer sets, thereby facilitating a
smoother functioning of the network.

Our objective here is to distinguish between signal and background, making it
a binary classification task. Signal events are thus categorized as 1, while
the Standard Model (SM) background events are denoted as 0. The optimization
goal is to achieve high accuracy in discriminating between signal and
background in the testing sample, followed by achieving an appropriate level
of significance171717Note that a good accuracy does not automatically
translate to a good significance as the number of background events is often
much larger than the number of signal events..

Following the training of the neural network, it is possible to assess the
significance of each feature in the analysis. Feature importances can be
evaluated on both global and local scales. Global feature importance assesses
the utility of a feature across the entire dataset, while local feature
importance pertains to a single prediction, such as an individual data point
or event. To interpret the model, we have chosen to utilize SHAP (SHapley
Additive exPlanations) [65], a local feature attribution technique based on
Shapley values. It quantifies the importance of a feature by determining how
effectively the model predicts a specific class when the feature is present
compared to when it is absent within the comprehensive set of features. For a
given feature, SHAP takes into account not only that feature but also the
potential interactions among the features, considering that correlations may
exist among them. It calculates the Shapley values $\phi_{i}$ using [65, 66]

$\displaystyle\phi_{i}=\sum_{S\subseteq N\backslash\\{i\\}}\
\frac{|S|!(M-|S|-1)!}{M!}\ [f_{x}(S\cup\\{i\\})-f_{x}(S)]\ .$ (17)

Here, $i$ represents the feature for which the SHAP value is being determined,
$N$ is the collection of all features, and $M$ denotes the total number of
features. The variable $S$ represents a subset of $N$ that excludes feature
$i$. The function $f_{x}$ corresponds to the model’s prediction. Consequently,
the algorithm calculates a weighted sum of the differences in model
predictions, considering the presence or absence of the $i^{th}$ feature for
all feasible combinations of the subset $S$.

###  6.3 NN Training

The construction of the ANN model utilizes the Keras library in conjunction
with TensorFlow for backend implementation. This is designed to classify the
contribution of signals versus Standard Model (SM) background events. For the
purpose of testing and validation, an equal number of samples are randomly
selected from the pools of signal and background data. The training set is
subjected to random shuffling, and the input data is subsequently supplied to
the network.

For our dataset, we generated 100,000 each of signal and SM background events,
the latter receiving contributions from processes like
$t\overline{t}t\overline{t}$, $t\overline{t}jj$, $t\overline{t}b\overline{b}$,
and $t\overline{t}W$, with proportions determined by their respective cross-
sections. The events are mixed randomly, and 20% of them are designated for
testing, while the remaining 80% is divided into two subsets. Within these
subsets, 80% is allocated for training, and the remaining 20% is used for
validation.

The activation function used in case of hidden layers is ReLU181818ReLU is a
piecewise linear function that will output the input directly if it is
positive, otherwise, it will output zero. (Rectifier Linear Unit)[67] and
softmax for the output layer. The Adam (Adaptive Moment Estimation)
optimizer[68] is used to minimize the loss-function between the input and
output. The Adam optimization algorithm is an extension of the stochastic
gradient descent that computes individual adaptive learning rates for
different parameters from estimates of first and second moments of the
gradients to update network weights iterative based in training data. Finally,
for the loss function, we have used the binary cross-entropy function, which
is defined as

$\displaystyle H(y,p)$ $\displaystyle=$
$\displaystyle-\frac{1}{N}\,\sum_{i=1}^{N}\,(y_{i}\cdot\log(p(y_{i}))+(1-y_{i})\cdot\log(1-p(y_{i}))).$
(18)

Here $y_{i}$ denotes the true label for the observables (either 0 or 1 for the
background and signal respectively), $p(y_{i})$ is the predicted probability
that the instance belongs to class 1 (signal) and $N$ represents the span of
the data-set.

Figure 6: The correlation plots between the observables for $\sqrt{s}$ = 14
TeV for signal (left) and background (right) respectively with $M_{G^{(1)}}$ =
4 TeV.

In Figure 6, we have illustrated the relationships among some of the
observables that serve as inputs to the machine learning network. The plot
reveals that certain observables exhibit modest correlations with each other,
making them suitable features for the discrimination training process.

However, there are other observables where the correlation is notably
pronounced. In these cases, the extent of correlation may vary between signal
and background. For instance, consider the correlation between $H_{T}$ and
$M_{12}$ (referred to as hT and m12 respectively in Figure 6). In this
particular scenario, the correlation between these two observables is
substantial for signal, while it is comparatively weaker for the background.
This difference can serve as an effective discriminator.

Our analysis also incorporates batch normalization [69], a technique that
normalizes the layers within the network to enhance training while maintaining
consistent accuracy levels. Notably, we have obtained similar results for both
the analyses, whether with or without batch normalization and therefore have
presented the figures and findings that include batch normalization. We also
experimented with standardization, a scaling method applied to various data
features. However, this did not yield any noticeable changes in the results.
In addition, we use the dropout regularisation technique [70] to prevent an
overfitting of the model. It works by randomly dropping out a certain fraction
(which depends on the dropout rate) of neurons in a layer during each forward
and backward pass. The details of the full ANN structure incorporated in the
analysis is given in Table 5.

Input |  Observables
---|---
Layers | dense layer : 50
dropout layer (0.3)
batch normalization layer
| dense layer : 20
| dropout layer (0.3)
| batch normalization layer
| dense layer : 5
| dropout layer (0.2)
Layers setting | hidden layer activation = relu
output layer activation = softmax
Compilation | loss = binary_crossentropy
optimizer = adam[68]
metric = accuracy
Table 5: ANN structure specifications.

Furthermore, we utilized a hyperparameter tuning technique called
GridSearchCV. This approach involves testing various combinations of
parameters provided through a list of dictionaries, including batch size and
the number of training epochs. The objective here is to determine the optimal
parameters for building the neural network, which can significantly impact its
performance.

###  6.4 ANN results

A network’s capability of accurately differentiating between signal and
background events is quantified through a metric known as the accuracy score.
In the present instance, the network has been trained for the discrimination
of signal events from background events, considering all background sources in
accordance with their respective cross-section ratios for a luminosity of 3000
fb-1. The outcome of this training demonstrates an accuracy level as high as
$\sim$89.5%. We have plotted the accuracy versus the number of epochs plot
upto 1000 epochs in Fig 7. Also shown is the variation in the loss over the
same number of epochs. As can be observed, the accuracy saturates for smaller
values of epochs and the difference between the training and validation
accuracy keeps on increasing with epochs. For more epochs, the validation
accuracy starts to reduce, as a result of overtraining. A similar pattern
follows for the loss as well.

Figure 7: The variation of model training and validation accuracy (loss) with
epochs on the left (right) hand side plot for $\sqrt{s}$ = 14 TeV
corresponding to $M_{G^{(1)}}$ = 4 TeV with batch size 60.

Figure 8 illustrates the SHAP values, as introduced in Section 6.2 and
described by Eq. 17. On the left plot’s $y$-axis, the feature names
(observables) are arranged in order of importance from top to bottom.
Meanwhile, the $x$-axis represents the SHAP value, with positive values
indicating contributions that increase the prediction and negative values
indicating contributions that decrease it. In the graph, each data point
corresponds to an original dataset entry, and its color signifies the value of
the associated feature. Red points represent high feature values, while blue
points represent low values. For instance, for the most important feature
$H_{T}$, we notice that the red points correspond to positive SHAP values,
suggesting that a higher $H_{T}$ value positively impacts the network’s
predictions compared to when $H_{T}$ is lower. This aligns with what we
observed in the cut-based analysis in Section 5.3 where increasing the $H_{T}$
threshold significantly reduced the background. On the other hand, considering
“tau321” (representing $\tau_{32}$ of the leading jet), a low value of this
observable improves the network’s performance. This is logical in the boosted
regime because the leading top can form a fatjet with three prongs, resulting
in a smaller $\tau_{32}$ value. Since we do not anticipate the background to
be highly boosted, a three-prong fatjet is less likely to occur, causing the
$\tau_{32}$ of the leading jet to be larger than the signal. Analogous
interpretations can be drawn for the other observables in a similar manner.

Figure 8: A summary plot showing the variation of SHAP values with the input
feature values for the top 20 variables from the signal versus all the
relevant backgrounds.

In addition to accuracy and feature importance, another essential evaluation
tool is the Receiver Operating Characteristic (ROC) curve. ROC curves offer a
visual representation of an algorithm’s performance, illustrating how well it
separates the two classes. The area under the ROC curve (AUC) is a valuable
performance metric, providing a single numerical value that quantifies the
model’s ability to distinguish between classes. In the context of this
analysis, the AUC-ROC curves are depicted in Figure 9. These curves offer
insights into the network’s discriminative power and provide a basis for
evaluating its performance in practical conditions.

In practical terms, the network’s performance can be further assessed by
calculating the signal significance. This measure accounts for the sensitivity
of the model in real-world scenarios, where the trade-off between signal and
background discrimination is crucial for making meaningful discoveries in
particle physics or similar fields. We achieve this through the optimisation
of the selection criteria for the output discriminant by carefully choosing a
cut-off point, denoted as $Y_{\rm cut}$, that maximizes the significance of
our analysis. The variation of the final significance with respect to $Y_{\rm
cut}$ is shown in the second plot of Figure 9. Notably, the exceptionally high
cross-section of the $t\overline{t}jj$ background process compels us to
contemplate an extreme value for $Y_{\rm cut}$.

Figure 9: AUC-ROC curve (left) and significance (right) respectively for
$\sqrt{s}$ = 14 TeV analysis with $M_{G^{(1)}}$ = 4 TeV corresponding to
accuracy of 89.5%.

A comparison between the current network-based approach and the traditional
cut-based technique performed earlier makes a strong case for the former. In
the context of the cut-based method, the final count yields approximately 19
signal events and 77 background events, which corresponds to significance less
than 2$\sigma$. However, when the artificial neural network (ANN) is employed,
we get 150 signal events and 1228 background events with significance of 4.05
at $Y_{\rm cut}$ = 99.5%. With even larger values of $Y_{\rm cut}$, the ANN
method yields a significantly higher significance of more than 4 as shown in
Figure 9. This signifies the network’s exceptional ability to enhance the
signal-to-noise discrimination and deliver improved results when compared to
the cut-based approach, where the signal significance is less pronounced.

##  7 Summary and Conclusion

There are a plethora of models where heavy colored gauge bosons exhibit
enhanced interactions with certain Standard Model (SM) fermions, most notably
the top quark. This phenomenon is commonly associated with the pursuit of a
dynamical explanation for Electroweak Symmetry Breaking (EWSB), especially in
generic models like those incorporating extended color symmetry with colorons
or axigluons as the mediators. For models constructed in higher dimensions,
such states arise naturally as Kaluza-Klein modes of the gluon. In particular,
within the bulk Randall-Sundrum scenarios, such KK-gluons have non-trivial
wavefunctions along the fifth (bulk) direction, typically resulting in
suppressed couplings with the light quark flavours. On the other hand, the
right-handed top-quark being localized close to the infrared brane, has an
enhanced coupling with the KK-gluons.

The large masses of such bosons renders pair-production cross sections to be
small. Similarly, single production cross section is small too, owing to the
suppressed coupling with a light quark-pair or a gluon-pair. Both these facts,
as well as the typically large SM backgrounds, thus, call for other modes.
Concentrating on the first of the KK-gluons (as they are well-separated),
namely, the $G^{(1)}$, a particularly interesting mode at the LHC is
associated production with a top pair, viz., $gg\to t\overline{t}G^{(1)}$,
with the KK-gluon subsequently going into a top-pair, thereby leading
preferentially to a four-top final state.

While the large $t\overline{t}G^{(1)}$ coupling facilitates this mode, it also
leads to the KK-gluon having a naturally large width, and the commonly used
narrow width approximation is often insufficient to capture its behavior
accurately. Given this, we need to include at least the one-loop renormalised
$G^{(1)}$ propagator.

The large mass of the $G^{(1)}$ typically results in both of the
$t\overline{t}$ pair coming off it to be highly boosted. Consequently, their
decay products tend to be collimated, resulting in a pair of top-fatjets. Of
the other two tops, we require one of them to decay hadronically and the other
leptonically. Thus, our final state of choice comprises of at least six jets
(of which two are top-like fatjets, and another two are $b$-tagged), one hard
and isolated lepton ($e^{\pm}/\mu^{\pm}$) and substantial missing transverse
momentum.

We begin our analysis with a conventional cut-based approach. A careful study
of the signal and background profiles shows us that only a handful of
kinematical variables are powerful discriminating tools. These are the
transverse momenta of the two leading jets, their individual jet masses, the
scalar sum of all the jet transverse momenta, and, to a reduced extent, the
transverse momentum of the lepton. Of course, the invariant mass of the two
leading jets is expected to be another powerful tool as, for the signal-
events, it is expected to peak near the $M_{G^{(1)}}$. However, this is
tempered to a great degree by the aforementioned very large width of the
$G^{(1)}$, and prevents reaching large significance values. The higher cross
sections of the background, specially the $t\overline{t}jj$, worsens things
further.

This challenge has enticed us to explore the application of machine learning
methods. We study here the most basic of them all, Artificial Neural Networks
(ANNs), to enhance the discrimination power. To expedite the convergence of
the neural network in this discrimination task, we have implemented some basic
cuts. The background sample is created by adding the relevant background
processes in proportions determined by their cross-sections. In order for the
network to effectively learn all the topological features of the signal and
background, we provide 16 low level observables and 18 derived observables as
inputs. Even though most of the observables play a role in this
discrimination, we find that $H_{T}$, number of $b$-jets, total number of
jets, invariant mass of the two leading jet-pairs ($m_{j_{1}j_{2}}$) and
$\tau_{32}$ of the leading jet are the five most influential features in the
process. The accuracy achieved by the network turns out to be $\sim 90\%$,
which is a testament of its excellent learning capability. Moreover, the
output discriminant exhibits significances in excess of 4$\sigma$, making it
superior to conventional cut-based analysis. Use of more sophisticated modern
machine-learning techniques would understandably push these limits higher,
rendering the exploration of the four-top channel examined here worthy of
further investigation at the LHC.

## Acknowledgments

KD acknowledges Council for Scientific and Industrial Research (CSIR), India
for JRF/SRF fellowship with the award letter no. 09/045(1654)/2019-EMR-I. LKS
acknowledges the UGC SRF fellowship and research Grant No. CRG/2018/004889 of
the SERB, India and Apex Project (theory), Institute of Physics (IoP),
Bhubaneswar, for the partial financial support. LKS also acknowledges SAMKHYA,
the High-Performance Computing Facility provided by the Institute of Physics
(IoP), Bhubaneswar.

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