aainstitutetext: Centre for Cosmology, Particle Physics and Phenomenology
(CP3),
Université Catholique de Louvain,
Chemin du Cyclotron, B-1348 Louvain la Neuve, Belgiumbbinstitutetext: Inter-
University Institute for High Energies (IIHE), Vrije Universiteit Brussel,
Pleinlaan 2, 1050 Brussels, Belgiumccinstitutetext: Dipartimento di Fisica e
Astronomia, Università di Bologna and INFN, Sezione di Bologna,
Via Irnerio 46, 40126 Bologna, Italyddinstitutetext: Department of Physics and
Astronomy, University of Manchester,
Oxford Road, Manchester M13 9PL, United Kingdom

# Complete SMEFT predictions for four top quark production at hadron colliders

Rafael Aoude a,b Hesham El Faham a,c Fabio Maltoni d Eleni Vryonidou
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study four top quark production at hadron colliders in the Standard Model
Effective Field Theory (SMEFT). We perform an analysis at the tree-level,
including all possible QCD- and EW-coupling orders and relevant dimension-six
operators. We find several cases where formally subleading terms give rise to
significant contributions, potentially providing sensitivity to a broad class
of operators. Inclusive and differential predictions are presented for the LHC
and a future $pp$ circular collider operating at $100$ TeV. We estimate the
sensitivity of different operators and perform a simplified chi-square fit to
set limits on SMEFT Wilson coefficients. In so doing, we assess the importance
of including subleading terms and differential information in constraining new
physics contributions. Finally, we compute the SMEFT predictions for the
double insertion of dimension-six operators and scrutinise the possible
enhancements to the sensitivity induced by a specific class of higher order
terms in the EFT series.

###### Keywords:

Top quark, QCD, SMEFT, Monte-Carlo, Collider physics

##  1 Introduction

In the last decade, the Large Hadron Collider (LHC) experiment at CERN has
tested our understanding of fundamental interactions up to several TeV’s of
energy. The unexpected success of the Standard Model (SM), on the one hand,
and its inherent incompleteness as a theory of nature, on the other hand, have
led to vigorous efforts by the experimental and theoretical communities to
study where new physics could lurk. In this endeavour, a unique role is played
by precision physics, where accurate theoretical predictions of SM processes
are compared with experimental measurements searching for deviations. The
upcoming third run of the LHC - characterised by a 4.5% increase in collision
centre of mass-energy, from 13 TeV to 13.6 TeV and a two-fold increase of
luminosity- will provide a new handle on rare phenomena (for a review of the
latest experimental measurements of $t\bar{t}t\bar{t}$ and future prospects at
the LHC, see Ref. Blekman:2022jag ). Among the rarest and most spectacular
processes at the LHC is the production of two pairs of top-antitop quarks,
i.e., four top quark production. This process is characterised by a tiny SM
cross-section at 13 TeV, of about 12 fb Frederix:2017wme , i.e., around five
orders of magnitude lower than that of $t\bar{t}$ production. However, despite
the tiny rates, $t\bar{t}t\bar{t}$ signatures are distinctive, leading to a
wealthy and energetic final state, which is challenging to mimic through other
processes. Therefore, the very high-$Q^{2}$ and low backgrounds offer a unique
opportunity for probing new physics Lillie:2007hd ; Pomarol:2008bh ;
Kumar:2009vs ; Deandrea:2014raa ; Berger:2011af ; Aguilar-Saavedra:2011mam ;
Beck:2015cga ; BhupalDev:2014bir ; Acharya:2009gb ; Gregoire:2011ka ;
Degrande:2010kt ; Cao:2021qqt ; Darme:2021gtt ; Banelli:2020iau ;
Englert:2019zmt .

Given its promise, precise predictions of four-top production have become
necessary. Next-to-leading order (NLO) corrections in QCD have been computed
first in Ref. Bevilacqua:2012em  and then also become available in event
generators Alwall:2014hca ; Maltoni:2015ena ; Jezo:2021smh . The complete NLO
predictions, including all possible QCD and electroweak (EW) orders, have been
calculated in Ref. Frederix:2017wme . They revealed a peculiar and unexpected
interplay between EW and QCD contributions, with significant contributions
with opposite signs arising from formally subleading terms. Significant
subleading contributions have also been observed in the case of new physics
contributions to four-top production Darme:2021gtt ; Degrande:2020evl .
Moreover, it has been suggested in the study of Ref. Cao:2016wib  that four-
top production may be a valuable probe of the top-Yukawa coupling, $y_{t}$, at
the tree-level.

In the SM, representative diagrams of the pure-QCD
$\mathscr{O}(\alpha_{s}^{2})$ four-top production are shown in fig. 1,
occurring through $gg$ and $q\bar{q}$ initial states. QCD-induced diagrams
typically provide the leading contribution. However, formally subleading
diagrams with insertions of EW couplings, i.e.
$\mathscr{O}(\alpha_{s}\alpha_{\mathrm{w}})$, can also be significant.
Examples of the latter are shown in fig. 2, where top quarks scatter through
the exchange of a Higgs boson or $Z/\gamma^{*}$.

$\bar{t}$$g$$\bar{t}$$g$$t$$t$$\bar{t}$$q$$t$$\bar{q}$$\bar{t}$$t$

Figure 1: Representative leading order tree-level Feynman diagrams of
$\mathscr{O}(\alpha_{s}^{2})$ for the $gg$-initiated (_left_) and the
$q\bar{q}$-initiated (_right_) SM four-top production at the LHC.

$\bar{t}$$g$$t$$g$$\bar{t}$$t$$H$$\bar{t}$$g$$t$$g$$\bar{t}$$t$$Z$$\gamma^{*}$

Figure 2: Representative diagrams of
$\mathscr{O}(\alpha_{s}\alpha_{\mathrm{w}})$ for the SM four-top production at
the LHC. The diagrams show the EW $tt\to tt$ scattering involving the exchange
of a Higgs boson (_left_) or a $Z$-boson/virtual photon (_right_).

These diagrams can contribute through their interference with QCD ones or
through their squares. In the SM at $\sqrt{s}=13$ TeV, it turns out that
contributions from the interference of this class of diagrams with the leading
QCD amplitude and from their squares are significant, reaching more than a
third of the leading tree-level QCD contributions. Nevertheless, these two
contributions come with opposite signs, and there is a significant
cancellation between them. This can be seen explicitly in table 7 of Ref.
Frederix:2017wme . These large cancellations at the tree-level also motivate a
high-order computation, including QCD and EW corrections, as discussed in Ref.
Frederix:2017wme . At one-loop order, additional cancellations occur between
different terms in the $\alpha_{s}$ and $\alpha_{\mathrm{w}}$ expansion, but
in general, higher-order corrections are dominated by $\alpha_{s}$ corrections
to the leading QCD term. It is worth noting that the size of various terms
varies significantly depending on the choice of the renormalisation scale.
Though it is clear from table 7 of Ref. Frederix:2017wme  that several
contributions are larger than what one would expect from the ratio of
$\alpha_{\mathrm{w}}/\alpha_{s}$. In summary, four-top production does not
necessarily submit to the “naive” $\alpha_{s}$ and $\alpha_{\mathrm{w}}$ power
counting. Therefore, a study of four-top production should be completed by
including all QCD and EW-induced terms in the computations. This consideration
constitutes the primary motivation behind the work presented in this paper.

The peculiar behaviour of the cross-section as a double series in $\alpha_{s}$
and $\alpha_{\mathrm{w}}$ for the SM process certainly motivates a detailed
study in the case of including new physics effects, particularly in the SMEFT
framework. The effective field theory approach assumes new physics to reside
at a high scale $\Lambda$ Weinberg:1978kz ; Buchmuller:1985jz ; Leung:1984ni .
In the SMEFT, the SM Lagrangian is augmented with higher-dimensions operators
built out of the SM fields and respecting the SM gauge symmetries, describing
short-distance interactions generated by new physics at high scales. The
pattern of such deformation depends on the details of the phenomena in the
ultra-violet (UV) region. Being unknown, one assumes all possible operators to
be there and studies their effects on low-energy observables. The beauty of
this framework is that it allows perturbative calculations to be performed
consistently order by order in the $1/\Lambda$ expansion. Such a powerful
approach provides a consistent and calculable framework in which the potential
deviations from the SM predictions can be encapsulated and predicted in type
and pattern. Studies in the context of the SMEFT at the LHC are an ongoing
effort in all SM sectors; the electroweak, the Higgs, the flavour, and the top
sectors. Global fits combining a broad set of publicly available data have
appeared Brivio:2019ius ; Hartland:2019bjb ; Ethier:2021bye ; Ellis:2020unq ;
Buckley:2015lku ; Miralles:2021dyw , indicating which directions (operators)
in the fits can be constrained and whether complementary information or new
strategies would be helpful. In the context of an EFT, four-top production is
exciting as it is the simplest process where top quark self-interactions could
be probed at the tree-level. In the SMEFT language, such interactions are
described by a set of dimension-six operators of the form
$\bar{\psi}\psi\bar{\psi}\psi$ operators with four top quark fields (left- or
right-handed). Other processes at colliders do not directly constrain these
operators at the tree-level. Therefore four-top production is naively expected
to be the first place to see their effects111 Proposals for constraining four-
top operators through loop effects have appeared in Refs. Degrande:2020evl ;
Banelli:2020iau ; Alasfar:2022zyr ; Dawson:2022bxd ..

This work considers all possible contributions of the SM and the SMEFT,
including all the dimension-six CP-even SMEFT operators that enter at the
tree-level. Part of our motivation is that in the SMEFT, the EFT interference
with the SM is expected to provide the leading cross-section contribution. As
the interference projects the kinematic and the colour structure of the SM
amplitudes, its size can change significantly from one operator to another. It
is also expected to vary depending on which contributions are included in the
SM, as different operators have different colour and chirality structures. As
mentioned previously, we retain all possible tree-level contributions at
different orders in QCD and EW couplings in our computations. Specifically, we
split the EW-induced contributions into the gauge and top-Yukawa ones and
determine the inclusive and differential predictions for the LHC and FCC-hh.
We organise our predictions as an expansion in $\alpha_{s}$, where each term
is expanded in the weak parameters, highlighting the potential significance of
the formally subleading terms.

To assess the reach of constraining the SMEFT Wilson coefficients (WCs) at
future colliders, we perform (_i_) a signal-strength-based projection study
for each operator at different collider energies obtaining theoretical limits
on the corresponding WCs; (_ii_) simplified chi-square
(${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$)
fits at different collider energies on selected sets of operators. In the
latter case, we also include differential information and assess its
constraining power compared to using only inclusive measurements.

Finally, we scrutinise the claim of Ref. Zhang:2017mls  on the enhanced EFT
sensitivity of four-top production to 2-heavy-2-light four-fermion operators
due to the contributions from double insertion. In Ref. Zhang:2017mls , it was
argued that though formally equivalent to single dimension-eight insertions,
in some UV models, double insertion could provide the dominant terms and four-
top production could compete with much more abundant top quark pair production
processes through enhancements scaling as $\sim(cE^{2}/\Lambda^{2})^{4}$.

This paper is structured as follows: in section 2, we describe the theoretical
tools for four-top production within the SMEFT framework, presenting the
operators’ definitions and the cross-section expansion in QCD and EW
couplings. The inclusive and differential predictions are presented in section
3 and section 4, respectively. The signal-strength projection study and the
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$
fits are presented in section 5 and section 6, respectively. We discuss the
results from the cross-section computation considering double EFT insertions
in section 7. The work is summarised and concluded in section 8.

##  2 SMEFT framework to four top quark production

###  2.1 Operators definitions

We compute the SMEFT contributions to four-top production using a specific
flavour assumption which singles out the top quark interactions,

$\displaystyle U(3)_{l}\times U(3)_{e}\times U(2)_{q}\times U(2)_{u}\times
U(3)_{d}\equiv U(2)^{2}\times U(3)^{3},$ (1)

where the subscripts refer to the five-fermion representations of the SM. This
minimal relaxation of the $U(3)^{5}$ group gives rise to top quark chirality-
flipping interactions, such as the dipole interactions and ones which modify
the top-Yukawa coupling. We use the notation and operator conventions of Refs.
Aguilar-Saavedra:2018ksv ; Degrande:2020evl  and study all operators in three
classes of dimension-six SMEFT: four-fermion (4F), two-fermion (2F), and
purely-bosonic (0F) operators, consistent with our flavour symmetry
assumption. We do not consider the two-fermion light quark operators as we
expect them to be better constrained in other production processes.

#### Four-fermion operators

Following the conventions and notation of Ref. Aguilar-Saavedra:2018ksv , the
four-fermion operators are defined as follows:

$\displaystyle\mathcal{Q}_{\scriptscriptstyle qq}^{\scriptscriptstyle
1(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}q_{j})(\bar{q_{k}}\gamma_{\mu}q_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle qu}^{\scriptscriptstyle
1(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}q_{j})(\bar{u_{k}}\gamma_{\mu}u_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle qd}^{\scriptscriptstyle
1(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}q_{j})(\bar{d_{k}}\gamma_{\mu}d_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle ud}^{\scriptscriptstyle
1(ijkl)}$
$\displaystyle=(\bar{u_{i}}\gamma^{\mu}u_{j})(\bar{d_{k}}\gamma_{\mu}d_{l}),$
${}^{\dagger}\mathcal{Q}_{\scriptscriptstyle quqd}^{\scriptscriptstyle
1(ijkl)}$ $\displaystyle=(\bar{q_{i}}u_{j})\epsilon(\bar{q_{k}}d_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle qq}^{\scriptscriptstyle
3(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}\tau^{I}q_{j})(\bar{q_{k}}\gamma_{\mu}\tau^{I}q_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle qu}^{\scriptscriptstyle
8(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}T^{A}q_{j})(\bar{u_{k}}\gamma_{\mu}T^{A}u_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle qd}^{\scriptscriptstyle
8(ijkl)}$
$\displaystyle=(\bar{q_{i}}\gamma^{\mu}T^{A}q_{j})(\bar{d_{k}}\gamma_{\mu}T^{A}d_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle ud}^{\scriptscriptstyle
8(ijkl)}$
$\displaystyle=(\bar{u_{i}}\gamma^{\mu}T^{A}u_{j})(\bar{d_{k}}\gamma_{\mu}T^{A}d_{l}),$
${}^{\dagger}\mathcal{Q}_{\scriptscriptstyle quqd}^{\scriptscriptstyle
8(ijkl)}$
$\displaystyle=(\bar{q_{i}}T^{A}u_{j})\epsilon(\bar{q_{k}}T^{A}d_{l}),$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle uu}^{\scriptscriptstyle(ijkl)}$
$\displaystyle=(\bar{u_{i}}\gamma^{\mu}u_{j})(\bar{u_{k}}\gamma_{\mu}u_{l}),$
(2)

where the notation $\mathcal{Q}$ indicates operators are given in the original
Warsaw basis Grzadkowski:2010es . In this work however, we use operators
aligned with the SMEFTatNLO Degrande:2020evl  conventions, hereafter referred
to as the “top-basis” and denoted by $\mathcal{O}$. The difference lies in the
slight modification of the four-fermion operators rendering them more suitable
for top quark physics, as well as normalising the operators
$\mathcal{O}_{\scriptscriptstyle tG}$ and $\mathcal{O}_{\scriptscriptstyle G}$
through the inclusion of an extra $g_{s}$ factor in their definitions. The
consequences of the latter normalisation are later discussed when presenting
the inclusive predictions in section 3. The translations of all four-fermion
operators from the Warsaw basis to the top-basis are given in table 5 of
appendix A. We also present the recent constraints on their corresponding
coefficients in table 6 of the same appendix based on the global analysis of
Ref. Ethier:2021bye .

#### Two-fermion and purely-bosonic operators

The set of potentially relevant two-fermion operators to four-top production
are defined as follows:

$\displaystyle\mathcal{Q}_{\scriptscriptstyle tB}$
$\displaystyle=i\big{(}\bar{Q}\tau^{\mu\nu}\,t\big{)}\,\tilde{\varphi}\,B_{\mu\nu}+\text{h.c.},$
$\displaystyle\mathcal{O}_{\scriptscriptstyle tW}$
$\displaystyle=i\big{(}\bar{Q}\tau^{\mu\nu}\,\tau_{\scriptscriptstyle
I}\,t\big{)}\,\tilde{\varphi}\,W^{I}_{\mu\nu}+\text{h.c.},$
$\displaystyle\mathcal{O}_{\scriptscriptstyle tG}$
$\displaystyle=ig{\scriptscriptstyle
S}\,\big{(}\bar{Q}\tau^{\mu\nu}\,T_{\scriptscriptstyle
A}\,t\big{)}\,\tilde{\varphi}\,G^{A}_{\mu\nu}+\text{h.c.},$
$\displaystyle\mathcal{O}_{\scriptscriptstyle t\varphi}$
$\displaystyle=\big{(}\varphi^{\dagger}\varphi-v^{2}/2\big{)}\bar{Q}t\tilde{\varphi}+\text{h.c.},$
$\displaystyle\mathcal{O}_{\scriptscriptstyle\varphi t}$
$\displaystyle=i\big{(}\varphi^{\dagger}\,\overset{\text{\scriptsize$\leftrightarrow$}}{D}_{\mu}\,\,\varphi\big{)}\big{(}\bar{t}\,\gamma^{\mu}\,t\big{)},$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle\varphi
Q}^{\scriptscriptstyle(1)}$
$\displaystyle=i\big{(}\varphi^{\dagger}\overset{\text{\scriptsize$\leftrightarrow$}}{D}_{\mu}\,\varphi\big{)}\big{(}\bar{Q}\,\gamma^{\mu}\,Q\big{)},$
$\displaystyle\mathcal{Q}_{\scriptscriptstyle\varphi
Q}^{\scriptscriptstyle(3)}$
$\displaystyle=i\big{(}\varphi^{\dagger}\overset{\text{\scriptsize$\leftrightarrow$}}{D}_{\mu}\,\tau_{I}\,\varphi\big{)}\big{(}\bar{Q}\,\gamma^{\mu}\,\tau^{I}\,Q\big{)}.$
(3)

For convenience and following the conventions of Degrande:2020evl ; Aguilar-
Saavedra:2018ksv  we consider the following linear combinations of Warsaw
operators’ coefficients:

$c_{tW}=C_{tW},\hskip
2.15277ptc_{tZ}=-\sin{\theta_{\mathrm{w}}}C_{tB}+\cos{\theta_{\mathrm{w}}}C_{tW},$
(4)

where we kept the notation $c_{i}$ for coefficients of operators written in
the top-basis while $C_{i}$ denotes coefficients in Warsaw basis. Similarly,
we use the linear combination $c^{(-)}_{\varphi Q}=C^{(1)}_{\varphi
Q}-C^{(3)}_{\varphi Q}$ instead of the singlet piece (this combination is
notated as cpQM in SMEFTatNLO while the triplet $c_{\varphi Q}^{(3)}$ as
cpQ3). The coefficient $c_{\varphi Q}^{(3)}$ is irrelevant to four-top
production since it modifies the $tWb$ vertex.

On the other hand, the relevant purely-bosonic operators in the top-basis are
defined as follows:

$\displaystyle\mathcal{O}_{\scriptscriptstyle\varphi G}$
$\displaystyle=\bigg{(}\varphi^{\dagger}\varphi-\frac{v^{2}}{2}\bigg{)}G_{A}^{\mu\nu}G^{A}_{\mu\nu},\qquad\qquad\mathcal{O}_{\scriptscriptstyle
G}=g_{s}f_{ABC}G^{A}_{\mu\nu}G^{B,\nu\rho}G^{C,\mu}_{\rho}.$ (5)

The latter is constrained by studies including multi-jet production
Krauss:2016ely ; Hirschi:2018etq . Bounds on the
$\mathcal{O}_{\scriptscriptstyle\varphi G}$ coefficient, as well as all two-
fermion coefficients in eq. 3, are given in table 7 of appendix A. Operators
that modify the Higgs couplings to the gauge bosons and those that enter via
fields’ redefinition have negligible contributions in four-top production.
Therefore, we omit them in what follows as we expect them to be constrained
much better in other processes.

Having defined the potential operators relevant to four-top production, we now
move to analyse the SM and EFT amplitudes in orders of QCD and EW couplings
and subsequently examine different terms contributing to the cross-section.

###  2.2 Leading order coupling expansion

In the presence of SMEFT operators, a generic scattering amplitude expanded in
the $1/\Lambda$ parameter can be written as follows:

$\displaystyle\mathcal{A}=\mathcal{A}_{\rm
SM}+\frac{1}{\Lambda^{2}}\mathcal{A}_{\rm(d6)}+\frac{1}{\Lambda^{4}}\big{(}\mathcal{A}_{\rm(d6)^{2}}+\mathcal{A}_{\rm(d8)}\big{)},$
(6)

leading to the decomposition of the partonic differential cross-section up to
$\mathscr{O}(\Lambda^{-4})$,

$\displaystyle d\sigma=d\sigma_{\rm SM}+\frac{1}{\Lambda^{2}}d\sigma_{\rm
int}+\frac{1}{\Lambda^{4}}\big{(}d\sigma_{\rm quad}+d\sigma_{\rm
dbl}+d\sigma_{\rm d8}\big{)}.$ (7)

The leading SMEFT contribution, $d\sigma_{\rm int}$, arises as the linear
interference between $\mathcal{A}_{\rm SM}$ and $\mathcal{A}_{\rm(d6)}$, while
the $d\sigma_{\rm quad}$ and $d\sigma_{\rm dbl}$ are the squared single-
insertion (also known as the quadratic), and double-insertion contributions,
respectively. All the $\mathscr{O}(\Lambda^{-4})$ contributions can be
schematically written in terms of the amplitudes,

$\displaystyle d\sigma_{\rm quad}\sim|\mathcal{A}_{\rm(d6)}|^{2},\qquad
d\sigma_{\rm dbl}\sim|\mathcal{A}_{\rm SM}\,\mathcal{A}_{\rm(d6)^{2}}|,\qquad
d\sigma_{\rm d8}\sim|\mathcal{A}_{\rm SM}\,\mathcal{A}_{\rm(d8)}|,$ (8)

where the latter is the contribution arising from amplitudes with a single
insertion of a dimension-eight operator interfering with the SM ones. The
construction of the SMEFT dimension-eight basis was explored in Murphy:2020rsh
; Li:2020gnx , however, the systematic treatment of those operators is beyond
the scope of this work. In this work, we study all the contributions arising
from dimension-six operators, including the particular case for which we
examine the double-insertion ones, $d\sigma_{\rm dbl}$, in section 7.

The SM differential cross-section can be expanded in orders of the QCD and EW
couplings,

$\displaystyle d\sigma_{\rm SM}=\sum_{n,m}\alpha_{s}^{n}\,\alpha_{\rm
w}^{m}\,d\sigma^{(n,m)}_{\rm
SM}=\sum_{i,j,k}\alpha_{s}^{i}\,\alpha^{j}\,\alpha_{t}^{k}\,d\sigma^{(i,j,k)}_{\rm
SM},$ (9)

with $\alpha_{\rm w}$ collectively representing $\alpha$ and $\alpha_{t}$, and

$\alpha_{s}=\frac{g_{s}^{2}}{4\pi},\qquad\alpha=\frac{e^{2}}{4\pi},\qquad\alpha_{t}=\frac{y_{t}^{2}}{4\pi}.$
(10)

The four-top production occurs via the $gg-$ and $q\bar{q}$-initiated
channels. Each of the amplitudes is a six-point diagram and thus has four
couplings, i.e. $(i+j+k)=2$, and so we expand the $gg-$ and
$q\bar{q}$-initiated SM amplitudes in terms of the QCD and EW couplings as
follows:

$\displaystyle\mathcal{A}^{(i,j,k)}_{{\rm SM},gg}$
$\displaystyle=\alpha_{s}^{2}\,\mathcal{A}^{(2,0,0)}_{{\rm
SM},gg}+\alpha_{s}\left(\,\alpha\,\mathcal{A}^{(1,1,0)}_{{\rm
SM},gg}+\alpha_{t}\,\mathcal{A}^{(1,0,1)}_{{\rm SM},gg}\right),$ (11a)
$\displaystyle\mathcal{A}^{(i,j,k)}_{{\rm SM},qq}$
$\displaystyle=\alpha_{s}^{2}\,\mathcal{A}^{(2,0,0)}_{{\rm
SM},qq}+\alpha_{s}\,\left(\alpha\,\mathcal{A}^{(1,1,0)}_{{\rm
SM},qq}+\alpha_{t}\,\mathcal{A}^{(1,0,1)}_{{\rm SM},qq}\right)$ (11b)
$\displaystyle+\left(\alpha^{2}\,\mathcal{A}^{(0,2,0)}_{{\rm
SM},qq}+\alpha^{3/2}\,\alpha_{t}^{1/2}\,\mathcal{A}^{(0,3/2,1/2)}_{{\rm
SM},qq}+\alpha\,\alpha_{t}\,\mathcal{A}^{(0,1,1)}_{{\rm SM},qq}\right),$

with the term containing half-integer couplings, i.e.
$\mathscr{O}(\alpha^{3/2}\,\alpha_{t}^{1/2})$, arising from diagrams
containing a Higgs boson coupling to a top quark via one top-Yukawa vertex and
a coupling to two EW bosons via one EW vertex. Each of the two $W$ bosons
couples with a fermion line.

Moving to the EFT case, as an example, we show the cross-section expansion for
one class of operators; the four-fermion ones. The insertions of a single
dimension-six four-fermion operator in the amplitudes of both production
channels are depicted in fig. 3.

$\bar{t}$$g$$\bar{t}$$g$$t$$t$$\bar{t}$$q$$t$$\bar{q}$$\bar{t}$$t$

Figure 3: Representative diagrams for four-top production with blobs
representing the one dimension-six EFT insertion, in the $gg$-initiated
production mode (_left_) and in the $q\bar{q}$-initiated production mode
(_right_).

The EFT linear interference cross-section can be decomposed in the same way as
the SM,

$\displaystyle d\sigma_{\rm int}=\sum_{n,m}\alpha_{s}^{n}\,\alpha_{\rm
w}^{m}\,d\sigma^{(n,m)}_{\rm
int}=\sum_{i,j,k}\alpha_{s}^{i}\,\alpha^{j}\,\alpha_{t}^{k}\,d\sigma^{(i,j,k)}_{\rm
int},$ (12)

where each of the expanded partial cross-section is a sum of contributions
from different WCs,

$d\sigma_{\rm int}^{(n,m)}=c_{[r]}d\sigma^{(n,m)}_{{\rm int}\,[r]}.$ (13)

The index $[r]$ runs over all possible dimension-six operators. For the
discussion, we write the EFT linear interference cross-section in terms of the
SM and EFT amplitudes,

$\displaystyle d\sigma_{\rm int}$ $\displaystyle=d\sigma_{{\rm
int},gg}+d\sigma_{{\rm int},qq}\sim 2\mathfrak{R}\left(\mathcal{A}_{{\rm
SM},gg}\,\mathcal{A}_{{\rm
EFT},gg}^{\dagger}\right)+2\mathfrak{R}\left(\mathcal{A}_{{\rm
SM},qq}\,\mathcal{A}_{{\rm EFT},qq}^{\dagger}\right),$ (14)

(we have omitted the PDF dependence for simplicity) with the four-fermion EFT
amplitudes reading

$\displaystyle\mathcal{A}^{(i,j,k)}_{{\rm EFT},gg,[\texttt{4F}]}$
$\displaystyle=\alpha_{s}\mathcal{A}^{(1,0,0)}_{{\rm EFT},gg\,[\texttt{4F}]},$
(15a) $\displaystyle\mathcal{A}^{(i,j,k)}_{{\rm EFT},qq\,[\texttt{4F}]}$
$\displaystyle=\alpha_{s}\mathcal{A}^{(1,0,0)}_{{\rm
EFT},qq\,[\texttt{4F}]}+\alpha\mathcal{A}^{(0,1,0)}_{{\rm
EFT},qq\,[\texttt{4F}]}+\alpha_{t}\mathcal{A}^{(0,0,1)}_{{\rm
EFT},qq\,[\texttt{4F}]}.$ (15b)

We can then write the $gg$ and the $q\bar{q}$-initiated interference cross-
section contributions induced by the four-fermion operators in terms of all
the QCD and EW coupling orders,

$\displaystyle d\sigma_{{\rm int},gg,\texttt{[4F]}}$
$\displaystyle=\alpha_{s}^{3}\,d\sigma^{(3,0,0)}_{{\rm
int},gg}+\alpha_{s}^{2}\left(\,\alpha\,d\sigma^{(2,1,0)}_{{\rm
int},gg}+\alpha_{t}\,d\sigma^{(2,0,1)}_{{\rm int},gg}\right).$ (16a)
$\displaystyle d\sigma_{{\rm int},qq,\texttt{[4F]}}$
$\displaystyle=\alpha_{s}^{3}\,d\sigma^{(3,0,0)}_{{\rm int},qq}$
$\displaystyle+\alpha_{s}^{2}\left(\alpha\,d\sigma^{(2,1,0)}_{{\rm
int},qq}+\alpha_{t}\,d\sigma^{(2,0,1)}_{{\rm int},qq}\right)$
$\displaystyle+\alpha_{s}\left(\alpha^{2}\,d\sigma^{(1,2,0)}_{{\rm
int},qq}+\alpha^{3/2}\,\alpha_{t}^{1/2}\,d\sigma^{(1,3/2,1/2)}_{{\rm
int},qq}+\alpha\alpha_{t}\,d\sigma^{(1,1,1)}_{{\rm
int},qq}+\alpha_{t}^{2}\,d\sigma^{(1,0,2)}_{{\rm int},qq}\right)$
$\displaystyle+(\alpha^{3})\,d\sigma^{(0,3,0)}_{{\rm
int},qq}+(\alpha^{5/2}\,\alpha_{t}^{1/2})\,d\sigma^{(0,5/2,1/2)}_{{\rm
int},qq}$ $\displaystyle+(\alpha^{2}\,\alpha_{t})\,d\sigma^{(0,2,1)}_{{\rm
int},qq}+(\alpha^{3/2}\,\alpha_{t}^{3/2})\,d\sigma^{(0,3/2,3/2)}_{{\rm
int},qq}+(\alpha\,\alpha_{t}^{2})\,d\sigma^{(0,1,2)}_{{\rm int},qq}.$ (16b)

##  3 Hierarchy of inclusive predictions

This section presents the numerical results from the complete LO SMEFT
predictions of the $t\bar{t}t\bar{t}$ production process at $\sqrt{s}=13$ TeV
for the LHC, and at $\sqrt{s}=100$ TeV for future circular $pp$ colliders. The
computations were performed via MadGraph5_aMC@NLO Alwall:2014hca ;
Frederix:2018nkq  with the use of the SMEFTatNLO model Degrande:2020evl , and
with the mass of the top quark, $m_{t}$, set to 172 GeV. Since
MadGraph5_aMC@NLO does not evolve the operator coefficients, and as
recommended, the factorisation ($\mu_{F}$), renormalization ($\mu_{R}$) scales
are fixed to 340 GeV$\sim(4m_{t})/2$222 The EFT renormalisation scale (mueft)
parameter of the SMEFTatNLO in MadGraph5_aMC@NLO is not relevant unless the
running of the EFT coefficients is included. We do not consider the running of
the EFT coefficients in this work.. The proton PDFs and their uncertainties
are evaluated employing reference sets and error replicas from the NNPDF3.1
NLO global fit NNPDF:2017mvq  in the five flavour scheme (5FS), in which the
bottom quark is taken to be massless. Unless otherwise explicitly mentioned,
no parton-level cuts are imposed. Before discussing the SMEFT results, we
first show the decomposition of the four-top SM cross-sections in table 1.

$\sqrt{s}$ | $\mathscr{O}(\alpha_{s}^{4})$ | $\mathscr{O}(\alpha_{s}^{3}\alpha)$ | $\mathscr{O}(\alpha_{s}^{3}\alpha_{t})$ | $\sum_{n,m}\mathscr{O}(\alpha_{s}^{2}\alpha^{n}\alpha^{m}_{t})$ | $\sum_{n,m}\mathscr{O}(\alpha_{s}\alpha^{n}\alpha^{m}_{t})$ | $\sum_{n,m}\mathscr{O}(\alpha^{n}\alpha^{m}_{t})$ | Inclusive
---|---|---|---|---|---|---|---
13 TeV | 6.15 | -1.44 | -0.58 | 2.33 | $\times$ | $\times$ | 6.46
100 TeV | 2570 | -313 | -197 | 753 | $\times$ | $\times$ | 2812

Table 1: Entries in each column correspond to the LO $t\bar{t}t\bar{t}$ SM
cross-section [fb] in line of $\alpha_{s}$ order, at the LHC ($\sqrt{s}=13$)
and FCC-hh ($\sqrt{s}=100$). For contributions at
$\mathscr{O}(\alpha_{s}^{N})$ with $N=0-2$, we sum all possible weak and
Yukawa coupling combinations and present the total cross-section at a given
order in $\alpha_{s}$. The notation “$\times$” denotes negligible
contributions. The column titled “Inclusive” shows the total cross-section.

The strengths of the linear interference of all the dimension-six SMEFT
operators belonging to the four-fermion, two-fermion, and purely-bosonic
classes are presented. The interference strength is the interference cross-
section in fb with the corresponding WC individually set to unity, and the
scale of new physics $\Lambda$ is fixed to 1(3) TeV, for the
$\sqrt{s}=13(100)$ TeV scenario. In presenting the results, the four-fermion
operators are categorised into two sub-classes; contact terms involving four
heavy quarks (4-heavy) and contact terms involving two heavy and two light
quarks (2-heavy-2-light). Respectively, those insertions are depicted by the
blobs shown in the _left_ and _right_ diagrams of fig. 3. On the other hand,
since not all the two-fermion and purely-bosonic operators are relevant to
$t\bar{t}t\bar{t}$ production, only the _contributing_ operators from these
classes are presented;

$\\{\mathcal{O}_{t\varphi},\mathcal{O}_{tZ},\mathcal{O}_{tW},\mathcal{O}_{tG},\mathcal{O}_{\varphi
Q}^{(-)},\mathcal{O}_{\varphi t},\mathcal{O}_{G},\mathcal{O}_{\varphi G}\\}.$
(17)

The total inclusive interference cross-section in the four-fermion case can be
defined as follows:

$\sigma_{\rm incl.}=\sigma_{3}+\sigma_{2}+\sigma_{1}+\sigma_{0},$ (18)

where $\sigma_{i}$ with $i=3,2,1,0$ denotes the contributions to $\sigma_{\rm
incl.}$ arising from terms with order $\alpha_{s}^{i}$ in the cross-section
expansion. For example, the $\sigma_{2}$ term denotes the interference cross-
section arising _only_ from the formally subleading terms in $\alpha_{s}$,
i.e. $\mathscr{O}(\alpha_{s}^{2}\alpha)$ and
$\mathscr{O}(\alpha_{s}^{2}\alpha_{t})$ in eq. 16a and eq. 16b, which can be
collectively written as follows:

$\sigma_{2}\equiv\alpha_{s}^{2}\left(\,\alpha\,\sigma^{(2,1,0)}_{{\rm
int}}+\alpha_{t}\,\sigma^{(2,0,1)}_{{\rm int}}\right).$ (19)

The interference strength is depicted in the heat maps presented in fig. 4-9,
the columns correspond to the operators’ coefficients. The top row shows the
total inclusive interference cross-section $\sigma_{\rm incl}$ labelled INCL,
while subsequent rows correspond to the separate contributions arranged in
order of $\alpha_{s}$, in line with the example of eq. 19.

Figure 4:  Depiction of the interference strength of the 4-heavy operators.
The columns denote the WCs in their UFO notation. The top row shows the total
inclusive predictions, i.e. summing all the QCD and EW-induced contributions.
Each of the subsequent rows indicates the summation of all terms at a given
order of $\alpha_{s}$ (an example is given in the main text). The predictions
are obtained at $c=1$, $\sqrt{s}=13$ TeV, and $\Lambda=1$ TeV. The SM LO
cross-section at $\sqrt{s}=13$ TeV is presented for reference. Figure 5:  Same
as fig. 4 but for the 2-heavy 2-light operators Figure 6:  Same as fig. 4 but
for the contributing operators (see eq. 17). Figure 7:  Same as fig. 4 but for
the FCC-hh scenario. The predictions are obtained at $c=1$, $\sqrt{s}=100$
TeV, and $\Lambda=3$ TeV. The SM LO cross-section at $\sqrt{s}=100$ TeV is
presented for reference. Figure 8:  Same as fig. 7 but for the 2-heavy 2-light
operators. Figure 9:  Same as fig. 7 but for the contributing operators.

####  $\mathbf{\sqrt{s}=13}$ TeV

Starting with the 4-heavy operators in fig. 4, we observe that for _all_ of
them, the dominant interference is the one arising from formally subleading
orders, $\sigma_{2}$. This observation contrasts the “naive” expectation that
leading (purely QCD-induced) terms would provide the highest contribution to
the cross-section through $\sigma_{3}$ and consequently highlights the
significance of the EW $tt\to tt$ scattering present in $t\bar{t}t\bar{t}$
production at LO (see fig. 2). The significance of such EW scattering in four-
top production has been pointed out in the NLO SM computation of Ref.
Frederix:2017wme . It is worth noting that such naive expectation not only
underestimates the interference strength of the 4-heavy operators but also
generates the ‘wrong’ sign of the interference structure. That is,
$\sigma_{2}$ for all the 4-heavy operators has the opposite sign of
$\sigma_{3}$. The former dictates the overall sign of the inclusive
predictions. Furthermore, the lower-order-$\alpha_{s}$ cross-sections, i.e.
$\sigma_{n}$ where $n<2$, are heavily suppressed, rendering the consideration
of cross-section contributions only down to $\sigma_{2}$ enough to attain
reliable predictions for this set of operators. Finally, for the 4-heavy
operators, the colour-singlets, i.e. $\mathcal{O}_{\scriptscriptstyle
QQ}^{\scriptscriptstyle 1}$, $\mathcal{O}_{\scriptscriptstyle
Qt}^{\scriptscriptstyle 1}$, and $\mathcal{O}_{\scriptscriptstyle
tt}^{\scriptscriptstyle 1}$, are observed to have a stronger interference
compared to the colour-octets, $\mathcal{O}_{\scriptscriptstyle
QQ}^{\scriptscriptstyle 8}$ and $\mathcal{O}_{\scriptscriptstyle
Qt}^{\scriptscriptstyle 8}$, we analyse this effect later in our discussion of
differential predictions.

Moving on to the 2-heavy-2-light operators, in fig. 5, we observe, except for
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$, the
interference strength in this class is dominated by the formally leading
$\sigma_{3}$ cross-section. This hints at the EW scattering effects being less
critical in the interference with $q\bar{q}$-initiated EFT amplitudes. The
insertions of the 2-heavy-2-light operators are only present in the
$q\bar{q}$-initiated production shown in the _right_ diagram of fig. 3.

Finally, the interference strength of the contributing operators is presented
in fig. 6. Due to the model normalisation in eq. 3 and eq. 5, it is
immediately apparent that only $\mathcal{O}_{\scriptscriptstyle tG}$ and
$\mathcal{O}_{\scriptscriptstyle G}$ have a non-vanishing $\sigma_{4}$. This
is expected because a contribution to the cross-section at this order in
$\alpha_{s}$ is not available for the other contributing operators; therefore,
their leading cross-sections are $\sigma_{3}$. In contrast to the
$\mathcal{O}_{\scriptscriptstyle tW}$ and $\mathcal{O}_{\scriptscriptstyle
tZ}$ dipoles, the contributing two-fermion operators,
$\mathcal{O}_{\scriptscriptstyle t\varphi}$,
$\mathcal{O}_{\scriptscriptstyle\varphi Q}^{\scriptscriptstyle(-)}$, and
$\mathcal{O}_{\scriptscriptstyle\varphi t}$, have formally subleading dominant
cross-sections. The $\mathcal{O}_{\scriptscriptstyle tG}$ operator even though
dominating at $\sigma_{4}$, has a non-negligible $\sigma_{2}$. Finally, and in
complete contrast to $\mathcal{O}_{\scriptscriptstyle G}$, the interference
strength of $\mathcal{O}_{\scriptscriptstyle\varphi G}$ tends to be inversely
proportional to orders in $\alpha_{s}$, in other words, proportional to the
number of EW propagators.

####  $\mathbf{\sqrt{s}=100}$ TeV

The only difference between the LHC and the FCC-hh computations is that we fix
the scale of new physics $\Lambda$ to 3 TeV for the latter. This is intended
to ensure a reliable expansion of the EFT series given the high collision
energy of FCC-hh. The interference strength at the FCC-hh scenario from the
4-heavy, 2-heavy-2-light, and the contributing operators are presented in fig.
7, fig. 8, and fig. 9, respectively. Apart from the expected scaling of the
cross-sections in the FCC-hh scenario, we see a similar pattern across the
board when comparing to the LHC study, albeit with some slight differences:
the $\sigma_{3}$ interference of $\mathcal{O}_{\scriptscriptstyle
Qt}^{\scriptscriptstyle 1}$ has an opposite sign in the 100 TeV scenario. The
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 1}$ and
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 1}$ operators have a
slightly dominant $\sigma_{2}$ in contrast to the LHC case where the dominant
cross-section is $\sigma_{3}$. Finally, for
$\mathcal{O}_{\scriptscriptstyle\varphi G}$, and while $\sigma_{1}$ is still
significant in parallel to the LHC scenario, the $\sigma_{3}$ interference is
slightly stronger in the 100 TeV scenario.

In summarising this section, we present table 2 in which all the operators
align with their most dominant cross-section contributions.

| 4H | 2L2H | 2F | 0F
---|---|---|---|---
$\sigma_{4}$ | $\times$ | $\times$ | $c_{tG}$ | $c_{G}$
$\sigma_{3}$ | - | $c_{Qq}^{83},c_{Qu}^{8},c_{tq}^{8},c_{Qd}^{8},c_{tu}^{8},c_{td}^{8},c_{Qq}^{81}$ | $c_{t\varphi},c_{tZ},c_{tW}$ | -
$c_{Qq}^{11},c_{Qu}^{1},c_{tq}^{1},c_{Qd}^{1},c_{tu}^{1},c_{td}^{1}$
$\sigma_{2}$ | $c_{QQ}^{8},c_{QQ}^{1},c_{Qt}^{8},c_{Qt}^{1},c_{tt}^{1}$ | $c_{Qq}^{31}$ | $c_{\varphi t},c_{\varphi Q}^{(-)}$ | -
$\sigma_{1}$ | - | - | - | $c_{\varphi G}$
$\sigma_{0}$ | - | - | - | -
Table 2: Indication of the most significant contribution to the total cross-
section of each operator at $\sqrt{s}=13$ TeV. Entries labelled $\times$
denote such coupling order is not allowed for the given class of operators.
The blue colour denotes operators with contributions not only dominant at this
given order in $\alpha_{s}$, but also other (higher or lower) orders in
$\alpha_{s}$ are significant enough that they can alter the total rate if not
considered.

Furthermore, we put together all operators featuring an unexpected enhancement
to their cross-sections from formally subleading terms,

$\textit{all}\,\,\text{4-heavy}\qquad\text{and}\qquad\\{\mathcal{O}_{Qq}^{3,1},\mathcal{O}_{t\varphi},\mathcal{O}_{tG},\mathcal{O}_{\varphi
Q}^{(-)},\mathcal{O}_{\varphi t},\mathcal{O}_{\varphi G}\\}.$ (20)

Hereafter, we refer to this group of operators as the _‘non-naive’_ ones. More
precisely, we define non-naive operators as ones for which _any_ of their
formally non-leading terms is significant in estimating their total
interference cross-section.

##  4 Differential predictions

This section presents the LHC and FCC-hh differential predictions for the set
of non-naive operators of eq. 20. The distributions are given in fig. 10-14.
Differential predictions for the rest of the operators are given in appendix
B. We present the distributions in the invariant mass bins of the four top
quark system, $m_{tttt}\sim\sqrt{s}$, for which we also include the SMEFT
diagonal333 These contributions correspond to the squared interference between
the same dimension-six operators. quadratic contributions. The input
parameters are the same as the ones used for the inclusive results. We show
pure SM predictions, SM summed to the linear EFT interference, and SM summed
to the linear EFT interference and the diagonal quadratic contributions.
Moreover, results are presented in every order of $\alpha_{s}$, e.g. INT201
indicates the interference contribution (INT) induced from
$\mathscr{O}(\alpha_{s}^{2}\alpha_{t})$ terms, where the first, second and
third digits denote the orders of $\alpha_{s}$, $\alpha$, and $\alpha_{t}$,
respectively. For orders ‘below’ the formal subleading one, we sum all EW-
induced contributions at this given $\alpha_{s}$-order. For example, 1XX
indicates summing all possible EW-induced contributions with one $\alpha_{s}$
coupling, in parallel to the notation used in section 3; 1XX
$\equiv\sigma_{1}$. The relative scale uncertainties are computed individually
from a nine-point renormalisation and factorisation scale variation around the
central scale of $340$ GeV for each EFT contribution. Contrary to inclusive
predictions, we here used WCs values extracted approximately from the global
fit of Ref. Ethier:2021bye , except for $\mathcal{O}_{\scriptscriptstyle G}$,
for which the value of its coefficient is specified in the corresponding plot.

####  $\mathbf{\sqrt{s}=13}$ TeV

One clear pattern in all of the 4-heavy operators’ predictions is the sizable
interference cross-section arising from the
$\mathscr{O}(\alpha_{s}^{2}\alpha)$ term and depicted by the blue line in the
second inset.

Figure 10:  Differential EFT predictions at $\sqrt{s}=13$ TeV shown in the
invariant mass bins of the four top quark system, $m_{tttt}$, for the set of
non-naive four-fermion operators, i.e. all 4-heavy operators and the
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ operator. The
values of the coefficients are extracted from the global fit study of Ref.
Ethier:2021bye , and $\Lambda$ is set to 1 TeV. The first inset displays the
relative scale uncertainties individually calculated for the SM, the
interference, and the squared EFT contributions. The second inset shows the
ratio of the interference at each order in $\alpha_{s}$ to the SM prediction.
The last inset shows the ratio of the total interference and the total squared
contributions to the SM.

Figure 11:  Same as fig. 10 but for the non-naive two-fermion and purely-
bosonic operators.

This observation corroborates what we observed in the inclusive results
section; formally subleading contributions are significant. For $\sigma_{2}$,
we find that the dominant contribution comes from the
$\mathscr{O}(\alpha_{s}^{2}\alpha)$ terms rather than those of
$\mathscr{O}(\alpha_{s}^{2}\alpha_{t})$. Moreover, and similarly to the
inclusive predictions, colour-singlets are observed to interfere with the SM
comparatively stronger than colour-octet ones. We had already mentioned this
pattern in section 3; however, we relegated the detailed discussion to this
section.

#### Interlude: on the interference pattern of colour-singlets and colour-
octets

It is expected that in formally QCD-dominated production processes, amplitudes
with colour-octet insertions would exhibit a stronger interference pattern
compared to colour-singlets. For instance, in $\bar{t}t$ production, colour-
singlet contributions vanish at the tree-level when considering only purely
QCD-induced SM amplitudes Brivio:2019ius . In four-top production, however,
the interference between colour-singlets and the QCD SM amplitudes is non-zero
due to the presence of more complicated colour structures, which allow the
top-anti-top pair to be in a colour-singlet state. This is evident at the
$\sigma_{3}$-level, where the colour-singlets interference can be of the same
order as that of the colour-octets. In addition, and as discussed previously,
the EW scattering effects in the $gg\to\bar{t}t\bar{t}t$ born-level
amplitudes, depicted in fig. 2, provide significant contributions to the
cross-section, in the SM and in the SMEFT. This explains the weaker
interference strength of colour-octets compared to colour-singlets in the set
of 4-heavy operators at the $\sigma_{2}$-level. The reason comes from the
different colour flow in the $tt$ $s$-channel scattering sub-amplitude; EFT
amplitudes with a colour-octet insertion would interfere with the formally
leading SM amplitudes where a gluon is the mediator of the $tt\to tt$
scattering. On the other hand, amplitudes with the insertion of a colour-
singlet operator are expected to interfere with the formally subleading SM
amplitudes where the $tt\to tt$ scattering happens via an EW mediator.

In contrast to the 4-heavy operators, the previously-mentioned insignificance
of EW scattering effects in the $q\bar{q}$-initiated production channel
explains the ‘typical’ stronger interference strength of the 2-heavy-2-light
colour-octets compared to colour-singlets. The only exception to this is the
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ operator.
Interestingly, the latter is also the _only_ 2-heavy 2-light operator in the
set of non-naive operators. As discussed above, this suggests that the
enhancement from formally subleading terms is indeed intertwined with the
‘unusual’ more potent interference from colour-singlets. The interference
pattern of colour-octets and -singlets in the set of 4-heavy operators is
summarised in fig. 12.

Figure 12:  The ratio of the linear EFT interference to the SM prediction for
the non-naive 4-heavy operators. Solid lines depict the
$\mathscr{O}(\alpha_{s}^{2}\alpha)$ contribution, while dashed lines depict
the $\mathscr{O}(\alpha_{s}^{3})$ one. Thick (thin) lines represent colour-
singlets (-octets).

Moving to the non-naive two-fermion and purely-bosonic operators, we observe a
different EFT structure compared to 4-heavy ones. EFT amplitudes with
insertions of four-fermion operators have effective contact terms as depicted
in fig. 3. The energy scaling of such amplitudes leads to consistent growth of
the quadratic contribution as a function of $\sqrt{s}$. This is seen in the
third inset of all the distributions in fig. 10. On the other hand, due to
their different energy scaling, the two-fermion operators exhibit suppressed
squared contributions and, in most cases, a decaying interference. Such effect
is apparent in the third insets of fig. 11, with a notable exception of the
$\mathcal{O}_{\scriptscriptstyle tG}$ operator. The latter receives
enhancement in its energy scaling from gluon field strength derivatives, hence
its different EFT structure: the linear contribution scales as $\sim 1/E^{3}$,
while the quadratic one scales as $\sim 1/E^{2}$. Such EFT structure is also
evident for $\mathcal{O}_{\scriptscriptstyle G}$ in fig. 24.

In summarising the LHC results, we note that the 4-heavy operators are the
most sensitive probes to four-top production. This can be deduced from their
sizable interference magnitude compared to the 2-heavy-2-light operators and
the two-fermion and purely-bosonic ones.

####  $\mathbf{\sqrt{s}=100}$ TeV

Despite the high collision energy of the FCC-hh computation, the increased
value of $\Lambda$, i.e. $\Lambda=3$ TeV, initially set to ensure a reliable
EFT expansion, significantly suppresses the magnitude of the EFT contributions
when compared to the ones from the LHC study. As a reminder, changing
$\Lambda$ from 1 to 3 TeV suppresses the interference contribution by a factor
of 9 and the quadratic one by a factor of $81$.

Figure 13:  Same as fig. 10 but for $\sqrt{s}=100$ TeV.

Figure 14:  Same as fig. 11 but for $\sqrt{s}=100$ TeV.

Nevertheless, and in exploiting the predictions of $m_{tttt}$ at higher
energies, we observe the expected energy growth of EFT contributions as a
function of $\sqrt{s}$; inherently due to the contact term nature of those
operators. Again, we observe colour-singlets interfering with the SM more
strongly than colour-octets, drawing parallels to the LHC predictions.
However, in contrast to the LHC predictions, for almost all 2-heavy-2-light
colour-octets presented in fig. 23 for the FCC-hh scenario and the choice of
$\Lambda$, the linear interference contributions dominate the quadratic ones.
Moving to two-fermion and purely-bosonic operators, and albeit a milder
quadratics growth for $\mathcal{O}_{\scriptscriptstyle G}$ at FCC-hh in fig.
25, we see a similar behaviour between the LHC and FCC-hh predictions.

##  5 Sensitivity projections at future colliders

Given the current evidence of $t\bar{t}t\bar{t}$ production amounting to
4.7$\sigma$ significance ATLAS-CONF-2021-013 , $t\bar{t}t\bar{t}$ is expected
to be discovered at the LHC with the Run III data. The four-top production is
induced mainly by gluons in the initial state rendering the $gg$-initiated
production dominating $\sim 87\%$ of the SM $t\bar{t}t\bar{t}$ cross-section
at 13 TeV and $\sim 99\%$ at 100 TeV. Since the main background contribution
to the $t\bar{t}t\bar{t}$ signal arises from $t\bar{t}W$ production
ATLAS:2018kxv , which proceeds only in the quark-initiated mode, an increase
in the collision energy can lead to an improvement of the signal-to-background
ratio of four-top production. Moreover, uncertainties polluting the
experimental measurement can be progressively reduced as a function of an
increasing integrated luminosity. The study of Ref. Azzi:2019yne  combined the
expected $t\bar{t}t\bar{t}$ experimental sensitivity at future LHC runs and
the state-of-art theoretical calculations Frederix:2017wme  to predict the
total uncertainty by which the $t\bar{t}t\bar{t}$ cross-section can be
determined. Such expected uncertainties are 102%, 58%, and 40%, at 95% CL, for
13, 14, and 27 TeV runs respectively with a corresponding integrated
luminosities of 300 fb-1, 3 ab-1, and 15 ab-1 Azzi:2019yne . In this section,
and using these estimated uncertainties, we reproduce the study of Ref.
Azzi:2019yne  and subsequently add predictions from FCC-hh for comparison. It
is worth noting that this study assumes the sensitivity of the effective
operators to be mainly induced from inclusive measurements. We impose an EFT
validity cut on the invariant mass of the four top quarks, $m_{tttt}<3$ TeV,
so the SMEFT predictions can be matched to UV models with higher energy
scales. The EFT validity cut is assumed not significantly to alter the
projected sensitivity.

To attain the projected sensitivity, we perform a scan over different values
of the WCs, computing the $t\bar{t}t\bar{t}$ signal strength, $\mu_{tttt}$, at
each point. The signal strength is defined as
$\mu_{tttt}=\sigma_{\mathrm{obs.}}^{tttt}/\sigma_{\mathrm{exp.}}^{tttt}$,
where $\sigma_{\mathrm{obs.}}^{tttt}$ is the obtained SMEFT cross-section
including the interference and the quadratic contributions, and
$\sigma_{\mathrm{exp.}}^{tttt}$ is the $t\bar{t}t\bar{t}$ cross-section
assuming no EFT contributions. The WC scans for all the 4-heavy operators
presented in fig. 15 show a significant EFT sensitivity enhancement at high
collision energies, in contrast to the 2-heavy-2-light operators, of which
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ operator’s
projection is presented in the _left_ panel of fig. 16. The reduced
sensitivity of 2-heavy-2-light operators at high energies is because the
$gg$-initiated production dominates over the $q\bar{q}$ one as the collision
energy increases. Furthermore, in the _right_ panel of fig. 16, we show the
scan of the $\mathcal{O}_{\scriptscriptstyle tG}$ operator’s WC, for it being
the most collision-energy-sensitive two-fermion operator in the set of
contributing operators. Finally, the projections for the purely-bosonic
operators $\mathcal{O}_{\scriptscriptstyle\varphi G}$ and
$\mathcal{O}_{\scriptscriptstyle G}$ are shown in fig. 17.

Figure 15: Four-top production signal strength as a function of the WC values
for all the 4-heavy operators. The EFT predictions include both linear
interference and quadratic contributions. The horizontal lines represent the
expected measurement at each collision energy derived from the expected total
uncertainty.

Figure 16: Same as fig. 15 but for the $\mathcal{O}_{\scriptscriptstyle
Qq}^{\scriptscriptstyle 3,1}$ (_left_) and $\mathcal{O}_{\scriptscriptstyle
tG}$ (_right_) operators.

Figure 17: Same as fig. 15 but for the $\mathcal{O}_{\scriptscriptstyle\varphi
G}$ (_left_) and $\mathcal{O}_{\scriptscriptstyle G}$ (_right_) operators.

We summarise the expected individual limits on the WCs of the 4-heavy
operators in table 3. In obtaining the limits, we used the previously-
mentioned expected total uncertainties at future LHC runs (represented by the
horizontal dashed line in the plots), keeping an estimate of 5% total
uncertainty on the $t\bar{t}t\bar{t}$ cross-section measurement at FCC-hh.
High collision energies will certainly aid in constraining the 4-heavy
effective coefficients through four-top production. On the other hand, we
expect other top quark processes to be more sensitive to the rest of the
operators.

$c_{i}$ | 13 TeV | 14 TeV | 27 TeV | 100 TeV
---|---|---|---|---
$c_{QQ}^{1}$ | [-2.2,3] | [-1.8,2.2] | [-1.2,1.8] | [-0.25,0.7]
$c_{QQ}^{8}$ | [-6.75,9] | [-5,7] | [-3.75,5.1] | [-1.0,2.25]
$c_{Qt}^{1}$ | [-2.6,2] | [-2,1.4] | [-1.4,1.1] | [-0.6,0.3]
$c_{Qt}^{8}$ | [-4.2,5.3] | [-3.2,4] | [-2.1,2.7] | [-0.45,1.05]
$c_{tt}^{1}$ | [-1.2,1.4] | [-0.7,1.2] | [-0.6,0.8] | [-0.15,0.35]
Table 3: Theoretical individual limits on the 4-heavy operators’ coefficients
for 13, 14, 27, and 100 TeV hadron colliders at the 95%CL level. Predictions
are obtained including both linear and quadratic contributions in SMEFT with
$\Lambda=1$ TeV.

##  6 Toy fits

In this section, we present limits on effective operators’ coefficients from
simplified
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$
individual fits in various collider scenarios: the LHC, FCC-hh and the HL-LHC.
We explore the impact of (_i_) subleading EW terms, (_ii_) differential
information and (_iii_) the collider energy on the WCs bounds.

#### Impact of subleading EW terms

We start by considering the relevance of the subleading terms in the
interference cross-section expansion of the 4-heavy operators. In fig. 18, the
individual limits on the 4-heavy coefficients at the FCC-hh are presented in
two cases: (I) with only QCD-induced (leading) terms taken into account, and
(II) when contributions to the cross-section from all tree-level terms in the
mixed QCD-EW expansion are included. For the SM prediction at the FCC-hh, we
use the results of table 1 with a 20% theoretical (systematic) uncertainty.
EFT predictions include only the linear interference contributions. For
simplicity, we assume the experimental measurement to be that of the SM cross-
section reported in table 1 with a 5% total (statistical and systematic)
uncertainty.

Figure 18: 95%CL limits on the 4-heavy operators’ coefficients at the FCC-hh
scenario from a
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$
fit. The limits are shown when only considering leading QCD terms and when
considering all the terms in the mixed QCD-EW cross-section expansion. The fit
uses the inclusive $t\bar{t}t\bar{t}$ cross-section, $\sigma_{tttt}$. EFT
predictions were obtained at the interference level.

The importance of the subleading terms is evident when considering _only_ the
contributions from linear interference. However, and since quadratic
contributions of four-heavy operators are _only_ QCD-induced, including them
in the fit would reduce the sensitivity to the subleading terms.

#### Impact of differential information

The HL-LHC will run at $\sqrt{s}=14$ TeV with 3 $\mathrm{ab}^{-1}$ of
integrated luminosity; therefore, it is expected to obtain differential
information for the four-top process experimentally. Motivated by the larger
impact of the EFT operators in the tails of distributions, as illustrated in
fig. 10, we examine the impact of adding the invariant mass distribution of
the four-top in our toy fit for the HL-LHC. Figure 19 displays the individual
limits for the same two cases used previously (QCD-only and mixed QCD-EW) and
compares the use of only inclusive information from $\sigma_{tttt}$ to when
also adding differential information in the fit from $m_{tttt}$. We use the
HL-LHC SM prediction calculated at LO, $\sigma_{tttt}^{\rm HL}=9.0$ fb, with a
20% theoretical uncertainty. The EFT predictions include the linear and
quadratic contributions. We assume the experimental measurement to be that of
the SM within the expected 28% experimental total uncertainty Azzi:2019yne ;
$\sigma_{tttt}^{\rm HL}=9.0\pm 2.52$ fb.

Figure 19: 95%CL limits on the 4-heavy operators’ coefficients at the HL-LHC
scenario from a
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$
fit. The limits are shown for when only considering leading QCD terms and when
considering all the terms, in using only inclusive information from
$\sigma_{tttt}$ and when adding differential information from $m_{tttt}$. EFT
predictions were obtained for the linear and quadratic contributions.

The $m_{tttt}$ distribution is organised in three bins: [600-1500],
[1500-2500], [2500-6000] GeV, with total experimental uncertainties amounting
to 28% for each of the first two bins, and 60% for the latter to account for
the degradation of the statistical uncertainty based on the number of events
expected in each bin. Even though very much simplified and not based on a
detailed analysis of how observables could provide most of the sensitivity,
our results indicate that differential information improves the sensitivity
and should be used whenever possible.

#### Comparison of different collider setups

To fully appreciate the impact of collider energy in constraining the relevant
coefficients, we compare the results from current LHC measurements with the
FCC-hh bounds. For simplicity, we only use the inclusive cross-section. The
limits obtained from the fit are presented in fig. 20. For both scenarios, EFT
predictions include the linear and quadratic contributions. For the LHC, we
use the SM prediction at NLO in QCD of Ref. Frederix:2017wme , and we fit the
theoretical predictions to the inclusive ATLAS ATLAS:2020hpj  and CMS
CMS:2019rvj  measurements. For the FCC-hh, we use the same theoretical and
experimental inputs used for the previous case of fig. 18.

Figure 20: Limits on all four-fermion and relevant operators used in this
study obtained from the
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}$
fit to the ATLAS ATLAS:2020hpj  and CMS CMS:2019rvj  inclusive measurements
and using the SM prediction of Ref. Frederix:2017wme  as well as FCC-hh
projections.

The results from this fit show the significant constraining power that the
FCC-hh will be able to provide for the SMEFT coefficients. Again, the effects
from the subleading terms are diluted by including the quadratic contributions
in the predictions. Finally, we note that it is expected that with the high-
energy reach of the FCC-hh, differential distributions extending well into the
multi-TeV range will become available and further improve the bounds beyond
those expected from the inclusive cross-section.

##  7 Double insertion

In this section, we critically assess Ref. Zhang:2017mls , where it was
suggested that 2-heavy-2-light operators could be better constrained in
$t\bar{t}t\bar{t}$ than in $t\bar{t}$ production. This suggestion was spurred
by the results of Ref. CMS:2017tec  reporting an upper limit on the
$t\bar{t}t\bar{t}$ cross-section to be 4.6 times that of the SM. Due to the
high-energy scale related to the $t\bar{t}t\bar{t}$ process, its cross-section
depending on the fourth power of the operators’ coefficients scales as
$\sim(cE^{2}/\Lambda^{2})^{4}$, an order that double insertion of dimension-
six operators can probe. Ref. Zhang:2017mls  argued these terms enhance the
EFT sensitivity of the 2-heavy-2-light operators to a level at which four-top
can compete with top pair production in constraining said operators.

Our study investigates the strength of the double-insertion contributions in
four-top production. In particular, we compare the EFT sensitivity from
double-insertion to that from the squared single-insertion of the same
2-heavy-2-light operator. As previously discussed, schematically, these
contributions at $\mathscr{O}(\Lambda^{-4})$ can be respectively written as
follows:

$\displaystyle d\sigma_{\rm dbl}\sim|\mathcal{A}_{\rm
SM}\,\mathcal{A}_{\rm(d6)^{2}}|\qquad\text{vs.}\qquad d\sigma_{\rm
quad}\sim|\mathcal{A}_{\rm(d6)}|^{2}.$ (21)

The Feynman diagrams depicting the amplitudes with two dimension-six EFT
insertions are shown in fig. 21.

$q$$t$$\bar{t}$$\bar{q}$$\bar{t}$$t$$q$$\bar{t}$$\bar{t}$$\bar{q}$$t$$t$

Figure 21: Representative diagrams of four-top production with two EFT
insertions represented by the shaded blobs.

As a first step, we proceed by reproducing the predictions of Ref.
Zhang:2017mls . We indeed observed that the double-insertion contributions
enhance the cross-section compared to the squared ones. However, this is only
true given the loose constraints on WCs, of order $\mathscr{O}(5-10)$, that
Ref. Zhang:2017mls  considered. Given the current comparatively stringent
bounds from the global study of Ref. Ethier:2021bye , we do not find an
enhanced EFT sensitivity due to double insertion. Our results are presented in
table 4 where we fixed the value of $c$ to unity and denoted amplitudes with
one EFT insertion with $\mathcal{A}_{1}$ and those with two insertions with
$\mathcal{A}_{2}$.

| 2-heavy 2-light at $c_{i}$=1 |
---|---|---
$\sqrt{s}=13$ TeV |  $\sqrt{s}=100$ TeV
$\mathcal{O}_{i}$ |  $|\mathcal{A}_{1}|^{2}$ [fb] |  $\sum_{k}\mathscr{O}(\mathcal{A}_{2})_{k}$ [fb] | ratio |  $|\mathcal{A}_{1}|^{2}$ [fb] |  $\sum_{k}\mathscr{O}(\mathcal{A}_{2})_{k}$ [fb] | ratio
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,8}$ | 0.27 | 0.01 | 0.04 | 6.40 | 0.40 | 0.06
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,8}$ | 0.28 | 0.05 | 0.18 | 6.36 | 0.63 | 0.10
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 8}$ | 0.21 | 0.03 | 0.14 | 5.34 | 0.50 | 0.09
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 8}$ | 0.34 | 0.06 | 0.18 | 8.44 | 0.76 | 0.09
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 8}$ | 0.13 | 0.03 | 0.23 | 3.13 | 0.35 | 0.11
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 8}$ | 0.17 | 0.03 | 0.18 | 3.97 | 0.41 | 0.10
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 8}$ | 0.10 | 0.02 | 0.20 | 2.18 | 0.27 | 0.12
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ | 1.84 | 0.15 | 0.08 | 46.98 | 5.49 | 0.12
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,1}$ | 1.84 | 0.08 | 0.04 | 47.35 | 0.81 | 0.02
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 1}$ | 1.14 | 0.06 | 0.05 | 29.94 | 2.83 | 0.09
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 1}$ | 1.80 | 0.14 | 0.08 | 46.54 | 6.33 | 0.14
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 1}$ | 0.70 | 0.08 | 0.11 | 17.55 | 2.15 | 0.12
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 1}$ | 1.11 | 0.04 | 0.04 | 29.10 | 2.48 | 0.09
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 1}$ | 0.68 | 0.05 | 0.07 | 17.44 | 1.79 | 0.10
Table 4: Cross-section predictions from the diagonal squared single-insertion
contributions at $\mathscr{O}(\Lambda^{-4})$ are denoted by
$|\mathcal{A}_{1}|^{2}$ and compared to the sum of all double-insertion
contributions, $\mathscr{O}(\mathcal{A}_{2})$, up to
$\mathscr{O}(\Lambda^{-8})$. The ratio column is that of double-insertion
contributions to squared single-insertion ones. The results are presented for
the LHC and FCC-hh collider setups.

We fixed $\Lambda$ to 1(3) TeV for the 13(100) TeV predictions, and summed all
contributions arising from double insertion up to $\mathscr{O}(\Lambda^{-8})$.
We see that for all the 2-heavy-2-light operators, the contributions from the
double insertion are negligible compared to ones from squared single-
insertion, namely $\mathscr{O}(10)$ smaller. A similar pattern is observed for
the FCC-hh case.

It is worth mentioning that while the squared dimension-six single-insertion
in the EFT expansion is invariant under field transformation, the term
corresponding to a double-insertion of dimension-six is not invariant unless
dimension-eight operators are taken into account.

##  8 Summary and conclusions

This work presented a complete analysis of four top quark production at hadron
colliders within the SMEFT framework. We have based our studies on predictions
at the tree-level, yet including all the possible QCD- and EW-coupling order
contributions, keeping gauge and top-Yukawa couplings separate. Observables
were computed in the SMEFT by considering linear, quadratic, and in specific
instances, double-insertion contributions of dimension-six operators. Within
the large set of SMEFT operators possibly contributing to four-top production,
we have identified a subclass, named _non-naive_ , consisting of all the four
heavy operators and the following subset of two-fermion and bosonic operators
$\\{\mathcal{O}_{Qq}^{3,1},\mathcal{O}_{t\varphi},\mathcal{O}_{tG},\mathcal{O}_{\varphi
Q}^{(-)},\mathcal{O}_{\varphi t},\mathcal{O}_{\varphi G}\\}$, i.e. the
operators whose leading contributions at the linear level to four-top
production cross-section arise from formally subleading terms in the QCD-EW
expansion.

We have then analysed the operators’ contributions to four-top production at
the LHC and the FCC-hh colliders. Three main conclusions can be drawn. First,
the 4-heavy operators provide the most significant contribution through the
$\mathscr{O}(\alpha_{s}^{2}\alpha_{\mathrm{w}})$ terms. The same happens for
the remaining six operators in the non-naive set. Second,
$\mathscr{O}(\alpha_{s}^{2}\alpha)$ is dominant compared to
$\mathscr{O}(\alpha_{s}^{2}\alpha_{t})$ coupling orders for the four-heavy
operators in the non-naive set. Third, from $t\bar{t}$ production, one would
naively expect that colour-octets would provide the dominant contributions.
However, we observe the opposite in $t\bar{t}t\bar{t}$ production; within the
non-naive set, the colour-singlets $c_{QQ}^{1},c_{Qt}^{1},c_{tt}^{1}$ have
larger linear interference cross-sections compared to their colour-octet
counterparts. The summary of these results can be found in table 2. Apart from
some slight differences associated with
$\mathcal{O}_{Qt}^{1},\mathcal{O}_{Qu}^{1},\mathcal{O}^{1}_{tq}$ and
$\mathcal{O}_{\varphi G}$, the pattern for all the other 19 operators remains
unchanged when the energy is increased to 100 TeV in the FCC-hh scenario. In
section 4, we have considered the four-top invariant mass differential
distributions, $m_{tttt}$, where the energy growth of each operator can be
studied. The results on the differential level corroborate the conclusions
drawn at the inclusive one. Ideally, a complete NLO calculation should be
performed as in Ref. Frederix:2018nkq , where significant (and exact)
accidental cancellations between NLO EW contributions were observed in the SM.
While the size of these cancellations or the lack thereof could change the
expectations of the size of the corrections based on simple coupling scaling,
we do not expect they would significantly alter the conclusions drawn here.

Looking ahead, we have considered in section 5 and section 6 the sensitivity
attainable at future colliders, i.e., at 13, 14 and 27 TeV for the LHC and the
FCC-hh at 100 TeV. As our previous LHC studies hinted, all the 4-heavy
operators have a sensitivity enhancement at high collision energies, in
contrast to the 2-heavy-2-light operators. Consequently, the expected limits
on their Wilson coefficients are the most stringent, especially for colour-
singlets.

We presented a study of the double insertion of dimension-six operators in
four-top production. The aim was to scrutinise the claim of enhanced
sensitivity in four-top production from multiple insertions of 2-heavy-2-light
operators. Given the current bounds on this set of operators, we find that the
sensitivity is not enhanced. This finding supports the conclusion that
2-heavy-2-light operators are better constrained elsewhere than in four-top
production.

We stress the importance of our results in summarising the four-top production
SMEFT predictions and analysing them in each order of the QCD-EW expansion,
underlining the significance of subleading terms for the four-heavy quark
operators. Moreover, within all the dimension-six SMEFT operators, we have
shown that four top quark production provides robust constraints on the four-
heavy coefficients.

Finally, our analysis motivates an effort towards a systematic study of
subleading effects in other processes, such as e.g. $t\bar{t}Z$ and, in
particular, $t\bar{t}W(+$jets). Once technically possible, such studies should
be upgraded to the NLO accuracy, including QCD and EW corrections, to control
the uncertainties fully.

## Acknowledgments

We thank Olivier Mattelaer, Ken Mimasu, Alexander Ochirov, Davide Pagani, and
Marco Zaro for valuable discussions. This work has received funding from the
European Union’s Horizon 2020 research and innovation program as part of the
Marie Skłodowska-Curie Innovative Training Network MCnetITN3 (grant agreement
no. 722104) and by the F.R.S.-FNRS under the “Excellence of Science” EOS be.h
project no. 30820817. R.A.’s research is funded by the F.R.S-FNRS project no.
40005600. E.V. has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 949451) and from a Royal Society University Research
Fellowship through grant URF/R1/201553. Computational resources have been
provided by the supercomputing facilities of the Université catholique de
Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif en
Fédération Wallonie Bruxelles (CÉCI) funded by the Fond de la Recherche
Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the
Walloon Region.

##  Appendix A Translations and constraints

Table 5 presents the definitions of the SMEFTatNLO 4F operators,
$\mathcal{O}_{i}$, in terms of the Warsaw basis coefficients. Respectively,
table 6 and table 7 present the bounds on the 4F and contributing operators
(except for $\mathcal{O}_{\scriptscriptstyle G}$) obtained from the global fit
of Ref. Ethier:2021bye .

2-heavy 2-light
---
$\mathcal{O}_{\scriptscriptstyle i}$ | UFO | Translation | $\mathcal{O}_{\scriptscriptstyle i}$ | UFO | Translation
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,1}$ | cQq11 | $\sum\limits_{i=1,2}[C_{qq}^{(1)}]^{ii33}+\frac{1}{6}[C_{qq}^{(1)}]^{i33i}+\frac{1}{2}[C_{qq}^{(3)}]^{i33i}$ | $\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,8}$ | cQq18 | $\sum\limits_{i=1,2}[C_{qq}^{(1)}]^{i33i}+3[C_{qq}^{(3)}]^{i33i}$
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ | cQq31 | $\sum\limits_{i=1,2}[C_{qq}^{(3)}]^{ii33}+\frac{1}{6}[C_{qq}^{(1)}]^{i33i}-\frac{1}{6}[C_{qq}^{(3)}]^{i33i}$ | $\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,8}$ | cQq38 | $\sum\limits_{i=1,2}[C_{qq}^{(1)}]^{i33i}-[C_{qq}^{(3)}]^{i33i}$
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 1}$ | ctu1 | $\sum\limits_{i=1,2}[C_{uu}]^{ii33}+\frac{1}{3}[C_{uu}]^{i33i}$ | $\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 8}$ | ctu8 | $\sum\limits_{i=1,2}2[C_{uu}]^{i33i}$
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 1}$ | ctd1 | $\sum\limits_{i=1,2(,3)}[C_{ud}^{(1)}]^{33ii}$ | $\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 8}$ | ctd8 | $\sum\limits_{i=1,2(,3)}[C_{ud}^{(8)}]^{33ii}$
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 1}$ | ctq1 | $\sum\limits_{i=1,2}[C_{qu}^{(1)}]^{ii33}$ | $\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 8}$ | ctq8 | $\sum\limits_{i=1,2}[C_{qu}^{(8)}]^{ii33}$
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 1}$ | cQu1 | $\sum\limits_{i=1,2}[C_{qu}^{(1)}]^{33ii}$ | $\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 8}$ | cQu8 | $\sum\limits_{i=1,2}[C_{qu}^{(8)}]^{33ii}$
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 1}$ | cQd1 | $\sum\limits_{i=1,2,(3)}[C_{qd}^{(1)}]^{33ii}$ | $\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 8}$ | cQd8 | $\sum\limits_{i=1,2,(3)}[C_{qd}^{(8)}]^{33ii}$
4-heavy
$\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 1}$ | cQQ1 | $2[C_{qq}^{(1)}]^{3333}-\frac{2}{3}[C_{qq}^{(3)}]^{3333}$ | $\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 8}$ | cQQ8 | $8[C_{qq}^{(3)}]^{3333}$
$\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 1}$ | cQt1 | $[C_{qu}^{(1)}]^{3333}$ | $\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 8}$ | cQt8 | $[C_{qu}^{(8)}]^{3333}$
$\mathcal{O}_{\scriptscriptstyle tt}^{\scriptscriptstyle 1}$ | ctt1 | $[C_{uu}^{(1)}]^{3333}$ |

Table 5:  The translation of four-fermion operators from the Warsaw basis to
the top-basis. The UFO column shows the notation of the corresponding WCs in
the SMEFTatNLO model.

2-heavy 2-light
---
UFO | $\mathscr{O}(\Lambda^{-2})$ | $\mathscr{O}(\Lambda^{-4})$ | UFO | $\mathscr{O}(\Lambda^{-2})$ | $\mathscr{O}(\Lambda^{-4})$
Individual | Marginalised | Individual | Marginalised | Individual | Marginalised | Individual | Marginalised
cQq11 | [-3.603,0.307] | [-8.047,9.400] | [-0.303,0.225] | [-0.354,0.249] | cQq18 | [-0.273,0.509] | [-2.258,4.822] | [-0.373,0.309] | [-0.555,0.236]
cQq31 | [-0.099,0.155] | [-0.163,0.296] | [-0.088,0.166] | [-0.167,0.197] | cQq38 | [-1.813,0.625] | [-3.014,7.365] | [-0.470,0.439] | [-0.462,0.497]
ctu1 | [-6.046,0.424] | [-15.565,15.379] | [-0.380,0.293] | [-0.383,0.331] | ctu8 | [-0.774,0.607] | [-16.952,0.368] | [-0.911,0.347] | [-1.118,0.260]
ctd1 | [-9.504,-0.086] | [-27.673,11.356] | [-0.449,0.371] | [-0.474,0.347] | ctd8 | [-1.458,1.365] | [-5.494,25.358] | [-1.308,0.638] | [-1.329,0.643]
ctq1 | [-0.784,2.771] | [-12.382,6.626] | [-0.205,0.271] | [-0.222,0.226] | ctq8 | [-0.396,0.612] | [-4.035,4.394] | [-0.483,0.393] | [-0.687,0.186]
cQu1 | [-0.938,2.462] | [-16.996,1.072] | [-0.281,0.371] | [-0.207,0.339] | cQu8 | [-1.508,1.022] | [-12.745,13.758] | [-1.007,0.521] | [-1.002,0.312]
cQd1 | [-0.889,6.459] | [-3.239,34.632] | [-0.332,0.436] | [-0.370,0.384] | cQd8 | [-2.393,2.042] | [-24.479,11.233] | [-1.615,0.888] | [-1.256,0.715]
4-heavy
cQQ1 | [-6.132,23.281] | [-190,189] | [-2.229,2.019] | [-2.995,3.706] | cQQ8 | [-26.471,57.778] | [-190,170] | [-6.812,5.834] | [-11.177,8.170]
cQt1 | [-195,159] | [-190,189] | [-1.830,1.862] | [-1.391,1.251] | cQt8 | [-5.722,20.105] | [-190,162] | [-4.213,3.346] | [-3.040,2.202]
ctt1 | [-2.782,12.114] | [-115,153] | [-1.151,1.025] | [-0.791,0.714] |

Table 6:  Bounds on four-fermion WCs from the global analysis of Ref.
Ethier:2021bye .

Contributing operators
---
UFO | $\mathscr{O}(\Lambda^{-2})$ | $\mathscr{O}(\Lambda^{-4})$ | UFO | $\mathscr{O}(\Lambda^{-2})$ | $\mathscr{O}(\Lambda^{-4})$
Individual | Marginalised | Individual | Marginalised | Individual | Marginalised | Individual | Marginalised
ctp | [-1.331,0.355] | [-5.739,3.435] | [-1.286,0.348] | [-2.319,2.797] | ctZ | [-0.039,0.099] | [-15.869,5.636] | [-0.044,0.094] | [-1.129,0.856]
ctW | [-0.093,0.026] | [-0.313,0.123] | [-0.084,0.029] | [-0.241,0.086] | ctG | [0.007,0.111] | [-0.127,0.403] | [0.006,0.107] | [0.062,0.243]
cpQM | [-0.998,1.441] | [-1.690,11.569] | [-1.147,1.585] | [-2.250,2.855] | cpt | [-2.087,2.463] | [-3.270,18.267] | [-3.028,2.195] | [-13.260,3.955]
cpG | [-0.002,0.005] | [-0.043,0.012] | [-0.002,0.005] | [-0.019,0.003] |  |  |  |  |

Table 7:  Bounds on the contributing two-fermion and purely-bosonic WCs from
the global analysis of Ref. Ethier:2021bye , except for the
$\mathcal{O}_{\scriptscriptstyle G}$ operator.

##  Appendix B Additional results for the LHC and FCC-hh

Table 8 and table 9 of this appendix present the LHC inclusive predictions for
the 4-heavy and 2-heavy 2-light four-fermion operators within their scale
uncertainties, respectively. Same is the case for table 10 but for the set of
contributing operators of eq. 17. Additional differential results at
$\sqrt{s}=13$ TeV are presented in fig. 22 for the four-fermion operators, and
in fig. 24 for non-four-fermion ones. Additional differential results for the
FCC-hh are presented in fig. 23 for the four-fermion operators, and in fig. 25
for non-four-fermion ones.

$\mathcal{O}_{i}$ |  $\mathscr{O}(\Lambda^{-2}):\sigma_{3}[\sigma_{2}]$[fb] |  $\sum\mathscr{O}(\Lambda^{-2})$[fb] |  $\sum\mathscr{O}(\Lambda^{-4})$[fb]
---|---|---|---
$\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 8}$ | 0.081 [-0.317]${}^{+54\%}_{-32\%}$ | -0.235${}^{+37\%}_{-25\%}$ | 0.121${}^{+45\%}_{-29\%}$
$\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 8}$ | 0.273 [-0.577]${}^{+54\%}_{-32\%}$ | -0.303${}^{+29\%}_{-22\%}$ | 0.354${}^{+45\%}_{-29\%}$
$\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 1}$ | 0.242 [-0.948]${}^{+54\%}_{-33\%}$ | -0.706${}^{+37\%}_{-25\%}$ | 1.086$(1)^{+46\%}_{-29\%}$
$\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 1}$ | -0.005 [0.725]${}^{+67\%}_{-61\%}$ | 0.720${}^{+41\%}_{-27\%}$ | 1.471$(2)^{+46\%}_{-29\%}$
$\mathcal{O}_{\scriptscriptstyle tt}^{\scriptscriptstyle 1}$ | 0.485 [-1.670]${}^{+54\%}_{-33\%}$ | -1.185$(1)^{+36\%}_{-24\%}$ | 4.339$(2)^{+46\%}_{-29\%}$
Table 8:  Inclusive predictions within relative scale uncertainties for 4-heavy operators (scales are given on $\sigma_{3}$ in the first column). $\sum\mathscr{O}(\Lambda^{-2})$ and $\sum\mathscr{O}(\Lambda^{-4})$ indicate total linear interference and total quadratic contributions, respectively. $\mathcal{O}_{i}$ |  $\mathscr{O}(\Lambda^{-2}):\sigma_{3}[\sigma_{2}]$[fb] |  $\sum\mathscr{O}(\Lambda^{-2})$[fb] |  $\sum\mathscr{O}(\Lambda^{-4})$[fb]
---|---|---|---
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,8}$ | 0.077 [-0.02]${}^{+42\%}_{-27\%}$ | 0.070${}^{+41\%}_{-27\%}$ | 0.274$(1)^{+29\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,8}$ | 0.278 [0.023]${}^{+43\%}_{-28\%}$ | 0.339${}^{+40\%}_{-26\%}$ | 0.275$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 8}$ | 0.202 [0.022]${}^{+43\%}_{-28\%}$ | 0.249${}^{+40\%}_{-26\%}$ | 0.211$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 8}$ | 0.315 [0.036]${}^{+43\%}_{-28\%}$ | 0.391${}^{+40\%}_{-26\%}$ | 0.335$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 8}$ | 0.115 [0.016]${}^{+44\%}_{-28\%}$ | 0.144${}^{+40\%}_{-26\%}$ | 0.129$(1)^{+31\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 8}$ | 0.178 [0.011]${}^{+43\%}_{-28\%}$ | 0.212${}^{+40\%}_{-26\%}$ | 0.167$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 8}$ | 0.101 [0.015]${}^{+44\%}_{-28\%}$ | 0.129${}^{+40\%}_{-26\%}$ | 0.103$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ | -0.038 [0.079]${}^{+41\%}_{-27\%}$ | 0.071${}^{+20\%}_{-16\%}$ | 1.841$(4)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,1}$ | -0.140 [0.016]${}^{+43\%}_{-28\%}$ | -0.113${}^{+47\%}_{-30\%}$ | 1.839$(4)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 1}$ | -0.083 [0.010]${}^{+41\%}_{-27\%}$ | -0.066${}^{+45\%}_{-29\%}$ | 1.137$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 1}$ | -0.131 [0.017]${}^{+41\%}_{-27\%}$ | -0.106${}^{+44\%}_{-29\%}$ | 1.799$(1)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 1}$ | -0.048 [0.002]${}^{+42\%}_{-27\%}$ | -0.049${}^{+41\%}_{-27\%}$ | 0.695$(1)^{+31\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 1}$ | -0.089 [0.022]${}^{+42\%}_{-27\%}$ | -0.056${}^{+52\%}_{-32\%}$ | 1.110$(4)^{+30\%}_{-21\%}$
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 1}$ | -0.051 [-0.011]${}^{+43\%}_{-28\%}$ | -0.065${}^{+40\%}_{-26\%}$ | 0.684$(1)^{+31\%}_{-21\%}$
Table 9:  Same as table 8 but for the 2-heavy 2-light operators. $\mathcal{O}_{i}$ |  $\mathscr{O}(\Lambda^{-2}):\sigma_{3}[\sigma_{2}]$[fb] |  $\sum\mathscr{O}(\Lambda^{-2})$[fb] |  $\sum\mathscr{O}(\Lambda^{-4})$[fb]
---|---|---|---
$\mathcal{O}_{\scriptscriptstyle tW}$ |  $\times$ [-0.233] | -0.220${}^{+53\%}_{-32\%}$ | 0.373${}^{+37\%}_{-24\%}$
$\mathcal{O}_{\scriptscriptstyle tZ}$ |  $\times$ [0.176] | 0.187${}^{+50\%}_{-31\%}$ | 0.264${}^{+37\%}_{-24\%}$
$\mathcal{O}_{\scriptscriptstyle tG}$ | 3.642$(1)$ [0.024]${}^{+68\%}_{-38\%}$ | 2.861$(1)^{*}$${}^{+75\%}_{-40\%}$ | 4.244$(2)^{+53\%}_{-32\%}$
$\mathcal{O}_{\scriptscriptstyle t\varphi}$ |  $\times$ [0.072] | -0.074${}^{+26\%}_{-20\%}$ | 0.012${}^{+40\%}_{-26\%}$
$\mathcal{O}_{\scriptscriptstyle\varphi Q}^{\scriptscriptstyle(-)}$ |  $\times$ [0.123] | -0.302*${}^{+35\%}_{-24\%}$ | 0.030${}^{+39\%}_{-26\%}$
$\mathcal{O}_{\scriptscriptstyle\varphi t}$ |  $\times$ [-0.114] | 0.307*${}^{+35\%}_{-24\%}$ | 0.030${}^{+40\%}_{-26\%}$
$\mathcal{O}_{\scriptscriptstyle G}$ | 1.633$(2)$ [0.113]${}^{+75\%}_{-40\%}$ | 1.715$(2)^{+75\%}_{-40\%}$ | 94.5$(33)^{+75\%}_{-39\%}$
$\mathcal{O}_{\scriptscriptstyle\varphi G}$ |  $\times$ [-0.107] | -0.480∗${}^{+41\%}_{-27\%}$ | 2.229$(1)^{+28\%}_{-20\%}$
Table 10:  Same as table 8 but for the contributing operators. The $\times$
denotes zero cross-section. The asterisk indicates the operator receives non-
negligible contributions at $\alpha_{s}$-orders lower than $\sigma_{2}$.

Figure 22:  Same as fig. 10 but for the rest of the four-fermion operators.

Figure 23:  Same as fig. 13 but for the rest of the four-fermion operators.

Figure 24:  Same as fig. 11 but for the rest of the contributing two-fermion
and purely-bosonic operators.

Figure 25:  Same as fig. 14 but for the rest of the contributing two-fermion
and purely-bosonic operators.

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