# Prospects for establishing limits on the SMEFT operators from the production
processes of three and four top quarks in hadron collisions

A.Aleshko  Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, 119991 Moscow, Russia  E.Boos  Skobeltsyn Institute of Nuclear
Physics, Lomonosov Moscow State University, 119991 Moscow, Russia  V.Bunichev
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
119991 Moscow, Russia  L.Dudko  Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University, 119991 Moscow, Russia

###### Abstract

Numerical simulations of processes of three and four top quark hadroproduction
are carried out in the SMEFT model framework. The simulated data are used to
derive expected theoretical constraints on Wilson coefficients of relevant
SMEFT operators of dimension six. Obtained limits for both cases are discussed
and compared in terms of processes sensitivity to possible BSM contribution.
Results show that operator $O_{tt}^{1}$ is better constrained by the process
of four top quark production, whereas other four operators $O_{QQ}^{1}$,
$O_{Qt}^{1}$, $O_{Qt}^{8}$ and $O_{QQ}^{8}$, are similarly constrained in
three and four top quark production processes. In all cases, the expected
limits taken from the simultaneous analysis of the production of three and
four top quarks are strengthened. The question of how kinematic cutoffs
motivated by partial unitarity affect the resulting constraints on the Willson
coefficients is addressed. It is shown that in all cases the limits are
getting somewhat worse if the kinematic cutoffs reflecting the unitarity are
applied.

##  1 Introduction

Currently, no experimental evidence of physics beyond the Standard Model (BSM)
has been observed. In the pursuit of the New Physics, researchers are inclined
to try “indirect” approaches, in which one seeks BSM manifestations in the
interactions of already known Standard Model particles. The conventional
assumption here is that New Physics is on a scale beyond our direct reach at
the moment, but should still manifest itself on lower scales in the form of
modified SM interactions, which can be measured and analyzed. A convenient
framework for such kind of analyzes is the Standard Model Effective Field
Theory (SMEFT) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] (see also reviews [11, 12] and
references therein). SMEFT approach is to parametrize BSM effects in a model-
independent way in terms of higher dimension gauge-invariant operators. If
only operators of dimension six are preserved, then the SMEFT Lagrangian reads
as follows:

$L=L_{SM}+\sum{\frac{c_{i}}{\Lambda^{2}}O^{d=6}_{i}},$ (1)

where $L_{SM}$ is the SM Lagrangian, $\Lambda$ \- hypothetical scale of the
BSM physics, $O^{d=6}_{i}$ \- local composite SMEFT operators of dimension
six, $c_{i}$ \- dimensionless Wilson coefficients. We consider here only terms
of dimension six since in the processes under study they are leading in the
expansion on inverse scale $\Lambda^{-1}$. In the SMEFT framework any
observable, in particular the cross-section, can be parametrized in the
following form:

$\sigma=\sigma_{SM}+\sum_{k}{\frac{c_{i}}{\Lambda^{2}}\sigma^{(1)}_{k}}+\sum_{j<=k}{\frac{c_{i}c_{k}}{\Lambda^{4}}\sigma^{(2)}_{k,j}},$
(2)

where $\sigma_{SM}$ is the SM value, $\sigma^{(1)}$ and $\sigma^{(2)}$ \-
coefficients, representing linear and quadratic (in terms of EFT coupling)
contributions of the SMEFT operators. Given the measurement of $\sigma$ and
having calculated values of $\sigma_{SM}$, $\sigma^{(1)}$ and $\sigma^{(2)}$,
one can estimate constrains on Wilson coefficients $c_{i}$. These constraints
can be used to calculate limits on physical parameters in various SM
extensions.

Search for four top quark production was performed and upper limits on its
cross section have been established [13, 14, 15] in proton-proton collisions
at $\sqrt{s}=13$ TeV. The process was observed recently with measured cross-
section $17.7^{+6.0}_{-5.4}$ fb [16] by CMS and $22.5^{+6.6}_{-5.5}$ fb [17]
by ATLAS experiments. The SM NLO QCD cross section at 14 TeV has been
calculated [18] for various choices of the factorisation and renormalisation
scales. In addition, the computation of NLO QCD and EW corrections in SM have
been carried out in [19] at 13 and 100 TeV energies. The SM cross section at
next-to-leading logarithmic accuracy including threshold non-logarithmic
corrections (so called NLL′) is now available for 13 and 13.6 TeV [20].
Complete tree level QCD and Electroweak analysis including contributions of
all five relevant SMEFT operators of dimension six was presented and expected
individual theoretical limits were given at various energies [21]. CMS and
ATLAS collaborations also presented experimental limits on corresponding
dimension six operators [13, 17]. One should stress that all the computations
of the four top quark production cross section at corresponding energies are
consistent with each other and with mentioned experimental measurements within
claimed uncertainties.

One can also consider processes with 3, 5, etc. top quarks in a final state
when searching for SMEFT manifestation beside the four top production. In the
current work, we consider the process of three top quark production and
compare its sensitivity to the contributions of the SMEFT operators with that
obtained in the relatively well-studied four top quark production process, and
also investigate how the sensitivity improves when both processes are taken
into account simultaneously.

Three top quark production has LO SM cross-section of 1.9 fb [22, 23] and has
not been experimentally observed yet. Nevertheless, one can estimate possible
constraints on relevant Wilson coefficients following from three top quark
production by comparing SM and SMEFT theoretical cross sections. We use the SM
cross section as an effective “measurement” and compare it with the SMEFT
cross section as given in Eq. 2. In order to compare expected sensitivities
obtained from the three and four top quark production processes, the same
analysis procedure is applied to both processes to determine the limits. The
expected limits obtained are also compared with currently known experimental
limits only for the production of four top quarks [13, 17].

One potentially serious problem arising in the SMEFT approach is that the
contributions of EFT operators grow too fast with energy. An important
consequence is that one should be very careful with the potential violation of
unitarity. In the current work, we investigate this problem by studying the
effect of the partial unitarity requirement on the limits of the extracted
Wilson coefficients.

Thus, the work has two main goals: to estimate the theoretically expected
limits on the Wilson coefficients obtained in the process of the production of
three and four top quarks and to study the impact of the unitary requirement.
The paper is organized as follows. Section 2 contains computation details,
cross sections and theory uncertainties. The partial unitarity limits are
discussed in Section 3. In Section 4 the methodology of obtaining limits has
been discussed. The final results and comparisons are presented in the
Summary.

##  2 Cross sections and theoretical uncertainties

###  2.1 Simulation details

In the present work, most of the calculations are conducted using the
MadGraph5_aMCNLO package [24]. Some of the results were also cross-checked
with the CompHEP package [25, 26]. For modeling with SMEFT operator
contributions, we use the SMEFTatNLO model [27]. Simulations at LO and NLO are
done using NNPDF31_lo_as_0118 and NNPDF31_nlo_as_0118_luxqed [28] PDF sets
respectively. The value of $\alpha_{s}$ is taken as follows from the PDF set.
The mass of the top quark is set to 172.5 GeV. To obtain
factorization/renormalization scale uncertainties, we perform calculations at
various characteristic scale points as implemented in MadGraph. All other
settings are taken as default in Madgraph unless stated otherwise.

###  2.2 Computations in Standard Model

Figure 1: Representative Feynman diagrams of four top quarks LO production in
the Standard Model.

Representative LO diagrams for four top SM production are depicted in figure
1. In our study we use NLO corrections to the cross section of four top quark
production as calculated in SM [18, 19]. The numerical value of the cross-
section strongly depends on the factorization/renormalization scale
$\mu_{F/R}$ and the PDF set chosen. In table 1 SM cross sections at
$\sqrt{s}=13$ TeV are presented. We explore different choices of
factorization/renormalization scales, which are often used for this process:
$H_{t}/2$, $2m_{top}$, and $m_{top}$. $H_{t}/2$ is defined as half of a scalar
sum of transverse momenta of all final state particles, while the $m_{top}$
corresponds to the fixed scale equal to the mass of the top quark. With the
current PDF set, the latter choice of scale provides a smaller K-factor, hence
we will use it for all subsequent computations.

Order | LO
---|---
Scale, $\mu_{F/R}$ | cross-section, $\sigma$ pb | $\delta_{\rm scale}$ | $\delta_{PDF}$
$m_{top}$ | 9.03 | +73% -39% |  $\pm$7.3%
$2m_{top}$ | 5.47 | +65.1% -36.9% |  $\pm$ 6.81%
$H_{t}/2$ | 3.93 | +60% -35% |  $\pm$ 6.4%
Order | QCD NLO
Scale, $\mu_{F/R}$ | cross-section, $\sigma$ pb | $\delta_{scale}$ | $\delta_{PDF}$
$m_{top}$ | 13.2 | +9.5% -20.1% |  $\pm$ 2.7%
$2m_{top}$ | 11.2 | +26.7% -25.0% |  $\pm$ 2.6%
$H_{t}/2$ | 8.4 | +29.6% -25.1% |  $\pm$ 2.5%
Table 1: Cross-sections of the four top quarks hadroproduction with
corresponding scale/PDF uncertainties.

Figure 2: Representative Feynman diagrams of three top quarks LO production in
the Standard Model.

The three top quark production is also been studied in some works [23, 29, 30,
22]. Representative diagrams for three top LO SM production are shown in
figure 2. A full set of diagrams can be found in [22]. The three top
production is also strongly dependent on the choice of QCD factorization
scale, the main consequence of which is a relatively large scale uncertainty.
To correctly compare the two processes we also want to use NLO corrections to
the cross section of three top quark production, similar to the four top quark
production. The calculation of triple top production with NLO corrections is
tricky due to the presence of resonant diagrams in the real part of the
correction. These are diagrams of type $pp\rightarrow tt\bar{t}t\bar{t}$ with
decay $t\rightarrow Wb$. LO cross-section for the production of four top
quarks is an order of magnitude greater than one for triple top production,
which therefore leads to spoilage of convergence of the perturbative series.
From a computational point of view, this means that the simulation will
produce very unstable results (or will not converge at all). To obtain precise
results, the issue should be considered in the same way as $tWb$ production
processes, which is beyond the scope of the current work. Therefore, for the
current exploratory goal of this paper, we simply use the general Diagram
Reduction procedure as implemented in the MadSTR plugin [31] for Madgraph.
Table 2 contains results for three and four top quarks SM production at c.o.m.
energies 13 and 14 TeV.

4 top production, LO
---
C.o.m. energy, TeV | SM cros.-sect. $\sigma_{SM}$, fb | scale uncert., % | PDF uncert., %
13 | 9.02 | 73.4 | 7.32
14 | 12.0 | 72.6 | 7.28
4 top production, NLO
13 | 13.2 | 20.1 | 2.5
14 | 17.8 | 20.3 | 2.5
3 top production, LO
C.o.m. energy, TeV | SM cros.-sect. $\sigma_{SM}$, fb | scale uncert., % | PDF uncert., %
13 | 1.16 | 30.1 | 7.1
14 | 1.5 | 29.2 | 6.64
3 top production, NLO
13 | 1.78 | 20.0 | 3.3
14 | 2.3 | 19.4 | 3.1

Table 2: SM cross-sections for process $pp\rightarrow t\bar{t}t\bar{t}$

###  2.3 SMEFT computations

The introduction of SMEFT operators adds new possible interaction vertices,
which modifies SM cross-sections. In this work following four-fermion SMEFT
operators were considered, which contribute to both three and four top quark
production processes:

$\displaystyle O_{tt}^{1}$
$\displaystyle=(\bar{t}_{R}\gamma^{\mu}t_{R})(\bar{t}_{R}\gamma_{\mu}t_{R}),$
(3) $\displaystyle O_{QQ}^{1}$
$\displaystyle=(\bar{Q}_{L}\gamma^{\mu}Q_{L})(\bar{Q}_{L}\gamma_{\mu}Q_{L}),$
$\displaystyle O_{Qt}^{1}$
$\displaystyle=(\bar{Q}_{L}\gamma^{\mu}Q_{L})(\bar{t}_{R}\gamma_{\mu}t_{R}),$
$\displaystyle O_{Qt}^{8}$
$\displaystyle=(\bar{Q}_{L}\gamma^{\mu}T^{A}Q_{L})(\bar{t}_{R}\gamma_{\mu}T^{A}t_{R}),$
$\displaystyle O_{QQ}^{8}$
$\displaystyle=(\bar{Q}_{L}\gamma^{\mu}T^{A}Q_{L})(\bar{Q}_{L}\gamma_{\mu}T^{A}Q_{L}),$

The SMEFT model introduces several additional parameters, namely the NP scale
$\Lambda$ and Wilson coefficients $c_{i}$ for each SMEFT operator. The scale
$\Lambda$ is conventionally set to 1 TeV, while $c_{i}$ is our ”parameters of
interest”, which we want to set bounds on.

The main idea of this work is to use the theoretical value of the cross-
section as an approximation for the experimental value in Eq. 2 to obtain
theoretical constraints on the Wilson coefficients of operators listed in Eq.
3.

To extract values for parameterization coefficients $\sigma^{(1)}$ and
$\sigma^{(2)}$ a series of simulations using SMEFT model was conducted. The
latest versions of Madgraph allow for direct computations of squared and
interference SMEFT terms, however, it is not advised, as computation of small
interference parts can potentially be unstable. Therefore following procedure
of coefficient extraction is utilized. Our approach is to calculate the cross-
section with two opposite values of $c_{i}$ (i.e. $c_{i}$ and $-c_{i}$). In
principle, one can take an arbitrary value of $c_{i}$, since it only affects
coefficients in the set of linear equations generated by 2, which when solved
for $\sigma^{(1)}$ and $\sigma^{(2)}$ provide the same values, regardless of
the choice. One should note, however, that due to the choice of $\Lambda$ the
value of $c_{i}$ is required to stay within the $[-4\pi,4\pi]$ range to ensure
the stability of the perturbation series. We tested several choices of $c_{i}$
and settled for the simple $c_{i}=\pm 1$ for the sake of clarity. Hence, to
obtain coefficients $\sigma^{(1)}$ and $\sigma^{(2)}$ we calculate cross-
section twice with $c_{i}=\pm 1$ and solve linear equation set obtained from
2.

The described procedure is conducted for all of the SMEFT operators listed in
3 as well as for both three and four top production processes. Thus obtained
values for $\sigma^{(1)}$ and $\sigma^{(2)}$ are used in section 4 to obtain
limits for Wilson coefficients $c_{i}$. However, before moving to the
calculation of restriction on Wilson coefficients, the important point to
discuss is the problem of potential violation of perturbative unitarity.

##  3 Optical theorem and perturbative unitarity

Effective operators lead to an increase in cross sections with an increase in
energy, which violates unitarity. For our calculations to be self-consistent,
we must check that we do not consider kinematic regions where perturbative
unitarity is violated. To estimate the admissible range of parameters, we
apply the optical theorem, which follows from the unitarity of the S matrix.
The optical theorem states that the imaginary part of the forward scattering
amplitude is proportional to the total cross section of the process:

$\displaystyle\sigma=\frac{1}{s}{\rm
Im}\left(A(\theta=0)\right)=\frac{16\pi}{s}\sum\limits^{\infty}_{l=0}(2l+1)|a_{l}|^{2},$
(4)

where $a_{l}$ – is the amplitude of the partial wave. Therefore,
Im$a_{l}=|a_{l}|^{2}$ and

$\displaystyle|\rm Re(a_{l})|^{2}+\left[\rm
Im(a_{l})-\frac{1}{2}\right]^{2}=\frac{1}{4}.$ (5)

$\displaystyle|\rm Re(a_{0})|<\frac{1}{2}.$ (6)

$\displaystyle
a_{0}=\frac{1}{16\pi\lambda}\left|\int\limits^{t_{+}}_{t_{-}}dt\cdot A\right|$
(7)

where $\lambda$ is the kinematic function of the triangle and $A$ is the
amplitude of the process.

Effective four-fermion operators are obtained by functional integration over
massive modes of intermediate vector fields. A four-particle vertex can be
formed from three-particle vertices of the interaction of two fermions with an
auxiliary massive vector field and propagators of this massive field. In this
case, the denominator of the auxiliary field propagator is replaced by a
constant equal to the scale of the new physics. The effective vertex of the
interaction of right-handed polarized fermions with the auxiliary vector field
has the form $\gamma(1+\gamma_{5})/\sqrt{2}$, and that of left-handed
polarized fermions - $\gamma(1-\gamma_{5})/\sqrt{2}$.

Using the Weyl representation for spinors and the method of helicity
amplitudes, for each case of an anomalous operator, the amplitudes $2\to 2$ of
$tt\to tt$ and $t\bar{t}\to t\bar{t}$ processes were calculated. It should be
noted that the $tt\to tt$ process includes t and u channel components, and the
$t\bar{t}\to t\bar{t}$ process includes s and t channel components, which were
taken into account in the calculation. For each process, out of 16 possible
helicity amplitudes, the one that gives the greatest contribution was chosen.
The resulting helicity amplitudes were expressed in terms of the Mandelshtam
invariant variables and integrated over the two-particle phase volume to
calculate the corresponding partial amplitudes.

Now let’s take a closer look at each of the listed processes. Thus, the
amplitude of the $tt\to tt$ process caused by the operator $O_{tt}^{1}$ is
formed by two interacting right-handed fermionic currents and consists of one
t-channel and one u-channel component. For simplicity, we fixed the angle
$\phi=0$. The helicity amplitudes of this process do not depend on the angle
$\theta$. To evaluate the applicability of the $O_{tt}^{1}$ operator, we chose
one of the dominant helicity amplitudes, which is proportional to the square
of the invariant mass s of the two top quarks. The remaining helicity
amplitudes are proportional to the t-quark mass, and their relative
contribution decreases with increasing s. Analytical expressions for the
dominant helicity amplitude and the partial amplitude calculated on its basis
are given in the Appendix in formulas (8) and (9), respectively. It can be
seen that the partial amplitude of the process is proportional to the factor
$(1+\beta)^{2}$, which tends to 4 for large s.

The process $tt\to tt$ with the operator $O_{QQ}^{1}$ is caused by the
interaction of two left-handed currents and its dominant helicity amplitude
looks similar to the case of the operator $O_{tt}^{1}$. The corresponding
expressions for the amplitudes are given in formulas (11) and (12).

For a process with the operator $O_{Qt}^{1}$ the picture becomes more
complicated. This process is caused by the interaction of the left and right
currents of massive fermions. To the t-channel and u-channel amplitude
components, two more are added, in which the right-handed and left-handed
three-particle vertices are swapped. The dominant helicity amplitude of the
process still does not depend on the angle $\theta$ and is directly
proportional to s (formula 13), however, due to a different combination of
initial and final polarizations of fermions, as well as a different
composition of the diagrams, the common factor of the dominant helicity
amplitude is now proportional to $(1+\beta^{2})$. This factor tends to 2 as s
increases, therefore the partial amplitude of the process $tt\to tt$ with the
operator $O_{Qt}^{1}$ (formula 14) is half the corresponding amplitude with
the operators $O_{tt}^{1}$ or $O_{QQ}^{1}$.

Now consider the process $tt\to tt$ with the operators $O_{QQ}^{8}$ and
$O_{Qt}^{8}$. The Lorentz structure of the corresponding amplitudes (formulas
14 and 17) is similar to the cases of the operators $O_{QQ}^{1}$ and
$O_{Qt}^{1}$, however, color factors $\lambda^{a}_{j,i}\lambda^{a}_{k,l}/4$
additionally appear in the diagrams. In order to estimate the level of maximum
influence of the operators $O_{QQ}^{8}$ and $O_{Qt}^{8}$, we took the largest
possible value of the color factor $\lambda^{a}_{j,i}\lambda^{a}_{k,l}/4=1/2$.
Thus, the obtained partial amplitudes for the process $tt\to tt$ with the
operators $O_{QQ}^{8}$ and $O_{Qt}^{8}$ are two times smaller than the
corresponding amplitudes with the operators $O_{QQ}^{1}$ and $O_{Qt}^{1}$.

The listed operators also contribute to the processes $t\bar{t}\to t\bar{t}$.
The amplitudes of this process include s-channel and t-channel components. The
dominant helicity amplitude of the process $t\bar{t}\to t\bar{t}$ with the
participation of the operator $O_{tt}^{1}$ (formula 19) has an angular
dependence and is proportional to the factor $-(1+\cos{\theta})$, which is
proportional to the Mandelstam variable u. Also, this helicity amplitude is
proportional to the factor $(1+\beta)$, which tends to 2 as the value of s
increases. Technically, the difference in the dominant helicity amplitudes for
the processes $tt\to tt$ and $t\bar{t}\to t\bar{t}$ is due to different
redistribution of momenta between the initial and final states, as well as
different combinations of initial and final polarizations. These differences
lead to the fact that at large s the partial amplitudes for the processes
$tt\to tt$ with the operators $O_{tt}^{1}$ and $O_{QQ}^{1}$ are 4 times larger
than the corresponding partial amplitudes for processes $t\bar{t}\to
t\bar{t}$. At the same time, for large values of s, partial amplitudes with
the operator $O_{Qt}^{1}$ for the process $t\bar{t}\to t\bar{t}$ coincide with
the corresponding partial amplitude of the process $t\bar{t}\to t\bar{t}$.

From the calculations performed, it is clear that the partial amplitudes for
the processes $t\bar{t}\to t\bar{t}$ do not exceed the partial amplitudes for
the processes $tt\to tt$, therefore, further, to estimate the limits of
applicability of the operators, we will use only partial amplitudes for $tt\to
tt$ processes.

It should be noted that the considered helicity dominant amplitudes do not
allow us to correctly estimate the contribution of processes with different
operators. For such a comparison, it is necessary to calculate the full square
of the amplitude modulus based on all the helicity amplitudes of the process.
We used the dominant helicity amplitudes only to estimate the limits of
applicability of the corresponding anomalous operators. We also note that the
use of pair production processes of top quarks to determine the unitary limit
of applicability of four-fermion operators provides a more conservative
estimate than directly using the processes of production of three or four top
quarks. We extend the values of the unitary limits obtained in the processes
$tt\to tt$ to all processes involving the four-fermion operators under study.
We used the obtained restrictions on the invariant mass of a pair of top
quarks to set cutoffs on the invariant mass of three and four top quarks in
the corresponding processes.

Based on the calculated partial amplitudes for the operators under study, we
plotted graphs of the boundary of perturbative unitarity, where the invariant
mass of a pair of top quarks is plotted along the x-axis, and the current
upper experimental limits for the Wilson coefficients of anomalous operators
are plotted along the y-axis. The intersection of the dotted lines
corresponding to the accuracy of the Wilson coefficient measurements with the
line of the unitarity boundary shows the value of the invariant mass of a pair
of top quarks above which the effective field theory approach does not work.
As experimental limits for the Wilson coefficients, their current values of
measurement accuracy in the processes of production of three top quarks are
taken.

Figure 3: Perturbative unitarity limit $a_{0}=\frac{1}{2}$(red line) for
various anomalous operators. Upper left: $O^{1}_{tt}$, upper right:
$O^{1}_{QQ}$, middle left: $O^{1}_{Qt}$, middle right: $O^{8}_{Qt}$, bottom:
$O^{8}_{QQ}$. The green zone corresponds to the allowed area. The dashed
horizontal line indicates the experimental limits on the corresponding Wilson
coefficients for different LHC energy values. Dashed vertical lines with an
arrow indicate the limits of the invariant mass of two t-quarks for
corresponding LHC energy values.

In Fig. 3, the red color marks the values at which the partial amplitude
$a_{0}=\frac{1}{2}$. The green color marks the zone of parameters allowed from
the point of view of perturbative unitarity. The dashed horizontal line
indicates the experimental limits on the corresponding Wilson coefficients for
different LHC operating modes. Dashed vertical lines with an arrow indicate
the limits of the invariant mass of two t-quarks for corresponding LHC modes.
When evaluating the unitarity limitation on the operators $O^{1}_{tt}$,
$O^{1}_{QQ}$, $O^{1}_{Qt}$, $O^{8}_{Qt}$ at the LHC energy equal 13 TeV, the
current values of the measurement accuracy of the Wilson coefficients obtained
in the CMC experiment [13] were used. At the same time, when estimating the
corresponding limitations for the LHC energy equal to 27 and 100 TeV, the
expected values of the measurement accuracy, obtained theoretically, were
used. For operator $O^{8}_{QQ}$ only theoretical values of measurement
accuracy were used.

From Fig. 3 it is clear that the lower the value of the upper experimental
limit on the Wilson coefficient of the operator, the higher the value of the
invariant mass of the process is allowed by the condition of perturbative
unitarity. At the same time, the upper limit of the Wilson coefficient of the
operator depends on the cross section of the process under study. The larger
the cross section of a hypothetical process involving an anomalous operator,
the greater the accuracy of its measurement is allowed. In the absence of
manifestations of new physics, this leads to lower values of the experimental
limits on the value of the Wilson coefficient.

For example, consider the behavior of unitary limits for 4t operators at the
LHC 13 TeV mode. (Fig. 3 top left) shows the unitary bound for the operator
$O^{1}_{tt}$. With an experimental limit on the interaction parameter of this
operator equal to 2 $TeV^{-2}$, we obtain an upper unitary limit on the
invariant mass of the process equal to 1.5 TeV. The same values are obtained
for the operator $O^{1}_{QQ}$ on (Fig. 3 upper right). For the operator
$O^{1}_{Qt}$ (Fig. 3 middle left) the picture changes. Since the partial
amplitude for this operator is two times less than for the operators
$O^{1}_{tt}$ and $O^{1}_{QQ}$, the line of the unitary boundary moves further
from the coordinate axes. At the same time, the cross section for processes
with four top quarks involving this operator is smaller than the corresponding
cross sections for the operators $O^{1}_{tt}$ and $O^{1}_{QQ}$, therefore the
accuracy of the experimental measurement of $C^{1}_{Qt}/\Lambda^{2}$ is more
rough and equal to 3.5 TeV-2. It turns out that the intersection of the rough
value of the experimental limit with the moved unitary boundary again gives a
unitary limit on the invariant mass equal to 1.5 TeV. This trend continues for
the $O^{8}_{Qt}$ operator (Fig. 3 middle right). The partial amplitude
corresponding to this operator is additionally two times smaller, and the
measurement accuracy $C^{8}_{Qt}/\Lambda^{2}$ is even rougher and is equal to
6.5 TeV-2. This leads to the fact that the unitary constraint on the invariant
mass of a process with this operator is again around the value of 1.5 TeV.
Thus, the total unitary limit for all considered 4t operators in the LHC 13
TeV mode is close to the value of 1.5 TeV.

It is also clear from (Fig. 3) that although changing the LHC operating mode
from 13 TeV to 27 TeV and 100 TeV leads to an expansion of the unitary limits
of operators, this expansion is not very significant and is equal to 2 TeV and
3 TeV, respectively. This situation is due to the fact that the cross sections
for processes with four top quarks are accumulated at invariant masses not
exceeding 10 TeV.

It should be noted that in the case of the considered 4-fermion operators, the
partial amplitude grows proportionally to s, which limits the use of these
operators already for the invariant mass of two top quarks equal to 1–3 TeV.
For comparison, the unitary limitation obtained for single top production with
anomalous operators contributing to the Wb [32] vertex, as well as with FCNC
operators [33], is at the level of 10 TeV and higher.

##  4 Setting limits on Wilson coefficients

Having the values of coefficients $\sigma^{(1)}$ and $\sigma^{(2)}$ and their
uncertainties, calculating constraints on $c_{i}$ is a matter of choice of a
suitable statistical model. In the statistical model, we compare the
calculated with Eq. 2 total cross section with the measured or SM expected
cross section, with corresponding uncertainties. We consider three types of
uncertainties. The first type is theory uncertainty which has been discussed
in Sec. 2, the numbers are provided in Tables 1, 2. The other two
uncertainties are experimental systematic and statistical uncertainties. Based
on the recent measurements of four top-quark production in CMS [16] and ATLAS
[17] experiments one can extrapolate the statistical uncertainty and estimate
experimental systematic relative uncertainty to be the same for all
calculations. This approach is more or less conservative since the analysis
methodology usually improves and one can expect smaller systematic
uncertainties in the future. Since the number of expected events $n\sim\sigma
L$ the extrapolation of statistical relative uncertainty can be taken as
$\delta(n1)/\delta(n2)=\sqrt{(\sigma_{2}L_{2})/(\sigma_{1}L_{1})}$. Based on
the CMS results [16] $\sigma_{\rm
4top}=17.7\rm^{+3.7}_{-3.5}(stat)^{+2.3}_{-1.9}(sys)$ fb we estimate
experimental uncertainties as listed in Table 3. The integrated luminosity (L)
for the calculations has been taken as 138 fb-1 for 13 TeV (available
experimental results), 3 ab-1 for 14 TeV (HL-LHC), 15 ab-1 for 27 TeV (HE-LHC)
and 25 ab-1 for 100 TeV (FCC).

| $pp\rightarrow 4top$ | $pp\rightarrow 3top+X$
---|---|---
|  $\delta sys$, % |  $\delta stat$, % |  $\delta sys$, % |  $\delta stat$, %
13 TeV | 13 | 24 | 13 | 76
14 TeV | 13 | 5 | 13 | 14
27 TeV | 13 | 0.6 | 13 | 2
100 TeV | 13 | 0.1 | 13 | 0.3
Table 3: Estimated experimental systematic and statistical uncertainties for
processes of three and four top quark production at energies of 13, 14, 27 and
100 TeV

###  4.1 Cross checks of the methodology

For the first fit, a statistical model based on the chi-square distribution
was used. Later, the results were validated using acknowledged EFTfitter [34]
and SMEFiT [35] packages. The first comparison between different statistical
models was done for linear terms in Eq. 2 where only $\sigma^{(1)}$ terms are
taken into account. Based on the calculated cross sections of the four and
three top-quarks production processes in SM and with EFT contribution one can
estimate the expected results with different statistical approaches. The
achieved with the fit 95% CL exclusion limits on the Wilson coefficients
$C_{k}/\Lambda^{2}(TeV^{-2})$ are shown in Table 4. Two processes
$pp\rightarrow 4top$ (4t) and $pp\rightarrow 3top+X$ (3t) at $\sqrt{s}=13$ TeV
are considered separately and as a combination in one statistical model. Two
types of statistical models are realized. The 1D approach with a variation of
only one Wilson coefficient in the statistical model, and the 5D approach with
a simultaneous variation of all five coefficients in the same model. The
combination of 4t and 3t processes has been considered in a dedicated
statistical model marked as (3+4t). For some of the couplings, there is no
sensitivity (flat posterior distribution) with linear terms in the range from
-150 to 150 for the value of the coefficient. In this case, the sign “-” is
shown in the table.

Stat. model | $C_{tt}^{1}$ | $C_{QQ}^{1}$ | $C_{Qt}^{1}$ | $C_{Qt}^{8}$ | $C_{QQ}^{8}$
---|---|---|---|---|---
$\chi^{2}$,1D,4t | [-27,27] | [-39,39] | - | [-28,28] | -
EFTfitter,1D,4t | [-27,27] | [-39,39] | - | [-28,28] | [-130,130]
SMEFiT,1D,4t | [-27,27] | [-39,39] | - | [-28,28] | [-130,130]
$\chi^{2}$,1D,3t | - | [-27,27] | [-64,64] | [-123,121] | [-43,43]
EFTfitter,1D,3t | - | [-27,27] | [-64,64] | [-123,121] | [-43,43]
SMEFiT,1D,3t | - | [-27,27] | [-64,64] | [-123,121] | [-43,43]
$\chi^{2}$,1D,3+4t | [-27,27] | [-22,22] | [-64,64] | [-27,27] | [-42,42]
EFTfitter,1D,3+4t | [-27,27] | [-22,22] | [-64,64] | [-27,27] | [-42,42]
SMEFiT,1D,3+4t | [-27,27] | [-22,22] | [-64,64] | [-27,27] | [-42,42]
EFTfitter,5D,4t | [-138,138] | [-141,141] | - | [-138,138] | -
SMEFiT,5D,4t | [-138,138] | [-141,141] | - | [-138,138] | -
EFTfitter,5D,3t | - | [-125,125] | - | - | -
SMEFiT,5D,3t | - | [-125,125] | - | - | -
EFTfitter,5D,3+4t | [-132,132] | [-123,123] | - | [-140,140] | -
SMEFiT,5D,3+4t | [-132,132] | [-123,123] | - | [-140,140] | -
Table 4: Comparison of the expected limits on $C_{k}/\Lambda^{2}\rm[TeV^{-2}]$
estimated for $pp\rightarrow 4top$ (4t) and $pp\rightarrow 3top+X$ (3t) cross
sections with only linear $\sigma^{(1)}$ EFT terms. The statistical models
with only one Wilson coefficient have marked 1D and the models with all five
coefficients considered simultaneously have marked 5D. The models with the
combination of 4t and 3t processes are marked 3+4t.

The comparison in Table 4 for the linear EFT terms demonstrates significant
improvement of the $C_{QQ}^{1}$, $C_{Qt}^{1}$ and $C_{QQ}^{8}$ limits for the
scenarios where triple top-quark production has taken into account. The limits
in Table 4 demonstrate very weak sensitivity of the linear EFT terms to
considered operators, the limits are far beyond the perturbative unitarity
limits of $4\pi$.

In the scenarios with quadratic EFT terms, where $\sigma^{(2)}$ in Eq. 2 are
taken into account, the limits are significantly tighter than with only linear
terms scenarios. The corresponding limits with quadratic terms are shown in
Table 5 for different statistical models at $\sqrt{s}=13$ TeV.

Stat. model | $C_{tt}^{1}$ | $C_{QQ}^{1}$ | $C_{Qt}^{1}$ | $C_{Qt}^{8}$ | $C_{QQ}^{8}$
---|---|---|---|---|---
$\chi^{2}$,1D,4t | [-1.3,1.2] | [-2.5,2.3] | [-2.2,2.2] | [-6.3,5.2] | [-5.6,5.5]
EFTfitter,1D,4t | [-1.1,1.1] | [-2.2,2.1] | [-2.0,2.0] | [-5.7,4.6] | [-5.1,4.9]
SMEFiT,1D,4t | [-1.1,1.1] | [-2.2,2.1] | [-2.0,2.0] | [-5.7,4.6] | [-5.0,4.8]
$\chi^{2}$,1D,3t | [-4.2,4.1] | [-2.9,3.2] | [-2.9,3.0] | [-6.0,6.3] | [-5.8,6.8]
EFTfitter,1D,3t | [-3.8,3.7] | [-2.5,2.9] | [-2.6,2.7] | [-5.4,5.6] | [-5.2,6.1]
SMEFiT,1D,3t | [-3.7,3.7] | [-2.5,2.9] | [-2.6,2.7] | [-5.3,5.6] | [-5.1,6.1]
$\chi^{2}$,1D,3+4t | [-1.3,1.2] | [-2.2,2.2] | [-2.1,2.1] | [-5.2,4.7] | [-4.8,5.0]
EFTfitter,1D,3+4t | [-1.1,1.1] | [-2.0,2.0] | [-1.8,1.9] | [-4.7,4.2] | [-4.3,4.5]
SMEFiT,1D,3+4t | [-1.1,1.0] | [-2.0,2.0] | [-1.8,1.8] | [-4.7,4.2] | [-4.2,4.5]
EFTfitter,5D,4t | [-0.94,0.92] | [-2.3,2.2] | [-1.7,1.6] | [-4.9,3.7] | [-5.2,5.1]
SMEFiT,5D,4t | [-0.95,0.90] | [-1.8,1.7] | [-1.6,1.6] | [-4.8,3.6] | [-4.2,4.0]
EFTfitter,5D,3t | [-3.3,3.7] | [-2.1,2.4] | [-2.2,2.5] | [-4.6,5.4] | [-4.3,5.5]
SMEFiT,5D,3t | [-3.1,3.0] | [-2.0,2.4] | [-2.1,2.2] | [-4.3,4.6] | [-4.2,5.1]
EFTfitter,5D,3+4t | [-0.98,0.92] | [-1.8,1.8] | [-1.5,1.6] | [-4.0,3.4] | [-3.9,4.0]
SMEFiT,5D,3+4t | [-0.95,0.90] | [-1.6,1.6] | [-1.5,1.5] | [-4.0,3.3] | [-3.5,3.7]
Table 5: Comparison of the expected limits on $C_{k}/\Lambda^{2}\rm[TeV^{-2}]$
estimated for $pp\rightarrow 4top$ (4t) and $pp\rightarrow 3top+X$ (3t) cross
sections with quadratic $\sigma^{(2)}$ EFT terms are taken into account. The
statistical models with only one Wilson coefficient have marked 1D and the
models with all five coefficients considered simultaneously have marked 5D.
The models with the combination of 4t and 3t processes are marked 3+4t.

The Tables 4, 5 demonstrate good agreement between SMEFiT, EFTfitter and
$\chi^{2}$ results. For the quadratic fit, the $\chi^{2}$ test provides a bit
wider limits. For the linear fit multi-dimensional variation leads to much
weaker limits than the one-dimensional statistical model which is comparable
with previous results with linear fit [36]. In the scenario with quadratic EFT
terms, the multi-dimensional statistical model leads to tighter limits than
the one-dimensional statistical model. Such behavior is comparable with
previous results for the quadratic fits [37]. In Fig. 4 the posterior
distributions of the probability density function for the Wilson coefficients
are provided for the one-dimensional (left plots) and multi-dimensional (right
plots) statistical models, such example has taken for four-top-quark
production at $\sqrt{s}=13$ TeV with quadratic EFT terms. The distributions
are calculated with the SMEFiT package. Due to quadratic terms, the
distributions in multi-dimensional models are sharper and the integral for
0.95 quantile is gaining faster. Depending on the uncertainties such behavior
can give tighter limits in multi-dimensional statistical model than in one-
dimensional.

Figure 4: Posterior distributions of probability density function for the
Wilson coefficients in 1D (left plots) and 5D (right plots) statistical
models, in the scenario with quadratic terms are taken into account.

The achieved upper limits in Table 5 for the scenarios with quadratic EFT
terms demonstrate rather similar sensitivity of four- and triple-top quark
production processes to the considered EFT operators. The combination of four-
and triple-top quark production in one statistical model leads to better
sensitivity, in general.

The theoretical constraints seem to coincide pretty well with experimental
limits obtained from four top-quark production in CMS [13] and ATLAS [17]
experiments. Since there is a good agreement in the results from EFTfitter and
SMEFiT packages, EFT limits in the next sections are shown only from the
SMEFiT package.

###  4.2 Electroweak and QCD contributions in triple top-quark production

The importance of electroweak contribution (EW) to the triple top quark
production processes has been shown in the article [22]. The EW diagrams have
the same rate as diagrams with gluon (QCD), also the interference between EW
and QCD diagrams is of almost the same rate as QCD or EW contribution and is
negative. If one takes into account only QCD diagrams (e.g. it is the default
setting in MadGraph) the total cross section will be almost correct due to the
cancellation of negative interference terms and EW contribution, but the
kinematic properties can be significantly different. In this subsection, we
compare how the inclusion of EW contribution changes the EFT limits in
comparison with the limits where only QCD contribution has been taken into
account. The EFT limits shown in Table 6 are calculated with triple top quark
production processes with only QCD diagrams (marked as QCD) and in the
complete case with QCD, EW diagrams and their interference (marked as QCD+EW).
Two statistical models are shown with 1D and 5D variations. The quadratic EFT
terms with $\sigma^{(2)}$ terms are taken into account.

model | $C_{tt}^{1}$ | $C_{QQ}^{1}$ | $C_{Qt}^{1}$ | $C_{Qt}^{8}$ | $C_{QQ}^{8}$
---|---|---|---|---|---
3t,QCD,1D | [-3.6,3.7] | [-2.6,2.5] | [-2.5,2.5] | [-6.3,4.8] | [-6.5,5.3]
3t,QCD+EW,1D | [-3.4,3.4] | [-2.3,2.7] | [-2.4,2.5] | [-5.0,5.2] | [-4.8,5.6]
3t,QCD,5D | [-3.0,3.0] | [-2.2,2.1] | [-2.0,2.1] | [-5.3,3.9] | [-5.5,4.2]
3t,QCD+EW,5D | [-2.8,2.8] | [-1.9,2.2] | [-1.9,2.1] | [-4.1,4.3] | [-3.8,4.7]
Table 6: Comparison of the expected limits on $C_{k}/\Lambda^{2}\rm[TeV^{-2}]$
estimated for $pp\rightarrow 3top+X$ (3t) cross sections with quadratic
$\sigma^{(2)}$ EFT terms. Two simulation models are considered, the only QCD
diagrams and QCD plus EW diagrams and their interference. The statistical
models with only one Wilson coefficient have marked 1D and the models with all
five coefficients considered simultaneously have marked 5D.

Direct comparison of the expected limits in Table 6 demonstrate notable
improvement in the sensitivity for the calculations with the correct
simulation of all QCD, EW and interference contributions.

###  4.3 Limits with unitarity bound cuts

The unitarity bounds considered in Sec. 3 have to be taken into account with
additional requirements for an invariant mass of all top-quark pairs. Such
additional requirements have been applied for the calculation of cross
sections and $\sigma^{(1)}$, $\sigma^{(2)}$ EFT terms. The achieved limits on
$C_{k}/\Lambda^{2}(TeV^{-2})$ are provided in Table 7 and are marked as (cut)
when the requirement has applied, for the comparison of the results without
unitarity bound requirements (nocut) are also shown.

model | $C_{tt}^{1}$ | $C_{QQ}^{1}$ | $C_{Qt}^{1}$ | $C_{Qt}^{8}$ | $C_{QQ}^{8}$
---|---|---|---|---|---
4t,nocut,1D | [-1.1,1.1] | [-2.2,2.1] | [-2.0,2.0] | [-5.7,4.6] | [-5.0,4.8]
4t,cut,1D | [-1.2,1.2] | [-2.4,2.3] | [-2.2,2.2] | [-6.8,5.0] | [-6.0,5.7]
3t,nocut,1D | [-3.7,3.7] | [-2.5,2.9] | [-2.6,2.7] | [-5.3,5.6] | [-5.1,6.1]
3t,cut,1D | [-4.3,4.2] | [-2.9,3.2] | [-3.1,3.2] | [-6.9,7.3] | [-6.4,7.7]
3+4t,nocut,1D | [-1.1,1.0] | [-2.0,2.0] | [-1.8,1.8] | [-4.7,4.2] | [-4.2,4.5]
3+4t,cut,1D | [-1.2,1.2] | [-2.2,2.2] | [-2.1,2.1] | [-5.8,4.8] | [-5.2,5.4]
4t,nocut,5D | [-0.95,0.90] | [-1.8,1.7] | [-1.6,1.6] | [-4.8,3.6] | [-4.2,4.0]
4t,cut,5D | [-1.0,1.0] | [-2.0,1.9] | [-1.8,1.9] | [-5.7,4.1] | [-4.6,4.4]
3t,nocut,5D | [-3.1,3.0] | [-2.0,2.4] | [-2.1,2.2] | [-4.3,4.6] | [-4.2,5.1]
3t,cut,5D | [-3.5,3.4] | [-2.3,2.7] | [-2.5,2.7] | [-5.6,6.1] | [-5.1,6.5]
3+4t,nocut,5D | [-0.95,0.90] | [-1.6,1.6] | [-1.5,1.5] | [-4.0,3.3] | [-3.5,3.7]
3+4t,cut,5D | [-1.0,1.0] | [-1.8,1.8] | [-1.7,1.7] | [-4.8,3.8] | [-4.1,4.3]
Table 7: Comparison of the expected limits on $C_{k}/\Lambda^{2}\rm[TeV^{-2}]$
estimated for $pp\rightarrow 4top$ (4t) and $pp\rightarrow 3top+X$ (3t) cross
sections with quadratic $\sigma^{(2)}$ EFT terms. The limits are shown for the
case of unitarity bound cuts applied at the simulation level (cut) and without
such cuts (nocut). The statistical models with only one Wilson coefficient
have marked 1D and the models with all five coefficients considered
simultaneously have marked 5D.

Expected limits in Table 7 achieved with additional unitarity bound cuts
obviously worse than without such cuts, but not significantly. Since the
unitarity is necessary requirement the simulation for the final results in the
next section obeys the calculated in Sec. 3 unitarity bound cuts.

The general comparison of the separate and combined results for the four and
three top quark production processes demonstrates significant improvement in
sensitivity for the linear EFT terms, especially for $O_{QQ}^{1}$,
$O_{Qt}^{1}$ and $O_{QQ}^{8}$ operators. In the scenarios with quadratic EFT
terms the Wilson coefficient of the operator $O_{tt}^{1}$ is better
constrained by the process of four top quark production. However, in the case
of other operators, the three top quark production apparently provides similar
limits. The combination of four and three top quark production processes gains
in sensitivity for most of the considered operators.

##  5 Summary

In the present study, numerical simulations of processes of three and four top
quark production were carried out. Simulation results are presented for both
the Standard model and dimension six SMEFT. For the latter case,
parametrization EFT terms $\sigma^{(1)}$ and $\sigma^{(2)}$ in Eq. 2 were
extracted and used to derive expected constraints on the Wilson coefficients
$C_{k}/\Lambda^{2}(TeV^{-2})$ of respective SMEFT operators. Theoretical
limits are obtained for both channels of three and four top quark production
and are shown in Table 8. Constraints, obtained from combined statistics of
both channels, were also presented. The results in summary Table 8 have been
calculated with taking into account EW contribution in three top quark
production at LO, QCD NLO contribution in four top quark production and with
applied partial unitarity requirements discussed in the text above. The
expected limits corresponds to the univariate variation of each Wilson
coefficient separately. Results show that in the case of operator
$O_{tt}^{1}$, better constraints on the corresponding Wilson coefficient are
obtained from the process of four top quark production. For other four
operators $O_{QQ}^{1}$, $O_{Qt}^{1}$, $O_{Qt}^{8}$ and $O_{QQ}^{8}$, both
processes have similar sensitivity. Combined statistics of three and four top
quark production processes provide the most precise constraints in all
considered cases. The sensitivity to triple top quark production in present
collider experiments is largely limited by statistical uncertainty. For the
future colliders, such as HL-LHC, the statistical uncertainty will be
comparable to or much smaller than the systematic uncertainty, and the
sensitivity to the BSM contribution in triple top quark production will be
limited by theoretical calculations and experimental uncertainties. In the
Table 8 a conservative estimation of the uncertainties is used for future
colliders, also, the partial unitarity bounds reduce the sensitivity for the
considered EFT operators. As a result, we do not observe a dramatic increase
in sensitivity of three and four top quark production processes to EFT
operators with a significant increases in energy of future colliders.

Overall, the production of three top quarks seems to be quite an interesting
target for BSM studies. Despite the lower cross-section (as compared to more
widely discussed four top quark production), it can potentially provide some
interesting opportunities for constraining the New Physics.

Energy, model | $C_{tt}^{1}$ | $C_{QQ}^{1}$ | $C_{Qt}^{1}$ | $C_{Qt}^{8}$ | $C_{QQ}^{8}$
---|---|---|---|---|---
13 TeV, 4t | [-1.2, 1.2] | [-2.4, 2.3] | [-2.2, 2.2] | [-6.8, 5.0] | [-6.0, 5.7]
13 TeV, 3t | [-4.3, 4.2] | [-2.9, 3.2] | [-3.1, 3.2] | [-6.9, 7.3] | [-6.4, 7.7]
13 TeV, 3+4t | [-1.2, 1.2] | [-2.2, 2.2] | [-2.1, 2.1] | [-5.8, 4.8] | [-5.2, 5.4]
14 TeV, 4t | [-1.1, 1.0] | [-2.1, 2.0] | [-1.9, 1.9] | [-5.8, 4.2] | [-5.2, 4.9]
14 TeV, 3t | [-2.5, 2.5] | [-1.6, 2.0] | [-1.8, 1.9] | [-3.9, 4.4] | [-3.7, 5.1]
14 TeV, 3+4t | [-1.1, 1.0] | [-1.5, 1.7] | [-1.5, 1.6] | [-3.8, 3.6] | [-3.5, 4.3]
27 TeV, 4t | [-0.90, 0.83] | [-1.7, 1.6] | [-1.6, 1.6] | [-4.9, 3.6] | [-4.4, 4.2]
27 TeV, 3t | [-2.0, 2.0] | [-1.3, 1.5] | [-1.4, 1.6] | [-3.3, 3.9] | [-2.7, 4.1]
27 TeV, 3+4t | [-0.88, 0.83] | [-1.2, 1.3] | [-1.3, 1.3] | [-3.2, 3.2] | [-2.6, 3.5]
100 TeV, 4t | [-0.68, 0.66] | [-1.3, 1.3] | [-1.2, 1.2] | [-3.8, 3.0] | [-3.7, 3.6]
100 TeV, 3t | [-1.3, 1.4] | [-0.89, 1.0] | [-1.0, 1.1] | [-2.1, 2.6] | [-1.8, 2.7]
100 TeV, 3+4t | [-0.67, 0.64] | [-0.85, 0.94] | [-0.93, 0.94] | [-2.1, 2.3] | [-1.8, 2.5]
Table 8: Comparison of the expected limits on $C_{k}/\Lambda^{2}\rm[TeV^{-2}]$
estimated for $pp\rightarrow 4top$ (4t) and $pp\rightarrow 3top+X$ (3t) cross
sections with quadratic $\sigma^{(2)}$ EFT terms. The limits are shown with
applied unitarity bound requirements. The results have been achieved in one
dimensional statistical model with a variation of each coefficient separately.

## Acknowledgments

This work was supported by the Russian Science Foundation [grant number
22-12-00152].

## Appendix A

Leading helicity and partial amplitudes for process $tt\to tt$:

operator case $O_{tt}^{1}$:

$\displaystyle A=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot 2\cdot
s\cdot(1+\beta)^{2}$ (8)

$\displaystyle a_{0}=\frac{1}{16\pi\cdot
s\cdot\beta}\left|\int\limits^{0}_{4M_{t}^{2}-s}dt\cdot
A\right|=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot\frac{s}{8\pi}\cdot\beta(1+\beta)^{2}$
(9)

where:

$\displaystyle\beta=\sqrt{1-\frac{4M_{t}^{2}}{s}}$ (10)

operator case $O_{QQ}^{1}$:

$\displaystyle A=\left(\frac{C_{QQ}^{1}}{\Lambda^{2}}\right)\cdot 2\cdot
s\cdot(1+\beta)^{2}$ (11)

$\displaystyle
a_{0}=\left(\frac{C_{QQ}^{1}}{\Lambda^{2}}\right)\cdot\frac{s}{8\pi}\cdot\beta(1+\beta)^{2}$
(12)

operator case $O_{Qt}^{1}$:

$\displaystyle A=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot 2\cdot
s\cdot(1+\beta^{2})$ (13)

$\displaystyle
a_{0}=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot\frac{s}{8\pi}\cdot\beta(1+\beta^{2})$
(14)

operator case $O_{Qt}^{8}$:

$\displaystyle
A=\left(\frac{C_{Qt}^{8}}{\Lambda^{2}}\right)\cdot\frac{\lambda^{a}_{j,i}\lambda^{a}_{k,l}}{4}\cdot
2\cdot s\cdot(1+\beta^{2})=\left(\frac{C_{Qt}^{8}}{\Lambda^{2}}\right)\cdot
s\cdot(1+\beta^{2})$ (15)

$\displaystyle
a_{0}=\left(\frac{C_{Qt}^{8}}{\Lambda^{2}}\right)\cdot\frac{s}{16\pi}\cdot\beta(1+\beta^{2})$
(16)

operator case $O_{QQ}^{8}$:

$\displaystyle
A=\left(\frac{C_{QQ}^{8}}{\Lambda^{2}}\right)\cdot\frac{\lambda^{a}_{j,i}\lambda^{a}_{k,l}}{4}\cdot
2\cdot s\cdot(1+\beta)^{2}=\left(\frac{C_{QQ}^{8}}{\Lambda^{2}}\right)\cdot
s\cdot(1+\beta)^{2}$ (17)

$\displaystyle
a_{0}=\left(\frac{C_{QQ}^{8}}{\Lambda^{2}}\right)\cdot\frac{s}{16\pi}\cdot\beta(1+\beta)^{2}$
(18)

Leading helicity and partial amplitudes for process $t\bar{t}\to t\bar{t}$:

operator case $O_{tt}^{1}$:

$\displaystyle A=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot 2\cdot
u\cdot\frac{(1+\beta)}{\beta}$ (19)

$\displaystyle
a_{0}=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot\frac{s}{16\pi}\cdot\beta^{2}(1+\beta)$
(20)

operator case $O_{QQ}^{1}$:

$\displaystyle A=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot 2\cdot
u\cdot\frac{(1+\beta)}{\beta}$ (21)

$\displaystyle
a_{0}=\left(\frac{C_{QQ}^{1}}{\Lambda^{2}}\right)\cdot\frac{s}{16\pi}\cdot\beta^{2}(1+\beta)$
(22)

operator case $O_{Qt}^{1}$:

$\displaystyle A=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot\left(2\cdot
u\cdot\frac{(1-\beta^{2})}{\beta^{2}}-2\cdot s\cdot(1+\beta^{2})\right)$ (23)

$\displaystyle
a_{0}=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot\left(\frac{s}{8\pi}\cdot\beta(1+\beta^{2})-\frac{s}{16\pi}\cdot\beta(1-\beta^{2})\right)$
(24)

operator case $O_{Qt}^{8}$:

$\displaystyle
A=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot\frac{\lambda^{a}_{j,i}\lambda^{a}_{k,l}}{4}\cdot\left(2\cdot
u\cdot\frac{(1-\beta^{2})}{\beta^{2}}-2\cdot s\cdot(1+\beta^{2})\right)$ (25)

$\displaystyle
a_{0}=\left(\frac{C_{Qt}^{1}}{\Lambda^{2}}\right)\cdot\left(\frac{s}{16\pi}\cdot\beta(1+\beta^{2})-\frac{s}{32\pi}\cdot\beta(1-\beta^{2})\right)$
(26)

operator case $O_{QQ}^{8}$:

$\displaystyle
A=\left(\frac{C_{tt}^{1}}{\Lambda^{2}}\right)\cdot\frac{\lambda^{a}_{j,i}\lambda^{a}_{k,l}}{4}\cdot
2\cdot u\cdot\frac{(1+\beta)}{\beta}$ (27)

$\displaystyle
a_{0}=\left(\frac{C_{QQ}^{8}}{\Lambda^{2}}\right)\cdot\frac{s}{32\pi}\cdot\beta^{2}(1+\beta)$
(28)

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