aainstitutetext: CERN, Theoretical Physics Department, Geneva 23 CH-1211,
Switzerlandbbinstitutetext: IFIC, Universitat de València and CSIC,
Spainccinstitutetext: DAMTP, University of Cambridge, Wilberforce Road,
Cambridge, CB3 0WA, United Kingdomddinstitutetext: INFN, Sezione di Roma,
Piazzale A. Moro 2, I-00185 Roma, Italyeeinstitutetext: Cavendish Laboratory,
University of Cambridge, Cambridge, CB3 0HE, United Kingdomffinstitutetext:
Department of Physics and Astronomy, University of Manchester, Oxford Road,
Manchester M13 9PL, United Kingdom

# Snowmass White Paper: prospects for the measurement of top-quark couplings

Gauthier Durieux b Abel Gutiérrez Camacho c Luca Mantani b,d Víctor
Miralles***Corresponding author<EMAIL_ADDRESS>b Marcos
Miralles López b María Moreno Llácer e René Poncelet f Eleni Vryonidou b
Marcel Vos

###### Abstract

In this contribution to the 2021 Snowmass community planning exercise that
informs the American strategy for particle physics, we present the prospects
for measurements of the top-quark couplings at future colliders. Projections
are presented for the high luminosity phase of the Large Hadron Collider and a
future Higgs/electroweak/top factory electron-positron collider. Results are
presented for the expected bounds on Wilson coefficients of the relevant SMEFT
operators from a global fit to the top physics sector.

##  1 Introduction

High-energy colliders provide a wealth of measurements of the rates,
asymmetries and differential cross sections of a large number of processes.
The Standard Model Effective Field Theory offers a systematic framework to
order all these data and search for subtle patterns of deviations from the
Standard Model. Fits of the dimension-six operator coefficients to projections
for future experiments moreover offer an excellent benchmark to understand the
constraining power of future data sets, without assuming ad-hoc scenarios for
beyond-the-Standard-Model phenomena.

Since the discovery of the top quark, the Tevatron and LHC experiments have
developed an extensive program of measurements of top-quark properties and
interactions. Several groups have performed global fits to the SMEFT operator
coefficients that affect the top-quark sector Brivio _et al._ (2020); Hartland
_et al._ (2019); Buckley _et al._ (2016) or the top/Higgs/electroweak sector
Ethier _et al._ (2021); Ellis _et al._ (2021).

Also bottom-quark production is experimentally distinguishable from other QCD
processes that lead to final states with jets, thanks to precise vertex
detectors and sophisticated flavour tagging algorithms. As the left-handed top
and bottom quarks are part of the same weak-isospin doublet, and share a
dependence on certain Wilson coefficients Durieux _et al._ (2019, 2018), we
consider both third-generation quarks together in this work, including the
bottom-quark operators that affect top-quark or otherwise relevant observables
in our fit.

The new collider facilities contemplated in the global road maps for particle
physics (see Ref. Eur (2020) for the 2020 update of the European strategy) are
expected to improve the measurement of top- and bottom-quark interaction rates
and properties. The remaining LHC program, including the high-luminosity
upgrade (HL-LHC), can provide precise rate measurements for rare top
production processes and can extend the classical hadron-collider top
production processes well into the boosted regime Azzi _et al._ (2019). The
main strength of an electron-positron collider operated above the top-quark
pair production threshold is that it enables an ultra-precise characterisation
of the top-quark electroweak (EW) interactions Amjad _et al._ (2015). Finally,
a new hadron or lepton collider that pushes the energy frontier to the 10 TeV
scale may open up a new kinematic regime, strongly enhancing the sensitivity
to four-quark operators Durieux _et al._ (2018).

In this contribution, we study the impact of the HL-LHC and several future
electron-positron collider scenarios on the top- and bottom-quark sectors of
the SMEFT. All the fits presented here have been performed using the HEPfit
package De Blas _et al._ (2020). We do not provide quantitative results for
the potential of the highest-energy options (FCChh, muon collider, advanced
electron-positron collider), but discuss some qualitative features.

##  2 SMEFT basis

Effects of new physics in the couplings of the top quark can be described as
effective interactions of SM particles at energies below a new physics
matching scale $\Lambda$. These effective interactions can be parameterised in
terms of a set of Wilson coefficients $C_{i}$ of dimension-six operators
$O_{i}$ in the effective Lagrangian,

$\mathcal{L}_{\text{eff}}=\mathcal{L}_{\text{SM}}+\left(\frac{1}{\Lambda^{2}}\sum_{i}\mathcal{C}_{i}O_{i}+\text{h.c.}\right)+\mathcal{O}\left(\Lambda^{-4}\right),$
(1)

where, for this work, the sum runs over the operators shown in Tables 1 and 2
that involve top and bottom quarks, as described below, and which can be
interpreted in terms of new physics mediators. This EFT preserves the Lorentz
and gauge symmetries of the SM, and operators with odd dimension are omitted
since they violate baryon or lepton number.

These dimension-six operators contribute to the collider observables that we
consider in this work. The leading-order contribution is given by the
interference of the dimension-six operators with the SM, which generates
linear terms in the operator coefficients divided by the square of the new
physics scale $\Lambda$. Quadratic contributions are generated by squaring the
amplitudes featuring a dimension-six operator insertion and lead to terms of
order $\Lambda^{-4}$, the same order as the interference of the SM with the
dimension-eight operators that we ignore in this work. Even if the known
quadratic terms are often included in SMEFT fits Aguilar-Saavedra _et al._ ,
we opt for a more conservative approach here and include only the linear ones.
For the two-quark operators similar constraints could be obtained while using
linear and linear plus quadratic terms, for most of the Wilson coefficients,
as shown in Ref. Miralles _et al._ (2022). For the four-quark operators the
inclusion of quadratic terms helps to eliminate the blind directions since
they reduce the possibility of having strong cancellations among the different
contributions. This effect can be observed when comparing the results of Ref.
Ellis _et al._ (2021) (where only linear terms are included) with the results
of Ref. Ethier _et al._ (2021) (where linear and quadratic terms are
considered). Note also that, considering only linear terms, we lose
sensitivity to the four-quark operators featuring a colour-singlet top
current, since they do not interfere with the dominant QCD amplitudes for pair
production. We refer to Ref. Brivio _et al._ (2020) for a detailed study on
the contributions of the top-quark operators to the observables included.

Coefficients fitted
---
2-quark | $C_{tG}$ | $C_{\varphi Q}^{3}$ | $C_{\varphi Q}^{-}=C_{\varphi Q}^{1}-C_{\varphi Q}^{3}$
$C_{\varphi t}$ | $C_{\varphi b}$ | $C_{tZ}=c_{W}C_{tW}-s_{W}C_{tB}$
– | $C_{t\varphi}$ | $C_{tW}$
4-quark | $C_{tu}^{8}=\sum\limits_{\scaleto{i=1,2}{4pt}}2C_{uu}^{(i33i)}$ | $C_{td}^{8}=\sum\limits_{\scaleto{i=1,2,3}{4pt}}C_{ud}^{8(33ii)}$ | $C_{Qq}^{1,8}=\sum\limits_{\scaleto{i=1,2}{4pt}}C_{qq}^{1(i33i)}+3C_{qq}^{3(i33i)}$
$C_{Qu}^{8}=\sum\limits_{\scaleto{i=1,2}{4pt}}C_{qu}^{8(33ii)}$ | $C_{Qd}^{8}=\sum\limits_{\scaleto{i=1,2,3}{4pt}}C_{qd}^{8(33ii)}$ | $C_{Qq}^{3,8}=\sum\limits_{\scaleto{i=1,2}{4pt}}C_{qq}^{1(i33i)}-C_{qq}^{3(i33i)}$
– | – | $C_{tq}^{8}=\sum\limits_{\scaleto{i=1,2}{4pt}}C_{uq}^{8(ii33)}$
2-quark 2-lepton | $C_{eb}$ | $C_{et}$ | $C_{lQ}^{+}=C_{lQ}^{1}+C_{lQ}^{3}$
$C_{lb}$ | $C_{lt}$ | $C_{lQ}^{-}=C_{lQ}^{1}-C_{lQ}^{3}$
– | – | $C_{eQ}$
Table 1: Here we present the Wilson coefficients that have been fitted in our
analysis in terms of those of Table 2. Those in first block are related with
the two-quark operators, those in the second block with the four-quark
operators and the last block is related with the two-quark two-lepton
operators. Relevant operators
---
Coefficient | Operator | Coefficient | Operator
$C_{\varphi Q}^{1}$ | $\left(\bar{Q}\gamma^{\mu}Q\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right)$ | $C_{\varphi Q}^{3}$ | $\left(\bar{Q}\tau^{I}\gamma^{\mu}Q\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}^{I}\\!\\!\\!_{\mu}\>\varphi\right)$
$C_{\varphi t}$ | $\left(\bar{t}\gamma^{\mu}t\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right)$ | $C_{\varphi b}$ | $\left(\bar{b}\gamma^{\mu}b\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right)$
$C_{t\varphi}$ | $\left(\bar{Q}t\right)\left(\epsilon\varphi^{*}\;\varphi^{\dagger}\varphi\right)$ | $C_{tG}$ | $\left(\bar{t}\sigma^{\mu\nu}T^{A}t\right)\left(\epsilon\varphi^{*}G_{\mu\nu}^{A}\right)$
$C_{tW}$ | $\left(\bar{Q}\tau^{I}\sigma^{\mu\nu}t\right)\left(\epsilon\varphi^{*}W_{\mu\nu}^{I}\right)$ | $C_{tB}$ | $\left(\bar{Q}\sigma^{\mu\nu}t\right)\left(\epsilon\varphi^{*}B_{\mu\nu}\right)$
$C_{qq}^{1(ijkl)}$ | $(\bar{q}_{i}\gamma^{\mu}q_{j})(\bar{q}_{k}\gamma_{\mu}q_{l})$ | $C_{qq}^{3(ijkl)}$ | $(\bar{q}_{i}\tau^{I}\gamma^{\mu}q_{j})(\bar{q}_{k}\tau^{I}\gamma_{\mu}q_{l})$
$C_{uu}^{(ijkl)}$ | $(\bar{u}_{i}\gamma^{\mu}u_{j})(\bar{u}_{k}\gamma_{\mu}u_{l})$ | $C_{ud}^{8(ijkl)}$ | $(\bar{u}_{i}\gamma^{\mu}T^{A}u_{j})(\bar{d}_{k}\gamma_{\mu}T^{A}d_{l})$
$C_{qu}^{8(ijkl)}$ | $(\bar{q}_{i}\gamma^{\mu}T^{A}q_{j})(\bar{u}_{k}\gamma_{\mu}T^{A}u_{l})$ | $C_{qd}^{8(ijkl)}$ | $(\bar{q}_{i}\gamma^{\mu}T^{A}q_{j})(\bar{d}_{k}\gamma_{\mu}T^{A}d_{l})$
$C_{lQ}^{1}$ | $\left(\bar{Q}\gamma_{\mu}Q\right)\left(\bar{l}\gamma^{\mu}l\right)$ | $C_{lQ}^{3}$ | $\left(\bar{Q}\tau^{I}\gamma_{\mu}Q\right)\left(\bar{l}\tau^{I}\gamma^{\mu}l\right)$
$C_{lt}$ | $\left(\bar{t}\gamma_{\mu}t\right)\left(\bar{l}\gamma^{\mu}l\right)$ | $C_{lb}$ | $\left(\bar{b}\gamma_{\mu}b\right)\left(\bar{l}\gamma^{\mu}l\right)$
$C_{eQ}$ | $\left(\bar{Q}\gamma_{\mu}Q\right)\left(\bar{e}\gamma^{\mu}e\right)$ | $C_{et}$ | $\left(\bar{t}\gamma_{\mu}t\right)\left(\bar{e}\gamma^{\mu}e\right)$
$C_{eb}$ | $\left(\bar{b}\gamma_{\mu}b\right)\left(\bar{e}\gamma^{\mu}e\right)$ | – | –
Table 2: Here we show the most relevant operators whose linear combinations
have been fitted in this work. The first block are two-quark operators, the
second block are four-quark operators and the last block are two-quark two-
lepton operators. In these operators $Q$ is the left-handed doublet of the two
heaviest quarks, the Latin letters are flavour indices, $\tau^{I}$ are the
Pauli matrices, $T^{A}=\lambda^{A}/2$ with $\lambda^{A}$ the Gell-Mann
matrices.

The number of operators involved in the SMEFT description is prohibitive if
one adopts the most general flavour structure. We focus on the operator
coefficients of the Warsaw basis Grzadkowski _et al._ (2010) (see also Refs.
Aguilar-Saavedra (2009); Zhang and Willenbrock (2011)) that involve at least
one top quark, as well as the bottom-quark operators that affect the
observables included in our study. Motivated by the minimal flavour violation
ansatz, a $U(2)_{q}\otimes U(2)_{u}\otimes U(2)_{d}$ symmetry is imposed among
the first two generations, as in the conventions proposed by the LHC Top
Working Group Aguilar-Saavedra _et al._ . The three lepton generations are
treated independently but, since we only include two-quark two-lepton
operators in $e^{+}e^{-}$ collider production processes, our analysis is only
sensitive to the degrees of freedom corresponding to the electron. We do not
include the CP-violating imaginary parts of the Wilson coefficients, nor
operators that lead to flavour-changing-neutral-current interactions.

The operator coefficients included in our analysis are listed in Table 1 and
the operators are defined in Table 2. The selected sub-set of operators
consists of three main blocks: the two-quark operators that modify top- and
bottom-quark electroweak couplings and the $t\bar{t}$-gluon vertex, the four-
quark operators of the type $q\bar{q}t\bar{t}$ (i.e. two light quarks and two
heavy quarks) and the two-lepton-two-heavy-quark operators of the type
$e^{+}e^{-}{t\bar{t}}$ and $e^{+}e^{-}b\bar{b}$. The four-quark operators are
best probed at hadron colliders, while $e^{+}e^{-}$ colliders can provide
better bounds on the two-lepton-two-quark operators. Both types of machines
can provide bounds on the two-fermion operators and a direct comparison is
possible for this set. As in Ref. Aguilar-Saavedra _et al._ we use the linear
combinations $O_{\varphi Q}^{-}\equiv O_{\varphi Q}^{1}-O_{\varphi Q}^{3}$ and
$O_{qZ}\equiv-\sin\theta_{W}O_{qB}+\cos\theta_{W}O_{qW}$, and, likewise,
$C_{lQ}^{-}\equiv C_{lQ}^{1}-C_{lQ}^{3}$, as indicated in Table 2.

##  3 Prospects for the HL-LHC

In this section, we discuss the projections for the HL-LHC top physics
measurements included in our fit.

The LHC is a top-quark factory that produces enormous samples of $t\bar{t}$
events and sizeable samples for rare top processes. The prospects for
measurements at the LHC for the full 3000 fb-1 expected after the high-
luminosity phase of the LHC are based on an extrapolation from current (run 2)
measurements. The measurements that form the basis for the HL-LHC projection
are listed in Table 3, the inputs used for each observable can be found in
App. A.

The projections of the measurements of rare top-quark production processes are
modelled on the S2 scenario used to predict the precision of Higgs coupling
measurements in Ref. Cepeda _et al._ (2019). This scenario envisages that many
statistical and experimental uncertainties scale as $1/\sqrt{L_{\text{int}}}$,
where $L_{\text{int}}$ is the integrated luminosity. For the complete HL-LHC
programme, experimental uncertainties are reduced by a factor 5. Theory and
modelling uncertainties are divided by two, with respect to today’s state of
the art. This assumes N2LO calculations will be achieved for rare associated
production processes, and that Monte Carlo modelling can significantly be
improved in the next decade.

To gain the maximal sensitivity to the EFT coefficients, differential
measurements are included in our analysis. As in Ref. Miralles _et al._
(2022), for the $pp\rightarrow t\bar{t}Z$ and $pp\rightarrow t\bar{t}\gamma$
processes, differential measurements as a function of the $Z$-boson and photon
$p_{T}$ are included, enhancing the sensitivity to $C_{tZ}$, in particular
Bessidskaia Bylund _et al._ (2016).

Process | Observable | $\sqrt{s}$ | $L_{\text{int}}$ | Experiment | SM | Ref.
---|---|---|---|---|---|---
$pp\rightarrow{t\bar{t}}$ |  $d\sigma/dm_{t\bar{t}}$ (15+3 bins) | 13 TeV | 140 fb-1 | CMS | Czakon _et al._ (2013) | Tumasyan _et al._ (2021)
$pp\rightarrow{t\bar{t}}$ |  $dA_{C}/dm_{t\bar{t}}$ (4+2 bins) | 13 TeV | 140 fb-1 | ATLAS | Czakon _et al._ (2013) | ATLAS Collaboration
$pp\rightarrow t\bar{t}H+tHq$ | $\sigma$ | 13 TeV | 140 fb-1 | ATLAS | LHC Higgs Cross Section Working Group () (D. de Florian et al.) | ATLAS Collaboration (2020a)
$pp\rightarrow t\bar{t}Z$ |  $d\sigma/dp_{T}^{Z}$ (7 bins) | 13 TeV | 140 fb-1 | ATLAS | Broggio _et al._ (2019) | Aad _et al._ (2021)
$pp\rightarrow t\bar{t}\gamma$ |  $d\sigma/dp_{T}^{\gamma}$ (11 bins) | 13 TeV | 140 fb-1 | ATLAS | Bevilacqua _et al._ (2018, 2019) | ATLAS Collaboration (2020b)
$pp\rightarrow tZq$ | $\sigma$ | 13 TeV | 77.4 fb-1 | CMS | CMS Collaboration (2018a) | CMS Collaboration (2019)
$pp\rightarrow t\gamma q$ | $\sigma$ | 13 TeV | 36 fb-1 | CMS | CMS Collaboration (2018b) | CMS Collaboration (2018b)
$pp\rightarrow t\bar{t}W$ | $\sigma$ | 13 TeV | 36 fb-1 | CMS | LHC Higgs Cross Section Working Group () (D. de Florian et al.); Frederix _et al._ (2018) | CMS Collaboration (2018c)
$pp\rightarrow t\bar{b}$ (s-ch) | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Aliev _et al._ (2011); Kant _et al._ (2015) | ATLAS and CMS Collaborations (2019)
$pp\rightarrow tW$ | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Kidonakis (2010a) | ATLAS and CMS Collaborations (2019)
$pp\rightarrow tq$ (t-ch) | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Aliev _et al._ (2011); Kant _et al._ (2015) | ATLAS and CMS Collaborations (2019)
$t\rightarrow Wb$ |  $F_{0}$, $F_{L}$ | 8 TeV | 20 fb-1 | LHC | Czarnecki _et al._ (2010) | ATLAS and CMS Collaborations (2020)
$p\bar{p}\rightarrow t\bar{b}$ (s-ch) | $\sigma$ | 1.96 TeV | 9.7 fb-1 | Tevatron | Kidonakis (2010b) | CDF and D0 Collaborations (2014)
$e^{-}e^{+}\rightarrow b\bar{b}$ |  $R_{b}$ , $A_{FBLR}^{bb}$ |  $\sim$ 91 GeV | 202.1 pb-1 | LEP/SLD | - | Schael _et al._ (2006)

Table 3: Measurements included in the EFT fit of the top-quark electroweak
sector. For each measurement, the process, the observable, the centre-of-mass
energy, the integrated luminosity and the experiment/collider are given. The
last two columns list the references for the predictions and measurements that
are included in the fit. LHC refers to the combination of ATLAS and CMS
measurements. In a similar way, Tevatron refers to the combination of CDF and
D0 results, and LEP/SLD to different experiments from those two accelerators.

For the top-quark pair production process, statistics is abundant and
measurements in the bulk already reach a precision of a few %. The inclusive
cross section measurement is currently limited to a 2% uncertainty Aaboud _et
al._ (2016) by the uncertainty on the integrated luminosity. This uncertainty
is expected to be reduced to approximately 1% at the HL-LHC Azzi _et al._
(2019). Currently, theory uncertainties of the N2LO calculation are at the
level of 3–4% for the inclusive cross section Czakon _et al._ (2013). These
might be reduced to roughly half with the calculation of the N3LO corrections
and the improvement of the proton PDFs. Even in that case, theory
uncertainties are likely to remain the limiting factor.

The $t\bar{t}$ charge asymmetry is a subtle effect at the LHC, but it brings
important information to EFT fits Zhang _et al._ (2012). As a ratio, it can be
precisely predicted Czakon _et al._ (2018). Modelling uncertainties play an
important role ATLAS Collaboration  and are likely to limit future progress in
the inclusive measurement. Therefore, a less aggressive scenario is adopted,
where all experimental systematic uncertainties are improved by a factor 1/2
and only the statistical uncertainty scales with $1/\sqrt{L_{\text{int}}}$.

Figure 1: The 95% probability bounds on the Wilson coefficients for dimension-
six operators that affect the top-quark production and decay measurements
listed in Table 3 after run 2 of the LHC (in dark red) and prospects for the
bounds expected after completion of the complete LHC program, including the
high-luminosity stage (in light red). Only linear terms proportional to
$\Lambda^{-2}$ are taken into account in the dependence of the observables on
the Wilson coefficients. The individual bounds obtained from a single-
parameter fit are shown as solid bars, while the global or marginalised bounds
obtained fitting all Wilson coefficients at once are indicated by the full
bars (shaded region in each bar). The correlations between the Wilson
coefficients obtained in the global fit can be found in App. B.

For top-quark pair production, differential measurements of the cross section
Tumasyan _et al._ (2021) and the charge asymmetry ATLAS Collaboration  as a
function of the invariant mass of the $t\bar{t}$ system are considered. A
promising avenue for progress is the boosted regime, where the sensitivity to
four-fermion operators increases considerably Perello Rosello and Vos (2016).
Measurements of the cross section and charge asymmetry for $t\bar{t}$ systems
produced at large invariant mass already play an important role in the
constraints on the four-fermion operators and their weight will increase if
measurements on bulk $t\bar{t}$ are limited by experimental or theoretical
systematic uncertainties. To take maximal advantage of this potential, the
range of the projections is extended further into the high-$m_{t\bar{t}}$ tail
than the current run 2 measurements. More details in the binning that has been
considered can be found in App. C.

The 95% probability bounds from a fit to the current data are shown as the
dark red bars in Fig. 1. The data include LHC run 2 and run 1 measurements and
several legacy results from the Tevatron, LEP, and SLD experiments. The light
red bars present the extrapolation to the complete HL-LHC program, with an
integrated luminosity of 3 ab-1.

Across the board, the HL-LHC program is expected to improve the bounds by a
factor of two to four with respect to the current run 2 limits, both for
individual bounds and global fit results. Exceptions are the individual bounds
on $C_{\varphi Q}^{-}$ and $C_{\varphi Q}^{3}$, that continue to depend on the
$Zb\bar{b}$ measurements at the $Z$-pole.

Generally, the progress envisaged in the S2 scenario is limited by the theory
and modelling uncertainties, while statistical and experimental uncertainties
are expected to be sub-dominant in nearly all measurements in the S2 scenario.
Therefore, improving the accuracy of fixed-order predictions beyond the factor
two envisaged in the S2 scenario, which already assumes significant advances
in the theoretical calculations, will lead to a direct improvement of the
sensitivity. This will, however, likely require N3LO precision for
$2\rightarrow 3$ processes with top quarks in the final state.

The boosted regime is indeed confirmed as one of the keys to improving bounds
on the operators that affect the top-quark pair production process. In
particular, the high-$m_{t\bar{t}}$ tail of the top-quark pair production
measurements provides a significant reduction in the allowed regions of the
four-quark operators, which shrink by a factor between two and five (depending
on the operator) thanks to the enhanced sensitivity in this regime and the
more pronounced improvement in the measurement. This effect is present even in
a fit that only includes the linear (${\cal O}(\Lambda^{-2}$)) terms in the
parameterisation of the EFT dependence.

The marginalised bounds on the four-fermion operators remain an order of
magnitude worse than the individual bounds after the HL-LHC, even if both
individual and global bounds improve considerably. This is due to unresolved
correlations between the coefficients. The same feature is observed in recent
fits to the top sector of the SMEFT Brivio _et al._ (2020); Hartland _et al._
(2019) and in global Higgs/EW/top fits Ethier _et al._ (2021); Ellis _et al._
(2021). Stricter limits can be obtained if the dimension-six-squared terms
proportional to $\Lambda^{-4}$ are included in the fit Ethier _et al._ (2021).

Two-quark two-lepton operators, omitted in this section, can also be probed at
the LHC. Dedicated signal regions, for instance with off-$Z$-peak dilepton
invariant masses in $pp\to t\bar{t}\ell^{+}\ell^{-}$ Durieux _et al._ (2015);
Chala _et al._ (2019); Sirunyan _et al._ (2021), would increase their
sensitivity.

##  4 Prospects for $e^{+}e^{-}$ colliders

In this section, we study the impact of precision measurements in
$e^{+}e^{-}\rightarrow b\bar{b}$ and $e^{+}e^{-}\rightarrow t\bar{t}$
production, using operating scenarios of the main circular and linear collider
concepts.

Prospects for the $e^{+}e^{-}\rightarrow b\bar{b}$ process are included that
are based on the full-simulation studies of the ILD concept Okugawa _et al._
(2019) at $\sqrt{s}=$ 250 GeV . The prospects are based on realistic estimates
of efficiency and acceptance, including the signal losses required to ensure a
robust calibration of the flavour tagging efficiency. The statistical
uncertainties on the measurements of the cross section and forward-backward
asymmetry are complemented by polarisation and flavour-tagging systematics.
For the $Z$-pole runs we use the projections for $R_{b}$ and $A_{FB}$ provided
by the FCCee and CEPC projects for their “TeraZ” runs at the $Z$-pole.

Figure 2: Comparison of current LHC constraints with HL-LHC ones, and those
deriving from ILC runs at 250, 500 and 1000 GeV. The limits on the
$q\bar{q}t\bar{t}$ and $C_{tG}$ coefficients are not shown, since the
$e^{+}e^{-}$ collider measurements considered are not sensitive to them, but
all operators are included in the global fit. The improvement expected from
the HL-LHC on these coefficients is shown in Fig. 1. The solid bars provide
the individual limits of the single-parameter fit and the shaded ones the
marginalised limits of the global fit. The correlations between the Wilson
coefficients obtained in the global fit can be found in App. B.

The $e^{+}e^{-}\rightarrow t\bar{t}$ process opens up for centre-of-mass
energies that exceed twice the top mass (i.e. $\sqrt{s}\gtrsim$ 350 GeV ) and
probes the electroweak couplings of the top quark at tree-level. Data taken
with different beam polarisations at linear colliders can be used to
distinguish the photon and $Z$-boson couplings Amjad _et al._ ; Amjad _et al._
(2015); Durieux _et al._ (2018); Abramowicz _et al._ (2019). At circular
colliders, a measurement of the final state polarisation using the semi-
leptonically decaying top quarks can also be used to separate the two
contributions Janot (2015). We base our prospects on the study of
statistically optimal observables defined at leading order on the
$e^{+}e^{-}\to t\bar{t}\rightarrow WbWb$ differential distribution Durieux _et
al._ (2018). This $WbWb$ final state also receives contribution from single
top production which become sizeable at high centre-of-mass energies.
Realistic acceptance, identification and reconstruction efficiencies are
estimated from full-simulation studies for the ILC and CLIC in Ref. Amjad _et
al._ ; Abramowicz _et al._ (2017). Since they were performed only for sub-set
of centre-of-mass energies and beam polarisations, overall efficiency factors
are extrapolated as a functions of the centre-of-mass energy. They drop
significantly for the TeV centre-of-mass energies of ILC and CLIC since a
degradation of top-selection and flavour-tagging capabilities is expected in
this regime.

In Fig. 2, the impact of runs at different centre-of-mass energies is
illustrated. The current bounds in brown are compared to HL-LHC ones in red.
The subsequent bars add data at
$\sqrt{s}=250~{}{\mathrm{Ge\kern-1.00006ptV}}\,$, 500 GeV and 1 TeV . The beam
polarisations and integrated luminosities of the different ILC stages are
summarised in Table 4. Only the electroweak operators are presented, as the
$e^{+}e^{-}$ data have the strongest impact there, but results corresponds to
a global analysis, including also the $q\bar{q}t\bar{t}$ operators and
$C_{tG}$.

The dark green bar shows that the “Higgs factory” run improves the bounds on
bottom-quark operators, including $C_{\varphi Q}^{3}$ and $C_{\varphi Q}^{-}$
and $C_{\varphi b}$. The improvement is especially pronounced for the
individual bounds. As expected, data above the top-quark pair production
threshold is required to improve the bounds on the top-quark operators.

Runs at two different centre-of-mass energies above the top-quark pair
production threshold are required to disentangle the $e^{+}e^{-}t\bar{t}$
operator coefficients from the two-fermion operator coefficients Durieux _et
al._ (2018). The two sets of operators have very different scaling with
energy: the sensitivity to four-fermion operators grows quadratically, while
it is constant or grows only linearly for two-fermion operators. In a fit to
data taken at a single centre of mass, linear combinations of their
coefficients remain degenerate and form blind directions. The combination of
runs at two different centre-of-mass energies effectively disentangles them
and provides global fit constraints close to the individual bounds

Figure 3: Comparison of the constraints expected from a combination of HL-LHC
and lepton collider data. The limits on the $q\bar{q}t\bar{t}$ and $C_{tG}$
coefficients are not shown, since the $e^{+}e^{-}$ collider measurements
considered are not sensitive to them, but all operators are included in the
global fit. The improvement expected from the HL-LHC on these coefficients is
shown in Fig. 1. The solid bars provide the individual limits of the single-
parameter fit and the shaded ones the marginalised limits of the global fit.
The correlations between the Wilson coefficients obtained in the global fit
can be found in App. B.

Several further processes are accessible to $e^{+}e^{-}$ colliders, but have
not been taken into account in this study. The top-quark Yukawa coupling can
be determined through the tree-level dependence of the associated
$e^{+}e^{-}\rightarrow t\bar{t}H$ production process. This requires runs with
a centre-of-mass energy above 500–550 GeV . At linear colliders, where the
luminosity grows with energy, there is a broad plateau up to about 1.5 TeV
where $e^{+}e^{-}\rightarrow t\bar{t}H$ is accessible. Based on full-
simulation studies of Ref. Price _et al._ (2015) we expect an improvement in
the constraint on $C_{t\varphi}$ by a factor two with respect to the HL-LHC.

Machine | Polarisation | Energy | Luminosity | Reference
---|---|---|---|---
ILC | P($e^{+}$, $e^{-}$):$(\pm 30\%,\,\mp 80\%)$ | 250 GeV | 2 ab-1 | A. Aryshev et al. () (editors)
500 GeV | 4 ab-1
1 TeV | 8 ab-1
CLIC | P($e^{+}$, $e^{-}$):$(0\%,\,\pm 80\%)$ | 380 GeV | 1 ab-1 | Robson and Roloff
1.4 TeV | 2.5 ab-1
3 TeV | 5 ab-1
FCC-$ee$ | Unpolarised | Z-pole | 150 ab-1 | Bernardi _et al._
240 GeV | 5 ab-1
350 GeV | 0.2 ab-1
365 GeV | 1.5 ab-1
CEPC | Unpolarised | Z-pole | 57.5 ab-1 | Bernardi _et al._
240 GeV | 20 ab-1
350 GeV | 0.2 ab-1
360 GeV | 1 ab-1
Table 4: Here we show the different working configurations for the future
$e^{+}e^{-}$ colliders.

In Fig. 3, we compare the bounds expected from the HL-LHC and from the final
stages of the CEPC, FCC-$ee$, ILC and CLIC. The centre-of-mass energies,
integrated luminosities and beam polarisations envisaged for each of these
projects are given in Table 4. The circular colliders (FCC-$ee$ and CECP)
operated at and slightly above the $t\bar{t}$ threshold are expected to
improve constraints on the bottom- and top-operators by factors 5 and 2 for
some two-fermion operators. Indeed, their “TeraZ” runs provide very
competitive bounds (individual ones, in particular) on two-fermion bottom-
operator coefficients. Their constraining power on four-fermion operators is,
however, limited by the energy reach. Since, at these colliders, the two runs
above the $t\bar{t}$-threshold are very close the two-fermion and four-fermion
operators are harder to disentangle. The global limits remain significantly
above the individual bounds.

The linear colliders (ILC and CLIC), operated at two centre-of-mass energies
above the $t\bar{t}$ threshold, can provide very tight bounds on all
operators. The bounds on four-fermion operators take advantage of the energy-
growing sensitivity and become very competitive if $e^{+}e^{-}$ collision data
at a centre-of-mass energy greater than 1 TeV is available. The ILC1000 and
CLIC3000 bounds of $\mathcal{O}(10^{-3})$ on the $e^{+}e^{-}t\bar{t}$
operators are by far the tightest top-sector SMEFT constraints that can be
achieved at any future collider considered in this work.222A muon collider or
advanced linear collider have the potential to improve these bounds further,
but quantitative projections for integrated luminosity and experimental
performance are currently not available.

##  5 Pushing the energy frontier

Several projects have been defined that extend the energy of colliders well
beyond the TeV scale. Collisions at a centre-of-mass energy of 10 TeV and
beyond could be achieved at a large (100 km circumference) hadron collider
Abada _et al._ (2019); Group (2015), at a linear electron-positron collider
implementing novel accelerating techniques Cros and Muggli , or at compact
circular muon colliders Stratakis _et al._ . The potential of these machines
for the SMEFT fit lies mainly in the energy-growing sensitivity to new
physics. In the top- and bottom-quark sectors of the SMEFT, the sensitivity to
four-fermion operators shows a strong increase Aguilar-Saavedra _et al._
(2015); Perello Rosello and Vos (2016); Englert _et al._ (2016); Durieux _et
al._ (2018). For a given measurement precision, bounds derived in higher-
energy collisions are therefore much stronger than those derived from
measurements at lower energy.

We illustrate the increased sensitivity with the dependence of the
differential cross section at high $m_{{t\bar{t}}}$ to $C_{tu}^{8}$ and
$C_{tG}$. At the LHC, the cross section measurement in the boosted regime
(with $m_{t\bar{t}}>$ 1.4 TeV ), yields the following relation with the Wilson
coefficient:

$\sigma(m_{t\bar{t}}>1.4~{}{\mathrm{Te\kern-1.00006ptV}}\,)=1.8~{}\mathrm{pb}\times[1+0.3\cdot
C_{tG}+0.1\cdot C_{tG}^{2}+0.1\cdot C_{tu}^{8}+0.3\cdot(C_{tu}^{8})^{2}+...]$
(2)

A 100 TeV pp collider has a seven times larger energy reach and one could
envisage a measurement with $m_{{t\bar{t}}}>$ 10 TeV that would have the
following dependence:

$\sigma(m_{t\bar{t}}>10~{}{\mathrm{Te\kern-1.00006ptV}}\,)=0.1~{}\mathrm{pb}\times[1+0.3\cdot
C_{tG}+1.8\cdot C_{tG}^{2}+3\cdot C_{tu}^{8}+256\cdot(C_{tu}^{8})^{2}+...]$
(3)

The increase in the factors that multiply the Wilson coefficients is very
clear for the quadratic term of $C_{tG}$ and for both the linear and quadratic
terms in $C_{tu}^{8}$ (and similarly for the other four-fermion operator
coefficients).333Note that here we are showing the impact on linear and
quadratic terms since many studies include both contributions. We remark that
in the quantitative analysis presented in this work only linear terms are
considered, as explained above.

We therefore expect that FCChh and SPPC measurements in the 10 TeV regime,
with precision comparable to that of current boosted measurements at the LHC,
could provide bounds that are a factor 20 sharper than the HL-LHC prospects,
bringing the constraints down from ${\cal
O}(1~{}{\mathrm{Te\kern-1.00006ptV}}\,^{-2})$ to ${\cal
O}(0.1~{}{\mathrm{Te\kern-1.00006ptV}}\,^{-2})$. This, of course, requires
that techniques be developed to efficiently trigger, select and reconstruct
events with highly boosted top quarks Kogler _et al._ (2019); Nachman _et al._
and that the experimental response and Monte Carlo modelling be controlled to
a similar level.

##  6 Conclusion

New energy-frontier colliders are expected to provide an important push for
top- and bottom-quark physics. We assess this potential in the framework of
the Standard Model Effective Field Theory, performing fits to measurement
projections in several scenarios.

For the study of rare top production processes at the High-Luminosity phase of
the LHC, we adopt the S2 scenario that is also used for Higgs measurements
Cepeda _et al._ (2019). It envisages a strong $(\propto
1/\sqrt{L_{\text{int}}})$ reduction of experimental uncertainties and a
reduction by a factor two for theory and modelling uncertainties. For the very
abundant top-quark pair production, a more conservative scenario is adopted.
Across the board, the HL-LHC is expected to improve bounds on Wilson
coefficients by a factor three with respect to the current results that are
dominated by LHC run 2. Global bounds on two-fermion operator coefficients are
expected to range between 0.03 TeV -2 and 6 TeV -2, while four-fermion
operator coefficients remain of order 1 due to strong degeneracies.

A new electron-positron collider can provide very tight bounds on bottom-quark
operators and — if operated above the ${t\bar{t}}$ production threshold — on
the top-quark ones. Constraints on the two-fermion operator coefficients that
affect the top- and bottom-quark electroweak couplings improve by up to two
orders of magnitude, reaching order 0.1 TeV -2 for most coefficients. With
runs at several energies above the $t\bar{t}$ production threshold the entire
sector can be tightly constrained. Operation at TeV centre-of-mass energies
would yield bounds of order $10^{-3}$ TeV -2 on the $e^{+}e^{-}t\bar{t}$
operator coefficients, making them the most precisely constrained operators in
the top sector of the SMEFT. The $Z$-pole run offers the best bounds on the
two-fermion bottom-quark operators.

An energy-frontier proton collider is expected to further improve the bounds
on four-fermion operators involving a pair of top quarks. While no systematic
and quantitative projections have been released, the increased sensitivity to
$q\bar{q}t\bar{t}$ operators in the highly boosted regime offers the potential
for an order-of-magnitude improvement with respect to the HL-LHC projections
at a 100 TeV proton collider.

## Acknowledgments

The work of VM is supported by the Italian Ministry of Research (MUR) under
the grant PRIN20172LNEEZ. EV 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. RP acknowledges the support
from the Leverhulme Trust and the Isaac Newton Trust, as well as the use of
the DiRAC Cumulus HPC facility under Grant No. PPSP226. The work of MV is
supported by the projects No. PGC2018-094856-B-100 (MICINN), PROMETEO-2018/060
(Generalitat Valenciana) and ILINKB20065 (CSIC). The work of MMLL is supported
by the projects SEJI-2020/037 (Generalitat Valenciana) and RYC2019-028510-I
(MICINN). The work of AGC is supported by the program JAEICU-21-IFIC-6 (CSIC).

##  Appendix A Appendix: Inputs included in the fits

### LHC and HL-LHC inputs

$pp\rightarrow t\bar{t}$ | Measured ($\text{fb}\cdot\text{GeV}^{-1}$) | SM ($\text{fb}\cdot\text{GeV}^{-1}$) | LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$) | HL-LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$)
---|---|---|---|---
theo. | exp. | theo. | exp.
stat. | sys. + mod. | tot. | stat. | sys. + mod. | tot.
$m_{t\bar{t}}:$ (250-400) | 344.2 | 328.09 | 42.68 | 1.604 | 13.19 | 13.29 | 21.34 | 0.3465 | 6.600 | 6.60
$m_{t\bar{t}}:$ (400-480) | 871.7 | 883.38 | 111.28 | 4.784 | 41.29 | 41.57 | 55.64 | 1.0335 | 20.65 | 20.67
$m_{t\bar{t}}:$ (480-560) | 542.8 | 545.13 | 72.26 | 4.737 | 23.07 | 23.55 | 36.13 | 1.0233 | 11.54 | 11.58
$m_{t\bar{t}}:$ (560-640) | 315.3 | 318.36 | 46.58 | 4.561 | 14.22 | 14.93 | 23.29 | 0.9853 | 7.110 | 7.18
$m_{t\bar{t}}:$ (640-720) | 180.9 | 182.33 | 24.41 | 4.331 | 8.395 | 9.446 | 12.21 | 0.9356 | 4.198 | 4.30
$m_{t\bar{t}}:$ (720-800) | 109.6 | 109.27 | 18.79 | 3.748 | 5.468 | 6.629 | 9.395 | 0.8097 | 2.734 | 2.85
$m_{t\bar{t}}:$ (800-900) | 62.51 | 64.83 | 10.94 | 2.253 | 3.026 | 3.772 | 5.470 | 0.4867 | 1.513 | 1.59
$m_{t\bar{t}}:$ (900-1000) | 36.97 | 36.59 | 6.877 | 1.550 | 1.988 | 2.521 | 3.439 | 0.3348 | 0.994 | 1.05
$m_{t\bar{t}}:$ (1000-1150) | 17.84 | 19.15 | 3.904 | 0.6554 | 0.9134 | 1.124 | 1.952 | 0.1416 | 0.4567 | 0.48
$m_{t\bar{t}}:$ (1150-1300) | 9.005 | 9.060 | 2.020 | 0.4129 | 0.5086 | 0.6551 | 1.010 | 0.08920 | 0.2543 | 0.27
$m_{t\bar{t}}:$ (1300-1500) | 3.815 | 4.256 | 1.113 | 0.2151 | 0.2024 | 0.2954 | 0.5565 | 0.04647 | 0.1012 | 0.11
$m_{t\bar{t}}:$ (1500-1700) | 1.769 | 1.777 | 0.5259 | 0.1441 | 0.1108 | 0.1817 | 0.2630 | 0.03113 | 0.0554 | 0.064
$m_{t\bar{t}}:$ (1700-2000) | 0.6941 | 0.6806 | 0.2203 | 0.06014 | 0.0397 | 0.0720 | 0.1101 | 0.01298 | 0.01985 | 0.0237
$m_{t\bar{t}}:$ (2000-2300) | 0.1905 | 0.23647 | 0.0994 | 0.03459 | 0.0170 | 0.0386 | 0.0497 | 0.007474 | 0.00850 | 0.0113
$m_{t\bar{t}}:$ (2300-2600)∗ | – | 0.12738 | 0.0640 | – | – | – | 0.0320 | 0.005416 | 0.00300 | 0.0072
$m_{t\bar{t}}:$ (2600-3000)∗ | – | 0.04184 | 0.0209 | – | – | – | 0.01045 | 0.002688 | 0.00103 | 0.0033
$m_{t\bar{t}}:$ (3000-3500)∗ | – | 0.01170 | 0.005851 | – | – | – | 0.002926 | 0.001271 | 0.000317 | 0.0015
$m_{t\bar{t}}:$ (3500-4000)∗ | – | 0.00329 | 0.001643 | – | – | – | 0.000822 | 0.000674 | 0.000097 | 0.00077

Table 5: We show the unfolded bin contents for the absolute parton-level
differential cross-section measurement. The correlations are shown in Fig. 4.

$pp\rightarrow t\bar{t}$ | Measured (%) | SM (%) | LHC Unc. (%) | HL-LHC Unc. (%)
---|---|---|---|---
theo. | exp. | theo. | exp.
stat. | sys. + mod. | tot. | stat. | sys. + mod. | tot.
$m_{t\bar{t}}:$ (0-500) | 0.45 | 0.55 | 0.0770 | 0.309 | 0.340 | 0.460 | 0.0385 | 0.0667 | 0.170 | 0.183
$m_{t\bar{t}}:$ (500-750) | 0.51 | 0.72 | 0.1008 | 0.219 | 0.210 | 0.304 | 0.0504 | 0.0474 | 0.105 | 0.115
$m_{t\bar{t}}:$ (750-1000) | 1.00 | 0.79 | 0.1106 | 0.533 | 0.460 | 0.704 | 0.0553 | 0.1152 | 0.230 | 0.257
$m_{t\bar{t}}:$ (1000-1500) | 1.69 | 0.96 | 0.1344 | 0.776 | 0.270 | 0.822 | 0.0672 | 0.1677 | 0.135 | 0.215
$m_{t\bar{t}}:$ (1500-2000)∗ | – | 1.01 | 0.1414 | – | – | – | 0.0707 | 0.5703 | 0.624 | 0.846
$m_{t\bar{t}}:$ (2000-2500)∗ | – | 1.10 | 0.1540 | – | – | – | 0.0770 | 1.4750 | 0.683 | 1.625
$m_{t\bar{t}}:$ (2500-3000)∗ | – | 1.85 | 0.2590 | – | – | – | 0.1295 | 5.4213 | 1.147 | 5.541

Table 6: We show the values included for the charged asymmetry. The
correlations are shown in Fig. 5.

$pp\rightarrow t\bar{t}Z$ | Measured ($\text{fb}\cdot\text{GeV}^{-1}$) | SM ($\text{fb}\cdot\text{GeV}^{-1}$) | LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$) | HL-LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$)
---|---|---|---|---
theo. | exp. | theo. | exp.
stat. | sys. | mod. | tot. | stat. | sys. | mod. | tot.
$p_{T}^{Z}:$ (0-40) | 1.47 | 2.21 | 0.263 | 0.53 | 0.23 | 0.21 | 0.615 | 0.132 | 0.114 | 0.050 | 0.105 | 0.163
$p_{T}^{Z}:$ (40-70) | 4.32 | 4.59 | 0.543 | 0.94 | 0.60 | 0.51 | 1.223 | 0.272 | 0.203 | 0.130 | 0.253 | 0.349
$p_{T}^{Z}:$ (70-110) | 4.24 | 4.60 | 0.555 | 0.75 | 0.54 | 0.36 | 0.993 | 0.278 | 0.162 | 0.117 | 0.182 | 0.270
$p_{T}^{Z}:$ (110-160) | 4.4 | 3.45 | 0.429 | 0.55 | 0.43 | 0.39 | 0.800 | 0.215 | 0.118 | 0.093 | 0.197 | 0.248
$p_{T}^{Z}:$ (160-220) | 1.75 | 2.05 | 0.261 | 0.31 | 0.15 | 0.13 | 0.371 | 0.131 | 0.067 | 0.033 | 0.066 | 0.100
$p_{T}^{Z}:$ (220-290) | 0.58 | 1.03 | 0.130 | 0.16 | 0.047 | 0.034 | 0.174 | 0.065 | 0.035 | 0.010 | 0.017 | 0.041
$p_{T}^{Z}:$ (290-400) | 0.56 | 0.59 | 0.071 | 0.11 | 0.055 | 0.057 | 0.132 | 0.036 | 0.023 | 0.012 | 0.029 | 0.038

Table 7: We show the unfolded bin contents for the absolute parton-level
differential cross-section measurement. The correlations are shown in Fig. 6.

$pp\rightarrow t\bar{t}\gamma$ | Measured ($\text{fb}\cdot\text{GeV}^{-1}$) | SM ($\text{fb}\cdot\text{GeV}^{-1}$) | LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$) | HL-LHC Unc. ($\text{fb}\cdot\text{GeV}^{-1}$)
---|---|---|---|---
theo. | exp. | theo. | exp.
stat. | sys. | mod. | tot. | stat. | sys. | mod. | tot.
$p_{T}^{\gamma}:$ (20-25) | 1.782 | 1.670 | 0.066 | 0.116 | 0.168 | 0.108 | 0.231 | 0.033 | 0.025 | 0.036 | 0.054 | 0.070
$p_{T}^{\gamma}:$ (25-30) | 1.328 | 1.183 | 0.040 | 0.089 | 0.052 | 0.092 | 0.138 | 0.020 | 0.019 | 0.011 | 0.046 | 0.051
$p_{T}^{\gamma}:$ (30-35) | 0.966 | 0.8663 | 0.0302 | 0.072 | 0.026 | 0.060 | 0.097 | 0.0151 | 0.016 | 0.0056 | 0.030 | 0.0342
$p_{T}^{\gamma}:$ (35-40) | 0.705 | 0.6616 | 0.0205 | 0.058 | 0.015 | 0.042 | 0.0733 | 0.0103 | 0.0125 | 0.0032 | 0.021 | 0.0248
$p_{T}^{\gamma}:$ (40-47) | 0.474 | 0.4790 | 0.0160 | 0.04 | 0.0096 | 0.048 | 0.0629 | 0.0080 | 0.0086 | 0.0021 | 0.024 | 0.0254
$p_{T}^{\gamma}:$ (47-55) | 0.333 | 0.3464 | 0.0094 | 0.031 | 0.0067 | 0.017 | 0.0360 | 0.0047 | 0.0067 | 0.0014 | 0.0085 | 0.0109
$p_{T}^{\gamma}:$ (55-70) | 0.221 | 0.2188 | 0.0056 | 0.019 | 0.0038 | 0.0081 | 0.0210 | 0.0028 | 0.0041 | 0.00082 | 0.0041 | 0.0058
$p_{T}^{\gamma}:$ (70-85) | 0.122 | 0.1286 | 0.0031 | 0.014 | 0.0026 | 0.0069 | 0.0158 | 0.0016 | 0.0030 | 0.00056 | 0.0035 | 0.0046
$p_{T}^{\gamma}:$ (85-132) | 0.060 | 0.06037 | 0.0017 | 0.005 | 0.0014 | 0.0068 | 0.0086 | 0.00084 | 0.0011 | 0.00029 | 0.0034 | 0.0036
$p_{T}^{\gamma}:$ (132-180) | 0.020 | 0.02373 | 0.00077 | 0.003 | 0.00044 | 0.00080 | 0.00314 | 0.00039 | 0.00065 | 0.000095 | 0.00040 | 0.00077
$p_{T}^{\gamma}:$ (180-300) | 0.009 | 0.00790 | 0.00028 | 0.00045 | 0.000085 | 0.0014 | 0.00144 | 0.00014 | 0.000097 | 0.000018 | 0.00068 | 0.00069

Table 8: We show the unfolded bin contents for the absolute parton-level
differential cross-section measurement. The correlations are shown in Fig. 7.

Process | Measured (fb) | SM (fb) | LHC Unc. (fb) | HL-LHC Unc. (fb)
---|---|---|---|---
theo. | exp. | theo. | exp.
stat. | sys. | mod. | tot. | stat. | sys. | mod. | tot.
$pp\rightarrow t\bar{t}H+tHq$ | 640 | 664.3 | 41.7 | 90 | 40 | 70.7 | 121.2 | 20.9 | 19.4 | 8.6 | 35.4 | 41.3
$pp\rightarrow t\bar{t}Z$ | 990 | 810.9 | 85.8 | 51.5 | 48.9 | 67.3 | 97.8 | 42.9 | 11.1 | 10.6 | 33.6 | 37.0
$pp\rightarrow t\bar{t}\gamma$ | 39.6 | 38.5 | 1.76 | 0.8 | 1.25 | 2.16 | 2.62 | 0.88 | 0.17 | 0.27 | 1.08 | 1.13
$pp\rightarrow tZq$ | 111 | 102 | 3.5 | 13.0 | 6.1 | 6.2 | 15.7 | 1.75 | 2.09 | 0.98 | 3.1 | 3.87
$pp\rightarrow t\gamma q$ | 115.7 | 81 | 4 | 17.1 | 21.1 | 21.1 | 34.4 | 2 | 1.9 | 2.3 | 10.6 | 11.0
$pp\rightarrow t\bar{t}W+\text{EW}$ | 770 | 647.5 | 76.1 | 120 | 59.6 | 73.0 | 152.6 | 38.1 | 13.1 | 6.5 | 36.5 | 39.4
$pp\rightarrow t\bar{b}$ (s-ch) | 4900 | 5610 | 220 | 784 | 936 | 790 | 1454 | 110 | 35 | 42 | 395 | 399
$pp\rightarrow tW$ | 23100 | 22370 | 1570 | 1086 | 2000 | 2773 | 3587 | 785 | 49 | 89 | 1386 | 1390
$pp\rightarrow tq$ (t-ch) | 87700 | 84200 | 250 | 1140 | 3128 | 4766 | 5810 | 125 | 51 | 140 | 2383 | 2390
$F_{0}$ | 0.693 | 0.687 | 0.005 | 0.009 | 0.006 | 0.009 | 0.014 | 0.003 | 0.0004 | 0.0003 | 0.004 | 0.004
$F_{L}$ | 0.315 | 0.311 | 0.005 | 0.006 | 0.003 | 0.008 | 0.011 | 0.003 | 0.0003 | 0.0002 | 0.004 | 0.004

Table 9: The data shown is the inclusive cross-section written in fb for all
the channels except for the $W$ Helicities ($F_{0}$ and $F_{L}$). Figure 4:
Correlation matrix included for the differential cross section of $pp\to
t\bar{t}$ extracted from Tumasyan _et al._ (2021). Cells are filled if the
correlation is higher than 10% in absolute value.  Figure 5: Correlation
matrix included for the differential measurements of the charged asymmetry of
$pp\to t\bar{t}$ extracted from ATLAS Collaboration . Cells are filled if the
correlation is higher than 10% in absolute value. Figure 6: Correlation matrix
included for the differential measurements of the charged asymmetry of $pp\to
t\bar{t}Z$ extracted from Aad _et al._ (2021). Cells are filled if the
correlation is higher than 10% in absolute value.  Figure 7: Correlation
matrix included for the differential measurements of the charged asymmetry of
$pp\to t\bar{t}\gamma$ extracted from ATLAS Collaboration (2020b). Cells are
filled if the correlation is higher than 10% in absolute value.

### Future $e^{+}e^{-}$ colliders inputs: $e^{+}e^{-}\rightarrow b\bar{b}$

Machine | Polarisation | Energy | Luminosity | Observable | Value
---|---|---|---|---|---
ILC | P($e^{+}$, $e^{-}$):$(-30\%,\,+80\%)$ | 250 GeV | 2 ab-1 | $\sigma_{b}$ | $3182\pm 4.930\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.6267\pm 0.000738$
500 GeV | 4 ab-1 | $\sigma_{b}$ | $693.5\pm 1.188\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.6194\pm 0.001140$
1 TeV | 8 ab-1 | $\sigma_{b}$ | $168.10\pm 0.311\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.617\pm 0.001342$
P($e^{+}$, $e^{-}$):$(+30\%,\,-80\%)$ | 250 GeV | 2 ab-1 | $\sigma_{b}$ | $960.6\pm 1.774\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.426\pm 0.00201$
500 GeV | 4 ab-1 | $\sigma_{b}$ | $203.9\pm 0.442\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.493\pm 0.00268$
1 TeV | 8 ab-1 | $\sigma_{b}$ | $49.21\pm 0.145$
$A_{\text{FB}}^{b}$ | $0.507\pm 0.00335$
CLIC | P($e^{+}$, $e^{-}$):$(0\%,\,+80\%)$ | 380 GeV | 1 ab-1 | $\sigma_{b}$ | $964.6\pm 2.03\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.680\pm 0.002$
1.5 TeV | 2.5 ab-1 | $\sigma_{b}$ | $57.8\pm 0.311\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.674\pm 0.005$
3 TeV | 5 ab-1 | $\sigma_{b}$ | $14.41\pm 0.0923\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.673\pm 0.008$
P($e^{+}$, $e^{-}$):$(0\%,\,-80\%)$ | 380 GeV | 1 ab-1 | $\sigma_{b}$ | $329.8\pm 1.01\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.479\pm 0.004$
1.5 TeV | 2.5 ab-1 | $\sigma_{b}$ | $19.61\pm 0.153\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.520\pm 0.011$
3 TeV | 5 ab-1 | $\sigma_{b}$ | $4.88\pm 0.0505$
$A_{\text{FB}}^{b}$ | $0.522\pm 0.015$
FCC | Unpolaried | Z-pole | 150 ab-1 | $\sigma_{b}$ | $8340800\pm 2449\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.23365\pm 0.000130$
240 GeV | 5 ab-1 | $\sigma_{b}$ | $1670\pm 2.51\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.584\pm 0.000679$
365 GeV | 1.5 ab-1 | $\sigma_{b}$ | $647.2\pm 1.341\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.591\pm 0.0016$
CEPC | Unpolaried | Z-pole | 57.5 ab-1 | $\sigma_{b}$ | $8340800\pm 1947\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.23365\pm 0.000130$
240 GeV | 20 ab-1 | $\sigma_{b}$ | $1670\pm 2.40\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.584\pm 0.000339$
365 GeV | 1 ab-1 | $\sigma_{b}$ | $647.2\pm 1.51\text{ fb}$
$A_{\text{FB}}^{b}$ | $0.591\pm 0.00200$

Table 10: Inputs included to restrict the bottom quark sector

### Future $e^{+}e^{-}$ colliders inputs: $e^{+}e^{-}\rightarrow t\bar{t}$

ILC500 |  Uncertainty Individual |  Uncertainty Global | Correlation
---|---|---|---
$C_{\varphi Q}^{-}$ | $C_{\varphi t}$ | $C_{tW}$ | $C_{tZ}$
$C_{\varphi Q}^{-}$ | 0.032 | $0.11$ | 1.0000 | 0.8221 | 0.9412 | 0.9381
$C_{\varphi t}$ | 0.032 | $0.11$ | 0.8221 | 1.0000 | 0.9379 | 0.9409
$C_{tW}$ | 0.0007 | $0.014$ | 0.9411 | 0.9379 | 1.0000 | 0.9986
$C_{tZ}$ | 0.0008 | $0.016$ | 0.9381 | 0.9409 | 0.9986 | 1.0000

Table 11: Inputs included to restrict the top-quark sector in the ILC working
at 500 GeV. These constrains come from an analysis with the optimal
observables from Ref. Durieux _et al._ (2018). The details of the ILC
configuration can be found in Tab. 4.

ILC500+ ILC1000 |  Uncertainty Individual |  Uncertainty Global | Correlation
---|---|---|---
$C_{\varphi Q}^{-}$ | $C_{\varphi t}$ | $C_{tW}$ | $C_{tZ}$ | $C_{lQ}^{-}$ | $C_{eQ}$ | $C_{lt}$ | $C_{et}$
$C_{\varphi Q}^{-}$ | 0.0268 | 0.0759 | 1.000 | 0.306 | 0.734 | 0.720 | -0.397 | 0.255 | 0.315 | -0.227
$C_{\varphi t}$ | 0.0268 | 0.0764 | 0.306 | 1.000 | 0.724 | 0.738 | 0.261 | -0.317 | -0.354 | 0.326
$C_{tW}$ | 0.00057 | 0.0079 | 0.734 | 0.724 | 1.000 | 0.991 | 0.120 | -0.160 | 0.178 | -0.084
$C_{tZ}$ | 0.00065 | 0.0090 | 0.720 | 0.738 | 0.991 | 1.000 | 0.093 | -0.225 | 0.125 | -0.129
$C_{lQ}^{-}$ | 0.00024 | 0.00041 | -0.397 | 0.261 | 0.120 | 0.093 | 1.000 | -0.279 | -0.180 | 0.184
$C_{eQ}$ | 0.00034 | 0.00053 | 0.255 | -0.317 | -0.160 | -0.225 | -0.279 | 1.000 | 0.204 | -0.055
$C_{lt}$ | 0.00029 | 0.00048 | 0.315 | -0.354 | 0.178 | 0.125 | -0.180 | 0.204 | 1.000 | -0.280
$C_{et}$ | 0.00025 | 0.00040 | -0.227 | 0.326 | -0.084 | -0.129 | 0.184 | -0.055 | -0.280 | 1.000

Table 12: Inputs included to restrict the top-quark sector in the ILC working
at 500 GeV and 1000 GeV. These constrains come from an analysis with the
optimal observables from Ref. Durieux _et al._ (2018). The details of the ILC
configuration can be found in Tab. 4.

CLIC |  Uncertainty Individual |  Uncertainty Global | Correlation
---|---|---|---
$C_{\varphi Q}^{-}$ | $C_{\varphi t}$ | $C_{tW}$ | $C_{tZ}$ | $C_{lQ}^{-}$ | $C_{eQ}$ | $C_{lt}$ | $C_{et}$
$C_{\varphi Q}^{-}$ | 0.065 | 0.127 | 1.000 | -0.097 | 0.599 | 0.585 | -0.312 | 0.185 | 0.305 | -0.256
$C_{\varphi t}$ | 0.065 | 0.128 | -0.097 | 1.000 | 0.593 | 0.604 | 0.310 | -0.249 | -0.266 | 0.259
$C_{tW}$ | 0.00125 | 0.0114 | 0.599 | 0.593 | 1.000 | 0.992 | 0.110 | -0.118 | 0.136 | -0.082
$C_{tZ}$ | 0.00144 | 0.0130 | 0.585 | 0.604 | 0.992 | 1.000 | 0.098 | -0.148 | 0.112 | -0.104
$C_{lQ}^{-}$ | 0.00012 | 0.00015 | -0.312 | 0.310 | 0.110 | 0.098 | 1.000 | -0.114 | -0.233 | -0.026
$C_{eQ}$ | 0.00019 | 0.00021 | 0.185 | -0.249 | -0.118 | -0.148 | -0.114 | 1.000 | -0.066 | -0.162
$C_{lt}$ | 0.00015 | 0.00018 | 0.305 | -0.266 | 0.136 | 0.112 | -0.233 | -0.066 | 1.000 | -0.131
$C_{et}$ | 0.00013 | 0.00015 | -0.256 | 0.259 | -0.082 | -0.104 | -0.026 | -0.162 | -0.131 | 1.000

Table 13: Inputs included to restrict the top-quark sector in CLIC. These
constrains come from an analysis with the optimal observables from Ref.
Durieux _et al._ (2018). The details of the CLIC configuration can be found in
Tab. 4.

FCC$-ee$ |  Uncertainty Individual |  Uncertainty Global | Correlation
---|---|---|---
$C_{\varphi Q}^{-}$ | $C_{\varphi t}$ | $C_{tW}$ | $C_{tZ}$ | $C_{lQ}^{-}$ | $C_{eQ}$ | $C_{lt}$ | $C_{et}$
$C_{\varphi Q}^{-}$ | 0.143 | 10.91 | 1.000 | -0.678 | -0.045 | -0.213 | -0.996 | 0.979 | 0.703 | -0.595
$C_{\varphi t}$ | 0.147 | 10.91 | -0.678 | 1.000 | -0.014 | -0.130 | 0.701 | -0.600 | -0.996 | 0.979
$C_{tW}$ | 0.00145 | 0.0418 | -0.045 | -0.014 | 1.000 | 0.610 | 0.051 | -0.116 | 0.017 | -0.090
$C_{tZ}$ | 0.00180 | 0.117 | -0.213 | -0.130 | 0.610 | 1.000 | 0.149 | -0.393 | 0.064 | -0.319
$C_{lQ}^{-}$ | 0.00293 | 0.381 | -0.996 | 0.701 | 0.051 | 0.149 | 1.000 | -0.959 | -0.721 | 0.632
$C_{eQ}$ | 0.00454 | 0.365 | 0.979 | -0.600 | -0.116 | -0.393 | -0.959 | 1.000 | 0.639 | -0.486
$C_{lt}$ | 0.00300 | 0.384 | 0.703 | -0.996 | 0.017 | 0.064 | -0.721 | 0.639 | 1.000 | -0.958
$C_{et}$ | 0.00427 | 0.358 | -0.595 | 0.979 | -0.090 | -0.319 | 0.632 | -0.486 | -0.958 | 1.000

Table 14: Inputs included to restrict the top-quark sector in the FCC$-ee$.
These constrains come from an analysis with the optimal observables from Ref.
Durieux _et al._ (2018). The details of the FCC$-ee$ configuration can be
found in Tab. 4.

CEPC |  Uncertainty Individual |  Uncertainty Global | Correlation
---|---|---|---
$C_{\varphi Q}^{-}$ | $C_{\varphi t}$ | $C_{tW}$ | $C_{tZ}$ | $C_{lQ}^{-}$ | $C_{eQ}$ | $C_{lt}$ | $C_{et}$
$C_{\varphi Q}^{-}$ | 0.172 | 11.55 | 1.000 | -0.683 | -0.044 | -0.230 | -0.995 | 0.975 | 0.714 | -0.584
$C_{\varphi t}$ | 0.178 | 11.50 | -0.683 | 1.000 | -0.012 | -0.141 | 0.712 | -0.591 | -0.995 | 0.975
$C_{tW}$ | 0.00174 | 0.0501 | -0.044 | -0.012 | 1.000 | 0.608 | 0.052 | -0.118 | 0.019 | -0.093
$C_{tZ}$ | 0.00217 | 0.137 | -0.230 | -0.141 | 0.608 | 1.000 | 0.163 | -0.423 | 0.070 | -0.345
$C_{lQ}^{-}$ | 0.00353 | 0.403 | -0.995 | 0.7120 | 0.052 | 0.163 | 1.000 | -0.951 | -0.735 | 0.628
$C_{eQ}$ | 0.00548 | 0.392 | 0.975 | -0.591 | -0.118 | -0.423 | -0.951 | 1.000 | 0.636 | -0.456
$C_{lt}$ | 0.00363 | 0.405 | 0.714 | -0.995 | 0.019 | 0.070 | -0.735 | 0.636 | 1.000 | -0.945
$C_{et}$ | 0.00515 | 0.382 | -0.584 | 0.975 | -0.093 | -0.345 | 0.628 | -0.456 | -0.950 | 1.000

Table 15: Inputs included to restrict the top-quark sector in CEPC. These
constrains come from an analysis with the optimal observables from Ref.
Durieux _et al._ (2018). The details of the CEPC configuration can be found in
Tab. 4.

##  Appendix B Appendix: Correlation matrices

In the following we show the correlation matrices obtained for the different
scenarios that we have considered.

Figure 8: Correlation matrix obtained for the global fit including the data of
the LHC, Tevatron and LEP.  Figure 9: Correlation matrix obtained for the
global fit including the data of the HL-LHC, Tevatron and LEP.  Figure 10:
Correlation matrix obtained for the global fit including the data of the HL-
LHC, Tevatron, LEP and ILC working at 250 GeV.  Figure 11: Correlation matrix
obtained for the global fit including the data of the HL-LHC, Tevatron, LEP
and ILC working at 250 GeV and 500 GeV.  Figure 12: Correlation matrix
obtained for the global fit including the data of the HL-LHC, Tevatron, LEP
and ILC working at 250 GeV, 500 GeV and 1000 GeV.  Figure 13: Correlation
matrix obtained for the global fit including the data of the HL-LHC, Tevatron,
LEP and the final stage of CLIC.  Figure 14: Correlation matrix obtained for
the global fit including the data of the HL-LHC, Tevatron, LEP and the final
stage of CEPC.  Figure 15: Correlation matrix obtained for the global fit
including the data of the HL-LHC, Tevatron, LEP and the final stage of FCC.

##  Appendix C Appendix: Binning for the differential measurements

The binning of the cross section measurement is based on the CMS measurement
of Ref. Tumasyan _et al._ (2021), combining analyses targetting the resolved
and boosted topologies. This analysis is available on HEPDATA under:
https://www.hepdata.net/record/ins1901295

The binning of the charge asymmetry measurement is based on the ATLAS run 2
analysis reported in Ref. ATLAS Collaboration .

In both cases, the $m_{t\bar{t}}$ range is extended with several further bins
to take advantage of the greater reach of the full HL-LHC programme and the
energy-growth of the sensitivity of some operators. The final binning is given
in Table 16. Bins that have changed with respect to Refs. Tumasyan _et al._
(2021); ATLAS Collaboration , and additional bins, are indicated with an
asterisk (*).

observable | binning
---|---
$\sigma$ vs. $m_{t\bar{t}}$ [GeV] | bin 1 | bin 2 | bin 3 | bin 4 | bin 5 | bin 6
| 250-400 | 400-480 | 480-560 | 560-640 | 640-720 | 720-800
| bin 7 | bin 8 | bin 9 | bin 10 | bin 11 | bin 12
| 800-900 | 900-1000 | 1000-1150 | 1150-1300 | 1300-1500 | 1500-1700
| bin 13 | bin 14 | bin 15 | bin 16 | bin 17 | bin 18
| 1700-2000 | 2000-2300 | 2300-2600∗ | 2600-3000∗ | 3000-3500∗ | 3500-4000∗
$A_{C}$ vs. $m_{t\bar{t}}$ [GeV] | bin 1 | bin 2 | bin 3 | bin 4 | bin 5 | bin 6
| 500-750 | 750-1000 | 1000-1500 | 1500-2000∗ | 2000-2500∗ | 2500-3000∗

Table 16: The binning for the cross section and charge asymmetry differential
measurements in $pp\rightarrow{t\bar{t}}$. Bins that differ from those used in
the run 2 analyses are indicated with an asterisk (*).

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