The Future of US Particle Physics

The Energy Frontier Report

2021 US Community Study
on the Future of Particle Physics

organized by
the APS Division of Particles and Fields

In memory of Meenakshi Narain,
a friend and a colleague, who passed away on January 1, 2023.
This report will be one of her enduring legacies,
and a result of her guidance, leadership and vision.
We will miss her.

###### Contents

  1. 0 Energy Frontier
    1. 1 Executive Summary
    2. 2 Big Questions in the Energy Frontier
    3. 3 Addressing the Big Questions with EF Colliders
    4. 4 Electroweak Sector of the Standard Model
      1. 1 Higgs and BSM physics
        1. Higgs present and future
        2. What can we learn about BSM physics from Higgs physics
      2. 2 Heavy-flavor and top-quark physics
        1. Top-quark mass
        2. Top-quark production processes
        3. Top-quark coupling measurements and EFT fits
        4. BSM physics from top-quark physics
        5. Heavy-flavor and top-quark physics summary
      3. 3 Electroweak precision physics and new physics constraints
      4. 4 EFT and new physics
        1. Multi-boson processes
        2. SMEFT global fits
    5. 5 QCD and Strong Interactions
      1. 1 Perturbative QCD
      2. 2 Non-perturbative QCD
      3. 3 Forward Physics
      4. 4 Heavy Ions
      5. 5 Cross-Cutting QCD
    6. 6 The physics beyond the Standard Model
      1. 1 Composite Higgs Models
      2. 2 SUSY Models
      3. 3 New Bosons, Heavy Resonances, and New Fermions
      4. 4 Long Lived Particles
      5. 5 Dark Matter
        1. Testing the traditional WIMP paradigm
        2. Beyond WIMP dark matter
        3. Dark matter complementarity
    7. 7 The Energy Frontier vision
      1. 1 The immediate-future Energy Frontier collider
      2. 2 The intermediate-future Energy Frontier collider
      3. 3 The long-term-future Energy Frontier collider
      4. 4 Opportunity for US as a site for a future Energy Frontier Collider
    8. 8 Acknowledgments
  2. 1 EF Appendix
    1. 1.A Detectors, reconstruction, simulation and data analysis
    2. 1.B Enabling the Energy Frontier research
    3. 1.C Timelines and Costs of Proposed Future Colliders

##  Chapter 0 Energy Frontier

Frontier Conveners: Meenakshi Narain1, Laura Reina2, Alessandro Tricoli3
Topical Group Conveners: Michael Begel3, Alberto Belloni4, Tulika Bose5,
Antonio Boveia6, Sally Dawson3, Caterina Doglioni7, Ayres Freitas8, James
Hirschauer9, Stefan Hoeche9, Yen-Jie Lee10, Huey-Wen Lin11, Elliot Lipeles12,
Zhen Liu13, Patrick Meade14, Swagato Mukherjee3, Pavel Nadolsky15, Isobel
Ojalvo16, Simone Pagan Griso17, Christophe Royon18, Michael Schmitt19,
Reinhard Schwienhorst11, Nausheen Shah20, Junping Tian21, Caterina Vernieri22,
Doreen Wackeroth23, Lian-Tao Wang24
Liaisons: Antonio Boveia6, Dmitri Denisov3, Caterina Doglioni7, Sergey
Gleyzer25, Meenakshi Narain1, Peter Onyisi26, Laura Reina2, Manuel Franco
Sevilla4, Maksym Titov27, Caterina Vernieri22, Daniel Whiteson28

1 Physics Department, Brown University, Providence, RI 02912, USA
2 Physics Department, Florida State University, Tallahassee, FL 32306-4350,
USA
3 Department of Physics, Brookhaven National Laboratory, Upton, NY, 11973,
U.S.A.
4 Department of Physics, University of Maryland, College Park, MD 20742-4111,
USA
5 Department of Physics, University of Wisconsin, 1150 University Avenue,
Madison, WI 53706-1390, USA
6 Department of Physics and Center for Cosmology & Astroparticle Physics, The
Ohio State University, Columbus, OH 43210, USA
7 School of Physics & Astronomy, University of Manchester, Manchester, United
Kingdom, and Fysiska institutionen, Lund University, Lund, Sweden
8 Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA
15260, USA
9 Fermilab, P.O. Box 500, Batavia, IL 60510, USA
10 The Massachusetts Institute of Technology, Department of Physics,
Cambridge, MA 02139, USA
11 Department of Physics & Astronomy, Michigan State University, East Lansing,
MI 48824, USA
12 Department of Physics & Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
13 School of Physics & Astronomy, University of Minnesota, Minneapolis, MN
55455, USA
14 C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony
Brook, NY 11794, USA
15 Department of Physics, Southern Methodist University, Dallas, TX 75275, USA
16 Department of Physics, Princeton University, Princeton, NJ 08554, U.S.A.
17 Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA
18 Department of Physics & Astronomy, The University of Kansas, Lawrence, KS
66045, USA
19 Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA
20 Department of Physics & Astronomy, Wayne State University, Detroit, MI
48202, USA
21 ICEPP, University of Tokyo, Japan
22 SLAC National Accelerator Laboratory, Stanford, CA 94309, USA
23 Department of Physics, University at Buffalo, Buffalo, NY 14260, USA
24 Physics Department, Enrico Fermi Institute and Kavli Institute for
Cosmological Physics, University of Chicago, IL 60637, USA
25 Department of Physics & Astronomy, University of Alabama, Tuscaloosa, AL
35487-0324, USA
26 Department of Physics, University of Texas, Austin, TX 78712, USA
27 IRFU, CEA, Université Paris-Saclay, France
28 Department of Physics & Astronomy, University of California, Irvine, CA
92697-4575, USA

###  1 Executive Summary

The Energy Frontier (EF) aims at investigating the fundamental physics of the
Universe at the highest energies or – equivalently – the shortest timescales
after the Big Bang. We investigate open questions and explore the unknown
using various probes to discover and characterize the nature of new physics,
through the breadth and multitude of collider physics signatures. While the
naturalness principle suggests new physics to lie at mass scales close to the
electroweak scale, in many cases direct searches for specific models have
placed strong bounds around 1-2 TeV. Thus, the energy frontier has moved
beyond the TeV scale and the exploration of the 10 TeV scale becomes crucial
to shed light on physics beyond the Standard Model (SM).

We need to use both energy reach and precision measurements to push beyond the
1 TeV scale in our exploration. The quest for new physics will be thus
conducted in a two-tier approach: 1) looking for indirect evidence of beyond-
the-Standard-Model physics (BSM) through precision measurements of the
properties of the Higgs boson and other SM particles, and 2) searching for
direct evidence of BSM physics at the energy frontier, reaching multi-TeV
scales.

The EF currently has a top-notch program with the LHC and the High Luminosity
LHC (HL-LHC) at CERN, which sets the basis for the EF vision. The EF supports
continued strong US participation in the success of the LHC, and the HL-LHC
construction, operations, computing and software, and most importantly in the
physics research programs, including auxiliary experiments.

The discussions on projects that extend the reach of the HL-LHC underlined
that preparations for the next collider experiments have to start now to
maintain and strengthen the vitality and motivation of the community.
Colliders are the ultimate tool to carry out such a program thanks to the
broad and complementary set of measurements and searches they enable. Several
projects have been proposed such as ILC, CLIC, FCC-ee, CEPC, Cool Copper
Collider (C3) or HELEN for e+e- Higgs Factories, and CLIC at 3 TeV centre-of-
mass energy, FCC-hh, SPPC and Muon Collider for multi-TeV colliders. For a
detailed discussion of timeline, cost, challenges of those accelerator
projects we refer to the Accelerator Frontier Integration Task-Force (ITF)
report [1] and Appendix 1.C. Dedicated fora were established across frontiers
to bring together diverse expertise in the study of future $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ colliders. Results from their studies are available in their
reports [2, 3] and have informed the studies presented in this report.

All proposed collider physics experiments need complex detectors as well as
software and computing infrastructure, with cutting-edge technologies to meet
their ambitious physics goals. The needs extend beyond generic R&D. Experience
from R&D and building previous experiments informs us that it takes about 10
years from CD-0 (Critical Decision - 0) to the end of construction of a
collider detector. For e+e- Higgs factories, immediate investment in targeted
detector R&D is needed at a high priority to explore innovative new
technologies and address specific detector challenges in preparation for
technical designs. The schedule for multi-TeV colliders and their detectors is
technically limited. Therefore for multi-TeV colliders, a long-term program to
study the accelerator and detector challenges and reduce overall costs for
construction is needed.

In summary, the EF supports a fast start for construction of an e+e- Higgs
factory (linear or circular), and a significant R&D program for multi-TeV
colliders (hadron and muon). The realization of a Higgs factory will require
an immediate, vigorous and targeted detector R&D program, while the study
towards multi-TeV colliders will need significant and long-term investments in
a broad spectrum of R&D programs for accelerators and detectors. These
projects have the potential to be transformative as they will push the
boundaries of our knowledge by testing the limits of the SM, and indirectly or
directly discovering new physics beyond the SM.

The US EF community has also expressed renewed interest and ambition to bring
back energy-frontier collider physics to the US soil while maintaining its
international collaborative partnerships and obligations.

The EF community proposes several parallel investigations over a time period
of ten years or more for pursuing its most prominent scientific goals, namely
1) supporting the full (3 - 4.5 ab-1) HL-LHC physics program, 2) proceeding
with a Higgs factory, and 3) planning for multi-TeV colliders at the energy
frontier.

The proposed plans in five year periods starting 2025 are given below.

For the five year period starting in 2025:

  1. 1.

Prioritize the HL-LHC physics program, including auxiliary experiments,

  2. 2.

Establish a targeted $e^{+}e^{-}$ Higgs factory detector R&D program,

  3. 3.

Develop an initial design for a first stage TeV-scale Muon Collider in the US,

  4. 4.

Support critical detector R&D towards EF multi-TeV colliders.

For the five year period starting in 2030:

  1. 1.

Continue strong support for the HL-LHC physics program,

  2. 2.

Support construction of an $e^{+}e^{-}$ Higgs factory,

  3. 3.

Demonstrate principal risk mitigation for a first stage TeV-scale Muon
Collider.

Plan after 2035:

  1. 1.

Continuing support of the HL-LHC physics program to the conclusion of archival
measurements,

  2. 2.

Support completing construction and establishing the physics program of the
Higgs factory,

  3. 3.

Demonstrate readiness to construct a first-stage TeV-scale Muon Collider,

  4. 4.

Ramp up funding support for detector R&D for energy frontier multi-TeV
colliders.

The EF community recognizes that its success critically depends on the
resources obtained by the Accelerator Frontier (AF), as there is a direct
linkage between the EF vision and advances in accelerator R&D. The EF
community strongly supports the AF in its proposal to establish an
$e^{+}e^{-}$ Higgs factory program, and start R&D for energy frontier multi-
TeV colliders with appropriate funding [4]. Moreover, the visibly strong
interdependence between the EF and the Theory Frontier is key to the success
of both frontiers, and EF supports a strong and well funded theory program
[5]. Contributions from Instrumentation Frontier [6] and the Computational
Frontiers [7] are key to the realization of the vision of the EF. In addition,
the collaboration with the Community Engagement Frontier [8] as well as the
cross-fertilization with other Frontiers such as the Rare Processes and
Precision Measurements [9], Cosmic [10], and Neutrino Physics [11] Frontiers
are important. Finally, to achieve its goal to advance the current and future
scientific endeavors, the EF community has a renewed focus on building and
advancing an inclusive workforce, with members from diverse backgrounds,
expertise, and career levels.

The vision of the EF community is presented more extensively in Sec. 7, and is
motivated by the physics studies that are summarized in Secs. 2 to 6. The
Appendices in Sec. 1 discuss the technological developments that will enable
the next generations of EF experiments, as well as opportunities and
challenges for the EF community, in terms of collaborations, vibrancy, and
diversity as well as needed investments. The estimated timelines and cost for
the next generation of large-scale projects in the EF are also provided.

###  2 Big Questions in the Energy Frontier

After decades of pioneering explorations and milestone discoveries, the
Standard Model (SM) of particle physics has been confirmed as the theory that
describes electroweak (EW) and strong interactions up to energies of a few
hundreds of GeV with great accuracy but it is also leaving some fundamental
questions unexplained. In this context, the EF aims at advancing the
investigation of some of the still open fundamental questions such as the
evolution of the early universe, the matter-antimatter asymmetry of the
universe, the nature of dark matter, the origin of neutrino masses, the origin
of the EW scale, and the origin of flavor dynamics, with a broad and strongly
motivated physics program that will push the exploration of particle physics
to the TeV energy scale and beyond. Our sharply focused agenda includes in-
depth studies of the SM as well as the exploration of physics beyond the SM to
discover new particles and interactions. The vision of the EF must keep its
focus on these big questions, and must provide opportunities to examine them
from as many angles as possible, while also continuing to pursue the
exploration of the unknown, a leading driver of the EF physics program. This
is the core of the EF program as pictorially illustrated in Fig. 1.

Collider Physics offers a unique opportunity to study a huge number of
phenomena and explore the connections between many of the fundamental
questions we want to answer. The big questions outlined at the center of Fig.
1 unambiguously require new concepts beyond the SM of particle physics. A
variety of processes can be used as probes (depicted as the smaller circles of
Fig. 1) to discover and then characterize the nature of the BSM physics at
play and energy-frontier colliders are instrumental to investigating such
phenomena via numerous distinct signatures (listed as the outermost text in
Fig. 1).

The multi-probe characteristic of high-energy colliders makes them a unique
tool in the fundamental quest for answers to the core unknowns of particle
physics. With a combined strategy of precision measurements and high-energy
exploration, future lepton colliders starting at energies as low as a few
hundreds GeV up to a few TeV can shed substantial light on some of these key
questions. Ultimately, it will be crucial to find a way to carry out
experiments at higher energies, directly probing new physics at the 10 TeV
energy scale and beyond.

Figure 1: Six categories of Probes and a multiple of Signatures accessible at
energy-frontier colliders to address the Big Questions that are at the center
of the EF pursuit.

Within this big picture, the activities of the Snowmass 2021 EF has been
structured around three main probing areas, broadly defined as EW Physics
(Higgs-boson physics, top-quark and heavy-flavor physics, electroweak gauge
bosons physics), Quantum Chromodynamics (QCD) and Strong Interactions
(precision QCD, hadronic structure and forward QCD, heavy ions), and BSM
physics (model-specific explorations, general explorations, dark matter at
colliders), which have focused on three main key questions, as presented
below.

  1. 1.

What can we learn about the origin of the EW scale and the EW phase transition
from an in-depth study of the the properties and interactions of SM particles?

     * The 10${}^{\mbox{th}}$ anniversary of the discovery of the Higgs boson [12, 13], a tremendous achievement for our field, is being celebrated this year, 2022. This discovery provides the last piece of the SM puzzle and at the same time gives a unique connection to new physics beyond the SM that we need to exploit. The multitude of studies relating to precision Higgs-boson measurements (mass, width, couplings) may help uncover the nature of physics above the EW scale. Together with the full spectrum of EW, top-quark, and flavor-physics precision measurements, these studies will greatly improve the constraining power of global fits of fundamental parameters. The expected large number of Higgs-boson events may lead to measurements of the shape of the Higgs potential. The study of the Higgs boson, may also give us insight into flavor physics and vice versa. Last but not least, this study may lead us to understand the implications for naturalness.

  2. 2.

What can we learn about the dynamics of strong interactions in different
regimes ?

     * The fundamental theory of strong interactions, QCD, plays a central role in the SM both as a confining gauge theory that can inspire a deeper understanding of quantum field theories, and at the same time as a ubiquitous crucial component of all measurements at current and future hadron and lepton colliders. Answering the big questions ahead of us intrinsically depends on exploring and understanding the complex phenomenology of strong interactions in different regimes. To harvest the revolutionary progress witnessed in perturbative QCD calculations and Monte Carlo event generators during recent years, residual systematic uncertainties, from parton-shower effects to hadronization models, will have to be understood and reduced. New ideas to describe the dynamics that drives internal jet structure and their evolution as a function of energy will have to be explored. First-principle calculation of parton densities will have to be benchmarked. The upcoming era, featuring the HL-LHC, Belle II, the Electron-Ion Collider (EIC), new advances in theory including in lattice QCD, and potentially a Higgs factory promises to be a golden age for QCD.

  3. 3.

How can we build a complete program of BSM searches which includes both model-
specific and model-independent explorations?

     * Models connect the high-level unanswered questions in particle physics (dark matter, EW naturalness, CP violation, etc) to specific phenomena, in a self-consistent way. They can be very predictive but model-dependent studies may fail to consider a broad range of new phenomena and search avenues. Many important questions need to be critically addressed. Which models should be considered and how can model parameter spaces be compared in a consistent way ? Can searches be conducted and interpreted in a model-agnostic way ? How can results from different experiments be compared in a model-independent way to ensure complementarity and avoid gaps in coverage ? Can future colliders chart new regions of model parameter space and also fill in gaps left by existing colliders ? What is the complementarity between precision measurements and direct searches ?

A detailed account of the work done by the EF Topical Groups in each of these
areas is presented in a series of reports [14, 15, 16, 17, 18], highlights of
which are illustrated in Secs. 4-6 of this report.

Finding answers generates more specific questions that will be considered in
the studies presented in this report and will be an important factor in
building a concrete vision for the future of particle physics explorations at
the energy frontier. Among others, important aspects that have emerged as most
relevant in determining future directions for the EF are the following: the
theoretical motivation behind specific future directions, the potential of
each future-collider proposal to provide substantial insights in answering the
key questions identified as the focus of EF as well as the breadth and
complementarity of their physics program, and the identification of what
collider and detector developments are necessary to fully pursue the desired
physics program of both precision measurements and searches for new physics.

The collider scenarios discussed in the following are summarized in Table 2
for Higgs factories and in Table 2 for multi-TeV colliders.

Table 1: Benchmark scenarios for Snowmass 2021 Higgs factory studies. Collider | Type | $\sqrt{s}$ | $\mathcal{P}[\%]$ | $\mathcal{L}_{\rm int}$
---|---|---|---|---
|  |  | $e^{-}/e^{+}$ |  ${\rm ab}^{-1}$ /IP
HL-LHC | pp | 14 TeV |  | 3
ILC & C3 | ee | 250 GeV | $\pm 80/\pm 30$ | 2
|  | 350 GeV | $\pm 80/\pm 30$ | 0.2
|  | 500 GeV | $\pm 80/\pm 30$ | 4
|  | 1 TeV | $\pm 80/\pm 20$ | 8
CLIC | ee | 380 GeV | $\pm 80/0$ | 1
CEPC | ee | $M_{Z}$ |  | 50
|  | 2$M_{W}$ |  | 3
|  | 240 GeV |  | 10
|  | 360 GeV |  | 0.5
FCC-ee | ee | $M_{Z}$ |  | 75
|  | 2$M_{W}$ |  | 5
|  | 240 GeV |  | 2.5
|  | 2 $M_{top}$ |  | 0.8
$\mu$-collider | $\mu\mu$ | 125 GeV |  | 0.02

Table 2: Benchmark scenarios for Snowmass 2021 multi-TeV collider studies. Collider | Type | $\sqrt{s}$ | $\mathcal{P}[\%]$ | $\mathcal{L}_{\rm int}$
---|---|---|---|---
|  | (TeV) | $e^{-}/e^{+}$ |  ${\rm ab}^{-1}$/IP
HE-LHC | pp | 27 |  | 15
FCC-hh | pp | 100 |  | 30
SPPC | pp | 75-125 |  | 10-20
LHeC | ep | 1.3 |  | 1
FCC-eh |  | 3.5 |  | 2
CLIC | ee | 1.5 | $\pm 80/0$ | 2.5
|  | 3.0 | $\pm 80/0$ | 5
$\mu$-collider | $\mu\mu$ | 3 |  | 1
|  | 10 |  | 10

###  3 Addressing the Big Questions with EF Colliders

While the existence of BSM physics is well established by observational
phenomena, and heavily suggested by theoretical considerations, the energy
scale at which it will manifest and its characteristics, e.g. the couplings to
known SM particles, are only indirectly constrained. For example Dark Matter
(DM) may be a thermal weakly-interacting massive particle (WIMP) with possible
extensions to the multi-TeV range, or it might have an extraordinarily low
mass. Naturalness suggests that the mass scale of new physics should be close
to the EW scale, but in concrete scenarios, such as supersymmetry, it can
reach the 10 TeV scale or higher given the measured Higgs-boson mass.
Furthermore it is quite possible that phenomena could show up unexpectedly at
unpredicted scales as we have seen historically in our field.

Depending on the mass scale of new physics and the type of collider, the
primary method for discovering new physics can vary. Investigation at the
energy frontier allows one to combine direct BSM searches with precision
measurements of observables sensitive to scales above the available center-of-
mass energy. Furthermore, the type of collider and detectors that are employed
will directly influence what signatures can be probed.

The fundamental lessons learned from the LHC thus far are that a Higgs-like
particle exists at 125 GeV and there is no other obvious and unambiguous
signal of BSM physics. This implies that either there is generically a gap to
the scale of new physics, or new physics must be weakly coupled to the SM, or
hidden in backgrounds at the LHC. The HL-LHC will either strengthen these
conclusions further or potentially point us in a particular direction for
discovery.

Figure 2: The direct coverage of various colliders in the schematic space of
coupling to the SM versus mass scale of BSM physics. “Higgs factory” and
“multi-TeV colliders” correspond to a generic option among the ones listed in
Table 2 and Table 2 respectively.

To understand how future colliders have complementary potential to unlock the
mysteries around these fundamental questions beyond what the LHC and HL-LHC
physics program can probe, it is illustrative to use a simplified picture as
depicted in Fig. 2. We can imagine that generic new physics lives in a 2D
parameter space governed by the coupling of new states to the SM and their
mass scale. If the center-of-mass energy of a collider is above the one of the
LHC, it can directly search for new states to higher mass scales. Higgs
factories have smaller center-of-mass energies than the LHC, and therefore do
not extend the direct mass reach beyond the LHC. However, by colliding leptons
they offer significantly reduced backgrounds and the ability for triggerless
readout, therefore they have the potential to probe new physics that is weakly
coupled to the SM. Additionally, even in the overlap region of Higgs factories
with the LHC, the former can be sensitive to new physics that is difficult to
discriminate from backgrounds at the LHC.

Beyond the direct search for new physics, a key program for the EF is the
precision measurement of SM predictions and parameters. Highly precise
measurements allow the probing of scales above the kinematic limit for direct
searches at colliders. This can be captured through Effective Field Theory
(EFT) techniques when there is a gap between the probed energy scale and the
scale of new physics. In EF studies, typically this is done by employing
specific EFTs, e.g. SMEFT or the more general HEFT formalisms. If $M$ is the
mass scale of new physics and $g_{BSM}$ is its generic coupling to the SM,
then often deviations in SM parameters, $\eta_{SM}$, which occur from
integrating heavy particles out at tree-level, scale at the leading order as

$\delta\eta_{SM}\sim g_{BSM}^{2}\frac{v^{2}}{M^{2}},$ (1)

for Higgs related parameters, where $v$ is the vacuum expectation value of the
Higgs, or in general as

$\delta\eta_{SM}\sim g_{BSM}^{2}\frac{E^{2}}{M^{2}},$ (2)

where it is assumed that the energy scale $E\ll M$ for the formalism to be
applicable. If new physics only creates loop level deviations in a SM
observable, then one can insert a loop factor $\sim 1/16\pi^{2}$ into Eqns. 1
and 2. Therefore depending on the precision achievable, as seen in Eqns. 1 and
2, mass scales larger than the direct reach can be probed. We can then overlay
these types of indirect collider searches, particularly relevant for Higgs
factories in Table 2, on our schematic space of BSM physics shown in Fig. 2,
as explicitely illustrated in Fig. 3. As can be seen in Fig. 3 the energy
versus precision trade-off crucially depends on the precision attainable.
Suggestively, we have shown a 1% precision often associated with parameter
measurements (except for e.g. the $HZZ$ coupling at Higgs factories), where
the scaling typically does not extend beyond the LHC without invoking strong
coupling. However, for quantities that are measured significantly more
precisely, e.g. $\lesssim 0.1\%$, at future Higgs-factory programs, such as
$M_{W}$, the reach can extend much further. This exact scaling in mass reach
depends on the type of BSM physics, and both Higgs parameters and EW
observables measured at Higgs factories are important for understanding
complementary measurements available at future multi-TeV colliders. The
precision that can ultimately be reached and in what types of observables
strongly motivates advances in detector technology, increases in luminosity,
use of polarization at lepton colliders, and improved theoretical
calculations. Moreover depending on the type of collider, for example at a
multi-TeV collider, the dichotomy between precision reach and energy reach can
potentially be bridged with the availability of large statistics for processes
as e.g. Higgs production if the environment can be fully controlled.

Figure 3: Kinematic reach of direct and indirect searches in the same
coupling-mass plane as in Fig. 2. The solid lines illustrate direct search
limits while the dashed lines represent indirect limits. Higgs Factories (HF)
can provide increased reach through indirect searches, benefiting from high-
precision measurements. A 1% precision measurement suggests a probed scale of
up to a few TeV in perturbative ultra-violet completion models if BSM physics
couples at tree-level to the observables of interest. For illustration the
potential LHC limit is also depicted. If new physics couples at loop-level
then the scale probed indirectly is lower. The ultimate precision, and what
scale the multi-TeV colliders can probe is collider specific.

There can be a multitude of phenomena studied at low masses, incompatible with
the EFT framework at those energies, that benefit from a reduced background
environment at an $e^{+}e^{-}$ Higgs factory. Additionally, even within one
collider, precision measurements and direct searches coexist and offer
multiple complementary probes. To move beyond scaling it is useful to give a
few examples of how direct and indirect complementarity play out within the
EF, as well as what are potential BSM features at low energy that the LHC may
be missing. The remaining of the EF report and the EF Topical Group reports
[14, 15, 16, 17, 18] will provide examples and more details, but we list a few
illustrative ones here below.

  * •

Higgs couplings and mass reach: the Higgs boson as the primary target for all
future EF colliders provides many examples of the interplay of direct and
indirect searches. For example in Two-Higgs Doublet Models, indirect Higgs-
boson precision measurements can be complemented by direct resonance searches,
EW precision, flavor physics and beyond. Numerous cases are discussed in
Section 1 and in the corresponding Topical Group report [14].

  * •

Supersymmetry is still a leading example of BSM physics, and while it is a
canonical example of direct searches it also can be tested indirectly in
numerous ways [18]. However as shown in Section 2 for pMSSM parameter scans,
indirect searches at Higgs factories do not exceed the typical region covered
by the HL-LHC. Further examples can be found in several of the EF Topical
Group reports [14, 18].

  * •

The search for anomalous Trilinear Gauge Couplings (aTGC) and anomalous
Quartic Gauge Couplings (aQGC) offer a particularly interesting example of the
interplay of several search strategies and measurements [16]. At lowest
dimension in the SMEFT expansion, deviations are only possible from BSM
physics at loop level from new EW charged states. Therefore in addition to
indirect multiboson measurements, direct searches for charged particles, which
generate the effects, are powerful probes. These can include long-lived
particle searches in the degenerate mass limit, or more canonical direct
searches that can also be complemented by precision measurements of Higgs-
boson properties when the splitting becomes larger. Furthermore there are
multiple ways to search for vector boson scattering such as in ultraperipheral
heavy-ion collisions, as discussed in Section 5 and in Ref. [17].

  * •

Higgs-to-invisible decays is an example of where there can be “holes” in the
LHC coverage even at low mass scales. A similar consideration can be made for
precision measurements of Higgs-boson decays into light quark flavors which
are difficult to detect at the LHC despite being copiously produced. These are
both further discussed in Section 1 and in the corresponding Topical Group
report [14].

In summary, EF colliders are the ultimate tool we possess to clarify the
fundamental nature of deviations observed either at collider experiments or at
low-energy experiments, thanks to their vast and unique physics programs.

The current landscape also provides notable examples of deviations in
precision measurements of SM parameters that are currently under scrutiny by
the particle physics community, and that can provide targets for direct BSM
searches. Most notably, anomalies in (semi)leptonic B-hadrons decay, including
hints of lepton flavor universality violations, the muon anomalous magnetic
moment ($g_{\mu}-2$), and the recent $W$-mass measurement might develop into
unambiguous BSM signals that call for direct exploration to clarify their
nature. Regardless the nature of the present and future anomalies, such
occurrences call for a flexible EF program that can develop and deploy a
collider able to directly and unambiguously probe energy scales of the order
of several TeV.

###  4 Electroweak Sector of the Standard Model

The investigation of EW interactions has played a crucial role in defining the
SM, predicting all its bosonic and fermionic constituents, and guiding to
their discovery over the last several decades. The discovery of the Higgs
boson has provided the last piece of evidence to establish the SM as a well-
defined quantum field theory and confirm its predictions, but it has also left
unanswered more fundamental questions on the nature of this particle, its
relation to defining the EW scale via the spontaneous breaking of the EW
symmetry, as well as its role in the origin of fermion masses and flavor
dynamics. Hence, focused precision studies of the properties of EW physics
still offer a unique handle to indirectly probe the kind of fundamental new
physics that could explain the origin of the EW scale and relate it to some of
the big open questions we outlined in Sec. 2.

The LHC and HL-LHC will continue exploring EW interactions at the Energy
Frontier with a broad program of measurements of Higgs-boson, EW gauge-boson
($W,Z$), and heavy-fermions ($t,b$, and also $c$ quark) properties and
interactions. Future colliders, both Higgs Factories and multi-TeV colliders,
will push the precision of such measurements to indirectly constrain effects
of new physics in the 10 TeV range or above and, at the highest energies,
provide direct access to such physics. In this section we review the current
status, future projections, and overall reach of EW physics studies
articulated into three sections on Higgs-boson physics (Sec. 1), top-quark and
heavy-flavor physics (Sec. 2), and EW gauge-boson physics (Sec. 3). The
current and future constraining power of global fits of EW observables in the
SM effective field theory is presented in Sec. 4.

####  1 Higgs and BSM physics

Over the past decade the LHC has fundamentally changed the landscape of high
energy particle physics through the discovery of the Higgs boson and the first
measurements of many of its properties. As a result of this, and no other
“discoveries” at the LHC, the questions surrounding the Higgs boson have only
become sharper and more pressing for planning the future of particle physics.

The SM is an extremely successful description of nature, with a basic
structure dictated by symmetry. Still, symmetry alone is not sufficient to
fully describe the microscopic world we explore, and even after specifying the
gauge and space-time symmetries, and number of generations, there are still 19
parameters undetermined by the SM (not including neutrino masses). Out of
these parameters 4 are intrinsic to the gauge theory description, the gauge
couplings and QCD theta angle. The other 15 parameters are intrinsic to the
Higgs sector or how other SM particles couple to the Higgs, illustrating its
paramount importance in the SM. In particular, the masses of all fundamental
particles, their mixing, CP violation, and the basic vacuum structure are not
predicted from symmetry principles of the SM and need to be derived from
experimental data to test the consistency of the SM itself. However, the
centrality of the Higgs boson goes far beyond just dictating the parameters of
the SM.

The Higgs boson is connected to some of our most fundamental questions about
the Universe. Its most basic role in the SM is to provide a source of
Electroweak Symmetry Breaking (EWSB). This effect can be formulated via the
Higgs potential in the following way:

$V(H)=-\mu^{2}(H^{\dagger}H)+\lambda(H^{\dagger}H)^{2}\,.$ (3)

Indeed, if the mass parameter in the potential (the term quadratic in the SM
complex scalar doublet $H$) simply had a positive sign rather than negative,
there would be no EWSB and our universe would not exist in its current form.
The explanations of why EWSB occurs and the presence of such a minus sign are
outside the realm of the SM Higgs boson. In other examples of spontaneous
symmetry breaking that we have seen manifest in our universe described by
Quantum Field Theory (QFT), there has been a dynamical origin. In principle
the Higgs could be a composite of some other strongly coupled dynamics, as we
have seen before in history for other particles. Alternatively, the Higgs
could be a fundamental scalar and EWSB could arise dynamically through its
interactions with other BSM fields. It is even possible that there could be
dynamical connections to cosmology or the anthropic principle. No matter what
the origin of EWSB is, it will leave imprints on the properties of the SM
Higgs boson itself. Moreover, answers to questions such as “Is the Higgs boson
composite or fundamental?” can have ramifications far beyond just the origin
of EWSB.

If the Higgs boson is a fundamental particle, it represents the first
fundamental scalar particle discovered in nature. This has profound
consequences both theoretically and experimentally. From our modern
understanding of QFT, fundamental scalars should not exist in the low energy
spectrum without an UV-sensitive fine tuning if the SM is an EFT of some more
fundamental theory. This is known as the naturalness or hierarchy problem.
From studying properties of the Higgs boson, one can hope to learn whether
there is some larger symmetry principle at work stabilizing the spectrum. For
example supersymmetry, neutral naturalness, or if the correct theory is a
composite Higgs model, the Higgs could be a pseudo-Goldstone boson.

Experimentally there are also a number of intriguing directions that open up
if the Higgs boson is a fundamental particle. The most straightforward
question is whether the Higgs boson is a unique scalar field in our universe,
or is it just the first of many. Additional scalars can always couple to the
Higgs at the renormalizable level, and depending on their symmetry properties
they can couple to gauge bosons or fermions as well (e.g. the more commonly
known Two Higgs Doublet Models). What this implies is that if the Higgs is not
unique, there are two complementary methods for investigating this: searching
directly for these new scalar states, or measuring their indirect effects on
the SM Higgs-boson properties. Also related to the fact that new scalars can
always leave imprints on the Higgs is that a fundamental Higgs particle is
special in QFT. Using only the SM Higgs field, one can construct the lowest-
dimension gauge- and Lorentz-invariant operator in the SM. This means that
generically if there are ‘Hidden” sectors beyond the SM sector (perhaps
related to DM), the Higgs boson is the most relevant portal to these sectors,
often referred to as the “Higgs Portal”. Whether these new sectors have a mass
scale well below or above the scale of EWSB implies drastically different
experimental observables. For light new sectors of the universe, this can
manifests itself as exotic Higgs decays, invisible Higgs decays, and shifts in
the Higgs total width, as can be probed well at $e^{+}e^{-}$ colliders. For
heavier hidden sectors, the observables are different and the highest energy
collider options are needed.

Another aspect of determining if the Higgs is a fundamental scalar particle
concerns whether the minimal Higgs potential is correct. If there are new BSM
particles that couple to the Higgs, the potential itself can receive
modifications. This is not solely a question about the potential, because its
form has repercussions for both our understanding of the early universe and
its ultimate fate. For the early universe, the SM predicts that the
electroweak symmetry should be restored at high temperatures. However,
depending on the actual form of the potential the question remains as to
whether there even was a phase transition let alone its strength.
Additionally, depending on its form, the Higgs potential can control the
future of our universe as our vacuum may be metastable. Furthermore a strong
EW first order phase transition can have implications for Baryogenesis as
well.

Finally, the Higgs boson is connected to some of the most puzzling questions
in the universe: flavor, mass, and CP violation. There are effectively two
types of interactions in the SM, gauge interactions and Higgs interactions.
Gauge interactions are tightly constrained and do not fundamentally
differentiate flavor. Higgs interactions govern all the important quantities
for flavor, mass, and CP violation in the SM. In particular, all problems
connected with flavor and CP – the origin of fermion masses, the origin of
neutrino masses, the origin of the neutrino mixing matrix and CKM angles –
ultimately require knowledge of the fundamental nature of the Higgs sector.
Otherwise, we are just fitting parameters without an understanding. The full
information that we need is only available at high energy by studying the
Higgs boson.

The fact that understanding the properties of the SM Higgs boson connects to
so many fundamental questions illustrates how central it is to the HEP
program. The connections briefly reviewed so far obviously can each be
expanded in greater detail, but to collect the various themes in a simple-to-
digest manner they are illustrated in Fig. 4 and examples of the interplay
between experimental observables and fundamental questions are given in Fig.
5. The generality of the concepts and questions posed in Fig. 4 could even
belie connections to additional fundamental mysteries. For example, the Higgs
portal could specifically connect to DM or other cosmological mysteries.

Figure 4: The Higgs boson as the keystone of the Standard Model is connected
to numerous fundamental questions that can be investigated by studying it in
detail through the many experimental probes illustrated in Fig. 5. Figure 5:
Examples of the interplay between experimental observables and fundamental
questions connected to the Higgs boson.

##### Higgs present and future

The LHC Run 2 with about 140 fb-1 of data analyzed is providing a wealth of
new measurements for the Higgs sector. The most recent Higgs-boson mass
measurements, from CMS and ATLAS, set its value to be 125.38$\pm$0.14 GeV [19]
and 124.92$\pm$0.21 GeV [20] respectively, using both the diphoton and $ZZ$
decay channels. The Higgs-boson mass is a free parameter in the SM and it is
now known to per-mille accuracy. We are entering the era of precision Higgs
physics, with some of the Higgs-boson couplings measurements approaching
$\mathcal{O}$(5-10)% precision. All the major production mechanisms of the SM
Higgs boson ($h$) have been observed at the LHC: gluon fusion (ggF), vector-
boson fusion (VBF), the associated production with a $W$ or $Z$ boson ($Wh$,
$Zh$), and the associated production with top quarks
($\mathrm{t\overline{t}h}$, th). All of these channels are precisely measured,
with the experimental sensitivity of some modes nearing the precision of
state-of-the-art theory predictions. Further details of the current LHC
measurements at ATLAS and CMS are contained within the Higgs-physics Topical
Group report [14].

A simultaneous fit of many individual production rate times branching-fraction
measurements is performed to determine the values of the Higgs-boson coupling
strength. The $\kappa$-framework defines a set of parameters ($\kappa_{X}$ for
$X=W,Z,\ldots$) that affect the Higgs-boson coupling strengths without
altering the shape of any kinematic distributions of a given process. SM
values ($\kappa_{X}\\!=\\!1$) are assumed for the coupling strength modifiers
of first-generation fermions, the other coupling strength modifiers are
treated independently. The results are shown in Fig. 6 for ATLAS and CMS. In
this particular fit, the presence of non-SM particles in the loop-induced
processes is parameterized by introducing additional modifiers for the
effective coupling of the Higgs boson to gluons, photons and $Z\gamma$,
instead of propagating modifications of the SM-particle couplings through the
loop calculations. In these results, it is also assumed that any potential
effect beyond the SM does not substantially affect the kinematic properties of
the Higgs boson decay products. The coupling modifiers are probed at a level
of uncertainty of 10%, except for $\kappa_{b}$ and $\kappa_{\mu}$ ($\approx
20\%$), and $\kappa_{Z\gamma}$ ($\approx 40\%$). Notably absent are couplings
to light fermions, many of whose SM values are out of reach for a LHC
measurement (i.e the up, down, strange, charm, and electron couplings) but
further details may be found in the Higgs-physics Topical Group report [14].

Figure 6: Left, ATLAS best-fit values and uncertainties for Higgs-boson
coupling modifiers per particle type with effective photon and gluon
couplings, the branching fraction to invisible (Bi), and undetected decays
(Bu) included as free parameters, and the measurement of the Higgs-boson decay
rate to invisible final states included in the combination [21]. Right, CMS
summary of the Higgs-boson couplings modifier best fit. The thick (thin) black
lines report the 1$\sigma$ (2$\sigma$) confidence intervals [22].

The scalar potential of the SM responsible for the EWSB mechanism, is
currently still very far from being probed. After EWSB, the SM scalar
potential (see Eq. 3) gives rise to cubic and quartic terms in the Higgs-boson
field, induced by the self-coupling term ($\lambda$). Within the SM, the
Higgs-boson self-coupling is fully predicted in terms of the Fermi coupling
constant and the Higgs-boson mass, which has been measured at per-mille level
accuracy by the ATLAS and CMS experiments [20, 19]. The Higgs self-coupling is
accessible through Higgs-boson pair production ($hh$) and inferred from
radiative corrections to single-Higgs production measurements. Measuring this
coupling is essential to shed light on the structure of the Higgs potential,
whose exact shape can have deep theoretical consequences.

The maximum value of the acceptance for the $gg\rightarrow hh$ process is
obtained for $\kappa_{\lambda}\sim 2$, where the cross section is at a
minimum. Here $\kappa_{\lambda}$ refers to the ratio of the measured value to
the predicted SM value of the Higgs self coupling and must be unity if the SM
is a complete theory. Measuring $\kappa_{\lambda}\neq 1$ would unambiguously
imply that there is some new physics beyond the SM. The corresponding
intervals where $\kappa_{\lambda}$ is observed (expected) to be constrained at
95% CL are listed in Table 3 for the main channels.

Final state | Collaboration | allowed $\kappa_{\lambda}$ interval at 95% CL
---|---|---
|  | observed | expected
$b\overline{b}b\overline{b}$ | ATLAS | -3.5 – 11.3 | -5.4 – 11.4
CMS | -2.3 – 9.4 | -5.0 – 12.0
$b\overline{b}\tau\tau$ | ATLAS | -2.4 – 9.2 | -2.0 – 9.0
CMS | -1.7 – 8.7 | -2.9 – 9.8
$b\overline{b}\gamma\gamma$ | ATLAS | -1.6 – 6.7 | -2.4 – 7.7
CMS | -3.3 – 8.5 | -2.5 – 8.2
comb | ATLAS | -0.6 – 6.6 | -1.0 – 7.1
CMS | -1.2 – 6.8 | -0.9 – 7.1
Table 3:  The observed and expected 95% CL intervals on $\kappa_{\lambda}$ for
the most sensitive individual final states analyzed for non-resonant $hh$
production at 13 TeV with about 126-139 fb-1 under the assumption of no $hh$
production. All other Higgs boson couplings are set to their SM values.
Constraints derived under different statistical assumptions are also available
in [23, 24, 25, 26, 27, 22, 28].

The planned HL-LHC, starting in 2029111This refers to the updated schedule
presented in January 2022 [29] will extend the LHC dataset by a factor of
$\mathcal{O}(10)$, and produce about 170 million Higgs bosons and 120 thousand
Higgs-boson pairs. This would allow an increase in the precision for most of
the Higgs-boson couplings measurements. The HL-LHC will dramatically expand
the physics reach for Higgs physics. Current projections are based on Run 2
results and some basic assumptions that some of the systematic uncertainties
will scale with luminosity and that improved reconstruction and analysis
techniques will be able to mitigate pileup effects. The studies also assume
that the theory uncertainty is reduced by a factor of 2 relative to current
values. Studies based on the 3000 fb-1 HL-LHC dataset estimate that we could
achieve $\mathcal{O}(2-4\%)$ precision on the couplings to $W$, $Z$, and third
generation fermions. But the couplings to $u$, $d$, and $s$ quarks will still
not be accessible at the LHC directly, while the charm-quark Yukawa is
projected to be directly constrained to $\kappa_{c}<1.75$ at the 95% CL [30].
The Higgs-boson self coupling is a prime target of the HL-LHC and current
rough projections claim the trilinear self-coupling will be probed with
$\mathcal{O}$(50%) precision. We will be able to exclude the hypothesis
corresponding to the absence of self-coupling at the 95% CL in these
projections for HL-LHC, but not to test the SM prediction [30].

Future colliders are charged with the challenging tasks of testing the SM
predictions of the Higgs-boson Yukawa couplings to light-flavor quarks, and
improving the precision on the LHC Higgs-coupling measurements. An
$e^{+}e^{-}$ Higgs factory or Muon Collider can measure these couplings with
smaller uncertainties than the HL-LHC due to a combination of knowing the
momentum of the incoming particles more precisely, smaller background
environments, and better detector resolutions. Indeed, at an $e^{+}e^{-}$
Higgs factory the precision can be enhanced by the availability of precise
calculations combined with much more democratic production rates: Higgs
production is roughly of the same order as other processes in $e^{+}e^{-}$
collisions, whereas the LHC must trigger and select Higgs events among
backgrounds that are multiple orders of magnitude larger. Moreover, tagging of
charm and strange quarks, as previously demonstrated at SLC/LEP, gives
effective probes for precision measurements of charm- and strange-quark Yukawa
couplings. The cleaner $e^{+}e^{-}$ environment aided by beam polarization
could become a sensitive probe to reveal more subtle phenomena [31]. For high-
energy Muon Colliders, the primary driver is the cleaner environment plus
increased statistics [32]. The measurement of the Higgs self-coupling demands
access to high-energy center-of-mass collisions to benefit from the larger
dataset of $hh$ pairs and is a major goal of all future colliders.

Figure 7: Projected relative Higgs-coupling measurements in $\%$ when combined
with HL-LHC results. All values assume no BSM decay modes. In addition, only
the following collider stages are shown: $3~{}ab^{-1}$ and two interaction
points (IPs), ATLAS and CMS, for the HL-LHC at 14 TeV, $2~{}ab^{-1}$ and 1 IP
at 250 GeV for ILC/C3, $20~{}ab^{-1}$ and 2 IP at 240 GeV for CEPC,
$1~{}ab^{-1}$ and 1 IP at 380 GeV for CLIC, and $5~{}ab^{-1}$ and 4 IPs at 240
GeV for FCC-ee. Note that the HL-LHC $\kappa_{hcc}$ projection uses only the
CMS detector and is an upper bound [30].

While all future colliders give strong contributions to the Higgs precision
program, the first stages of $e^{+}e^{-}$ Higgs factories are particularly
compelling since they can all be constructed in the near future if funding is
available, while other collider options require significantly more R&D.
Studies for the five current $e^{+}e^{-}$ Higgs factory proposals—ILC [33],
C3[31], CEPC [34], CLIC [35], and FCC-ee [36]—demonstrate that experiments at
these facilities can achieve high precision. Despite their different
strategies, all these proposals lead to very similar projected uncertainties
on the Higgs-boson couplings when the colliders are run at the same energies.
The higher luminosity proposed for circular $e^{+}e^{-}$ colliders is
compensated by the advantages of polarization at linear colliders, yielding
very similar projected sensitivity for the precision of Higgs couplings [37,
38].

We show the projected sensitivity for the first stages of proposed
$e^{+}e^{-}$ colliders combined with HL-LHC projections in Fig. 7. These
results are done in the so-called “kappa-0” framework, which does not allow
for BSM decays of the Higgs boson, and are then combined with projections for
results from HL-LHC. It is clear that the dominant improvement from HL-LHC
results is in the couplings to $b$’s, $c$’s, and $\tau^{\prime}s$, along with
extremely precise measurements of the Higgs interactions with $W$ and $Z$
bosons. The future lepton colliders not only can significantly improve on the
knowledge of the coupling to the charm quark, but potentially also the
coupling to the strange quark, with possible future detector advances, and may
even set relevant direct bounds on Higgs couplings to up and down quarks. A
dedicated run at the Higgs pole by the FCC-ee has the possibility to measure
the coupling of the Higgs to electrons, which would be an important
verification of the SM. Therefore there are subtle differences in the various
$e^{+}e^{-}$ Higgs factories and in some cases further study is needed to
understand how real the differences are.

Measuring the Higgs couplings can be viewed as part of a global program of
fitting to BSM physics in the EFT framework. In this approach, Higgs
interactions are connected to processes without Higgs bosons through the EFT
operators, the so-called “Higgs without Higgs” events. The $\kappa$ framework,
where the kinematic structure of the Higgs interactions is assumed to be
identical to the SM, can be seen as a simplified metric for understanding the
capabilities of future colliders for Higgs studies alone, and the only
possibility when BSM physics affects Higgs properties at scales not validly
described by an EFT approach. In these cases a combination of $\kappa$ fits
and other observables can be more useful. The dedicated EFT analysis shown in
Section 4 combines information from the Higgs sector with information from
precision EW measurements, diboson production, and top-quark measurements,
including kinematic information, to attempt to gain a deeper understanding of
the underlying physics.

Beyond couplings to fermions and gauge bosons, the HL-LHC can constrain the
Higgs boson width indirectly from the $ZZ\rightarrow$ 4 lepton channel, with a
projected measurement of $\Gamma_{h}=4.1^{+.7}_{-.8}$ MeV, corresponding to
roughly a $17\%$ accuracy[30]. The indirect measurement of the Higgs-boson
width can be sensitive to the assumption that there is no new BSM physics
contributing to the width. However, it is more akin to an absolute coupling
normalization and can be viewed as part of the larger “Higgs without Higgs”
framework. BSM physics that invalidates these measurements is not generic, but
further complementary information from other colliders is desired.

One distinct advantage of the lepton colliders is the possibility to obtain
extremely precise and relatively model-independent measurements of the Higgs
boson width. The measurement of the width can confirm the SM prediction given
that it can be very sensitive to the scale of new physics. The fully
reconstructed $Z$ boson in the final state along with the well determined
4-momenta of the initial state leptons in the $Zh$ process allows for a clean
determination of the Higgs-boson kinematics regardless of the Higgs decay
channel. The full FCC-ee program (combined with HL-LHC) allows for a $1\%$
measurement of the Higgs-boson width. Using a SMEFT fit, the ILC finds similar
results for the full program, but with just the initial 250 GeV run, a $2\%$
measurement on the total width can be obtained. A Muon Collider running at
$\sqrt{s}=125$ GeV can obtain a model independent measurement of the Higgs-
boson width at the 68$\%$ level of $2.7\%~{}(1.7\%)$ with $5~{}\mbox{
fb}^{-1}(20~{}\mbox{fb}^{-1})$ by using a line-shape measurement [39]. A high-
energy Muon Collider should obtain a similar order of magnitude precision
using the indirect methods employed at the LHC with the same theoretical
assumptions, and the FCC-hh could in principle also use these methods with
further study.

collider |  Indirect-$h$ | $hh$ | combined
---|---|---|---
HL-LHC [40] | 100-200% | 50% | 50%
ILC250/C3-250 [33, 31] | 49% | $-$ | 49%
ILC500/C3-550 [33, 31] | 38% | 20% | 20%
ILC1000/C3-1000 [33, 31] | 36% | 10% | 10%
CLIC380 [35] | 50% | $-$ | 50%
CLIC1500 [35] | 49% | 36% | 29%
CLIC3000 [35] | 49% | 9% | 9%
FCC-ee [36] | 33% | $-$ | 33%
FCC-ee (4 IPs) [36] | 24% | $-$ | 24%
FCC-hh [41] | - | 3.4-7.8% | 3.4-7.8%
$\mu$(3 TeV) [39] | - | 15-30% | 15-30%
$\mu$(10 TeV) [39] | - | 4% | 4%
Table 4:  Sensitivity at 68% probability on the Higgs cubic self-coupling at
the various future colliders. Values for the indirect single Higgs
determinations below the first line are taken from [42]. The values quoted
here are combined with an independent determination of the self-coupling with
uncertainty 50% from the HL-LHC.

A measurement of the Higgs self-coupling is out of reach of Run 3 of the LHC
and requires either a larger dataset, or/and a higher collision energy. The
self coupling can be measured by the direct production of $hh$ pairs, or
inferred indirectly through the contribution of the Higgs self-coupling to
loop corrections to the single-Higgs rate. However, for the indirect
measurement to be relevant, it requires that new physics contributions
dominate only the triple Higgs coupling shift. While this can naively be
accounted for in a SMEFT fit, in realistic models this is much more difficult
[43].

The projected sensitivities to the Higg-boson self-coupling at the various
future colliders are presented in Table 4. These correspond to projections for
a single experiment except for the ’combined’ results which for the HL-LHC
correspond to the combined projections of ATLAS and CMS experiments. We see
that this is an extremely challenging measurement at all colliders. Since the
measurement is limited by the small number of $hh$ events, the measurement
improves with the higher energy colliders. The indirect measurement improves
with the luminosity of the lepton colliders since it is extracted from single-
Higgs production. In principle measurements at different center of mass
energies can be used to disentangle the indirect effects of shifts in the
triple Higgs couplings, however it also depends on the assumptions of what
types of other operators can contribute.

The ATLAS and CMS experiments have determined that the observed 125-GeV mass
boson has the Higgs boson quantum numbers $j^{PC}=0^{++}$. Small violations of
CP symmetry in the $hVV$ ($V=W,Z$) and $hf{\overline{f}}$ couplings are still
allowed and are an important target of future experimental measurements.
Hadron colliders provide essentially the full spectrum of possible
measurements sensitive to CP violation in the Higgs boson interactions. Most
processes other than the Higgs gluon interactions could be studied at an
$e^{+}e^{-}$ collider, especially with the beam energy above the
$t{\overline{t}}h$ threshold. Future $e^{+}e^{-}$ colliders are expected to
provide comparable sensitivity to HL-LHC in $hf{\overline{f}}$ couplings, and
potentially higher sensitivity in $hZZ$ couplings. A Muon Collider operating
at the Higgs boson pole allows to measure the CP structure of the $h\mu\mu$
vertex with the beam polarization.

Figure 8: A snapshot of future Higgs precision measurements of SM quantities
based on the order of magnitude for the fractional uncertainties with the
range defined through the geometric mean. In this figure the first stages of
each $e^{+}e^{-}$ Higgs factory are shown in combination with the Hl-LHC, as
well as the HL-LHC separately. The Higgs factories are defined as those listed
in Section 3 of the EF Report, excluding the 125 GeV Muon Collider whose
timescale is in principle longer term. The specific precision associated to
each coupling can be found in the Higgs-physics Topical Group report [14] and
references therein. A * is put on the ILC measurements for the strange-quark
Yukawa coupling to single it out as a new measurement proposed during Snowmass
2021, and is shown in Fig 13. The ? symbol is used in the case where an
official study has not yet been performed, for example in the case of strange
tagging for CLIC, FCC-ee, and CEPC. This does not mean that they cannot
achieve a similar precision, but it is yet to be demonstrated whether based on
their detector concepts the measurements is worse or can be improved.  Figure
9: A snapshot of future Higgs precision measurements of SM quantities based on
the order of magnitude for the fractional uncertainties with the range defined
through the geometric mean. In this figure the ultimate reach of the final
stages of all Higgs factories and multi-TeV colliders are shown in combination
with the HL-LHC results, as well as the HL-LHC separately. All benchmarks and
stages are defined in Section 3 of the Energy Frontier Report. The specific
precision associated to each coupling can be found in the Higgs-physics
Topical Group report [14] and references therein. A * is put on the ILC
measurements for the strange-quark Yukawa coupling to single it out as a new
measurement proposed during Snowmass 2021, and shown in Fig 13. The ? symbol
is used in the case where an official study has not yet been performed. It
does not connotate that a given collider should be worse than similar ones,
but simply that whether it is better or worse based on detector design has not
been demonstrated.

We have presented an overview of the types of precision measurements of Higgs-
boson properties that can be done at future colliders, specific numbers for
certain observables at all colliders have been shown, and general coupling
fits in the case of the first-stage $e^{+}e^{-}$ Higgs factories were shown in
Fig. 7. However, it is important to understand that there are more observables
than discussed here and to consider how high-energy colliders fit into the
Higgs precision program. A comprehensive study is presented in the Higgs-
physics Topical Group report [14]. However, instead of displaying all info as
a large table or bar chart, we will conclude this section by displaying a
quantitative coarse grained version of all Higgs precision results in Fig. 8
and Fig. 9. As can be seen in Fig. 7 and Fig. 8, all first-stage Higgs
factories have a very similar reach. However, it also emphasizes that there
are still a large number of missing pieces to the Higgs puzzle after only
first-stage Higgs factories. In Fig. 9, all stages of future colliders that
are discussed in Section 3, are combined with the HL-LHC. As we see in this
coarse graining, all colliders are compelling, and there is significant
progress through Higgs factories and multi-TeV collider proposals compared to
the HL-LHC program. In particular with the additional stages we start to see
differences in the various proposed collider programs for Higgs physics.
Linear $e^{+}e^{-}$ colliders begin to demonstrate advantages especially in
the measurement of the Higgs self coupling compared to circular $e^{+}e^{-}$
colliders, whereas the circular colliders can potentially measure the electron
Yukawa coupling. Furthermore, multi-TeV colliders such as the Muon Collider or
FCC-hh extend the knowledge of the Higgs boson even further, albeit on
different timescales. However, despite the enormous progress we expect from
the currently proposed future colliders in testing the SM Higgs boson sector
with high precision, the research for further improvements will have to
continue. More specifically, the measurements of certain couplings, e.g. the
light-quark Yukawas or the quartic self-coupling of the Higgs boson, that are
challenging at the future colliders proposed in the Snowmass 2021 proceedings,
motivate a continuing research and development.

##### What can we learn about BSM physics from Higgs physics

The ultimate goal of precision Higgs physics is to learn about new physics at
high scales, or to find portals to new physics that could be present at the EW
scale or below. As discussed earlier from an EFT context, the generic scale
associated with precision Higgs physics at future colliders typically extends
up to a few TeV.

To go further requires the understanding of the interplay between UV models
and Higgs physics. Given that the mapping of fundamental physics questions to
Higgs direct and indirect observables is difficult to fully organize
comprehensively, the topical report instead focused on specific types of
models and observables: Higgs Singlets, Higgs Doublets (including Flavor),
Loop-level deviations, and Higgs Exotic Decays. Fundamental questions of
course can be related to all of these types of models and is done so in the
Higgs-physics Topical Group report [14]. Other connections to fundamental
questions are also emphasized in other parts of the EF report, for example
whether the Higgs boson is an elementary or composite particle is investigated
in Section 1.

Given that many of the model-dependent topics have been covered extensively
for years, we first wish to highlight some of the results that are new
compared to the recent European Strategy Update [44]:

  * •

The phenomenology of a strong electroweak phase transition is significantly
more nuanced than previously envisioned. It can manifest through shifts in the
Higgs cubic coupling, but could still occur without any currently or far
future measurable deviation in this coupling [45, 46, 47]. Deviations in all
types of observables are also possibly correlated with the phase transition,
including exotic Higgs decays [48].

  * •

Flavored phenomenology is much richer than previously explored. Flavor-
violating decays have now richer possibilities and models [49, 50]. Flavor-
preserving deviations in consistent light-quark Yukawa couplings now also
exist [51, 52], and there are studies for direct probes of this at
$e^{+}e^{-}$ colliders and related resonance probes from the LHC and other
colliders [53].

  * •

Singlet phenomenology, a canonical example in BSM Higgs phenomenology, can be
quite a bit more varied, including the introduction of scalar resonances
decaying to particles with different masses. These searches were further
explored [54, 55, 56].

  * •

There are now viable models of triple-Higgs production at the HL-LHC and
beyond [55, 57, 52, 58]. The measurement of the quartic coupling should now be
considered a standard part of BSM Higgs phenomenology and triple-Higgs and
quartic-Higgs measurements should be pursued at future colliders.

We now give a sampling of results based on UV complete models, starting with
the simplest extension of the Higgs sector of the SM with an additional scalar
singlet $S$. Despite the simplicity of this type of models, such results
display a wide range of phenomenology and connections to fundamental physics
questions. For example, with a single degree of freedom from a real scalar,
one can connect to the EW phase transition and thereby models of baryogenesis.
This Higgs portal can then be connected to dark sectors and DM, or can be
viewed as a proxy for models of neutral naturalness. The existence of a new
scalar then also applies to the question of whether or not the Higgs is unique
and modifies the Higgs potential. This can have implications for the stability
of our universe. Thus, the rich phenomenology of the real singlet scalar is
quite extensive. One can add additional scalar, i.e. a singlet complex scalar,
or even more. The phenomenology can be further complicated and projections
onto a two-dimensional plane are not sufficient. In particular, because the
masses of the various singlets can be varied, resonance decays have a much
wider range of phenomenology. Figure 10 illustrates this possibility, where we
see that a rate larger than that for SM di-Higgs production ($h_{1}h_{1}$ in
Fig. 10) is possible [54].

Figure 10: Production of a pair of Higgs bosons in the complex singlet model.
In this figure $h_{1}$ is the SM Higgs boson, and $h_{2}$, $h_{3}$ are new
scalars with $2m_{h3}<m_{h2}$. The maximum rates allowed by current LHC data
are shown in black and projections for $\sqrt{s}=14$ TeV in red [54].

Two Higgs Doublet Models (2HDMs) provide the next simplest extension after
scalar singlets to the Higgs sector. They are particularly interesting because
they allow for a new state with SM gauge charge that can also acquire a vacuum
expectation value while naturally allowing for small EW precision corrections.
The new doublet allows for additional Higgs bosons beyond the observed 125 GeV
CP-even neutral scalar $h(125)$, namely, an additional CP-even neutral scalar
$H$, one CP-odd Higgs boson $A$, and a pair of charged Higgs bosons $H^{\pm}$.
Restricting ourselves to the standard types of 2HDM still allows for an
enormous range of phenomenology.

The standard parametrization of the physics is done in terms of a ratio of the
vacuum expectation values of the 2HDM states, $\tan\beta$, and a mixing angle
$\cos(\beta-\alpha)$ as well as the masses of the various scalar eigenstates.

Precision Higgs measurements probe the model parameter space as demonstrated
in Fig. 11 [59] and the improvement at lepton colliders for moderate
$\tan\beta$ is apparent. Figure 11 (right) demonstrates the ability of a high-
energy Muon Collider to probe the parameter space of the 2HDM models. We note
that the region of moderate $\tan\beta$ is best probed by $B$ decays. Limits
from direct searches for the heavier Higgs bosons of the 2HDM are from HL-LHC
studies. For high $\tan\beta$, the decay of the heavier Higgs boson to
$\tau^{+}\tau^{-}$ provides a stringent limit, as seen in Fig. 13 [60].

Figure 11: Limits on the parameters of a 2HDM from precision Higgs-coupling
measurements. Left: Limits from future $e^{+}e^{-}$ colliders combined with
limits from the HL-LHC. The allowed region is to the right of the curves.
[59]. Right: Limits from a 3 TeV Muon Collider. The TeV scale refers to the
mass of the heavier Higgs boson in the 2HDM.

Figure 12: Capability of HL-LHC to probe the scalar sector of the 2HDM.

Figure 13: A 2HDM with non standard Yukawa couplings, in the case of an
enhanced coupling to the strange quark.

A wider range of phenomenology is allowed in the 2HDMs. In particular 2HDMs do
not have to be restricted to the usual four types of natural flavor conserving
models. In Fig. 13 an example of a Spontaneous Flavor Violation (SFV) 2HDM
exemplifies the wide range of phenomenology associated to direct and indirect
searches, as well as the new techniques proposed at the ILC for tagging
strange quarks directly. These models and measurements can also be further
extended into relevant bounds on up and down quark Yukawas as also shown in
the ILC white paper [53].

There are of course numerous connections to deeper questions and additional
models covered in the Higgs physics Topical Group report [14], but at a
general level even Fig. 5 does provide important lessons. The first is that
many observables map to fundamentally different questions related to the Higgs
boson. It is, therefore, non-trivial to connect observables related to Higgs
physics with fundamental questions. This has been referred to as the “Higgs
Inverse Problem”, in analogy with the previously coined LHC inverse problem
for BSM physics. Examples of this are given in the Higgs-physics [14] as well
as in the EW physics [16] Topical Group reports. The second important lesson,
alluded to in Fig. 5, is that Higgs related observables do not just fall into
the standard $\kappa$ or EFT fits. If there are any deviations in Higgs
couplings, or differential measurements etc., there must be new physics that
couples to the Higgs boson which gives origin to it. How it can be searched
for is an ever expanding program and depends on the mass scale of new physics
and collider energy. As mentioned earlier in the context of Higgs width
measurements, there is an ever expanding program of “Higgs without Higgs”
measurements and other types of differential probes being discovered. Suffice
it to say, even 10 years after the Higgs discovery, we are still in the
earliest stages of fully exploiting the potential connection of the Higgs
boson to BSM physics.

####  2 Heavy-flavor and top-quark physics

The top quark plays a special role in the EW sector of the SM, with a Yukawa
coupling ($y_{t}$) of order unity ($y_{t}=\sqrt{2}m_{t}/v\approx 1$, where
$m_{t}$ is the top-quark mass and $v$ is the vacuum expectation value of the
Higgs field) introducing large quadratic corrections to the Higgs-boson mass,
and affecting the stability of the EW vacuum [61]. The top-quark sector is
therefore a crucial component of precision EW tests and particularly relevant
in searches for BSM physics. Figure 14 illustrates the different topics that
are addressed through studying top quarks. As the heaviest of all elementary
particles, the top quark is relevant for understanding the origin of the
Higgs-boson mass and all quark masses. Precise measurements of the masses of
the top quark (see Section 2) together with the Higgs boson and the $W$ boson
provide a stringent test of the EW sector of the SM. In addition, the top
quark decays before it can hadronize, making it the only bare quark that can
be studied directly, including at high momenta (see Section 11). Top-quark
production (see Section 2) and decay kinematic information constrain top-quark
EW couplings (see Section 3) and the CKM element $V_{tb}$. Searches for
flavor-changing neutral currents (FCNC) and CP violation focus on the top-
quark couplings. Direct searches for new particles and interactions look for
top-quark partners, SUSY, and high-mass resonances decaying to top quarks (see
Section 6). Studies of top-quark production at the highest energies (multi-TeV
colliders) probe models of compositeness (see Section 1). The abundance of top
quarks at the LHC makes them ideal for detector calibration of bottom-quark
tagging and bottom- and light-quark jet energy calibration. Top-quark
production processes constitute an important background in many precision
measurements and searches. Lepton colliders operating at or above the top-
quark production threshold provide significantly improved measurements of the
top-quark mass and its couplings. Precision measurements of top-quark and
bottom-quark production at lepton colliders are the inputs needed to
significantly improve the sensitivity of third generation and global EFT fits,
examples of which are presented in Section 3. Moreover, running at the $Z$
pole at $e^{+}e^{-}$ colliders allows access to high-precision measurements of
heavy-quark individual vector and axial-vector couplings. Very precise
measurements of the $Z\to b\overline{b}$ rate can be of particular interest
for top-quark precision physics given the special relationship of the bottom
and top quarks in the SM and BSM models. The prospects for measurements of
$Z\to b\overline{b}$ EW precision observables and of EW couplings
$g_{Z,(L,R)}^{bb}$ from a SMEFT global analysis at various future colliders
are shown in Table 8 and Fig. 21, respectively.

Figure 14: Illustration of the different aspects of the top-quark physics
program.

##### Top-quark mass

The top-quark mass, $m_{t}$, is one of the most important parameters of the SM
and relevant as an input for precise predictions and for the understanding of
SM properties such as the stability of the spontaneously broken vacuum state.
Top-quark loop corrections impact the mass of the $W$ boson, e.g. a top-quark
mass change of 100 MeV changes the $W$ boson mass by 1 MeV [62]. Thus, given
the expected precision on the $W$ mass at the HL-LHC and at future lepton
colliders, precisions of better than 500 MeV and 50 MeV are required for top-
quark mass measurements at the HL-LHC and at future lepton colliders,
respectively, for precision EW fits [63, 64] (see also Section 3). Since
isolated quarks cannot be observed, the top-quark mass is not physical, but a
renormalization-scheme-dependent quantity. This scheme dependence can only be
well-defined and controlled for mass-sensitive observables that are calculable
in perturbation theory (at least at the NLO level). The currently most precise
top-quark mass determinations at the LHC are obtained from the direct
reconstruction of the top-quark decay products (jets and leptons). Analytic
perturbative QCD calculations are not available for the kinematic distribution
of these objects, therefore the top-quark mass is extracted from comparisons
to predictions by Monte Carlo (MC) event generators (so-called mass from decay
or MC mass). It is estimated that interpreting the mass from decay in a well-
defined scheme (like the $\overline{\text{MS}}$ scheme) has an uncertainty of
about 500 MeV [65, 66]. The current and expected precision for measurements of
the mass from decay are compared to the projections from Snowmass 2013 in Fig.
15 (left). The recent measurements significantly improved on the projections
from 2013.

Figure 15: (Left) Comparison of top-quark mass measurements from top decay (MC
mass) at the Tevatron and the LHC, and projections for future LHC sensitivity
and for future mass sensitivity from a top threshold scan at a lepton
collider. The interpretation uncertainty for the mass from decay is not
included [15]. (Right) Comparison of projected $\overline{\rm MS}$ top-quark
mass determinations from decay and pole-mass measurements at the LHC and HL-
LHC and from a PS mass measurement at a future lepton collider. The dashed-
dotted lines show the approximate uncertainty in interpreting the mass from
decay as $\overline{\rm MS}$ top-quark mass [65, 66] (labeled
“Interpretation”) or the combined uncertainty from theory and the conversion
to the $\overline{\rm MS}$ scheme (labeled “Theo.+trans.”).

Top-quark mass measurements in a well-defined scheme by contrast are based on
comparisons of perturbative calculations of differential and total cross
sections with measurements that are unfolded to the parton level (so-called
indirect mass or pole mass). Current measurements for a top-quark mass
measurement performed in the pole mass scheme have reached a precision of 1.2
GeV [67, 68, 69]. The uncertainty is currently dominated by theoretical
uncertainties, in particular due to Parton Distribution Functions (PDFs). For
the projection, it is assumed here that the theory uncertainty can be reduced
through dedicated PDF fits in $t\overline{t}$ events and through higher-order
calculations of the $t\overline{t}j$ process.

The ultimate precision in the top-quark mass will be reached in a scan of the
top-quark production threshold at a lepton collider (here denoted as PS mass).
The corresponding expected precision for the PS mass for different
$e^{+}e^{-}$ collider options is shown in Table 5. The overall uncertainty is
expected to be limited by systematic uncertainties, in particular in the
theoretical predictions, including the uncertainty on the strong coupling
constant $\alpha_{s}$. The top-quark width is similarly measured with the
highest precision at a lepton collider, in combination with the top-quark
mass, in an energy scan of the top-production threshold. The Yukawa coupling
of the top quark $y_{t}$ can be measured in the same scan from the plateau
above the top-production threshold [70].

$\delta m_{t}^{{\rm PS}}$ [MeV] | ILC | CLIC | FCC-ee
---|---|---|---
${\cal L}[\mbox{fb}^{-1}]$ | 200 | 100 [200] | 200
Statistical uncertainty | 10 | 20 [13] | 9
Theoretical uncertainty (QCD) | 40 – 45
Parametric uncertainty $\alpha_{s}$ | 26 | 26 | 3.2
Parametric uncertainty $y_{t}$ HL-LHC | 5
Non-resonant contributions | $<40$
Experimental systematic uncertainty | 20 – 30 | 11 – 20
Total uncertainty | 40 – 75
Table 5: Anticipated statistical and systematic uncertainties in the
measurement of the threshold mass, $m_{t}^{\mathrm{PS}}$, from a threshold
scan around 350 GeV obtained with a one-dimensional fit of the top-quark mass,
keeping $\Gamma_{t}$, yt, and $\alpha_{s}$ fixed. CLIC assumes a lower
integrated luminosity than the other facilities. For comparison, the
statistical precision achievable with 200 $\mbox{fb}^{-1}$ for CLIC is also
given. It should be noted that the results shown for ILC and FCC-ee assume a
8-point scan with a compressed energy range which improves sensitivity for
$m_{t}^{{\rm PS}}$ at the expense of $y_{t}$ sensitivity. For the standard
10-point scan assumed for CLIC the statistical uncertainties would be 12 and
10 MeV for ILC and FCC-ee, respectively. The uncertainty due to the current
world average for $\alpha_{s}$ is shown for ILC and CLIC, while for FCC-ee,
the run at the $Z$ pole (Tera-$Z$) will reduce this uncertainty significantly.
Concrete studies for CEPC are not yet available, but it can be assumed that
uncertainties are similar as for FCC-ee.

To be able to compare the projections for the different top-quark mass
measurements, they have been converted to projections for the
$\overline{\text{MS}}$ top-quark mass, shown in Fig. 15 (right). The dash-
dotted lines show the approximate uncertainty in interpreting the mass from
decay as a $\overline{\text{MS}}$ top-quark mass [65, 66] or the combined
uncertainty from theory and the conversion to the $\overline{\text{MS}}$
scheme (known to ${\cal O}(\alpha_{s}^{4})$ in QCD) in case of the pole and PS
mass measurements.

##### Top-quark production processes

Top quarks are produced copiously at hadron colliders in many different
production modes, $t\overline{t}$, single top, and both modes in association
with other quarks and bosons. Modeling the different processes requires
precision higher-order QCD calculations with heavy quarks. The cross sections
for the production of top-quark pairs [71], as well as top-quark pairs in
association with various other particles are shown in Fig. 16 [15]. Measuring
all of these processes is possible with high precision at the HL-LHC. Even the
process with the lowest production cross section, the production of four top
quarks, can be measured with an uncertainty of about 20% at the HL-LHC, and
should be measurable to a few percent at higher-energy hadron colliders
(including FCC-hh) [72, 73]. This production mode has the highest energy
threshold of all top-quark-related SM processes studied at hadron colliders.
It is sensitive to $y_{t}$ and BSM interactions, for example contact
interactions [72]. The measurements of $t\overline{t}H$ production directly
probe the top Yukawa coupling. The measurements of $t\overline{t}Z$,
$t\overline{t}\gamma$ have intrinsic value, as they form the first constraints
on the EW couplings of the top quark. Precision measurements of the processes
shown in Fig. 16 and the corresponding single top-quark processes are utilized
in global EFT fits. It should be possible to reach a precision of O(1%) for
$t\overline{t}$ production, larger for other processes. This requires careful
calibration, improved modeling of top-quark and other processes, and improved
theory calculations; currently the scale uncertainty on
$\sigma(t\overline{t})$ is about 3% at NNLO QCD with NNLL soft-gluon
resummation [74], while the PDF uncertainty is about 3% [75]. Extending the
differential top-quark measurements to high, multi-TeV transverse momenta
gives sensitivity to 4-fermion interactions involving the third generation.

Figure 16: Total cross sections as a function of the center-of-mass energy
$\sqrt{s}$ at $pp$ colliders for (left) $t\overline{t}$ production at LO, NLO,
N2LO, and approximate N3LO [71] and for (right) various $pp\to t\overline{t}X$
processes at NLO. The $pp\rightarrow t\overline{t}$ cross section is also
shown for reference. Light objects are also required to have $p_{T}>25$ GeV
and $|\eta|<2.5$, while jets are clustered using the anti-$k_{T}$ algorithm
with $R=0.4$.

At lepton colliders running at or above the top-quark production threshold
allows for high-precision measurements of $t\overline{t}$ production and of
the couplings of the top-quark to the $Z$ boson. These measurements, as well
as the corresponding measurements of the production of bottom-quark pairs at
similar precision, will allow to significantly extend the sensitivity of
global EFT fits, see Fig. 17. Producing $t\overline{t}$ in association with
the Higgs, $W$ or $Z$ bosons requires significantly higher center-of-mass
energies at a lepton collider.

##### Top-quark coupling measurements and EFT fits

The measurements of cross sections of top-quark production processes provide
important inputs to global EFT fits [76]. Differential production measurements
and studies of top-quark decay and top-quark final state correlations provide
further constraints [77]. At hadron colliders, these include top-quark pair
and single top-quark production processes, and associated production, measured
differentially. The precision of the measurements is limited by systematic
uncertainties, the largest of which are due to jet energy calibration and QCD
modeling of the top-quark final states. At lepton colliders, the final state
can be fully reconstructed, and most measurements are effectively background-
free, in contrast to hadron colliders. The lepton collider measurements
constrain all EW couplings of the top quark. The gain in precision is
particularly pronounced in the couplings to neutral bosons that are tested
directly in $e^{+}e^{-}\to t\overline{t}$ production. The coupling to the $W$
boson is constrained strongly already in helicity fraction measurements at the
Tevatron and the LHC and the improvement compared to HL-LHC is not as dramatic
(see Section 3). Figure 17 shows the reach of the HL-LHC and the improvement
that can be expected from adding lepton collider data to a global EFT fit of
the Wilson coefficients relevant for top-quark couplings. The fit uses cross
sections for various top-quark production processes (Section 2) and angular
correlations at the HL-LHC, and optimal variables at the lepton collider.

Figure 17: Comparison of the constraints expected from a combination of HL-LHC
and lepton collider data on Wilson coefficients for EFT operators relevant to
top-quark couplings, see Section 3. The solid bars provide the individual
limits of the single-parameter fit and the shaded ones the marginalised limits
of the global fit. The details of the fit can be found in Refs. [78, 79].

##### BSM physics from top-quark physics

Besides indirect searches for BSM physics in the EFT framework, the top quark
is also a sensitive probe in direct searches for new physics. BSM models which
aim to protect the Higgs boson mass from large quadratic corrections beyond
the EW energy scale are often closely connected to the top quark, such as low-
energy SUSY. For example, in [80] the correlation of the spins of the two top
quarks in $t\overline{t}$ production at the LHC are used to constrain SUSY
models with top squarks with masses close to the top-quark mass (and small
neutralino mass). Figure 19 shows the projected limit for a 30 GeV-wide
”corridor” in stop mass ($m(\tilde{t})$) and neutralino mass ($m(\chi_{0})$)
around the top-quark mass ($m(\tilde{t})-m(\chi_{0})-m(t)|<30$ GeV) [80]. The
width of the corridor corresponds to the experimental resolution and the
region where direct stop searches are not sensitive because of the large
$t\overline{t}$ background. The limits expected for the HL-LHC are a factor
two (at low $m(\tilde{t})$) to ten (at high $m(\tilde{t})$) better than the
Run 2 limits in this region. The predicted SUSY stop pair production cross
section in this region is between 10 pb and 100 pb, meaning the entire area
will be excluded.

As the heaviest fermion, it is also expected to play a central role in models
of compositeness, together with the Higgs boson [81]. Examples of the
phenomenology of these type of models in the top-quark sector are the
occurrence of new fermionic resonances (top partners), anomalous 4-top quark
production, and modified top Yukawa and top-EW couplings. For example, Fig. 19
compares the reach of future colliders in the plane of mass scale and coupling
$g*$ for a model of total right-handed top-quark compositeness, giving rise to
sizeable 4-top Wilson coefficients [82].

Figure 18: Limit on the cross section for SUSY stop production in the
compressed region where the stop mass ($m(\tilde{t})$) is close to the
neutralino mass ($m(\chi_{0})$), $m(\tilde{t})-m(\chi_{0})=175$ GeV [80].
Every point in this plot is excluded [80].

Figure 19: Exclusion (2-$\sigma$) sensitivity projections for compositeness
models at future colliders as labeled, for models where both the Higgs boson
and the top quark with right-handed couplings are composite. Plot based on
Refs. [81, 82].

Top-quark interactions are also an excellent probe of FCNC involving the third
generation of leptons. Searches for FCNC interactions in single top-quark
production at hadron colliders (sensitive to gluon FCNC interactions) and
lepton colliders (sensitive to photon and $Z$ boson FCNC interactions) take
advantage of needing lower center-of-mass energy to produce one top quark
rather than two. The large samples of top quarks collected at the LHC and
expected at the HL-LHC allow for searches in the top-quark decay (sensitive to
photon, $Z$ boson, and Higgs FCNC interactions). The limits on the top decay
branching ratios are around $10^{-4}$ with the Run 2 dataset, these will be
improved to around $10^{-5}$ at the HL-LHC. Lepton colliders are sensitive to
FCNC couplings of the top quark to the photon and the $Z$ boson, especially at
energies below the $t\overline{t}$ production threshold [83, 70]. The
production of a single top quark together with an up or charm quark provides a
unique final state signature. Combining runs at multiple center-of-mass
energies provides additional sensitivity, especially at the highest energies
reached in $e^{+}e^{-}$ only by CLIC [84]. This is an area where a Muon
Collider might also provide additional sensitivity.

##### Heavy-flavor and top-quark physics summary

Table 6 compares a few top-quark measurements between different future
collider options. Each of the measurements can be improved at future colliders
beyond the precision at the HL-LHC. Significantly improving the precision of
the top-quark Yukawa coupling beyond the 2-4% uncertainty expected at the HL-
LHC [30] and measuring the top-Higgs coupling in $t\overline{t}H$ events
requires a high-energy lepton collider at a center-of-mass energy of at least
500 GeV or a hadron collider. The projected top-quark mass uncertainty at
lepton colliders is obtained from Table 5, taking the lower end of each
uncertainty range. The precision of the coupling measurements to the SM bosons
will all be significantly improved at a lepton collider operating at or above
the top-production threshold. The four-top coupling can be probed at hadron
colliders, or at lepton colliders running at sufficiently high energies.
Searches for FCNC via the $Z$ boson or photon are done in top-quark decays at
hadron colliders, the sensitivity is significantly extended at lepton
colliders operating as a Higgs factory. These projections also rely on a
significant theoretical effort to meet the tremendous challenges in providing
theory predictions at the required level of precision.

Parameter | HL-LHC | ILC 500 | FCC-ee | FCC-hh
---|---|---|---|---
$\sqrt{s}$ [TeV] | 14 | 0.5 | 0.36 | 100
Yukawa coupling $y_{t}$ (%) | 3.4 | 2.8 | 3.1 | 1.0
Top mass $m_{t}$ (%) | 0.10 | 0.031 | 0.025 | –
Left-handed top-$W$ coupling $C_{\phi Q}^{3}$ (TeV-2) | 0.08 | 0.02 | 0.006 | –
Right-handed top-$W$ coupling $C_{tW}$ (TeV-2) | 0.3 | 0.003 | 0.007 | –
Right-handed top-$Z$ coupling $C_{tZ}$ (TeV-2) | 1 | 0.004 | 0.008 | –
Top-Higgs coupling $C_{\phi t}$ (TeV-2) | 3 | 0.1 | 0.6 |
Four-top coupling $c_{tt}$ (TeV-2) | 0.6 | 0.06 | – | 0.024
Table 6: Anticipated precision of top-quark Yukawa coupling and mass
measurements, and of example EFT Wilson coefficient for the top-quark coupling
to $W$, $Z$ and Higgs bosons, as well as a four-top Wilson coefficient. The
reaches of the CEPC and SPPC are expected to mirror those of the FCC-ee and
the FCC-hh respectively.

####  3 Electroweak precision physics and new physics constraints

The precise measurement of physics observables and the test of their
consistency within the SM are an invaluable approach, complemented by direct
searches for new particles, to determine the existence of physics beyond the
SM.

Indirect searches for new physics, which exploit off-shell and loop
contributions of new particles, allow one to explore a much wider range of
energy scales than those probed by direct searches in specific BSM scenarios.
Such indirect BSM effects are typically inversely proportional to some power
of the mass scale of the new degrees of freedom, so that high precision is
crucial for probing large energy scales. The achievable precision of an
experiment is determined by the statistics of the collected data sample, the
experimental and theoretical systematic uncertainties, and their correlations.

##### Electroweak precision physics

The current precision for a few selected electroweak precision pseudo-
observables (EWPOs) is listed in Table 7. The HL-LHC with integrated
luminosity of 3000 fb-1 can make improved measurements of certain EWPOs. The
effective weak mixing angle can be extracted from measurements of the forward-
backward asymmetry in Drell-Yan production, $pp\to\ell^{+}\ell^{-}$
($\ell=e,\mu$), while the $W$-boson mass can be determined from measurements
of $pp\to\ell\nu$. Both measurements crucially depend on precise knowledge of
PDFs and theory input for QCD and EW corrections, where the SM has to be
assumed for the latter.

EWPO Uncertainties | Current | HL-LHC
---|---|---
$\Delta m_{W}$ (MeV) | 12 / 9.4† | 5
$\Delta m_{t}$ (GeV) | 0.6* | 0.2
$\Delta\sin\theta^{\ell}_{\mbox{\scriptsize eff}}$ ($\times 10^{5}$) | 13 | $<10$

† The recent $W$ mass measurement from CDF with 9.4 MeV precision [85] has not
yet been included in the global average [86].

*  This value includes an additional uncertainty due to ambiguities in the top mass definition (see [15] for more details).

Table 7: The current precision of a few selected EWPOs, based on data from
LEP, SLC, Tevatron and LHC [86], and expected improvements from the HL-LHC
[30]. $\Delta$ ($\delta$) stands for the absolute (relative) uncertainty.

Future high-luminosity $e^{+}e^{-}$ colliders can be used to study the masses
and interactions of electroweak bosons to much higher precision than before.
We here focus on four collider proposals: ILC [87, 88, 33], CLIC [89, 90],
FCC-ee [91, 36], and CEPC [92, 93]. For ILC, CLIC and FCC-ee, we use the run
scenarios and integrated luminosities in Table 2, whereas for CEPC the 50 MW
upgrade is assumed, which corresponds to 100 ab-1 on the $Z$ pole, 6 ab-1 at
the $WW$ threshold, and 1 ab-1 at the $t\overline{t}$ threshold [93]. For ILC
also the Giga-$Z$ option with 100 fb-1 on the $Z$ pole is considered. [Note
that a $Z$-pole run is also proposed as a possible option for CLIC [94].]
Table 8 summarizes the achievable precision for a range of EWPOs.

Quantity | current | ILC250 | ILC-GigaZ | FCC-ee | CEPC | CLIC380
---|---|---|---|---|---|---
$\Delta\alpha(m_{Z})^{-1}\;(\times 10^{3})$ | 17.8∗ | 17.8∗ |  | 3.8 (1.2) | 17.8∗ |
$\Delta m_{W}$ (MeV) | 12∗ | 0.5 (2.4) |  | 0.25 (0.3) | 0.35 (0.3) |
$\Delta m_{Z}$ (MeV) | 2.1∗ | 0.7 (0.2) | 0.2 | 0.004 (0.1) | 0.005 (0.1) | 2.1∗
$\Delta m_{H}$ (MeV) | 170∗ | 14 |  | 2.5 (2) | 5.9 | 78
$\Delta\Gamma_{W}$ (MeV) | 42∗ | 2 |  | 1.2 (0.3) | 1.8 (0.9) |
$\Delta\Gamma_{Z}$ (MeV) | 2.3∗ | 1.5 (0.2) | 0.12 | 0.004 (0.025) | 0.005 (0.025) | 2.3∗
$\Delta A_{e}\;(\times 10^{5})$ | 190∗ | 14 (4.5) | 1.5 (8) | 0.7 (2) | 1.5 (2) | 60 (15)
$\Delta A_{\mu}\;(\times 10^{5})$ | 1500∗ | 82 (4.5) | 3 (8) | 2.3 (2.2) | 3.0 (1.8) | 390 (14)
$\Delta A_{\tau}\;(\times 10^{5})$ | 400∗ | 86 (4.5) | 3 (8) | 0.5 (20) | 1.2 (20) | 550 (14)
$\Delta A_{b}\;(\times 10^{5})$ | 2000∗ | 53 (35) | 9 (50) | 2.4 (21) | 3 (21) | 360 (92)
$\Delta A_{c}\;(\times 10^{5})$ | 2700∗ | 140 (25) | 20 (37) | 20 (15) | 6 (30) | 190 (67)
$\Delta\sigma_{\rm had}^{0}$ (pb) | 37∗ |  |  | 0.035 (4) | 0.05 (2) | 37∗
$\delta R_{e}\;(\times 10^{3})$ | 2.4∗ | 0.5 (1.0) | 0.2 (0.5) | 0.004 (0.3) | 0.003 (0.2) | 2.5 (1.0)
$\delta R_{\mu}\;(\times 10^{3})$ | 1.6∗ | 0.5 (1.0) | 0.2 (0.2) | 0.003 (0.05) | 0.003 (0.1) | 2.5 (1.0)
$\delta R_{\tau}\;(\times 10^{3})$ | 2.2∗ | 0.6 (1.0) | 0.2 (0.4) | 0.003 (0.1) | 0.003 (0.1) | 3.3 (5.0)
$\delta R_{b}\;(\times 10^{3})$ | 3.1∗ | 0.4 (1.0) | 0.04 (0.7) | 0.0014 ($<0.3$) | 0.005 (0.2) | 1.5 (1.0)
$\delta R_{c}(\times 10^{3})$ | 17∗ | 0.6 (5.0) | 0.2 (3.0) | 0.015 (1.5) | 0.02 (1) | 2.4 (5.0)
Table 8: EWPOs at future $e^{+}e^{-}$ colliders: statistical error (estimated
experimental systematic error). $\Delta$ ($\delta$) stands for absolute
(relative) uncertainty, while * indicates inputs taken from current data [86].
See Refs. [42, 95, 96, 97, 33, 93].

The table separately lists the expected statistical and experimental
systematic uncertainties. Note that the latter are based on assumptions about
future performance improvements that cannot be substantiated at this time.
Uncertainties due to the physics modeling affect all collider proposals
equally. As part of the Snowmass 2021 process, a consistent set of assumptions
is being used and applied uniformly. See EF04 Topical Group report for more
details [16].

The impact of these estimated future precision measurements on the indirect
determination of the Higgs-boson and top-quark mass is illustrated in Fig. 20.
The dependence on $m_{H}$ and $m_{t}$ appears in loop corrections to the SM
theory predictions for $Z$ coupling parameters and the $W$ mass, and their
agreement with direct measurements of these masses is a highly non-trivial
test of the SM.

Figure 20: Indirect sensitivity to $m_{H}$ and $m_{t}$ for a fit of SM theory
predictions to current and projected future data for electroweak precision
tests ($W$ mass and $Z$-pole quantities). For comparison, the direct
measurement precision is also shown (on the scale of the plot the width of the
$m_{H}$ band is not visible). The light (dark) shaded areas depict 95% (68%)
confidence level regions. For the future collider scenarios it is assumed that
the central values coincide with the SM expectations.

For “canonical” electroweak precision measurements (Z-pole, $WW$ threshold),
circular $e^{+}e^{-}$ colliders (FCC-ee, CEPC) have in general a higher
sensitivity than linear colliders (ILC, CLIC) due to the high luminosity at
center-of-mass energies below 200 GeV. Beam polarization at the linear
colliders improves their sensitivity and can help to control systematics. In
particular, for a linear collider run on the $Z$ pole, beam polarization would
enable measurements of the asymmetry parameters $A_{f}$ with a precision that
is only a factor of a few worse than for circular colliders, in spite of
several orders of magnitude larger statistics for $Z$-pole physics at circular
colliders.

For many of the most precisely measurable precision observables at linear
colliders, the most significant source of experimental systematics stems from
the polarization calibration. For the circular colliders, on the other hand,
modeling uncertainties for hadronic final states appear to be the dominant
systematic error source.

To exploit the full potential of the anticipated precision of any future
$e^{+}e^{-}$ collider, theory inputs are needed on multiple fronts. Accurate
MC tools for the simulation of QED and QCD radiation are crucial for the
evaluation of acceptance effects, and theory calculations including higher-
order effects are needed for the prediction of irreducible backgrounds. For
the interpretation of electroweak precision measurements, one needs to compare
the measured values to their expectation within the SM, which requires multi-
loop theory computations. For the anticipated experimental precision FCC-ee,
CEPC, ILC or CLIC, the current state of the art of theory calculations needs
to be extended by at least one order of perturbation theory, i.e. N2LO/NLL
contributions for MC tools and backgrounds, and N3LO and partial N4LO
contributions for the SM predictions. See EF04 Topical Group report for more
details [16].

The SM predictions also rely on other SM parameters as inputs, such as the
electromagnetic coupling at the weak scale, $\alpha(m_{Z})$, the top-quark
mass, $m_{t}$, and the strong coupling, $\alpha_{s}$. The current
uncertainties for these parameters would severely limit the possibility for
future high-precision studies, and thus it is necessary to perform improved
measurements of these quantities at the future $e^{+}e^{-}$ colliders. See
sections 2 and 1 for more information.

EWPO | Current | Projected param. error
---|---|---
uncertainties | param. error | Scenario 1 | Scenario 2
$\Delta m_{W}$ (MeV) | 5 | 2.8 | 0.6
$\Delta\Gamma_{Z}$ (MeV) | 0.5 | 0.3 | 0.1
$\Delta\sin^{2}\theta^{\ell}_{\mbox{\scriptsize eff}}\;(\times 10^{5})$ | 4.2 | 3.7 | 1.1
$\Delta A_{\ell}\;(\times 10^{5})$ | 30 | 25 | 7.5
$\delta R_{\ell}\;(\times 10^{3})$ | 6 | 3.2 | 1.3

Input par. uncertainties |  $\Delta m_{t}$ [GeV] |  $\Delta m_{H}$ [GeV] |  $\Delta m_{Z}$ [MeV] | $\Delta(\Delta\alpha)$ | $\Delta\alpha_{s}$
---|---|---|---|---|---
Current | 0.6 | 0.17 | 2.1 | $10^{-4}$ | $9\times 10^{-4}$
Scenario 1 | 0.3 | 0.02 | 0.8 | $10^{-4}$ | $5\times 10^{-4}$
Scenario 2 | 0.05 | 0.01 | 0.1 | $3\times 10^{-5}$ | $2\times 10^{-4}$

Table 9: Impact of uncertainties of SM input parameters on the prediction of a
few selected EWPOs in top table (see Ref. [98]). Current uncertainties are
compared to two future scenarios in the bottom table.

The impact of the uncertainties of the SM input parameters on the
interpretation of electroweak precision measurements is illustrated in Table
9. In addition to the current measurement precision, two future scenarios are
considered, where Scenario 1 assumes improvements from a Higgs factory with
moderate luminosity spent on the $Z$ pole and no $t\overline{t}$ running,
whereas Scenario 2 displays the full potential of achievable precision at
future $e^{+}e^{-}$ colliders. Note that the dependence of the predictions for
$\Gamma_{Z}$ and $R_{\ell}$ on $\alpha_{s}$ are to a certain extent circular,
since these quantities would be used for the extraction of the strong coupling
constant at future $e^{+}e^{-}$ colliders [99].

Experiments at lower-energy $e^{+}e^{-}$ colliders, lepton-proton colliders,
or neutrino scattering facilities can deliver complementary information about
electroweak quantities. For instance, a beam polarization upgrade of
SuperKEKB/Belle II can enable a measurement of the electroweak mixing angle
via tests of vector coupling universality in neutral currents with
unprecedented precision. Electron-proton colliders such as EIC, LHeC and FCC-
eh could provide information about the running electroweak mixing angle at
scales of a few tens of GeV and enable the separate determination of up- and
down-quark electroweak couplings with high precision.

A Muon Collider with center-of-mass energy $\sqrt{s}\approx 91$ GeV [100]
would also be very interesting for electroweak precision measurements, but
more studies are needed.

####  4 EFT and new physics

##### Multi-boson processes

The SM predicts the existence of multi-boson interactions, which give rise to
final states with two or three bosons. Anomalies in the rate and kinematics of
these final states can be indicative of new physics not currently described in
the SM. Such anomalies can be parametrized through modifications of the
strength or form of the SM multi-boson vertices. A newer approach consists in
using EFT operators of dimension six or above, where measurements of multi-
boson processes can be recast as determinations of the Wilson coefficients of
these operators.

It shall be noted that the sensitivity to BSM effects, or, in other terms, the
upper limits to the Wilson coefficients of new operators, scale with a power
of the center-of-mass energy, thus making multi-TeV colliders the ideal tools
for studying these final states. At this time, the most promising avenues for
reaching multi-TeV energies are proton-proton colliders or $\mu^{+}\mu^{-}$
colliders.

High-energy ($>1$ TeV) lepton colliders are effectively boson colliders. The
total cross section for many production processes is dominated by vector-boson
fusion (VBF) and/or vector-boson scattering (VBS) contributions. However, for
studies of BSM effects at very high invariant masses, non-VBF processes become
typically more dominant.

At multi-TeV lepton colliders, multiple electroweak gauge-boson production is
ubiquitous, and new theoretical tools are needed for calculating and
simulating these effects (e.g. theory modeling of EW PDF and fragmentation,
initial-state - ISR - and final-state - FSR- radiation).

Plentiful experimental results with multi-boson final states are available.
Both the ATLAS and CMS collaborations have measured di-boson [101, 102, 103,
104, 105, 106, 107, 108, 109], tri-boson processes [110, 111, 112, 113], as
well as VBF/VBS processes [114, 115, 116, 117, 118, 119, 120, 121, 122, 123,
124], which are characterized by a $VVjj$ final state. Di-boson final states
include $W^{+}W^{-}$, same-sign $W^{\pm}W^{\pm}$, $WZ$, $ZZ$, $Z\gamma$. Tri-
boson final states include $W\gamma\gamma$, $Z\gamma\gamma$, $WV\gamma$ (where
$V=W,Z$), and $WVV^{\prime}$ (where $V,V^{\prime}=W,Z$).

Bounds on new physics have been determined in the language of anomalous gauge-
boson couplings (aGCs) [101, 104, 105, 106, 109] and effective operators [102,
120, 121, 106, 122, 107, 108, 123, 113]. The latter is theoretically preferred
since it provides a consistent power counting and allows one to implement
theoretical consistency constraints. In these studies, only one or two
aGCs/operators are allowed to be non-zero at the same time, i.e. no full
aGC/SMEFT analysis has been performed.

The most up-to-date limits on gauge-coupling anomalies are available at Refs.
[125, 126]. Expected limits at the end of the HL-LHC and potential HE-LHC runs
are reported in Ref. [127].

There are ongoing studies for the determination of aGCs or multi-gauge boson
SMEFT operators at future hadron and lepton colliders (see e.g. Ref. [128,
129]). Preliminary results indicate that a 100-TeV $pp$ collider or a multi-
TeV $\mu^{+}\mu^{-}$ collider would have strongly enhanced sensitivity to
these BSM effects compared to the LHC, but more work is needed to obtain a
comprehensive picture.

##### SMEFT global fits

Assuming new physics scales are significantly higher than the EW scale, EFTs
provide a model-independent prescription that allows us to put generic
constraints on new physics and to study and combine large sets of experimental
data in a systematically improvable QFT approach. All new physics effects are
represented by a set of higher dimensional operators which consist of only the
SM fields and respect the SM gauge symmetries.

The EFT approach has some features that are of particular interest for
studying precision EW physics, for instance: it provides a well-defined
theoretical framework that enables the inclusion of radiative corrections for
both the SM and BSM parts; and the synergies between different precision EW
measurements can be explored globally so that a comprehensive picture of the
constraints on new physics can be drawn. However, the EFT approach also has
some practical limitations since it has in principle an infinite number of
degrees of freedom, and it is only an adequate description if the new physics
scales are larger than the experimentally reachable energies. In a realistic
global EFT fit, various flavor assumptions and truncations to the lowest order
of relevant operators often have to be applied.

A model-independent parametrization of the new-physics reach of different
colliders is given by the SMEFT framework, where the SM is extended by higher-
dimensional operators, with the leading contribution to most observables
furnished by dimension-6 operators. Several subsets of such dimension-6
operators have been investigated in a series of global fits across a large
number of observables: (a) operators contributing to electroweak gauge-boson
interactions; (b) operators contributing to Higgs interactions; (c) operators
contrbuting to top-quark interactions; and (d) operators contributing to four-
fermion contact interactions.

For Snowmass 2021, the global EFT fit for the European Study Group (ESG) [42]
has been extended in a few directions [78]: the consistent implementation of
full EFT treatment in $e^{+}e^{-}\to W^{+}W^{-}$ using optimal observables;
the new inclusion of a large set of 4-fermion operators; a more complete set
of operators that are related to top-quarks. In all the fits, operators for
third-generation fermions are treated independently, without assuming flavor
universality, and in some cases even universality between the first two
generations has been lifted. However, no flavor-changing operators were
included in the analysis. The projections of the uncertainties of required
input observables are provided by EF01 for Higgs related observables [14],
EF03 for top-quark related observables [15], EF04 for $W/Z$ related
observables [16], and the Rare Process and Precision Measurements Frontier for
a set of low-energy measurements [9]. Care has been taken to ensure that the
various inputs are consistent and based on similar assumptions, e.g. by using
extrapolations to compare inputs from two different $e^{+}e^{-}$ colliders.

Figure 21 displays the result of the global EFT fit for the subset of
operators that affect Higgs and EW observables. Instead of showing the
projected constraints on the Wilson coefficients of the operators considered,
they have been translated into constraints on the effective Higgs and gauge-
boson couplings. See EF04 Topical Group report for more details [16].

Figure 21: Precision reach on Higgs and electroweak effective couplings from
an SMEFT global analysis of the Higgs and EW measurements at various future
colliders. The wide (narrow) bars correspond to the results from the
constrained-$\Gamma_{H}$ (free-$\Gamma_{H}$) fit. The HL-LHC and LEP/SLD
measurements are combined with all future lepton collider scenarios. For
$e^{+}e^{-}$ colliders, the high-energy runs are always combined with the low
energy ones. For the ILC, the (upper edge of the) triangle mark shows the
results for which a Giga-$Z$ run is also included. For the Muon Collider,
three separate scenarios are considered. The subscripts in the collider
scenarios denote the corresponding integrated luminosity of the run in ${\rm
ab}^{-1}$.

Generally, future lepton colliders have the best reach for many of the
aforementioned operators. Circular $e^{+}e^{-}$ colliders have the highest
sensitivity to EW operators, due to the large statistical precision of
$Z$-pole and $WW$-threshold measurements. All lepton colliders ($e^{+}e^{-}$
and $\mu^{+}\mu^{-}$) are comparable in their reach for Higgs operators,
although a multi-TeV Muon Collider cannot constrain exotic Higgs decays in a
model-independent way, and the combination with a run on the $s$-channel Higgs
resonance would be required for this purpose. Since some of the same operators
contribute to $Z$-pole precision observables, as well as to $HZ$, $WW$, and
$ZZ$ pair production cross sections, the operator constraints extracted from
the latter can be improved by performing a combined fit with $Z$-pole data.
This effect is more significant for circular $e^{+}e^{-}$ colliders than for
linear $e^{+}e^{-}$ colliders, since for the latter beam polarization helps to
disentangle the contributions of different operators in $HZ/WW/ZZ$ pair
production processes.

Figure 22: Precision reach on a subset of 4-fermion operators from a SMEFT
global fit at various future lepton colliders. ”LEP+SLC+SLD” represents
current measurements which are always combined in the future collider
scenarios. The horizontal white line for ILC illustrates the global fit
results when the pole observables from its Giga-$Z$ option are included.

Figure 22 shows a selection of results for a fit that combines a set of Higgs
and EW operators with 4-fermion operators. The latter are better constrained
at linear $e^{+}e^{-}$ than circular $e^{+}e^{-}$ machines, taking advantage
of the higher energy reach and beam polarizations222For now the global fit for
4-fermion operators did not include Muon Colliders.. A recent analysis of the
sensitivity of Muon Colliders to new 4-fermion interactions can be found in
Ref. [81]. Low-energy measurements (from fixed-target neutrino and electron
scattering, tau and meson decays) are needed to close the fit for four-fermion
operators. For complete results of the combined fit with 4-fermion operators,
see Ref. [16].

Not only four-fermion operators, but also top-quark electroweak operators are
best constrained at lepton colliders with $\sqrt{s}\geq 500$ GeV, and
measurements at two or more values of $\sqrt{s}$ are crucial for breaking
degeneracies. Many constraints on top-quark operators are improved by
combining $e^{+}e^{-}$ and (HL-)LHC inputs and exploiting synergies between
them. See EF04 Topical Group report for more information [16].

Figure 23: 95% CL exclusion reach of different colliders on four-fermion
contact interactions from the operator $O_{2B}$ (numbers for ESG are taken
from Ref. [130]). Figure 24: 95% CL exclusion reach of different colliders on
the leptoquark model parameters. Only future $e^{+}e^{-}$ scenarios with
energies below the $t\overline{t}$ threshold have been considered since the
analysis did not include any top-quark observables.

The results from global fits can be also interpreted in terms of constraints
on simple BSM benchmark models. Figure 23 is one example for a scenario where
a flavor-universal 4-fermion contact interaction is introduced, which can be
described by the single operator $O_{2B}$ [131]. A second example is shown in
Fig. 24, which extends the SM by two leptoquark fields, an SU(2) singlet with
Yukawa coupling $\lambda_{1}$ and mass $M_{1}$, and an SU(2) triplet with
Yukawa coupling $\lambda_{3}$ and mass $M_{3}$ [132, 133]. This model
generates several 4-fermion interactions, and the bounds on the latter can be
translated to bounds on the model parameters shown in the figure, where the
constraining power of future $e^{+}e^{-}$ colliders is manifest. More examples
on composite Higgs models and $Z^{\prime}$ models are also discussed in the
BSM Topical Group report [18].

###  5 QCD and Strong Interactions

The fundamental theory of strong interactions, QCD, plays a unique role in the
Standard Model. Being a confining gauge theory, it is an interesting quantum
field theory to study in its own right. It is also a crucial tool to enable
discovery at every high-energy collider. It predicts a rich panoply of
phenomena associated with both perturbative and non-perturbative dynamics of
the strong interactions. Continued success of the high-energy and nuclear
physics research program hinges on an improved understanding of both regimes,
as well as the dynamical transition between them.

Future SM measurements and new physics searches will allow the exploration of
new kinematic regions, such as very high transverse momentum and very forward
rapidities, where large scale hierarchies may induce hitherto unseen QCD
effects. The upcoming era — featuring the HL-LHC, Belle II, the EIC, new
advances in theory including lattice QCD, and potentially a Higgs factory —
will be a new golden age for QCD, easily rivaling the 1990’s when the
Tevatron, HERA, and LEP were all operating.

Measurements of jet, heavy-quark, and top-quark cross sections at the HL-LHC
will test perturbation theory to an unprecedented level, and constrain PDFs
[134] as well as the strong coupling $\alpha_{s}$ [99]. Accurate prediction of
QCD radiative effects will remain a key factor in precision measurements of
the $W$-boson and top-quark masses, and of the weak mixing angle at hadron
colliders, as discussed in Sec. 3. Versatile techniques are refined at the LHC
to analyze the internal particle composition of hadronic jets [135, 136].
These techniques are widely employed to minimize the impact of pileup, probe
fundamental and emergent properties of the strong force, enhance precision of
measurements of highly-Lorentz-boosted particles, and extend the sensitivity
of searches for new particles (cf. Sec. 11) [137].

Detection of the decay products of far-forward hadrons at the proposed Forward
Physics Facility (FPF) at the HL-LHC (Sec. 3) would enable QCD studies and BSM
searches in a novel high-energy regime [138, 139]. The FPF would observe
neutrinos of all flavors, and possibly new particles, on its own or in
coincidence with ATLAS, leading to deeper understanding of small-$x$ dynamics,
forward heavy-flavor production, neutrino scattering in the TeV range, and the
structure of nuclear matter.

Due to their QCD-neutral initial state, $e^{+}e^{-}$ colliders offer the
cleanest environment to study QCD dynamics. Belle II will perform various
measurements at low and medium energies [140], such as the cross section for
$e^{+}e^{-}\to\text{hadrons}$ (in particular two pions), from which the
leading-order hadronic contribution to the anomalous magnetic moment of the
muon can be extracted. Unprecedented statistical precision combined with low
backgrounds will enable highly impactful studies of factorization, QCD
evolution, and multidimensional correlation functions, which will help to
constrain MC models of hadronization for the HL-LHC program. There has been
much progress since LEP in understanding hadronic final states at $e^{+}e^{-}$
colliders, driven by techniques to analyze jet substructure and energy
correlation functions. Measurements of event shapes will enable precision
extractions of $\alpha_{s}$ [141, 142]. These colliders offer a particular
advantage of producing pure samples of gluon jets via the process
$e^{+}e^{-}\to hZ$, with $Z$ decaying to leptons, and the Higgs boson decaying
to $gg$ [143]. Better understanding of $b$-quark showering and hadronization,
as well as $b$-quark production by secondary gluons, will be important: these
are leading sources of systematic uncertainty in the measurement of the
$b$-fraction in hadronic decays ($R_{b}$) and the $b$ forward-backward
asymmetry in $Z$ decays [144].

Proposed Muon Colliders offer a physics reach for discoveries that is similar
to that of proposed high-energy hadron colliders. While operating at few-TeV
energies, the Muon Colliders maintain appealing experimental aspects of lepton
collider environments, e.g. a lack of pileup and underlying event. Advanced
pileup mitigation techniques studied at the LHC could provide versatile
handles to remove beam-induced background contamination during reconstruction
[145, 146, 147, 148].

The EIC physics program [149], dedicated to exploration of hadronic matter,
has strong synergies with the HL-LHC and FPF. The EIC can precisely measure
the hadronic structure in deep inelastic scattering (DIS) through both
neutral- and charged-current reactions, to perform spin-dependent three-
dimensional tomography of nucleons and various ion species using highly-
polarized beams for electrons and light ions. With its variable center-of-mass
energy and excellent detection of final hadronic states, the EIC can resolve
the flavor composition of unpolarized proton PDFs in the kinematic region of
relevance for BSM searches at the HL-LHC, but at QCD scales of only a few
(tens of) GeV. The EIC program will stimulate the development of theoretical
and numerical tools for QCD at the interface between particle and nuclear
physics.

Proposed lepton-hadron colliders operating in the TeV energy range (Muon-Ion
Collider [150], LHeC [151, 152], FCC-eh [153]) would be both machines for
subpercent-level measurements of $\alpha_{s}$, nucleon structure, EW and Higgs
couplings, as well as discovery machines to search for new physics such as
compositeness and leptoquarks. Future hadron-hadron colliders, including the
FCC-hh [41] operating at 100 TeV, would open unprecedented opportunities for
precision measurements in perturbative and non-perturbative QCD. Their physics
program would require innovative developments both in particle detection and
QCD theory (Sec. 29), such as parton distributions for electroweak bosons,
predictions for boosted final states inside jets, and new types of event
generators.

Table 10: Summary of the Les Houches precision wish-list for hadron colliders [154]. HTL stands for calculations in heavy top limit, VBF* stands for structure function approximation. process | known | desired
---|---|---
$pp\to h$ | N3LOHTL, N2LO${}_{\rm QCD}^{(t)}$, N(1,1)LO${}^{\rm(HTL)}_{{\rm QCD}\otimes{\rm EW}}$ | N4LOHTL (incl.), N2LO${}_{\rm QCD}^{(b,c)}$
$pp\to h+j$ |  N2LOHTL, NLOQCD, N(1,1)LOQCD⊗EW |  N2LOHTL $\\!\otimes\,$NLOQCD + NLOEW
$pp\to h+2j$ | NLOHTL $\\!\otimes\,$LOQCD | N2LOHTL $\\!\otimes\,$NLOQCD + NLOEW,
| N3LO${}_{\rm QCD}^{(\rm VBF^{*})}$ (incl.), N2LO${}_{\rm QCD}^{(\rm VBF^{*})}$, NLO${}_{\rm EW}^{(\rm VBF)}$ | N2LO${}_{\rm QCD}^{(\rm VBF)}$
$pp\to h+3j$ |  NLOHTL, NLO${}_{\rm QCD}^{(\rm VBF)}$ |  NLOQCD + NLOEW
$pp\to Vh$ | N2LOQCD + NLOEW, NLO${}_{gg\to HZ}^{(t,b)}$ |
$pp\to Vh+j$ |  N2LOQCD |  N2LOQCD \+ NLOEW
$pp\to hh$ | N3LOHTL $\\!\otimes\,$NLOQCD | NLOEW
$pp\to h+t\overline{t}$ |  NLOQCD + NLOEW, N2LOQCD (off-diag.) |  N2LOQCD
$pp\to h+t/\overline{t}$ | NLOQCD | N2LOQCD, NLOQCD + NLOEW
$pp\to V$ |  N3LOQCD, N(1,1)LOQCD⊗EW, NLOEW |  N3LOQCD + N(1,1)LOQCD⊗EW, N2LOEW
$pp\to VV^{\prime}$ | N2LOQCD + NLOEW , + NLOQCD ($gg$) | NLOQCD ($gg$,massive loops)
$pp\to V+j$ | N2LOQCD + NLOEW | hadronic decays
$pp\to V+2j$ |  NLOQCD + NLOEW , NLOEW |  N2LOQCD
$pp\to V+b\overline{b}$ | NLOQCD | N2LOQCD + NLOEW
$pp\to VV^{\prime}+1j$ |  NLOQCD + NLOEW |  N2LOQCD
$pp\to VV^{\prime}+2j$ | NLOQCD (QCD), NLOQCD + NLOEW (EW) | Full NLOQCD + NLOEW
$pp\to W^{+}W^{+}+2j$ | Full NLOQCD + NLOEW |
$pp\to W^{+}W^{-}+2j$ | NLOQCD + NLOEW (EW component) |
$pp\to W^{+}Z+2j$ | NLOQCD + NLOEW (EW component) |
$pp\to ZZ+2j$ | Full NLOQCD + NLOEW |
$pp\to VV^{\prime}V^{\prime\prime}$ |  NLOQCD, NLOEW (w/o decays) |  NLOQCD + NLOEW
$pp\to W^{\pm}W^{+}W^{-}$ | NLOQCD \+ NLOEW |
$pp\to\gamma\gamma$ |  N2LOQCD + NLOEW |  N3LOQCD
$pp\to\gamma+j$ | N2LOQCD + NLOEW | N3LOQCD
$pp\to\gamma\gamma+j$ |  N2LOQCD + NLOEW, + NLOQCD ($gg$ channel) |
$pp\to\gamma\gamma\gamma$ | N2LOQCD | N2LOQCD + NLOEW
$pp\to 2$ jets |  N2LOQCD, NLOQCD + NLOEW |  N3LOQCD + NLOEW
$pp\to 3$ jets | N2LOQCD + NLOEW |
$pp\to t\overline{t}$ |  |  N2LOQCD (w/ decays)+ NLOEW (w/o decays)
---
NLOQCD + NLOEW (w/ decays, off-shell effects)
N2LOQCD
|  N3LOQCD
---
$pp\to t\overline{t}+j$ |  |  NLOQCD (w/ decays, off-shell effects)
---
NLOEW (w/o decays)
|  N2LOQCD + NLOEW (w/ decays)
---
$pp\to t\overline{t}+2j$ |  |  NLOQCD (w/o decays)
---
|  NLOQCD + NLOEW (w/ decays)
---
$pp\to t\overline{t}+Z$ |  |  NLOQCD + NLOEW (w/o decays)
---
NLOQCD (w/ decays, off-shell effects)
|  N2LOQCD + NLOEW (w/ decays)
---
$pp\to t\overline{t}+W$ |  |  NLOQCD + NLOEW (w/ decays, off-shell effects)
---
|  N2LOQCD + NLOEW (w/ decays)
---
$pp\to t/\overline{t}$ |  |  N2LOQCD*(w/ decays)
---
NLOEW (w/o decays)
|  N2LOQCD + NLOEW (w/ decays)
---
$pp\to tZj$ |  |  NLOQCD + NLOEW (w/ decays)
---
|  N2LOQCD + NLOEW (w/o decays)
---

####  1 Perturbative QCD

Precision calculations. Perturbative precision calculations are crucial for
measurements of SM parameters and a key ingredient for the reliable estimation
of SM backgrounds to new physics searches. They also serve as an input to
precision simulations in modern MC event generators for collider physics [5].
There has been significant recent progress in the computation of QCD radiative
corrections [155, 156, 157, 154]. Several groups have used different
approaches to achieve the first $2\rightarrow 3$ N2LO calculations of hadron
collider processes. There have also been significant steps forward in the
development of improved infrared subtraction schemes including methods to deal
with higher-multiplicity processes at N2LO. The computation of full SM
corrections has seen major improvements as well [154]. A summary of the state
of the art and targets for future measurements is shown in Table 10. The
listed calculations correspond to the next frontier in cross section
calculations for a multitude of processes, and their completion will elevate
sensitivity of tests of the Standard Model and new physics searches at the HL-
LHC by providing more accurate predictions for signal and background
scattering processes. This Les Houches precision wish-list has served as a
summary and repository for the higher-order QCD and EW calculations relevant
for high-energy colliders, providing a crucial link between theory and
experiment.

Strong coupling. The strong coupling, $\alpha_{s}$, is a fundamental parameter
of the SM and the least well known of its gauge couplings. The uncertainty on
$\alpha_{s}$ will be one of the limiting factors in many measurements
including Higgs couplings at the HL-LHC. BSM physics can also impact
extractions of $\alpha_{s}$ in different ways and introduce tensions between
their results. The relative uncertainty in the current world average, assuming
no new physics or systematic discrepancies among extractions, is 0.8% and,
within the next decade, can be reduced to $\approx$ 0.4% (Table 11) [99]. This
requires completing the necessary pQCD calculations and control of various
commensurate factors to the same level. Many lattice QCD (LQCD) methods have
been developed to extract $\alpha_{s}$, and to provide systematic checks of
these LQCD methods [158, 159, 160, 161]. There are proposals to apply similar
checks to phenomenological determinations [162, 163, 99].

Table 11: Summary of current and projected future (within the decade ahead or, in parentheses, longer time scales) uncertainties in the $\alpha_{s}(m_{Z})$ extractions used today to derive the world average of $\alpha_{s}$ [99].  |  Relative $\alpha_{s}(m_{Z})$ uncertainty
---|---
Method | Current | Near (long-term) future
(1) Lattice | $0.7\%$ | $\approx\,0.3\%~{}(0.1\%)$
(2) $\tau$ decays | $1.6\%$ | $<1.\%$
(3) $Q\overline{Q}$ bound states | $3.3\%$ | $\approx\,1.5\%$
(4) DIS & global PDF fits | $1.7\%$ |  $\approx\,1\%$ (0.2%)
(5) $e^{+}e^{-}$ jets & evt shapes | $2.6\%$ |  $\approx\,1.5\%$ ($<1$%)
(6) Electroweak fits | $2.3\%$ |  ($\approx\,0.1\%$)
(7) Standalone hadron collider observables | 2.4% | $\approx\,1.5\%$
World average | $0.8\%$ |  $\approx\,0.4\%$ (0.1%)

The FCC-ee, which combines $3\times 10^{12}$ $Z$ bosons decaying hadronically
at the $Z$ pole, and a $\sqrt{s}$ calibration to tens of keV accuracy [164],
would provide measurements with unparalleled precision. The FCC-ee extraction
of $\alpha_{s}(m_{Z})$ with 0.1% uncertainty [165] will enable searches for
small deviations from SM predictions that could signal the presence of new
physics contributions. Dedicated high-luminosity $e^{+}e^{-}$ runs at the $Z$
pole would also enable further precision tests of QCD through the study of the