renormalization group running of the bottom-quark mass [166, 167]. Future $ep$
collider experiments would also provide many opportunities for precision
determinations of $\alpha_{s}$. The EIC [168, 149] and EicC [169] will provide
new high-luminosity data that could lead to a few percent uncertainty level
[149]. The LHeC [151, 152] would provide hadronic final-state observables
covering a considerably larger range than was possible at HERA. It could
determine $\alpha_{s}$ from inclusive DIS data alone, something not feasible
with HERA data, and an experimental uncertainty possibly reduced to 0.2%
[152].

Jet substructure. Quark- and gluon-initiated jets are used in measurements of
$\alpha_{s}$, the extraction of universal objects within factorized QCD, and
for tuning parton-shower MC generators. They are statistically distinguishable
due to their different fragmentation processes and can be separated using new
tools such as jet substructure [170, 171, 172, 173, 135, 174, 136, 175].
Charm- and bottom-quark jets, such as in the $h\rightarrow b\overline{b}$ and
$h\rightarrow c\overline{c}$ final states, can be effectively separated from
other jets due to the long lifetime of the heavy quark and the mass of the
heavy-flavored hadrons [176, 177, 178, 179, 180, 181, 182]. Identifying these
types of jets is a standard benchmark for development of new classical and
machine learning-based jet taggers and can help enhance certain BSM signals
[183, 184, 155]. Tagging has not realized its full potential due to large
uncertainties in the modeling of gluon jets [183]. High purity samples of
gluon jets provided by future lepton colliders through the process
$\ell^{+}\ell^{-}\to h[\to gg]Z[\to\ell\ell]$ would significantly change this
situation [143].

Jet substructure techniques are usually applied to identify Lorentz-boosted
massive particles such as $W$, $Z$, $h$ bosons, top quarks, and BSM particles
in complex final states. Many collider scenarios also result in $h$, $W$, and
$Z$ bosons radiating off of very high energy jets (“Weak-strahlung”). There
may also be top quarks produced via gluon splitting to $t\overline{t}$ within
a jet that originates from light quarks or gluons. Identification of these
signatures will be crucial for future high-energy colliders [137].
Unconventional signatures include cases where jets are composed of leptons and
hadrons, only leptons, only photons, hadrons and missing transverse energy
etc. In addition to the jet kinematics and substructure, the jet timing
information [185] and other information can be used for classification.
Examples include jets containing one or more hard leptons [186, 187, 188, 189,
190], displaced vertices [191, 188, 192], hard photons [193, 194], or
significant missing transverse momentum [195, 196]. Some of these anomalous
signatures are already starting to be explored at the LHC [197, 198, 199, 200,
201].

The jet substructure program has led to the introduction of techniques that
systematically remove low-energy soft radiation [170, 171, 172, 173, 202, 203]
and can significantly reduce the dependence of observables on non-perturbative
QCD effects. For a generic infrared and collinear safe observable, one can
measure its “groomed” counterpart, which will be infrared and collinear safe.
Although these observables are theoretically cumbersome to compute, they can
be very useful, for example for measurements of $\alpha_{s}$.

Finer calorimeter granularity [204, 205], more hermetic coverage of tracking
detectors, and precise timing information are expected to improve substructure
measurements. In particular, at future Muon Colliders, such detectors in
addition to detectors that specifically target the identification of particles
originating from beam background, can reduce the impact of beam background on
jet substructure observables.

New observables. Measurements of the flow of radiation, traditionally studied
using event shapes or energy correlation functions, provide some of the most
informative tests of QCD [206]. Energy correlators exhibit simple structures
in perturbation theory [207, 208, 209, 210, 211, 212]. Their measurements at
future colliders would provide remarkable insights into the dynamics of jets
and hadronization [213, 206].

Modern measurements rely strongly on the use of particle flow and tracking
information. However, only observables that are completely inclusive over the
spectrum of final states can be computed purely from perturbation theory. The
non-perturbative input needed for theoretical predictions of track-based
observables is universal and can be parametrized by so-called “track
functions” [214, 215], which describe the fraction of energy carried by
charged particles from a fragmenting quark or gluon. Recently it has been
shown how to compute jet substructure observables at high precision by
incorporating track functions [216, 217]. Such calculations give promise for
precision jet substructure measurements at the HL-LHC. Track functions could
be measured precisely at the ILC and other future $e^{+}e^{-}$ colliders.

Table 12: Top part: PDF-focused topics explored in Snowmass 2013 [218] and 2021 studies [134]. Bottom part: a selection of new critical tasks to develop a new generation of PDFs that meet the targets of the HL-LHC physics program. TOPIC |  STATUS, Snowmass’2013 |  STATUS, Snowmass’2021
---|---|---
Achieved accuracy of PDFs |  N2LO for evolution, DIS and vector boson production |  N2LO for all key processes; N3LO for some processes
PDFs with NLO EW contributions |  MSTW’04 QED, NNPDF2.3 QED |  LuXQED and other photon PDFs from several groups; PDFs with leptons and massive bosons
PDFs with resummations |  Small x (in progress) |  Small-x and threshold resummations implemented in several PDF sets
Available LHC processes to determine nucleon PDFs |  $W/Z$, single-incl. jet, high-$p_{T}$ $Z,$ $t\overline{t}$, $W+c$ production at 7 and 8 TeV |  \+ $t\overline{t}$, single-top, dijet, $\gamma/W/Z+$jet, low-Q Drell Yan pairs, … at 7, 8, 13 TeV
Current, planned & proposed experiments to probe PDFs |  LHC Run-2 |
DIS: LHeC |  LHC Run-3, HL-LHC |
DIS: EIC, LHeC, MuIC, … |  |
Benchmarking of PDFs for the LHC |  PDF4LHC’2015 recommendation in preparation |  PDF4LHC’21 recommendation issued
Precision analysis of specialized PDFs |  |  Transverse-momentum dependent PDFs, nuclear, meson PDFs
NEW TASKS in the HL-LHC ERA
Obtain complete N2LO and N3LO predictions for PDF-sensitive processes |  Improve models for correlated systematic errors |  Find ways to constrain large-x PDFs without relying on nuclear targets
Develop and benchmark fast N2LO interfaces |  Estimate N2LO/N3LO theory uncertainties |  New methods to combine PDF ensembles, estimate PDF uncertainties, deliver PDFs for applications

####  2 Non-perturbative QCD

Parton distribution functions in the nucleon. An overwhelming number of
theoretical predictions for hadron colliders require PDFs [219, 220, 221, 222,
223, 224, 225, 226, 227, 228, 229], the non-perturbative functions quantifying
probabilities for finding quarks and gluons in hadrons in high-energy
scattering processes. PDFs contribute to precise measurements of the QCD
coupling constant, heavy-quark masses, weak boson mass, and electroweak
flavor-mixing parameters. PDFs often introduce the dominant source of
uncertainty in collider experiments, such as in the CDF II high-statistics
measurement of the $W$-boson mass [85]. Reducing these uncertainties requires
continuous benchmarking and improvements of the theoretical framework [230,
231]. The accurate knowledge of PDFs is also critical in searches for BSM
interactions.

Table 12 illustrates the progress made on the PDF determinations since the
previous Snowmass Summer Study in 2013 [218]. Details are presented in the
Snowmass PDF whitepaper [134]. The N2LO QCD accuracy has become the standard
for the modern nucleon PDFs, with N3LO being on the horizon within the next
decade. In addition, some PDF sets for precision physics include photon PDFs
and QCD resummations. Current PDF predictions for parton luminosities agree
within uncertainties at invariant masses $30\lesssim m_{X}\lesssim 10^{3}$
GeV, relevant e.g. for Higgs and gauge-boson production, but in the gluon
sector (gluon-gluon and gluon-quark parton luminosities), differences are seen
at large masses [134]. These differences are a consequence of both methodology
and data sets included in PDF fits. The available PDF ensembles account for a
combination of experimental, theoretical, and methodological uncertainties
[232] in different ways, and, as a result, the provided PDFs can differ. The
PDF-dependent cross sections can differ as well by the amounts exceeding the
missing N3LO contributions.

The bottom part of Table 12 lists new tasks for the PDF analysis that emerge
in the era of precision QCD. While the most precise N2LO or even N3LO
theoretical cross sections should be preferably used, the accuracy of the
theoretical predictions in the fits also depends on other factors. For complex
N2LO/N3LO calculations, their fast approximate implementations (such as fast
N2LO interfaces) must be developed. Propagation of correlated systematic
errors into PDFs is a challenging task that requires combined efforts of
experimentalists and theorists. Control of uncertainties requires, in
particular, to either fit the experiments that are minimally affected by the
unknown factors (for example, to include cross sections only on proton, rather
than on nuclear targets), or to estimate the associated uncertainties from
these factors directly in the fit. The PDF uncertainties must representatively
reflect all PDF behaviors compatible with the fitted data [233]. PDFs must be
developed for a wide range of users in a format that optimizes for accuracy,
versatility, and speed across a broad range of applications — a highly non-
trivial task.

Figure 25: Examples of projections for PDF uncertainties in the HL-LHC era.
Left: Uncertainties for N2LO Higgs production via gluon fusion at
$\sqrt{s}=14$ TeV obtained with published PDF4LHC15 N2LO PDFs [75] (green
band) and after additional constraints are imposed on these PDFs using
simulated HL-LHC data in two scenarios (red and blue bands) [234]. Right: 90%
CL uncertainty ellipses for N2LO predictions for $gg\to h$ and $t\overline{t}$
production at the LHC with $\sqrt{s}=14$ TeV obtained using CT18 N2LO PDFs
[225] and after imposing simulated constraints from inclusive DIS at the EIC
[149].

Recent studies [234, 149] provide projections using various techniques for the
reduction of PDF uncertainties under anticipated near-future theoretical and
experimental developments. As an illustration, the left panel of Fig. 25
compares the current PDF uncertainty for $gg\to h$ production and its
reduction when simulated HL-LHC measurements are included in the conservative
(scen A) and optimistic (scen C) scenarios, using PDF4LHC15 N2LO PDFs [75] as
the baseline. The right panel shows an analogous projection for the reduction
of the PDF uncertainty on the SM Higgs and $t\overline{t}$ cross sections at
the LHC upon including the simulated measurements in DIS at the EIC, this time
using the CT18 N2LO framework [225]. The ability of the LHC measurements to
reduce the PDF uncertainty critically depends on the control of systematic
effects. Deep inelastic scattering and hadroproduction at the EIC will
constrain the PDFs for the LHC high-mass BSM searches most directly and
without systematic or new-physics factors relevant at the LHC.

Predicting hadron structure in lattice QCD. As lattice QCD techniques advance
in computations of PDFs from first principles, unpolarized phenomenological
PDFs in the nucleon serve as important benchmarks for testing lattice QCD
methods [235, 236]. Namely, precisely determined phenomenological PDFs in the
nucleon serve as a reference to validate lattice and non-perturbative QCD
calculations. The combination of the observation-driven PDF analysis and
lattice QCD is thus especially promising and drives related studies of three-
dimensional structure of baryons and mesons, including dependence on
transverse momentum and spin. Figure 26 (left) shows the impact of lattice QCD
calculations on a quantity affecting precision measurements at hadron
colliders — the difference between the strange quark and antiquark PDFs. Such
novel calculations can significantly constrain quantities that are difficult
to assess with conventional PDF estimates. Figure 26 (right) illustrates that
recent lattice QCD calculations are now able to predict quark PDFs. Lattice
QCD is most powerful in predicting various QCD charges and distributions of
partons carrying 10% of the hadron’s energy or more.

Figure 26:  Left: Impact of constraints from lattice QCD (black dashed area)
on the difference between strange quark and antiquark PDFs in a CT18As N2LO
fit [237]. The red (green) error bands correspond to the current (reduced by
50%) lattice QCD errors. Right: determination of a quark PDF in a pion using a
combination of experimental and lattice QCD data, and including threshold
resummation [238].

A Snowmass whitepaper [239] summarizes rapid advances in lattice QCD
calculations of PDFs and other QCD functions. New experiments and facilities
will pursue exploration of the three-dimensional structure described by
transverse-momentum–dependent distributions (TMDs) as well as generalized
parton distributions (GPDs) – hybrid momentum- and coordinate-space
distributions that bridge the conventional form factors and collinear PDFs.
These experiments will match the ongoing theoretical advancements that open
doors to many previously unattainable predictions, from the $x$ dependence of
collinear nucleon PDFs to TMDs [240, 241, 242, 243, 244] and related functions
[245, 246, 247, 248, 249], GPDs [250, 251, 252, 253], and higher-twist terms –
a progress that was not envisioned as possible during the 2013 Snowmass study.

There remain challenges to be overcome in lattice calculations, such as
reducing the noise-to-signal ratio, extrapolating to the physical pion mass,
and increasing hadronic boosts to suppress systematic uncertainties.
Computational resources place significant limitations on the achievable
precision, as sufficiently large and fine lattices are necessary to suppress
finite-size and higher-twist contaminating contributions. New ideas can bypass
these limitations. With sufficient support, lattice QCD can fill in the gaps
where the experiments are difficult or not yet available, improve the
precision of global fits, and provide better SM inputs to aid new-physics
searches across several HEP frontiers.

Hadronization and fragmentation functions. Hadronization is a process of
formation of detected final-state hadrons from partons. Governed by non-
perturbative dynamics, hadronization cannot be calculated analytically and is
elusive in lattice calculations. Accurate description of hadronization is,
however, absolutely indispensable for predicting production of hadronic
states.

Fragmentation functions (FFs) – the probabilities for producing hadrons from
fragementing partons – are instrumental for LHC experiments, as well as for
extracting the spin-(in)dependent nucleon structure [254] at Belle II and the
EIC. The Belle II program will emphasize investigation of full
multidimensional dependency of FFs with complex final states, such as
dihadrons or polarized hyperons, in order to map out their factorization
universality properties and kinematic dependencies. Tagging on such final-
state degrees of freedom allows more targeted access to the hadron structure
in semi-inclusive deep inelastic scattering (SIDIS) experiments, e.g. at JLab
and the EIC [255, 256, 257, 258], where TMD PDFs and FFs will be the primary
means to investigate the mechanism of hadronization in a 3D-picture [259].

Models of hadronization for the full event are necessary to predict background
and signal processes in various BSM searches. Currently, modeling of
backgrounds originating from light-quark fragmentation is mainly performed by
MC event generators. Tuning those generators to a precision needed for
discovery requires a model for correlated production of multiple hadrons that
can only be verified with clean $e^{+}e^{-}$ hadroproduction data. High-
luminosity data of this kind from Belle II, combined with data at other
energies from LEP and SLD, will provide a large lever arm in energy to
confidently extrapolate the hadronization model from low to LHC energies and
beyond.

Where neither the single-hadron FFs nor the MC event generators are sufficient
to model multi-particle fragmentation, semi-analytical approaches to account
for particle correlations gain more recognition. There have been significant
recent efforts to define many-particle correlation observables that may obey
factorization theorems, be sensitive to hadronization dynamics, can be
interpreted within hadronization models (e.g. a QCD string model), and might
be computed from QCD first principles in the future. Such correlation
observables are already a focus at the LHC [260].

Accurate knowledge of FFs in $e^{+}e^{-}$ collisions (especially, the gluon FF
[261]) is of utmost importance for having an accurate “QCD vacuum” baseline to
quantify “QCD medium” modifications of hadronization in proton-nucleus and
nucleus-nucleus collisions [262, 263]. Looking toward the future, an FCC-ee
would offer a clean radiation environment to systematically explore color
reconnection, hadronization, and multiparticle correlations (in spin, color,
space, momenta) — the non-perturbative final-state effects that currently rely
on phenomenological MC modeling and may limit the ultimate accuracy at hadron-
hadron colliders. In the $pp$ case, the color reconnection effects may
constitute 20–40% of the ultimately achievable uncertainty in the measurement
of the top mass [264] and can also impact limits in CP-violation searches in
$H\to W^{+}W^{-}$ hadronic decays. At the FCC-ee, $e^{+}e^{-}\to W^{+}W^{-}\to
q_{1}\overline{q}_{2}q_{3}\overline{q}_{4}$ and $e^{+}e^{-}\to t\overline{t}$
processes, with the top quarks decaying and hadronizing closely to one
another, would directly probe the color reconnection and interactions between
hadronic strings [265, 266, 267, 268].

####  3 Forward Physics

The LHC experiments opened access to a wide range of forward and diffractive
processes, driving advances in relevant QCD theory, such as charting the gluon
at very low $x$, revealing dynamics at high partonic densities, and testing MC
models for forward hadron production. Understanding small-$x$ dynamics in $pp$
collisions, already important at the (HL-)LHC [269, 127], is crucial for any
future higher-energy $pp$ collider such as FCC-hh [270, 271, 272, 273], where
standard electroweak processes, e.g. $W$ and $Z$ production, are dominated by
low–$x$ dynamics, and an accurate calculation of the Higgs production cross
section requires accounting for BFKL resummation [274, 275, 276, 277] or
partonic saturation [278].

Diffraction. Some configurations of final states with high forward
multiplicities, as well as those with the absence of energy in the forward
region (so-called rapidity gaps), in elastic, diffractive, and central
exclusive processes originate from purely non-perturbative reactions, while
others can be explained in terms of multi-parton chains or extensions of
perturbative QCD such as the BFKL formalism. These processes are interesting
for the exploration of electroweak and BSM physics. Understanding the elastic
cross section and diffraction better, and probing models for Odderon [279,
280, 281, 282, 283, 284, 285] and Pomeron exchange, will be key studies at the
HL-LHC, EIC, and any future hadron collider. Further progress in this
fundamental area requires the combination of experimental measurements,
including at the EIC and FPF, and theoretical work. The FPF also allows
exploration of BFKL evolution and gluon saturation.

Physics opportunities at the CERN Forward Physics Facility. Given its unique
configuration, the FPF [138, 139] would extend the coverage of the LHC
measurements (notably, the LHCb) at small $x$ by almost two orders of
magnitude at low $Q$, reaching down to $x\simeq 10^{-7}$ (Fig. 27). In its
proposed main configuration, the FPF will detect far-forward neutrinos,
produced from charm meson decays in a main LHC detector, by DIS on a tungsten
target. Therefore, FPF measurements would provide a bridge between the physics
program at the HL-LHC and that of a higher-energy $pp$ collider. Successful
interpretation of FPF measurements will require a coordinated program
including forward production at LHCb [286, 287, 288, 289], large-$x$ charge
current DIS at EIC [149], and small-$x$ scattering at the HL-LHC and future
DIS facilities such as the Muon-Ion Collider [150] and LHeC [152]. In turn,
with future experiments designed to detect neutrino interactions with energies
between several hundreds of GeV and a few TeV, an energy range that has not
been precisely probed for any neutrino flavor, the FPF will significantly
extend accelerator cross section measurements and will provide improved
predictions for key astroparticle physics processes, such as ultra-high energy
neutrino-nucleus and cosmic ray interaction cross sections.

Figure 27 also demonstrates that the FPF will be sensitive to very high–$x$
kinematics and in particular the intrinsic charm component of the proton
[290]. While charm production in $pp$ collisions is dominated by gluon–gluon
scattering, in the presence of a non–perturbative charm PDF in the proton, the
charm-gluon initial state may be dominant for forward $D$-meson production.
FPF measurements, as part of a broader physics program including LHCb and the
EIC, would provide complementary handles on high-$x$ intrinsic charm. At the
small-$x$ end, FPF observations would reduce the currently large uncertainties
on the expected flux of prompt neutrinos arising from the decays of charm
mesons produced in cosmic-ray collisions in the atmosphere [291, 292, 293,
294]. These represent an important background for astrophysical neutrinos at
neutrino telescopes such as IceCube [295] and KM3NET [296].

Figure 27: Left: The production and detection processes for forward $D$-meson
production at the HL-LHC followed by their decay into neutrinos falling within
the FPF acceptance. Right: The $(x,Q)$ regions (red ovals) that can be
accessed at the FPF via this process.

Neutrino-induced deep inelastic scattering (DIS) mediated by charged current
(CC), such as that at the FPF, provides essential flavor separation in global
PDF fits [297] that is not achieved solely in neutral-current (NC) DIS. Hence
FPF data can complement other planned experiments, such as the EIC. The
coverage for CC DIS on nuclei at the FPF in Fig. 27 broadly overlaps with that
for charged-lepton CC DIS at the EIC [149, 298]. Analogous information from
previous neutrino-induced DIS measurements on nuclear targets (NuTeV [299],
NOMAD [300], CCFR [301], CHORUS [302]) plays the key role in global PDF fits
of nucleon and nuclear PDFs (with the two related via nuclear corrections).
Inclusive CC DIS and, especially, semi-inclusive charm production in CC DIS
are the primary channels to probe the PDFs for strange quarks and antiquarks.
The strange-quark PDF offers insights about the non-perturbative proton
structure [303], while it also drives much of the PDF uncertainty in $W$ boson
mass measurements at the LHC [304]. On the experimental side, determination of
the (anti-)strangeness PDF has been a hot topic, as the different experimental
results somewhat favor different shapes for $s-$quark PDFs [305, 306, 225,
307]. The elevated PDF uncertainty from fitting such inconsistent experiments
propagates into various pQCD predictions [308, 134, 99].

####  4 Heavy Ions

The relativistic heavy-ion (HI) program prioritizes the studies of the quark-
gluon plasma (QGP), the partonic structure of nuclei, collectivity in small
collision systems [309, 310, 311, 312, 313], and nuclear electromagnetic (EM)
interactions [314, 315]. The HI studies have been a valuable part of the LHC
physics program, with a variety of cross-cutting connections to the other
areas of LHC physics summarized in Sec. 5. Reference [316] presents a detailed
plan for the HI measurements at the HL-LHC. These measurements will greatly
benefit from planned detector upgrades for ALICE, ATLAS, and CMS. ALICE FoCal,
a forward calorimeter that will allow us to probe parton distributions down to
lower Bjorken-$x$ values than are previously accessible [317]. The increased
pseudorapidity acceptance of charged particle tracking will be a boon to bulk
particle detection. The upgraded Zero Degree Calorimeters (ZDCs) [318, 319]
will improve triggering and identification for ultraperipheral collisions (see
Sec. 5). The addition of time-of-flight particle identification capability in
the CMS timing detector [320] will allow differentiation among low-momentum
pions, kaons, protons, and charmed states. The planned major upgrade of the
ALICE detector for HL-LHC Run 5 (ALICE 3 [321]) will enable far-reaching
studies of the QGP properties. The HL-LHC HI program is synergistic with the
HI programs at RHIC and the EIC. The sPHENIX detector [322, 323, 324] at RHIC
will start HI data-taking in 2023 with the aim to collect quality heavy-flavor
meson, quarkonium, and jet data in Au-Au collisions.

Hard probes. High momentum-transfer interactions between partons in the
nuclear medium serve as short-distance probes of the QGP. They can reveal the
impact of QGP on color charges of fast-moving partons or slow-moving heavy
quarks, commonly resulting in the attenuation of the jets [325, 326, 327, 328,
329, 330, 331, 332, 333, 334, 335, 336, 337, 338], or modifications of the jet
substructure [339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350,
351], referred to as jet quenching. ALICE, ATLAS, and CMS measurements will
significantly reduce statistical and systematic uncertainties in key
measurements of medium modification of light or heavy quark jets produced in
association with $\gamma$, $Z$, or $\mathrm{D^{0}}$ [324, 352]. Studies of
medium dependence of jet quenching at the HL-LHC would be facilitated by
collecting large $pp$ samples of events with the same signatures and low
pileup. By comparing the LHC and RHIC data, we aim to constrain the
temperature dependence of the transport coefficients of QGP.

The main HL-LHC experiments will also increase capabilities for detection of
heavy-flavor (HF) mesons [353, 354, 355, 356, 357, 358, 359, 360, 361] and
quarkonia [362, 363, 364, 365, 366, 367, 368, 369]. Heavy quarks, $c$ and $b$,
are unique, slow-moving QGP probes produced during the early stages of the
hard collision. In the medium, they radiate less energy than light flavors due
to the dead-cone effect. Observations of HF mesons elucidate mechanisms for
their thermalization, Debye color screening and recombination [370, 371, 372]
inside the QGP. Measurements of the $p_{\mathrm{T}}$ dependence of the
quarkonium nuclear modification factors ($R_{\mathrm{AA}}$) will discern
whether high-$p_{\mathrm{T}}$ quarkonium formation is driven by the Debye
screening mechanism or by energy loss in the medium [373]. Improved elliptic
flow measurements of charm mesons in p-Pb [374] and of HF decay muons [375]
and $\Upsilon(1S)$ [373] in Pb-Pb collisions will provide insights both on the
degree of thermalization of HF quarks at low $p_{\mathrm{T}}$ and on the
recombination of bottomonia from deconfined $b$ quarks in the QGP. Production
of strange $B$ mesons and charm baryons in $pp$ and Pb-Pb collisions [373]
will investigate the interplay between the predicted enhancement of strange
quark production and the quenching mechanism for beauty quarks, and on the
contribution of recombination of HF quarks with lighter quarks to
hadronization in HI collisions. Precise measurements of beauty mesons in p-Pb
collisions [373] will help to establish the relative contributions of
hadronization and nuclear-matter effects. Interpretation of these HL-LHC
observations will be facilitated by the sPHENIX measurements of HF hadron
spectra, particle multiplicities, and azimuthal anisotropy at lower energy,
allowing especially to strongly constrain the HF diffusion coefficient and its
temperature dependence.

Hadronic structure. Abundant production of light (anti-)nuclei at ALICE opens
promising opportunities. As in the cases of light nuclei and charmonium, the
statistical hadronization or coalescence assumption can render a unique
insight into the structure (e.g. tetraquark or molecular state) of exotic
hadrons, such as $X(3872)$ studied by LHCb in high-multiplicity $pp$
collisions [376] and by CMS in Pb+Pb collisions [377]. Those initial
measurements will be followed up with high-statistics data in LHC Runs 3 and
4. In LHC Run 5, the ALICE 3 detector would precisely measure multi-charm
baryons, expanding the hadronization studies in Runs 3 and 4. ALICE 3 would be
a perfect tool to study the formation of light nuclei, hyper-nuclei, super-
nuclei, and properties of exotic states, e.g. $X(3872)$ and the newly
discovered $T_{cc}^{+}$.

Collectivity of small and large systems will be studied with high-statistics
samples of pp, pPb, and PbPb events at the HL-LHC. The pivotal upgrades of
ATLAS and CMS trackers will extend observations of charged particles to a wide
pseudorapidity range ($|\eta|<4$). For small systems, we expect a crucial
improvement in the determination of the created system’s size by measuring the
Hanbury-Brown-Twiss (HBT) radii [375]. Azimuthally-dependent femtoscopy will
measure the spatial ellipticity of the medium at freeze-out, e.g., to find the
normalized second-order Fourier component of the transverse HBT radius as a
function of the flow’s magnitude. The extended $\eta$ acceptance in Run 4 will
refine the characterization of the rapidity dependence of factorization
breaking. Better measurements of the forward-backward multiplicity correlation
and multi-particle cumulants will deepen our understanding of the early-stage
medium fluctuations [375].

Figure 28:  Prerequisites for achieving percent-level accuracy in QCD
calculations.

####  5 Cross-Cutting QCD

Successes across many areas depend on future developments at the intersections
of QCD and other domains. To take advantage of precise perturbative QCD
calculations, commensurate advances must be achieved in accuracy of long-
distance QCD and electroweak contributions, event generation, error analysis.
These tasks require collaborations of experimentalists and theorists, model-
builders and QCD experts, and, more broadly, support for the QCD
infrastructure that adapts theoretical tools for experimental analyses
according to standard protocols. This subsection presents examples of such
cross-cutting issues from the QCD physics report [17].

Comprehensive uncertainty estimates. Measurements of the $W$-boson mass and
other LHC precision observables are affected by many systematic uncertainties
arising from both the experimental and theoretical sides. NNLO PDFs used in
many analyses are sensitive to modeling of experimental systematics that is
not under full control in the PDF fits [134]. Accuracy of PDFs can be
increased if experimental groups publish more complete models of systematic
uncertainties [378]. To estimate these uncertainties, high-order calculations
for complex final states must be adapted to be fast and practical [379, 380,
381]. New types of complexity issues emerge in comparisons of multiparametric
models with many parameters to very large data samples expected at the LHC Run
3 and HL-LHC. Undetected biases due to non-representative exploration of
contributing systematic factors must be watched for, as has been recently
demonstrated on an example of a PDF global fit [233]. Figure 28 illustrates
that elevating the accuracy of QCD calculations to one percent requires both
individual precise theoretical calculations as well as accurate supporting
theoretical infrastructure that would allow to explore exhaustively the
relevant systematic factors. Reaching this target also requires agreed-upon
standards and practices for accuracy control at all stages of the analyses.

QCD in new physics searches and SMEFT fits. The energy reach of many BSM
searches at the HL-LHC depends on the interplay between precision calculations
of matrix elements and PDF analyses. At masses above a few TeV, production of
BSM particles requires large momentum fractions $x$, where PDF uncertainties
remain large due to a mix of influences from low statistics, nuclear
corrections, higher-twist contributions, intrinsic heavy-quark components.
Either forward particle production at the LHC or, often more cleanly, DIS at
the EIC can constrain PDFs in the large-$x$ region relevant for the TeV mass
range.

Figure 29:  Left: The 95% confidence level bounds on the plane of the Wilson
coefficients considered in Ref. [382] obtained using either fixed SM PDFs
(blue) or conservative SM PDFs that do not include high-energy data (green).
PDF uncertainties are included in the solid lines and not included in the
dashed lines. Results are compared to those obtained in a simultaneous fit of
SMEFT and PDFs, when the PDFs are allowed to vary when varying the values of
the Wilson coefficients (orange). Right: Compilation of exclusion ALPs limits
obtained by different studies [316, 383, 384], with limits from $\gamma\gamma$
scattering at ATLAS [385], CMS [386], the projected performance with ALICE 3
detector [387], and exclusive $\gamma\gamma$ production with tagged forward
protons at ATLAS and CMS-TOTEM [388, 389].

Searches for deviations from SM examined in the language of EFT (e.g. SMEFT
analyses) is an active research area (see Sec. 4 and EF04 report [16]).
Although the proton structure parametrized by PDFs is intrinsically a low-
energy input and should in principle be separable from the imprints of SMEFT
operators, the complexity of the LHC environment might intertwine them [382,
390, 391, 392, 393, 394]. As illustrated in the left Fig. 29, fitting the
SMEFT coefficients and PDFs together generally results in different
constraints than when the fixed PDFs are used. Constraints on either the PDFs
or EFT operators in low-energy experiments, such as the EIC, where some or all
new physics contributions are absent, can be crucial for disentangling the
SM/BSM degeneracies at the (HL-)LHC, especially for spin-dependent EFT
operators [395, 396].

Ultraperipheral ion collisions (UPCs) are electromagnetic interactions of
relativistic heavy ions, occurring when the nuclei pass by with impact
parameter $b>2R_{A}$, where $R_{A}$ is the nuclear radius [397, 398]. UPCs are
energy-frontier for collisions associated with photons since these photon
fields around ions interact with the opposing nucleus, in a photonuclear
interaction, or with each other, in a $\gamma\gamma$ collision. Quantitative
predictions for exclusive processes and/or small momentum fractions probed in
the UPCs require coordinated advancements in perturbative QCD, nuclear, and
electroweak theory, serving as an example of cross-cutting connections. In
addition, the UPCs can be used to search for BSM phenomena. ALICE, ATLAS, and
CMS have a suite of planned measurements of photonuclear reactions with
upgraded detectors. LHCb is well positioned for exclusive production studies
in UPCs in the forward direction. Exclusive photonuclear production of vector
mesons, jets, and heavy quarks at the highest center-of-mass energies yields a
tool to study nuclear parton densities (especially the gluon density) at
momentum fractions down to $x\gtrsim 10^{-6}$, where saturation may be most
pronounced . The expected performance at HL-LHC and the future plan have been
detailed in Refs. [399, 400]. In the $\gamma\gamma$ mode with tagging on two
intact initial hadrons, the LHC can reach unprecedented sensitivities to
quartic anomalous couplings, such as $\gamma\gamma\gamma\gamma$, $\gamma\gamma
WW$, $\gamma ZZ$, $\gamma\gamma\gamma Z$, $\gamma\gamma t\overline{t}$ only to
quote a few [401, 402, 403, 404, 405, 406, 407, 408, 389]. The final states in
$\gamma\gamma$ interactions at the LHC are remarkably clean (like at LEP),
resulting in sensitivities to quartic anomalous couplings and to the
production of axion-like particles (ALPs) at high masses that is better by 2-3
orders of magnitude than in usual LHC searches. Figure 29 (right) illustrates
the sensitivity to ALPs in $pp$ and heavy-ion interactions at the LHC, with
the coupling plotted as a function of the particle’s mass [384]. Finally, the
production of an electron and muon pairs via $\gamma\gamma$ interactions are
of great interest since they give access to $\tau$ pairs, which are sensitive
to BSM physics, including $\tau$ dipole moment, lepton compositeness or
supersymmetry [409].

Detectors and QCD theory for hadron colliders. Detectors for a possible 100
TeV hadron-hadron collider must implement innovative designs to achieve the
necessary precision when measuring the SM processes, while also precisely
reconstructing multi-TeV physics objects. They should be able to seek massive
hyperboosted objects inside highly collimated hadronic jets and to deal with a
factor-of-five larger pileup than at the HL-LHC. In theoretical predictions, a
predictive BFKL-like QCD formalism will be necessary to quantify parton
scattering at momentum fractions as low as $10^{-7}$. Electroweak gauge bosons
$W$ and $Z$, leptons, and top quarks will be copiously produced in a mix of
QCD and EW interactions and will need to be included into the PDFs together
with quarks and gluons [410]. Detailed studies of hadronic and electromagnetic
showers will be needed in the next few decades to achieve the best predictions
at future hadron collider energy and luminosity.

###  6 The physics beyond the Standard Model

There are abundant reasons why physics beyond the SM of particle physics is
likely and, in some cases, unavoidable. Such reasons are connected to the
fundamental questions that lie at the core of the energy frontier program, as
we discussed in Sec. 2, answering which is among the highest priorities of
particle physics. Current and future experiments at the energy frontier offer
unique capabilities to explore many of these questions.

In this context, energy frontier explorations follow three main leads:

  1. 1.

Studies of phenomena that have been observed but for which a fundamental
explanation is still lacking. These include:

     * •

What is the fundamental composition of Dark Matter?

     * •

What is the additional source of CP violation needed to explain the matter-
antimatter asymmetry observed in the universe?

     * •

Possible observations of BSM physics referred to broadly as Anomalies.

  2. 2.

Following guiding principles forming the basis of the successful stories
behind the SM and, more generally, of modern theoretical physics. These may
offer us insight on where the theoretical framework is “hinting” for a more
complete description of Nature, such as:

     * •

Naturalness.

     * •

The structure of flavor.

  3. 3.

Search for possible new phenomena that might not fit in the simplest
theoretical extensions of the SM and for which, as history has shown many
times, particle physics should maintain a wide open view for. This includes
addressing questions like:

     * •

Are there new interactions or new particles around or above the electroweak
scale?

     * •

Is lepton universality violated?

     * •

Are there long-lived or feebly-interacting particles which have evaded
traditional BSM searches?

     * •

Broadly speaking, how can we reduce biases in our searches and conduct them in
a more model-independent way?

Two main theoretical approaches in exploring BSM physics can be commonly
identified. The first consists in seeking self-consistent theories that aim to
address the questions above and can significantly boost our understanding of
the fundamental laws of Nature. These well-motivated models of BSM physics,
such as SUSY and Composite Models, are self-consistent to high-energy scales,
and are excellent test cases for exploring possible experimental signatures
and their interrelation. Looking beyond these prominent models, the landscape
of possible experimental and theoretically-motivated models and signatures is
very large. In this approach, well-defined but incomplete theories extend
specific areas without the expectation of full self-consistency. These
simplified models or portal models are in some cases simplifications of
complete theories. It is not practical nor useful to try to be exhaustive in
projecting the scientific output of projects targeting all such models.
Instead, we focus on a representative set of models and signatures that are
deeply connected with the fundamental questions above and represent a wide
range of physics that can be explored at the energy frontier. Such an approach
has the advantage of providing a manageable framework where different
experimental results can be easily compared and, eventually, mapped into the
parameter space of complete theories. However, the drawback is that those have
intrinsically a larger degree of arbitrariness and should be viewed as simpler
guiding frameworks for the more general exploration of BSM physics.

In this section, we summarize and chose a few representative benchmark models
and scenarios from the BSM Topical Group report [18]. These benchmarks include
DM-driven considerations, as well as exciting recent development on long-lived
particles. We discuss their implications for the current and future collider
programs.

####  1 Composite Higgs Models

Determining whether the Higgs boson is an elementary or composite particle
remains a fundamental open question. The idea of a composite Higgs boson is
attractive because it avoids the theoretical challenges associated with
explaining the relatively small mass of a fundamental scalar particle. A
composite Higgs boson requires a new strong gauge interaction whose coupling
becomes strong above the TeV scale and binds together new elementary
constituents. These constituents inevitably form not only the Higgs boson, but
also many other bound states, much like the structure seen with QCD dynamics.
In such models, the Higgs boson is the lightest bound state similar to the
pion of QCD, protected by an approximate global symmetry, and observing other
heavier resonances above the Higgs boson mass at the compositeness scale would
be a tell-tale sign of Higgs compositeness. Current searches generically
constrain the lowest lying resonances to be heavier than the TeV scale with
lower mass limits in the range of 1-3 TeV for resonances with spin 1/2, 1, and
2. In addition to the direct production of new resonances, Higgs compositeness
would also cause deviations in the couplings of the Higgs boson to gauge
bosons and the (composite) top quark. These deviations are inversely
proportional to the scale of compositeness and therefore require precision
measurements for detection.

The phenomenology of Composite Higgs models is mainly governed by two
parameters: the mass (compositeness) scale $m_{*}$ and the coupling $g_{*}$,
which sets the scale of the couplings in the EFT Lagrangian. The strongly
interacting model is expected to have $g_{*}>1$ couplings, while unitarity
requires $g_{*}<4\pi$. The Wilson Coefficients, defined in Ref. [42], can be
all parameterized in terms of this mass scale and coupling, modulo order 1
factors. Different colliders have complementary sensitivities to the various
operators ($C_{\phi}$, $C_{2w}$, $C_{w}$); these are summarized in the BSM
Topical Group report [18].

Figure 30: Exclusion (2-$\sigma$) sensitivity projections for future
colliders. Plot based on Refs. [411, 81].

Sensitivity to a toy Composite Higgs model from several future colliders is
shown in Fig. 30. Curves from HL-LHC, FCC-ee/eh/hh, and CLIC are taken from
Ref. [412]. Each Wilson coefficient has a different mass-dependence of the
coupling sensitivity. Sensitivity to $C_{\phi}$ arises primarily through
precision measurements of Higgs couplings; sensitivity to $C_{2w}$ arises from
measurements of high energy Drell-Yan events; and sensitivity to $C_{w}$ more
broadly comes from electroweak precision fits. Also shown is the direct search
sensitivity for a triplet vector $\rho$ resonance at FCC-hh. The sensitivity
from the 10 TeV Muon Collider is taken from studies of the tree level process
$\mu^{+}\mu^{-}\rightarrow hh\nu\overline{\nu}$ [81], which provides good
sensitivity for $C_{w}$ and $C_{2w}$, but not $C_{\phi}$. Sensitivity to
$C_{\phi}$ from Higgs coupling measurements at a Muon Collider [413] are
expected to be competitive, but are not shown here. We can also see the
complementarity between direct resonance searches and the precision
measurements on the SMEFT operators in this figure. This implies that if
future discoveries point to signals of composite models, we will be able to
use a whole class of operators to interpret other direct searches and pin down
the underlying theory.

####  2 SUSY Models

Supersymmety (SUSY) is a symmetry that extends the SM fields by adding a set
of partner fields with the same Yukawa couplings and gauge quantum numbers but
different spins. An extended Higgs sector is also required for SUSY. The
motivations for this symmetry include that it results in the unification of
gauge and Yukawa couplings at high energies, radiative effects directly lead
to electroweak symmetry breaking, and in some versions it naturally contains a
DM candidate. Furthermore SUSY appears in low-energy realizations of grand
unified theories and superstrings, that allow for a consistent quantization of
gravity. SUSY, however, cannot be an unbroken symmetry of nature, because
particles with the same mass but different spin are not observed. Instead SUSY
is assumed to be broken by a set of soft SUSY-breaking terms. These terms
govern the masses of the predicted SUSY partner particles.

There are many specific models within the SUSY framework. The sensitivity
studies presented here focus on $R$-parity conserving decays in the minimal
supersymmetric standard model (MSSM). In the MSSM, there is a lightest
supersymmetric particle (LSP) which is protected from decay by $R$-parity and
therefore contains a dark matter candidate. There are many other models
including those that violate $R$-parity in different ways, and the next-to-
MSSM (NMSSM) which includes an additional singlet to dynamically generate the
dimensionful $\mu$-term.

The Higgs-boson mass in the MSSM is strongly constrained but receives
logarithmic corrections due to stop-squark loops. To achieve the observed
Higgs mass, these loops need to be above order 1 TeV which is the scale just
being reached by the LHC (see Fig. 31). Conversely, for a small mixing between
the two stop squarks and $\tan\beta\gg 1$, the stop-quarks mass can be at most
5-10 TeV [414].

This wide variety of phenomena and the fact that it is widely studied make the
MSSM a good context to make comparison plots between different collider
scenarios. Figure 31 shows the comparative sensitivity in the MSSM for a
representative set of key points in the model space. It includes large mass
splittings for stop squarks, which are strongly produced, and two example
weakly-produced scenarios. The first weak production example is a classic
Wino-Bino model with large mass splitting, and the second is a Higgsino model
with a small mass splitting motivated by naturalness considerations. The
relevance of these plots goes beyond the SUSY context. The relative
sensitivity to weak and strong, large and small mass-splittings are
representative of what sensitivity might be observed in other models with new
states. The studies focus on $R$-parity conserving SUSY where there is a
stable lightest-supersymmetric state that is only weakly interacting. This is
a challenging scenario, particularly for hadron colliders which have pile-up
effects, a range of parton-collision energies, and reduced resolution and
information about the momentum conservation in the beam direction, however
there are some $R$-parity-violating and other scenarios which are at least as
challenging and remain significantly unconstrained by the LHC. The plot show
the 95% exclusion limits. For discovery, the sensitivity at a hadron collider
would be lower, while at a lepton collider it would be quite similar.

Figure 31: Comparison of 95% exclusion SUSY sensitivities at different
colliders for a representative set of scenarios, including small and large
mass splittings for stop squarks, which are strongly produced, a large mass
splitting Wino-Bino model, and a small mass splitting Higgsino model. The
limits come from a combination of dedicated studies and extrapolations based
on the collider reach program [415]. The hashed gray band indicates the range
of estimates in the case where both a dedicated study and Run-2 extrapolation
are available. Current expected limits from the LHC are shown as vertical
lines. For the ILC limits (also relevant for other $e^{+}e^{-}$ colliders, not
shown) there are indirect constraints from precision $e^{+}e^{-}\rightarrow
f\overline{f}$ measurements [416]

The range of possible SUSY models is vast. Even within the MSSM, there are
many parameters and the complex interplay can lead to different signatures.
One way to understand this complex space is to construct a Monte Carlo scan
over the parameter space. For this purpose, a good candidate is the pMSSM,
which reduces the 120-parameter MSSM space to 19 free parameters, specified at
the EW scale, based on assumptions related to current experimental constraints
(including those from flavor, CP violation, and EW symmetry breaking) rather
than details of the SUSY breaking mechanism. Then with the scan points, the
masses of particles, the relevant couplings, and impacts on precision
measurements, rare processes, and cosmology can be studied.

Figure 32 shows the dependence of the $h(125)\rightarrow b\overline{b}$
branching fraction on the mass of the psuedo-scalar Higgs $m_{A}$ and
$\tan\beta$, the ratio of the up and down vacuum expectation values [417]. The
branching fraction is reported in terms of the coupling modifier $\kappa_{b}$
(ratio to the SM coupling). The plot shows the fraction of pMSSM scan points
with $\kappa_{b}$ within $1\%$ of the SM expectation of unity, where the range
of $1\%$ is chosen to approximately reflect the 95% CL corresponding to the
0.48% precision on $\kappa_{b}$ expected from a combination of precision
measurements at FCC-ee, FCC-eh, and FCC-hh [42]. Expected 95% CL exclusions
from direct searches for pseudoscalar Higgs boson ($A$) at the HL-LHC and FCC-
hh are overlaid for reference; points to the left of the lines are excluded.
Exclusions at low $\tan\beta$ are obtained from studies of $A\rightarrow
bb/tt$ [418], and those at high $\tan\beta$ come from projections for
$A\rightarrow\tau^{+}\tau^{-}$ [418, 30]. As is evident in the plot, direct
searches for $A$ at the HL-LHC are expected to provide better sensitivity to
the MSSM than the highest precision measurements of $\kappa_{b}$, which shows
the strongest MSSM-related deviation of any Higgs coupling parameter.

Figure 32: The fraction of pMSSM scan points with $\kappa_{b}$ within $1\%$ of
the SM expectation of unity as a function of $\tan\beta$ and $M_{A}$. The
range of $1\%$ is chosen to approximately reflect the 95% CL corresponding to
the 0.48% precision on $\kappa_{b}$ expected from the FCC-ee/eh/hh combination
[42]. Expected 95% CL exclusions from HL-LHC [418, 30] and FCC-hh [418] are
overlaid for reference. White bins include no scan points generated by the
Markov chain Monte Carlo (McMC) procedure. Gray bins include scan points
generated by the McMC, but rejected at a later step because of lack of
consistency with current precision measurements and direct searches.

####  3 New Bosons, Heavy Resonances, and New Fermions

Direct searches for new states beyond specific models provide vital
information in our explorations for new physics. The new states could appear
as heavy resonances, such as new bosons and new fermions, well-motivated from
the model-building perspectives. In this section, we chose the case of new
bosons as an example. Various representative examples of studies for current
and future facilities are discussed in the BSM Topical Group report [18]. New
heavy vector bosons are often regarded as the standard candle for BSM
searches. The canonical example is a $Z^{\prime}$ boson, which is a neutral
vector particle coupling to a SM fermion and antifermion. From the
phenomenological perspective, $Z^{\prime}$ searches are generally
characterized by the production coupling, the decay coupling, and the
resonance mass, where the decay coupling is typically traded for the branching
fraction to the desired final state. A coupling vs. mass framework for
$Z^{\prime}$ searches [419, 420] helps distill the $Z^{\prime}$ resonance
signal from disparate ultraviolet models into the minimal new physics
parameter space relevant for resonance searches at colliders. This framework
also enables direct comparison of experimental reach across different collider
proposals, including a comparison of $e^{+}e^{-}$, $pp$, and $\mu^{+}\mu^{-}$
colliders as well as other collider options.

Two specific $Z^{\prime}$ models studied in the many Snowmass 2021
contributions include the universal $Z^{\prime}$ model and the Sequential
Standard Model (SSM). The universal $Z^{\prime}$ model features a $Z^{\prime}$
boson with unit charges for all SM fermions, hence its universal designation.
The sequential standard model (SSM) $Z^{\prime}$ boson follows the same
coupling pattern of the SM $Z$ boson, and is the benchmark model most commonly
used by experimental searches. Figure 33 compares the sensitivity to a
universal $Z^{\prime}$ at different colliders [421, 411]. Included in the
figure are new Snowmass 2021 results for the Muon Collider: it shows that a
Muon Collider at $\sqrt{s}=3$ TeV is competitive with other colliders, with
sensitivity nearly identical to ILC at $\sqrt{s}=1$ TeV. A Muon Collider at
$\sqrt{s}=10$ TeV has the highest mass reach for a universal $Z^{\prime}$ with
large couplings $g_{Z^{\prime}}$, probing masses $M_{Z^{\prime}}>100$ TeV,
although only indirectly. A Muon Collider at $\sqrt{s}=10$ TeV is sensitive to
smaller couplings than the other colliders, with the exception of FCC-hh,
which has the highest sensitivity from direct searches within the mass region
$M_{Z^{\prime}}<28$ TeV but can directly probe up to about 45 TeV. In this
respect the FCC-hh and a 10-TeV Muon Collider will play a very complementary
role. Lepton colliders have an edge in sensitivity when the boson is so heavy
that only indirect effects can be measured, arising from the fact that in the
signal kinematic distributions the lepton collider experiments benefit from
relatively smaller systematic uncertainties. We can also see in this figure
the complementarity between direct resonance searches and precision
measurements of the SMEFT operators. Direct resonance searches allows us to go
to small couplings within accessible energies at lepton colliders. Series of
chiral determination of the BSM interference effects can also enable us to
extract the new resonance’s interaction structure [422, 33, 423].

Figure 33: Coupling versus mass reach at 95% CL for $e^{+}e^{-}$ colliders
(CEPC, ILC, CLIC and FCC-ee), $pp$ colliders (HL-LHC, HE-LHC and FCC-hh), an
electron-proton collider (FCC-eh) [411], and the Muon Collider [421].

All the different Snowmass contributions related to this topic can be
organized into a summary table, see Table 13, to enable an illustrative
comparison between the various $Z^{\prime}$ models and current and possible
collider scenarios. To enable the comparison and focus on the mass reach of
the different colliders, we adopted the $g_{Z}^{\prime}=0.2$ coupling
parameter for the universal $Z^{\prime}$ model, since it roughly aligns with
the mass reach for the SSM $Z^{\prime}$ model in the resonance channels
studied.

At first glance, this table shows the obvious correlation that higher center
of mass collider energy reaches higher values in $Z^{\prime}$ mass, where the
orders of magnitude spanned in collider energy pay off in orders of magnitude
in $Z^{\prime}$ mass reach. This is justified since the resonance signal is
assured when the $Z^{\prime}$ boson is within the kinematic reach of the
collider. Moreover, for a given operating point of a collider, we see that the
two $Z^{\prime}$ model benchmarks have very comparable results, which reflects
the fact that the underlying charge assignments of SM fermions to the
$Z^{\prime}$ currents only differ by $\mathcal{O}(1)$ factors, and so these
results would be broadly applicable in other models where $Z^{\prime}$ bosons
couple to all SM fermions, such as in gauged $B-L$ models. For more fermion-
specific models, such as $L_{\mu}-L_{\tau}$ or gauged baryon number, which are
equally relevant to the model benchmarks shown in Table 13, the distinction
between the different colliders becomes dramatically more important since the
$Z^{\prime}$ resonance would be produced via a tree-level coupling in some
colliders while only produced via a kinetic mixing coupling or a loop-induced
coupling in others. As a first estimate, the corresponding reach for a point
of comparison to Table 13 would then adopt a coupling suppressed by a loop
factor when the model does not couple to the initial partons at tree-level.

In Table 13 the relationship between the $Z^{\prime}$ mass reach at 95% CL and
the mass reach at 5$\sigma$ depends on the collider type and final state. The
two sensitivities are roughly equal for dilepton final states at $pp$
colliders, because the $Z^{\prime}$ peak is beyond the highest masses of the
dilepton continuum background from electroweak production via Drell-Yan, a
convincing and background-free exclusion or discovery. For dijet final states
at $pp$ colliders, the direct searches for a $Z^{\prime}$ dijet mass bump has
a 95% CL mass reach that is roughly 20-30% larger than the $5\sigma$ mass
reach, because here the continuum background is larger from strong production
of dijets via QCD. Finally, lepton colliders search within the kinematic
distributions of fermion pairs for the indirect effects of a $Z^{\prime}$,
with huge backgrounds at di-fermion masses significantly lower than the
$Z^{\prime}$ pole mass, resulting in a 95% CL mass reach that is roughly
60-100% larger than the $5\sigma$ mass reach. Therefore, Table 13 illustrates
both the power of lepton colliders for indirect discovery of new physics, and
the subsequent necessity of a higher energy to directly produce and confirm
that new physics.

Machine | Type | $\mathbf{\sqrt{s}}$ | $\mathbf{\int Ldt}$ | Source |  $\mathbf{Z^{\prime}}$ Model | $\mathbf{5\sigma}$ | 95% CL
---|---|---|---|---|---|---|---
|  | (TeV) | (ab-1) |  |  | (TeV) | (TeV)
|  |  |  | RH [424] |  $Z^{\prime}_{SSM}\rightarrow$ dijet | 4.2 | 5.2
HL-LHC | $pp$ | 14 | 3 | ATLAS [425] | $Z^{\prime}_{SSM}\rightarrow l^{+}l^{-}$ | 6.4 | 6.5
|  |  |  | CMS [426] | $Z^{\prime}_{SSM}\rightarrow l^{+}l^{-}$ | – | 6.8
|  |  |  | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 6
ILC250, CLIC380 | $e^{+}e^{-}$ | 0.25 | 2 | ILC [427] | $Z^{\prime}_{SSM}\rightarrow f^{+}f^{-}$ | 4.9 | 7.7
or FCC-ee |  |  |  | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 7
HE-LHC | $pp$ | 27 | 15 | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 11
|  |  |  | ATLAS [425] | $Z^{\prime}_{SSM}\rightarrow e^{+}e^{-}$ | 12.8 | 12.8
ILC | $e^{+}e^{-}$ | 0.5 | 4 | ILC [427] | $Z^{\prime}_{SSM}\rightarrow f^{+}f^{-}$ | 8.3 | 13
|  |  |  | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 13
CLIC | $e^{+}e^{-}$ | 1.5 | 2.5 | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 19
Muon Collider | $\mu^{+}\mu^{-}$ | 3 | 1 | IMCC [421] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | 10 | 20
ILC | $e^{+}e^{-}$ | 1 | 8 | ILC [427] | $Z^{\prime}_{SSM}\rightarrow f^{+}f^{-}$ | 14 | 22
|  |  |  | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 21
CLIC | $e^{+}e^{-}$ | 3 | 5 | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 24
|  |  |  | RH [424] |  $Z^{\prime}_{SSM}\rightarrow$ dijet | 25 | 32
FCC-hh | $pp$ | 100 | 30 | EPPSU [411] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | – | 35
|  |  |  | EPPSU [428] | $Z^{\prime}_{SSM}\rightarrow l^{+}l^{-}$ | 43 | 43
Muon Collider | $\mu^{+}\mu^{-}$ | 10 | 10 | IMCC [421] | $Z^{\prime}_{Univ}(g_{Z^{\prime}}=0.2)$ | 42 | 70
Table 13: List the operating point and mass reach, for 5$\sigma$ discovery and
95% CL exclusion, of the SSM $Z^{\prime}$ model taken from Refs. [424, 428,
425, 426, 427], and the mass reach of the universal $Z^{\prime}$ model with a
coupling $g_{Z^{\prime}}=0.2$ from Refs. [421, 411], determined from Fig. 33.

Searches for light but very weakly coupled new particles are motivated by a
variety of new physics scenarios. A prime example is the ALP. While ALPs have
a rich phenomenology, including prompt and long-lived signatures, one of the
main phenomenological target at experiments is the ALP coupling to two
photons, allowing a smooth transition between traditional QCD axion and ALP
parameters. Figure 34 shows the results of Snowmass 2021 studies on the
sensitivity to ALP, including the case of the Muon Collider [39, 429] compared
with other colliders [36, 412] in the $m_{a}$ vs. $g_{a\gamma\gamma}$ plane.
For ALP decays to diphotons, a 10 TeV Muon Collider is the most sensitive to
high ALP masses $m_{a}>200$ GeV, from vector boson fusion production processes
($VV\rightarrow a\rightarrow\gamma\gamma$). FCC-ee has the best sensitivity in
the medium mass range $1\lesssim m_{a}<100$ GeV, from the associated
production process ($Z\rightarrow\gamma a$), thanks to the potentially very
large integrated luminosity expected at the $Z$ pole in circular $e^{+}e^{-}$
colliders. In the more near term, the best collider limits on ALPs coupling to
photons over the range $m_{a}\approx 0.1-100$ GeV will be set by exploiting
photon-photon collisions in ultraperipheral interactions of heavy-ions during
the HL-LHC phase [430, 384]. It is worth noting that the ALP is typically
expected to have non-suppressed coupling to gluons, in particular in its
connection to the Strong CP puzzle of QCD [431, 432]. Having gluonic couplings
changes the considerations for the search channels and the performance at
different facilities appreciably (see recent phenomenological studies [433,
434, 139]).

The sensitivity to dijet resonances at $pp$ colliders was explored during
Snowmass 2021 as discussed in Refs. [424, 36, 435]. The process,
$pp\rightarrow X\rightarrow 2\mbox{ jets}$, is an essential benchmark of
discovery capability of $pp$ colliders and is sensitive to a variety of models
of new physics at the highest mass scales. The sensitivity to a dijet
resonance is mainly determined by its cross section. The study considered
strongly produced models, those with large production cross sections, that
include scalar diquarks, colorons and excited quarks. At the highest resonance
masses these strongly produced models can only be observed at a $pp$ collider,
as lepton colliders can only produce diquarks and excited quarks in pairs at
significantly lower masses. Also considered are weakly produced models, with
production cross sections that are roughly two orders of magnitude smaller,
that include $W^{\prime}$s, $Z^{\prime}$s and Randall-Sundrum gravitons, which
can also be observed at lepton colliders as previously discussed.

Figure 34:  The ALP coupling in the diphoton channel $g_{a\gamma\gamma}$
versus 95% CL mass reach is shown for multiple colliders [36, 412], including
Snowmass 2021 studies on the Muon Collider [39, 429] (orange) at $\sqrt{s}=3$
TeV (dashed) and 10 TeV (solid) as well as from the DUNE near detector [436]
with liquid Argon technology (dark red) and gaseous Argon technology (dark
yellow).

####  4 Long Lived Particles

Particles with long lifetime arise in many generic BSM models. The space of
signatures for these long-lived particles (LLP) signatures is very rich and
complicated, ranging from exotic-looking tracks to heavy stable charged
particles to various types of displaced objects (e.g. vertices, jets,
leptons). Here we highlight two examples. More benchmark cases, results, and
discussions can be found in the BSM Topical Group report [18].

The first example is that of LLPs that are electrically charged and can be
produced by many different models. In the case of one particular signature, if
the charged LLP decays within the detector, the LLP could produce a
disappearing track signature if it decays to neutral and/or very soft
particles that cannot be reconstructed. Disappearing tracks are particularly
motivated in models of SUSY and dark matter.

Figure 35 shows the projected reach of disappearing track signatures at the
HL-LHC [30], HE-LHC [437], LE-FCC [411], FCC-ee [411], CEPC [411], CLIC [438],
ILC [439], FCC-eh [153], FCC-hh [440], and several high energy Muon Colliders
[441], assuming a pure Higgsino with its natural mass splitting. Further
discussion on these constraints and their implications for dark-matter can be
found in the section on dark matter. The sensitivities are driven by many
factors, and in particular, the proximity of the tracking system to the
interaction points and low pile-up environment could help enhance them.

Figure 35: Overview of the sensitivity to the pure Higgsino, assuming its
natural mass splitting, for various future colliders. Figure adapted from Ref.
[441].

Many new physics models with an hidden sector could lead to long-lived
signatures and displaced vertices. Here we use a simplified heavy neutral
lepton (HNL) model, motivated by the neutrino mass model building and the
seesaw mechanism, as an example to show the different coverage of displaced
signatures in the current and future experiments. For more details on
different models and search coverage, see the discussion in the BSM Topical
Group report [18]. Extensions to the SM that account for neutrinos masses
typically incorporate heavy neutrinos that are “sterile” with very small
mixings to SM neutrinos, and they have masses much larger than the eV scale.
Neutral leptons with masses on the MeV scale or higher are referred to as
_heavy_ neutral leptons (HNLs).

Figure 36:  Constraints and future sensitivities for HNLs with mass $M$ and
mixing $U_{\mu}^{2}$ with muon neutrinos (summed over three HNL flavours).
_Medium gray:_ Constraints on the mixing of HNLs from past experiments [442,
443, 444, 445, 446, 447, 448, 449, 450, 451, 452]. _Colourful lines:_
Estimated sensitivities of the main HL-LHC detectors [453, 454, 455]) and NA62
[456], with the sensitivities of selected planned or proposed experiments
[457, 458, 459, 460, 461, 462], as well proposed future colliders [463, 464,
465, 466, 467, 468, 469, 470, 471, 472]. _Green band:_ Indicative lower bound
on the total HNL mixing $U_{e}^{2}+U_{\mu}^{2}+U_{\tau}^{2}$ from the
requirement to explain the light neutrino oscillation data [473] when varying
the lightest neutrino mass and marginalising over light neutrino mass
orderings. _Light gray:_ Lower bound on $U_{\mu}^{2}$ from BBN [474, 475].
More details can be found in Refs. [18, 476]

In the timescale of HL-LHC and beyond, many proposed experiments could offer
discovery potential for Type-1 Seesaw HNLs, particularly in the low-
mass/small-coupling region where long-lived searches will be required. Figure
36, adapted from [463, 477], shows the expected reach of experiments such as
FASER2, MATHUSLA, CODEXb, and future colliders. To guide the eye, the “type-I
seesaw” line indicates the approximate parametric scaling associated with a
simplified model with just a single neutrino flavor. Realistic three-
generation models can populate the experimentally accessible regions in Fig.
36.

####  5 Dark Matter

The existence of dark matter (DM) is some of the most concrete evidence for
particle physics beyond the Standard Model. However, little is known about DM
beyond its gravitational effects. One of the central questions of particle and
astroparticle physics is “What is the nature of dark matter and how does it
interact with ordinary matter?”.

Observing non-gravitational interactions of DM would bring us closer to
answering this question. Many theoretical hypotheses foresee interactions
between the DM and the SM to explain the measured relic dark matter density in
the universe. Such DM-SM interactions are also key to terrestrial searches for
dark matter. Since DM-SM interactions are generally feeble (as a consequence
of DM’s darkness), DM produced in SM-particle collisions would escape
detection at collider experiments. Nevertheless, these invisible particles can
be discovered when collisions appear to fail to conserve transverse momentum,
leading to missing transverse momentum in a hermetic detector.

While finding the connection between new invisible particles at colliders and
astrophysical DM requires an observation of DM in astrophysics
experiments333The search for dark matter must be conducted in synergy between
different Frontiers, using multiple probes and assumptions. This is discussed
in the Snowmass 2021 cross-frontier report on DM complementarity [478]., to
verify the new particle’s cosmological connection, collider experiments have
unique abilities to study DM-SM interactions and the properties of DM in
detail, including discovering new high energy particles that mediate the
interactions. The historical discovery of other invisible particles –
neutrinos – offers an example of such synergy: an observation of neutron
decays at low energies required high energy colliders for the study of
neutrino interactions and the discovery and study of the weak force mediators.

Following this example, here we focus on DM search benchmarks where the SM-DM
interaction is mediated by either an existing or a new particle, which in turn
decays into invisible (DM) particles and can also decay back into SM particles
allowing for a discovery in visible final states. Collider experiments can
also investigate theories of complex dark sectors and the wealth of particles
they hypothesize.

##### Testing the traditional WIMP paradigm

Among the possible theories of particle DM, WIMPs have been the most prominent
target of experiments. These are also theories to which colliders are
especially well-suited. In the traditional WIMP scenario, DM consists of a
single particle of mass between roughly 1 GeV and 100 TeV. As a thermal relic,
its mass and coupling to the SM allow them to reach a common thermal
equilibrium in the early universe and control the present abundance of dark
matter, largely independent of the universe’s initial conditions and
evolutionary history. WIMP candidates feature in many theories with
connections to other EW-scale new physics, including supersymmetry, the
archetype for the WIMP idea [479].

Since the 2013 Snowmass report, experiments at LHC Run 2, as well as at
underground facilities and with astrophysical observations, have made
significant progress in ruling out some possible WIMP parameters.
Nevertheless, others remain a compelling target for DM searches at colliders
and beyond (see, among numerous others, Refs. [480, 481, 482, 483, 484, 485,
486], for example).

Broadly speaking, WIMPs can be classified according to the way in which the DM
particle coupled with the Standard Model. Here we discuss some of the most
widely-studied model categories.

###### Minimal WIMPs.

One simple, predictive WIMP scenario identifies the DM particle as the
lightest member of an EW multiplet. Examples are the Higgsino (a Dirac fermion
doublet) and the Wino (a Majorana fermion triplet) in supersymmetry. More
general cases have also been considered [487, 488]. In these models, the
interaction strengths are fixed by the SM gauging couplings, and the only free
parameter, the mass of the dark matter particle, $m_{\chi}$, is chosen to give
the observed thermal relic abundance [489]. These “thermal” targets are
typically in the TeV range [490, 491, 492, 493, 494, 495] and it is critical
to probe them at future high-energy colliders.

Figure 37 summarizes projections for $2\sigma$ exclusions from future
colliders for Higgsino and Wino DM. New studies of the reach of a high energy
Muon Collider [496, 497, 441, 498, 485, 499, 500] were developed within
Snowmass 2021 in addition to the studies within the Briefing Book for the
European Strategy for Particle Physics Update [412].

At hadron colliders, the dominant signature of WIMP DM is often (but not
always) large missing transverse momentum recoiling against SM particles (e.g.
jets) [501, 437]. At lepton colliders, other signatures involving SM EW gauge
bosons and leptons come into play [496, 485, 497, 499, 500]. These signals are
relatively insensitive to the splitting between masses of the EW multiplet
and, hence, more robust against variations beyond the minimal scenario. The
loop-induced mass splitting among the components of the EW multiplet also
results in a disappearing track signature which can enhance the reach but is
more sensitive to the mass splitting and detector backgrounds.

The basic lesson from Fig. 37 is that high energy colliders, such as a hadron
collider with $E_{\rm CM}\simeq 100$ TeV or a Muon Collider with $E_{\rm
CM}\simeq 10$ TeV, can definitively test these scenarios. High energy
$e^{+}e^{-}$ colliders, with energies up to 3 TeV, can cover lower-mass
regions.

Figure 37: A summary of the reach of future colliders for simple WIMPs from
four search strategies, as indicated in the legend. For comparison, the
reaches of the direct and indirection detectors are also included (orange bars
at top). For lepton colliders where a detailed study is not available, the
kinematic limit $m_{\chi}=0.5\times E_{\rm CM}$ is used to indicate potential
reach; Muon-collider studies suggest this is likely to be an overestimate.
Hadron-collider projections are from [440, 502], while lepton-collider
projections are from projections in [496, 441, 497].

###### Higgs mediation.

DM could also couple to the SM via portals, which is a direct coupling via
gauge-invariant operators. The Higgs boson provides a prime example: as a
spin-0 particle, this ‘Higgs portal’ allows a renormalizable coupling with the
DM that can have a sizable effects on SM Higgs properties. Searches at
colliders are powerful probes of the Higgs portal. For example, DM production
would enhance tiny rate of invisible decays of the Higgs predicted by the SM,
provided the DM mass is less than half the Higgs mass. Precision measurements
of the Higgs couplings, another main objectives of a future collider, would
also contribute to probe the Higgs portal scenario. Future prospects for the
Higgs portal were studied in the European Strategy physics Briefing Book [412]
and are discussed in the BSM Topical Group report [18].

Models involving a larger extension of the scalar sector can also be probed
with Higgs measurements and BSM Higgs searches. Example of such extensions are
the Inert Doublet Model, where an extra scalar doublet provides a DM
candidate, and a 2HDM where an additional pseudoscalar has direct couplings to
DM. The HL-LHC and lepton colliders are expected to be sensitive to large
parts of the parameter space of these models, as can be seen in studies
targeting specific signatures [30, 503, 504].

###### BSM mediation.

Another category of models involves DM interacting with the SM only via one or
more new bosons. Although this category encompasses a huge set of possible
models, it is likely that only a few new particles will be relevant in the
early phase of a discovery. The relevant phenomenology can be captured by
simplified models of collider DM production.

The simplified models used to design present LHC searches, described in Ref.
[505], are inspired by the vector and scalar bosons of the SM. They involve a
single fermionic DM species and a mediator particle, either a vector
($Z^{\prime}$) with pure vector or axial-vector couplings to SM and DM
fermions, or a scalar or pseudoscalar boson with Yukawa couplings to the SM
and DM fermions. This involves 4 to 6 free parameters such as the masses of
the DM and the mediator, and their couplings to the SM and DM.

Contributions to Snowmass 2021 have significantly extended the projections
made for these models by the recent European Strategy Update Briefing Book
[412]. Prior results, originally focusing on relatively large coupling values
appropriate for LHC Run 2 [506, 507], have now been extended to arbitrary
coupling values [508]. This provides a better picture of how future
experiments would improve upon current searches and complement direct-
detection and indirect-detection experiments, especially when the dark matter
mass is less than a few GeV. The new studies of coupling dependence are
especially useful in understanding the effect of different coupling values
assumed for the signal at colliders [509], while direct detection only depends
on a specific combination of the couplings and the mass of the mediator.

(a)

(b)

(c)

(d)

Figure 38: Top: Projected exclusion limits on the couplings $g_{q}$
LABEL:sub@subfig:gqscan-dmLight, $g_{\chi}$ LABEL:sub@subfig:gchiscan-dmLight,
and $g_{l}$ LABEL:sub@subfig:glscan-dmLight for a vector mediator at the HL-
LHC and FCC-hh (for quark and DM couplings only). The result is shown as a
function of the mediator mass $m_{med}$; the mass of the DM candidate is fixed
to 1 GeV in all cases. The coupling on the $y$ axis is varied while the other
two couplings are fixed: in LABEL:sub@subfig:gqscan-dmLight, $g_{\chi}$=1.0
and $g_{l}$=0.0; in LABEL:sub@subfig:gchiscan-dmLight, $g_{q}$=0.1 and
$g_{l}$=0.0; and in LABEL:sub@subfig:glscan-dmLight, $g_{q}$=0.25 and
$g_{\chi}$=1.0. The arrows in the lower edge of the contours indicate that
other searches for lower mass mediators that are normally performed at
colliders could be sensitive to these models, but are not shown because the
inputs received focused on the highest mediator masses only. It is worth
noting that the lower bounds for both HL-LHC and FCC-hh for mediator
(resonance) searches shown in the figure can be significantly improved by
using non-standard data taking techniques such as data scouting/trigger-level
analysis described in Section 1.A and in Refs. [510, 511] specifically for
dijet resonances. Bottom: Example of the effects on the HL-LHC exclusion
limits in $\sigma_{\mathrm{SI}}$ for the monojet search
LABEL:sub@subfig:couplingscan-dd-monojet when varying the quark coupling,
$g_{q}$. The dark matter coupling is held fixed to $g_{\mathrm{DM}}=1$ and
there is no coupling to leptons. Limits from existing direct detection
experiments [512, 513, 514, 515] are shown for context (a recent re-analysis
of the DarkSide-50 data [516] has subsequently become available).

The top panels of Fig. 38 show the projections for future hadron colliders for
couplings to the SM fermions and DM fermion. These are derived from the
projections for missing-momentum signatures of the mediator (e.g. jet plus
missing transverse momentum, or mono-jet) as well as searches for the visible
decays (e.g. dijet and dilepton resonances). Observing both visible and
invisible mediator decays would provide complementary views of DM-SM
interactions. Hadron colliders with a higher center-of-mass energy can reach
higher mediator and dark matter masses. Searches for invisible particles
should be able to constrain models with much smaller couplings than current
searches, especially at mediator masses below the TeV. These results can also
be recast in terms of dark-photon benchmark models used by the Rare Processes
and Precision Measurements Frontier [9], as discussed in the next Section.

The reach of future colliders for these simplified models is dependent on the
interaction strength between the mediator and the SM, especially the quark
coupling of the mediator, which governs its production. It is expected that
lepton colliders will also strongly constrain models with lepton couplings,
especially in the case of polarized beams [517].

In order to understand when direct and indirect detection experiments can
confirm that collider signals are connected to DM, the European Strategy
Briefing Book compared collider and direct/indirect detection experiments,
highlighting where a simultaneous discovery would be possible.

Following those studies, the searches shown in the top row panels of Fig. 38
can be reparameterized as limits on the DM-nucleon cross section for direct
detection searches, as shown in Fig. 38b. As expected from Refs. [506, 507],
as future collider searches reach smaller SM-DM and SM-SM couplings, their
coverage in the DM-nucleon cross section plane improves. When the coupling
sensitivity limit approaches, collider projections gradually disappear.

The sensitivity of collider searches also depends on the ratio of the DM to
mediator mass. Searches at colliders are most sensitive when the mediator can
be produced directly from SM particle collision, and when the mediator is much
heavier than the DM particle. However, a strength of collider experiments is
that constraints on the dark interactions for these models can also be
obtained when the DM is too heavy to be produced directly. In BSM mediation
(as well as EFT) models, different mass hierarchy hypotheses are plausible, so
different mass ratios as well as different couplings should be tested. We
refer to the following section for a specific choice of a benchmark for a
vector mediator, motivated by thermal relic DM.

The lessons from these plots remain similar to Ref. [412]. When particle DM is
discovered in a direct- or indirect-detection search, Fig. 38b illustrate – in
a necessarily model-dependent fashion for the specific simplified model
considered – when collider searches for invisible particles would also be
sensitive to production of the mediator. In roughly these regions, both types
of searches would have complementary discovery potential, as discovery at a
direct-detection experiment would be combined with further study of the type
and properties of the DM-SM interaction at a hadron collider.

These figures also indicate where collider searches for invisible particles
would add unique sensitivity to the coverage from the other DM experiments.
Nevertheless, even in these regions, other experiments would be essential to
confirm that the invisible particles are indeed associated with galactic dark
matter.

##### Beyond WIMP dark matter

As the astrophysical evidence for dark matter does not point to a particular
mass scale for DM particles, there are many other intriguing DM hypotheses
beyond the WIMP that can be probed at colliders and neighboring facilities.
These viable possibilities include extending the thermal relic paradigm to
lighter DM (below 1 GeV), provided it has feeble couplings to SM, or more
complex dark sectors signatures that can also include long-lived particles.
Here, we will highlight some representative beyond-WIMP DM models and the
prospects for their discovery at future colliders and complementary
facilities.

###### Portal models for DM: dark photon and dark Higgs.

The experimental characteristics of feebly-coupled, low-mass thermal relic DM
particles can be represented by a small set of relevant interactions, mediated
by new portal particles, akin to the simplified models mentioned above. One
such portal particle is a new, low-mass vector boson, commonly called dark
photon, that couples to DM and to the SM via a new electromagnetic-like
interaction (kinetic mixing). A dark photon benchmark model has been adopted
by the Rare Processes and Precision Measurements Frontier [9] to compare the
sensitivity of accelerator experiments to thermal DM below a GeV. Colliders
can also probe such a dark photon in a complementary way.

The left panel of Fig. 39 reinterprets present and future collider searches
for vector-mediated DM in terms of the parameters of this dark-photon
benchmark, for a pseudo-Dirac DM particle with a mass below a TeV. It shows
that the HL-LHC dataset is needed to reach the thermal relic milestone for DM
masses above roughly 100 GeV, while FCC-hh is needed to cover the remaining
parameter space444This model assumes that the DM particle constitutes the
entire astrophysical DM abundance. For the thermal relic in all such simple
models, this is not a strict requirement if other processes not present in
minimal model enhance or deplete the DM abundance.. High-energy collider
experiments are therefore needed, together with accelerator and B-factory
experiments, to explore the full mass range for DM in this scenario.

(a)

(b)

Figure 39: LABEL:sub@subfig:darkPhoton Comparison of the expected constraints
for the dark photon model, starting from the reinterpretation of the results
[518] in terms of the simplified vector mediator model (LHC), and for HL-LHC,
FCC-hh and Belle-II 20/fb [519, 520] in terms of the dark photon model and
LABEL:sub@subfig:flavorfulScalarResonances existing constraints and future
prospects for the up-specific scalar mediator $S$ in the renormalizable
completion described in [521, 522]. Each figure also depicts the parameter
combinations that would reproduce a thermal relic.

Although not immediately obvious from the formulation of minimal benchmarks
[521, 522, 523], many portal models require both low and high mass particles
to be self-consistent. New TeV-scale particles at loop-level are needed to
generate the kinetic mixing interaction that characterizes dark-photon models.
These particles present discovery opportunities at colliders such as HL-LHC or
the FCC complex, and could be inferred from precision measurements at lepton
colliders [523]. For instance, if a light scalar singlet (a dark Higgs)
decaying to light DM couples predominantly to one SM fermion family, then new
heavy states are required to keep the model self-consistent at higher energy
scales [521, 522, 524, 51]. In this example, the Higgs-boson mass and the DM
mass are both of the order of a GeV, while the other particles needed to
complete the model have masses of the order of a TeV. Figure 39b shows
estimates for current and future collider searches for these new particles.

Similar considerations are generically applicable to different types of portal
models, encouraging complementary probes for the lower-energy phenomenology of
this model and higher-energy particles required for the completion. Light
portal particles (e.g. light dark Higgs, dark photon) with feeble couplings
can be discovered at Rare and Precision Frontier experiments [9], while the
higher-energy particles can be discovered at future hadron or Muon Colliders.
A corresponding discovery in low-threshold Cosmic Frontier experiments [10]
would determine the cosmological nature of the dark matter particle.

###### More complex dark sectors.

Taken further, dark sectors could reach or exceed the complexity of the
Standard Model, for example in the class of Hidden Valley models [525] where a
portal particle feebly couples ordinary matter to a more complex dark
spectrum. Dark sector models can include a large number of states with
different masses and lifetimes, where a viable DM candidate arises from the
uncharged states of the dark sector [526, 527, 528, 529]. These lead to
qualitatively different signals with respect to WIMP dark matter.

Alternate mechanisms to obtain the DM abundance as well as new forces can
point to new dark matter candidates and parameter regions. In particular, new
QCD-like confining forces within the dark sector (a ‘dark QCD’ [530]) can give
rise to stable dark hadrons that can be viable DM candidates. These dark
hadrons can be embedded in dark showers that could be discovered at present
and future colliders [531]. A dark-sector–confining phase transition can also
lead to new viable masses for dark matter [532]. As for the other models,
connecting collider signals to cosmology requires other
experiments555Additional information on the dark sector spectrum can come from
the lattice community, see e.g. [533].

As these models offer myriad ways to connect colliders to cosmology via the
dynamics of a complicated dark sector, future efforts to explore such a
connection are needed to identify regions of parameter space that are
cosmologically interesting, and to help contextualise future discoveries in
multiple Frontiers.

###### Dark sector discoveries at facilities co-located with EF colliders: the
FPF example.

The signatures generated by feebly-coupled models often include particles with
long lifetimes, as discussed earlier. Covering these signatures at colliders
requires both general-purpose and highly-optimized experiments. In particular,
portal models and beyond-WIMP models are targets for dedicated facilities and
experiments designed to use existing high-energy beams and supplemented by
information recorded at traditional collider experiments. Collisions of high
energy beams produce a large, forward flux of secondary particles, including
light DM of dark-sector particles, that are best investigated by dedicated
experiments in facilities downstream from the standard detectors.

An example of such a facility is the FPF, proposed to be installed in the long
shutdown between the LHC Run 3 and the start of the HL-LHC [138, 139]. In
particular, the proposed FLArE [534] experiment would be sensitive to thermal-
relic DM mediated by a dark photon, and experiments such as FASER2 [458] and
FORMOSA [535] would target a number of DM and dark-sector scenarios. The
opportunities for DM discoveries (as well as of neutrino measurements relevant
to cosmic DM experiments) offered by the FPF would further enhance the HL-LHC
DM and dark-sector search program, especially since development and
construction can be performed in parallel.

Future colliders offer opportunities for similar compelling experiments for DM
physics. Smaller-scale experiments can be optimized to cover a number of
beyond-WIMP scenarios that complement and enhance what can be discovered at
the larger, general-purpose experiments, while exploiting the same colliding
beams.

##### Dark matter complementarity

The search for DM has become a major science driver across all experimental
frontiers (CF, EF, RF and NF) and the cross-cutting AF, IF, UF and TF
frontiers. Discovering its microscopic physics requires collaboration within
and across all of these communities, and a variety of experiments at different
scales. The first step toward a unified strategy, understanding how the
efforts of each community complement the others, is discussed in a separate
community whitepaper, the Snowmass 2021 Dark Matter Complementarity report
[478]. This document, developed by the combined effort of these communities,
sketches opportunities for cooperation, lays out a road map for a coherent
strategy and presents case studies for discovery.

###  7 The Energy Frontier vision

The EF community has proposed several opportunities for pursuing its
scientific goals, among them the most prominent ones are Higgs-boson factories
and multi-TeV colliders at the Energy Frontier. These projects have the
potential to be truly transformative as they will push the boundaries of our
knowledge by testing the limits of the SM and directly discovering new physics
beyond the SM. 666The EF vision, as outlined in the following, has been
formulated from the input received directly from the Energy Frontier community
during the Snowmass process, including EF wide meetings and workshops, Topical
Group meetings, and Agorá events on future colliders. Full documentation of
these activities can be found on the Snowmass 2021 EF wiki page [536].

The EF aims to facilitate US leadership in an innovative, comprehensive, and
international program of collider physics. The timescales to fully realize the
EF vision extend to the end of this century, and the ultimate goals can only
be realized if our actions foster a vibrant, diverse, and intellectually rich
US EF community. Maintaining and strengthening such a community is only
possible if our plans reflect the aspirations of and provide a rich set of
opportunities for Early Career physicists.

During the coming decade it is essential to complete the highest priority
recommendation of the last community planning exercise (P5) and to fully
realize the scientific potential of the HL-LHC by collecting and analyzing at
least 3 ab-1 of integrated luminosity. The precision electroweak measurements
possible at an $e^{+}e^{-}$ collider (Higgs factory) would greatly extend and
complement the scientific results provided by the HL-LHC. The EF endorses
making the investments now to enable US leadership in a Higgs factory and
start the construction for a Higgs factory in parallel with HL-LHC operations.
To realize this goal, the US EF community needs to expand the R&D, in
collaboration with the international community on the detector and accelerator
technologies which will be required for a Higgs factory. In addition, the
global HEP community should consider opening a dialogue for a US-based site
for a future $e^{+}e^{-}$ collider.

The next step in our exploration of the fundamental properties of matter
requires the exploration of the multi-TeV energy scale. Any deviations
observed at HL-LHC or an $e^{+}e^{-}$ Higgs factory would strongly motivate
such a program. Two possible and potentially complementary paths forward
appear most promising to develop the capability to explore the energy-scale
frontier: 1) a 100 TeV (or higher) hadron collider, and 2) a high-energy Muon
Collider. The community proposes the US (in collaboration with international
partners) embark on a R&D program addressing high-priority critical aspects of
accelerators to reach these high energies, and to develop the detector
technologies needed to withstand the complex backgrounds and high radiation
environments envisioned for these two types of future colliders.

Thus, the energy frontier believes that it is essential to complete the HL-LHC
program, to support construction of a Higgs factory, and to ensure the long-
term viability of the field by developing a multi-TeV energy frontier facility
such as a Muon Collider or a hadron collider.

A key role in the success of the US EF at the HL-LHC, at future $e^{+}e^{-}$
colliders (Higgs factories) as well as at future multi-TeV colliders is played
by the Theory Frontier: model building, precision calculations and simulations
are necessary for precision measurements and searches of new physics. The
theory community must be adequately funded to support the success of the EF
community as a whole. In addition, the EF community thrives on collaborating
with other frontiers within HEP, and has relied on cross-fertilization
opportunities available via the interdependence with various fields and
opportunities, see Sec. 1.B.

It is essential for the US and the global HEP community to develop an
integrated plan for future colliders to pursue the reach of our ultimate goal
of uncovering new particles, new forces, and unveiling more fundamental laws
of nature.

####  1 The immediate-future Energy Frontier collider

The immediate future is the HL-LHC. The physics case for this program rests on
its ability (1) to extend the direct search for new elementary particles, (2)
to measure the couplings of the Higgs boson at a level that is sensitive to
corrections from physics beyond the SM, (3) to demonstrate the presence of a
self-coupling of the Higgs boson, (4) to measure the couplings of the top
quark at a level that is sensitive to corrections from physics beyond the SM,
and (5) to extend our understanding of QCD and strong interactions by
improving the precision of the corresponding relevant measurements.

The results from the HL-LHC set the basis for any vision of the future of the
EF program. The EF scientific program at the LHC is thriving and Run 3 will
triple the current integrated luminosity. The HL-LHC is scheduled to start
operation in 2029 and to increase the integrated luminosity by another factor
of 5 over the following decade. The HL-LHC program spans a very wide range of
physics topics, where some legacy results are expected to remain the most
sensitive for a long period of time after the end of the HL-LHC data-taking.
The unprecedented data set accumulated at the HL-LHC will allow the study of
SM phenomena that remain elusive so far because of their small rates, and to
extend the reach of searches for new processes beyond the SM. These data will
boost the potential of the experiments to make direct discoveries which may
revolutionize the human understanding of nature. These results rely both on
innovative detector technology, and a major theoretical undertaking in
reducing the expected systematics.

The study of the Higgs-boson self-interaction is one of the primary goals of
the HL-LHC due to its role in cosmological theories, involving, for example,
the vacuum stability. Higgs-boson pair production is a flagship measurement,
with a projected evidence of a di-Higgs signature at the 4$\sigma$ level, by
the end of HL-LHC running. Among all the Higgs-boson self-interaction terms,
the trilinear self-interaction is the only one in the reach of the HL-LHC and
it is parametrized by the coupling strength, which can be measured with a
sensitivity of $50\%$.

Studying the properties of the Higgs boson is a key mission of the HL-LHC.
Uncertainties in the signal strength modifier and coupling measurements in the
main production and decay channels will reduce from their current levels of
10-50% to less than 5%, moving Higgs-boson measurements into the regime of
precision physics, and allowing for spotting deviations from the SM. Rare
processes such as Higgs-boson decays to $c\overline{c}$, which are
challenging, and currently at the level of about 7.6 times the SM cross
section, will become accessible at the HL-LHC. The sensitivity to the H
$\rightarrow$ invisible branching ratio would reduce from the current
$\sim$20% to a few percent, approaching the SM prediction of 0.1%. For the
heavy Higgs bosons predicted by BSM theories with extended scalar sectors, the
reach would increase to masses up to 1 TeV.

The HL-LHC will also extend the sensitivity in direct searches for BSM
particles. For example, in SUSY, the reach for gluinos and squarks will
increase by up to 1-2 TeV, while chargino, neutralino and slepton reach will
increase by up to 0.5-1 TeV. This will allow to make a more conclusive
statement on the naturalness hypothesis. The reach for new resonances decaying
to SM particles will extend on average by 2-3 TeV. Moreoever, HL-LHC will
provide the ability of studying resonance decays to lighter BSM particles,
such as in the case of $Z^{\prime}$ decays to charginos, which are barely
accessible at the LHC, due to typically small branching ratios. Another set of
dedicated searches will look for dark matter, typically in final states with
invisible dark matter and visible mono-X and increase the current sensitivity.
Sensitivity to long-lived particles will be especially enhanced at the HL-LHC
due to improved and innovative detectors. The reach for Higgsinos via a
disappearing track search will increase to masses up to 350 GeV. Sensitivity
to long lived neutralinos decaying to photon and graviton is expected to
increase to masses of 700 GeV, improving the reach in short $c\tau$ and at
high masses. Additionaly, dedicated displaced muon reconstruction will improve
cross section reach for smuons or dark photons by 1-2 orders of magnitude for
different values of lifetime with respect to Run 3.

Expected measurements of QCD interactions, e.g. in jet, photon as well as in
$W$ and $Z$ boson productions, will considerably improve the understanding of
PDFs at low and high momentum fraction $x$, which are critical for carrying
out the rich physics program of precision Higgs-boson measurements as well as
BSM searches. Heavy Ion studies at the HL-LHC will include measurements of
parton densities in broad kinematic range and search for saturation, precision
measurement of macroscopic long wavelength QGP properties, developing a
unified picture of collectivity across colliding systems, assessing
microscopic parton dynamics underlying QGP properties, and performing
precision QED and BSM physics, for example in ultraperipheral collisions.

The international collaborations at the LHC recognize the importance of the
Snowmass 2021 process to the HEP community in the US and beyond. Continued
strong US participation is in particular critical and essential to the success
of the HL-LHC physics program, including the Phase-2 detector upgrades, the
HL-LHC data-taking operations, and the physics analyses based on the HL-LHC
dataset.

Additionally, auxiliary experiments and facilities are proposed to take
advantage in far forward kinematic regions. Forward physics facilities allow
to further extend the breadth of the HL-LHC physics: they can study regions of
parameter phase space for BSM, for example in LLPs and DM searches, that would
otherwise remain uncovered, and can perform novel QCD and neutrino
measurements in the very forward region.

We note that cross domain collaboration and studies with the EIC measurements
(under the purview of the DOE nuclear physics) identified the synergies and
complementarities in many physics studies, experimental technologies, analysis
techniques, and theoretical tools.

The US HEP community is heavily involved in ATLAS, CMS, LHCb, and ALICE, and
it contributes to aspects of the LHC accelerator infrastructure. More than
half of the US HEP community is involved in the LHC. Over the last years, US
institutions have graduated about 100 PhDs/year based on research carried out
with LHC data. During the last decade, ATLAS and CMS have together published
more than 2000 scientific papers in peer reviewed journals. This has had a
significant impact on the advancement of the field, and is an unprecedented
achievement compared to any previous collider experiment. In addition, the
PhDs apply the valuable skills they learn during their research to many
domains of science and industry, which is an essential positive economic
return of the LHC program. The vibrant scientific program of the HL-LHC will
continue this tradition and provide excellent training environment to the next
generation of students and postdoctoral researchers. Given the broad portfolio
of HL-LHC, similar number of student cohorts, i.e. about 100 PhDs/year from
the US are expected to benefit from the HL-LHC program.

In conclusion, our highest immediate priority accelerator and project is the
HL-LHC, the successful completion of the detector upgrades, operations of the
detectors at the HL-LHC, data taking and analysis, including the construction
of auxiliary experiments that extend the reach of HL-LHC in kinematic regions
uncovered by the detector upgrades.

At the same time, the time scales for realizing what comes next require also
an effort to advance preparations for the next collider of the intermediate
future during this time frame. This is reflected in the discussion on needs
and timelines in Sec. 1.

####  2 The intermediate-future Energy Frontier collider

The intermediate future is the an $e^{+}e^{-}$ Higgs factory, either based on
a linear or a circular collider. The physics case for this program rest on its
ability to (1) measure the couplings of the Higgs boson to the sub-percent
level and discern the pattern of modifications from beyond the Standard Model,
(2) search for exotic Higgs decays due to a ”Higgs portal” to a hidden sector
of forces, (3) measure the parameters of the Standard Model, including the
$Z$, $W$, top, and Higgs-boson masses to very high precision, and to provide
stringent tests of couplings in the electroweak sector, (4) measure the
electroweak couplings of the top quark at a level that can clearly reveal
corrections from beyond the Standard Model, and (5) perform precision
measurements of QCD phenomena which are testing-grounds of QFT in both
perturbative and non-perturbative regimes, and provide complementary
information relevant to cosmology and BSM physics.

The $e^{+}e^{-}$ colliders are the vehicle that will enable a high-precision
physics program in the EW sector by increasing the precision of SM
measurements. The physics case for an $e^{+}e^{-}$ Higgs factory is compelling
and the program is possible essentially with current technology. The various
proposed facilities have a strong core of common physics goals that
underscores the importance of realizing at least one such collider somewhere
in the world. A timely implementation of a Higgs factory is important, as
there is considerable US support for initiatives that can be achieved on a
time scale relevant for early career physicists.

Circular $e^{+}e^{-}$ colliders will be implemented in stages as electroweak,
flavor, QCD, Higgs-boson, and top-quark factories by spanning the energy range
from the $Z$ pole (and below) up to the top-quark pair threshold and beyond.
The highlights of such colliders are the highest luminosities at the $Z$ pole,
$WW$ threshold, and Zh energies (the “intensity frontier”); the exquisite beam
energy calibration at the $Z$ pole and $WW$ threshold; the possibility of
center-of-mass energy monochromatization at $\sqrt{s}=m_{h}$; the
compatibility with four interaction points; and the cleanest experimental
environment. Circular $e^{+}e^{-}$ colliders are therefore excellent Higgs
factories: they produce over a million Higgs bosons in three years at 240 GeV
center-of-mass energy. They provide the most precise determination of the
Higgs-boson coupling to the $Z$ boson, and of the Higgs-boson width (and
mass), and they could provide the opportunity for the discovery of the Higgs-
boson self-coupling and for a first measurement of the electron Yukawa
coupling. These colliders are also much more than Higgs factories. At the $Z$
pole, the Tera-$Z$ factory, with several trillions $Z$ produced, offers
opportunities for a comprehensive set of electroweak measurements with the
best prospects for precision, such as $Z$-boson mass and width (10’s of keV),
effective weak mixing angle (few 10-6), a direct determination of the
electromagnetic coupling constant, which allow sensitivity to mass scales up
to 70 TeV to be reached. In addition, a Tera-$Z$ factory has comprehensive
programs for QCD physics, e.g. the most precise measurement of the strong
coupling constant, flavor and rare decay physics, e.g. the search for lepton-
flavor violation, as well as direct searches for heavy neutral leptons, axion-
like particles, and other feebly coupled dark matter particle candidates.
Collisions at the $WW$ threshold will allow the most precise determination of
the $W$ mass (300 keV) and width (1 MeV), while running at the top-pair
threshold, will provide the best prospect for the top mass measurement.

Linear $e^{+}e^{-}$ colliders are primarily aimed at precision measurement of
the Higgs-boson properties with the aim to potentially flag violations of the
SM. The linear $e^{+}e^{-}$ colliders will also run at center-of-mass energies
covering the production thresholds of the $Z$ boson, $WW$ pairs, $Zh$ pairs,
$\rm{t\overline{t}}$ pairs, and $tH$ and Higgs-boson pair production. The
center-of-mass energy can be chosen flexibly depending on new discoveries at
the LHC, or elsewhere. The $e^{+}e^{-}$ linear collider will use polarized
electron and positron beams to enhance signal reactions and allow the
measurement of helicity-dependent observables, multiplying the physics output
per unit of luminosity. Beam polarization also enables the suppression of
backgrounds and provides cross-checks for the control of systematic errors. In
electroweak measurements, beam polarization gives direct access to $Z$ left-
right asymmetries, which are very sensitive precision probes. For example, in
a dedicated run at the $Z$ pole, it allows a measurement of
$\sin^{2}\theta_{w}$ at the level of $10^{-5}$, comparable to the TeraZ
capability. The nominal $e^{+}e^{-}$ collider physics program begins with
running at a center of mass energy of 250 GeV. At this energy the total cross
section and the branching ratios for all Higgs decays can be determined,
including decays to invisible final states. It provides the potential to
search for exotic Higgs decays. These measurements will improve our knowledge
of Higgs-boson couplings by a large margin over HL-LHC results. A global fit
using an EFT framework allows for a determination of the $hbb$ couplings to
1%, the $hWW$ and $hZZ$ to 0.7%, and all other important Higgs boson couplings
to close to 1%. These levels of precision are sufficient to be sensitive to
new physics beyond the reach of direct searches at the LHC. The second step in
the linear $e^{+}e^{-}$ collider program would be an energy upgrade to
$\sim$600 GeV. This would improve the precision of all measurements from the
250 GeV running by a factor two. Beyond the improved precision of the Higgs-
boson couplings, the couplings of the top quark can be explored and it is
possible to search for pair production of new particles produced in EW
interactions that are difficult to discover at LHC. At $\sqrt{s}$ of 500 GeV,
the Higgs pair production can be observed and the Higgs-boson trilinear
coupling can be measured with a precision of 22%.

As the physics case for a Higgs factory is strong, the US should participate
in global efforts to construct an $e^{+}e^{-}$ collider. Planning the center-
of-mass energy of this collider is important, and planned upgrades of the
center-of-mass energy will provide access to a broader spectrum of Standard
Model physics, including top-quark physics, which is an important component of
the energy frontier physics program. The accelerator R&D should aim at
developing sustainable facilities for Higgs research at a reasonable cost. To
enable the realization of a Higgs factory in the shortest possible timescale,
a targeted program on detector R&D for Higgs factories should be supported
with an immediate start and with initial emphasis on studies that are
applicable across proposed facilities. The critical detector R&D areas have
been identified as 4D tracking and vertexing, i.e. providing precision time
and spatial measurements in a single detector unit, low-mass detectors, e.g.
monolithic detectors that embed the electronics in the sensing elements,
wireless data transmission technologies, implementation of advanced AI/ML
algorithms in on-detector electronics, dual calorimetry among others, see Sec.
1.A.

The Energy Frontier also supports the possibility of a Higgs factory in the
US. Given global uncertainties, consideration should also be given to the
timely realization of a possible domestic Higgs factory, in case none of the
currently proposed global options are realized.

####  3 The long-term-future Energy Frontier collider

In the long-term future we envision a collider that probes the multi-TeV
energy scales, i.e. up to about hundred TeV center-of-mass energies. Its
physics case rests on its ability to (1) produce the fundamental particles
that generate the mechanism of electroweak symmetry breaking, (2) produce
particles with flavor-dependent couplings to quarks and leptons, (3) search
for thermal dark matter particles into the region of strong coupling in the
dark sector, and (4) in general explore the unknown at the highest possible
energy scale.

A 10-TeV scale Muon Collider with sufficient integrated luminosity provides an
indirect energy reach similar or better to that of a 100 TeV proton-proton
collider (e.g. FCC-hh, SppC), although its direct reach is limited to 10 TeV.
The similarity between high energy lepton colliders (effectively $W-W$
colliders) and hadron circular colliders (effectively gluon-gluon colliders)
is outstanding. Studies indicate that both the muon and hadron colliders have
similar reach and can significantly constrain scenarios motivated by the
naturalness principle. The 100 TeV hadron collider will have an advantage when
it comes to searching for colored states, while the Muon Collider naturally
has better access to EW states. Multi-TeV muon colliders will have the benefit
of excellent signal to background ratios, taking advantage of the vector boson
fusion production processes. As the multi-TeV colliders are planned for after
the Higgs factory, they will benefit from the precision studies of the Higgs-
boson properties in understanding the possible scale of new physics. One of
the key measurements from the multi-TeV colliders is the one of the Higgs
self-coupling to a precision of a few percent, and the scanning of the Higgs
potential.

This program to enable new physics insight into higher scales is currently
limited by technological readiness. In addition to the most prominent projects
that have been proposed, i.e. hadron and Muon Colliders, other auxiliary
proposals include high-energy $e^{+}e^{-}$ colliders using plasma wakefield or
structure advanced acceleration. All of these proposals require substantial
accelerator R&D.

A vigorous R&D program into accelerator and detector technologies will be
critical to position the US and international community at the forefront of
this research on a long term. This R&D program must specifically enable
instrumentation research that goes beyond current projects, so that the
detector technology will be available to make use of these high-energy
colliders once they can be built. The critical detector R&D areas have been
identified as 4-dimensional tracking with precision timing detectors, small
area silicon pixels in a high radiation environment. Besides that, particle
flow calorimeters with hybrid segmented crystal, and fiber readout may offer a
better alternative to silicon, see Sec. 1.A.

With the start of the Snowmass 2021 studies, there has been a surge in the
interest of the US and the international community for the Muon Collider
option because of advances in technology and analysis methodologies. There is
synergy with the formation of the International Muon Collider Collaboration
hosted at CERN. About a third of the contributed papers in the EF are on Muon
Colliders. Since the last Snowmass study in 2013, there has been substantial
progress in understanding the physics case, the detector requirements, as well
as developing novel techniques to address the major beam induced backgrounds.

The investment in detector and collider R&D for hadron and muon facilities,
and planning for discussion of siting options for Muon Colliders have to start
now, and run in parallel with the HL-LHC and any $e^{+}e^{-}$ precision
electroweak program. Enabling this future, also requires strong input from
every area of the theoretical community to understand the discovery potential
of such colliders.

####  4 Opportunity for US as a site for a future Energy Frontier Collider

Our vision for the EF can only be realized as a worldwide program, and CERN as
host of the LHC has been the focus of EF activities for the past couple of
decades. In order for scientists from all over the world to buy into the
program, the program has to consider siting future accelerators anywhere in
the world. The US community has to continue to work with the international
community on detector designs and develop extensive R&D programs, and the
funding agencies (DOE and NSF) should vigorously fund such programs (as
currently the US is severely lagging behind).

The US community has expressed a renewed ambition to bring back EF collider
physics to the US soil, while maintaining its international collaborative
partnerships and obligations, for example with CERN. The international
community also realizes that a vibrant and concurrent program in the US in EF
collider physics is beneficial for the whole field, as it was when Tevatron
was operated simultaneously as LEP.

The US EF community proposes to develop plans to site an $e^{+}e^{-}$ collider
in the US. A Muon Collider remains a highly appealing option for the US, and
is complementary to a Higgs factory. For example, some options which are
considered as attractive opportunities for building a domestic EF collider
program are:

  * •

A US-sited linear $e^{+}e^{-}$ (ILC/CCC) Collider

  * •

Hosting a 10 TeV range Muon Collider

  * •

Exploring other $e^{+}e^{-}$ collider options to fully utilize the Fermilab
site

Planning to proceed in multiple parallel prongs may allow us to better adapt
to international contingencies and eventually build the next collider sooner.
Such a strategy will also help develop a robust long term plan for the global
HEP community. Therefore, there are requests to assess the cost of siting a
linear $e^{+}e^{-}$ Higgs factory collider option, and a multi-TeV Muon
Collider in the US. In addition, to realize a successful US $e^{+}e^{-}$
linear collider program, cost reduction options, and targeted accelerator R&D,
e.g. for the Cool Copper Collider (CCC), is very important in the near term.

Some of the options listed above capitalize on the existing facilities, and on
expertise in key areas of accelerator and detector R&D at Fermilab. Among
other sites, Fermilab is proposed as an ideal one for a Muon Collider with a
center-of-mass energy reach at the desirable 10-TeV scale. The synergy with
the existing/planned accelerator complex and the neutrino physics program at
Fermilab is an additional stimulus for such investment of effort. A roadmap of
the accelerator R&D timeline, see Ref. [537] and Sec. 1.C, indicates that a 3
TeV Muon Collider is possible by 2045, though technically limited. A set of
Muon Collider design options, with one of the siting options being at
Fermilab, should be considered as an integral part of a global discussion for
siting and selecting an international Muon Collider.

A goal should be to prepare a document summarizing design for the Fermilab-
sited Muon Collider in time for the next Snowmass. Investment in this effort
will reinvigorate the US high-energy collider community and enable much needed
global progress towards possible discoveries at the next energy frontier.

The proposals and R&D efforts to address the innovative detector developments
for Higgs factories are well underway globally and many challenges are
resolved. Bold “new” projects such as a linear $e^{+}e^{-}$ CCC, and a Muon
Collider will offer the next generation some challenges to rise to. It will
inspire more young people from the US to join HEP and in the long term help
with strengthening the vibrancy of the field. Realizing our ultimate goal will
require significant funding and government support. The community feels that
there is potential to raise funds and obtain government buy-in for a future
collider project located in the US. However, funding is not all that is
needed. We also need a future program that continues to inspire the next
generation of high energy physicists, and one that entices the next generation
of graduate students to choose high energy physics as their field.

###  8 Acknowledgments

We acknowledge the input from the EF community-at-large in Spring/Summer 2022
(included verbatim in Ref. [538]) and in particular Alain Blondel, James Brau,
Joel Butler, Sekhar Chivukula, Dmitri Denisov, Sarah Eno, Stephen Gourlay,
Paul Grannis, Christophe Grojean, Tao Han, Patrick Janot, Andy Lankford, Lucie
Linssen, Michael Peskin, Chris Rogers, Vladimir Shiltsev, Jianchun Wang for
their help in developing and formulating the vision of the Energy Frontier. We
also thank the convenors of the e+e- Collider Forum, Maria Chamizo Llatas,
Sridhara Dasu, Ulrich Heintz, Emilio Nanni, John Power and Stephen Wagner, as
well as the convenors of the Muon Collider Forum, Derun Li, Diktys Stratakis,
Kevin Black, Sergo Jindariani, Fabio Maltoni and Patrick Meade. Last but not
least, we thank the Department of Energy for supporting the March 2022 EF
workshop.

##  Chapter 1 EF Appendix

This appendix presents the requirements, needs and challenges for the detector
developments for future collider physics, including detection technologies,
software and computing, event simulations and analysis techniques as well
prospects for Monte Carlo event generators. In addition it discusses
opportunities and challenges that the EF community will encounter in future EF
projects, e.g. attraction and mentoring of young minds, diversity and
inclusion, needed investments etc. The appendix ends presenting the timeline
and expected costs of the major EF projects that were discussed in Snowmass
2021.

##  Appendix 1.A Detectors, reconstruction, simulation and data analysis

Enabling particle detection techniques, data computation and analysis methods,
as well as precision calculations, event generators, detector and accelerator
simulations are critical for any future project at the energy frontier.

Different collider projects have different requirements on detector layout and
performance. For example, $e^{+}e^{-}$ colliders require detectors with high
granularity and low material budget, while hadron colliders have additional
requirements on radiation tolerance for detectors to be placed close to the
beam pipe. The detector design directly impacts the computational resources
and affects the data analysis methodology. Needs for software and computing
resources should be accounted for at the start of the detector design stage to
inform an optimal design strategy and ensure the best possible physics
performance of reconstruction algorithms. Requirements for computational
resources are set by detector design parameters such as number of readout
channels, luminosity, trigger system, data compression as well as the
experimental computing model and the analysis methods. The sophistication of
artificial intelligence and machine learning (AI/ML) techniques is growing
ever more, and collider physics should continue to play a pioneering role in
this expanding field of research by capturing and leading such developments.
As we experienced at the LHC experiments, AI/ML techniques have become
necessary tools to perform precision measurements and expand the reach of
searches for new physics. We expect AI/ML techniques will play even greater
roles at the HL-LHC and future colliders The direction of current research
suggests that such techniques will be employed in future experiments not only
in offline data analyses but also in online data selection and reconstruction,
making their accuracy and reliability even more critical [539, 7].

Most measurements at high-energy colliders require the understanding of
radiative corrections on observables. A close collaboration between theory and
experiments is mandatory to design observables that have a reduced exposure to
high-order effects and exploit detector capabilities as well as the latest
developments in precision calculations and Monte Carlo simulations. A
fundamental tool to design detectors, develop software and perform data
analysis in all experiments is the Monte Carlo simulation of final state
particles entering the detector. Essential software packages need to be
supported to produce Monte Carlo samples, including physics generators and the
Geant4 detector simulation toolkit. The computational needs for such
simulations strongly depend on the size of the MC samples, the complexity of
the detectors, the precision and accuracy needed for the physics models
embedded in these tools, as well as the data formats. For future collider
feasibility and design studies, accelerator modeling is also of fundamental
importance [540, 7].

Some of the most relevant aspects of needs for detector, method and tool
development and research for the coming decade will be discussed in this
subsection.

#### Detectors

Particle detectors are key to address science challenges and their development
is based on our understanding of fundamental laws of physics. Therefore, there
is a “virtuous cycle” between fundamental laws of nature and enabling detector
concepts and techniques. Such virtuous cycle must remain strong and unbroken
in order to lead to greater physics discoveries. In this context, detectors in
high-energy physics face a huge variety of operating conditions and employ
technologies that are often deeply intertwined with developments in industry.
The environmental credentials of detectors are also increasingly in the
spotlight.

At the EF, one can distinguish two major drivers for detector R$\&$D: detector
upgrades towards future hadron and Muon Colliders and development of advanced
technologies for the future $e^{+}e^{-}$ machines.

The detectors for the next $e^{+}e^{-}$ Higgs factory must provide excellent
precision and efficiency for all basic signatures, i.e. electrons, photons,
muon and tau leptons, hadronic jets, and missing energy over an extensive
range of momenta. The tracking resolutions should enable high-precision
reconstruction of the recoil mass in the $e^{+}e^{-}\rightarrow Zh$ process
for instance. These inherently very accurate physics requirements ask for
integrated concepts with unprecedented precision, minimal power consumption
and ultra-light structures. The demands for high resolution (granularity) and
low material budget on one hand, and low power consumption on the other hand,
exceed significantly what is the state-of-the-art today. Such leaps in
performance can only be achieved by entering new technological territory in
detector R$\&$D. To quote just some examples, several new concepts for silicon
sensor integration, such as monolithic devices, are being pursued for pixel
vertex detectors, new micro-pattern gas amplification detectors (MPGD) are
explored for tracking and muon systems, and the particle-flow approach to
calorimetry promises to deliver unprecedented jet energy resolution [6].

The Basic Research Needs for High Energy Physics Detector Research &
Development document (BRN in short) [541] compiles a list of requirements for
transformative and innovative technologies at the next generation of energy
frontier experiments focused on precision Higgs and SM physics, and searches
for BSM phenomena, such as (1) low-mass, highly-granular tracking detectors
and (2) highly-granular calorimeters, both with high-precision timing
capabilities.

Emerging novel vertex and tracking detector technologies are the vital
backbone for the success at a future electron-positron machines. These will
operate in an environment with high (continuous or bunched) beam currents, a
minimum distance from beam axis of about 20 mm, a requirement of $<5~{}\mu m$
single point resolution, high granularity ($<30\times 30~{}\mu m^{2}$), power
dissipation ($<50~{}mW/cm^{2}$), low mass ($\sim 0.1~{}\%$ of $X_{0}$, or 100
$\mu m$ Si-equivalent per layer). The clear challenge is unprecedented spatial
resolution, to be achieved with ultra-small pixels, and thus extremely low
material budget. Very thin detector assemblies are mandatory, while providing
high stiffness and stability. The tracking resolutions should enable high-
precision reconstruction of the recoil mass in the $e^{+}e^{-}\rightarrow Zh$
process, as shown in Table 1, and allow very efficient $b$ and $c$ tagging and
tau-lepton identification through the reconstruction of secondary vertices.
The detector solenoid magnetic field must be limited to 2 T for the Tera-$Z$
operation at future circular electron-positron colliders, to avoid a blow up