Higgs Factory Considerations

J.A. Bagger1, B.C. Barish2,3, S. Belomestnykh4, P.C. Bhat4, J.E. Brau5, M.
Demarteau6, D. Denisov7, S.C. Eno8, C.G.R. Geddes9, P.D. Grannis10, A.
Hutton11, A.J. Lankford12, M.U. Liepe13, D.B. MacFarlane14, T. Markiewicz14,
H.E. Montgomery11, J.R. Patterson13, M. Perelstein13, M.E. Peskin14, M.C.
Ross14, J. Strube15,5, A.P. White16, G.W. Wilson17

1Johns Hopkins University, 2Caltech, 3U.C. Riverside, 4Fermilab, 5University
of Oregon, 6ORNL, 7Brookhaven Lab, 8University of Maryland, 9LBNL, 10Stony
Brook University, 11Jefferson Lab, 12U.C. Irvine, 13Cornell University,
14SLAC, 15PNNL, 16University of Texas, Arlington, 17University of Kansas

> ABSTRACT
>
>
>
>
> We discuss considerations that can be used to formulate recommendations for
> initiating a lepton collider project that would provide precision studies of
> the Higgs boson and related electroweak phenomena.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  1 Introduction

The International Committee on Future Accelerators (ICFA) [1] has called for
the construction of an $e^{+}e^{-}$ collider as the next global accelerator
project after the LHC [2]. Such a collider would complement the discoveries of
the LHC by carrying out precision measurements of the heaviest Standard Model
(SM) particles and discovering potential new phenomena, so as to further our
understanding of fundamental physics.

This call was repeated and amplified in the 2020 update of the European
Strategy for Particle Physics [3], which states “An electron-positron Higgs
factory is the highest-priority next collider.”

During the Snowmass 2021 process, a set of potential lepton colliders that
would operate in the energy region from the $Z$ boson mass to the TeV scale
was considered by the Implementation Task Force (ITF) of the Accelerator
Frontier group [4] and the relevant parameters were collected. In parallel,
the Snowmass Energy Frontier group has examined the physics goals to be
addressed. These colliders have a common goal of producing large samples of
Higgs bosons, although they can also be operated to target other physics
goals.

In this White Paper, written by members of the Americas Linear Collider
Committee (ALCC), we take the ITF list and develop considerations that could
be used to make a recommendation on the optimum strategy for realizing a
future lepton collider. Our many years of engagement with the technologies and
physics requirements of these colliders provides the basis for our undertaking
this report. Despite the ALCC emphasis on the International Linear Collider
(ILC) and the Compact Linear Collider (CLIC) over the past several years, we
are motivated by the desire to investigate the physics associated with the
Higgs boson and related precision tests, for whichever lepton collider can be
realized. The criteria for a recommendation include physics requirements,
collider design characteristics, and more general considerations. They also
imply questions of siting such a machine: we believe that it is appropriate to
consider whether a Higgs factory might be built in the US with international
cooperation.

We refrain from making a recommendation for the optimal route to a Higgs
factory. Such a recommendation is expected to be made by the Particle Physics
Project Prioritization Panel (P5), to be formed subsequent to the report of
the Snowmass 2021 workshop. Thus, this paper is written to offer help for the
Snowmass evaluation of Higgs factories in its report and in the anticipated P5
recommendations.

In Section 2 we summarize the history of Higgs collider activity over the past
several years. Section 3 lists the potential projects considered by the ITF.
In Section 4 we identify a set of considerations based on the physics goals of
a lepton collider. In Section 5 we discuss the technical considerations for
the collider projects themselves and continue in Section 6 with more general
considerations. In each of these three Sections we first identify a set of
considerations which we think are the most important. These are followed by
additional criteria, listed without prioritization, that could be taken into
account. These objective considerations can clarify the situation but, perhaps
surprisingly, will not lead to a clear conclusion. We thus find it useful to
discuss two further topics. Section 7 addresses the role that the United
States could serve, either as a host for a Higgs Factory or as participant in
a project elsewhere. Section 8 discusses the potential evolution of future
facilities, the consideration of costs and the need for global coherence.

##  2 History

In 2013, ICFA formed the Linear Collider Collaboration (LCC) “to coordinate
global research and development efforts for next-generation particle physics
colliders”. The Linear Collider Board (LCB), comprised of members from Asia,
Europe and the Americas, was charged by ICFA to oversee the LCC. The Americas
Linear Collider Committee was subsequently formed to provide a liaison for
physicists and funding agencies in the US and Canada with the LCC/LCB and the
global communities. At the time ALCC was formed the only colliders on the
horizon were linear (the ILC and CLIC) rather than circular, hence the ALCC
name. The most technologically advanced possibility was the ILC whose
technical design report was completed in 2013 [5]. Physicists in Japan
proposed that the ILC be sited in Japan with support from nations in Europe,
the Americas and Asia. The LCC focused much of its attention on R&D on
remaining topics for ILC, developing plans for a proposed site and attempting
to obtain Japanese government approval for the ILC.

Meanwhile the CLIC design based on RF accelerating gradients derived from
deceleration of a high current drive beam was furthered through a series of
subsystem tests. These now establish the technical feasibility for a first
implementation as a 380 GeV collider on a technically limited schedule of the
mid 2030’s [6].

In 2020, the LCC reached the end of its mandate. Given the prospect of
Japanese hosting of the ILC, ICFA created the International Development Team
(IDT), whose role was to make preparations for the ILC Pre-Laboratory in
Japan. The mission of the Pre-Laboratory [7] would be to finalize the ILC
engineering design and to define the organization and global funding
arrangements. However, support for the ILC within the Japanese government and
the timeline for creation of the Pre-Laboratory remain uncertain.

Based on the very successful CERN experience of the sequential LEP and LHC
colliders in the same tunnel, new circular $e^{+}e^{-}$ Higgs factories have
been proposed with sufficiently large circumference to keep radiative losses
for $e^{\pm}$ beams manageable, and to allow 100-TeV scale $pp$ colliders with
achievable magnet technologies. The recent European Study Update (ESPPU)
favored a strategy in which a repeat of the LEP/LHC pattern would result in a
Future Circular Collider sequence (FCC-ee, FCC-hh) following after the HL-LHC.
This recommendation was accepted by the CERN Council and a FCC Feasibility
Study, to report to the next ESPP in four to five years, is under way. A
similar approach has been proposed in China with the sequential CEPC
$e^{+}e^{-}$ and SppC $pp$ colliders.

Work toward a muon collider that could operate at the Higgs resonance as well
as at higher energies intensified in the past year and the CERN-led
International Muon Collider Collaboration has been formed.

Given the long-stated goals of ICFA and the increasing appreciation throughout
the world of the importance of precision studies of the Higgs boson, it is
natural now to consider the broad range of facilities and possible sites to do
this physics.

##  3 Potential Higgs factory projects

The candidate projects considered by the ITF are:

A. Conventional circular $e^{+}e^{-}$ colliders, with follow-on 100 TeV-class
$hh$ colliders

  * •

FCC-ee (at CERN) [8, 9]

  * •

CEPC (in China) [10]

B. Linear $e^{+}e^{-}$ colliders of varying degrees of maturity and ambitions
to reach TeV-scale energies

  * •

ILC [11]

  * •

CLIC [12]

  * •

C3 [13, 14]

C. Energy recovery has been demonstrated in linacs, including examples with
superconducting RF of modest energy scale. This group of colliders includes
both circular and linear energy-recovery machines. A recent European Roadmap
for Accelerator R&D [15] deemed them to be considerably less mature than their
more conventional siblings.

  * •

CERC (circular FCC-like collider with energy recovery) [16]

  * •

ReLiC (Opposing linacs that alternately accelerate and decelerate electrons
and positrons) [17]

  * •

ERLC (two parallel linacs in each side with the energy from decelerating one
beam feeding the beam that is accelerating) [18]

D. Two Fermilab site filler Higgs factories at about 250 GeV have been
motivated by a desire to reawaken interest in a potential US re-entry into the
collider field.

  * •

Circular $e^{+}e^{-}$ FNAL site filler [19]

  * •

Linear $e^{+}e^{-}$ FNAL site filler [20]

E. All the machines above are based on electron-positron collisions. The
alternative of a muon collider is the sole member of this group. Interest in
such a machine has waxed and waned. Its pursuit was shelved by the 2014 P5
Report but was recently re-established as a possible future direction in the
ESPPU report.

  * •

Muon Collider [21, 22]

##  4 Physics considerations

A set of physics questions for study at lepton colliders operating in the
energy range from the $Z$-pole to the TeV region is being discussed in detail
by the Energy Frontier group for Snowmass 2021. These will provide the primary
scientific guidelines for making recommendations on a particular facility. It
is important to note that the scientific questions to be addressed at a lepton
collider are quite well defined; the discoveries of the recent hadron
colliders have provided clear targets of opportunity at known energies of
operation. The physics program is not a green field exploration for most of
its targets.

We believe that the questions that must be addressed with the highest priority
are:

P1. Precision measurement of Higgs couplings to SM fermions and gauge bosons
A Higgs factory should make precision measurements of the couplings and total
width of the Higgs boson. Their measurement, compared with the precisely
predicted Standard Model values, gives an incisive way to seek new physics and
provides a strong indication of its nature.

P2. Measurement of Higgs self-couplings
The Higgs self-coupling establishes the form of the Higgs potential. Any
deviation from the Standard Model form is an incontrovertible sign of new
physics.

P3. Sensitivity to rare or non-SM Higgs decays
Direct measurement of Higgs decays to invisible particles or other exotic
final states will help pin down the nature of new physics. The ability to
measure the total width at a lepton collider offers another avenue to sense
new physics.

All lepton colliders considered by the ITF do address the above criteria
associated with the Higgs boson properties, but differ in their capabilities
for exploring other related questions. There are thus additional physics
criteria which differentiate candidate facilities, based upon their ability to
study other massive particles and to infer the presence of new physics beyond
the SM. Among these other criteria are:

P4. Discovery potential for new non-SM physics

P5. Ability to directly measure top electroweak and Yukawa couplings

P6. Sensitivity to new physics through precision measurement of loop effects

P6a. Precision top mass and width measurements

P6b. Improvement of precision of $Z$-pole parameters

P6c. Improved $W$ mass measurement

P7. Ability to improve precision of the strong coupling constant

##  5 Considerations relating to Higgs factory technical issues

Each of the Higgs factories discussed in the ITF report is capable of
operation below and/or above the energy for Higgs production, although
sometimes with substantial modification. Possible further-future extensions of
the energy reach of linear collider facilities to many TeV include those using
the very high accelerating gradients provided by laser or beam-driven plasmas
or structure wake fields to extend energy reach to many TeV.

We judge the highest priority considerations to be:

T1. Range of possible operating energies and ease of changing energy
Ideally, a new facility would be capable of addressing physics questions over
a range of energies.

T2. Annual integrated luminosity recorded by all experiments
To the extent that the precision of measurements is controlled by statistical
uncertainty, the integrated luminosity delivered to the experiments is of
prime importance.

T3. Upgradability to higher energy and luminosity
Searches for new particles or deviations from the SM-predicted precision
measurements are enhanced by operating at energies in the TeV range or by
increases in luminosity.

T4. Extent and cost of remaining R&D
The extent to which a proposed facility is ready for approval and construction
start can be measured by the number and criticality of needed R&D projects
still remaining, their estimated cost and, importantly, their likely duration.
In some cases the R&D may be absolutely critical to confirm feasibility.

More specific technical criteria which could differentiate among potential
projects include:

T5. Ability to operate at the top pair threshold

T6. Ability to run at the $Z$ pole

T7. Ability to run at the $WW$ threshold

T8. Collision energy stability and calibration precision at all operating
energies

T9. Beam position stability and luminosity calibration precision

T10. Ability to control beam-related backgrounds at the IR

T11. Ability to provide independent confirmations of new discoveries

T12. Ability to provide polarized electrons

T13. Ability to provide polarized positrons

T14. Possibility to reconfigure as a $e^{-}e^{-},e^{-}\gamma$ or
$\gamma\gamma$ collider

T15. Possibility to reconfigure as a $hh$ or $ep$ collider

T16. Opportunities for beam dump experiments, long lived particles, etc.

T17. Need for, and scientific utility of, technology demonstrators

##  6 Considerations relating to more general project issues

The final category of criteria for developing recommendations is more general
than those in the preceeding sections, but is perhaps even more important. The
most important of these in our view are:

G1. Construction cost
The construction cost of any of the accelerator and detector facilities under
consideration is sufficiently large as to require international collaboration,
and will be the dominant factor in securing agreement to undertake a project.

G2. Possible start date of physics
From the physics point of view it is of course desirable to start a new
collider as soon as possible. The start date is controlled to a large extent
by the cost and the suite of major scientific projects already planned or in
construction by the Host nation or region. For example, the FCC-ee at CERN is
constrained by the operation of the HL-LHC. Funding for a new collider in the
US would be constrained by the construction of DUNE/LBNF. In the US, the
timing would also depend on the sequence of major projects from the several
offices within the DOE Office of Science. The need to coordinate the timing of
the funds from many participating nations may introduce further constraints.
The time required to conduct needed R&D (Section 5, item T4) can also affect
the start date.

G3. Sustainability and operating costs
The environmental sustainability and cost to operate the facility are major
considerations for the acceptance of a new collider. This need was highlighted
in the EPPSU [3] and amplified in a study on sustainability for the ILC in
Japan [23].

G4. Suitability as basis for follow-on facilities
Many projects in the past have been made possible by the existence of
infrastructure built for a previous facility. The nature of scientific
research tells us that the discoveries of today will dictate the needs for
future research. And it is almost always the case that the next steps will be
more extensive and expensive than those that came before. So the effective re-
use of prior infrastructure is essential.

G5. Broader impacts on society through technological innovations
Although fundamental scientific questions motivate the new facilities,
governments also gauge their desirability on the basis of benefits to the
wider society. New technologies with wider capabilities that are enabled by
prospective scientific projects are a strong selling point for their approval.

Other general considerations for recommendations include:

G6. Compactness, extent of surface disruption, ability to obtain land use
agreements

G7. Environmental and radiation concerns

G8. Synergistic use of a facility for other science/engineering studies

##  7 Domestic US Considerations

For almost a decade, the US focus has been on the ILC in Japan, based upon the
recognition that it is the most technically advanced, and that Europe and the
US presently have compelling near-term priorities. The situation has now
changed, with the lack of progress in securing government support for ILC in
Japan, the maturation of drive-beam RF for a linear collider at CERN, and the
development of attractive circular $e^{+}e^{-}$ collider proposals in Europe
and China. In this more fluid situation, it will be important for the US to
work toward finding a solution that leads to the execution of at least one of
the projects somewhere in the world. As the “Americas” Linear Collider
Committee, it would be negligent not to discuss the possibility of US hosting.
In this section, we point out the positive aspects, although we recognize the
hurdles to be surmounted in realizing a US-hosted facility.

###  7.1 US contributions to accelerator projects

The US has a long record of technical advances in accelerator technology that
positions it well to lead a project either at home or elsewhere in the world.
The US expertise in accelerator science and in building large accelerator
facilities is longstanding and deep, with major projects undertaken in the DOE
offices of High Energy Physics, Nuclear Physics, and Basic Energy Sciences.
The Global Design Effort that led to the technical design report for the ILC
was based on the leadership by US physicists and engineers in many areas of
accelerator science and technology. This experience, and the flexibility shown
by members of the US community, gives an excellent springboard for leadership
of a new Higgs factory collider.

The core technology for the ILC, superconducting radio frequency (SRF)
acceleration, has matured and expanded in the US and worldwide. The circular
colliders considered in the ITF report would also use SRF cavities. CEBAF at
Jefferson Lab upgraded its energy threefold over its design due to its use of
SRF acceleration. The DOE Office of Science has invested in such
infrastructure at its labs, enabling the construction of SNS at Oak Ridge,
LCLS-II and LCLS-II-HE at SLAC, FRIB at Michigan State University, and now
PIP-II at Fermilab. The Electron Ion Collider project currently in preparation
at BNL relies, like other modern circular machines, on SRF acceleration.

SLAC has been a world leader of high-gradient normal-conducting RF
acceleration for decades. The SLAC linac, the SLC, the FACET test facility and
the engagement in CLIC R&D have recently led to the development of the C3
(Cool Copper Collider) proposal for a Higgs factory. Variations of this
technology are used at Paul Scherrer Institute (SwissFEL), Trieste (FERMI) and
SLAC (LCLS) free electron lasers, and are leading to the development of
compact accelerators for a variety of uses. The path to maturity for CLIC, and
even more so for C3, is longer than that for the ILC, but there has been
considerable excitement within the accelerator community for these
technologies. These options have the potential for increased accelerating
gradients over the SRF option, hence a reduced linear collider footprint and
potential cost savings.

The US has a history of pioneering work in circular accelerator construction
at the largest scales with the Tevatron, RHIC and the LHC. The development of
powerful magnets for circular colliders that was pioneered in the US was key
contributions to the LHC. The realization of energy-recovery techniques has
been refined and applied in US projects. The US expertise in beam dynamics has
enabled the physics programs at CESR, PEP-II, the Tevatron, the Main Injector,
and SLC. Much of the development of a muon collider has been accomplished by
the US-based Muon Accelerator Project (MAP) [24].

It is easy to see how the US could play a lead role, at home or abroad, in a
linear or circular $e^{+}e^{-}$ collider based on either superconducting or
normal conducting RF.

In its previous incarnation, P5 made the recommendation to shelve the Muon
Collider efforts. They were judged to be diverting the community and required
large investments that were difficult to provide given the commitments to the
LHC and those emerging for the neutrino program. Nevertheless, we note that
the European community, having raised the question of a Muon Collider and the
idea to contemplate a demonstrator design for the next ESPPU, is looking to
the US for experienced help.

###  7.2 US contributions to collider detectors

The US has been at the forefront in building and operating detectors for
colliders for decades. The detectors at hadron or $ep$ colliders at the ISR,
Tevatron, HERA and LHC were built with the leadership of US physicists,
covering all of the major subsystems. Detectors at $e^{+}e^{-}$ colliders have
evolved from those at SPEAR, PEP and CESR, through those at SLC, LEP, KEK-B
and PEP-II, and now on to the Higgs factories. US physicists were the
spokespersons or top-level detector and physics leaders in most of these
experiments.

In the past decade or more, US physicists have taken lead roles in the designs
of both of the ILC detectors [25], and have played the dominant role in the
SiD consortium. Innovative concepts for lepton collider detectors continue to
surface and although the funding for lepton collider detector R&D in the US
has been minimal over the past decade, the depth of expertise in the US
community is great. The ILC detector designs have been taken over with modest
modifications to detectors being proposed at CLIC, FCC-ee and CEPC.

US physicists have led the world-wide articulation of the lepton collider
physics program, and have contributed significant new ideas that will broaden
its scope. Many of the software tools developed for the study of lepton
collider physics processes and the collider backgrounds have been initiated by
the US community.

There can be no doubt that US physicists will be important leaders for the
physics and detector programs at a lepton collider hosted either domestically
or abroad.

###  7.3 R&D needs

Test facilities at ANL, BNL, Cornell, FNAL, LBNL and SLAC offer a wide range
of opportunities for R&D on new methods and technologies for accelerators.
These facilities will promote advances in plasma and dielectric structure
wakefield acceleration, new RF techniques, and development and test of new SRF
technology. Their operation fosters the training of accelerator scientists
over a broad range of topics, giving the basis for undertaking major new
projects.

Whatever choice is made for a future Higgs factory, we believe that the US
should be a leading participant. This implies the need for substantial R&D for
technical areas (including detectors) to which the US could contribute, and
also for demonstrator projects for new collider concepts or major subsystems.
In the case of some proposed facilities, much of this R&D has already been
done, but for others, most remains. For the ILC, a first demonstrator project
was the European XFEL, and a second is LCLS-II. The choice of scale for a
demonstrator needs to balance the desirability of having a machine which is
useful scientifically with the desire to invest as minimally as possible in
this R&D phase. A general rule of thumb for such R&D funding is about 10% of
the total project cost. A discussion of the need for enhanced funding for
Higgs factory R&D can be found in a White Paper submitted for Snowmass 2021
[26].

##  8 Global considerations

###  8.1 Evolution of facilities

As indicated by criterion G4 above, a new project that lends itself naturally
to extension to further facilities is attractive in creating a basis for the
present while enabling a future path in a relatively cost-effective manner.
Many of the possible Higgs factories mentioned in Section 3 could be imagined
to allow new opportunities through follow-on upgrades for many decades.

The circular colliders are based on a large tunnel, sized for a very high
energy hadron-hadron collider. The tunnel size is dictated by the goal of 100
TeV $pp$ collisions using foreseeable magnet technology. However, the hadron
machine requires magnets that do not currently exist and will require
considerable further R&D. The hadron-hadron and electron-positron circular
colliders are linked in several ways. The large tunnel needed for the hadron
collider facilitates the $e^{+}e^{-}$ collider with technology that is
relatively well proven. But since the full energy hadron machine needs major
advances in magnet technology, an $e^{+}e^{-}$ precursor is natural for
filling the time gap that may occur in achieving them, thus maximizing the
physics exploitation of the tunnel.

A linear collider Higgs factory can be extended in length, or its acceleration
gradient upgraded, to increase the energy. In these scenarios, the tunnel is a
major reusable component that can be filled with quite different RF technology
from that used in the initial incarnation. Advances in SRF offer energy
extension up to 3 TeV or beyond [27]. A change at some point to a CLIC or C3
technology might also provide a 3 TeV horizon. If very high gradient plasma
acceleration can be developed as a viable alternative; the linear tunnels
could offer the site for even further improvements in collision energy.

The ERL-based Higgs factories offer a longer term evolution in energy and
luminosity that could be achieved through upgrades based on either the
circular or linear collider infrastructures of Groups A or B. Their relevance
for the present discussion depends on the anticipated window to advance the
required R&D. Of course, without support, that R&D will not happen.

Many challenges remain before a muon collider could be realized and a
relatively long period for R&D is needed. Such a facility would necessarily
require novel systems for a high-power proton driver, muon capture, cooling
and subsequent acceleration. It is possible that a first step of a muon-based
facility could be a high intensity neutrino factory. In any case the front end
systems for muon production and capture could be the springboard for later
expansion to higher energy.

###  8.2 Costs

The total cost of the facility is the dominating consideration for the
multiple nations needed to join together in building a new collider. The
question of how much cost is acceptable depends on many factors, including the
expected scientific impact, the technological benefits accruing from the
project, the broader societal sense of its importance, and political
considerations of international cooperative ventures. The acceptable cost in
the US could also depend on the range of potential partners from within the
broader DOE and NSF programs. If a new facility were to be based in the US
with some contributions from other regions, one could look to the scale of
current or recent projects such as EIC [28], PIP-II [29] or LBNF/DUNE [30].
The final cost of NASA’s James Webb Space Telescope of about $10B (US
accounting), with foreign contributions at the level of around 5%, represents
an upper limit for the cost of a new US-based project that will be difficult
for HEP to reach. For US participation in projects abroad, the precedents
include LHC ($531M in FY2008 for the initial accelerator and detector
construction), ITER (US share of 9.1% of a total cost which has risen to over
$20B), and the expectation that US participation in ILC in Japan would require
about a contribution of at least 10%, but perhaps up to 25%, of the full
project cost of about $7.5B (Japanese/European accounting) for the initial
collider, excluding detectors. These comparisons must of course be made with
care due to the differences in cost accounting practices in different regions,
differing definitions of what is included in the cost of different projects,
and fluctuating exchange rates (it can be preferable to compare costs in
different nations using Purchasing Power Parity [31]).

###  8.3 Global coherence

Up until fairly recently, comparable new high energy physics facilities have
been built in different regions of the world that have often provided
independent confirmation of results. PEP-II, PETRA and TRISTAN all gave
similar opportunities. Several B-factories (PEP-II, KEK-B and CESR) were
built. The CERN Sp$\overline{\rm p}$S and the Fermilab Tevatron, or RHIC and
the LHC ion collider plowed similar furrows. Only when the projects became as
large and costly as the LHC did the world enter the stage where only one
facility of a given type could be envisioned.

It is not realistic to imagine that scientific advisory panels alone are
sufficient to insist on a rational array of complementary facilities in
different parts of the world, thus optimizing the global scientific payoff.
But it is nevertheless worthwhile for such panels to explain how their
recommendations fit into the global picture and to enunciate the goal of
developing a coherent world plan.

##  9 Conclusions

We have attempted in this paper to discuss in an even-handed manner the issues
that we imagine would need to be considered when attempting to chart a path
forward for Higgs physics. Our discussion has been rooted in the experience in
HEP over the past 40 years or so. We embarked on this report with the goal of
an objective assessment of the relative performance of different machines.
Such a comparison is complicated by the scope of physics beyond that
prescribed as mandatory, which differs for the proposed options. This led us
to the necessity of considering aspects such as the possibility of a domestic
option and/or the global particle physics roadmap. From the outset, we
intended to lay out the considerations without making a choice. Our
discussions reinforced that intent. We leave the resolution of the conundrum
to the upcoming P5 panel, HEPAP and ICFA.

##  10 Acknowledgments

We are indebted to Thomas Roser and the Accelerator Frontier Implementation
Task Force for the careful description of the various colliders appropriate
for the study of the Higgs boson. The talks and discussions in the “Snowmass
Agora on Future Colliders” organized by the Snowmass 2021 Energy and
Accelerator Frontier working groups and Fermilab have been an important
contribution to our understanding of the various potential projects. We
gratefully acknowledge the support of the US Department of Energy and the US
National Science Foundation.

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