††thanks:  0000-0002-3143-1976

# Physics at Future Colliders:
the Interplay Between Energy and Luminosity

Zhen Liu<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA Lian-Tao Wang<EMAIL_ADDRESS>Department of Physics, University of Chicago, Chicago, IL 60637, USA

###### Abstract

In this note, as an input to the Snowmass studies, we provide a broad-brush
picture of the physics output of future colliders as a function of their
center of mass energies and luminosities. Instead of relying on precise
projections of physics reaches, which are lacking in many cases, we mainly
focused on simple benchmarks of physics yields, such as the number of Higgs
boson produced. More detailed considerations for lepton colliders are given
since there have been various recent proposals. A brief summary for hadron
colliders based on a simple scaling estimate of the physics reaches is also
included.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

###### Contents

  1. I Introduction
  2. II Lepton colliders: sub-TeV case
  3. III Lepton collider: TeV and Beyond
  4. IV High energy proton proton collider
  5. V Summary
  6. A Scaling for the reach at hadron colliders

##  I Introduction

There have been many proposals of future colliders [1, 2, 3, 4, 5, 6, 7, 8,
9]. As an input to the Snowmass study, we discuss the connection between
luminosity and physics reach at future lepton colliders and hadron colliders
as a function of the center-of-mass energy. This is a vast topic. In-depth
studies, with extensive simulation taking into account accelerator and
detector design details, are needed to generate a full-fledged and precise
answer. This is beyond the scope of this short note. Moreover, the detailed
designs have not been carried out for some of the more recent proposals
included in the Snowmass discussion. At the same time, the current proposals
(and the corresponding studies) contain specific run energies and
luminosities. It would be helpful to have a sketch of the physics reach beyond
those benchmark points, which would help consider variations of the existing
proposals and new ones. In this note, we focus on giving such an overview
without a specific focus on any proposals.

We consider a broad range of potential collision energies, from 80 GeV to 20
TeV for lepton colliders and up to 100 TeV for pp colliders. Each potential
future collider will have a broad physics program. To cover all of them, if
not impossible, will go much beyond the scope of this note. There has been
much discussion about the physics cases of future colliders in recent years.
One outcome is the emergence of a relatively small set of significant physics
drivers based on essential questions to be pursued in the beyond-LHC era. Very
broadly speaking, they can be put into the following three categories: 1)
Understanding the property of the Higgs boson to an unprecedented accuracy; 2)
Direct search of new physics resonances near and above weak scale. This is
partly motivated by the hierarchy problem but covers new physics beyond that;
3) Testing the WIMP Dark matter paradigm. The proposed future colliders have
broader physics programs, and each one of them has its own special strengths.
At the same time, these three physics drivers are shared by all of these
proposals.

Even after narrowing our focus onto these categories and drawing on available
studies as much as possible, it is impossible to cover the reach in the range
of energy and types of machines of interest here. We will content ourselves to
offer a high-level view, with ballpark (crude) estimates based on
simplifications and approximate extrapolations. We will lay out the detail of
our estimates in each physics topic.

To put the capabilities of future colliders in perspective, we will show a set
of targets for physics output, such as the number of Higgs bosons to be
produced per year. These are based on the collection of available studies and
simple extrapolations and should be thought of as ballpark estimates. Our
results are presented so that it is easy to scale to new targets. Many of the
projections will be updated due to the Snowmass studies in the near future,
and we will update our targets accordingly. We will also overlay these targets
with the proposed run plan of the various future colliders. We note that many
run plans are still being finalized in the Snowmass processes. For
definiteness, we will use the plan presented at a series of Agora meetings
[10, 11, 12, 13, 14] for this version of the white paper. We will update the
figures with the finalized numbers. Part of the motivation of this work is our
discussion with the Implementation Task Force in the Snowmass studies on
presenting the physics yields of various future collider proposals.

##  II Lepton colliders: sub-TeV case

Leptons colliders below TeV often have their primary physics goals as
precision measurements of the properties of the SM particles. An obvious focus
is the Higgs boson. Many proposed experimental programs exists in this energy
range, such as CEPC [1], FCC-ee [2], ILC [4], CLIC-380 [5]. Meanwhile, many
new possibilities are also under consideration, such as a 125 GeV Muon
collider [6], $C^{3}$ [7, 15], ReliC [8] and CERC [9]. The main functions of
this class of machines are 1) Higgs factory; 2) Z factory; 3) WW, threshold
and beyond; 4) $t{\bar{t}}$, threshold and beyond. In this section, we discuss
them in turn.

Figure 1: Top: Required luminosity for lepton colliders for various physics
goals in the energy range of 80 GeV to 1 TeV. The required instantaneous
luminosities assume one detector. For experiments with multiple concurrent
detectors running, one can multiply its luminosity by the number of detectors
to obtain the effective physics yields. The run plan of the colliders often
has various stages to achieve different precision goals. Hence, the
luminosities goals displayed here are “ballpark” figures which loosely
correspond to run plans. The actual physics output for each stage of a
proposed collider can be worked out from this figure straightforwardly.
Bottom: We overlaid the instantaneous luminosities of various proposed lepton
colliders in this energy range. The run plans are taken from Snowmass Agora
presentations [11, 10, 12].

The summary plot is shown in Fig. missing 1. We will discuss the corresponding
calculations and physics details later in this section. We select six
different (minimal) “target” or “benchmark” cases for lepton colliders in this
energy range to demonstrate the required luminosity. These six target cases
are centered around Higgs physics (single Higgs and double Higgs productions),
electroweak physics ($WW$, $Z$ and EFT), and top physics ($t\bar{t}$).
Depending on the center-of-mass energy of the lepton colliders, the required
luminosities vary mostly in the range from $10~{}\,{\rm fb}^{-1}$ per year to
$10~{}\,{\rm ab}^{-1}$ per year. In the bottom panel of Fig. missing 1, we
overlaid the run plans of many proposals currently under consideration in the
Snowmass process. We can see from Fig. missing 1 that most proposed colliders
can cover all these cases reasonably comfortably. Hence, there is room for
relaxing luminosities requirements at early stages if constrained by other
considerations while still achieving some physics goals.

It is important to note the impact of beam polarization here. Beam
polarization generally can help reach the same physics goals with lower
luminosity. The signal cross-section can increase in the favorable
configuration, and the background might also be suppressed. Further, if one
can also flip the beam polarization, new physics can be accessed more readily
by eliminating various systematic uncertainties through asymmetry observables
[16, 17, 18]. However, the plot will be jam-packed if we also consider
different polarization configurations, defeating the starting point of this
study to provide a simple, quantitative, and schematic understanding of the
luminosity requirement for future colliders. Hence, we choose to show the
cases with unpolarized beams only. In this regard, the designed luminosity of
machines, such as ILC or CLIC, have an effective luminosity higher than
plotted here, possibly by a few tens of percent. For instance, the ILC-
designed beam polarization of $(+0.8,-0.3)$ will yield 40% more Higgs bosons
than unpolarized beams [4].

Next, we discuss each physics goal and the corresponding collider performance
in detail.

Single Higgs and Higgs coupling precision.

One of the main goals of low energy lepton colliders is to function as a Higgs
factory and measure the Higgs couplings with unprecedented precision. HL-LHC
can measure some of the Higgs couplings to an accuracy of a few percent.
Hence, a meaningful target for a Higgs factory would be to push the accuracy
higher by one order of magnitude to the per mil level. New physics, at scale
$\Lambda$, will generically modify the Higgs coupling at the level of

$\displaystyle\delta=c\frac{v^{2}}{\Lambda^{2}}$ (1)

A per mil level measurement allows us to probe new physics at the level of a
few TeV. Numerous detailed studies have supported this generic expectation
[19, 20].

For a specific Higgs coupling, obtaining the projected accuracy at a given
collider requires detailed studies by combining all the relevant channels,
considering possible degeneracies, and incorporating realistic accelerator and
detector conditions. While this has been done for some of the existing
proposals [21, 1, 4, 2, 5], it is not possible to accurately extrapolate to a
broader range of energies and luminosities. At the same time, conservatively,
$10^{6}$ Higgs boson would at least be needed even to have a chance of
measuring $HZZ$ Higgs coupling to a per mil level. Again, this expectation is
borne out by the studies, and current proposals all roughly have $10^{6}$
Higgs bosons as a target. Given the importance of the Higgs precision program,
we anticipate most sub TeV colliders spend at least five years on Higgs runs.
Hence, we set the first target of one million Higgs bosons in five years.

We show in Fig. missing 1 the required luminosity for single Higgs production
(red line). Typical $e^{+}e^{-}$ machines access such physics through the $ZH$
associated production, reaching a minimal luminosity requirement with the
center-of-mass energy around 240-250 GeV. Beyond $ZH$ associated production,
the $WW$-fusion starts to dominate the production at around $E_{\rm CM}=400$
GeV. Its importance is already revealed even at somewhat lower energies where
it is not the dominant production process. For instance, some circular lepton
colliders such as the FCC-ee and the CEPC plan to run at around 360 GeV
center-of-mass energy. Running at this energy, the $WW$-fusion production of
Higgs provides complementary information about Higgs couplings and help reduce
the correlations between measurement relying on $ZH$ production. Linear
colliders can access the Higgs physics at higher center-of-mass energies. For
example, the CLIC would start at $E_{\rm CM}=380$ GeV. The $WW$-fusion and
$ZH$ associated production would both play essential roles. The detailed
comparison of the Higgs precision physics reach can be found in various
studies, e.g., Ref. [20]. There is also a proposal for a 125 GeV muon collider
Higgs factory. At this collider, the production mode for Higgs differs from
this plot, and hence we did not show it here. For $10^{6}$ Higgs within 5
years, one needs about $14\,{\rm fb}^{-1}/{\rm yr}$ ($1.4\times 10^{32}~{}{\rm
cm}^{-2}{\rm s}^{-1}$) with a resonance scan strategy developed in Ref. [6].
Beyond Higgs coupling precision, the lepton collider Higgs program also
provides intriguing opportunities in probing Higgs exotic decays [22, 23],
complementary to the hadron collider program [24, 25].

Double Higgs and Higgs self-coupling.

Another important measurement is the Higgs self-coupling. We distinguish it
from the discussion above since it requires good statistics in the double-
Higgs final state. The extraction of the Higgs self-coupling requires
consistent treatment of all relevant couplings that could affect the $HH$
process at the lepton collider, as many single Higgs couplings can also affect
the rates. Many studies have shown that a TeV lepton collider could extract
Higgs self-coupling at around 10% level. Beyond the double Higgs production,
lepton colliders could also access Higgs self-coupling through loop-induced
processes. Ref. [26] shows that in a consistent SMEFT framework, one can
extract the Higgs self-coupling to around 40% without the $HH$ processes at
low energy Higgs factories. To set a target for the $HH$ process, we show the
required luminosity for $5\times 10^{3}$ $HH$ within 5 years. This calculation
included the $ZHH$ associated production, VBF $HH$ production, and
$t\bar{t}HH$ productions. The former two processes are dominant at an energy
below and above around 800 GeV, respectively.

Precision $Z$.

Circular $e^{-}e^{+}$colliders offer the possibilities of a high statistics
$Z$ factory. Such a $Z$ factory also has a broad physics program. First of
all, as demonstrated by LEP-I and SLC, a $Z$-pole program brings in a suite of
electroweak precision observables. LEP-I produced about $10^{7}$ $Z$ bosons.
To be significantly better, a new $Z$-factory would need to produce at least
$10^{9}$ $Z$ bosons, dubbed “Giga Z”.

The precision electroweak program is an essential input for other physics
quantity extractions, such as the Higgs and top quark couplings. Studies [27,
28] show that a “Giga Z” provides sufficient precision to not significantly
hinder the physics extraction of other physics programs, making it a good
benchmark. A further improvement would require significantly better
measurements of other input parameters of the electroweak precision fit,
motivating runs at the $WW$ and $t\bar{t}$ thresholds (discussed later), and
increasing the Z-pole statistics.

The $Z$-pole physics as precision electroweak physics can also be accessed in
other processes, though less efficiently. For instance, through the “radiative
return” process, one can also measure the $Z$-boson couplings to various final
states, and through the EFT contact operators, one can probe four-fermions
interactions effectively. In Fig. missing 1, we show the luminosity
requirement to produce $10^{9}$ (nearly) on-shell $Z$ boson around $Z$-pole
and through the “radiative return” process. We also include other
contributions of diboson processes, but for the region of interest, they are
never dominant. The radiative return calculation is based upon the framework
and beam structure-function specified in Ref. [29, 30, 31].

In addition to the electroweak precision measurements, a Z factory is both a
$\tau$ factory and a $b$ factory. Current proposals of Z factories, such as
FCC-ee and CEPC, often set $10^{12}$ Z as a target. This can further improve
the sensitivity to rare decays and the potential for them as $\tau$ and $b$
factories [1, 2], which can be competitive to other $\tau$-charm factories
[32] and b factories [33] running at much lower energies. Moreover, it
produces $\tau$ and $b$ at a much more significant boost, which can be
beneficial in specific channels. Rare Z decay also offers a great opportunity
in probing a broad range of dark sector models [34]. The reach-scale linearly
with the total number of $Z$s for distinct final states that have a negligible
background.

Precision $WW$ threshold physics.

$WW$ production near the threshold is another important physics channel at
lepton colliders for electroweak precision physics. $W$ mass measurement is
crucial for interpreting the electroweak precision observable on the $Z$-pole.
The precision of the W mass measurement by colliders has an error bar of
$\delta m_{W}\simeq 12$ MeV [35]. The recent result reported by the CDF
collaboration improve the precision to $\delta m_{W}=9.4$ MeV [36] (with an
intriguing deviation from the SM electroweak fit and other collider
measurements [35, 37, 38, 39, 40]). The HL-LHC is expected to improve this
further to about $\delta m_{W}\simeq 8$ MeV by the end of HL-LHC [41, 42]. A
$WW$ threshold (with precision better than a few MeV ) scan would be a
decisive W mass measurement. It is also indispensable for realizing the full
potential of electroweak precision measurement at a Tera-$Z$ factory [27, 1,
2].

At the same time, many new physics generate deviations in the $W$ boson
couplings, such as the TGC. This can leads to signals in the $WW$ final state.
Since the effect of such new physics tends to grow with energy, it would be
beneficial to have good statistics in the WW final states at higher energies.
Based on these, about $10^{6}$ $WW$s would be a good target for running close
to the threshold. In Fig. missing 1, we show the luminosity requirement for
$10^{6}$ $WW$s for one year of running. One can also achieve precision
measurements, e.g., on TGC, via $WW$ pairs beyond threshold scan. In this
sense, the $WW$ measurement for EFT can be simultaneously realized in the
Higgs factory and higher energy runs.

Precision Top quark physics.

The motivation to have a good scan of the $t{\bar{t}}$ threshold is similar to
that for the $WW$ threshold since top mass is a crucial input to the
electroweak precision fit. Hadronic uncertainties will limit the top quark
mass measurement at hadron colliders to $\delta m_{t}\sim 10^{2}$ MeV.
Threshold scan with order $10^{5}$ $t\bar{t}$ pairs can push the precision on
the top mass down to around 10 MeV [43]. Many new physics models also predict
deviations in top quark gauge couplings. Due to large backgrounds and
theoretical uncertainties, the HL-LHC will have limited sensitivities to top
quark gauge couplings. The lepton colliders could probe the top gauge
couplings and top EFT operators to a good precision through direct pair
production and their angular correlations. To achieve 10 MeV top mass
precision and sub-percent level top quark coupling precision, we set a target
of $10^{5}$ $t\bar{t}$ per year, shown in the orange curve in Fig. missing 1.
The $s$-channel Drell-Yan-like processes drive the production of low-energy
lepton colliders. When we move to high-energy lepton colliders, the production
will be driven more by the VBF process [44, 45].

EFT.

Lepton colliders offer a clean environment for the search for new physics. At
the same time, the direct production of the new physics particles is limited
by the center-of-mass energy. Hence, the direct production of high-scale
physics is unlikely to be a significant physics driver in the range of
energies considered here.

On the other hand, integrating-out new physics leads to higher dimensional EFT
operators. Such EFT operators will modify the SM couplings on the order shown
in Eq. (missing) 1. Hence, precision measurement of Higgs, $Z$, $W$, and top
couplings discussed above probe such new physics. At the same time, in the
appropriate channels, the new physics effect can grow with energy as
$(E/\Lambda)^{2}$. Hence, it can be beneficial to measure them at high
energies. The EFT probe is different in its sensitivity scaling with collider
energy compared to the on-shell precision measurement. A typical example of
such new physics is a heavy $Z^{\prime}$ resonance. Integrating out such a
resonance will generate a dimension-6 four-Fermi operator, which in term
modifies the high energy distribution of $\ell^{+}\ell^{-}\rightarrow
f\bar{f}$ 111Heavy vector resonances can also have other final states such as
$WW$ and $Zh$. The scaling of the reach in these channels would be similar..
For valid momentum expansion of the EFT, at linear order, for a vector-vector
four-fermion $e^{+}e^{-}f\bar{f}$ operator
($\bar{e}\gamma_{\mu}e\bar{f}\gamma^{\mu}f$) with coefficient $c_{\rm EFT}$
and scale $\Lambda^{2}$,

$\Delta\sigma(e^{+}e^{-}\rightarrow f\bar{f})\propto\frac{c_{\rm
EFT}}{\Lambda^{2}},$ (2)

where $c_{\rm EFT}$ parameterizes the coupling between the new physics and the
SM fermions. Meanwhile, the SM background scales as

$\sigma(e^{+}e^{-}\rightarrow f\bar{f})\propto\frac{1}{E^{2}}.$ (3)

Hence, for the fixed target scale of EFT sensitivity, assuming dominance of
statistical uncertainty, the required luminosity scales as $1/E^{2}$. In Fig.
missing 1 we set a EFT scale of $\Lambda/\sqrt{c_{\rm EFT}}=10$ TeV to
demonstrate the required luminosity generic $10$-TeV scale generic new physics
in EFT. Specifically in Fig. missing 1 and Fig. missing 2, we choose $f$ to be
bottom quark.

##  III Lepton collider: TeV and Beyond

Beyond sub-TeV lepton colliders, the proposals of realizing multi-TeV, even
multi-tens of TeV lepton colliders through various techniques show great
promise in exploring high energy particle physics. These exciting
possibilities include CLIC [5] (and envisioned CLIC-like upgrade possibilities
for lepton colliders such as ILC and $C^{3}$ [15]), muon collider [46], plasma
weak field acceleration [47], etc. In this section, we focus on high energy
lepton colliders with $E_{\rm CM}$ in the range of 1 - 20 TeV222The upper
limit of the energy is mainly set by what’s being proposed during the Snowmass
process. The lesson we learned here can also be applied to higher energy
lepton colliders, with the targets extrapolated in a straightforward manner. .
Here, the primary goal would be searching for heavy new physics resonances. At
the same time, high energy lepton collider can still contribute to the
measurement of the Higgs coupling, such as Higgs precision coupling
measurements, top Yukawa coupling, and Higgs self-coupling. Since
$e^{-}e^{+}$and $\mu^{-}\mu^{+}$ colliders have very similar reaches in this
range of energies, we do not distinguish them.

Figure 2: Required luminosity for lepton colliders for various physics goals
in the energy range 1 TeV to 20 TeV, assuming one detector for each
experiment. For high-energy lepton colliders, while it is possible to have a
staged approach to run at different energies, such plannings are still at very
early stages. Hence we assumed a uniform run time of 10 yrs. The luminosity
requirement for 5$\sigma$ discovery of the benchmark DM scenarios Higgsino and
Wino are shown in purple lines. The disappearing track search dominates the
Dark Matter discovery luminosity curves. The exception is for Higgsino with
lepton collider energy below 6 TeV, where the boost of the Higgsino is
insufficient to generate the signal, and the missing energy search dominates
the search; for details, see Refs. [48, 49]. Bottom: We overlaid the
instantaneous luminosities of various proposed lepton colliders in this energy
range. The run plans are again taken from Snowmass Agora presentations [11,
10, 12, 14].

The summary of our results are shown in Fig. missing 2. In the following, we
will discuss the additional physics cases and considerations beyond those of
the lower energy lepton colliders discussed in the previous section II.

Single and Double Higgs.

The Higgs boson precision program is an essential component of a high-energy
lepton collider as well. The high center-of-mass energy allows various new
opportunities around precision Higgs physics. First, one can produce more
Higgs bosons in the VBF dominant processes at high energy lepton colliders.
Hence, we show two benchmark targets of $10^{6}$ and $10^{7}$ Higgs bosons,
shown in solid red and dashed red lines in Fig. missing 2. We include various
single and double Higgs production modes to determine such target luminosity
curves. However, the result is unchanged if we only include VBF WW-fusion
Higgs production in the region of interest. From the lineshape, we can see
that as one goes to higher center-of-mass energy, the required luminosity is
lower due to the logarithmic enhancement in such processes. One should note
that the Higgs boson at high energy lepton collider might be associated with a
more significant boost, and the combinatorics background can be larger.

The large center-of-mass energy of high energy lepton colliders also enables
copious production of the double Higgs processes, even triple Higgs [50, 51].
Measuring these processes enables us to determine the Higgs potential further.
Here in Fig. missing 2 we set a target of $10^{5}$ double Higgs process, which
should be sufficient for one to extract Higgs self-coupling at the percent
level. Again, the double Higgs process is dominated by the VBF process, and we
can see the required luminosity is high.

EFT.

Compared with the lower energy lepton colliders, running at higher energies
can extend the reach to new physics encoded in SMEFT even further by measuring
its contribution to high energy distributions. As argued in the previous
section, for a typical four-fermi contact interaction, the required luminosity
for a fixed target scale of $\Lambda/c_{\rm EFT}$ scales as $1/E^{2}$. For
high-energy lepton colliders, our target scale would be more prominent. In
Fig. missing 2, we set two target scales; one is fixed at 100 TeV, and the
other is a factor of 10 higher than the center-of-mass energy. We show the
required luminosities in solid and dashed blue curves in these two setups.

Top partners, $T^{\prime}$.

The hierarchy problem is undoubtedly one of the most prominent physics
motivations for new physics beyond the Standard Model, and addressing it is a
leading physics driver for future colliders. Among the new physics particle
associated with the hierarchy problem, the top partner is probably the most
important one due to the significant role the top quark played in the dynamics
of the electroweak symmetry breaking. It is one of the leading physics targets
for the upcoming runs of the LHC and will continue to be so for future
colliders in the coming decades. Hence, we will use the top partner as a
benchmark to characterize the capability of high-energy lepton colliders in
new physics searches.

High energy lepton colliders are optimal in searching for top partners. In
most models of top partners, it has either the same or similar quantum numbers
as the SM top quarks. Hence, it can be produced copiously at the lepton
colliders through their EW gauge interactions in the Drell-Yan-like process.
Since the decay of the top partners typically gives energetic and visible
final states, we expect they can be identified and separated from the
background in the lepton collider environment. With small statistics of 20
signal events, one should be able to discover them. We expect the reach of the
top partner at the lepton collider should be close to the kinematic threshold.
With this in mind, we set a target top partner mass to be at 90% of the pair
production kinematic threshold, $2m_{T^{\prime}}=0.9\times E_{\rm CM}$, and
show the required luminosity for a generic scalar or fermionic top partners
with dashed and solid yellow curves in Fig. missing 2. Any top partner with a
mass less than this target should be discoverable for the given luminosity.
The production rate will be slightly higher with the removal of threshold
suppression. In principle, a top partner with a mass at the weak scale could
have a sizable background from the SM processes, depending on its primary
decay channel. However, this possibility is highly disfavored with the current
LHC search results.

WIMP Dark matter.

Testing the WIMP (Weakly Interacting Massive Particle) paradigm of dark matter
is another main physics driver for future colliders. Among various possible
candidates, the minimal model would be dark matter as a member of an
electroweak multiplet. Such dark matter multiplet can be produced copiously at
the high energy lepton colliders. However, unlike in the case of top partners,
the signal is more difficult to detect. One has to rely on relatively soft
objects, and additional radiated SM particles in the event to identify the
signal. The background would be significant at lepton colliders. It is not
possible to determine the reach from simple estimates. Fortunately, there have
been relatively detailed studies in this case [48, 49, 52, 53, 54]. The
signals can be categorized into general and inclusive missing energy searches.
One can look for the mono-photon, mono-$Z$, mono-$W$, and even mono-lepton at
lepton colliders to hunt for WIMP dark matter. The Drell-Yan-like pair
productions dominate the WIMP dark matter production. However, for WIMP dark
matter mass somewhat smaller than the kinematic threshold, the VBF process can
dominate. The VBF process produces different kinematics and allows for
interesting and clean signals such as mono-lepton or VBF di-leptons [48, 49].

The fermionic electroweak doublet and triplet are particularly important among
the WIMP candidates. They are also the primary DM candidates in the Minimal
Supersymmetric Standard Model, known as pure-Higgsino and pure-Wino,
respectively. Beyond the inclusive missing energy searches, one can rely on
the somewhat long-live charged state in the multiplet. A typical lifetime in
the rest frame of the pure Higgsino and pure Wino is around $0.02$ ns and
$0.2$ ns and thermal dark matter target mass of 1.1 TeV and 2.8 TeV,
respectively. One can search for the corresponding “disappearing track”
signature. The availability of this signal is somewhat model dependent as it
is very sensitive to the mass splitting with the electroweak multiplet.
However, if available, the disappearing track empowers a much higher signal to
background ratio and hence a more effective search.

We show the required luminosity as a function of the lepton collider center-
of-mass energy for Higgsino and Wino in solid and dashed purple lines,
respectively. At low center-of-mass energy, the search sensitivity is driven
by the inclusive missing energy searches, as the boost of the pair-produced
charged Higgsino is low. Hence, the lifetime is not long enough to support a
high efficiency of the “disappearing track” signatures. Beyond 6 TeV center-
of-mass energy, the search sensitivities are driven by the disappearing track
searches. Different lepton colliders will have different sources of background
for such a signature, and future studies will be of particular importance with
concrete beam and detector designs. We can see from the figure that a 10 TeV
lepton collider could achieve the Higgsino and Wino dark matter goal robustly.

##  IV High energy proton proton collider

Figure 3: Luminosity requirements for high energy hadron colliders, for
possible initial states (gg, red; $q\bar{q}$ blue). One set of goals, the
optimal case, is to have the ultimate reach scale linearly with the center-of-
mass energy. This would require the luminosity scale as $E_{\rm CM}^{2}$,
similar to the lepton collider. At the same time, to get most of the increase
in the reach, say $70\%$ of the optimal reach, the required luminosity is much
lower. This is a consequence of the scaling of parton luminosity as a function
of the parton center-of-mass energy. The scaling of the reach shown here is
done by statistics and using parton luminosity [55]. We used the reach at the
HL-LHC as a reference point for the extrapolation. For the gluon-gluon initial
state dominated process, we assumed the reach of the mass of new physics at
the HL-LHC of 3 TeV (approximately 1.5 TeV for pair production). This could be
similar to the case, for example, the stop. For $q\bar{q}$ initiated
processes, we assumed a reach at the HL-LHC of 1 TeV (which would be about 500
GeV for pair production). This would be similar to the electroweak states.
Changing the assumption of HL-LHC reach will give rise to some differences but
affect the qualitative feature. The luminosities of several proposed future
hadron colliders, taken from Agora presentations [13], are shown.

High energy proton-proton colliders [3, 13], with $E_{\rm cm}$ up to the range
of 100 TeV, is a promising way of exploring the energy frontier. Here, the
physics targets for specific proposals have been carefully studied [56, 3],
including the luminosities needed to achieve them [57]. In this section, we
focus on the interplay between luminosity and energy for a broader range of
energies.

It is more challenging to make projections for the hadron colliders,
especially for searches dominated by systematics. In this case, the prospects
depend sensitively on the assumption of potential improvement in these
systematics. Some of these can undoubtedly be improved with the accumulation
of data, such as those channels dominated by the statistics of a sideband. At
the same time, a lot could depend on the details of the accelerator and
detector.

At the same time, it is possible to make some rough estimates for searches
based on the behavior of parton luminosity and statistics [55]. This method,
and a discussion of the qualitative behavior of the reach, are summarized in
Appendix A. By the nature of this estimate, we will not be able to take into
account subtle kinematic effects. The main factor which determines the reach
is the composition of the initial state that dominates the production. For new
physics charged under color, typically, the gluon-gluon initial state would
give the dominant contribution. For example, this is the case for the stop in
supersymmetry333A possible exception is diquark production, to which valance
quark gives the dominant contribution.. At the same time, $q\bar{q}$ initial
state dominants the contribution to new physics, which are neutral under
color, both for the case of single production (such as $Z^{\prime}$) or pair
production (such as electroweak multiplets).

Figure 4: Mass reach at high energy hadron colliders in comparison with the
reach at HL-LHC [57]. The case of gg initial state is used as an example.
During the initial stage of the run, there is a rapid gain of reach for a
relatively small amount of integrated luminosity.

Our result is presented in Fig. missing 3. In the following, we make a couple
of observations.

  * •

Given two colliders with the center of mass energies $E_{1}$ and $E_{2}$, the
corresponding reach of the masses of a particular new physics particle are
denoted as $M_{1}$ and $M_{2}$, respectively. Ideally (the optimal case), one
would like to have the reach of new physics scale linear with the center-of-
mass energy, $M_{2}/M_{1}=E_{2}/E_{1}$. With a mild assumption on the
production rate scaling, it is straightforward to see we need ${\cal
L}_{2}/{\cal L}_{1}=E^{2}_{2}/E^{2}_{1}$ (see Appendix A), which can be seen
in the dashed curves in Fig. missing 3. This could be a large step in
luminosity increase for a large increase of the center-of-mass energy.

  * •

Due to the nature of the parton luminosity as a function of parton center-of-
mass energy, there is a rapid gain in reach for a relatively small amount of
luminosity. This can also be seen in the sharp rises during the early stages
of the run, shown in Fig. missing 4. Hence, to achieve a somewhat lower goal
for the mass reach enhancement, for example, $70\%$ of $E_{2}/E_{1}$, one
needs a significantly smaller amount of data, shown as the solid lines in Fig.
missing 3.

  * •

The parton luminosity for the gluon-gluon initial state falls as a function of
the parton center-of-mass energy faster than the $q\bar{q}$ initial state. As
a result, the luminosity needed for obtaining a significant fraction of the
maximal reach in the gluon-gluon initial state-dominated processes is less
than that of processes dominated by the $q\bar{q}$ processes (see Appendix A).
This is also shown in Fig. missing 3 as the difference between the solid red
and blue lines.

##  V Summary

In this white paper, we provide a broad-brush picture of the physics output of
future colliders as a function of their center of mass energies and
luminosities. The key results for colliders at different energies can be found
in Fig. missing 1, Fig. missing 2, and Fig. missing 3. This is an input to
Snowmass studies, in particular for the Implementation Task Force, which helps
present the physics yields of various future collider proposals.

###### Acknowledgements.

The authors thank Xing Wang for helpful discussion and Matthew Low for help in
preparing the results for hadron colliders. We would like to thank the members
of the Implementation Task Force for their discussions and suggestions. ZL is
supported by the U.S. Department of Energy (DOE) under grant No. DE-SC0022345.
LTW is supported by the DOE grant DE-SC0013642. ZL acknowledges the support of
the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP), where the
final stage of this work is conducted, which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s
Excellence Strategy-EXC-2094-390783311.

##  Appendix A Scaling for the reach at hadron colliders

The basic feature of luminosity needed as a function of $E_{\rm CM}$ can be
understood as follows. Consider two colliders with center-of-mass energies
$E_{1}$ and $E_{2}$, with integrated luminosities ${\cal L}_{1}$ and ${\cal
L}_{2}$, respectively. The reach in certain channel are $M_{1}$ and $M_{2}$
for the two colliders. Since the parton luminosities scale as a power law of
$\tau=\hat{s}/E^{2}_{\rm CM}$, under the assumption that one needs the same
number of signal events to obtain a limit or see an excess, we have 444Here we
assume the partonic matrix element squared is independent of $\sqrt{\hat{s}}$,
which is true for a broad class of processes, e.g., typical $2\rightarrow 2$
process through marginal operators, resonance productions with fixed couplings
(total width of the resonance proportional to mass).

$\displaystyle\frac{1}{M_{1}^{2}}\frac{1}{\tau_{1}^{a}}{\cal
L}_{1}\simeq\frac{1}{M_{2}^{2}}\frac{1}{\tau_{2}^{a}}{\cal L}_{2},$ (4)

where $\tau_{1,2}=M^{2}_{1,2}/E^{2}_{1,2}$. 555Here we assume the same scale
of the signal rate and background rate, as a function of partonic center of
mass energy $\sqrt{\hat{s}}\sim M$. If we take HL-LHC as a base point for
extrapolation, $E_{1}=14$ TeV and ${\cal L}_{1}=3$ ab-1. $M_{1}$ would be the
maximal reach achievable at the HL-LHC. $a$ is typically in the range of
$3\sim 5$. With this, we obtain

$\displaystyle\left(\frac{\tau_{2}}{\tau_{1}}\right)^{2+2a}\simeq\frac{E_{1}^{2}}{E_{2}^{2}}\frac{{\cal
L}_{2}}{{\cal L}_{1}}.$ (5)

If we want to achieve the maximal reach, i.e., the reach scale with the
center-of-mass energy, then $\tau_{2}=\tau_{1}$. In this case, the luminosity
needed is proportional to the $E_{\rm CM}^{2}$, as expected. On the other
hand, if we would like only to achieve a fraction $x$ (say $x=70\%$) of the
maximal reach, we need a factor of $x^{2+2a}$ less luminosity. This is
significantly less if $a$ is large. In this case, the process with initial
state parton luminosity with a smaller $a$ would need more luminosity, as in
the case of the $q\bar{q}$ in comparison with $gg$ initial states. Of course,
this approximation is crude and ignores a couple of effects. First of all,
naively, $\tau_{1}$ would cancel in this extrapolation, as the result will
only depend on fraction $x$. However, $a$ is not a constant, it depends on
$\tau$. Therefore, the actual reach at HL-LHC for a particular process
($\tau_{1}$) affects the needed luminosity. Hence, treating $a$ as a constant
at most only conveys the qualitative feature of the scaling. In addition, the
parton luminosity has a logarithmic dependence on the overall energy scale,
which we have ignored in this scaling argument (although it has been taken
into account in the numerical result presented in Fig. missing 3).

## References

  * [1] CEPC Study Group Collaboration, M. Dong et al., “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” arXiv:1811.10545 [hep-ex].
  * [2] FCC Collaboration, A. Abada et al., “FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2,” Eur. Phys. J. ST 228 no. 2, (2019) 261–623.
  * [3] FCC Collaboration, A. Abada et al., “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 no. 4, (2019) 755–1107.
  * [4] ILC Collaboration, I. Adachi et al., “The International Linear Collider: Report to Snowmass 2021,” arXiv:2203.07622 [physics.acc-ph].
  * [5] R. Franceschini et al., “The CLIC Potential for New Physics,” CERN Yellow Rep. Monogr. 3/2018 (12, 2018) , arXiv:1812.02093 [hep-ph].
  * [6] J. de Blas, J. Gu, and Z. Liu, “Higgs Precision at a 125 GeV Muon Collider,” arXiv:2203.04324 [hep-ph].
  * [7] M. Bai et al., “C3: A ”Cool” Route to the Higgs Boson and Beyond,”  10, 2021.  arXiv:2110.15800 [hep-ex].
  * [8] V. N. Litvinenko, N. Bachhawat, M. Chamizo-Llatas, Y. Jing, F. Méot, I. Petrushina, and T. Roser, “The ReLiC: Recycling Linear $e^{+}e^{-}$ Collider,” arXiv:2203.06476 [hep-ex].
  * [9] V. N. Litvinenko, N. Bachhawat, M. Chamizo-Llatas, F. Meot, and T. Roser, “CERC - Circular e+e- Collider using Energy-Recovery Linac,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07358 [physics.acc-ph].
  * [10] “Snowmass agora on future colliders: Circular e+e- colliders.” https://indico.fnal.gov/event/52534/.
  * [11] “Snowmass agora on future colliders: Linear e+e- colliders.” https://indico.fnal.gov/event/52161/.
  * [12] “Snowmass agora on future colliders: Muon colliders.” https://indico.fnal.gov/event/53010/.
  * [13] “Snowmass agora on future colliders: Circular pp and ep colliders.” https://indico.fnal.gov/event/53473/timetable/.
  * [14] “Snowmass agora on future colliders: Advanced colliders.” https://indico.fnal.gov/event/52161/.
  * [15] S. Dasu et al., “Strategy for Understanding the Higgs Physics: The Cool Copper Collider,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07646 [hep-ex].
  * [16] T. Han, P. Langacker, Z. Liu, and L.-T. Wang, “Diagnosis of a New Neutral Gauge Boson at the LHC and ILC for Snowmass 2013,” arXiv:1308.2738 [hep-ph].
  * [17] N. Craig, J. Gu, Z. Liu, and K. Wang, “Beyond Higgs Couplings: Probing the Higgs with Angular Observables at Future e+ e- Colliders,” JHEP 03 (2016) 050, arXiv:1512.06877 [hep-ph].
  * [18] P. Janot, “Top-quark electroweak couplings at the FCC-ee,” JHEP 04 (2015) 182, arXiv:1503.01325 [hep-ph].
  * [19] J. Gu, H. Li, Z. Liu, S. Su, and W. Su, “Learning from Higgs Physics at Future Higgs Factories,” JHEP 12 (2017) 153, arXiv:1709.06103 [hep-ph].
  * [20] J. de Blas et al., “Higgs Boson Studies at Future Particle Colliders,” JHEP 01 (2020) 139, arXiv:1905.03764 [hep-ph].
  * [21] F. An et al., “Precision Higgs physics at the CEPC,” Chin. Phys. C 43 no. 4, (2019) 043002, arXiv:1810.09037 [hep-ex].
  * [22] Z. Liu, L.-T. Wang, and H. Zhang, “Exotic decays of the 125 GeV Higgs boson at future $e^{+}e^{-}$ lepton colliders,” Chin. Phys. C 41 no. 6, (2017) 063102, arXiv:1612.09284 [hep-ph].
  * [23] M. Carena, J. Kozaczuk, Z. Liu, T. Ou, M. J. Ramsey-Musolf, J. Shelton, Y. Wang, and K.-P. Xie, “Probing the Electroweak Phase Transition with Exotic Higgs Decays,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.08206 [hep-ph].
  * [24] D. Curtin et al., “Exotic decays of the 125 GeV Higgs boson,” Phys. Rev. D 90 no. 7, (2014) 075004, arXiv:1312.4992 [hep-ph].
  * [25] M. Cepeda, S. Gori, V. M. Outschoorn, and J. Shelton, “Exotic Higgs Decays,” arXiv:2111.12751 [hep-ph].
  * [26] S. Di Vita, G. Durieux, C. Grojean, J. Gu, Z. Liu, G. Panico, M. Riembau, and T. Vantalon, “A global view on the Higgs self-coupling at lepton colliders,” JHEP 02 (2018) 178, arXiv:1711.03978 [hep-ph].
  * [27] J. Fan, M. Reece, and L.-T. Wang, “Possible Futures of Electroweak Precision: ILC, FCC-ee, and CEPC,” JHEP 09 (2015) 196, arXiv:1411.1054 [hep-ph].
  * [28] J. de Blas, G. Durieux, C. Grojean, J. Gu, and A. Paul, “On the future of Higgs, electroweak and diboson measurements at lepton colliders,” JHEP 12 (2019) 117, arXiv:1907.04311 [hep-ph].
  * [29] M. Greco, T. Han, and Z. Liu, “ISR effects for resonant Higgs production at future lepton colliders,” Phys. Lett. B 763 (2016) 409–415, arXiv:1607.03210 [hep-ph].
  * [30] S. Jadach and R. A. Kycia, “Lineshape of the Higgs boson in future lepton colliders,” Phys. Lett. B 755 (2016) 58–63, arXiv:1509.02406 [hep-ph].
  * [31] S. Jadach, B. F. L. Ward, and Z. Was, “Coherent exclusive exponentiation for precision Monte Carlo calculations,” Phys. Rev. D 63 (2001) 113009, arXiv:hep-ph/0006359.
  * [32] X.-D. Shi, X.-R. Zhou, X.-S. Qin, and H.-P. Peng, “A fast simulation package for STCF detector,” JINST 16 no. 03, (2021) P03029, arXiv:2011.01654 [physics.ins-det].
  * [33] Belle-II Collaboration, W. Altmannshofer et al., “The Belle II Physics Book,” PTEP 2019 no. 12, (2019) 123C01, arXiv:1808.10567 [hep-ex]. [Erratum: PTEP 2020, 029201 (2020)].
  * [34] J. Liu, L.-T. Wang, X.-P. Wang, and W. Xue, “Exposing the dark sector with future Z factories,” Phys. Rev. D 97 no. 9, (2018) 095044, arXiv:1712.07237 [hep-ph].
  * [35] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [36] CDF Collaboration, T. Aaltonen et al., “High-precision measurement of the W boson mass with the CDF II detector,” Science 376 no. 6589, (2022) 170–176.
  * [37] J. de Blas, M. Pierini, L. Reina, and L. Silvestrini, “Impact of the recent measurements of the top-quark and W-boson masses on electroweak precision fits,” arXiv:2204.04204 [hep-ph].
  * [38] J. Gu, Z. Liu, T. Ma, and J. Shu, “Speculations on the W-Mass Measurement at CDF,” arXiv:2204.05296 [hep-ph].
  * [39] J. Fan, L. Li, T. Liu, and K.-F. Lyu, “$W$-Boson Mass, Electroweak Precision Tests and SMEFT,” arXiv:2204.04805 [hep-ph].
  * [40] E. Bagnaschi, J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, “SMEFT Analysis of $m_{W}$,” arXiv:2204.05260 [hep-ph].
  * [41] Gfitter Group Collaboration, M. Baak, J. Cúth, J. Haller, A. Hoecker, R. Kogler, K. Mönig, M. Schott, and J. Stelzer, “The global electroweak fit at NNLO and prospects for the LHC and ILC,” Eur. Phys. J. C 74 (2014) 3046, arXiv:1407.3792 [hep-ph].
  * [42] M. Awramik, M. Czakon, A. Freitas, and G. Weiglein, “Precise prediction for the W boson mass in the standard model,” Phys. Rev. D 69 (2004) 053006, arXiv:hep-ph/0311148.
  * [43] F. Simon, “Scanning Strategies at the Top Threshold at ILC,” in International Workshop on Future Linear Colliders.  2, 2019.  arXiv:1902.07246 [hep-ex].
  * [44] T. Han, Y. Ma, and K. Xie, “High Energy Leptonic Collisions and Electroweak Parton Distribution Functions,” Phys. Rev. D 103 (2021) 031301, arXiv:2007.14300 [hep-ph].
  * [45] A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz, and X. Zhao, “Vector boson fusion at multi-TeV muon colliders,” JHEP 09 (2020) 080, arXiv:2005.10289 [hep-ph].
  * [46] C. Aimè et al., “Muon Collider Physics Summary,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07256 [hep-ph].
  * [47] M. Fuchs et al., “Snowmass Whitepaper AF6: Plasma-Based Particle Sources,”  3, 2022.  arXiv:2203.08379 [physics.acc-ph].
  * [48] T. Han, Z. Liu, L.-T. Wang, and X. Wang, “WIMPs at High Energy Muon Colliders,” arXiv:2009.11287 [hep-ph].
  * [49] T. Han, Z. Liu, L.-T. Wang, and X. Wang, “WIMP Dark Matter at High Energy Muon Colliders $-$A White Paper for Snowmass 2021,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07351 [hep-ph].
  * [50] T. Han, D. Liu, I. Low, and X. Wang, “Electroweak couplings of the Higgs boson at a multi-TeV muon collider,” Phys. Rev. D 103 no. 1, (2021) 013002, arXiv:2008.12204 [hep-ph].
  * [51] M. Chiesa, F. Maltoni, L. Mantani, B. Mele, F. Piccinini, and X. Zhao, “Measuring the quartic Higgs self-coupling at a multi-TeV muon collider,” JHEP 09 (2020) 098, arXiv:2003.13628 [hep-ph].
  * [52] R. Capdevilla, F. Meloni, R. Simoniello, and J. Zurita, “Hunting wino and higgsino dark matter at the muon collider with disappearing tracks,” arXiv:2102.11292 [hep-ph].
  * [53] S. Bottaro, D. Buttazzo, M. Costa, R. Franceschini, P. Panci, D. Redigolo, and L. Vittorio, “Closing the window on WIMP Dark Matter,” Eur. Phys. J. C 82 no. 1, (2022) 31, arXiv:2107.09688 [hep-ph].
  * [54] S. Bottaro, A. Strumia, and N. Vignaroli, “Minimal Dark Matter bound states at future colliders,” JHEP 06 (2021) 143, arXiv:2103.12766 [hep-ph].
  * [55] “The collider reach.” http://collider-reach.web.cern.ch/collider-reach/.
  * [56] M. Ahmad et al., “CEPC-SPPC Preliminary Conceptual Design Report. 1. Physics and Detector,”.
  * [57] I. Hinchliffe, A. Kotwal, M. L. Mangano, C. Quigg, and L.-T. Wang, “Luminosity goals for a 100-TeV pp collider,” Int. J. Mod. Phys. A 30 no. 23, (2015) 1544002, arXiv:1504.06108 [hep-ph].