ultimately the worst-case scenario; even a modest splitting in the higgsino
multiplet would make it possible to significantly suppress backgrounds beyond
those considered here, and once sufficient phase space becomes available for
on-shell decays within the higgsino multiplet, bounds should approach the
$m_{\tilde{h}}\sim 0.9\sqrt{s}/2$ reach corresponding to distinctive final
states. In this limit, percent tuning becomes accessible at $\sqrt{s}\gtrsim
2$ TeV, per mille tuning at $\sqrt{s}\gtrsim 6$ TeV, and per myriad tuning at
$\sqrt{s}\gtrsim 20$ TeV.

Now we turn to the stop, whose contribution to the tuning of the weak scale at
leading-logarithmic order is

$\Delta_{\tilde{t}}\simeq\frac{3y_{t}^{2}}{4\pi^{2}}\frac{m_{\tilde{t}}^{2}}{m_{h}^{2}}\log\frac{\Lambda}{m_{\tilde{t}}}\,,$
(45)

where $\Lambda$ is a UV scale at which SUSY breaking is communicated; here we
conservatively take $\Lambda=10\times m_{\tilde{t}}$. Here 10% tuning
corresponds to $m_{\tilde{t}}\sim 950$ GeV, percent tuning to
$m_{\tilde{t}}\sim 3$ TeV, and per mille tuning to $m_{\tilde{t}}\sim 9.5$
TeV.

Barring a high degree of degeneracy between the lightest stop and its decay
products, the final state is quite distinctive and we can anticipate reach
scaling as $m_{\tilde{t}}\sim 0.9\sqrt{s}/2$. To validate this expectation, we
perform a parton-level analysis for two simplified stop-top-neutralino models:
one in which the stop is an $SU(2)_{L}$ singlet denoted $\tilde{t}_{R}$, the
other in which it is part of an $SU(2)_{L}$ doublet denoted $\tilde{t}_{L}$.
In both models the neutralino $\chi$ is taken to be bino-like. To ensure the
relevant production processes are not contaminated by contributions from heavy
superpartners, we construct both models in FeynRules using the $t$-channel
dark matter framework [134] with couplings set to their supersymmetric values.
We then simulate pair production of the stop in both muon annihilation and VBF
in Whizard, followed by the decay of the stop to a top quark and neutralino.
We generate samples for a range of stop masses up to $\sqrt{s}/2$, keeping the
neutralino effectively massless with $m_{\chi}=1\text{ GeV}$. We consider the
leading background to be VBF $t\bar{t}$ production, for which the $W^{+}W^{-}$
fusion contribution yields a $t\bar{t}$ \+ missing energy final state closely
mimicking the signal. For simplicity, we consider perfect reconstruction of
the top quarks in the final state. We then separately consider two possible
cuts to separate signal from background: (1) a cut on the $M_{C}$ variable
[135] constructed from the four-momenta of the $t$ and $\bar{t}$,

$\displaystyle
M_{C}=\sqrt{2\big{(}E_{1}E_{2}+\vec{p}_{1}\cdot\vec{p}_{2}+m_{t}^{2}\big{)}}\,,$
(46)

for which the signal has an endpoint at $M_{C}^{\rm max}=m_{\tilde{t}}$
assuming a massless neutralino, and (2) a cut on the missing transverse
momentum $\not{p}_{T}$. For each signal benchmark, we determine the $M_{C}$
and $\not{p}_{T}$ cuts that maximize $S/\sqrt{B}$, translating it into the
exclusion reach shown in Fig. 16 corresponding to $S/\sqrt{B}=2$ under either
the optimistic or the conservative luminosity scaling. For the right-handed
stop, the cross section for VBF pair production is generally too small to be a
meaningful contribution to the total signal rate, while for the left-handed
stop both muon annihilation and VBF production play an interesting role. In
each case, the $\not{p}_{T}$ cut alone is sufficient to provide sensitivity
all the way up to $m_{\tilde{t}}\sim\sqrt{s}/2$, save for the largest values
of $\sqrt{s}$ where some degradation of the limit is observed under
conservative luminosity assumptions due to the significant size of the
$t\bar{t}+\nu\bar{\nu}$ background at these energies. Of course, this highly
simplified analysis neglects a host of relevant effects associated with the
decay and reconstruction of top quarks, energy resolution, forward
reconstruction, and beam-induced backgrounds, but it supports the supposition
that significant kinematic separation of signal and background is possible,
and similar performance is likely achievable with an optimized selection using
all kinematic information.

Figure 16: Exclusion reach on a stop-top-neutralino simplified model as a
function of $\sqrt{s}$ and $m_{\tilde{t}}$ with $m_{\chi}=1$ GeV, assuming
“optimistic” (solid) and “conservative” (dashed) integrated luminosity scaling
as detailed in the body of the text. Left: Limit on pair production of an RH
stop in muon annihilation. Center: Limit on pair production of an LH stop in
muon annihilation. Right: Limit on pair production of an LH stop in vector
boson fusion.

As such, we consider the stop mass reach of muon colliders operating at
various values of $\sqrt{s}$ assuming that the reach scales as
$m_{\tilde{t}}\sim 0.9\times\sqrt{s}/2$. As a starting point, the anticipated
$2\sigma$ exclusion reach at the HL-LHC is $m_{\tilde{t}}\sim 1.7$ TeV, which
suggests that a muon collider operating at $\sqrt{s}\gtrsim 4$ TeV is required
to compete with LHC limits. Reaching percent-level tuning requires
$\sqrt{s}\gtrsim 6$ TeV, while per mille requires $\sqrt{s}\gtrsim 20$ TeV and
per myriad $\sqrt{s}\gtrsim 66$ TeV. Ultimately, the fine-tuning reach as a
function of $\sqrt{s}$ is comparable to the worst-case higgsino scenario with
only Standard Model radiative splittings, the one loop suppression of the stop
contribution to fine tuning competing with the more challenging backgrounds
faced by the higgsino.

Apart from considerations of fine tuning, the stop mass is also central to the
supersymmetric prediction of the observed Higgs mass. Famously, accommodating
$m_{h}\sim 125$ GeV in the minimal supersymmetric extension of the Standard
Model without significant mixing between the stop gauge eigenstates suggests
that the stops lie around or above $\sim 5\text{ - }10$ TeV; see, e.g., [136]
for a review. A high-energy muon collider operating at $\sqrt{s}=30$ TeV could
cover the typical scale of the stop mass suggested by the observed Higgs mass
at large values of $\tan\beta$, subject to further dependence on mixing angles
and the remaining sparticle spectrum. At moderate values of $\tan\beta$, the
stops can be heavier, $\sim 100\text{ - }1000$ TeV in well-motivated models.
However, in such scenarios the electroweakinos may be accessible (e.g., via
the searches discussed in Sec. 4.2), and the lightest sfermions, such as the
right-handed stau, may be an order of magnitude lighter than the stops and
could be directly accessible. Thus, the full range of searches for
superparticles could cover a substantial portion of the parameter space
motivated by the measured Higgs mass.

Finally, we turn to the gluino, which evades our reach estimates because it
does not carry electroweak quantum numbers. Nonetheless, it may still be
produced by a variety of higher-order processes, including gluon splitting
from $q\bar{q}$ production. The gluino contribution to the tuning of the weak
scale at leading logarithmic order scales as

$\Delta_{\tilde{g}}\simeq\frac{\alpha_{s}y_{t}^{2}}{\pi^{3}}\frac{m_{\tilde{g}}^{2}}{m_{h}^{2}}\log^{2}\frac{\Lambda}{m_{\tilde{g}}}$
(47)

Again taking $\Lambda=10\times m_{\tilde{g}}$, 10% tuning corresponds to
$m_{\tilde{g}}\sim 3$ TeV, percent tuning to $m_{\tilde{g}}\sim 9.5$ TeV, and
per mille tuning to $m_{\tilde{g}}\sim 30$ TeV.

To estimate the production rate, we consider the process
$\mu^{+}\mu^{-}\rightarrow\tilde{g}\tilde{g}+q\bar{q}$ with modest phase space
cuts, as detailed in the Appendix. Although the final state is quite
distinctive – presumably zero background is achievable with high signal
efficiency – the ultimate reach of a muon collider is quite sensitive to the
integrated luminosity due to the relatively small cross section. In Fig. 17,
we plot the discovery reach (assuming zero background, i.e., 5 signal events)
for both optimistic and conservative luminosity assumptions. The optimistic
luminosity assumption leads to a mass reach scaling with collider energy,
while the conservative luminosity assumption leads to a relatively constant
reach for $\sqrt{s}\gtrsim 10$ TeV.

Figure 17: Gluino discovery reach from
$\mu^{+}\mu^{-}\rightarrow\tilde{g}\tilde{g}+q\bar{q}$ as a function of
$\sqrt{s}$ , assuming “optimistic” (solid) and “conservative” (dashed)
integrated luminosity scaling as detailed in the body of the text.

As a starting point, the anticipated $2\sigma$ exclusion reach at the HL-LHC
is $m_{\tilde{g}}\sim 3.2$ TeV, which suggests that the discovery reach of a
muon collider could just reach the edge of LHC limits for $\sqrt{s}\gtrsim 10$
TeV and 10 ab-1. The optimistic luminosity scaling paints a rosier picture,
providing sensitivity to $m_{\tilde{g}}\sim 20$ TeV by $\sqrt{s}=100$ TeV.
Ultimately, the gluino reach lags behind the fine-tuning sensitivity of
higgsino and stop reach, but nonetheless a muon collider is comparable to or
better than the LHC depending on the c.m. energy and integrated luminosity.

Taken together, these projections suggest that a high-energy muon collider
operating at $\sqrt{s}\sim 20\text{ - }30$ TeV is capable of probing a natural
supersymmetric explanation of the weak scale beyond the per mille level, with
meaningful sensitivity to both of the states most important for understanding
the scale and origin of electroweak symmetry breaking: the higgsinos and
stops. Setting aside fine tuning, which may or may not be a sharp guide to the
scale of new physics, a collider operating at $\sqrt{s}\sim 30$ TeV could
reach the scale suggested by the Higgs mass prediction of the MSSM, regardless
of the the stop sector mixing.

#####  4.3.1.2 The gravitino

Although impressive in reach, the potential for a high-energy muon collider to
probe Standard Model superpartners represents a continuation of the already-
vast search program currently under way at the LHC. But a muon collider offers
more than incremental progress in the search for supersymmetry; it would be
perhaps the first collider with the potential to directly discover
supersymmetry through its universal feature, the goldstone fermion of
spontaneous supersymmetry breaking.

A universal prediction of spontaneous breaking of supersymmetry in the rigid
limit is the presence of a massless Majorana fermion, the goldstino. When
gravity effects are taken into account, the goldstino is eaten through super-
Higgs mechanism by the spin 3/2 gravitino [137], which obtains a mass

$m_{3/2}=\frac{F}{\sqrt{3}M_{\text{Pl}}}\,,$ (48)

where $\sqrt{F}$ is the scale of SUSY breaking and $M_{\text{Pl}}=2.4\times
10^{18}\text{ GeV}$. The scale of the superpartners is set schematically by
the supersymmetry-breaking contributions

$m_{\text{soft}}\sim\frac{g_{\text{eff}}^{2}}{16\pi^{2}}\sqrt{F}+m_{3/2}\,,$
(49)

The first term in (49) is the gauge mediation contribution, with
$g_{\text{eff}}$ an effective coupling encoding suppressions or enhancements
in the soft masses which are model dependent. For instance for the gaugino
masses one typically has $g_{\text{eff}}^{2}\sim
g_{\text{SM}}^{2}\,s_{g}\,N_{\text{mess}}\,\frac{\sqrt{F}}{M_{\text{mess}}}$,
where $s_{g}\lesssim 1$ parametrizes possible gaugino screening effects. The
second term in (49) is the gravity mediation contribution. We have assumed
that a single supersymmetry-breaking scale controls both. Low-energy
supersymmetry breaking models are those in which the gauge mediation
contribution dominates, such that the gravitino is the lightest supersymmetric
particle (LSP) in the spectrum and hence cosmologically stable unless
$R$-parity is broken.

In low-energy supersymmetry breaking scenarios, the stable gravitino is
dominantly produced in the early universe through gluon-gluino scattering
processes as computed in Refs. [138, 139, 140, 141]. The gravitino yield can
be written as

$Y_{3/2}^{\text{UV}}=C_{\text{UV}}\frac{m_{\tilde{g}}^{2}T_{\text{r.h.}}}{m_{3/2}^{2}M_{\text{Pl}}}\,,\qquad\text{with}\qquad
C_{\text{UV}}=\frac{45\sqrt{5}f_{3}}{8\pi^{13/2}g_{*}^{3/2}}\simeq 4\times
10^{-5}\,,$ (50)

where $m_{\tilde{g}}$ is the gluino soft mass and $T_{\text{r.h.}}$ is the
reheating temperature. Requiring that the gravitino yield does not overclose
the universe typically results in strong constraints on the maximal reheating
temperature in these scenarios [142, 143, 144, 145].

An interesting allowed region is the so called _ultralight gravitino window_
where $T_{\text{r.h.}}$ is required to be high enough to make the gravitino
thermalize in the early universe. Requiring
$Y_{3/2}^{\text{UV}}>Y_{\text{eq}}$ with $Y_{\text{eq}}=1.8\times 10^{-3}$
gives

$T_{\text{r.h.}}>\frac{45m_{3/2}^{2}M_{\text{Pl}}}{m_{\tilde{g}}^{2}}\simeq
9\text{ MeV}\left(\frac{m_{3/2}}{16\text{ eV}}\right)^{2}\left(\frac{2\text{
TeV}}{m_{\tilde{g}}}\right)^{2}\,.$ (51)

Since the gravitino freezes out when it is still relativistic, the matter
power spectrum is going to be damped at small scales [146]. In Ref. [147] a
combination of CMB data from WMAP and Lyman-$\alpha$ forest data was used to
set an upper bound on the gravitino mass (and the SUSY-breaking scale)

$m_{3/2}<16\text{ eV}\quad\Longleftrightarrow\quad\sqrt{F}<260\text{ TeV}\,.$
(52)

This bound will presumably be improved with current Planck data and even
further with future cosmological surveys.

Given the upper bound on the SUSY-breaking scale, the gluino can be made to
lie above the 2 TeV reach of the LHC provided

$g_{\text{eff}}^{2}>1.2\times\left(\frac{m_{\tilde{g}}}{2\text{
TeV}}\right)\left(\frac{260\text{ TeV}}{\sqrt{F}}\right)\,.$ (53)

following the parametrics in Eq. (49). We refer to [148, 149] for explicit
models which realize a heavy superpartner spectrum with such a low SUSY-
breaking scale. If we are agnostic about UV completions, the perturbativity of
$g_{\text{eff}}$ gives a lower bound on $\sqrt{F}$ which is a loop factor
below 260 TeV. The gravitino window can then be defined as

$10^{-3}\text{ eV}<m_{3/2}<16\text{ eV}\quad\Longleftrightarrow\quad 2.5\text{
TeV}<\sqrt{F}<260\text{ TeV}\,,$ (54)

where the lower bound is given by the current LHC bound on the gluinos plus
perturbativity of the SM couplings, while the upper bound is from cosmological
constraints on warm dark matter.

Within the light gravitino window defined in Eq. 54, we can safely assume that
the superpartner spectrum is decoupled and the gravitino interactions with the
SM are described by a universal EFT with couplings controlled by the
supersymmetry breaking scale only [150, 151, 152]. Collider searches for
direct gravitino production can hence provide a robust limit on the
supersymmetry breaking scale, independent of the specific details of the
superpartner spectrum. Traditionally, this has not been an emphasis of the
supersymmetry search program at hadron colliders because direct searches for
colored superpartners always exceed the sensitivity of direct gravitino
searches. In contrast, at high-energy lepton colliders the reduced background
and rapid growth of signal cross sections with $\sqrt{s}$ makes this a
competitive channel for the discovery of supersymmetry.

Figure 18: Background cross section for the $\gamma+\text{MET}$ final state as
a function of the c.m. energy $\sqrt{s}$. The black line is the total
background cross section in pb while the red and blue lines show the separate
contributions of $Z\gamma$ and $W^{+}W^{-}$ respectively.

Here, we explore the sensitivity of a muon collider to gravitino pair
production in the mono-photon final state via the production mode
$\mu^{+}\mu^{-}\to\tilde{G}\tilde{G}\gamma$. The analytic cross section for
this process in the gravitino EFT has been computed in Ref. [151]; it includes
ISR photon topologies from contact interactions between leptons and
gravitinos, as well as contact interactions involving leptons, gravitinos and
a photon. In the limit of soft and collinear photon emission the cross section
reads

$\sigma(\mu^{+}\mu^{-}\to\tilde{G}\tilde{G}\gamma)=\frac{\alpha
s^{3}}{160\pi^{2}F^{4}}\left[\frac{247}{60}+2\log\left(\frac{2E_{\text{min}}^{\gamma}}{\sqrt{s}}\right)\right]\log\left(\frac{1-\cos\theta_{\text{min}}}{1+\cos\theta_{\text{min}}}\right)\,,$
(55)

where $\sqrt{s}$ is the c.m. energy, $E^{\gamma}_{\text{min}}$ is the minimal
energy of the photon, and $\theta_{\text{min}}$ is the minimal photon angle
with respect to the beam direction. In the following numerical analysis we
will employ the complete analytic formula, but we can use the above equation
to get some reasonable estimates for the reach in terms of $\sqrt{F}$.

We assume minimal cuts on the photon ($E^{\gamma}_{\text{min}}>50$ GeV,
$|\eta_{\gamma}|<2.4$). The SM background at high $\sqrt{s}$ is dominated by
$W^{+}W^{-}$ fusion as can be seen in Fig. 18, and it is roughly constant and
equal to $2$ pb for $\sqrt{s}\gtrsim 3$ TeV. We further define the signal
region by requiring the photon energy not to exceed the endpoint of the signal
process, $E^{\gamma}_{\rm max}=\sqrt{s}/2$. With these cuts, the $2\sigma$
sensitivity to $\sqrt{F}$ using Eq. 55 is

$\sqrt{F}\lesssim 61.7\text{ TeV}\left(\frac{\mathcal{L}}{1000\text{
ab}^{-1}}\right)^{1/16}\left(\frac{\sqrt{s}}{100\text{
TeV}}\right)^{3/4}\left(4.8+\log\left[\frac{\sqrt{s}}{100\text{
TeV}}\right]\right)^{1/8}\,,$ (56)

where we have only included statistical errors. The corresponding numerical
result is displayed in Fig. 19. In the same figure, we show the decrease in
the reach assuming a $1\%$ systematic error on both signal and background.
Given that the signal cross section grows as the 6${}^{\text{th}}$ power of
the c.m. energy, the energy increase is the most beneficial in increasing the
sensitivity to high values of $\sqrt{F}$.

We conclude that a future high energy muon collider can almost certainly push
up the lower bound of the ultralight gravitino window by one order of
magnitude. An improvement of the cosmological bounds with respect to the ones
derived from WMAP data could allow the light gravitino window to be completely
closed in the future. More broadly, this exemplifies the ability of a high-
energy muon collider to directly probe the mechanism of supersymmetry
breaking.

Figure 19:  Estimated reach of the gravitino search interpreted as a
constraint on the supersymmetry breaking scale $\sqrt{F}$. The solid black
lines denote limit contours assuming only statistical errors:
$S/\sqrt{S+B}=1.96$. The dashed blue lines denote limit contours that include
a systematic error: $S/\sqrt{S+B+\epsilon^{2}(S^{2}+B^{2})}=1.96$ with
$\epsilon=1\%$. The points O1$\dots$O5 indicate the optimistic benchmarks of
Table 1, starting from $\sqrt{s}=10\text{ TeV}$ and with increasing center-of-
mass energies.

#### 4.3.2. The “big” hierarchy problem: compositeness

We next turn our attention to the competing solution to the “big” hierarchy
problem, compositeness. Though compositeness may take many guises, its
realizations all share a set of common features given the discovery of a light
Standard Model-like Higgs with an apparent mass gap. The expectation is that
fermionic top partners should exist at a scale $m_{T}\sim y_{t}f$
parametrically below the scale of compositeness $m_{*}=g_{*}f$ at which a host
of resonances appear. See e.g. [153] for an excellent overview.

Perhaps the strongest tests of compositeness come from its distinctive
imprints on the electroweak sector of the SM, including modification of Higgs
couplings [154]. To the extent that the tuning associated with vacuum
misalignment is generally on the order of $\Delta\sim f^{2}/v^{2}$ and the
leading deviations from Standard Model predictions can be captured by
irrelevant operators suppressed by $f^{2}$ or $(g_{*}/g)^{2}f^{2}$, tests of
such deviations typically provide the most powerful constraints. Here, high-
energy muon colliders enjoy a particular advantage, leveraging the energy
growth of these deviations without the considerable backgrounds of $pp$
counterparts. As demonstrated in [38], this allows a high-energy muon collider
to access Higgs compositeness scales well above any other future collider
project. For example, sensitivity to operators such as

$\displaystyle\mathcal{O}_{H}=\frac{1}{2}\left(\partial_{\mu}|H|^{2}\right)^{2}\qquad\text{and}\qquad\mathcal{O}_{W}=\frac{ig}{2}\left(H^{\dagger}\sigma^{a}\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\leftrightarrow$}\over{D^{\mu}}}H\right)D^{\nu}W_{\mu\nu}^{a}\,,$
(57)

allows a muon collider operating at $\sqrt{s}=10$ TeV to probe compositeness
scales as high as $m_{*}\sim 45$ TeV with 10 ab-1, while a collider operating
at $\sqrt{s}=30$ TeV can probe compositeness up to $m_{*}\sim 140\text{ TeV}$
with 90 ab-1 [38]. The sensitivity at these two energies covers fine tuning
associated with vacuum misalignment from the percent level to the per mille
level.

To the extent that it is possible to decrease the tuning associated with
vacuum alignment, sensitivity to fermionic top partners provides a
complementary probe. Here the test of fine tuning is expected to be
qualitatively similar to that of supersymmetric tops, with two relative
advantages: for fermions the $s$-channel production cross section is only
linearly suppressed by velocity near threshold, and in the absence of
additional structure they typically decay directly to visible and highly
distinct final states. Following the fine-tuning estimates in the
supersymmetric case, reaching percent-level tuning requires $\sqrt{s}\gtrsim
6$ TeV, while per mille requires $\sqrt{s}\gtrsim 20$ TeV and per myriad
$\sqrt{s}\gtrsim 66$ TeV. As in the case of supersymmetry, a high-energy muon
collider operating at $\sqrt{s}\sim 20\text{ - }30$ TeV would decisively test
compositeness as a natural explanation for the weak scale.

#### 4.3.3. The “little” hierarchy problem

As we have seen, a high-energy muon collider operating at tens of TeV could
provide satisfying coverage of known solutions to the “big” hierarchy problem.
It is no less suited to constraining or discovering solutions to the “little”
hierarchy problem, whose subtle signatures will remain largely untouched by
the LHC. There are a plethora of such solutions which operate by relying on
novel field theoretical [155, 156, 157, 158, 159, 98, 160, 99, 161, 162, 163,
164] or cosmological [125, 165, 166, 167, 168] ingredients. Here we will focus
on solutions involving “neutral naturalness” such as the Twin Higgs [98] or
Hyperbolic Higgs [162], in which the partner particles are entirely neutral
under the Standard Model. Such models feature (at least) four possible avenues
to discovery at colliders: Higgs coupling deviations from mixing between
scalars; direct production of SM singlet partner particles; displaced vertices
from exotic Higgs decays; and direct production of the “radial mode”
associated with spontaneous breaking of the discrete symmetry.

In general, the Higgs coupling deviations are a subset of those expected in
composite models, including most notably the oblique operator
$\mathcal{O}_{H}$ defined in Eq. 57. As with composite models, constraints on
$\mathcal{O}_{H}$ translate directly into bounds on $f^{2}/v^{2}$, a typical
measure of the fine tuning. The results of [38] suggest sensitivity to $f\sim
16v$ at $\sqrt{s}=10$ TeV and $f\sim 46v$ at $\sqrt{s}=30$ TeV, probing the
tuning of the weak scale between the percent and per mille levels within the
framework of the Twin Higgs. The prospects for discovery via direct production
of partner particles are considerably weaker. Although the precise reach
depends on whether these partner particles are fermions (as in the Twin Higgs)
or bosons (as in the Hyperbolic Higgs), the qualitative sensitivity can be
inferred from the reach for the “nightmare scenario” of a
$\mathbb{Z}_{2}$-symmetric singlet scalar summarized in Sec. 4.1 and App. A,
extending at most to the hundreds of GeV.

A more exotic collider signature of models of neutral naturalness is the
prediction of displaced decays of dark particles into Standard Model states
[126]. The scalar mixing produces a non-zero branching ratio of Higgses into
neutral partner states, which gives a portal for energy to be transferred to
the partner sector at a collider. Some of the produced partner sector states
are unstable to decaying through an off-shell Higgs into Standard Model
states, bearing out “Hidden Valley” type phenomenology [169, 170]. This
produces spectacular signatures – for example vertices displaced from the
beamline from which Standard Model jets appear – and this allows search
strategies with very low background [171, 124]. The prospects for probing
these signatures in neutral naturalness models at future lepton colliders have
been studied in the context of Higgs factories [172, 173], and a branching
ratio reach is projected which is competitive with or better than the LHC
forecasts. Dedicated study for a higher-energy muon collider has yet to be
performed.

The prospects for discovering the “radial mode” are much stronger than for
other partner particles. These can be obtained by interpreting the model
independent results of Fig. 14 in the Twin Higgs, where the mixing angle
scales as $\sin\gamma\sim g_{*}v/m_{\phi}$. Concretely, the Higgs potential in
the simplest realization of the Twin Higgs can be written as

$V=\lambda_{*}\bigg{(}|H_{A}|^{2}+|H_{B}|^{2}-\frac{f_{0}^{2}}{2}\bigg{)}^{2}+\kappa\Big{(}|H_{A}|^{4}+|H_{B}|^{4}\Big{)}+\sigma_{\rm
soft}f^{2}\,|H_{A}|^{2}\,.$ (58)

The SM Higgs sector is extended by the addition of the twin Higgs $H_{B}$, a
singlet under the SM and doublet under a mirror electroweak gauge group
$SU(2)_{B}$. The twin Higgs is coupled to the SM Higgs $H_{A}$ via a portal
coupling $\lambda_{*}$. With only this quartic included, the potential
linearly realizes an $SO(8)$ symmetry, spontaneously broken to $SO(7)$ at the
scale $f$. The radiative stability of the construction is ensured by a
$\mathbb{Z}_{2}$ symmetry between the SM and the mirror sector, which is
softly broken by $\sigma_{\rm soft}$ to allow for $f>v$ and a viable
phenomenology.131313Here for simplicity, we set the $\mathbb{Z}_{2}$-breaking
quartic to zero. Having it non-zero will only impact the phenomenology at
small $m_{\phi}$, allowing the Higgs mass constraint to be satisfied at large
$f$ and small $m_{\phi}$. See [174, 28] for a discussion. The $SO(8)$-breaking
quartic $\kappa$ receives IR contributions from (mirror) top loops
$\kappa\simeq(3y_{t}^{4}/8\pi^{2})\log m_{*}/m_{t}$. Finally, $f_{0}\simeq f$
up to corrections of order $\kappa/\lambda_{*}$.

After the spontaneous breaking of the $SO(8)$ symmetry, we are left with two
real scalars in the spectrum: the SM-like Higgs $h$ and the radial mode $\phi$
with masses $m_{\phi}^{2}\simeq 2\lambda_{*}f^{2}$ and $m_{h}^{2}\simeq
4\kappa v^{2}$ in the limit $\lambda_{\ast}\gg\kappa\,,\sigma_{\rm soft}$.
This shows that the Higgs mass is already of the correct size given the
typical size of the irreducible contributions to $\kappa$. The requirement to
reproduce the electroweak scale $v$ and the Higgs mass $m_{h}$ fixes 2 out of
the 4 free parameters in Eq. (58). The remaining two are chosen to be ($f$,
$m_{\phi}$) while the mixing between the twin Higgs and the SM Higgs is
predicted, and it scales as $\sin\gamma\simeq v/f$ in the $m_{\phi}\gg m_{h}$
limit.

Figure 20: The same limits on a singlet scalar Higgs as shown in Fig. 14,
interpreted in terms of the reach on the sigma-model scale $f$ in the context
of a Twin Higgs model.

In Twin Higgs models, the fine tuning of the electroweak scale generically
scales as

$\frac{\delta
m_{h}^{2}}{m_{h}^{2}}\simeq\frac{3y_{t}^{4}}{2\pi^{2}\lambda_{*}}\left(\frac{m_{*}}{m_{h}}\right)^{2}\log\frac{m_{*}}{m_{t}}\
,$ (59)

where $m_{*}$ should be interpreted as the cutoff of the potential in Eq.
(58), where colored states are expected to be necessary to stabilize the scale
$f$ from radiative corrections. Fixing the scale of the colored states
$m_{*}$, the fine tuning of the electroweak scale in Twin Higgs models is
parametrically reduced with respect to that of conventional supersymmetry or
composite scenarios by $y_{t}^{2}/\lambda_{\ast}$; see, e.g., [174, 175]. The
gain in fine tuning is limited by the perturbativity requirement on the linear
potential in Eq. (58), which roughly requires $\lambda_{*}\lesssim 4\pi$,
allowing colored states as heavy as 5 TeV with fine tuning on the order of 1%.
The implications of a muon collider’s sensitivity to the radial mode are shown
in Fig. 20, reinterpreting the reach illustrated in Fig. 14 in terms of the
Twin Higgs parameter space. Keeping the scale of the colored states fixed, the
fine tuning of the electroweak scale goes like $1/\lambda_{*}$, making the
sliver of parameter space with largest $\lambda_{*}$ (or equivalently with
largest $m_{\phi}/f$) the most appealing.

Apart from fine tuning considerations, Fig. 20 illustrates the correlation
between direct searches and Higgs coupling deviations in the Twin Higgs
parameter space. Interestingly, a high energy muon collider of
$\sqrt{s}=14\text{ TeV}$ will be able to fully probe the parameter space
corresponding to a % deviation in Higgs couplings ($\sin\gamma\simeq 0.1$) by
means of direct searches for the extra scalar. This may be taken as further
motivation to reach such a high c.m. energy at a future collider. Conversely,
moving into the region with small $\lambda_{*}$, the deviations in the Higgs
couplings will be difficult to observe, but direct production at a muon
collider could still cover a large portion of this parameter space.

## 5\. Complementarity

Given the time required to achieve first collisions at a future collider, a
number of planned and proposed experiments capable of extending our
sensitivity to indirect signs of new physics could lead to orders of magnitude
improvement of current limits. Among others, experiments searching for
electric dipole moments, anomalous flavor violation beyond the Standard Model,
and stochastic gravitational wave backgrounds will probe scales ranging from
tens to hundreds of TeV on relevant timescales.141414There are already hints
that something interesting might be going with the muon, which could have
exciting potential implications for a future muon collider, e.g., see recent
studies regarding the muon $g-2$ [39, 40, 41, 42, 43], $B$ meson [44], and $K$
meson [45] anomalies. Ultimately, a signature at any of these experiments
would provide an indication of new physics at a scale amenable to further
exploration. This motivates asking what energies and luminosities would be
required for a muon collider to directly test the origin of indirect signals,
providing another set of sharp goalposts.

###  5.1 EDMs

Electric dipole moments of elementary particles offer a nearly background-free
probe of new physics beyond the Standard Model, since all CP-violating SM
effects are accompanied by flavor-violating spurions and give rise to
extremely small EDMs. Recently, substantial progress has been made in
experimental searches for EDMs using paramagnetic molecules. For instance, by
studying the polar ThO molecule, the ACME collaboration has set a bound on the
electron EDM of $|d_{e}|\leq 1.1\times 10^{-29}\,e\,\mathrm{cm}$ at 90%
confidence [176]. Rapid experimental progress towards the use of atoms and
molecules, including novel approaches using polyatomic molecules, is likely to
improve the electron EDM sensitivity to at least $10^{-32}\,e\,\mathrm{cm}$ in
the coming decade (see Refs. [177, 178] and references therein). These
approaches will also offer novel opportunities to probe hadronic EDMs.

Figure 21: One possibility for testing the physics associated with a Barr-Zee
type contribution to a lepton EDM at a future muon collider. Left: The two-
loop Barr-Zee contribution to a lepton EDM. Right: A $\gamma\gamma\to hh$
process at a muon collider, sensitive to loops of charged particles that
couple to the Higgs. The dotted blue box shows that both processes probe the
same underlying physics.

The discovery of an electric dipole moment could definitively prove the
existence of new physics beyond the SM. In a typical model, weakly-coupled
$k$-loop physics, at a mass scale $M$ with CP-violating phase
$\delta_{\text{CP}}$, could generate an electron EDM of the size

$d_{e}\sim\sin(\delta_{\text{CP}})\frac{e\,m_{e}}{M^{2}}\left(\frac{\alpha}{4\pi}\right)^{k}\simeq
10^{-32}\,e\,{\rm cm}\sin(\delta_{\text{CP}})\times\begin{cases}\left(1~{}{\rm
PeV}/M\right)^{2}&\text{for }k=1\\\\[5.0pt] \left(20~{}{\rm
TeV}/M\right)^{2}&\text{for }k=2\end{cases}\,\,.$ (60)

This is only a rough, order-of-magnitude guide, but it shows that an EDM
experiment could provide the first discovery of the effects of new physics
beyond the reach of current colliders. A one-loop EDM of this size could
arise, for example, from sleptons with masses of order PeV and order one CP-
violating phases [179, 180], or from lighter sleptons with smaller CP-
violating phases. A two-loop EDM could arise, for instance, from chargino
loops in split SUSY [181, 182]. More generally, any new particles interacting
with electroweak gauge bosons and the Higgs can produce a 2-loop EDM through
Barr-Zee diagrams [183], see the left panel of Fig. 21.

An EDM discovery would tell us that new physics exists, but would tell us very
little about the nature of the new physics. Colliders will have a crucial role
to play, by providing more insight on the new particles responsible for the
EDM and allowing us to determine which extension of the SM explains the
effect. A one-loop EDM would be associated with new particles carrying
electric charge which could be directly pair-produced at a muon collider if
they are kinematically within reach, e.g., for sleptons we would search for
$\mu^{+}\mu^{-}\to{\tilde{\ell}}^{+}{\tilde{\ell}}^{-}$. The range of
possibilities at two loops is broader (e.g., [184, 185]), but the case of the
Barr-Zee diagram offers a particularly appealing target. It involves new
electroweak particles which could be pair produced directly. However, it also
implies that these particles alter the interactions between Higgs and gauge
bosons, as illustrated in Fig. 21. Since high-energy lepton colliders are
electroweak gauge boson colliders, they offer a unique prospect to directly
probe the same underlying electroweak interactions that generate the EDM, via
precision studies of processes like $\gamma\gamma\to hh$. We leave a detailed
assessment of the prospects for EDM/collider complementarity along these lines
for future work.

###  5.2 Flavor

One of the biggest puzzles in the SM is the pattern of fermion masses and
mixings. Both the quark and lepton sectors have significant mass hierarchies,
whereas the mixing matrices take a very different form in the two sectors. We
expect that at high energies, where the flavor pattern of the SM is
established, there may be much larger rates of flavor-changing processes than
the SM predicts. This is a strong motivation for searching for flavor-
violating processes at high-energy colliders.

Conversely, some of our most stringent bounds on physics beyond the SM come
from low- or medium-energy precision tests of flavor-changing processes, e.g.,
$K-{\bar{K}}$ or $D-{\bar{D}}$ mixing [186]. Of particular interest, in the
context of a muon collider, are precision tests of charged lepton flavor
violation processes like $\mu\to e\gamma$, $\mu\to 3e$, $\tau\to 3\mu$, or
$\mu$-to-$e$ conversion within atomic nuclei. As with EDMs, these processes
are expected to be much more strongly constrained in the coming decades than
they are at present, due to a number of currently operating or planned future
experiments [187]. These experiments can indirectly probe physics at energies
of 10s of TeV or even higher. A high-energy muon collider can probe the same
physics (e.g., through direct searches for flavor-changing processes like
$\mu^{+}\mu^{-}\to\tau^{\pm}\mu^{\mp}$), or can help to elucidate the
underlying mechanism of flavor violation by directly producing new particles
with flavor-violating interactions, such as the mixed slepton production
process $\mu^{+}\mu^{-}\to{\tilde{\ell}}_{i}^{+}{\tilde{\ell}}_{j}^{-}$.
Below, we will give first estimates of the physics reach for both of these
scenarios. We will see that there is a powerful complementarity between
precision lepton flavor experiments and high-energy muon collider searches.

#### 5.2.1. Lepton-flavor violating contact interactions

Belle has set a limit of $\textrm{Br}(\tau\to 3\mu)<2.1\times 10^{-8}$ [188],
and this bound will be improved to $3.5\times 10^{-10}$ by Belle II [189].
These bounds can be contrasted with constraints on the $\mu\to 3e$ branching
ratio, which is currently constrained to be $1.0\times 10^{-12}$ by SINDRUM
[190], but will be dramatically improved, eventually to a sensitivity $\sim
1\times 10^{-16}$, by the Mu3e experiment [187].

We parametrize the four-fermion operators relevant for the $\tau\to 3\mu$
decay via

$\mathcal{L}\supset V_{LL}^{\tau
3\mu}\big{(}\bar{\mu}\gamma^{\mu}P_{L}\mu\big{)}\big{(}\bar{\tau}\gamma_{\mu}P_{L}\mu\big{)}+V_{LR}^{\tau
3\mu}\big{(}\bar{\mu}\gamma^{\mu}P_{L}\mu\big{)}\big{(}\bar{\tau}\gamma_{\mu}P_{R}\mu\big{)}+\big{(}L\leftrightarrow
R\big{)}+\textrm{h.c.}\,,$ (61)

with an equivalent set for the $\mu\to 3e$ decay. In what follows, we will
assume all the $\tau 3\mu$ coefficients are equal:

$V_{LL}^{\tau 3\mu}=V_{LR}^{\tau 3\mu}=V_{RL}^{\tau 3\mu}=V_{RR}^{\tau
3\mu}=\frac{c^{\tau 3\mu}}{\Lambda^{2}}$ (62)

where $c^{\tau 3\mu}$ is a dimensionless coefficient and $\Lambda$ is to be
interpreted as the scale of new physics. Setting the Wilson coefficients to
unity, we see that the current Belle (future Belle II) constraint above
translates into a bound of $\Lambda>14.7\text{ TeV}$ ($\Lambda>40.9\text{
TeV}$), far below the corresponding constraints from SINDRUM (with $\tau 3\mu$
replaced by $\mu 3e$), which gives $\Lambda>273\text{ TeV}$. Generically,
though, one might expect new physics violating lepton flavor to manifest with
hierarchies similar to the pattern of hierarchies in the SM lepton Yukawas, in
which case the bounds from $\tau$ and $\mu$ might be more comparable. We will
discuss this more in what follows.

At a muon collider, these same operators can be probed directly via
$\mu^{+}\mu^{-}\to\ell^{+}_{i}\ell^{-}_{j}$. We will focus on $\mu\tau$
production, since it can be compared directly with the sensitivity from tau
decays. Our analysis closely follows an analogous study at an $e^{+}e^{-}$
collider in Ref. [191]. With only the four-fermion operator insertion, the
rate for $\mu^{+}\mu^{-}\to\mu\,\tau$ grows as $s$. Taking $c^{\tau
3\mu}/\Lambda^{2}=1/(50\text{ TeV})^{2}$, the cross section grows from
$0.059\text{ fb}$ at a $3\text{ TeV}$ muon collider to $66.1\text{ fb}$ at
$\sqrt{s}=100\text{ TeV}$.

Ignoring detector effects, there are two primary backgrounds for the flavor-
violating process in the SM: $\tau^{+}\tau^{-}$ production, where (at least)
one of the taus decays to a muon and a neutrino and
$\mu^{+}\mu^{-}\to\mu\,{\nu}_{\mu}\,\tau\,{\nu}_{\tau}$ production via
intermediate $W$ bosons. These backgrounds can be significantly suppressed
with simple cuts on the muon energy and the missing 3-momentum. In particular,
we demand that the most energetic muon in the event has at least $90\%$ of the
beam energy and that the direction of the total missing 3-momentum vector be
at least $170^{\circ}$ from the 3-momentum of the most energetic muon. These
cuts suppress the $\tau^{+}\tau^{-}$ background by a factor $\sim 300$, and
the $W^{+}W^{-}$ background by a factor $\sim 10$, depending slightly on the
c.m. energy.

| $N_{\text{post-cuts}}$
---|---
$\sqrt{s}$ (TeV) |  $\mu^{+}\mu^{-}\to\mu\,\tau$ |  $\mu^{+}\mu^{-}\to\mu\,\tau\,\nu_{\mu}\,\nu_{\tau}$ |  $\mu^{+}\mu^{-}\to\tau^{+}\tau^{-}$
$.125$ | 0.0948 | 30.8 | $3.42\times 10^{4}$
$3$ | $53.3$ | $6.32\times 10^{3}$ | 40.4
$6$ | $212$ | $3.26\times 10^{3}$ | 9.52
$14$ | $1.14\times 10^{3}$ | $1.14\times 10^{3}$ | 0.138
$100$ | $5.73\times 10^{4}$ | 60.9 | 0.0312
Table 5: Number of signal and background events after kinematic cuts and
estimating the loss of signal efficiency due to initial state radiation for
$1\text{ ab}^{-1}$ of data and $c^{\tau 3\mu}/\Lambda^{2}=1/(50\text{
TeV})^{2}$.

The signal process, on the other hand, passes these cuts with near perfect
efficiency. Kinematically, there is no loss of signal events, and the only
degradation is due to initial state radiation. This leads to a $\sim 10\%$
reduction at $\sqrt{s}=3\text{ TeV}$, slowly increasing to a $13\%$ reduction
at $\sqrt{s}=100\text{ TeV}$. The number of signal and background events after
applying the kinematic cuts and initial state radiation degradation for
$1\text{ ab}^{-1}$ of data with $c^{\tau 3\mu}/\Lambda^{2}=1/(50\text{
TeV})^{2}$ are shown in Table 5. We find a signal-to-background ratio $\sim 1$
to be achievable at a $14\text{ TeV}$ machine, and this dramatically rises to
$\sim 10^{3}$ at a $100\text{ TeV}$ machine. The resulting bounds, assuming
integrated luminosities of $1\text{ ab}^{-1}$ at $0.125,3,6,14$, and
$100\text{ TeV}$ are shown in Fig. 22. It is clear that even a $3\text{ TeV}$
machine would be able to set a direct bound at the same level as the future
Belle II sensitivity, and this constraint can be improved by up to $\sim 2$
orders of magnitude at a higher energy machine.

In Fig. 22, these results are compared to the constraints on the analogous
4-fermion operator in the $\mu\to 3e$ decay with various ansatz regarding
flavor violation. The diagonal lines show the expected relationship between
the two Wilson coefficients assuming (i) flavor anarchy (all coefficients
$\sim 1$), (ii) Minimal Leptonic Flavor Violation (MLFV) [192], (iii) the
Wilson coefficients scale like the square root of the Yukawa couplings of the
leptons involved in the flavor violation, and (iv) the Wilson coefficients
scale like the product of the same Yukawa couplings.151515 One should be
cautious that if a flavorful ansatz is used, the inferred scale from the
$100\text{ TeV}$ bounds may be low enough that an effective field theory
description is no longer valid.

While the muon decay sets the strongest limits assuming anarchical
coefficients, we see that a $14\text{ TeV}$ muon collider could set a bound
comparable to the current SINDRUM limit in the case of MLFV, and would be
comparable to the Stage-I Mu3e sensitivity if the coefficients scale like the
square root of the Yukawa couplings. In the extreme case, where the Wilson
coefficients behave like the product of the two Yukawas, even a $3\text{ TeV}$
muon collider would provide a bound complementary to the final Mu3e
sensitivity, with higher energy machines improving this bound by orders of
magnitude.

Figure 22: Summary of muon collider and precision constraints on flavor-
violating 3-body decays. The colored horizontal lines show the sensitivity to
the $\tau 3\mu$ operator at various energies, all assuming $1\text{ ab}^{-1}$
of data. The dashed horizontal (vertical) lines show the current or expected
sensitivity from $\tau\to 3\mu$ ($\mu\to 3e$) decays for comparison. The
diagonal black lines show the expected relationship between different Wilson
coefficients with various ansatz for the scaling of the flavor-violating
operators (e.g., “Anarchy” assumes that all Wilson coefficients are
$\mathcal{O}(1)$).

In addition to the $\tau 3\mu$ operators considered here, we expect roughly
similar sensitivity to the $\mu^{+}\mu^{-}\to\mu^{\pm}e^{\mp}$ process, as
well as to the processes such as $\mu^{+}\mu^{-}\to\tau^{\pm}e^{\mp}$ that
violate lepton flavor by two units. Overall, we see that a muon collider would
be capable of directly probing flavor-violating interactions that are quite
complementary to future precision constraints.

#### 5.2.2. Direct probes of lepton-flavor violation in the MSSM

Charged lepton flavor violation in the MSSM arises as a result of the soft-
breaking terms in the slepton mass matrix having non-diagonal entries in the
basis where the SM lepton Yukawas are diagonal. In this case, the physical
sleptons will be mixtures of different flavors, and their interactions with
leptons and neutralinos/charginos will be flavor-violating. These flavor
violating interactions lead to processes such as rare muon decays or muon-to-
electron conversion at loop level, and thus, low energy experiments can
indirectly probe these interactions with sensitivities extending beyond the
TeV scale, depending on the flavor structure of the theory [179, 193]. A high-
energy muon collider, on the other hand, would not only be capable of
producing superpartners at high masses, but would also provide direct
measurements of the lepton-flavor violating processes that would complement
these low-energy probes and provide detailed insight into the mechanism of
supersymmetry breaking.

For simplicity, we will consider a simplified scenario where the effects of
all scalar superpartners except for $\tilde{e}_{R}$ and $\tilde{\mu}_{R}$
decouple. In this case, the slepton mixing reduces to a $2\times 2$ problem
with slepton-mass squared matrix

$\mathcal{M}^{2}_{\tilde{\ell},RR}=\begin{pmatrix}\Delta_{RR,11}&\widetilde{m}^{2}_{E,12}\\\
\widetilde{m}^{2}_{E,12}&\Delta_{RR,22}\end{pmatrix}\,,$ (63)

where the diagonal terms are the sum of both soft-SUSY-breaking scalar masses
($\widetilde{m}^{2}_{E}$) and $D$-terms as well as terms dictated by
supersymmetry, and we have assumed the off-diagonal soft-breaking terms are CP
conserving. This mass matrix can be diagonalized via a unitary matrix $U_{R}$
to yield mass eigenstates $m^{2}_{\tilde{e}_{1}},m^{2}_{\tilde{e}_{2}}$ with
the mixing angle given by

$\frac{1}{2}\sin(2\theta_{R})=\frac{\widetilde{m}^{2}_{E,12}}{m^{2}_{\tilde{e}_{1}}-m^{2}_{\tilde{e}_{2}}}\,.$
(64)

We will further consider the situation where the lightest supersymmetric
particle is a pure bino with mass $M_{1}$ and assume the other neutralinos can
be ignored. With $m_{\tilde{\ell}}>M_{1}$, the sleptons decay directly to a
lepton and bino, and the latter will appear as missing momentum.

#####  5.2.2.1 Nearly-degenerate sleptons:

As a first benchmark scenario we consider the situation where the selectron
and smuon are nearly degenerate in mass. This situation is well-motivated from
gauge-mediated supersymmetry breaking scenarios, and also leads to a strong
suppression of the lepton-flavor violation via a “super-GIM” mechanism,
allowing the superpartners to be relatively light. Such a scenario was
previously studied in the context of $e^{+}e^{-}$ collisions in Ref. [194],
but for relatively light superpartners. A high-energy muon collider would
allow similar tests with a substantially more impressive mass reach.

In the limit of a small mass splitting, the parameter governing the amount of
flavor violation is given by

$(\delta_{RR})^{12}\equiv\frac{\widetilde{m}^{2}_{E,12}}{\sqrt{\Delta_{RR,11}\Delta_{RR,22}}}\simeq\frac{1}{2}\frac{\Delta
m^{2}}{\bar{m}^{2}}\sin(2\theta_{R})\,,$ (65)

where we have introduced the average mass squared, $\bar{m}^{2}$ and mass-
splitting $\Delta m^{2}=m^{2}_{\tilde{e}_{1}}-m^{2}_{\tilde{e}_{2}}$.

Following [194], we can compute the probability that a gauge eigenstate
$\tilde{\mu}$ decays into a final state with an electron – including the
interference effects when $\bar{m}\Gamma$ and $\Delta m^{2}$ are of similar
size – and thus find the cross section for the flavor violating process
$\mu^{+}\mu^{-}\to\tilde{e}^{+}_{1,2}\,\tilde{e}^{-}_{1,2}\to\mu^{\pm}\,e^{\mp}\,\chi_{1}^{0}\,\chi_{1}^{0}$
as a function of the mass splitting and mixing angle. The results are shown in
Fig. 23 for several choices of slepton and bino mass and at several different
several of mass energies alongside the current bounds from $\mu\to e\gamma$
from MEG [195] and the expected sensitivities from several future experiments
[196, 197, 198].

Figure 23: Contours (in dashed gray) of the cross section in the $\delta
m^{2}/\bar{m}^{2}$ vs. $\sin(2\theta_{R})$ plane for the flavor violating
process $\mu^{+}\mu^{-}\to\mu^{\pm}\,e^{\pm}\,\chi^{0}_{1}\,\chi^{0}_{1}$ for
the nearly-degenerate slepton scenario described in the text. The bounds from
current and future precision searches are overlaid in blue and green, along
with red dashed and dotted curves showing the expected reach with 1 and
$5\text{ ab}^{-1}$ of data. The different panels correspond to benchmark
choices of c.m. energy, and superpartner masses.

The primary background for these flavor-violating processes at a lepton
collider is production of the different flavor final states and missing energy
via intermediate $W$ bosons. The total cross section for this background,
including branching ratios, is $52\text{ fb}$ at a $6\text{ TeV}$ collider
($15/0.6\text{ fb}$ at a $14/100\text{ TeV}$ machine), but the kinematics of
this process are quite different from the slepton-pair production signal of
interest. Moreover, for the flavor-violating scenarios at hand, it is likely
that the relevant slepton and neutralino masses will have already been
measured from the corresponding flavor-conserving processes (for details on
how this can be done, see e.g., Refs. [199, 200]). With the slepton masses
known, it is then possible to fully reconstruct the two final state neutralino
momenta by requiring that the neutralino and lepton momenta satisfy the
slepton mass-shell constraint, along with conservation of energy and momentum.
In general, these conditions will be impossible to satisfy for the background
events, and indeed, we find in simulation that only $\sim 1/500$ background
events can reconstruct the neutralino momenta while satisfying conservation of
energy, while $\sim 98\%$ of the signal events reconstruct the momenta
successfully.

In Fig. 23, we show the $5\sigma$ discovery reach for the three benchmark
scenarios assuming $1\text{ ab}^{-1}$ ($5\text{ ab}^{-1}$) of data in red
dashed (dotted) lines, assuming the efficiencies just quoted apply at all
masses and energies. We see that a $6\text{ TeV}$ muon collider would probe a
great deal of parameter space complementary to the Stage-II Mu2e bounds in the
case of $1\text{ TeV}$ nearly-degenerate sleptons. More energetic colliders
would be able to measure parameters that are beyond the reach of even the
PRISM/PRIME future sensitivity.

#####  5.2.2.2 A single light slepton:

As an alternative scenario, we consider a situation with only one slepton
within reach of the collider that is a nearly pure right-handed selectron, but
with some small mixing with a heavier right-handed smuon, i.e.,
$\delta_{RR}^{12}>0$. We can then examine the cross sections for the flavor
violating process as a function of $\delta_{RR}^{12}$ and $M_{1}$. The results
are shown in Fig. 24, where we again superimpose the current and future
sensitivities from the low-energy experiments.

Figure 24: Similar to Fig. 23, but in the $(\delta_{RR})_{12}$ vs. $M_{1}$
plane for the single light slepton scenario.

The backgrounds for this scenario are the same as in the nearly-degenerate
case already described, and we assume that the same efficiencies for removing
the background described above can be achieved for all neutralino masses and
collider energies here as well. The resulting $1$ and $5\text{ ab}^{-1}$
discovery reaches are again shown as dashed and dotted red lines,
respectively. In this situation, a $6\text{ TeV}$ collider would cover much of
the parameter space probed by MEG II, except when the neutralino is close to
the light slepton mass, but would fall short of the eventual Mu2e Stage II
sensitivity. A higher energy muon collider, however, would surpass the Mu2e
sensitivity, and could measure flavor violating insertions as small as a few
$\times~{}10^{-3}$ for light sleptons at 4 TeV and 20 TeV with a muon collider
of 14 and 100 TeV respectively. Overall, these benchmarks illustrate that muon
colliders would have impressive capabilities for not only discovering
superpartners, but measuring their flavor structure in detail.

###  5.3 Gravitational waves

The central importance and capabilities of a future collider must be seen
within the context of the broader experimental efforts in particle physics. We
have already discussed the connections between possible discoveries in
precision flavor and CP violation experiments, direct and indirect dark matter
detection, as well as searches for new light fields and dark forces. On the
cosmological front, we are entering an era in which dramatic new forms of
“fossil” evidence for BSM physics may be found, within stochastic
gravitational wave backgrounds (SGWB) [201, 202] and within primordial non-
Gaussianities in Large Scale Structure (LSS) and high-redshift 21-cm 3D “maps”
[203]. While discoveries in any of these non-collider experiments would be
spectacular, powerful new colliders would provide the “gold standard”
laboratory conditions to corroborate, connect, extend, and analyze their full
significance.

First order cosmological phase transitions that could occur due to extensions
of the SM or within dark sectors could be powerful sources of SGWB, while
possibly providing the non-equilibrium conditions needed for generating matter
asymmetries. The peak SGWB frequency, after redshifting from the time of
production in the very early universe, is given by

$\omega\simeq 0.03\,{\rm
mHz}\frac{\beta}{H_{\textrm{PT}}}\frac{T_{\textrm{PT}}}{\rm TeV}\,,$ (66)

where $T_{\textrm{PT}}$ is the temperature immediately after the phase
transition, $1/\beta$ is essentially its duration, and $H_{\textrm{PT}}$ is
the Hubble expansion rate during this era [204, 205]. Typically, one expects
$\beta\sim{\cal O}(10\text{ - }100)H_{\textrm{PT}}$. Fortuitously then, for
$T_{\textrm{PT}}$ in the BSM-motivated range, TeV - $100$ TeV, we can expect
SGWB in roughly the mHz - Hz range accessible to proposed gravitational wave
detectors such as LISA, BBO, and DECIGO. If a SGWB from a phase transition is
detected, it would be critical to piece together the information in its
frequency spectrum with the complementary microphysics accessible within
collider experiments to whatever extent possible. Quite plausibly, these
elements relate to extensions of the SM Higgs sector. A high energy muon
collider provides a balance of potential to probe the Higgs sector, to create
very massive BSM states related to the phase transition, and given its clean
environment to possibly produce and diagnose a small number of events
resulting from the presence of a dark sector in case the SGWB originated
there.

Let us illustrate the interplay between gravitational wave detection and a
muon collider by one of the central questions of particle physics: Is the
Higgs boson truly elementary, or a composite of new strongly-coupled confined
constituents? In the latter case, the compositeness would greatly mitigate the
electroweak hierarchy problem (see [206, 153] for review). We would then
expect to see excited composites at high-energy colliders as well as a
deconfinement-to-confinement phase transition at gravitational wave detectors.
This physics is usually discussed in its more theoretically tractable AdS/CFT
dual formulation as a 5D Randall-Sundrum I (RS1) model [207], translating to
the collider phenomenology of Kaluza-Klein excitations and the transition
between the black-brane horizon and IR-brane phases of RS1 [208, 209, 210,
211, 212]. For a very recent work refining phase transition dynamics and for
references, see [213]. For example, if LISA were to detect a SGWB with peak
frequency $0.5$ mHZ and gravitational power spectrum of $3\times 10^{-8}$ at
the peak, in the RS1 framework this would correspond to a supercooled phase
transition with $T_{\textrm{PT}}\simeq 1.3$ TeV, with Kaluza-Klein resonances
in the (multi-)TeV regime accessible to muon colliders.161616Given a high
degree of supercooling, say by a factor of $10^{3}$, we are assuming that
bubble collisions dominantly source the SWGB signal and that these are
captured by the “envelope approximation”. This approximation can be invalid in
some regimes, but its replacements would not invalidate the general
complementarity of the collider and gravitational wave experiments. See the
discussion reviewed in [214]. If say a $2$ TeV “radion” spin-$0$ excitation
were discovered at a muon collider via vector boson fusion, its mass and
cross-section would independently determine $T_{\textrm{PT}}$, giving a
valuable cross-check of the underlying physics. Further, it would bound the
heavy Kaluza-Klein spectrum to begin below $40$ TeV. If say a spin-$2$ Kaluza-
Klein graviton with mass $20$ TeV is eventually discovered in associated
production with the radion, the cross-section for this process (along with
cross-sections for radion processes) together with the frequency and power
spectrum from the SGWB data would allow us to extract the small “critical
exponent” responsible for generating the large electroweak hierarchy, in this
example $\epsilon=0.1$. We emphasize that this central parameter of the
composite dynamics would be difficult to extract experimentally without
combining collider and gravitational wave data. All the relevant measurements
would be challenging, but worthy ambitions for future experiments.

Upcoming precision LSS and 21-cm surveys offer the potential to detect heavy
particle production and propagation during inflation, imprinted on the non-
Gaussian bispectrum in distinctive non-local effects (non-analytic in co-
moving momenta). This field of “cosmological collider physics” is sensitive to
particle masses of order the inflationary Hubble scale or even somewhat higher
[215, 216]. (For very recent work and references, see [217]). We do not as yet
know the scales of inflation. If new particles are discovered in cosmological
non-Gaussianities, they may lie far above the reach of terrestrial colliders,
in which case they would give complementary information to what we learn from
even a powerful muon collider. But there are two scenarios in which they could
give us a (pre)view of collider-accessible physics: (i) if the inflationary
Hubble scale is of order $100$ TeV or less, then obviously “cosmological
collider physics” may directly be within reach of future terrestrial
colliders; (ii) even if the inflationary Hubble scale is orders of magnitude
above the TeV scale, there is a “heavy-lifting” mechanism [218] whereby the
particles seen in non-Gaussianities were given inflationary scale masses
through strong curvature effects, but such effects are negligible today so
that the particles may now be within terrestrial collider reach.

## 6\. Summary and Future Directions

The goal of this work is to paint the physics case of a high-energy muon
collider with a broad brush, emphasizing the sense in which such a collider is
positioned to answer the many questions posed or sharpened by the discovery of
the Higgs. The broad outlines of this case are drawn by the physics of both
muon annihilation and vector boson fusion, which in tandem provide compelling
rates for Standard Model and beyond-the-Standard Model processes across a
range of energies. Relative to recent work highlighting the significance of
vector boson fusion, we have emphasized the value of muon annihilation as a
discovery mode for sufficiently distinctive new physics. To characterize the
rich physics of the initial state, we surveyed descriptions of the virtual
electroweak gauge boson content of high-energy muons ranging from the
Effective Vector Approximation to electroweak PDFs. As a practical matter, we
emphasized the sense in which the simple “leading log PDF” is sufficient to
capture most of the qualitative physics, and briefly explored the impact of
finite $W$ and $Z$ masses on electroweak PDFs.

Turning to the physics case itself, we highlighted the potential of a muon
collider to illuminate various aspects of electroweak symmetry breaking, dark
matter, and naturalness. The measurement of the top Yukawa coupling provided a
particularly sharp test case for probing the physics of electroweak symmetry
breaking, in which deviations from the Standard Model prediction lead to
significant changes in the rates for longitudinal vector boson fusion into
$t\bar{t}$ pairs. This is sufficient for a muon collider at or above
$\sqrt{s}=14$ TeV to test the top Yukawa coupling beyond the expected
sensitivity of the LHC. The increase in reach provided by polarized beams
exemplifies their potential value. As a second test case relevant to the Higgs
potential and the electroweak phase transition, we proposed two avenues for
constraining the so-called “nightmare scenario” in which the electroweak phase
transition is strengthened by a light, $\mathbb{Z}_{2}$-symmetric singlet
scalar that only couples to the Higgs. In addition to the canonical search for
pair production of the scalar in missing energy final states, we noted the
possibility of again using longitudinal vector boson fusion into $t\bar{t}$
pairs, this time looking for a feature in the invariant mass distribution
associated with the scalar threshold. In both channels, a collider operating
above $\sqrt{s}=10$ TeV can cover much of the motivated parameter space. We
further considered direct searches for additional Higgs bosons, focusing on
the case of a singlet scalar mixing with the Higgs. Here the abundant
production of such scalars via vector boson fusion would allow a collider
operating above $\sqrt{s}=10$ TeV to probe mixing angles whose indirect
imprints (in the form of Higgs coupling deviations) lie beyond the reach of
any proposed collider, and a collider operating above $\sqrt{s}=30$ TeV to far
exceed the mass reach of a 100 TeV $pp$ collider.

The implications of a muon collider for dark matter are exemplified by its
coverage of “minimal dark matter” models, in which the dark matter particle
resides in an electroweak multiplet whose interactions with SM gauge bosons
can generate the observed abundance. We highlighted two classes of search
strategies for these multiplets, using either a missing mass or a disappearing
track signature. Depending on the integrated luminosity, a high-energy muon
collider operating at $\sqrt{s}=10$ or 14 TeV can discover smaller electroweak
representations (such as an $SU(2)$ doublet or triplet) at their thermal
targets, while a collider operating at $\sqrt{s}=30$ \- $100$ TeV can cover
the thermal targets for higher electroweak representations as well.

We illustrated the potential of a muon collider to discover or constrain
natural explanations of the weak scale by considering the reach for both
solutions to the “big” hierarchy problem (supersymmetry or compositeness) and
the “little” hierarchy problem (represented here by the Twin Higgs). Within
the context of supersymmetry, searches for the higgsino and the stop would
allow a muon collider operating at $\sqrt{s}\gtrsim 20$ TeV to comprehensively
cover tuning beyond the per mille level, and collisions at $\sqrt{s}=30$ TeV
would reach stop masses consistent with the observed Higgs mass even in the
absence of significant stop mixing. Despite its lack of electroweak quantum
numbers, even the gluino could be probed beyond LHC limits leveraging
radiation off of $q\bar{q}$ pairs at $\sqrt{s}\gtrsim$ 10 TeV. In addition to
these conventional channels, a muon collider can provide the first direct
collider test of the supersymmetry-breaking sector itself, achieving
sensitivity to direct production of the gravitino consistent with LHC bounds
on Standard Model superpartners, provided a low scale of supersymmetry
breaking. The coverage of compositeness is comparable, with the addition of
powerful indirect tests coming from the energy growth of irrelevant operators
parameterizing mixing of the Higgs and composite states. Even solutions to the
“little” hierarchy problem without conventional LHC signatures, such as the
Twin Higgs, lie within reach of a muon collider. In addition to its
considerable sensitivity to indirect effects, a high-energy muon collider
would be able to directly access the radial mode of these models and
decisively confirm or falsify them as a viable explanation of the weak scale.

A host of other experiments will indirectly probe physics as high as the PeV
scale in the years preceding the first beams at a high-energy muon collider.
To this end, we have considered the complementarity between a muon collider
and potential signals of new physics in electric dipole moments, flavor
violation, and gravitational waves. Of particular interest are precision tests
of charged lepton flavor violation in processes such as $\mu\rightarrow
e\gamma,\mu\rightarrow 3e,\tau\rightarrow 3\mu$ and $\mu$-to-$e$ conversion.
Here we have explored the detailed reach of a muon collider for both indirect
sources of lepton flavor violation (such as flavor-violating four-fermion
operators) and direct sources (such as flavor-violating slepton interactions
in the MSSM), in both cases finding that muon colliders below $\sqrt{s}\sim
10$ TeV provide complementary sensitivity to experiments such as Mu2e and
Mu3e, while more energetic colliders are capable of probing parameter space
beyond the reach of future proposals.

Broadly speaking, we find that a muon collider operating at tens of TeV and
tens of ab-1 is capable of surpassing the indirect reach of proposed
$e^{+}e^{-}$ Higgs factories and the direct reach of proposed 100 TeV $pp$
colliders, covering a broad swathe of motivated physics beyond the Standard
Model and probing explanations for potential signals in experiments across the
many frontiers of particle physics. Perhaps the most compelling gains stand to
be made by colliders operating between $\sqrt{s}=14$ \- $30$ TeV, for which
most of the questions posed by the discovery of the Higgs boson may be
decisively answered.

Needless to say, many of the projections made in this work are naive in light
of the significant uncertainties and many unresolved challenges facing both
accelerators and detectors. Nonetheless, we hope that they provide a
qualitative guide to the energies and luminosities that would position a
future muon collider as a comprehensive successor to the LHC, pinpointing a
variety of directions that merit more careful study. Beyond characterizing the
reach in conventional benchmarks for a future collider program, we have
identified a number of opportunities uniquely suited to a muon collider,
including direct tests of low-energy supersymmetry breaking. Key outstanding
questions include the performance and prospects of searches involving missing
energy, which are central to the coverage of dark matter, the electroweak
phase transition, and supersymmetry breaking; the invariant mass resolution in
heavy Standard Model final states, essential to making the most of the
abundant opportunities provided by the fusion of longitudinal vector bosons;
and the feasibility of fully instrumenting the forward region, which will
shape the set of available observables and the composition of signal
processes.

Much remains before us. But the muons are calling, and we must go.

## Acknowledgments

We are grateful to Ken Van Tilburg for collaboration during the early stages
of this work. We thank Kaustubh Agashe, Tao Han, Sergo Jindariani, Fabio
Maltoni, and Dave Soper for useful conversations.

The work of I. Banta, T. Cai, N. Craig, G. Koszegi, A. McCune, and T. Trott is
supported by the U.S. Department of Energy under the grant DE-SC0011702. The
work of D. Buttazzo and A. Tesi is supported by MIUR under grant number
2017L5W2PT. The work of T. Cohen is supported by the U.S. Department of Energy
under grant number DE-SC0011640. The work of J. Fan is supported by the U.S.
Department of Energy under grant number DE-SC-0010010. The work of I. Garcia
Garcia is funded by the Gordon and Betty Moore Foundation through Grant
GBMF7392, and in part by the National Science Foundation under Grant No. NSF
PHY-1748958. The work of S. Homiller, Q. Lu, and M. Reece was supported in
part by the DOE Grant DE-SC0013607 and by the Alfred P. Sloan Foundation Grant
No. G-2019-12504. The work of P. Meade is supported in part by the National
Science Foundation under Grant No. NSF PHY-1915093. The work of A. Mariotti is
supported in part by the SRP HEP-VUB and by the EOS-be.h n.30820817. D.
Sutherland has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 754496. The work of L.-T. Wang is supported by the DOE grant DE-
SC0013642. The work of S. Koren is funded by a Mafalda and Reinhard Oehme
Postdoctoral Fellowship. The work of M. Ekhterachian and R. Sundrum is
supported by the NSF grant PHY-1914731.

##  Appendix A Simplified Models

In order to benchmark the physics potential of muon colliders over a range of
energies, we compute the rates for various processes using representative
simplified models. Simplified models are defined in FeynRules using a
combination of public and custom model files. Wherever possible, we simulate
processes in both Whizard [219, 220, 221] and Madgraph5 [222]. We group the
majority of processes into three categories: “annihilation” when states are
produced directly in $\mu^{+}\mu^{-}$ collisions; “neutral VBF” when states
are produced via vector boson fusion from combinations of electroweak vectors
carrying zero total charge; and “charged VBF” when states are produced via
vector boson fusion from combinations of electroweak vectors carrying nonzero
total charge. For the most part, we present results in annihilation and
neutral VBF channels. For both charged and neutral VBF processes, we exclude
contributions from on-shell $W$ and $Z$ bosons by imposing appropriate
invariant mass cuts on the final state, as in [32]. In what follows, we
typically do not display simulated VBF cross sections for mass points close to
threshold given the strong sensitivity to phase space cuts; as noted in the
text, annihilation production dominates near threshold. Where relevant, we
also consider QCD “bremsstrahlung” processes, in which a gluon is radiated off
a hard final state quark and then “splits” into the strongly-interacting final
state of interest; this is particularly relevant for gluino production in
supersymmetric models.

To validate the results of our simulations, we compute a selection of
representative cross sections analytically using a combination of FeynArts,
FormCalc, and FeynCalc. We compute select annihilation cross sections
explicitly, while for select VBF cross sections we compute the “partonic”
cross sections explicitly and obtain total inclusive cross sections via
numerical convolution with the LL luminosity functions detailed in Sec. 3.

###  A.1 Standard Model

We begin by exploring some SM rates as a function of $\sqrt{s}$. This provides
some interesting benchmarks on its own, and also allows one to get a sense of
the raw rate for backgrounds that are relevant when estimating the new physics
potential of a muon collider.

For SM interactions, we treat the first and second generation fermions as
massless and we use a diagonal CKM matrix for simplicity. We choose to work in
terms of the measured value $\alpha(m_{Z})^{-1}=127.9$, at the cost of having
$m_{W}=79.8$ GeV, which is calculated using the tree-level relations in terms
of $G_{F},\alpha$. We take $m_{t}=173$ GeV and $m_{h}=125$ GeV. Cross sections
for representative processes computed using Whizard are illustrated in Fig.
25, and we find excellent agreement with Madgraph5. Unsurprisingly, as
$\sqrt{s}$ is increased the $s$-channel rates decrease as a power law, while
the VBF processes grow logarithmically.

Figure 25: Cross sections for representative annihilation (dashed) and VBF
(solid) SM processes as a function of $\sqrt{s}$.

###  A.2 Supersymmetry

Perhaps the best studied simplified models are motivated by the Minimal
Supersymmetric Standard Model (MSSM). For supersymmetric processes, we use a
variety of FeynRules models. For gluino pair production we use the default
MSSM model in FeynRules, keeping the gluino light while decoupling all other
sparticles. The cross section for $\tilde{g}\tilde{g}$ production resulting
from the “bremsstrahlung” of a gluon off of a $q\bar{q}$ pair is shown in Fig.
26. For stop pair production we use the $t$-channel dark matter FeynRules
model [134] with couplings set to their supersymmetric values. Cross sections
for the production of the $SU(2)$ singlet $\tilde{t}_{R}$ and $SU(2)$ doublet
$\tilde{Q}_{3}$ via annihilation and VBF are shown in Figs. 27 and 28. For
slepton pair production, we use the default MSSM model in FeynRules,
alternately keeping either $\tilde{\tau}_{R}$ or $\tilde{L}_{3}$ light while
decoupling all other sparticles. Cross sections for the production of the
$SU(2)$ singlet $\tilde{\tau}_{R}$ and $SU(2)$ doublet $\tilde{L}_{3}$ via
annihilation are shown in Fig. 29. For higgsino pair production, we use the
default MSSM model in FeynRules, keeping the components $\chi_{1}^{\pm}$ and
$\chi_{1,2}^{0}$ light while decoupling all other sparticles and taking care
to keep the neutralino and chargino mixing matrices appropriately aligned.
Cross sections for both chargino and neutralino pair production via
annihilation and VBF are shown in Fig. 30. For wino pair production, we use
the default MSSM model in FeynRules, keeping the components $\chi_{1}^{\pm}$
and $\chi_{1}^{0}$ light while decoupling all other sparticles and taking care
to keep the neutralino and chargino mixing matrices appropriately aligned.
Cross sections for both chargino and neutralino pair production via
annihilation and VBF are shown in Fig. 31. Across all processes, we see the
same trend as with the Standard Model processes: when both $s$-channel and VBF
production is possible, the former dominates at low $\sqrt{s}$ while the
latter takes over for high $\sqrt{s}$. Another interesting point to note is
that the squark production rate is significantly larger than the gluino rate,
which implies that gluino production could be dramatically impacted when
considering a more complete model that includes both gluinos and squarks.

Figure 26: Cross section for $\tilde{g}\tilde{g}$ production resulting from
the “bremsstrahlung” of a gluon off of a $q\bar{q}$ pair as a function of
$m_{\tilde{g}}$ and $\sqrt{s}$, computed using Whizard.

Figure 27: Cross section for $\tilde{t}_{R}\tilde{t}_{R}^{*}$ production via
annihilation (left) and VBF (right) as a function of $m_{\tilde{t}}$ and
$\sqrt{s}$, computed using Whizard.

Figure 28: Cross section for $\tilde{Q}_{3}\tilde{Q}^{*}_{3}$ production via
annihilation (left) and VBF (right) as a function of $m_{\tilde{Q}}$ and
$\sqrt{s}$, computed using Whizard.

Figure 29: Cross section for $\tilde{\tau}_{R}\tilde{\tau}_{R}^{*}$ production
(left) and $\tilde{L}_{3}\tilde{L}^{*}_{3}$ production (right) via
annihilation as a function of $m_{\tilde{t}}$ and $\sqrt{s}$, computed using
Whizard.

Figure 30: Cross section for $\tilde{h}_{1}^{0}\tilde{h}_{2}^{0}$ and
$\tilde{h}^{+}\tilde{h}^{-}$ production as a function of $m_{\tilde{h}}$ and
$\sqrt{s}$ via annihilation (top) and VBF (bottom), computed using Whizard.

Figure 31: Cross section for $\tilde{W}^{+}\tilde{W}^{-}$ production via
annihilation (top left) and VBF (top right), $\tilde{W}^{\pm}\tilde{W}^{0}$
production via VBF (bottom) as a function of $m_{\tilde{W}}$ and $\sqrt{s}$,
computed using MadGraph5.

###  A.3 Vector-like quarks

Next we turn to vector-like quarks, an important example that arises in
scenarios ranging from global symmetry approaches to the hierarchy problem to
parity solutions to the strong CP problem. We restrict our considerations to
the gauge interactions of an $SU(3)$ triplet, $SU(2)$ singlet, hypercharge
$+\frac{2}{3}$ Dirac fermion $T$, which we implement in FeynRules. Cross
sections for $T\overline{T}$ production via annihilation and VBF are shown in
Fig. 32, computed by convolving partonic amplitudes obtained via FeynArts and
FormCalc with the LL luminosity functions detailed in Sec. 3.

Figure 32: Cross section for $T\overline{T}$ production via annihilation
(left) and VBF (right) as a function of $m_{T}$ and $\sqrt{s}$, computed by
convolving partonic amplitudes obtained via FeynArts and FormCalc with the LL
luminosity functions detailed in Sec. 3.

###  A.4 Higgs portal

Next, we turn to one of the models that has implications for baryogenesis,
dark matter, and models of neutral naturalness. Furthermore, this model is
simultaneously very simple to state while being very difficult to discover.
Specifically, we extend the SM using a $\mathbb{Z}_{2}$ symmetric singlet $S$
that couples via the Higgs portal:

$\mathcal{L}=\frac{1}{2}(\partial
S)^{2}-\frac{1}{2}M_{S}^{2}S^{2}-\lambda_{HS}\,S^{2}|H|^{2}\,.$ (67)

We omit a possible singlet quartic, which does not influence the
phenomenology, and assume $M_{S},\lambda_{HS}$ are such that $S$ is stabilized
at the origin and the $\mathbb{Z}_{2}$ symmetry is unbroken. The physical mass
of the singlet in the broken phase is

$\displaystyle m_{S}^{2}=M_{S}^{2}+\lambda_{HS}v^{2}\,.$ (68)

As written, the singlet is absolutely stable and may constitute a dark matter
candidate. It also may be rendered unstable by explicit $\mathbb{Z}_{2}$
breaking, leading to prompt or long-lived signatures.

The differential cross section for $s$-channel production of $SS$ in
association with a $Z$ boson is [223]

$\displaystyle\frac{\text{d}\sigma}{\text{d}t\,\text{d}u}(\mu^{+}\mu^{-}\rightarrow
ZSS)$
$\displaystyle=\frac{\big{[}\big{(}-\frac{1}{2}+s_{W}^{2}\big{)}^{2}+s_{W}^{4}\big{]}\lambda_{HS}v^{2}m_{Z}^{2}}{16(2\pi)^{3}s}\bigg{(}\\!\frac{e}{s_{W}c_{W}}\\!\bigg{)}^{4}\bigg{[}\frac{s+(t-m_{Z}^{2})(u-m_{Z}^{2})/m_{Z}^{2}}{(s-m_{Z}^{2})^{2}+\Gamma_{Z}^{2}m_{Z}^{2}}\bigg{]}$
$\displaystyle\hskip
50.0pt\times\left[\frac{1}{s+t+u-m_{Z}^{2}-m_{h}^{2}}\right]^{2}\left[1-\frac{4m_{S}^{2}}{s+t+u-m_{Z}^{2}}\right]^{1/2}\,,$
(69)

where the Mandelstam variables are defined in the standard way for $2\to 2$
processes such that $s+t+u=m_{Z}^{2}+Q^{2}$, where $Q^{2}$ is the square of
the invariant mass of the two singlets in the final state. At large $s$, this
leads to a cross section falling as $1/s^{2}$. This motivates considering a
variety of VBF-like processes, including $W^{+}W^{-}$ fusion into $SS$ and
$HSS$ final states, as well as $WZ/W\gamma$ fusion into the $WSS$ final state.
We implement the model in FeynRules and compute cross sections primarily in
Whizard, finding good agreement with both MadGraph5 and the result of
convolving the partonic cross sections with our LL PDFs. Cross sections for
all four processes as a function of $m_{S}$ and $\sqrt{s}$ are shown in Fig.
33.

Figure 33: Cross section for various Higgs portal production modes as a
function of $m_{S}$ and $\sqrt{s}$ computed using Whizard, including
$s$-channel $SS+Z$ production (upper left), VBF $SS$ production (upper right),
VBF $SS+h$ production (lower left), and VBF $SS+W$ production (lower right).
As a benchmark, we take $\lambda_{HS}=1$ here.

#### A.4.1. Missing mass analysis

Given the central importance of the singlet scalar Higgs portal in BSM
scenarios ranging from electroweak baryogenesis to neutral naturalness, we
carry out a series of simplified analyses to benchmark the reach of dedicated
searches for $S$, assuming that $S$ is detector stable. These analyses cover
three different production modes: $W^{+}W^{-}$ fusion into an $SS$ pair in
association with an ISR photon; $W^{+}W^{-}$ fusion into an $SS$ pair in
association with a Higgs boson; and $s$-channel production of an $SS$ pair in
association with a $Z$ boson. In each case and for each benchmark value of
$\sqrt{s}$, we simulate 100k background events and 10k signal events for each
of $m_{S}=75,100,150,225,350,500,750,1000\text{ GeV}$ in Whizard.

#####  A.4.1.1 SS$\bm{\;+\;\gamma}$ analysis

For this signal, we consider the production of $SS$ via $W^{+}W^{-}$ fusion
accompanied by an ISR photon, $\mu^{+}\mu^{-}\rightarrow
SS+\nu\bar{\nu}+\gamma$, where $S$ is treated as invisible. The primary SM
background for this final state is
$\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}+\gamma$, i.e., production of a single
photon via $W^{+}W^{-}$ fusion. We require that the ISR photon falls within
the detector acceptance, here defined as $10^{\circ}<\theta<170^{\circ}$, and
employ a “missing mass” strategy in which the photon energy is required to lie
between

$50\,{\rm GeV}<E_{\gamma}<\frac{s-4m_{S}^{2}}{2\sqrt{s}}\,.$ (70)

The background spectrum is harder than the signal spectrum, so the lower bound
on the photon energy serves primarily to avoid singular regions of phase
space. The upper bound on the photon energy reflects the lower bound on the
missing mass distribution.

The integrated luminosity required to exclude a singlet scalar at 95% CL using
this analysis is shown in Fig. 34.

Figure 34: Left: Projected 95% CL exclusion limits as a function of $m_{S}$
and $\lambda_{HS}$ from the $SS+\gamma$ final state for various energy and
luminosity benchmarks described in the text. The dashed line corresponds to a
singlet scalar acquiring mass entirely from electroweak symmetry breaking.
Right: The integrated luminosity $\mathcal{L}$ in units of ab-1 required to
exclude a singlet scalar whose mass $m_{S}$ is due entirely to electroweak
symmetry breaking at 95% CL in the $SS+\gamma$ final state.

#####  A.4.1.2 SS$\bm{\;+\;}$h analysis

For this signal, we consider the production of $hSS$ via $W^{+}W^{-}$ fusion,
$\mu^{+}\mu^{-}\rightarrow SS+h+\nu\bar{\nu}$, where $S$ is treated as
invisible. The primary SM background for this final state is
$\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}+h$, i.e., production of a single Higgs
via $W^{+}W^{-}$ fusion. We require the Higgs boson falls within the detector
acceptance, here defined as $10^{\circ}<\theta<170^{\circ}$, and employ a
“missing mass” strategy in which the Higgs energy is required to lie between

$E_{\rm min}<E_{h}<\frac{s-4m_{S}^{2}+m_{h}^{2}}{2\sqrt{s}}\,.$ (71)

The signal spectrum is harder than the background spectrum, so the lower bound
$E_{\rm min}$ on the Higgs energy is chosen for each value of $m_{S}$ to
maximize $S/\sqrt{B}$. The upper bound on the Higgs energy reflects the lower
bound on the missing mass distribution.

The integrated luminosity required to exclude a singlet scalar at 95% CL using
this analysis is shown in Fig. 35.

Figure 35: Left: Projected 95% CL exclusion limits as a function of $m_{S}$
and $\lambda_{HS}$ from the $SS+h$ final state for various energy and
luminosity benchmarks described in the text. The dashed line corresponds to a
singlet scalar acquiring mass entirely from electroweak symmetry breaking.
Right: The integrated luminosity $\mathcal{L}$ in units of ab-1 required to
exclude a singlet scalar whose mass $m_{S}$ is due entirely to electroweak
symmetry breaking at 95% CL in the $SS+h$ final state.

#####  A.4.1.3 SS$\bm{\;+\;}$Z analysis

For this signal, we consider $s$-channel production of $ZSS$
$\mu^{+}\mu^{-}\rightarrow SS+Z$, where $S$ is treated as invisible. The
primary SM background for this final state is $\mu^{+}\mu^{-}\rightarrow
Z+\nu\bar{\nu}$, which has contributions from both $W^{+}W^{-}$ fusion and
$\mu^{+}\mu^{-}$ annihilation. We require that the $Z$ boson falls within the
detector acceptance, here defined as $10^{\circ}<\theta<170^{\circ}$, and
employ a “missing mass” strategy in which the $Z$ energy is required to lie
between

$E_{\rm min}<E_{Z}<\frac{s-4m_{S}^{2}+m_{Z}^{2}}{2\sqrt{s}}\,.$ (72)

The signal spectrum is harder than the background spectrum, so the lower bound
$E_{\rm min}$ on the $Z$ energy is chosen for each value of $m_{S}$ to
maximize $S/\sqrt{B}$. The upper bound on the $Z$ energy reflects the lower
bound on the missing mass distribution. At higher energies, the optimal value
of $E_{\rm min}$ leads to background acceptance of less than $10^{-4}$ for
lower values of $m_{S}$, in which case we generate additional exclusive
background samples to improve Monte Carlo statistical error.

The integrated luminosity required to exclude a singlet scalar at 95% CL using
this analysis is shown in Fig. 36.

Figure 36: Left: Projected 95% CL exclusion limits as a function of $m_{S}$
and $\lambda_{HS}$ from the $SS+Z$ final state for various energy and
luminosity benchmarks described in the text. The dashed line corresponds to a
singlet scalar acquiring mass entirely from electroweak symmetry breaking.
Right: The integrated luminosity $\mathcal{L}$ in units of ab-1 required to
exclude a singlet scalar whose mass $m_{S}$ is due entirely to electroweak
symmetry breaking at 95% CL in the $SS+Z$ final state.

#####  A.4.1.4 Missing mass combination

To benchmark the combined sensitivity of these three searches, we consider the
naive combination that results from adding their significance in quadrature.
The integrated luminosity required to exclude a singlet scalar at 95% CL using
the combination of analyses is shown in Fig. 37.

Figure 37: Left: Projected 95% CL exclusion limits as a function of $m_{S}$
and $\lambda_{HS}$ from the naive combination of $SS+\gamma/h/Z$ final states
for various energy and luminosity benchmarks described in the text. The dashed
line corresponds to a singlet scalar acquiring mass entirely from electroweak
symmetry breaking. Right: The integrated luminosity $\mathcal{L}$ in units of
ab-1 required to exclude a singlet scalar whose mass $m_{S}$ is due entirely
to electroweak symmetry breaking at 95% CL from the naive combination of
$SS+\gamma/h/Z$ final states.

#### A.4.2. Invariant mass analysis

An alternative strategy to probe the Higgs portal singlet scenario is to
search for the virtual effects of $S$ through its influence on the Higgs
sector, rather than looking for this state on-shell as described in Sec.
A.4.1. At 1-loop order, the simplified model of Eq. 67 includes modifications
in both the Higgs 2-point function, as well as the couplings between the Higgs
and SM degrees of freedom. As already noted in Sec. 4.1, this kind of
tampering in the Higgs sector will affect the behavior of scattering
amplitudes involving longitudinally polarized gauge bosons, given the role of
the Higgs in maintaining perturbative unitarity of the SM. In this section, we
illustrate how the resulting features in the differential cross section for
the process $\mu^{+}\mu^{-}\rightarrow t\bar{t}+X$ may be leveraged to probe
this model.

The 1-loop correction to the Higgs 2-point function in the model of Eq. 67
includes both mass and wave-function renormalization effects, and it is given
by

$\Sigma_{2}(p^{2})=\frac{(\lambda_{HS}v)^{2}}{16\pi^{2}}\int_{0}^{1}\text{d}x\,\log\left(\frac{m_{S}^{2}-p^{2}x(1-x)}{m_{S}^{2}-m_{h}^{2}x(1-x)}\right)-(p^{2}-m_{h}^{2})\delta
Z\ ,$ (73)

where

$\delta
Z=-\frac{(\lambda_{HS}v)^{2}}{16\pi^{2}}\int_{0}^{1}\text{d}x\,\frac{x(1-x)}{m_{S}^{2}-m_{h}^{2}x(1-x)}\,,$
(74)

and we have imposed renormalization conditions such that the Higgs propagator
has a single pole at $p^{2}=m_{h}^{2}$ with residue $i$. Wavefunction
renormalization further affects SM Higgs couplings, which are modified by a
universal factor of $(1+\delta_{*})$, with $\delta_{*}\simeq\delta Z/2$.

At the partonic level, the processes most sensitive to these modifications in
the Higgs sector are those involving longitudinally polarized $W$ and $Z$
bosons in the initial state, with $W^{+}_{L}W^{-}_{L}\rightarrow t\bar{t}$
being the most relevant given the enhanced $W$ content of the muons. This
results in a distinctive kinematic feature in the differential cross section
for the process $\mu^{+}\mu^{-}\rightarrow t\bar{t}+X$ that peaks at a scale
$m_{t\bar{t}}\simeq 2m_{S}$, where $m_{t\bar{t}}$ is the invariant mass of the
$t\bar{t}$ pair. For illustration, Fig. 38 shows the fractional deviation in
$\text{d}\sigma/\text{d}m_{t\bar{t}}$ with respect to the SM for a collision
center-of-mass energy $E_{\rm CM}=14$ TeV, and a singlet mass $m_{S}=750$ GeV
that results entirely from electroweak symmetry breaking. The maximum size of
the deviation can be of order $\sim 1\%$. Our calculation includes the 1-loop
effects described in the previous paragraph, but is otherwise performed at
tree-level. This approximation is justified as long as $\lambda_{HS}$ is
larger than any of the couplings of the SM, which is always case when the
singlet mass arises entirely from its coupling to the Higgs and provided
$m_{S}>m_{t}$.

Figure 38: Fractional deviation in the differential cross section for the
process $\mu^{+}\mu^{-}\rightarrow t\bar{t}+X$ as a result of the radiative
corrections present in the model of Eq. 67 (see Eqs. (73)-(74) and surrounding
discussion). The center-of-mass-energy of the colliding muons is taken to be
$14$ TeV, and the mass of the singlet $m_{S}=750$ GeV, all of which arises
from electroweak symmetry breaking. The solid line includes both charged and
neutral intermediate VBF processes, whereas the dashed line includes only
contributions from $W^{+}W^{-}$ fusion, as would be appropriate if background
events could be eliminated by identifying the outgoing muons. The vertical
dashed line corresponds to $m_{t\bar{t}}=2m_{s}=1.5$ TeV, at which the peak of
the kinematic feature in the differential cross section takes place.

An estimate of the integrated luminosity required to exclude a singlet whose
mass is due entirely to electroweak symmetry breaking at 95% CL using this
analysis is shown in Fig. 39. The number of signal and background events have
been computed by integrating the differential cross section over a mass window
spanning the range between $90\%$ and $150\%$ of $2m_{S}$. The corresponding
$S/\sqrt{B}$ ratio has been computed for $m_{S}=225,350,500,750$, and $1000$
GeV and linearly interpolated for intermediate masses.171717For a center-of-
mass energy of the incoming muons $E_{\rm CM}=1$ TeV, we only estimate the
required luminosity for $m_{S}=225$ and $350$ GeV, and linearly interpolate
between the two. For the larger values of $m_{S}$, it is not possible for the
deviation in the differential cross section to reach its peak at
$m_{t\bar{t}}=2m_{S}$, as the singlet never becomes on-shell, and therefore
the method discussed here is not applicable.

Figure 39: Estimate of the integrated luminosity required to exclude a Higgs
portal singlet $S$ whose mass arises entirely from electroweak symmetry
breaking, for various center-of-mass energies. Left: $t\bar{t}$ production
through intermediate neutral and charged VBF is treated as background. Right:
Only $t\bar{t}$ production through charged VBF is treated as background, under
the assumption that the outgoing muons accompanying neutral VBF can be
identified.

Although a more sophisticated analysis would be required to draw robust
conclusions, the results in Fig. 39 suggest that this search strategy could
improve on the individual channels described in Sec. A.4.1, and may be
competitive with their combination. The analysis could be further improved by
considerations related to the angular distribution of the outgoing $t\bar{t}$
pair, or through the use of polarized beams. More importantly, our discussion
illustrates how muon colliders offer qualitatively new possibilities to search
for new physics, by taking advantage of both their high energy reach as well
as their underlying identity as gauge boson colliders.

###  A.5 Hidden valleys

To benchmark the sensitivity of muon colliders to Hidden Valley scenarios
[169, 170], we consider a particularly minimal realization in which the vector
current of a SM-neutral Dirac fermion $\chi$ is coupled to the muon vector
current of the Standard Model via a dimension-6 contact term:

$\mathcal{L}=i\bar{\chi}\not{D}\chi-
m_{\chi}\bar{\chi}\chi+\frac{c}{\Lambda^{2}}(\bar{\mu}\gamma^{\mu}\mu)(\bar{\chi}\gamma_{\mu}\chi)\,.$
(75)

We implement this model in FeynRules and compute the cross section using
MadGraph5. The cross section for $\chi\bar{\chi}$ production as a function of
$m_{\chi}$ and $\sqrt{s}$ is shown in Fig. 40 for $\Lambda=100$ TeV and $c=1$.

Figure 40: Cross section for $\chi\bar{\chi}$ production in the hidden valley
simplified model as a function of $m_{\chi}$ and $\sqrt{s}$ for $\Lambda=100$
TeV and $c=1$, computed using MadGraph5.

###  A.6 Axion-like particles

As our final example, we consider a simplified model in which an axion-like
particle $a$ couples to electroweak field strengths. We use the ALPsEFT
FeynRules model file documented in [224], taking the Lagrangian to be

$\mathcal{L}\supset\frac{1}{2}(\partial
a)^{2}-\frac{1}{2}m_{a}^{2}a^{2}-\frac{c_{\tilde{B}}}{f_{a}}aB_{\mu\nu}\tilde{B}^{\mu\nu}-\frac{c_{\tilde{W}}}{f_{a}}aW^{a}_{\mu\nu}\tilde{W}^{a,\mu\nu}\,.$
(76)

Cross sections for VBF production of $a$ as a function of $m_{a}$ and
$\sqrt{s}$ are shown in Fig. 41 for $f_{a}=100$ TeV and
$(c_{\tilde{B}},c_{\tilde{W}})=(1,0)$ [left] and $(0,1)$ [right].

Figure 41: Cross section for VBF production of $a$ in the axion-like particle
simplified model as a function of $m_{a}$ and $\sqrt{s}$ for $f_{a}=100$ TeV,
computed using MadGraph5. Left: $c_{\tilde{B}}=1,c_{\tilde{W}}=0$. Right:
$c_{\tilde{B}}=0,c_{\tilde{W}}=1$.

## References

  * [1] ATLAS Collaboration, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys.Lett. B716 (2012) 1–29, arXiv:1207.7214 [hep-ex].
  * [2] CMS Collaboration, S. Chatrchyan et al., “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys.Lett. B716 (2012) 30–61, arXiv:1207.7235 [hep-ex].
  * [3] P. Bambade et al., “The International Linear Collider: A Global Project,” arXiv:1903.01629 [hep-ex].
  * [4] CLICdp, CLIC Collaboration, T. Charles et al., “The Compact Linear Collider (CLIC) - 2018 Summary Report,” CERN Yellow Rep. Monogr. 1802 (2018) 1–98, arXiv:1812.06018 [physics.acc-ph].
  * [5] FCC Collaboration, A. Abada et al., “FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2,” Eur. Phys. J. ST 228 (2019) no. 2, 261–623.
  * [6] CEPC Study Group Collaboration, M. Dong et al., “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” arXiv:1811.10545 [hep-ex].
  * [7] FCC Collaboration, A. Abada et al., “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 (2019) no. 4, 755–1107.
  * [8] M. Ahmad et al., “CEPC-SPPC Preliminary Conceptual Design Report. 1. Physics and Detector,”.
  * [9] G. I. Budker, “Accelerators and colliding beams,” Conf. Proc. C 690827 (1969) 33–39.
  * [10] V. V. Parkhomchuk and A. N. Skrinsky, “Ionization Cooling: Physics and Applications,” AIP Conf. Proc. 352 (1996) 7–9.
  * [11] D. Neuffer, “Principles and Applications of Muon Cooling,” Part. Accel. 14 (1983) 75–90.
  * [12] D. Neuffer, “Multi-TeV muon colliders,” AIP Conf. Proc. 156 (1987) 201–208.
  * [13] V. D. Barger, M. S. Berger, J. F. Gunion, and T. Han, “s channel Higgs boson production at a muon muon collider,” Phys. Rev. Lett. 75 (1995) 1462–1465, arXiv:hep-ph/9504330.
  * [14] V. D. Barger, M. S. Berger, J. F. Gunion, and T. Han, “Higgs Boson physics in the s channel at $\mu^{+}\mu^{-}$ colliders,” Phys. Rept. 286 (1997) 1–51, arXiv:hep-ph/9602415.
  * [15] C. M. Ankenbrandt et al., “Status of muon collider research and development and future plans,” Phys. Rev. ST Accel. Beams 2 (1999) 081001, arXiv:physics/9901022.
  * [16] J.-P. Delahaye et al., “Enabling Intensity and Energy Frontier Science with a Muon Accelerator Facility in the U.S.: A White Paper Submitted to the 2013 U.S. Community Summer Study of the Division of Particles and Fields of the American Physical Society,” in Community Summer Study 2013: Snowmass on the Mississippi.  8, 2013.  arXiv:1308.0494 [physics.acc-ph].
  * [17] J.-P. Delahaye et al., “A Staged Muon Accelerator Facility For Neutrino and Collider Physics,” in 5th International Particle Accelerator Conference, p. WEZA02.  6, 2014.  arXiv:1502.01647 [physics.acc-ph].
  * [18] R. D. Ryne et al., “Design Concepts for Muon-Based Accelerators,” in 6th International Particle Accelerator Conference, p. WEPWA057.  2015\.
  * [19] K. Long, D. Lucchesi, M. Palmer, N. Pastrone, D. Schulte, and V. Shiltsev, “Muon colliders to expand frontiers of particle physics,” arXiv:2007.15684 [physics.acc-ph].
  * [20] MICE Collaboration, T. A. Mohayai, “First Demonstration of Ionization Cooling in MICE,” in 9th International Particle Accelerator Conference, p. FRXGBE3.  2018\.  arXiv:1806.01807 [physics.acc-ph].
  * [21] MICE Collaboration, V. Blackmore, “Recent results from the study of emittance evolution in MICE,” in 9th International Particle Accelerator Conference, p. TUPML067.  2018\.  arXiv:1806.04409 [physics.acc-ph].
  * [22] MICE Collaboration, M. Bogomilov et al., “Demonstration of cooling by the Muon Ionization Cooling Experiment,” Nature 578 (2020) no. 7793, 53–59, arXiv:1907.08562 [physics.acc-ph].
  * [23] M. Antonelli, M. Boscolo, R. Di Nardo, and P. Raimondi, “Novel proposal for a low emittance muon beam using positron beam on target,” Nucl. Instrum. Meth. A807 (2016) 101–107, arXiv:1509.04454 [physics.acc-ph].
  * [24] J. P. Delahaye, M. Diemoz, K. Long, B. Mansoulié, N. Pastrone, L. Rivkin, D. Schulte, A. Skrinsky, and A. Wulzer, “Muon Colliders,” arXiv:1901.06150 [physics.acc-ph].
  * [25] 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].
  * [26] E. Eichten and A. Martin, “The Muon Collider as a $H/A$ Factory,” Phys. Lett. B 728 (2014) 125–130, arXiv:1306.2609 [hep-ph].
  * [27] N. Chakrabarty, T. Han, Z. Liu, and B. Mukhopadhyaya, “Radiative Return for Heavy Higgs Boson at a Muon Collider,” Phys. Rev. D 91 (2015) no. 1, 015008, arXiv:1408.5912 [hep-ph].
  * [28] D. Buttazzo, D. Redigolo, F. Sala, and A. Tesi, “Fusing Vectors into Scalars at High Energy Lepton Colliders,” JHEP 11 (2018) 144, arXiv:1807.04743 [hep-ph].
  * [29] P. Bandyopadhyay and A. Costantini, “Obscure Higgs boson at Colliders,” Phys. Rev. D 103 (2021) no. 1, 015025, arXiv:2010.02597 [hep-ph].
  * [30] T. Han, S. Li, S. Su, W. Su, and Y. Wu, “Heavy Higgs Bosons in 2HDM at a Muon Collider,” arXiv:2102.08386 [hep-ph].
  * [31] W. Liu and K.-P. Xie, “Probing electroweak phase transition with multi-TeV muon colliders and gravitational waves,” arXiv:2101.10469 [hep-ph].
  * [32] 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].
  * [33] T. Han, Z. Liu, L.-T. Wang, and X. Wang, “WIMPs at High Energy Muon Colliders,” arXiv:2009.11287 [hep-ph].
  * [34] 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].
  * [35] 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].
  * [36] 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 (2021) no. 1, 013002, arXiv:2008.12204 [hep-ph].
  * [37] L. Di Luzio, R. Gröber, and G. Panico, “Probing new electroweak states via precision measurements at the LHC and future colliders,” JHEP 01 (2019) 011, arXiv:1810.10993 [hep-ph].
  * [38] D. Buttazzo, R. Franceschini, and A. Wulzer, “Two Paths Towards Precision at a Very High Energy Lepton Collider,” arXiv:2012.11555 [hep-ph].
  * [39] R. Capdevilla, D. Curtin, Y. Kahn, and G. Krnjaic, “A Guaranteed Discovery at Future Muon Colliders,” arXiv:2006.16277 [hep-ph].
  * [40] D. Buttazzo and P. Paradisi, “Probing the muon g-2 anomaly at a Muon Collider,” arXiv:2012.02769 [hep-ph].
  * [41] R. Capdevilla, D. Curtin, Y. Kahn, and G. Krnjaic, “A No-Lose Theorem for Discovering the New Physics of $(g-2)_{\mu}$ at Muon Colliders,” arXiv:2101.10334 [hep-ph].
  * [42] N. Chen, B. Wang, and C.-Y. Yao, “The collider tests of a leptophilic scalar for the anomalous magnetic moments,” arXiv:2102.05619 [hep-ph].
  * [43] W. Yin and M. Yamaguchi, “Muon $g-2$ at multi-TeV muon collider,” arXiv:2012.03928 [hep-ph].
  * [44] G.-y. Huang, F. S. Queiroz, and W. Rodejohann, “Gauged $L_{\mu}{-}L_{\tau}$ at a muon collider,” arXiv:2101.04956 [hep-ph].
  * [45] G.-y. Huang, S. Jana, F. S. Queiroz, and W. Rodejohann, “Probing the $R_{K^{(*)}}$ Anomaly at a Muon Collider,” arXiv:2103.01617 [hep-ph].
  * [46] N. Bartosik, A. Bertolin, L. Buonincontri, M. Casarsa, F. Collamati, A. Ferrari, A. Ferrari, A. Gianelle, D. Lucchesi, N. Mokhov, and et al., “Detector and physics performance at a muon collider,”Journal of Instrumentation 15 (May, 2020) P05001–P05001. http://dx.doi.org/10.1088/1748-0221/15/05/P05001.
  * [47] N. V. Mokhov and S. I. Striganov, “Detector background at muon colliders,” arXiv:1204.6721 [physics.ins-det].
  * [48] C. CMS, “A MIP Timing Detector for the CMS Phase-2 Upgrade,” Tech. Rep. CERN-LHCC-2019-003. CMS-TDR-020, CERN, Geneva, Mar, 2019.  https://cds.cern.ch/record/2667167.
  * [49] ATLAS Collaboration Collaboration, “Technical Proposal: A High-Granularity Timing Detector for the ATLAS Phase-II Upgrade,” Tech. Rep. CERN-LHCC-2018-023. LHCC-P-012, CERN, Geneva, Jun, 2018.  https://cds.cern.ch/record/2623663.
  * [50] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, and B. Mistlberger, “High precision determination of the gluon fusion Higgs boson cross-section at the LHC,” JHEP 05 (2016) 058, arXiv:1602.00695 [hep-ph].
  * [51] LHC Higgs Cross Section Working Group Collaboration, D. de Florian et al., “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector,” arXiv:1610.07922 [hep-ph].
  * [52] R. Boughezal, J. M. Campbell, R. K. Ellis, C. Focke, W. Giele, X. Liu, F. Petriello, and C. Williams, “Color singlet production at NNLO in MCFM,” Eur. Phys. J. C 77 (2017) no. 1, 7, arXiv:1605.08011 [hep-ph].
  * [53] J. Chen, T. Han, and B. Tweedie, “Electroweak Splitting Functions and High Energy Showering,” JHEP 11 (2017) 093, arXiv:1611.00788 [hep-ph].
  * [54] T. Han, Y. Ma, and K. Xie, “Quark and Gluon Contents of a Lepton at High Energies,” arXiv:2103.09844 [hep-ph].
  * [55] C. von Weizsacker, “Radiation emitted in collisions of very fast electrons,” Z. Phys. 88 (1934) 612–625.
  * [56] E. Williams, “Nature of the high-energy particles of penetrating radiation and status of ionization and radiation formulae,” Phys. Rev. 45 (1934) 729–730.
  * [57] S. Dawson, “The Effective W Approximation,” Nucl. Phys. B249 (1985) 42–60.
  * [58] J. Gao, “Probing light-quark Yukawa couplings via hadronic event shapes at lepton colliders,” JHEP 01 (2018) 038, arXiv:1608.01746 [hep-ph].
  * [59] W. Altmannshofer, J. Brod, and M. Schmaltz, “Experimental constraints on the coupling of the Higgs boson to electrons,” JHEP 05 (2015) 125, arXiv:1503.04830 [hep-ph].
  * [60] 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].
  * [61] M. A. Valdivia Garcia, A. Faus-Golfe, and F. Zimmermann, “Towards a Monochromatization Scheme for Direct Higgs Production at FCC-ee,”. https://cds.cern.ch/record/2159683.
  * [62] D. d’Enterria, “Higgs physics at the Future Circular Collider,” PoS ICHEP2016 (2017) 434, arXiv:1701.02663 [hep-ex].
  * [63] S. Homiller and P. Meade, “Measurement of the Triple Higgs Coupling at a HE-LHC,” JHEP 03 (2019) 055, arXiv:1811.02572 [hep-ph].
  * [64] M. L. Mangano, G. Ortona, and M. Selvaggi, “Measuring the Higgs self-coupling via Higgs-pair production at a 100 TeV p-p collider,” Eur. Phys. J. C 80 (2020) no. 11, 1030, arXiv:2004.03505 [hep-ph].
  * [65] LHC Higgs Cross Section Working Group Collaboration, A. David, A. Denner, M. Duehrssen, M. Grazzini, C. Grojean, G. Passarino, M. Schumacher, M. Spira, G. Weiglein, and M. Zanetti, “LHC HXSWG interim recommendations to explore the coupling structure of a Higgs-like particle,” arXiv:1209.0040 [hep-ph].
  * [66] LHC Higgs Cross Section Working Group Collaboration, J. R. Andersen et al., “Handbook of LHC Higgs Cross Sections: 3. Higgs Properties,” arXiv:1307.1347 [hep-ph].
  * [67] J. de Blas et al., “Higgs Boson Studies at Future Particle Colliders,” JHEP 01 (2020) 139, arXiv:1905.03764 [hep-ph].
  * [68] DELPHES 3 Collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi, “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 02 (2014) 057, arXiv:1307.6346 [hep-ex].
  * [69] N. Bartosik et al., “Preliminary Report on the Study of Beam-Induced Background Effects at a Muon Collider,” arXiv:1905.03725 [hep-ex].
  * [70] N. Bartosik et al., “Detector and Physics Performance at a Muon Collider,” JINST 15 (2020) no. 05, P05001, arXiv:2001.04431 [hep-ex].
  * [71] M. Selvaggi, “Talk at mdi studies meeting of the muon collider collaboration.” https://indico.cern.ch/event/957299/contributions/4023467/attachments/2106044/3541874/delphes_card_mucol_mdi%20.pdf.  2020-09-21.
  * [72] G. W. Foster and N. V. Mokhov, “Backgrounds and detector performance at a 2 $\times$2 TeV $\mu^{+}\mu^{-}$ collider,” AIP Conf. Proc. 352 (1996) 178–190.
  * [73] N. V. Mokhov, Y. I. Alexahin, V. V. Kashikhin, S. I. Striganov, and A. V. Zlobin, “Muon Collider Interaction Region and Machine-Detector Interface Design,” Conf. Proc. C 110328 (2011) 82–84, arXiv:1202.3979 [physics.acc-ph].
  * [74] N. V. Mokhov and S. I. Striganov, “Detector Background at Muon Colliders,” Phys. Procedia 37 (2012) 2015–2022, arXiv:1204.6721 [physics.ins-det].
  * [75] H. Abramowicz et al., “Higgs physics at the CLIC electron–positron linear collider,” Eur. Phys. J. C 77 (2017) no. 7, 475, arXiv:1608.07538 [hep-ex].
  * [76] M. Boronat, J. Fuster, I. Garcia, E. Ros, and M. Vos, “A robust jet reconstruction algorithm for high-energy lepton colliders,” Phys. Lett. B750 (2015) 95–99, arXiv:1404.4294 [hep-ex].
  * [77] M. Boronat, J. Fuster, I. Garcia, P. Roloff, R. Simoniello, and M. Vos, “Jet reconstruction at high-energy electron-positron colliders,” Eur. Phys. J. C78 (2018) no. 2, 144, arXiv:1607.05039 [hep-ex].
  * [78] L. Linssen, A. Miyamoto, M. Stanitzki, and H. Weerts, “Physics and detectors at clic: Clic conceptual design report,” 2012.
  * [79] F. An et al., “Precision Higgs physics at the CEPC,” Chin. Phys. C 43 (2019) no. 4, 043002, arXiv:1810.09037 [hep-ex].
  * [80] D. Egana-Ugrinovic, S. Homiller, and P. Meade, “Aligned and Spontaneous Flavor Violation,” Phys. Rev. Lett. 123 (2019) no. 3, 031802, arXiv:1811.00017 [hep-ph].
  * [81] D. Egana-Ugrinovic, S. Homiller, and P. R. Meade, “Higgs bosons with large couplings to light quarks,” Phys. Rev. D 100 (2019) no. 11, 115041, arXiv:1908.11376 [hep-ph].
  * [82] W. Altmannshofer, J. Eby, S. Gori, M. Lotito, M. Martone, and D. Tuckler, “Collider Signatures of Flavorful Higgs Bosons,” Phys. Rev. D 94 (2016) no. 11, 115032, arXiv:1610.02398 [hep-ph].
  * [83] D. Curtin et al., “Exotic decays of the 125 GeV Higgs boson,” Phys. Rev. D 90 (2014) no. 7, 075004, arXiv:1312.4992 [hep-ph].
  * [84] D. Dicus and V. Mathur, “Upper bounds on the values of masses in unified gauge theories,” Phys. Rev. D 7 (1973) 3111–3114.
  * [85] B. W. Lee, C. Quigg, and H. Thacker, “The Strength of Weak Interactions at Very High-Energies and the Higgs Boson Mass,” Phys. Rev. Lett. 38 (1977) 883–885.
  * [86] B. W. Lee, C. Quigg, and H. Thacker, “Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass,” Phys. Rev. D 16 (1977) 1519.
  * [87] M. Veltman, “Second Threshold in Weak Interactions,” Acta Phys. Polon. B 8 (1977) 475.
  * [88] M. S. Chanowitz and M. K. Gaillard, “The TeV Physics of Strongly Interacting W’s and Z’s,” Nucl. Phys. B261 (1985) 379–431.
  * [89] T. Appelquist and M. S. Chanowitz, “Unitarity Bound on the Scale of Fermion Mass Generation,” Phys. Rev. Lett. 59 (1987) 2405. [Erratum: Phys.Rev.Lett. 60, 1589 (1988)].
  * [90] B. Henning, D. Lombardo, M. Riembau, and F. Riva, “Measuring Higgs Couplings without Higgs Bosons,” Phys. Rev. Lett. 123 (2019) no. 18, 181801, arXiv:1812.09299 [hep-ph].
  * [91] M. Cepeda et al., Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC, vol. 7, pp. 221–584.  12, 2019.  arXiv:1902.00134 [hep-ph].
  * [92] T. Hahn, “Generating Feynman diagrams and amplitudes with FeynArts 3,” Comput. Phys. Commun. 140 (2001) 418–431, arXiv:hep-ph/0012260.
  * [93] R. Mertig, M. Bohm, and A. Denner, “Feyn Calc: Computer algebraic calculation of Feynman amplitudes,” Comput. Phys. Commun. 64 (1991) 345–359.
  * [94] V. Shtabovenko, R. Mertig, and F. Orellana, “New Developments in FeynCalc 9.0,” Comput. Phys. Commun. 207 (2016) 432–444, arXiv:1601.01167 [hep-ph].
  * [95] V. Shtabovenko, R. Mertig, and F. Orellana, “FeynCalc 9.3: New features and improvements,” Comput. Phys. Commun. 256 (2020) 107478, arXiv:2001.04407 [hep-ph].
  * [96] D. Curtin, P. Meade, and C.-T. Yu, “Testing Electroweak Baryogenesis with Future Colliders,” JHEP 11 (2014) 127, arXiv:1409.0005 [hep-ph].
  * [97] P. Meade and H. Ramani, “Unrestored Electroweak Symmetry,” Phys. Rev. Lett. 122 (2019) no. 4, 041802, arXiv:1807.07578 [hep-ph].
  * [98] Z. Chacko, H.-S. Goh, and R. Harnik, “The Twin Higgs: Natural electroweak breaking from mirror symmetry,” Phys. Rev. Lett. 96 (2006) 231802, arXiv:hep-ph/0506256 [hep-ph].
  * [99] N. Craig, S. Knapen, and P. Longhi, “Neutral Naturalness from Orbifold Higgs Models,” Phys. Rev. Lett. 114 (2015) no. 6, 061803, arXiv:1410.6808 [hep-ph].
  * [100] N. Craig, H. K. Lou, M. McCullough, and A. Thalapillil, “The Higgs Portal Above Threshold,” JHEP 02 (2016) 127, arXiv:1412.0258 [hep-ph].
  * [101] U. Ellwanger, C. Hugonie, and A. M. Teixeira, “The Next-to-Minimal Supersymmetric Standard Model,” Phys. Rept. 496 (2010) 1–77, arXiv:0910.1785 [hep-ph].
  * [102] S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, “Singlet Higgs phenomenology and the electroweak phase transition,” JHEP 08 (2007) 010, arXiv:0705.2425 [hep-ph].
  * [103] J. R. Espinosa, T. Konstandin, and F. Riva, “Strong Electroweak Phase Transitions in the Standard Model with a Singlet,” Nucl. Phys. B854 (2012) 592–630, arXiv:1107.5441 [hep-ph].
  * [104] D. E. Morrissey and M. J. Ramsey-Musolf, “Electroweak baryogenesis,” New J. Phys. 14 (2012) 125003, arXiv:1206.2942 [hep-ph].
  * [105] A. V. Kotwal, M. J. Ramsey-Musolf, J. M. No, and P. Winslow, “Singlet-catalyzed electroweak phase transitions in the 100 TeV frontier,” Phys. Rev. D94 (2016) no. 3, 035022, arXiv:1605.06123 [hep-ph].
  * [106] G. Kurup and M. Perelstein, “Dynamics of Electroweak Phase Transition In Singlet-Scalar Extension of the Standard Model,” Phys. Rev. D96 (2017) no. 1, 015036, arXiv:1704.03381 [hep-ph].
  * [107] D. Buttazzo, F. Sala, and A. Tesi, “Singlet-like Higgs bosons at present and future colliders,” JHEP 11 (2015) 158, arXiv:1505.05488 [hep-ph].
  * [108] S. Alipour-Fard, N. Craig, S. Gori, S. Koren, and D. Redigolo, “The second Higgs at the lifetime frontier,” JHEP 07 (2020) 029, arXiv:1812.09315 [hep-ph].
  * [109] R. Franceschini et al., “The CLIC Potential for New Physics,”CERN Yellow Rep. Monogr. 3/2018 (12, 2018) , arXiv:1812.02093 [hep-ph].
  * [110] CLICdp Collaboration, P. Roloff, U. Schnoor, R. Simoniello, and B. Xu, “Double Higgs boson production and Higgs self-coupling extraction at CLIC,” Eur. Phys. J. C 80 (2020) no. 11, 1010, arXiv:1901.05897 [hep-ex].
  * [111] M. Cirelli, N. Fornengo, and A. Strumia, “Minimal dark matter,” Nucl. Phys. B 753 (2006) 178–194, arXiv:hep-ph/0512090.
  * [112] M. Cirelli and A. Strumia, “Minimal Dark Matter: Model and results,” New J. Phys. 11 (2009) 105005, arXiv:0903.3381 [hep-ph].
  * [113] S. D. Thomas and J. D. Wells, “Phenomenology of Massive Vectorlike Doublet Leptons,” Phys. Rev. Lett. 81 (1998) 34–37, arXiv:hep-ph/9804359.
  * [114] M. R. Buckley, L. Randall, and B. Shuve, “LHC Searches for Non-Chiral Weakly Charged Multiplets,” JHEP 05 (2011) 097, arXiv:0909.4549 [hep-ph].
  * [115] M. Ibe, S. Matsumoto, and R. Sato, “Mass Splitting between Charged and Neutral Winos at Two-Loop Level,” Phys. Lett. B 721 (2013) 252–260, arXiv:1212.5989 [hep-ph].
  * [116] L. Di Luzio, R. Gröber, J. F. Kamenik, and M. Nardecchia, “Accidental matter at the LHC,” JHEP 07 (2015) 074, arXiv:1504.00359 [hep-ph].
  * [117] E. Del Nobile, M. Nardecchia, and P. Panci, “Millicharge or Decay: A Critical Take on Minimal Dark Matter,” JCAP 04 (2016) 048, arXiv:1512.05353 [hep-ph].
  * [118] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO].
  * [119] K. M. Belotsky, M. Khlopov, S. Legonkov, and K. Shibaev, “Effects of new long-range interaction: Recombination of relic heavy neutrinos and antineutrinos,” Grav. Cosmol. 11 (2005) 27–33, arXiv:astro-ph/0504621.
  * [120] J. Hisano, S. Matsumoto, M. Nagai, O. Saito, and M. Senami, “Non-perturbative effect on thermal relic abundance of dark matter,” Phys. Lett. B 646 (2007) 34–38, arXiv:hep-ph/0610249.
  * [121] M. Cirelli, A. Strumia, and M. Tamburini, “Cosmology and Astrophysics of Minimal Dark Matter,” Nucl. Phys. B 787 (2007) 152–175, arXiv:0706.4071 [hep-ph].
  * [122] H. An, M. B. Wise, and Y. Zhang, “Effects of Bound States on Dark Matter Annihilation,” Phys. Rev. D 93 (2016) no. 11, 115020, arXiv:1604.01776 [hep-ph].
  * [123] A. Mitridate, M. Redi, J. Smirnov, and A. Strumia, “Cosmological Implications of Dark Matter Bound States,” JCAP 05 (2017) 006, arXiv:1702.01141 [hep-ph].
  * [124] J. Liu, Z. Liu, and L.-T. Wang, “Enhancing Long-Lived Particles Searches at the LHC with Precision Timing Information,” Phys. Rev. Lett. 122 (2019) no. 13, 131801, arXiv:1805.05957 [hep-ph].
  * [125] P. W. Graham, D. E. Kaplan, and S. Rajendran, “Cosmological Relaxation of the Electroweak Scale,” Phys. Rev. Lett. 115 (2015) no. 22, 221801, arXiv:1504.07551 [hep-ph].
  * [126] N. Craig, A. Katz, M. Strassler, and R. Sundrum, “Naturalness in the Dark at the LHC,” JHEP 07 (2015) 105, arXiv:1501.05310 [hep-ph].
  * [127] N. Craig, “The State of Supersymmetry after Run I of the LHC,” in Beyond the Standard Model after the first run of the LHC.  9, 2013.  arXiv:1309.0528 [hep-ph].
  * [128] S. Dimopoulos and G. F. Giudice, “Naturalness constraints in supersymmetric theories with nonuniversal soft terms,” Phys. Lett. B 357 (1995) 573–578, arXiv:hep-ph/9507282.
  * [129] A. Pomarol and D. Tommasini, “Horizontal symmetries for the supersymmetric flavor problem,” Nucl. Phys. B 466 (1996) 3–24, arXiv:hep-ph/9507462.
  * [130] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, “The More minimal supersymmetric standard model,” Phys. Lett. B 388 (1996) 588–598, arXiv:hep-ph/9607394.
  * [131] C. Brust, A. Katz, S. Lawrence, and R. Sundrum, “SUSY, the Third Generation and the LHC,” JHEP 03 (2012) 103, arXiv:1110.6670 [hep-ph].
  * [132] M. Papucci, J. T. Ruderman, and A. Weiler, “Natural SUSY Endures,” JHEP 09 (2012) 035, arXiv:1110.6926 [hep-ph].
  * [133] R. Barbieri and G. F. Giudice, “Upper Bounds on Supersymmetric Particle Masses,” Nucl. Phys. B 306 (1988) 63–76.
  * [134] C. Arina, B. Fuks, and L. Mantani, “A universal framework for t-channel dark matter models,” Eur. Phys. J. C 80 (2020) no. 5, 409, arXiv:2001.05024 [hep-ph].
  * [135] D. R. Tovey, “On measuring the masses of pair-produced semi-invisibly decaying particles at hadron colliders,” JHEP 04 (2008) 034, arXiv:0802.2879 [hep-ph].
  * [136] P. Draper and H. Rzehak, “A Review of Higgs Mass Calculations in Supersymmetric Models,” Phys. Rept. 619 (2016) 1–24, arXiv:1601.01890 [hep-ph].
  * [137] S. Deser and B. Zumino, “Broken Supersymmetry and Supergravity,” Phys. Rev. Lett. 38 (1977) 1433–1436.
  * [138] M. Bolz, A. Brandenburg, and W. Buchmuller, “Thermal production of gravitinos,” Nucl. Phys. B 606 (2001) 518–544, arXiv:hep-ph/0012052. [Erratum: Nucl.Phys.B 790, 336–337 (2008)].
  * [139] J. Pradler and F. D. Steffen, “Thermal gravitino production and collider tests of leptogenesis,” Phys. Rev. D 75 (2007) 023509, arXiv:hep-ph/0608344.
  * [140] J. Pradler and F. D. Steffen, “Constraints on the Reheating Temperature in Gravitino Dark Matter Scenarios,” Phys. Lett. B 648 (2007) 224–235, arXiv:hep-ph/0612291.
  * [141] V. S. Rychkov and A. Strumia, “Thermal production of gravitinos,” Phys. Rev. D 75 (2007) 075011, arXiv:hep-ph/0701104.
  * [142] T. Moroi, H. Murayama, and M. Yamaguchi, “Cosmological constraints on the light stable gravitino,” Phys. Lett. B 303 (1993) 289–294.
  * [143] M. Kawasaki and T. Moroi, “Gravitino production in the inflationary universe and the effects on big bang nucleosynthesis,” Prog. Theor. Phys. 93 (1995) 879–900, arXiv:hep-ph/9403364.
  * [144] T. Moroi, “Effects of the gravitino on the inflationary universe,” other thesis, 3, 1995.
  * [145] L. J. Hall, J. T. Ruderman, and T. Volansky, “A Cosmological Upper Bound on Superpartner Masses,” JHEP 02 (2015) 094, arXiv:1302.2620 [hep-ph].
  * [146] E. Pierpaoli, S. Borgani, A. Masiero, and M. Yamaguchi, “The Formation of cosmic structures in a light gravitino dominated universe,” Phys. Rev. D 57 (1998) 2089–2100, arXiv:astro-ph/9709047.
  * [147] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese, and A. Riotto, “Constraining warm dark matter candidates including sterile neutrinos and light gravitinos with WMAP and the Lyman-alpha forest,” Phys. Rev. D 71 (2005) 063534, arXiv:astro-ph/0501562.
  * [148] A. Hook and H. Murayama, “Low-energy Supersymmetry Breaking Without the Gravitino Problem,” Phys. Rev. D 92 (2015) no. 1, 015004, arXiv:1503.04880 [hep-ph].
  * [149] A. Hook, R. McGehee, and H. Murayama, “Cosmologically Viable Low-energy Supersymmetry Breaking,” Phys. Rev. D 98 (2018) no. 11, 115036, arXiv:1801.10160 [hep-ph].
  * [150] A. Brignole, F. Feruglio, and F. Zwirner, “On the effective interactions of a light gravitino with matter fermions,” JHEP 11 (1997) 001, arXiv:hep-th/9709111.
  * [151] A. Brignole, F. Feruglio, and F. Zwirner, “Signals of a superlight gravitino at $e^{+}e^{-}$ colliders when the other superparticles are heavy,” Nucl. Phys. B 516 (1998) 13–28, arXiv:hep-ph/9711516. [Erratum: Nucl.Phys.B 555, 653–655 (1999)].
  * [152] A. Brignole, F. Feruglio, M. L. Mangano, and F. Zwirner, “Signals of a superlight gravitino at hadron colliders when the other superparticles are heavy,” Nucl. Phys. B 526 (1998) 136–152, arXiv:hep-ph/9801329. [Erratum: Nucl.Phys.B 582, 759–761 (2000)].
  * [153] G. Panico and A. Wulzer, The Composite Nambu-Goldstone Higgs, vol. 913.  Springer, 2016.  arXiv:1506.01961 [hep-ph].
  * [154] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, “The Strongly-Interacting Light Higgs,” JHEP 06 (2007) 045, arXiv:hep-ph/0703164 [hep-ph].
  * [155] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, “Electroweak symmetry breaking from dimensional deconstruction,” Phys. Lett. B 513 (2001) 232–240, arXiv:hep-ph/0105239.
  * [156] N. Arkani-Hamed, A. G. Cohen, T. Gregoire, and J. G. Wacker, “Phenomenology of electroweak symmetry breaking from theory space,” JHEP 08 (2002) 020, arXiv:hep-ph/0202089.
  * [157] N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire, and J. G. Wacker, “The Minimal moose for a little Higgs,” JHEP 08 (2002) 021, arXiv:hep-ph/0206020.
  * [158] N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, “The Littlest Higgs,” JHEP 07 (2002) 034, arXiv:hep-ph/0206021.
  * [159] P. J. Fox, A. E. Nelson, and N. Weiner, “Dirac gaugino masses and supersoft supersymmetry breaking,” JHEP 08 (2002) 035, arXiv:hep-ph/0206096.
  * [160] G. Burdman, Z. Chacko, H.-S. Goh, and R. Harnik, “Folded supersymmetry and the LEP paradox,” JHEP 02 (2007) 009, arXiv:hep-ph/0609152.
  * [161] N. Craig, S. Knapen, and P. Longhi, “The Orbifold Higgs,” JHEP 03 (2015) 106, arXiv:1411.7393 [hep-ph].
  * [162] T. Cohen, N. Craig, G. F. Giudice, and M. Mccullough, “The Hyperbolic Higgs,” JHEP 05 (2018) 091, arXiv:1803.03647 [hep-ph].
  * [163] C. Cheung and P. Saraswat, “Mass Hierarchy and Vacuum Energy,” arXiv:1811.12390 [hep-ph].
  * [164] T. Cohen, N. Craig, S. Koren, M. Mccullough, and J. Tooby-Smith, “Supersoft Top Squarks,” Phys. Rev. Lett. 125 (2020) no. 15, 151801, arXiv:2002.12630 [hep-ph].
  * [165] N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, H. D. Kim, and D. Pinner, “Solving the Hierarchy Problem at Reheating with a Large Number of Degrees of Freedom,” Phys. Rev. Lett. 117 (2016) no. 25, 251801, arXiv:1607.06821 [hep-ph].
  * [166] A. Hook and G. Marques-Tavares, “Relaxation from particle production,” JHEP 12 (2016) 101, arXiv:1607.01786 [hep-ph].
  * [167] M. Geller, Y. Hochberg, and E. Kuflik, “Inflating to the Weak Scale,” Phys. Rev. Lett. 122 (2019) no. 19, 191802, arXiv:1809.07338 [hep-ph].
  * [168] C. Csáki, R. T. D’Agnolo, M. Geller, and A. Ismail, “Crunching Dilaton, Hidden Naturalness,” Phys. Rev. Lett. 126 (2021) 091801, arXiv:2007.14396 [hep-ph].
  * [169] M. J. Strassler and K. M. Zurek, “Echoes of a hidden valley at hadron colliders,” Phys. Lett. B 651 (2007) 374–379, arXiv:hep-ph/0604261.
  * [170] T. Han, Z. Si, K. M. Zurek, and M. J. Strassler, “Phenomenology of hidden valleys at hadron colliders,” JHEP 07 (2008) 008, arXiv:0712.2041 [hep-ph].
  * [171] 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 (2017) no. 6, 063102, arXiv:1612.09284 [hep-ph].
  * [172] S. Alipour-Fard, N. Craig, M. Jiang, and S. Koren, “Long Live the Higgs Factory: Higgs Decays to Long-Lived Particles at Future Lepton Colliders,” Chin. Phys. C 43 (2019) no. 5, 053101, arXiv:1812.05588 [hep-ph].
  * [173] Z. S. Wang and K. Wang, “Physics with far detectors at future lepton colliders,” Phys. Rev. D 101 (2020) no. 7, 075046, arXiv:1911.06576 [hep-ph].
  * [174] A. Katz, A. Mariotti, S. Pokorski, D. Redigolo, and R. Ziegler, “SUSY Meets Her Twin,” JHEP 01 (2017) 142, arXiv:1611.08615 [hep-ph].
  * [175] R. Contino, D. Greco, R. Mahbubani, R. Rattazzi, and R. Torre, “Precision Tests and Fine Tuning in Twin Higgs Models,” Phys. Rev. D96 (2017) no. 9, 095036, arXiv:1702.00797 [hep-ph].
  * [176] ACME Collaboration, V. Andreev et al., “Improved limit on the electric dipole moment of the electron,” Nature 562 (2018) no. 7727, 355–360.
  * [177] W. B. Cairncross and J. Ye, “Atoms and molecules in the search for time-reversal symmetry violation,” Nature Rev. Phys. 1 (2019) no. 8, 510–521.
  * [178] N. R. Hutzler, “Polyatomic Molecules as Quantum Sensors for Fundamental Physics,” Quantum Sci. Technol. 5 (2020) 044011, arXiv:2008.03398 [physics.atom-ph].
  * [179] W. Altmannshofer, R. Harnik, and J. Zupan, “Low Energy Probes of PeV Scale Sfermions,” JHEP 11 (2013) 202, arXiv:1308.3653 [hep-ph].
  * [180] D. McKeen, M. Pospelov, and A. Ritz, “Electric dipole moment signatures of PeV-scale superpartners,” Phys. Rev. D87 (2013) no. 11, 113002, arXiv:1303.1172 [hep-ph].
  * [181] N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice, and A. Romanino, “Aspects of split supersymmetry,” Nucl. Phys. B709 (2005) 3–46, arXiv:hep-ph/0409232 [hep-ph].
  * [182] G. F. Giudice and A. Romanino, “Electric dipole moments in split supersymmetry,” Phys. Lett. B634 (2006) 307–314, arXiv:hep-ph/0510197 [hep-ph].
  * [183] S. M. Barr and A. Zee, “Electric Dipole Moment of the Electron and of the Neutron,” Phys. Rev. Lett. 65 (1990) 21–24. [Erratum: Phys.Rev.Lett. 65, 2920 (1990)].
  * [184] C. Cesarotti, Q. Lu, Y. Nakai, A. Parikh, and M. Reece, “Interpreting the Electron EDM Constraint,” JHEP 05 (2019) 059, arXiv:1810.07736 [hep-ph].
  * [185] G. Panico, A. Pomarol, and M. Riembau, “EFT approach to the electron Electric Dipole Moment at the two-loop level,” JHEP 04 (2019) 090, arXiv:1810.09413 [hep-ph].
  * [186] G. Isidori, Y. Nir, and G. Perez, “Flavor Physics Constraints for Physics Beyond the Standard Model,” Ann. Rev. Nucl. Part. Sci. 60 (2010) 355, arXiv:1002.0900 [hep-ph].
  * [187] A. Baldini et al., “A submission to the 2020 update of the European Strategy for Particle Physics on behalf of the COMET, MEG, Mu2e and Mu3e collaborations,” arXiv:1812.06540 [hep-ex].
  * [188] K. Hayasaka et al., “Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Produced $\tau^{+}\tau^{-}$ Pairs,” Phys. Lett. B 687 (2010) 139–143, arXiv:1001.3221 [hep-ex].
  * [189] Belle-II Collaboration, W. Altmannshofer et al., “The Belle II Physics Book,” PTEP 2019 (2019) no. 12, 123C01, arXiv:1808.10567 [hep-ex]. [Erratum: PTEP 2020, 029201 (2020)].
  * [190] SINDRUM Collaboration, U. Bellgardt et al., “Search for the Decay $\mu^{+}\to e^{+}e^{+}e^{-}$,” Nucl. Phys. B 299 (1988) 1–6.
  * [191] B. Murakami and T. M. Tait, “Searching for lepton flavor violation at a future high energy $e^{+}e^{-}$ collider,” Phys. Rev. D 91 (2015) 015002, arXiv:1410.1485 [hep-ph].
  * [192] V. Cirigliano, B. Grinstein, G. Isidori, and M. B. Wise, “Minimal flavor violation in the lepton sector,” Nucl. Phys. B 728 (2005) 121–134, arXiv:hep-ph/0507001.
  * [193] S. A. R. Ellis and A. Pierce, “Impact of Future Lepton Flavor Violation Measurements in the Minimal Supersymmetric Standard Model,” Phys. Rev. D 94 (2016) no. 1, 015014, arXiv:1604.01419 [hep-ph].
  * [194] N. Arkani-Hamed, H.-C. Cheng, J. L. Feng, and L. J. Hall, “Probing lepton flavor violation at future colliders,” Phys. Rev. Lett. 77 (1996) 1937–1940, arXiv:hep-ph/9603431.
  * [195] MEG Collaboration, A. Baldini et al., “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C 76 (2016) no. 8, 434, arXiv:1605.05081 [hep-ex].
  * [196] MEG II Collaboration, A. Baldini et al., “The design of the MEG II experiment,” Eur. Phys. J. C 78 (2018) no. 5, 380, arXiv:1801.04688 [physics.ins-det].
  * [197] Mu2e Collaboration, L. Bartoszek et al., “Mu2e Technical Design Report,” arXiv:1501.05241 [physics.ins-det].
  * [198] Mu2e Collaboration, F. Abusalma et al., “Expression of Interest for Evolution of the Mu2e Experiment,” arXiv:1802.02599 [physics.ins-det].
  * [199] ECFA/DESY LC Physics Working Group Collaboration, J. Aguilar-Saavedra et al., “TESLA: The Superconducting electron positron linear collider with an integrated x-ray laser laboratory. Technical design report. Part 3. Physics at an e+ e- linear collider,” arXiv:hep-ph/0106315.
  * [200] A. Freitas, H.-U. Martyn, U. Nauenberg, and P. Zerwas, “Sleptons: Masses, mixings, couplings,” in International Conference on Linear Colliders (LCWS 04), pp. 939–946.  9, 2004.  arXiv:hep-ph/0409129.
  * [201] C. Caprini and D. G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,” Class. Quant. Grav. 35 (2018) no. 16, 163001, arXiv:1801.04268 [astro-ph.CO].
  * [202] N. Christensen, “Stochastic Gravitational Wave Backgrounds,” Rept. Prog. Phys. 82 (2019) no. 1, 016903, arXiv:1811.08797 [gr-qc].
  * [203] P. D. Meerburg et al., “Primordial Non-Gaussianity,” arXiv:1903.04409 [astro-ph.CO].
  * [204] M. Kamionkowski, A. Kosowsky, and M. S. Turner, “Gravitational radiation from first order phase transitions,” Phys. Rev. D 49 (1994) 2837–2851, arXiv:astro-ph/9310044.
  * [205] S. J. Huber and T. Konstandin, “Gravitational Wave Production by Collisions: More Bubbles,” JCAP 09 (2008) 022, arXiv:0806.1828 [hep-ph].