*mps*

# Review of Neutral Naturalness

Brian Batell  Pittsburgh Particle Physics, Astrophysics, and Cosmology Center,
Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, USA  Matthew Low  Pittsburgh Particle
Physics, Astrophysics, and Cosmology Center, Department of Physics and
Astronomy,
University of Pittsburgh, Pittsburgh, USA  Ethan T. Neil  Department of
Physics, University of Colorado, Boulder, Colorado 80309, USA  Christopher B.
Verhaaren  Department of Physics and Astronomy,
Brigham Young University, Provo, UT, 84602 USA

###### Abstract

The hierarchy between the mass of the Higgs boson and larger mass scales
becomes ever more puzzling as experiments push to higher energies. Neutral
naturalness is the umbrella term for symmetry-based explanations for these
hierarchies whose quark symmetry partners are not charged under the
$SU(3)_{c}$ color gauge group of the Standard Model. Though the first
manifestations of this idea predate the physics runs of the Large Hadron
Collider, since the Higgs discovery this paradigm has grown and developed to
include a wide variety of models, connections to intriguing collider signals,
various dark matter candidates, intersections with astrophysics and cosmology,
motivations for lattice studies, and ties to neutrinos and flavor. In this
review we orient the reader within this growing literature and reveal
interesting directions for further study.

###### pacs:

††preprint: PITT-PACC-2205

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

###### Contents

  1. I Executive Summary
  2. II Introduction
  3. III Models
    1. III.1 The Twin Higgs and its Variants
      1. III.1.1 UV Completions of the Twin Higgs
    2. III.2 Neutral Naturalness and SUSY
    3. III.3 Neutral Naturalness and pNGB Higgs
    4. III.4 Reoccurring Themes
  4. IV Collider Phenomenology
    1. IV.1 Higgs Physics
    2. IV.2 Other Portals to SM Neutral Sectors
    3. IV.3 New SM Charged States
    4. IV.4 Quirks
  5. V Connections to Lattice Gauge Theory
  6. VI Connections to Dark Matter
    1. VI.1 Mirror Twin Higgs Dark Matter
    2. VI.2 Fraternal Twin Higgs Dark Matter
    3. VI.3 Exotic Dark Matter
  7. VII Connections to Astrophysics and Cosmology
    1. VII.1 Cosmological Signals
    2. VII.2 Cosmological Tensions
    3. VII.3 Phase Transitions, Gravitational Waves, and Baryogenesis
    4. VII.4 Mirror Stars
  8. VIII Connections to Neutrinos
  9. IX Connections to Flavor

##  I Executive Summary

As experiments have reached ever higher energies without clear evidence of
deviations from the Standard Model (SM), the so-called hierarchy problem
between the electroweak scale and higher physics energy scales has become
increasingly acute. A particularly interesting explanation of Higgs
naturalness is positing a new symmetry which relates the SM quarks to
colorless partners, which has become known as neutral naturalness. These are
primarily bottom-up constructions that characterize the new partner particles
and interactions only up to the few TeV scale and require completion at high
energies. Typically, these completions include new colored states with masses
of order a few TeV, motivating future high-energy colliders to thoroughly test
Higgs naturalness due to new symmetries.

Realizations of neutral naturalness typically include a hidden, dark, or
secluded sector of particles that are related to at least some of the SM
fields by a discrete symmetry. Candidates for the cosmological dark matter,
either as a single particle species or from a collection of stable particles,
are often obvious consequences of this structure. This discrete symmetry also
relates the SM’s $SU(3)_{c}$ color group to a distinct hidden-sector $SU(3)$
gauge group. Consequently, neutral naturalness generically leads to rich dark
sectors, with phenomenological features such as dark showers, semivisible
jets, quirks, and long-lived particles. Useful non-perturbative inputs for
studying this strongly-coupled hidden $SU(3)$ sector can be obtained from
lattice gauge theory calculations of the spectrum of bound states, matrix
elements, and other observables. Because at least some of the hidden-sector
particles are essential to explaining the mass of the Higgs boson, the Higgs
is a robust link between the sectors. Signals of these connections can appear
in deviations from the SM Higgs couplings and exotic Higgs decay modes that
can be probed at future Higgs factories.

While the number of novel realizations of the idea of neutral naturalness
continues to grow, the majority of past work has focused on the twin Higgs
framework. This pertains to higher-energy (UV) completions, collider
phenomenology, dark matter candidates, cosmological and astrophysical signals,
and intersections with neutrino and flavor physics. That this narrower focus
has led to such a rich variety of signals and overlaps with other aspects of
beyond the SM physics is impressive, but also points the way to further
exploration through other neutral naturalness models. Such explorations may
lead to new connections to higher-scale physics that applies to the SM flavor
structure, the strong CP puzzle, or gauge unification. New dark matter
candidates, including those associated with UV completions as well as lighter,
non-WIMP, candidates may also be discovered.

Neutral naturalness connections across the range of new phenomena are
particularly fascinating due to the discrete symmetries that relate hidden-
sector particles to SM counterparts. This assortment of concrete models
accommodates a variety of new signals at the intensity frontier, energy
frontier, and cosmic frontier which are correlated with each other and,
through discrete symmetry, with the known properties of the SM. This makes
models of neutral naturalness compelling benchmarks for organizing simplified
model signals in addition to suggesting new strategies for discovering new
physics.

##  II Introduction

Despite its phenomenal experimental successes, the SM can only be considered
as a low-energy, effective field theory (EFT) description of microphysics. As
typically accepted, the SM does not specify how the neutrinos obtain mass nor
explain the observed asymmetry between matter and antimatter in the Universe.
The cosmological dark matter cannot be composed of SM fields and any structure
that explains the structure of quark and lepton flavor or the absence of
measurable CP violation in QCD must include physics beyond the SM. In addition
to these known unknowns, other particles and interactions may be essential to
Nature’s structure, even if we have not yet recognized what role they play.

Extensions of the SM are likely tied to an energy scale above which new
particles and interactions are manifest. Gravity is the most famous example,
whose detailed quantum interactions are tied to the Planck scale, which is of
order $10^{19}$ GeV. Within the SM the Higgs boson is unique: it is the only
elementary scalar field. As such, the generic EFT expectation is that the
vacuum expectation value (VEV) of the Higgs field (and hence the mass of the
Higgs boson) is set by the largest scale it interacts with, including the
Planck scale. Therefore, within the SM the hierarchy between the Planck scale
and the weak scale is mysterious. Indeed, concrete calculations of the Higgs
mass within SM extensions have borne out the EFT expectation.

The hierarchy between the Higgs mass and other scales may emerge from a
symmetry. If the Higgs is a pseudo-Nambu-Goldstone boson (pNGB) of an
approximate global symmetry, then its mass is controlled by the symmetry
breaking parameter. If nature manifests supersymmetry (SUSY), then the chiral
symmetry that protects fermion masses can be transferred to bosons like the
Higgs. In general, any symmetry that makes the Higgs insensitive to higher
mass scales is only as effective as the largest parameter that breaks the
symmetry. Therefore, to meaningfully address the hierarchy problem the top
quark, whose coupling to the Higgs is largest, must transform under the new
symmetry. The other particles that, along with the top quark, fill out a
representation of this symmetry are referred to as top partners.

The Large Hadron Collider (LHC) has provided a wealth of data with which the
ATLAS and CMS collaborations have sought for various types of top partners. As
a hadron machine, the LHC is well equipped to discover the new colored
particles predicted by many symmetry-based solutions to the hierarchy problem.
The LHC sensitivity to such partners varies from model to model, but in
general colored top partners have been excluded at energies below a couple of
TeV. While the gap between the top-partner bounds and the Higgs mass is small
compared to its separation from the Planck scale, it is still large enough
that these models must be tuned somewhat to produce the correct Higgs mass.
This is sometimes referred to as the little hierarchy problem.

Beginning in 2005 Chacko  _et al._ (2006a), symmetries were found that explain
these little hierarchies up to a few TeV _without_ colored top partners.
Direct production of colorless top partners at a hadron collider is typically
challenging, so the limits on such particles are weaker. Because the color-
neutral top-partners masses can be lower, the tuning is reduced. In other
words, the models are more natural. Consequently, experimental probes of Higgs
naturalness require even higher-energy colliders Curtin and Saraswat (2016).

All known realizations of (color) neutral naturalness feature a discrete
symmetry that relates SM fields to colorless counterparts. Though these
partner fields are neutral under SM color, the quark partners are charged
under an $SU(3)$ gauge group that is related to SM color by the discrete
symmetry. Consequently, neutral naturalness motivates certain classes of rich
hidden sectors, with a variety of particles and forces. What is more, the rich
structure of these hidden sectors is related (at least in part) to the SM by
the discrete symmetry. This leads to correlations between precision SM
measurements and dark sector structure.

The Higgs always links the SM to these new sectors. The colorless partners
must couple to the Higgs with SM field-like strength in order to protect the
Higgs mass. This ensures the Higgs sector acts as a portal between the SM and
the new sector. However, the manifestations of this connection can vary from
one concrete model to another, as do other phenomenological consequences. The
purpose of our review is to map this rich diversity so that new researchers
can more easily investigate and extend them.

This review is organized by topic so that readers may easily navigate to the
aspects of neutral naturalness they are most interested in. This leads to some
redundancy, as many works are relevant to multiple sections. We begin, in Sec.
III, with an overview of the various specific realizations of neutral
naturalness. The twin Higgs scenario is emphasized due to the growing body of
modifications that have been made to its basic structure as well as the
several UV completions that have been suggested. We then broadly classify
models as being related to SUSY or to pNGB Higgs constructions.

The following section, IV, considers the various collider signals that have
been associated with these models. In this section the organization is by
collider signature rather than by model. We then, in Sec. V, discuss results
from lattice gauge theory that have been important to understanding the
phenomenology of these models. As yet unknown quantities that can improve the
predictive power of phenomenological studies are also mentioned.

The rich dark sectors of neutral naturalness are connected to the cosmological
dark matter in Sec. VI. Here again the twin Higgs scenario has been most
thoroughly investigated, though the possibilities related to other frameworks
are intriguing. Other connections to astrophysics and cosmology, including
gravitational waves, baryogenesis, and the Hubble tension, are discussed in
Sec. VII. Intersections with neutrinos and flavor physics are outlined in
Secs. VIII and IX, respectively. These relationships are less explored, but
offer interesting correlations between precision SM measurements and hidden-
sector observables.

##  III Models

The broad ideas of neutral naturalness have been realized in several specific
ways. Many of them center on the twin Higgs framework. However, the other
familiar symmetry-based solutions to the hierarchy problem, supersymmetry and
pNGB Higgs models, also have many neutral natural realizations.

###  III.1 The Twin Higgs and its Variants

Neutral naturalness began with the mirror twin Higgs Chacko  _et al._ (2006a)
model, which also exemplifies many of the characteristics that are common to
most constructions.111There are twin Higgs models that do not exhibit neutral
naturalness; the top partners carry color Chacko  _et al._ (2006b); Goh and
Krenke (2007); Batra and Chacko (2009); Yu (2017). The same is true of other
models that are similar in construction to many neutral naturalness models
Kats  _et al._ (2017). To the SM with its $\left[SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\right]_{A}$ gauge structure is added a twin sector
with identical structure, that is, a gauge structure of $\left[SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\right]_{B}$. The twin character of this sector is
manifest in a discrete $Z_{2}$ symmetry that relates the couplings of the two
sectors. In addition to this mirror set-up the Higgs sectors of the two copies
are assumed to satisfy an approximate global symmetry. For instance, one can
arrange the visible and twin sector Higgs doublets, $H_{A}$ and $H_{B}$
(respectively), in terms of a four-plet of a global $SU(4)$

$\mathcal{H}=\left(\begin{array}[]{c}H_{A}\\\ H_{B}\end{array}\right)~{}.$ (1)

If this approximate symmetry is spontaneously broken down to $SU(3)$ at some
scale $f$ then seven pNGBs are produced.222We have used the $SU(4)\to SU(3)$
breaking pattern for simplicity. However, it is preferable in strongly-coupled
UV completions (see Sec. III.3) to use $SO(8)\to SO(7)$ (which also produces 7
pNGBs) as this preserves the custodial $SU(2)$ symmetry. Because of the
$SU(2)_{L}\times U(1)_{Y}$ gauging in each sector, six of the pNGBs are eaten
by electroweak gauge fields. This leaves one physical pNGB that is associated
with the observed Higgs boson. As a pNGB, the Higgs mass is controlled by the
approximate symmetry of the potential.

The SM fermions are not extended into representations of $SU(4)$, so one might
worry that Yukawa interactions $\lambda_{f}\overline{f}fH$ would break the
symmetry and produce large one-loop corrections to the scalar potential.
However, the discrete $Z_{2}$ symmetry that relates the SM fermions to the
twin fermions ensures that the quadratically divergent contributions to the
Higgs mass from the SM and twin fermions cancel. In effect, the SM and twin
fermions interact with the scalar sector as part of accidental $SU(4)$
multiplets.

This may be understood schematically as follows. The dominant one-loop
contributions to the $\left|H_{A}\right|^{2}$ and $\left|H_{B}\right|^{2}$
operators are from the visible-sector and twin-sector top quarks, $t_{A}$ and
$t_{B}$, respectively. These contributions are

$\frac{3\lambda_{t_{A}}^{2}}{8\pi^{2}}\Lambda^{2}_{\text{UV}}\left|H_{A}\right|^{2}~{},\
\ \ \text{and}\ \ \ \
\frac{3\lambda_{t_{B}}^{2}}{8\pi^{2}}\Lambda^{2}_{\text{UV}}\left|H_{B}\right|^{2}~{},$
(2)

where $\Lambda_{\text{UV}}$ is the UV cutoff of the effective theory,
typically taken to be a few TeV. The $Z_{2}$ symmetry between the sectors
ensures that $\lambda_{t_{A}}=\lambda_{t_{B}}\equiv\lambda_{t}$. This means
that the sum of these two contributions is

$\frac{3\lambda_{t}^{2}}{8\pi^{2}}\Lambda_{\text{UV}}^{2}\left(\left|H_{A}\right|^{2}+\left|H_{B}\right|^{2}\right)=\frac{3\lambda_{t}^{2}}{8\pi^{2}}\Lambda_{\text{UV}}^{2}\left|\mathcal{H}\right|^{2}~{}.$
(3)

This result is $SU(4)$ symmetric and, therefore, cannot contribute to the mass
of any pNGB of the broken $SU(4)$ symmetry. It is important to note that this
one-loop effect does not prevent the NGBs from getting a mass. The $SU(2)_{L}$
gauge couplings and the Yukawa coupling to fermions break the global $SU(4)$
and generate a $\delta(|H_{A}|^{4}+|H_{B}|^{4})$ potential term. This gives
the Higgs a $m_{h}\sim f\sqrt{\delta}$ mass where
$\delta\sim\ln\Lambda_{\text{UV}}/f$ depends logarithmically on the cutoff.

In general, we can write the scalar potential as Barbieri  _et al._ (2005)

$\displaystyle V_{H}=$
$\displaystyle-\mu^{2}\left|\mathcal{H}\right|^{2}+\lambda\left|\mathcal{H}\right|^{4}$
$\displaystyle+m^{2}\left(\left|H_{A}\right|^{2}-\left|H_{B}\right|^{2}\right)+\delta\left(\left|H_{A}\right|^{4}+\left|H_{B}\right|^{4}\right)~{},$
(4)

with the first line preserving the global $SU(4)$, while the second line does
not. For the Higgs to be a pNGB the parameters of the second line must be
small compared to those of the first. Small tree-level couplings are stable at
one-loop due to the discrete twin symmetry. One finds that contributions to
the symmetry breaking potential terms are at most logarithmically sensitive to
$\Lambda_{\text{UV}}$. This scenario can be extended to include not just a
single twin sector, but many copies of the SM Foot and Volkas (2007).

When the scalar potential in Eq. (4) has $\delta>0$ and $m=0$ Barbieri  _et
al._ (2005) both $H_{A}$ and $H_{B}$ obtain equal VEVs and, consequently,
equal couplings to SM and twin fields. These circumstances produce couplings
of the observed Higgs boson to SM fields that are only half their SM value,
contrary to LHC measurements. However, when $m^{2}\neq 0$ the twin symmetry is
softly broken, allowing $\langle H_{B}\rangle>\langle H_{A}\rangle$. The
resulting Higgs couplings are closer to the SM predictions while preserving
the one-loop protection of the Higgs mass. Choosing $m^{2}$ to give the
correct Higgs VEV does constitute a tuning of the model, but typically a mild
one, less tuned than 10%. This tuning can also be reduced in various ways,
such as by inducing electroweak symmetry breaking through the tadpole of an
auxiliary sector Harnik  _et al._ (2017).

The logarithmic sensitivity in the Higgs potential parameters is now augmented
by the $Z_{2}$ breaking. This can affect the twin Higgs protection at the two-
loop level if, for instance, the SM and twin top Yukawa couplings run
differently. Consequently, constructions like the twin Higgs typically only
address the “little” hierarchy problem. They provide a significant improvement
in fine-tuning, but must be completed by some other structure above the few
TeV scale.

These gains in naturalness withstand significant variations from the mirror
twin set-up. The so-called fraternal model Craig  _et al._ (2015a) makes
$Z_{2}$ breaking a main feature, only twinning the SM’s third generation and
allowing for modest deviations from the twin equality in some couplings. In
effect, the model keeps only the minimal ingredients required for a twin-
Higgs-like protection of the Higgs mass. In a similar vein, if the twin top
quark is taken to be vector-like then even the third generation leptons can be
removed without generating anomalies Craig  _et al._ (2016a). Rather than
removing entire generations from the mirror twin model, simply allowing
significant $Z_{2}$ breaking between the fermion Yukawa couplings (other than
the top quark) is enough to eliminate some cosmological concerns Barbieri  _et
al._ (2016). In weakly-coupled UV completions, a hard breaking of the $Z_{2}$
in the quartic terms of the scalar potential can further reduce the need for
fine-tuning Katz  _et al._ (2017).

Other efforts seek to completely preserve the discrete symmetry or explain how
it breaks. In Csáki  _et al._ (2020) the twin symmetry is preserved by much of
the model (additional top-partner multiplets do play a role) leading to
electroweak symmetry breaking with the correct hierarchy of VEVs, but without
$Z_{2}$ breaking in the gauge and top quark sectors of the model. Vector-like
leptons can also be introduced in each sector to produce a radiative Higgs
potential that spontaneously breaks the $Z_{2}$ Jung (2019). Spontaneous
breaking can also be induced by simply including a new scalar field in each
sector whose potential is similar to Eq. (4) but with $\delta<0$. This leads
to a completely $Z_{2}$ breaking vacuum with the VEV only in one sector. Such
scalars can have other nonzero quantum numbers like hypercharge Batell and
Verhaaren (2019); Liu and Weiner (2019) or color Liu and Weiner (2019); Batell
_et al._ (2020) which lead to a variety of twin sectors by breaking twin gauge
symmetries or giving additional masses to twin fermions.

Two Higgs doublet extensions of the SM have also been combined with the twin
Higgs idea Chacko  _et al._ (2006c, 2019). This larger scalar sector can be
used to explain the origin of $Z_{2}$ breaking. This includes spontaneously
breaking the twin $Z_{2}$ symmetry Beauchesne  _et al._ (2016), radiative
breaking Yu (2016a), and tadpole or quartic induced breaking Yu (2016b). Other
models address neutrinos masses in conjunction with a spontaneous breaking of
the discrete symmetry Liu and Weiner (2019); Feng and Yu (2020). The
additional Higgs doublet can be combined with a flavorful Yukawa structure to
address the SM flavor puzzle Altmannshofer and Maddock (2020). The additional
doublet can also make mirror neutrons an asymmetric dark matter candidate
Beauchesne (2020).

####  III.1.1 UV Completions of the Twin Higgs

As mentioned above, the twin Higgs mechanism only addresses the little
hierarchy problem. It is therefore important to construct realistic UV
completions of the twin Higgs that address the big hierarchy problem.
Additionally, one may then consider higher energies properties such as may be
related to the SM flavor structure or other puzzles. Weakly-coupled UV
completions are typically supersymmetric. These completions can be constructed
to eliminate the fine-tuning of supersymmetric quartic couplings Falkowski
_et al._ (2006); Chang  _et al._ (2007), but simpler constructions Craig and
Howe (2014) can retain small fine-tuning and preserve other virtues of SUSY
models such as gauge coupling unification.

Within these supersymmetric UV completions other aspects of twin Higgs models
can be explored unambiguously. In particular, various ways to further reduce
fine-tuning have been found, including a hard breaking of the twin symmetry
Katz  _et al._ (2017) and new $D$-term contributions to the $SU(4)$ symmetric
quartic Badziak and Harigaya (2017a, b, 2018). A “turtle” construction is a UV
extension of the twin Higgs, which can also precede a SUSY completion Asadi
_et al._ (2019), again reducing fine-tuning. Supersymmetric completions of the
twin Higgs can also be extended to solve the strong CP problem Albaid  _et
al._ (2015).

There are also several strongly-coupled UV completions of the twin Higgs
framework. These include composite Higgs Low  _et al._ (2015); Barbieri  _et
al._ (2015), holographic constructions Geller and Telem (2015), and universal
extra dimensions Burdman and Dias (2007). The EFT form factors of composite
twin Higgs models are determined by the discrete symmetry, differing from
other scenarios. This framework can also accommodate anarchic flavor with
significantly less fine-tuning Csaki  _et al._ (2016). Precision electroweak
constraints primarily apply to the Higgs quartic and are satisfied with only
moderate tuning Contino  _et al._ (2017).

Clearly, the twin Higgs scenario can be incorporated within various high scale
designs. In some cases a particular completion need not be selected, while for
others a specific high-energy construction is essential. Interestingly, a
renormalization group improved analysis of the twin Higgs effective potential
shows that the Higgs mass can be a largely UV independent prediction Greco and
Mimouni (2016).

###  III.2 Neutral Naturalness and SUSY

In contrast to the supersymmetric UV completions of the twin Higgs discussed
above, SUSY can also be the fundamental symmetry that protects the Higgs mass,
although it manifests in less familiar ways. These models share many
characteristics with folded SUSY Burdman  _et al._ (2007), so we outline its
broad aspects in some detail.

The essential idea behind folded SUSY is quite simple. Very schematically,
SUSY connects bosonic and fermionic degrees of freedom. Roughly, we can
consider that the top quark is extended to a SUSY multiplet

$t\Rightarrow\left.\left(\begin{array}[]{c}t\\\
\widetilde{t}\end{array}\right)\right\downarrow\text{SUSY}~{},$ (5)

where $\widetilde{t}$ denotes the scalar stop. This scalar partner to the top
quark is a manifestation of the SUSY structure in a given model. This symmetry
can also explain why the Higgs mass is insensitive to higher mass scales.
Having two copies of these particles would also preserve Higgs naturalness;
the cancellation between fermonic and bosonic contributions to the Higgs mass
renormalization would simply happen twice in exactly the same way.

The authors of folded SUSY observed a generalization of this doubling idea,
where each copy is charged under a different color group. If we label SM color
as $SU(3)_{A}$ then we can label the copy (or folded color) as $SU(3)_{B}$ and
write the top sector particles as

$\overset{\text{SU}(3)_{A}\times\text{SU}(3)_{B}\hskip
14.22636pt}{\left.\overrightarrow{\left(\begin{array}[]{cc}t_{A}&t_{B}\\\
\widetilde{t}_{A}&\widetilde{t}_{B}\end{array}\right)}\hskip
9.95845pt\right\downarrow}\text{SUSY}~{}.$ (6)

This extension keeps the Higgs natural through the SUSY structure. In this
construction the $A$ and $B$ sector fields are assumed to be related by a
discrete $Z_{2}$ exchange symmetry to ensure the coupling structure is the
same in each sector.

But what if the SM colored stop $\widetilde{t}_{A}$ and the folded top quark
$t_{B}$ were removed from the low-energy theory? Then the Higgs remains
natural because the contributions from the SM top and the folded stop combine
in precisely the same way as in standard SUSY, the fact that they are charged
under different color groups is irrelevant. In this scenario the folded stops
are much more difficult to produce at hadron colliders, as they do not carry
SM color. Note that in this case no true SUSY is manifest in the low-energy
theory as the fermions and bosons belong to different SUSY multiplets. The
protection of the Higgs mass results from an accidental SUSY realized by the
low-energy theory.

In folded SUSY the unwanted fields are removed using an additional spatial
dimension. The extra dimension is taken to be a finite, flat interval between
two 4D branes. It is also compactified on $S_{1}/Z_{2}$, making it an
orbifold, from which folded SUSY takes its name. The 5D superfields with
$\mathcal{N}=1$ SUSY can be usefully viewed as $\mathcal{N}=2$ SUSY in 4D. The
$\mathcal{N}=2$ SUSY is broken to $\mathcal{N}=1$ on each brane by the
boundary conditions each bulk field has there. In the original folded-SUSY
model the boundary conditions of the $A$ sector colored scalars and $B$ sector
colored fermions are chosen such that their Kaluza-Klein tower has no zero-
mode. This effectively eliminates them from the low-energy theory below the
Kaluza-Klein scale. The gauginos are also projected out of the low-energy
spectrum.

The $\mathcal{N}=1$ SUSY preserved by the field boundary conditions on each
brane can be _different_. This means that in the low-energy theory SUSY is
completely broken by the Scherk-Schwarz Scherk and Schwarz (1979a, b)
mechanism. Crucially, the complete breaking of SUSY by the mismatch of the
$\mathcal{N}=1$ SUSYs on the two branes is nonlocal. Consequently, corrections
to the brane localized Higgs potential are finite and calculable. Thus, the
simple concept of a SUSY-like protection of the Higgs mass through stops that
are neutral to SM color is concretely realized.

Similar to the twin Higgs framework, extra dimensional constructions come with
a UV cutoff that is typically not too far above the TeV scale. As shown in
Craig and Lou (2014), however, the extra dimension can be deconstructed while
preserving the natural virtues of folded SUSY. An estimate of the two-loop
calculation of the folded-SUSY Higgs potential indicates that it may not
produce the correct vacuum without modification. A simple change is to vary
the Scherk-Schwarz twist away from its maximum value Cohen  _et al._ (2016).
This can ensure electroweak symmetry breaking occurs at one-loop. The tuning
in folded SUSY can also be improved by making the gauginos bifundementals of
the two $A$ and $B$ color groups Gherghetta  _et al._ (2016).

The squark fields of folded SUSY and its variations are color neutral, but
carry SM electroweak charges. Two frameworks have been discovered that have
scalar top partners that are complete SM gauge singlets. The tripled-top
scenario Cheng  _et al._ (2018a, 2019) demonstrates that the folded-SUSY-like
cancellation can be accomplished in 4D using _two_ copies of the top sector.
In this construction the right-handed stops of each copy act as top partners,
so they are not charged under $SU(2)_{L}$. The hypercharge of the top-sector
copies may be freely specified, including the choice of scalar top partners
with no hypercharge, which are SM gauge singlets. The Hyperbolic Higgs model
Cohen  _et al._ (2018) obtains SM singlet scalar top partners from a 5D set-
up. In contrast to the approximate $U(4)$ symmetry of the scalar potential of
twin Higgs models, this scalar potential enjoys an approximate hyperbolic
symmetry of $U(2,2)$. The kinetic terms of the scalar fields are not invariant
under $U(2,2)$, but the form of the scalar potential is sufficient for SM
gauge singlet scalars to cancel the divergent loop contributions of SM quarks.

###  III.3 Neutral Naturalness and pNGB Higgs

Neutral naturalness began as a pNGB construction. The twin Higgs scenario
posits that the observed Higgs boson is the pNGB of an approximate $SU(4)$
symmetry. However, the neutral natural pNGB possibilities include a rich
variety of realizations. There is a class of theories that may be seen as the
direct generalization of the twin Higgs, called orbifold Higgs models Craig
_et al._ (2015b, c). From this perspective the twin Higgs gauge structure,
$SU(3)_{A}\times SU(2)_{A}\times SU(3)_{B}\times SU(2)_{B}\times Z_{2}$,
results from the orbifold $\left[SU(6)\times SU(4)\right]/Z_{2}$. This
provides an origin for the discrete symmetry that relates the gauge and Yukawa
couplings in the two sectors which is essential to the one-loop protection of
the Higgs. What is more, this orbifolding method can be used with other
symmetry groups to produce different numbers and varieties of hidden sectors
whose SM singlet top partners all cooperate to keep the Higgs natural.

There are other pNGB Higgs models that resemble folded SUSY in their set-up.
The usual pNGB Higgs scenario assumes an approximate global symmetry is
spontaneously broken at some scale $f$ and the Higgs is one of the resulting
pNGBs. The top quark is typically taken to belong to a representation of the
larger global symmetry. The new fields required to fill out the multiplet are
the top partners.

For an $SU(3)$ global symmetry we have the schematic form

$\left(\begin{array}[]{c}t\\\
b\end{array}\right)=q_{L}\Rightarrow\left.\left(\begin{array}[]{c}q_{L}\\\
T\end{array}\right)\right\downarrow SU(3)~{},$ (7)

where $T$ is the top-partner field. The protection of the Higgs mass works
just as well when it happens twice, including new fields charged under a new
color group

$\overset{\text{SU}(3)_{A}\times\text{SU}(3)_{B}\hskip
14.22636pt}{\left.\overrightarrow{\left(\begin{array}[]{cc}q_{L_{A}}&q_{L_{B}}\\\
T_{A}&T_{B}\end{array}\right)}\hskip 9.95845pt\right\downarrow}SU(3)~{}.$ (8)

The one-loop protection persists even if the $T_{A}$ and $q_{L_{B}}$ fields
are removed from the low-energy theory. In this case the top partner $T_{B}$
is not charged under SM color, though it would carry SM hypercharge. This is
the essence of the Quirky Little Higgs model Cai  _et al._ (2009), where the
unwanted fields are removed by the brane boundary conditions in an extra
dimensional space, similar to folded SUSY. This symmetry breaking pattern also
plays a role in some Dark Top constructions Poland and Thaler (2008), though
differences allow the top partner to be a stable dark matter candidate. The
neutral top partners can also be right-handed neutrinos Batell and McCullough
(2015) for various symmetry breaking patterns.

While the $SU(3)/SU(2)$ construction is in some sense minimal, it does not
provide the Higgs with a custodial $SU(2)$ symmetry that is highly motivated
by precision electroweak measurements. Orthogonal groups can ensure the
custodial form of Higgs structure. The minimal realization is $SO(5)/SO(4)$ Xu
_et al._ (2020, 2019) where boundary conditions in a warped extra dimension
are employed to lift unwanted fields from the low-energy spectrum.

While the $SO(5)/SO(4)$ symmetry breaking pattern preserves the custodial
symmetry, it differs from the twin Higgs construction in how the top partners
relate to the top quark. In the twin Higgs set-up, which can be thought of as
$SO(8)/SO(7)$, the top and its partner can be related by a discrete symmetry
because they are embedded within the $SO(8)$ in similar ways. This cannot be
done when the symmetry breaking pattern is $SO(5)/SO(4)$. The smallest
orthogonal group structure that allows the top quark and its partner to be
related by a twin parity is $SO(6)/SO(5)$ Serra and Torre (2018); Csáki  _et
al._ (2018), which has also been extended into the UV holographically Dillon
(2019).

Though it allows for twin parity, the $SO(6)/SO(5)$ symmetry breaking patter
also produces another pNGB which must be removed from the low-energy theory.
Another construction which allows twin parity, leads to a custodial symmetry,
and an embedding of both SM and twin $SU(2)_{L}$ is $SO(7)/G_{2}$ Serra  _et
al._ (2019). When $SO(7)$ is spontaneously broken to $G_{2}$ seven pNGBs are
produced, but six are eaten by the SM and twin $SU(2)_{L}$ gauge bosons,
leaving behind a single physical Higgs, just as in the twin Higgs. The
breaking pattern of $SO(7)/SO(6)$ leads to custodial symmetry for the pNGB
Higgs while the additional pNGBs become a viable complex scalar dark matter
candidate Ahmed  _et al._ (2020a).

Models employing a $SO(2N)/\left[SO(N)\times SO(N)\right]$ breaking pattern
have also been investigated Durieux  _et al._ (2022). These models naturally
produce a hierarchy between the Higgs VEV $v$ and the global symmetry breaking
scale $f$. In most pNGB models such a hierarchy is required to agree with
Higgs data, but is a source of tuning. These so-called Twin Gegenbaur
constructions agree with current Higgs measurements without additional tuning.

###  III.4 Reoccurring Themes

A few characteristics are common to most concrete manifestations of neutral
naturalness. First, the existence of multiple sectors related by a discrete
symmetry. In many cases, but not all, the number of sectors is two and the
symmetry is $Z_{2}$. The sectors may differ from each other in significant
ways, but aspects of the third generation, more specifically the top quark
sector, must have nearly identical couplings to the Higgs. Thus, the Higgs is
an essential portal between the sectors; there may be others depending on the
model.

The top partners, and often other fields, in the hidden sector are charged
under a QCD-like gauge group which is related to SM QCD by the discrete
symmetry or a larger symmetry structure. However, the spectrum of particles
can be quite different in the hidden sector, often with fewer light states.
This can produce a hidden confinement scale that is significantly higher than
the SM equivalent and a different spectrum of bound states.

Most of the models discussed are only effective theories up to the scale of a
few TeV. Several UV completions have been discussed for twin Higgs models.
Specific composite Higgs completions of pNGB Higgs constructions have also
been developed. In other cases definite UV completions have not been
specified. However, new colored states (and a variety of other signals) are
generically expected at the few TeV scale.

Clearly, neutral naturalness models motivate rich hidden sectors and, as
discussed in Sec. VI, provide many possible dark matter candidates. The
discrete symmetry typically relates the dark matter to known SM fields and
couplings. Consequently, the dark matter signals can be connected to the
naturalness of the Higgs.

Beyond dark matter, these hidden sectors present a structure at least as rich
and varied as the SM. While the hidden sectors are neutral under SM color,
they may share other gauge interactions. Therefore, searches for new
electroweak states are highly motivated by these constructions.

##  IV Collider Phenomenology

The primary characteristic of neutral naturalness is that there are no new
colored states up to the few TeV scale. Consequently, some of the most
powerful hadron collider searches for new states do not apply. However, there
are many other signatures of neutral naturalness that may appear at present
day and future colliders. Indeed, it has been argued Curtin and Saraswat
(2016) that the combination of current machines along with future lepton and
hadron colliders will be able to thoroughly probe symmetry-based solutions to
the hierarchy problem.

###  IV.1 Higgs Physics

The most robust prediction of neutral naturalness is the existence of new
states that couple to the Higgs. Consequently, examining the Higgs for
indications of these structures is highly motivated. In pNGB Higgs models the
tree-level couplings of the Higgs to SM fields are modified, which already
constrains such theories Burdman  _et al._ (2015); Thrasher (2017). The
deviations from SM couplings can also have significant effects on, for
instance, rare top quark decays Bie  _et al._ (2021). In supersymmetric
theories, or general bottom-up models with scalar top partners, the tree-level
Higgs couplings may be unaffected. However, loop-level couplings, such as
$h\gamma\gamma$ Fan and Reece (2014) and $hZ\gamma$ Archer-Smith  _et al._
(2021) (or even a loop-level contribution to $hZZ$ Craig  _et al._ (2013)),
can be modified by the contribution of the new particles to the loop
calculations.

The Higgs often acquires new, or exotic, decay modes. For example, in the
mirror twin Higgs model the Higgs couples to all twin quarks and leptons. The
strengths of these couplings and the mass of the twin fermions are both
controlled by the ratio of the SM Higgs VEV to the twin Higgs VEV, $v/f$. This
quantity is also directly proportional to the tuning of the model, making both
Higgs coupling deviations and the invisible Higgs width direct probes of
naturalness. Even when these states are to heavy for the an on-shell Higgs to
decay into them, they can be produced through an off-shell Higgs Craig  _et
al._ (2016b); Goncalves  _et al._ (2018); Gonçalves  _et al._ (2018).

Most models require that some of the hidden-sector particles are charged under
a hidden $SU(3)$ gauge group. In many scenarios the fields charged under
hidden color are much heavier than the hidden confinement scale, implying that
the lowest lying hidden hadrons are glueballs Craig  _et al._ (2015a). The
lightest glueball has the right quantum numbers to mix with the Higgs,
$0^{++}$, and does so through multi-loop processes Juknevich  _et al._ (2009);
Juknevich (2010). This can give the Higgs a small coupling to this glueball
state, which can also decay, after traversing a macroscopic distance, back
into SM fields. Despite the uncertainties of glueball hadronization Curtin
_et al._ (2022), these exotic displaced decays of the Higgs Curtin and
Verhaaren (2015); Csaki  _et al._ (2015) can be powerful probes of neutral
natural frameworks.

There can be other scalars, besides the observed 125 GeV Higgs, in the Higgs
sector, such as the twin Higgs of twin Higgs models or the heavy Higgses of
SUSY models. These scalars often have couplings to both SM and hidden-sector
fields, making them an interesting portal between the sectors. These motivate
specific heavy scalar targets at current and future colliders Buttazzo  _et
al._ (2015); Ahmed (2018); Buttazzo  _et al._ (2018) and can be used in
conjunction with other collider signals to prove such scalars belong to a
neutral naturalness scenario Chacko  _et al._ (2018a). This may be
accomplished through prompt heavy Higgs signals, and often with powerful
complementarity through displaced heavy Higgs decays Kilic  _et al._ (2019);
Alipour-Fard  _et al._ (2020).

###  IV.2 Other Portals to SM Neutral Sectors

While the Higgs is a robust portal to the hidden sector, there are several
other interesting ways that the sectors may be linked. The most obvious is by
SM gauge bosons, if the hidden sectors include states with SM gauge charges.
These scenarios are considered in the following section, IV.3.

In the case of SM neutral hidden sectors, like in the twin Higgs, the visible
and hidden sectors can still be joined through several possible portals. These
analyses have, so far, focused on the twin Higgs set-up, so the term twin
sector is appropriate. When twin hypercharge is gauged then there may be a
kinetic mixing with the SM hypercharge gauge field. Within the low-energy
theory such mixing requires at least four-loops Chacko  _et al._ (2006a);
Koren and McGehee (2020), which can be consistent with unbroken twin
hypercharge Davidson  _et al._ (2000); Vogel and Redondo (2014). However, for
kinetic mixing large enough to be relevant at colliders the twin hypercharge
boson must have a mass. In this scenario both the twin photon and twin $Z$ can
be discovered at current or future hadron colliders Chacko  _et al._ (2019).

The Higgs and hypercharge portals rely on fields with discrete symmetry
partners, the twin Higgs or twin hypercharge bosons. However, if there are
fields that the discrete symmetries take back to themselves, these can also
serve as portals between the sectors. Such singletons—completely neutral
fields with no $Z_{2}$ partners—may be scalar, fermions, or vectors and can
produce distinct collider signals Bishara and Verhaaren (2019); Liu and Weiner
(2019). For example, strongly-coupled UV completions of the twin Higgs may
include a spontaneously broken scaling symmetry, producing a light dilaton
which connects the visible and twin sectors Ahmed  _et al._ (2020b).

###  IV.3 New SM Charged States

Though the defining characteristic of neutral natural models is colorless top
partners, specific realizations often include new states charged under the SM
gauge groups. For instance, non-SUSY UV completions of the twin Higgs predict
exotic quarks charged under SM color in the few to tens of TeV mass range,
which can be thoroughly probed at the LHC and future colliders Cheng  _et al._
(2016). Other TeV scale states are SM color neutral but carry electroweak
charge. If the discrete twin symmetry is preserved in the UV then these states
may be out of reach of the LHC, but can be probed by future machines Cheng
_et al._ (2017). It may also be that the new electroweak charged states are
significantly lighter the colored states. In this case the LHC can produce
them at a meaningful rate. The twin color force draws the produced particles
into bound states whose resonant decays can be discovered at the LHC Cheng
_et al._ (2017); Li  _et al._ (2018).

Supersymmetric models can also lead to novel electroweak states. Folded-SUSY
sleptons Burdman and D’Agnolo (2015) carry electroweak charges and can be
discovered effectively at the LHC. In tripled-top constructions the $Z$ portal
can be the dominant connection to hidden-sector states Cheng  _et al._ (2019,
2022) with electroweak charges. These states are often bound by the hidden
color force, which in some cases leads to “quirky” dynamics, which we focus on
in the following section.

###  IV.4 Quirks

Neutral natural models typically include a confining gauge group in the hidden
sector. While the mirror twin Higgs model includes states at or below the
hidden confining scale, this appears to be more the exception than the rule.
In all frameworks with electroweak charged partner particles the bounds on new
states are at the 100 GeV scale or above Egana-Ugrinovic  _et al._ (2018).
However, the hidden strong coupling is related by a discrete symmetry to the
SM strong coupling in the TeV range. This typically leads to a hidden
confinement scale in the neighborhood of a few GeV scale. Models like these,
with no particles charged under the confining gauge group at or below the
confinement scale, are said to exhibit quirky Kang and Luty (2009) dynamics.

Without light quarks, which can be pair produced to fragment tubes of
confining color flux, states much heavier than the confinement scale evolve as
though connected by a string of constant tension. These quirky bound states
must shed energy until they reach low angular momentum configurations, after
which they can decay efficiently. In the case of folded SUSY the folded squark
bound states dominantly decay to a $W\gamma$ final state Burdman  _et al._
(2008), also updated to LHC resonance searches at 8 TeV Burdman  _et al._
(2015). These and other SM vector decay modes can be used to detect pNGB Higgs
models as well, such as $SO(7)/SO(6)$ Ahmed  _et al._ (2020a). The SM exhibits
a cancellation in $W\gamma$ production which can be used to strengthen these
searches Capdevilla  _et al._ (2020). While the final decay product is the
most conspicuous signal of these quirky states, the pattern of radiation they
produce as they de-excite can also lead to signals in the underlying event
Harnik and Wizansky (2009).

In addition to decays into SM vectors, the quirks or squirks have a
significant branching fraction into hidden gluons. Thus, heavy quirks can
produce showers of glueballs Burdman and Lichtenstein (2018); Curtin  _et al._
(2022). Some fraction of these glueballs, the $0^{++}$ states, have displaced
decays back into the SM, leading to striking signatures at current and
upcoming colliders Chacko  _et al._ (2016). The quirky nature of excited state
de-excitation can also play a significant role in determining the lifetime of
these new states. When the top partners are SM singlets or only carry
$SU(2)_{L}$ charge the slow de-excitation can lead to displaced signals, as
demonstrated in Cheng  _et al._ (2018a) for some tripled-top models. While the
quirk scenario is more general than neutral naturalness, these models point to
a particularly motivated region of quirky models. As such, they serve as
motivation for developing quirk collider searches Farina and Low (2017);
Knapen  _et al._ (2017); Li  _et al._ (2020a, b).

##  V Connections to Lattice Gauge Theory

As discussed above, a recurring theme in neutral naturalness models is the
appearance of a QCD-like $SU(3)$ gauge group in the hidden sector. Lattice
gauge theory calculations can provide direct estimates of masses and matrix
elements for hidden-sector hadrons, which may otherwise be very difficult to
estimate reliably.

A common situation for the hidden strong sector is that all of the hidden
quarks are heavy compared to the confinement scale. In this case, the low-
energy dynamics of the theory are well approximated by a “pure gauge” theory
with no dynamical fermions at all. The mass spectrum of glueball states in the
pure-gauge limit has been well-studied on the lattice, both for $SU(3)$
Morningstar and Peardon (1999); Chen  _et al._ (2006) and in the large-$N_{c}$
limit Lucini  _et al._ (2004, 2010). Lattice calculations can also provide
matrix elements, such as the glueball-to-vacuum matrix element which
determines glueball decay rates; these are generically more difficult to study
than the glueball mass, but some results are available in Chen  _et al._
(2006); Meyer (2009).

In the limit that their masses are very large, the properties of the heavy
“quirk” states can also be studied in pure-gauge theory as static quarks. The
string tension $\sigma$, which characterizes the potential between static
quarks, has been studied in pure gauge theory and in the large-$N_{c}$ context
by Lucini  _et al._ (2005, 2012). These studies, as well as investigations of
string-breaking effects such as Bulava  _et al._ (2019) (for QCD at near-
physical quark masses) or lattice calculations of parton distributions
Constantinou  _et al._ (2022), may also be of interest in modeling hidden
hadronization effects. References Lucini  _et al._ (2005, 2012) also study the
finite-temperature confining phase transition in the context of large-$N_{c}$;
in a neutral naturalness model, such a transition would be realized as a
cosmological phase transition in the early Universe.

Away from the pure-gauge limit, there is a substantial amount of existing
lattice QCD work done at heavier-than-physical quark masses. These results are
well-suited to being adapted for use in neutral naturalness models where the
hidden quarks are not so heavy compared to the confinement scale that they can
be completely ignored. A detailed survey and compilation of some of these
lattice results is given in DeGrand and Neil (2020). See also Ref. Hatton  _et
al._ (2021) for studies of bottomonium which can be directly repurposed for
use in neutral naturalness models with very heavy hidden-sector quark masses.

In principle, lattice calculations could also be used to study the ultraviolet
completions of composite twin Higgs or other pNGB Higgs neutral naturalness
models. There is already a substantial literature on lattice calculations and
composite Higgs models, see Svetitsky (2018); Witzel (2019); Drach (2020) for
recent reviews. This connection is most readily made for purely gauge-fermion
UV completions, such as those discussed in Batra and Chacko (2009); Serra and
Torre (2018); extensions of work on fermionic UV completions for general
composite Higgs theories Barnard  _et al._ (2014); Ferretti and Karateev
(2014) to the case of composite neutral naturalness could be particularly
useful.

##  VI Connections to Dark Matter

The bulk of neutral naturalness dark matter studies stem from the twin Higgs
framework. We divide these into those more closely aligned to the mirror set-
up and those that assume a fraternal starting point. After discussing twin
Higgs dark matter we describe some of the other possibilities that have begun
to be explored.

###  VI.1 Mirror Twin Higgs Dark Matter

As described in Sec. III there are a variety of possible particle spectra in
the twin sector and consequently many possibilities for stable twin particles.
With the correct interactions and abundance, a stable twin particle could
comprise part or all of the dark matter. In the fully $Z_{2}$ symmetric mirror
twin Higgs model the list of stable twin particles includes the twin photon,
the twin electron, twin neutrinos, and twin atoms. The twin photon, twin
electron, and twin neutrinos are not sufficiently cold, while self-
interactions of twin atoms are too large to make up the dark matter. Thus, in
addition to the model building reasons for breaking the $Z_{2}$ symmetry, a
viable dark matter candidate also requires $Z_{2}$ breaking.

In the mirror twin set-up, if the twin neutron were lighter than the twin
proton, it would be stablilized by twin baryon number, yielding a viable dark
matter prospect. Its abundance can be set by an asymmetry in the twin baryons
making the twin neutron a natural asymmetric dark matter candidate Farina
(2015); Barbieri  _et al._ (2016, 2017). The asymmetry in baryon number and
twin baryon number could have a common origin Farina  _et al._ (2016); Kilic
_et al._ (2021). In the simplest case, the twin neutron $n_{B}$ would have a
mass $m_{n_{B}}\approx(\Omega_{\rm DM}/\Omega_{\text{baryon}})m_{n}\approx
5~{}{\rm GeV}$ Farina (2015). If one allows the masses of the twin up quark,
twin down quark, and twin electron to freely vary then the twin neutron, twin
Hydrogen atom, and twin Helium atom are all possible dark matter candidates,
depending on the hierarchy among masses Barbieri  _et al._ (2016, 2017). When
the twin quark masses are taken above the twin QCD scale the preferred mass
range for the dark matter is $1-30~{}{\rm GeV}$ Barbieri  _et al._ (2016).
Obtaining a twin neutron that is lighter than the twin proton can be
accomplished by expanding the Higgs sector to a mirror twin two Higgs doublet
model Beauchesne (2020) or adding a pair of vector-like quarks with up-type
quantum numbers Beauchesne and Kats (2021).

Whether the dark matter is composed of twin neutrons, twin Hydrogen, or twin
Helium, it would interact with visible nucleons via the Higgs portal. Above 10
GeV the cross section is detectable with $\sigma_{\rm DD}\gtrsim
10^{-47}~{}{\rm cm}^{2}$ Barbieri  _et al._ (2016, 2017), while below a mass
of 10 GeV this cross section falls below the neutrino floor Farina (2015). For
these low masses, introducing kinetic mixing between twin hypercharge and
visible hypercharge allows for additional interactions between the dark matter
and visible nucleons. For a kinetic mixing of $\epsilon\sim 10^{-11}$ twin
Hydrogen or twin Helium dark matter could be detected through nuclear recoils
and twin electrons could be detected through electron recoils Chacko  _et al._
(2021). Kinetic mixing also allows for the possibility that the twin Hydrogen
accumulates in the center of stars. Even in small amounts, the twin Hydrogen
could modify the luminosity function of white dwarfs at observable levels
Curtin and Setford (2021).

Other than the twin neutron, there are a few more exotic possibilities that
exist within the mirror twin Higgs model. If the masses of the twin leptons
and twin photon are raised, the lightest twin particle becomes the twin pion.
When the twin pion has a mass of a few hundred MeV the correct relic abundance
can be achieved via the SIMP mechanism Hochberg  _et al._ (2019).
Alternatively, the mirror electron, at a few MeV, could also be the dark
matter with an abundance set by freeze-in if a kinetic mixing of $\epsilon\sim
10^{-12}$ between the visible and twin hypercharges is present Koren and
McGehee (2020).

As discussed in Sec. III.1.1 at energies above the $\sim 10~{}{\rm TeV}$ scale
the twin Higgs model needs to be UV completed. The most common UV completions
employ supersymmetry or a composite Higgs. In a supersymmetric UV completion,
in addition to the visible electroweakinos, the twin electroweakinos are also
possible dark matter candidates. One example is the twin bino with its freeze-
out abundance determined by annihilation to twin fermions Badziak  _et al._
(2020). The twin stau is another candidate with analogous annihilation
channels to the visible stau Badziak  _et al._ (2022).

###  VI.2 Fraternal Twin Higgs Dark Matter

The fraternal twin Higgs model automatically includes several dark matter
candidates. In this model there is only a single generation of twin leptons,
the twin $\tau$ and twin $\nu$, at least one of which is stabilized by twin
lepton number. When twin hypercharge is not gauged, the twin $\tau$ freezes
out via interactions mediated by the twin $W$ and is a viable WIMP for masses
between roughly between 60 GeV and 150 GeV Garcia Garcia  _et al._ (2015a);
Craig and Katz (2015). If one allows for the twin $\nu$ mass to be comparable
to the twin $\tau$, then the dark matter becomes multicomponent, consisting of
the twin $\tau$, twin $\nu$, and possibly also the twin $W$ Garcia Garcia  _et
al._ (2015a). Weakly gauging twin hypercharge opens up another annihilation
channel for the twin $\tau$ which lowers the preferred mass range to about
$1-20~{}{\rm GeV}$ Craig and Katz (2015). In the fraternal twin model the twin
$\tau$ unavoidably interacts with the Higgs leading to tension with current
direction detection bounds Akerib  _et al._ (2017); Cui  _et al._ (2017);
Aprile  _et al._ (2018). With an additional set of hypercharged scalars the
relation between the twin $\tau$ mass and direct detection cross section can
be broken allowing for rates below those of the neutrino floor Curtin  _et
al._ (2021).

Small variations on the fraternal twin Higgs set-up can lead to interesting
experimental signals and model building. If the first and second generation of
twin quarks are re-introduced, the lightest twin particles can be long-lived
pion-like mesons. The annihilation of the twin $\tau$s in indirection
detection leads to a QCD-like shower within the twin QCD sector ending either
with stable twin pions or visible photons Freytsis  _et al._ (2016).
Alternatively, if twin electromagnetism is broken, the twin photon obtains a
mass and the twin $\tau$ and twin $\nu$ can mix. After this mixing, the
dominantly twin $\nu$ state can be the dark matter, between 0.1 and 10 GeV,
where the relic density can be set by either co-annihilation or co-scattering,
depending on the mass of the twin $\tau$ and twin photon Cheng  _et al._
(2018b).

With only a single generation of twin quarks, the lightest twin baryon is the
spin$-3/2$ baryon $\Delta_{B}$ with the quantum numbers of three twin $b$
quarks $(b_{B}b_{B}b_{B})$. When this state is lighter than the twin $\tau$,
it is a viable asymmetric dark matter candidate both when $m_{b_{B}}$ is
parametrically above the twin QCD scale Terning  _et al._ (2019) or below the
twin QCD scale Garcia Garcia  _et al._ (2015b). When twin hypercharge is
gauged, the lightest neutral state is a twin atom which consists of a
$\Delta_{B}$ bound with a twin $\tau$. This $\Delta_{B}$ asymmetry and the
visible baryon asymmetry can be generated simultaneously, provided there is a
sufficiently large kinetic mixing Feng and Yu (2020).

With a little additional structure, the fraternal twin Higgs model contains
the correct moving pieces to implement either partially acoustic dark matter
(PAcDM) or quasi-acoustic dark matter (QuAcDM) Prilepina and Tsai (2017). For
PAcDM, the dark matter consists of a component that acts like traditional cold
dark matter and a second sub-dominant component that interacts efficient with
dark radiation. This interaction suppresses the matter power spectrum at small
scales. The twin baryon $\Delta_{B}$ acts as the cold dark matter while the
twin $\tau$ can comprise the sub-dominant dark matter component provided that
the twin $U(1)_{L}$ is gauged to mediate efficient interactions between the
twin $\tau$ and twin $\nu$. QuAcDM, on the other hand, allows all species of
DM to couple, slightly inefficiently, to dark radiation, also suppressing the
matter power spectrum at small scales. Gauging $U(1)_{B-L}$ realizes this set-
up where the $\Delta_{B}$ and twin $\tau$ couple to the twin $\nu$ via the
weakly gauged $U(1)_{B-L}$.

###  VI.3 Exotic Dark Matter

Additional particles introduced into twin models can also be the dark matter.
Adding a singlet scalar $S$ that couples to both Higgs doublets via
$S^{2}(H_{A}^{2}+H_{B}^{2})$ adds a scalar portal on top of the twin Higgs
model. The dark matter in this case is the scalar $S$ and the twin mechanism,
in addition to its usual advantages, now additionally lowers the direct
detection cross section, re-opening viable masses for the $S$ roughly above
100 GeV Curtin and Gryba (2021).

Twin models have also been proposed where the twin color $SU(3)$ is broken to
$SU(2)$. This breaking causes the twin quarks to split into an $SU(2)$ doublet
and $SU(2)$ singlet. The $SU(2)$ singlet that results from the twin $b$ quark
is a viable asymmetric dark matter candidate Kilic  _et al._ (2021).

Dark matter has been explored in neutral naturalness models other than the
twin Higgs. The dark top model cancels the leading top quark divergences with
3 species of a fermionic top partner Poland and Thaler (2008). In the twin
model there are 3 species of top partner due to the twin $SU(3)$ group while
in dark top models other group structures are possible. One example is the
coset $SU(6)_{C}\times SU(3)_{L}/SU(6)_{C}\times SU(2)_{L}$ where the left-
handed top quark can be embedded in a $({\bf 6},{\bf\bar{3}})$ of the global
group which contains a $({\bf 3},{\bf 1},{\bf 2})$, the SM top, and a $({\bf
1},{\bf 3},{\bf 1})$, the dark top, where the first two labels are the
$SU(3)\times SU(3)$ subgroup of $SU(6)_{C}$ and the third label is
$SU(2)_{L}$. Without the additional structure of a twin Higgs model, the dark
top is stable and thus a dark matter candidate. When the dark top is an
electroweak singlet, its primary interactions with the Standard Model are
through the Higgs which mediates both freeze-out and direct detection. For
dark tops that are partly electroweak doublets, there is a possible $Z$
interaction which greatly increases both the annihilation cross section and
direct detection rate.

In expanded cosets, the additional pNGBs that accompany the Higgs multiplet
could be the dark matter. The coset $SO(7)/SO(6)$ is an example where the
$SO(6)$ contains the electroweak group and an additional unbroken $U(1)$ Ahmed
_et al._ (2020a). This breaking leads to six pNGBs including the Higgs
multiplet and a complex scalar $\chi$ that is stablized by the $U(1)$. The
scalar $\chi$ interacts with the Higgs both through higher-order Goldstone
interactions and through interactions with heavy fermions. The momentum-
dependence of the Goldstone interactions suppresses the direct detection rate
relative to the typical Higgs-mediated rate.

##  VII Connections to Astrophysics and Cosmology

In addition to a variety of potential dark matter candidates, the presence of
a rich hidden sector in many neutral naturalness models has myriad
implications for astrophysics and cosmology.

###  VII.1 Cosmological Signals

One of the most notable challenges for the mirror twin Higgs model is the
contribution of the light twin particles to the energy density, as measured by
the parameter $\Delta N_{\rm eff}$. This is defined via

$\rho_{\rm rad}=\rho_{\gamma}+\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{\rm
eff}\,\rho_{\gamma},$ (9)

where $\rho_{\rm rad}$ is the total energy density in radiation,
$\rho_{\gamma}$ is the energy density of visible photons, and $N_{\rm
eff}=3.044+\Delta N_{\rm eff}$. For free streaming radiation $\Delta N_{\rm
eff}\leq 0.30$ Aghanim  _et al._ (2020). Without $Z_{2}$ breaking, three twin
neutrinos would contribute $\approx 3$ and the twin photon would contribute
$\approx 4.4$ for a total contribution of $\Delta N_{\rm eff}=7.4$ which is
strongly excluded. Including $Z_{2}$ breaking decreases this to $\Delta N_{\rm
eff}=5.6$ because when the visible and twin sectors decouple there are more
degrees of freedom in the visible thermal bath which will slightly heat the
visible sector radiation. The two sectors are kept in thermal equilibrium by
the Higgs portal interaction which decouples around 3 GeV Barbieri  _et al._
(2005).

Within the mirror twin Higgs framework, there are a few approaches to reducing
the contribution to $\Delta N_{\rm eff}$. The first is to asymmetrically
reheat the visible and twin sectors such that the temperature of the twin
sector is lower. Since $\rho_{\rm rad}\propto T^{4}$, a slight reduction in
the twin sector temperature is enough to satisfy observational constraints.
Adding a right-handed neutrino $N_{A}$ (and its twin $N_{B}$) can successfully
heat the visible sector preferentially Chacko  _et al._ (2017). The heavy
neutrinos dominate the energy density, then decay after the visible and twin
sectors decouple, but before well-probed processes, like Big Bang
nucleosynthesis, occur. The decays are dominantly to the visible sector
because of the mass difference between the gauge bosons of each sector. A
scalar singlet $X$ could also be used instead of a right-handed neutrino Craig
_et al._ (2017). With heavy right-handed neutrinos and a new colored scalar
(and its twin), baryogenesis can be achieved along with asymmetric reheating
Beauchesne and Kats (2021).

Additional portals between the visible and twin sectors can allow entropy in
the twin sector to return to the visible sector. For example, the neutrino
portal allows the twin neutrino, when sufficiently heavy, to undergo a three-
body decay into three visible neutrinos Liu and Weiner (2019).

Another approach to reducing $\Delta N_{\rm eff}$ is to adjust the decoupling
temperature between the visible and twin sectors. While the decoupling
temperature cannot be increased without giving up the twin mechanism, it can
be lowered if additional interactions are included to keep the sectors in
thermal equilibrium longer. If the decoupling temperature is below the twin
QCD phase transition (and above the visible QCD phase transition) then a
larger number of degrees of freedom in the visible thermal bath heat the
visible sector radiation substantially more Csaki  _et al._ (2017). A small
mixing between the visible and twin neutrinos is sufficient to reduce $\Delta
N_{\rm eff}$ to $\approx 1$ and additional model building can bring it into
agreement with the bound $\Delta N_{\rm eff}\leq 0.30$.

Yet a different approach to reducing $\Delta N_{\rm eff}$ leverages the fact
that the Higgs interacts weakly with the neutrinos. Therefore, if the twin
fermions are taken to be very heavy then the twin neutrinos decouple from the
thermal bath before the visible and twin sectors decouple Harigaya  _et al._
(2020). Since the twin neutrinos leave the thermal bath earlier, their
temperature is lower, reducing their contribution to $\Delta N_{\rm eff}$. The
contribution from the twin photons can be reduced by including a mass and
kinetic mixing with the visible photons so that their entropy is transferred
to the visible photons.

In the mirror twin Higgs model, the presence of light twin electrons, twin
protons, and twin photons results in twin baryon acoustic oscillations,
analogous to the baryon acoustic oscillations in the visible sector Chacko
_et al._ (2018b). These twin baryon acoustic oscillations occur prior to twin
recombination and both suppress structure formulation and leave an oscillatory
pattern in the matter power spectrum. The oscillations in the matter power
spectrum only occur at small scales (corresponding to modes that are inside
the horizon prior to twin recombination) and the inclusion of a twin sector,
with a sufficiently low temperature and relative energy density, is fully
consistent with existing Cosmic Microwave Background and large-scale structure
data Bansal  _et al._ (2021). Future measurements of the matter power spectrum
have the potential to observe twin baryon acoustic oscillations.

Without the twin electrons, twin protons, or twin photons, the fraternal twin
Higgs model alone does not suppress the matter power spectrum. A suppression
can be achieved in the fraternal twin Higgs model by gauging either twin
$U(1)_{B-L}$ or twin $U(1)_{L}$ Prilepina and Tsai (2017). This additional
force mediates interactions between some, or all, of the twin matter and the
radiation-like twin neutrinos. This coupling of the dark matter to radiation
suppresses the growth of density perturbations.

###  VII.2 Cosmological Tensions

At present, there are two notable cosmological discrepancies. The first is a
tension between the inferred value of the Hubble constant from Planck Aghanim
_et al._ (2020) and the measured late-time value Riess  _et al._ (2021). The
late-time value of Hubble is larger by $4-5\sigma$. The second is a tension
between the inferred value of $S_{8}$ (the amplitude of matter fluctuations at
scales of roughly $8h^{-1}~{}{\rm Mpc}$) from Planck Aghanim  _et al._ (2020)
and the value measured from weak lensing and galatic cluster surveys
Hildebrandt  _et al._ (2020). The value from Planck is larger by $2-3\sigma$.

One attractive feature of the mirror twin Higgs model is that it naturally
addresses these tensions simultaneously. Reference Bansal  _et al._ (2021)
showed that the twin photon and twin neutrinos provide enough additional
radiation to increase the value of the Hubble constant while the twin baryons
comprising a subcomponent of the dark matter suppresses the matter power
spectrum at small scales. It is estimated that the fraternal twin Higgs model
(with either twin $U(1)_{B-L}$ or twin $U(1)_{L}$ gauged) might similarly
address these tensions Prilepina and Tsai (2017).

###  VII.3 Phase Transitions, Gravitational Waves, and Baryogenesis

It is very interesting to consider the potential consequences of the various
early Universe phase transitions expected in twin Higgs models. Reference
Schwaller (2015) presents the basic conditions required for a strong first
order phase transition in confining $SU(N)$ hidden sectors and studies the
associated gravitational wave signals and observational prospects. In
particular, it is found that a strong first order mirror QCD phase transition
and corresponding observable gravitational wave signal is expected in
fraternal twin Higgs models since no light quark flavors are present in the
spectrum. Complementing this work, a detailed study of the electroweak and
global $U(4)$ symmetry breaking phase transitions in mirror twin Higgs models
was presented in Ref. Fujikura  _et al._ (2018). They indicate that these
phase transitions are generically not first order, which dampens the prospects
for electroweak baryogenesis and gravitational wave signatures associated with
these phase transitions.

Other works examine the possibility that electroweak symmetry may not be
restored at high temperature in twin Higgs models. Reference Kilic and
Swaminathan (2016) demonstrates that in mirror twin Higgs models, while the
one-loop quadratic contributions to the Higgs potential at finite temperature
from the SM degrees of freedom are cancelled by their same spin partners (a
consequence of the twin Higgs mechanism), subleading corrections remain which
drive the potential to a symmetry-restored phase at high temperatures. On the
other hand, Ref. Matsedonskyi (2021) points out that if the Yukawa couplings
of twin fermions other than those of the twin top are larger than their SM
counterparts, the electroweak symmetry may not be restored at scales below the
global symmetry breaking scale $f\sim{\rm TeV}$. This could have important
implications for electroweak baryogenesis.

Besides electroweak baryogenesis, several works propose mechanisms of
baryogenesis, leptogenesis, and cogenesis of visible and dark matter
asymmetries through the CP-violating out-of-equilibrium decays of heavy
states. Reference Farina  _et al._ (2016) explores a simple model of twin
baryogenesis, in which asymmetries in the twin and visible sectors are
generated through the out-of-equilibrium decay of a TeV scale particle charged
under both baryon and twin baryon number. The resulting twin baryons can serve
as an asymmetric dark matter candidate and explain the relic abundance
provided the twin baryon mass is around 5 GeV, which can be realized with the
introduction of suitable sources of $Z_{2}$ breaking. As another example, Ref.
Earl  _et al._ (2020) studies a model in which light Dirac neutrino masses are
generated through a see-saw mechanism connecting the visible and mirror
sectors. Leptogenesis occurs through the decays of the heavy singlet Dirac
neutrinos to the SM leptons and mirror leptons. See also Refs. Feng and Yu
(2020); Beauchesne and Kats (2021); Kilic  _et al._ (2021) for interesting
related mechanisms of baryogenesis or leptogenesis.

###  VII.4 Mirror Stars

In the simplest realizations of mirror twin Higgs models, the $Z_{2}$ symmetry
implies the presence of both electromagnetism and nuclear physics in the
mirror sector. It is therefore natural to speculate about astrophysical
compact objects built out of mirror matter, i.e., mirror stars Curtin and
Setford (2020a, b, 2021); Howe  _et al._ (2021). References Curtin and Setford
(2020a, b) discuss the astrophysical signatures of such mirror stars. If the
mirror and visible sector photons interact through a small kinetic mixing,
visible matter may be captured in mirror star cores. This visible sector
component of the mirror star—the “SM nugget”—is then heated through
interactions with mirror matter. This may present discoverable electromagnetic
signatures in both optical frequencies, due to bremsstrahlung cooling
processes, and in X-rays, due to “Thompson conversion” of mirror photons to
ordinary photons. A detailed search strategy for directly observing mirror
stars using Gaia observations is presented in Ref. Howe  _et al._ (2021).
Furthermore, Ref. Hippert  _et al._ (2021) explores the formation and
structure of mirror neutron stars using realistic equations of state adapted
from those describing ordinary neutron stars. The observational prospects for
mirror neutron stars using gravitational waves and binary pulsars are also
discussed.

As these studies illustrate, the presence of rich hidden sectors in neutral
naturalness theories may lead to highly novel astrophysical systems and
signatures. Given the exciting program of current and planned astronomical
surveys, further investigations in this direction would be timely.

##  VIII Connections to Neutrinos

Neutral naturalness models can impact neutrino physics in a variety of ways,
including through novel mechanisms for neutrino masses, new leptogenesis
scenarios, connections with the dark radiation problem, and explanations of
oscillation anomalies and other experimental discrepancies. First, in many
neutral naturalness scenarios such as the twin Higgs, there are a host of new
fermions that are neutral under the SM gauge symmetries. With suitable
symmetry breaking mechanisms, such fermions can marry with SM neutrinos,
effectively serving as right-handed neutrinos and generating neutrino masses
through a see-saw mechanism. Along these lines, Ref. Batell and McCullough
(2015) proposed that the neutral top partners, responsible for cancelling the
dominant corrections to the Higgs potential, may simultaneously function as
right-handed neutrinos. In this framework, naturalness considerations robustly
predict a TeV-scale see-saw, and large neutrino Yukawa couplings may be
achieved through the inverse or linear see-saw mechanisms along with an
associated rich phenomenology. Another interesting possibility for neutrino
mass generation arises in models with fermionic singletons—completely neutral
fields with no $Z_{2}$ partners—which can connect the visible and mirror
lepton sectors Bishara and Verhaaren (2019). There is clearly a wide scope for
further studies of novel neutrino mass mechanisms within models of neutral
naturalness.

The dynamics underlying neutrino masses may also have important consequences
for cosmology in the mirror twin Higgs framework, as discussed in Sec. VII. In
particular, the challenges associated with a large number of relativistic
degrees of freedom in the mirror twin Higgs model can be circumvented in
models with a low-scale ($\sim$GeV) see-saw model, as described in Ref. Chacko
_et al._ (2017). In this scenario, the right-handed neutrinos freeze-out with
a large thermal abundance and later become non-relativistic due to the
cosmological expansion, eventually dominating the energy density. After the SM
and mirror sectors decouple, the right-handed neutrinos preferentially decay
to SM states, increasing the energy density in the visible sector relative to
that of the mirror sector, thereby alleviating the bounds from dark radiation.
Reference Csaki  _et al._ (2017) also investigated the impact of a low-scale
see-saw on twin cosmology, demonstrating that mixing between the mirror and SM
neutrinos can maintain thermal equilibrium between the two sectors until
temperatures below the mirror QCD phase transition, thus lowering the
prediction for $\Delta N_{\rm eff}$. Interesting scenarios of leptogenesis
within the twin Higgs framework have also been put forth in Refs. Earl  _et
al._ (2020); Feng and Yu (2020), and it would be of great interest to expand
on these investigations; see Sec. VII.3 for further discussion.

In a separate direction, neutral naturalness models can potentially provide
explanations for experimental anomalies associated with the neutrino sector.
For example, Ref. Bai  _et al._ (2016) speculates that three twin neutrinos
may potentially explain the LSND and MiniBooNE anomalies while evading the
otherwise stringent constraints from disappearance experiments. As another
illustration, Ref. Aristizabal Sierra and Fong (2018) proposes an explanation
of the EDGES 21-cm absorption signal that relies on a relic population of
mirror sector neutrinos that produce mirror photons through their decays. The
mirror photons are then resonantly converted to visible sector photons,
enhancing the 21-cm absorption signal. Finally, in Refs. Liu  _et al._ (2020);
Liu and Zhou (2021) a potential resolution of the muon anomalous magnetic
moment discrepancy in see-saw extensions of twin Higgs models is explored.

##  IX Connections to Flavor

There are potentially far reaching implications for flavor physics in neutral
naturalness models, although the existing literature has only scratched the
surface of the subject. An interesting illustration comes from the
consideration of flavor in composite twin Higgs models, as in the study of
Ref. Csaki  _et al._ (2016). In ordinary composite Higgs models, the
attractive hypothesis of quark flavor anarchy generically leads to stringent
bounds on the scale of the composite resonances, which in turn requires a
rather severe tuning of the composite Higgs potential, typically at the per-
mille level. On the other hand, in composite twin Higgs models, the composite
resonances can be freely raised to scales of order 10 TeV without incurring
significant tuning, since the twin partners (and not the visible sector
composite resonances) cancel the dominant quantum corrections to the Higgs
potential. Raising the composite resonances then significantly weakens, though
does not entirely eliminate, constraints from flavor, allowing a scenario with
per-cent level tuning.

Also motivated by the flavor puzzle and the little hierarchy problem, Ref.
Altmannshofer and Maddock (2020) studied a twinned flavorful two-Higgs doublet
model, in which one Higgs doublet generates the masses of the third generation
while a second doublet is responsible for masses of the light generations.
This scenario can provide a partial understanding of the hierarchical patterns
observed in the fermion masses and also predicts novel signatures associated
with the heavy scalar doublet in both flavor physics and at colliders.

While the examples above primarily concern flavor physics in the visible
sector, there are also good motivations to consider the nature of fermion
flavor in the mirror sector in twin Higgs models. The fraternal twin Higgs
scenario Craig  _et al._ (2015a) represents perhaps the most dramatic breaking
of $Z_{2}$ symmetry in the the fermion flavor sector, with all but the third
generation twin fermions removed from the spectrum. This scenario is well
motivated from the a bottom-up naturalness perspective, and has important
consequences for phenomenology and cosmology, as discussed above in Secs.
III.1,VI.2,VII. Beyond the fraternal scenario, Ref. Barbieri  _et al._ (2016)
considers a more modest breaking of the $Z_{2}$ exchange symmetry in the
Yukawa couplings (except of course in the top Yukawa). This has several
interesting implications, including the radiative generation of $Z_{2}$
breaking in the Higgs potential, which provides the correct vacuum alignment,
the alleviation of bounds from too much dark radiation, and several potential
dark matter candidates in the form of mirror baryons and leptons. Following
this work, the authors proposed an effective theory that simultaneously
explains the fermion mass hierarchies as in e.g., Froggatt-Nielsen models, and
also realizes the $Z_{2}$ breaking in the Yukawa couplings through spontaneous
$Z_{2}$ breaking between the visible and mirror flavor symmetries Barbieri
_et al._ (2017). Novel spontaneous twin gauge symmetry breaking patterns can
also lead to new flavor structures in the twin fermion sector Batell and
Verhaaren (2019); Liu and Weiner (2019), with an array of associated
phenomenological signatures.

It would be interesting to further explore solutions to the flavor puzzle
within the neutral naturalness paradigm, particularly in the setting of
concrete UV completions that permit extrapolation between the TeV scale and
high scales where flavor is generated. Likewise, a worthy goal is to
understand to what extent the structure of flavor in the hidden sectors of
neutral naturalness theories can be tied through symmetry to visible sector
observables at colliders and in precision measurements.

###### Acknowledgements.

The work of B.B. and M.L. is supported by the U.S. Department of Energy under
grant No. DE–SC0007914. E.N. is supported by the U.S. Department of Energy
under grant No. DE-SC0010005.

## References

  * Chacko  _et al._ (2006a) Z. Chacko, H.-S. Goh, and R. Harnik, Phys. Rev. Lett. 96, 231802 (2006a), arXiv:hep-ph/0506256 .
  * Curtin and Saraswat (2016) D. Curtin and P. Saraswat, Phys. Rev. D 93, 055044 (2016), arXiv:1509.04284 [hep-ph] .
  * Chacko  _et al._ (2006b) Z. Chacko, H.-S. Goh, and R. Harnik, JHEP 01, 108 (2006b), arXiv:hep-ph/0512088 .
  * Goh and Krenke (2007) H.-S. Goh and C. A. Krenke, Phys. Rev. D 76, 115018 (2007), arXiv:0707.3650 [hep-ph] .
  * Batra and Chacko (2009) P. Batra and Z. Chacko, Phys. Rev. D 79, 095012 (2009), arXiv:0811.0394 [hep-ph] .
  * Yu (2017) J.-H. Yu, Phys. Rev. D 95, 095028 (2017), arXiv:1612.09300 [hep-ph] .
  * Kats  _et al._ (2017) Y. Kats, M. McCullough, G. Perez, Y. Soreq, and J. Thaler, JHEP 06, 126 (2017), arXiv:1704.03393 [hep-ph] .
  * Barbieri  _et al._ (2005) R. Barbieri, T. Gregoire, and L. J. Hall, (2005), arXiv:hep-ph/0509242 .
  * Foot and Volkas (2007) R. Foot and R. R. Volkas, Phys. Lett. B 645, 75 (2007), arXiv:hep-ph/0610013 .
  * Harnik  _et al._ (2017) R. Harnik, K. Howe, and J. Kearney, JHEP 03, 111 (2017), arXiv:1603.03772 [hep-ph] .
  * Craig  _et al._ (2015a) N. Craig, A. Katz, M. Strassler, and R. Sundrum, JHEP 07, 105 (2015a), arXiv:1501.05310 [hep-ph] .
  * Craig  _et al._ (2016a) N. Craig, S. Knapen, P. Longhi, and M. Strassler, JHEP 07, 002 (2016a), arXiv:1601.07181 [hep-ph] .
  * Barbieri  _et al._ (2016) R. Barbieri, L. J. Hall, and K. Harigaya, JHEP 11, 172 (2016), arXiv:1609.05589 [hep-ph] .
  * Katz  _et al._ (2017) A. Katz, A. Mariotti, S. Pokorski, D. Redigolo, and R. Ziegler, JHEP 01, 142 (2017), arXiv:1611.08615 [hep-ph] .
  * Csáki  _et al._ (2020) C. Csáki, C.-S. Guan, T. Ma, and J. Shu, JHEP 12, 005 (2020), arXiv:1910.14085 [hep-ph] .
  * Jung (2019) T. H. Jung, Phys. Rev. D 100, 115012 (2019), arXiv:1902.10978 [hep-ph] .
  * Batell and Verhaaren (2019) B. Batell and C. B. Verhaaren, JHEP 12, 010 (2019), arXiv:1904.10468 [hep-ph] .
  * Liu and Weiner (2019) D. Liu and N. Weiner, (2019), arXiv:1905.00861 [hep-ph] .
  * Batell  _et al._ (2020) B. Batell, W. Hu, and C. B. Verhaaren, JHEP 08, 009 (2020), arXiv:2004.10761 [hep-ph] .
  * Chacko  _et al._ (2006c) Z. Chacko, Y. Nomura, M. Papucci, and G. Perez, JHEP 01, 126 (2006c), arXiv:hep-ph/0510273 .
  * Chacko  _et al._ (2019) Z. Chacko, C. Kilic, S. Najjari, and C. B. Verhaaren, Phys. Rev. D 100, 035037 (2019), arXiv:1904.11990 [hep-ph] .
  * Beauchesne  _et al._ (2016) H. Beauchesne, K. Earl, and T. Grégoire, JHEP 01, 130 (2016), arXiv:1510.06069 [hep-ph] .
  * Yu (2016a) J.-H. Yu, Phys. Rev. D 94, 111704 (2016a), arXiv:1608.01314 [hep-ph] .
  * Yu (2016b) J.-H. Yu, JHEP 12, 143 (2016b), arXiv:1608.05713 [hep-ph] .
  * Feng and Yu (2020) W.-Z. Feng and J.-H. Yu, (2020), arXiv:2005.06471 [hep-ph] .
  * Altmannshofer and Maddock (2020) W. Altmannshofer and B. Maddock, (2020), arXiv:2003.01320 [hep-ph] .
  * Beauchesne (2020) H. Beauchesne, JHEP 09, 048 (2020), arXiv:2007.00052 [hep-ph] .
  * Falkowski  _et al._ (2006) A. Falkowski, S. Pokorski, and M. Schmaltz, Phys. Rev. D 74, 035003 (2006), arXiv:hep-ph/0604066 .
  * Chang  _et al._ (2007) S. Chang, L. J. Hall, and N. Weiner, Phys. Rev. D 75, 035009 (2007), arXiv:hep-ph/0604076 .
  * Craig and Howe (2014) N. Craig and K. Howe, JHEP 03, 140 (2014), arXiv:1312.1341 [hep-ph] .
  * Badziak and Harigaya (2017a) M. Badziak and K. Harigaya, JHEP 06, 065 (2017a), arXiv:1703.02122 [hep-ph] .
  * Badziak and Harigaya (2017b) M. Badziak and K. Harigaya, JHEP 10, 109 (2017b), arXiv:1707.09071 [hep-ph] .
  * Badziak and Harigaya (2018) M. Badziak and K. Harigaya, Phys. Rev. Lett. 120, 211803 (2018), arXiv:1711.11040 [hep-ph] .
  * Asadi  _et al._ (2019) P. Asadi, N. Craig, and Y.-Y. Li, JHEP 02, 138 (2019), arXiv:1810.09467 [hep-ph] .
  * Albaid  _et al._ (2015) A. Albaid, M. Dine, and P. Draper, JHEP 12, 046 (2015), arXiv:1510.03392 [hep-ph] .
  * Low  _et al._ (2015) M. Low, A. Tesi, and L.-T. Wang, Phys. Rev. D 91, 095012 (2015), arXiv:1501.07890 [hep-ph] .
  * Barbieri  _et al._ (2015) R. Barbieri, D. Greco, R. Rattazzi, and A. Wulzer, JHEP 08, 161 (2015), arXiv:1501.07803 [hep-ph] .
  * Geller and Telem (2015) M. Geller and O. Telem, Phys. Rev. Lett. 114, 191801 (2015), arXiv:1411.2974 [hep-ph] .
  * Burdman and Dias (2007) G. Burdman and A. G. Dias, JHEP 01, 041 (2007), arXiv:hep-ph/0609181 .
  * Csaki  _et al._ (2016) C. Csaki, M. Geller, O. Telem, and A. Weiler, JHEP 09, 146 (2016), arXiv:1512.03427 [hep-ph] .
  * Contino  _et al._ (2017) R. Contino, D. Greco, R. Mahbubani, R. Rattazzi, and R. Torre, Phys. Rev. D 96, 095036 (2017), arXiv:1702.00797 [hep-ph] .
  * Greco and Mimouni (2016) D. Greco and K. Mimouni, JHEP 11, 108 (2016), arXiv:1609.05922 [hep-ph] .
  * Burdman  _et al._ (2007) G. Burdman, Z. Chacko, H.-S. Goh, and R. Harnik, JHEP 02, 009 (2007), arXiv:hep-ph/0609152 .
  * Scherk and Schwarz (1979a) J. Scherk and J. H. Schwarz, Phys. Lett. B 82, 60 (1979a).
  * Scherk and Schwarz (1979b) J. Scherk and J. H. Schwarz, Nucl. Phys. B 153, 61 (1979b).
  * Craig and Lou (2014) N. Craig and H. K. Lou, JHEP 12, 184 (2014), arXiv:1406.4880 [hep-ph] .
  * Cohen  _et al._ (2016) T. Cohen, N. Craig, H. K. Lou, and D. Pinner, JHEP 03, 196 (2016), arXiv:1508.05396 [hep-ph] .
  * Gherghetta  _et al._ (2016) T. Gherghetta, M. Nguyen, and Z. Thomas, Phys. Rev. D 94, 115008 (2016), arXiv:1610.00342 [hep-ph] .
  * Cheng  _et al._ (2018a) H.-C. Cheng, L. Li, E. Salvioni, and C. B. Verhaaren, JHEP 05, 057 (2018a), arXiv:1803.03651 [hep-ph] .
  * Cheng  _et al._ (2019) H.-C. Cheng, L. Li, E. Salvioni, and C. B. Verhaaren, JHEP 11, 031 (2019), arXiv:1906.02198 [hep-ph] .
  * Cohen  _et al._ (2018) T. Cohen, N. Craig, G. F. Giudice, and M. Mccullough, JHEP 05, 091 (2018), arXiv:1803.03647 [hep-ph] .
  * Craig  _et al._ (2015b) N. Craig, S. Knapen, and P. Longhi, Phys. Rev. Lett. 114, 061803 (2015b), arXiv:1410.6808 [hep-ph] .
  * Craig  _et al._ (2015c) N. Craig, S. Knapen, and P. Longhi, JHEP 03, 106 (2015c), arXiv:1411.7393 [hep-ph] .
  * Cai  _et al._ (2009) H. Cai, H.-C. Cheng, and J. Terning, JHEP 05, 045 (2009), arXiv:0812.0843 [hep-ph] .
  * Poland and Thaler (2008) D. Poland and J. Thaler, JHEP 11, 083 (2008), arXiv:0808.1290 [hep-ph] .
  * Batell and McCullough (2015) B. Batell and M. McCullough, Phys. Rev. D 92, 073018 (2015), arXiv:1504.04016 [hep-ph] .
  * Xu  _et al._ (2020) L.-X. Xu, J.-H. Yu, and S.-H. Zhu, Phys. Rev. D 101, 095014 (2020), arXiv:1810.01882 [hep-ph] .
  * Xu  _et al._ (2019) L.-X. Xu, J.-H. Yu, and S.-H. Zhu, (2019), arXiv:1905.12796 [hep-ph] .
  * Serra and Torre (2018) J. Serra and R. Torre, Phys. Rev. D 97, 035017 (2018), arXiv:1709.05399 [hep-ph] .
  * Csáki  _et al._ (2018) C. Csáki, T. Ma, and J. Shu, Phys. Rev. Lett. 121, 231801 (2018), arXiv:1709.08636 [hep-ph] .
  * Dillon (2019) B. M. Dillon, Phys. Rev. D 99, 115008 (2019), arXiv:1806.10702 [hep-ph] .
  * Serra  _et al._ (2019) J. Serra, S. Stelzl, R. Torre, and A. Weiler, JHEP 10, 060 (2019), arXiv:1905.02203 [hep-ph] .
  * Ahmed  _et al._ (2020a) A. Ahmed, S. Najjari, and C. B. Verhaaren, JHEP 06, 007 (2020a), arXiv:2003.08947 [hep-ph] .
  * Durieux  _et al._ (2022) G. Durieux, M. McCullough, and E. Salvioni, (2022), arXiv:2202.01228 [hep-ph] .
  * Burdman  _et al._ (2015) G. Burdman, Z. Chacko, R. Harnik, L. de Lima, and C. B. Verhaaren, Phys. Rev. D 91, 055007 (2015), arXiv:1411.3310 [hep-ph] .
  * Thrasher (2017) K. Thrasher, (2017), arXiv:1705.01472 [hep-ph] .
  * Bie  _et al._ (2021) S.-Y. Bie, G.-L. Liu, and W. Wang, Chin. Phys. C 45, 013106 (2021), arXiv:2009.04858 [hep-ph] .
  * Fan and Reece (2014) J. Fan and M. Reece, JHEP 06, 031 (2014), arXiv:1401.7671 [hep-ph] .
  * Archer-Smith  _et al._ (2021) P. Archer-Smith, D. Stolarski, and R. Vega-Morales, JHEP 10, 247 (2021), arXiv:2012.01440 [hep-ph] .
  * Craig  _et al._ (2013) N. Craig, C. Englert, and M. McCullough, Phys. Rev. Lett. 111, 121803 (2013), arXiv:1305.5251 [hep-ph] .
  * Craig  _et al._ (2016b) N. Craig, H. K. Lou, M. McCullough, and A. Thalapillil, JHEP 02, 127 (2016b), arXiv:1412.0258 [hep-ph] .
  * Goncalves  _et al._ (2018) D. Goncalves, T. Han, and S. Mukhopadhyay, Phys. Rev. Lett. 120, 111801 (2018), [Erratum: Phys.Rev.Lett. 121, 079902 (2018)], arXiv:1710.02149 [hep-ph] .
  * Gonçalves  _et al._ (2018) D. Gonçalves, T. Han, and S. Mukhopadhyay, Phys. Rev. D 98, 015023 (2018), arXiv:1803.09751 [hep-ph] .
  * Juknevich  _et al._ (2009) J. E. Juknevich, D. Melnikov, and M. J. Strassler, JHEP 07, 055 (2009), arXiv:0903.0883 [hep-ph] .
  * Juknevich (2010) J. E. Juknevich, JHEP 08, 121 (2010), arXiv:0911.5616 [hep-ph] .
  * Curtin  _et al._ (2022) D. Curtin, C. Gemmell, and C. B. Verhaaren, (2022), arXiv:2202.12899 [hep-ph] .
  * Curtin and Verhaaren (2015) D. Curtin and C. B. Verhaaren, JHEP 12, 072 (2015), arXiv:1506.06141 [hep-ph] .
  * Csaki  _et al._ (2015) C. Csaki, E. Kuflik, S. Lombardo, and O. Slone, Phys. Rev. D 92, 073008 (2015), arXiv:1508.01522 [hep-ph] .
  * Buttazzo  _et al._ (2015) D. Buttazzo, F. Sala, and A. Tesi, JHEP 11, 158 (2015), arXiv:1505.05488 [hep-ph] .
  * Ahmed (2018) A. Ahmed, JHEP 02, 048 (2018), arXiv:1711.03107 [hep-ph] .
  * Buttazzo  _et al._ (2018) D. Buttazzo, D. Redigolo, F. Sala, and A. Tesi, JHEP 11, 144 (2018), arXiv:1807.04743 [hep-ph] .
  * Chacko  _et al._ (2018a) Z. Chacko, C. Kilic, S. Najjari, and C. B. Verhaaren, Phys. Rev. D 97, 055031 (2018a), arXiv:1711.05300 [hep-ph] .
  * Kilic  _et al._ (2019) C. Kilic, S. Najjari, and C. B. Verhaaren, Phys. Rev. D 99, 075029 (2019), arXiv:1812.08173 [hep-ph] .
  * Alipour-Fard  _et al._ (2020) S. Alipour-Fard, N. Craig, S. Gori, S. Koren, and D. Redigolo, JHEP 07, 029 (2020), arXiv:1812.09315 [hep-ph] .
  * Koren and McGehee (2020) S. Koren and R. McGehee, Phys. Rev. D 101, 055024 (2020), arXiv:1908.03559 [hep-ph] .
  * Davidson  _et al._ (2000) S. Davidson, S. Hannestad, and G. Raffelt, JHEP 05, 003 (2000), arXiv:hep-ph/0001179 .
  * Vogel and Redondo (2014) H. Vogel and J. Redondo, JCAP 02, 029 (2014), arXiv:1311.2600 [hep-ph] .
  * Bishara and Verhaaren (2019) F. Bishara and C. B. Verhaaren, JHEP 05, 016 (2019), arXiv:1811.05977 [hep-ph] .
  * Ahmed  _et al._ (2020b) A. Ahmed, B. M. Dillon, and S. Najjari, JHEP 02, 124 (2020b), arXiv:1911.05085 [hep-ph] .
  * Cheng  _et al._ (2016) H.-C. Cheng, S. Jung, E. Salvioni, and Y. Tsai, JHEP 03, 074 (2016), arXiv:1512.02647 [hep-ph] .
  * Cheng  _et al._ (2017) H.-C. Cheng, E. Salvioni, and Y. Tsai, Phys. Rev. D 95, 115035 (2017), arXiv:1612.03176 [hep-ph] .
  * Li  _et al._ (2018) L. Li, E. Salvioni, Y. Tsai, and R. Zheng, Phys. Rev. D 97, 015010 (2018), arXiv:1710.06437 [hep-ph] .
  * Burdman and D’Agnolo (2015) G. Burdman and R. T. D’Agnolo, (2015), arXiv:1512.00040 [hep-ph] .
  * Cheng  _et al._ (2022) H.-C. Cheng, L. Li, and E. Salvioni, JHEP 01, 122 (2022), arXiv:2110.10691 [hep-ph] .
  * Egana-Ugrinovic  _et al._ (2018) D. Egana-Ugrinovic, M. Low, and J. T. Ruderman, JHEP 05, 012 (2018), arXiv:1801.05432 [hep-ph] .
  * Kang and Luty (2009) J. Kang and M. A. Luty, JHEP 11, 065 (2009), arXiv:0805.4642 [hep-ph] .
  * Burdman  _et al._ (2008) G. Burdman, Z. Chacko, H.-S. Goh, R. Harnik, and C. A. Krenke, Phys. Rev. D 78, 075028 (2008), arXiv:0805.4667 [hep-ph] .
  * Capdevilla  _et al._ (2020) R. M. Capdevilla, R. Harnik, and A. Martin, JHEP 03, 117 (2020), arXiv:1912.08234 [hep-ph] .
  * Harnik and Wizansky (2009) R. Harnik and T. Wizansky, Phys. Rev. D 80, 075015 (2009), arXiv:0810.3948 [hep-ph] .
  * Burdman and Lichtenstein (2018) G. Burdman and G. Lichtenstein, JHEP 08, 146 (2018), arXiv:1807.03801 [hep-ph] .
  * Chacko  _et al._ (2016) Z. Chacko, D. Curtin, and C. B. Verhaaren, Phys. Rev. D 94, 011504 (2016), arXiv:1512.05782 [hep-ph] .
  * Farina and Low (2017) M. Farina and M. Low, Phys. Rev. Lett. 119, 111801 (2017), arXiv:1703.00912 [hep-ph] .
  * Knapen  _et al._ (2017) S. Knapen, H. K. Lou, M. Papucci, and J. Setford, Phys. Rev. D 96, 115015 (2017), arXiv:1708.02243 [hep-ph] .
  * Li  _et al._ (2020a) J. Li, T. Li, J. Pei, and W. Zhang, Phys. Rev. D 102, 056006 (2020a), arXiv:1911.02223 [hep-ph] .
  * Li  _et al._ (2020b) J. Li, T. Li, J. Pei, and W. Zhang, Eur. Phys. J. C 80, 651 (2020b), arXiv:2002.07503 [hep-ph] .
  * Morningstar and Peardon (1999) C. J. Morningstar and M. J. Peardon, Phys. Rev. D 60, 034509 (1999), arXiv:hep-lat/9901004 .
  * Chen  _et al._ (2006) Y. Chen _et al._ , Phys. Rev. D 73, 014516 (2006), arXiv:hep-lat/0510074 .
  * Lucini  _et al._ (2004) B. Lucini, M. Teper, and U. Wenger, JHEP 06, 012 (2004), arXiv:hep-lat/0404008 .
  * Lucini  _et al._ (2010) B. Lucini, A. Rago, and E. Rinaldi, JHEP 08, 119 (2010), arXiv:1007.3879 [hep-lat] .
  * Meyer (2009) H. B. Meyer, JHEP 01, 071 (2009), arXiv:0808.3151 [hep-lat] .
  * Lucini  _et al._ (2005) B. Lucini, M. Teper, and U. Wenger, JHEP 02, 033 (2005), arXiv:hep-lat/0502003 .
  * Lucini  _et al._ (2012) B. Lucini, A. Rago, and E. Rinaldi, Phys. Lett. B 712, 279 (2012), arXiv:1202.6684 [hep-lat] .
  * Bulava  _et al._ (2019) J. Bulava, B. Hörz, F. Knechtli, V. Koch, G. Moir, C. Morningstar, and M. Peardon, Phys. Lett. B 793, 493 (2019), arXiv:1902.04006 [hep-lat] .
  * Constantinou  _et al._ (2022) M. Constantinou _et al._ , (2022), arXiv:2202.07193 [hep-lat] .
  * DeGrand and Neil (2020) T. DeGrand and E. T. Neil, Phys. Rev. D 101, 034504 (2020), arXiv:1910.08561 [hep-ph] .
  * Hatton  _et al._ (2021) D. Hatton, C. T. H. Davies, J. Koponen, G. P. Lepage, and A. T. Lytle, Phys. Rev. D 103, 054512 (2021), arXiv:2101.08103 [hep-lat] .
  * Svetitsky (2018) B. Svetitsky, EPJ Web Conf. 175, 01017 (2018), arXiv:1708.04840 [hep-lat] .
  * Witzel (2019) O. Witzel, PoS LATTICE2018, 006 (2019), arXiv:1901.08216 [hep-lat] .
  * Drach (2020) V. Drach, PoS LATTICE2019, 242 (2020), arXiv:2005.01002 [hep-lat] .
  * Barnard  _et al._ (2014) J. Barnard, T. Gherghetta, and T. S. Ray, JHEP 02, 002 (2014), arXiv:1311.6562 [hep-ph] .
  * Ferretti and Karateev (2014) G. Ferretti and D. Karateev, JHEP 03, 077 (2014), arXiv:1312.5330 [hep-ph] .
  * Farina (2015) M. Farina, JCAP 11, 017 (2015), arXiv:1506.03520 [hep-ph] .
  * Barbieri  _et al._ (2017) R. Barbieri, L. J. Hall, and K. Harigaya, JHEP 10, 015 (2017), arXiv:1706.05548 [hep-ph] .
  * Farina  _et al._ (2016) M. Farina, A. Monteux, and C. S. Shin, Phys. Rev. D 94, 035017 (2016), arXiv:1604.08211 [hep-ph] .
  * Kilic  _et al._ (2021) C. Kilic, C. B. Verhaaren, and T. Youn, Phys. Rev. D 104, 116018 (2021), arXiv:2109.03248 [hep-ph] .
  * Beauchesne and Kats (2021) H. Beauchesne and Y. Kats, JHEP 12, 160 (2021), arXiv:2109.03279 [hep-ph] .
  * Chacko  _et al._ (2021) Z. Chacko, D. Curtin, M. Geller, and Y. Tsai, JHEP 11, 198 (2021), arXiv:2104.02074 [hep-ph] .
  * Curtin and Setford (2021) D. Curtin and J. Setford, JHEP 03, 166 (2021), arXiv:2010.00601 [hep-ph] .
  * Hochberg  _et al._ (2019) Y. Hochberg, E. Kuflik, and H. Murayama, Phys. Rev. D 99, 015005 (2019), arXiv:1805.09345 [hep-ph] .
  * Badziak  _et al._ (2020) M. Badziak, G. Grilli Di Cortona, and K. Harigaya, Phys. Rev. Lett. 124, 121803 (2020), arXiv:1911.03481 [hep-ph] .
  * Badziak  _et al._ (2022) M. Badziak, G. G. di Cortona, K. Harigaya, and M. Łukawski, (2022), arXiv:2202.10488 [hep-ph] .
  * Garcia Garcia  _et al._ (2015a) I. Garcia Garcia, R. Lasenby, and J. March-Russell, Phys. Rev. D 92, 055034 (2015a), arXiv:1505.07109 [hep-ph] .
  * Craig and Katz (2015) N. Craig and A. Katz, JCAP 10, 054 (2015), arXiv:1505.07113 [hep-ph] .
  * Akerib  _et al._ (2017) D. S. Akerib _et al._ (LUX), Phys. Rev. Lett. 118, 021303 (2017), arXiv:1608.07648 [astro-ph.CO] .
  * Cui  _et al._ (2017) X. Cui _et al._ (PandaX-II), Phys. Rev. Lett. 119, 181302 (2017), arXiv:1708.06917 [astro-ph.CO] .
  * Aprile  _et al._ (2018) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO] .
  * Curtin  _et al._ (2021) D. Curtin, S. Gryba, J. Setford, D. Hooper, and J. Scholtz, (2021), arXiv:2106.12578 [hep-ph] .
  * Freytsis  _et al._ (2016) M. Freytsis, S. Knapen, D. J. Robinson, and Y. Tsai, JHEP 05, 018 (2016), arXiv:1601.07556 [hep-ph] .
  * Cheng  _et al._ (2018b) H.-C. Cheng, L. Li, and R. Zheng, JHEP 09, 098 (2018b), arXiv:1805.12139 [hep-ph] .
  * Terning  _et al._ (2019) J. Terning, C. B. Verhaaren, and K. Zora, Phys. Rev. D 99, 095020 (2019), arXiv:1902.08211 [hep-ph] .
  * Garcia Garcia  _et al._ (2015b) I. Garcia Garcia, R. Lasenby, and J. March-Russell, Phys. Rev. Lett. 115, 121801 (2015b), arXiv:1505.07410 [hep-ph] .
  * Prilepina and Tsai (2017) V. Prilepina and Y. Tsai, JHEP 09, 033 (2017), arXiv:1611.05879 [hep-ph] .
  * Curtin and Gryba (2021) D. Curtin and S. Gryba, JHEP 08, 009 (2021), arXiv:2101.11019 [hep-ph] .
  * Aghanim  _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Chacko  _et al._ (2017) Z. Chacko, N. Craig, P. J. Fox, and R. Harnik, JHEP 07, 023 (2017), arXiv:1611.07975 [hep-ph] .
  * Craig  _et al._ (2017) N. Craig, S. Koren, and T. Trott, JHEP 05, 038 (2017), arXiv:1611.07977 [hep-ph] .
  * Csaki  _et al._ (2017) C. Csaki, E. Kuflik, and S. Lombardo, Phys. Rev. D 96, 055013 (2017), arXiv:1703.06884 [hep-ph] .
  * Harigaya  _et al._ (2020) K. Harigaya, R. Mcgehee, H. Murayama, and K. Schutz, JHEP 05, 155 (2020), arXiv:1905.08798 [hep-ph] .
  * Chacko  _et al._ (2018b) Z. Chacko, D. Curtin, M. Geller, and Y. Tsai, JHEP 09, 163 (2018b), arXiv:1803.03263 [hep-ph] .
  * Bansal  _et al._ (2021) S. Bansal, J. H. Kim, C. Kolda, M. Low, and Y. Tsai, (2021), arXiv:2110.04317 [hep-ph] .
  * Riess  _et al._ (2021) A. G. Riess _et al._ , (2021), arXiv:2112.04510 [astro-ph.CO] .
  * Hildebrandt  _et al._ (2020) H. Hildebrandt _et al._ , Astron. Astrophys. 633, A69 (2020), arXiv:1812.06076 [astro-ph.CO] .
  * Schwaller (2015) P. Schwaller, Phys. Rev. Lett. 115, 181101 (2015), arXiv:1504.07263 [hep-ph] .
  * Fujikura  _et al._ (2018) K. Fujikura, K. Kamada, Y. Nakai, and M. Yamaguchi, JHEP 12, 018 (2018), arXiv:1810.00574 [hep-ph] .
  * Kilic and Swaminathan (2016) C. Kilic and S. Swaminathan, JHEP 01, 002 (2016), arXiv:1508.05121 [hep-ph] .
  * Matsedonskyi (2021) O. Matsedonskyi, JHEP 04, 036 (2021), arXiv:2008.13725 [hep-ph] .
  * Earl  _et al._ (2020) K. Earl, C. S. Fong, T. Gregoire, and A. Tonero, JCAP 03, 036 (2020), arXiv:1903.12192 [hep-ph] .
  * Curtin and Setford (2020a) D. Curtin and J. Setford, Phys. Lett. B 804, 135391 (2020a), arXiv:1909.04071 [hep-ph] .
  * Curtin and Setford (2020b) D. Curtin and J. Setford, JHEP 03, 041 (2020b), arXiv:1909.04072 [hep-ph] .
  * Howe  _et al._ (2021) A. Howe, J. Setford, D. Curtin, and C. D. Matzner, (2021), arXiv:2112.05766 [hep-ph] .
  * Hippert  _et al._ (2021) M. Hippert, J. Setford, H. Tan, D. Curtin, J. Noronha-Hostler, and N. Yunes, (2021), arXiv:2103.01965 [astro-ph.HE] .
  * Bai  _et al._ (2016) Y. Bai, R. Lu, S. Lu, J. Salvado, and B. A. Stefanek, Phys. Rev. D 93, 073004 (2016), arXiv:1512.05357 [hep-ph] .
  * Aristizabal Sierra and Fong (2018) D. Aristizabal Sierra and C. S. Fong, Phys. Lett. B 784, 130 (2018), arXiv:1805.02685 [hep-ph] .
  * Liu  _et al._ (2020) G.-L. Liu, F. Wang, and W. Wang, (2020), arXiv:2012.09986 [hep-ph] .
  * Liu and Zhou (2021) G.-L. Liu and P. Zhou, (2021), arXiv:2101.00607 [hep-ph] .