P3H-23-034, TTP23-018

#  Dark Higgs Bosons at Colliders

Torben Ferber1, Alexander Grohsjean2, and Felix Kahlhoefer3

1Institute of Experimental Particle Physics (ETP),
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
2Institute of Experimental Particle Physics,
University of Hamburg, 22761 Hamburg, Germany
3Institut for Theoretical Particle Physics (TTP),
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Large Hadron Collider (LHC) has confirmed the Higgs mechanism to be
responsible for generating mass in the Standard Model (SM), making it
attractive to also consider spontaneous symmetry breaking as the origin of
mass for new particles in a dark sector extension of the SM. Such a dark Higgs
mechanism may in particular give mass to a dark matter candidate and to the
gauge boson mediating its interactions (called dark photon). In this review we
summarise the phenomenology of the resulting dark Higgs boson and discuss the
corresponding search strategies with a focus on collider experiments. We
consider both the case that the dark Higgs boson is heavier than the SM Higgs
boson, in which case leading constraints come from direct searches for new
Higgs bosons as well missing-energy searches at the LHC, and the case that the
dark Higgs boson is (potentially much) lighter than the SM Higgs boson, such
that the leading sensitivity comes from electron-positron colliders and fixed-
target experiments. Of particular experimental interest for both cases is the
associated production of a dark Higgs boson with a dark photon, which
subsequently decays into SM fermions, dark matter particles or long-lived dark
sector states. We also discuss the important role of exotic decays of the SM-
like Higgs boson and complementary constraints arising from early-universe
cosmology, astrophysics and direct searches for dark matter in laboratory
experiments.
_Keywords:_ Dark Matter at Colliders, Fixed-Target Experiments, New Light
Particles, Multi-Higgs Models, Beyond Standard Model Phenomenology

###### Contents

  1. 1 Introduction
  2. 2 Terminology and outline
  3. 3 Phenomenology of dark Higgs bosons
    1. 3.1 Dark Higgs production
    2. 3.2 Dark Higgs decays
    3. 3.3 Dark Higgs decay length
  4. 4 Constraints from observations of the SM-like Higgs boson
    1. 4.1 Signal strength
    2. 4.2 Invisible decays
    3. 4.3 Future projections and proposed experiments
  5. 5 Direct dark Higgs searches
    1. 5.1 Collider limits for $m_{\phi}>m_{h}$
    2. 5.2 Collider limits for $10~{}\mathrm{\,Ge\kern-1.00006ptV}\leq m_{\phi}\leq m_{h}$
    3. 5.3 Collider limits for $50~{}\mathrm{\,Me\kern-1.00006ptV}\leq m_{\phi}\leq 5~{}\mathrm{\,Ge\kern-1.00006ptV}$
    4. 5.4 Hints for dark Higgs bosons in hyperon decays
    5. 5.5 Future projections and proposed experiments
  6. 6 Searches for decays into dark photons
    1. 6.1 Decays of the dark Higgs boson
    2. 6.2 Decays of the SM-like Higgs boson
    3. 6.3 Future projections and proposed experiments
  7. 7 Searches for invisible dark Higgs decays
    1. 7.1 Heavy dark Higgs bosons ($m_{\phi}>m_{h}$)
    2. 7.2 Light dark Higgs bosons ($m_{\phi}\ll m_{h}$)
    3. 7.3 Future projections and proposed experiments
  8. 8 Searches for dark Higgs bosons in events with missing energy
  9. 9 Proposed searches for dark Higgs bosons in events with displaced vertices
  10. 10 Constraints from cosmology, astrophysics and non-collider experiments
    1. 10.1 Cosmological and indirect detection constraints
    2. 10.2 Astrophysical and direct detection constraints
  11. 11 Outlook

##  1 Introduction

Following the discovery of a Standard Model (SM)-like Higgs boson at the Large
Hadron Collider (LHC), the ATLAS and CMS collaborations have been able to map
out in detail its properties and find consistency with the assumption that it
couples to other SM particles proportionally to their mass [1, 2]. These
results demonstrate in a compelling way that the Higgs mechanism is
responsible for generating the masses of elementary particles. Nevertheless,
the origin of the majority of mass in the universe remains a mystery, because
it is not in the form of known particles but in a completely new form called
dark matter (DM). If DM is also composed of elementary particles, we can hope
to detect these particles in the laboratory or to infer their properties from
astrophysical observations. Many models predicting observable signatures have
been proposed and a worldwide effort to carry out the corresponding
measurements is under way, see Ref. [3] for a recent review.

A particularly intriguing possibility is that DM particles obtain their mass
in a way that is reminiscent of electroweak symmetry breaking, i.e. by
coupling to a complex scalar field, which obtains a vacuum expectation value
(vev) that spontaneously breaks a new gauge symmetry. This mechanism is
referred to as the _dark Higgs mechanism_ , and the resulting scalar boson is
called the dark Higgs boson.111To the best of our knowledge, the term was
first proposed in Ref. [4], although very similar models had already been
proposed earlier, see e.g. Refs. [5, 6, 7]. In many regards the model that we
consider is identical with the Hidden Abelian Higgs Model (HAHM) proposed in
Ref. [8]. The key implication of this idea is that DM particles do not arise
in isolation, but together with additional scalar and vector bosons, forming
an entire _dark sector_.

The presence of a dark Higgs boson and a new gauge symmetry beyond the SM
provide exciting new possibilities to study the phenomenology of DM. Not only
do the additional states and interactions offer new ways for DM particles to
be produced in the laboratory, but they themselves become an object worthy of
study. Indeed, even if DM particles are too heavy or too weakly coupled to be
found in the laboratory, we may be able to produce and detect dark Higgs
bosons. This is because scalar fields generally mix with each other through
the scalar potential. The dark Higgs boson hence inherits some properties of
its SM counterpart, and in turn modifies the experimental signatures of the
latter. In other words, the discovery of the SM-like Higgs boson at the LHC
may pave the path also for the discovery of dark Higgs bosons.

In this review we discuss the properties of dark Higgs bosons and present an
overview of existing experimental result. We start with the simplest case, in
which the dark Higgs boson is the only accessible state beyond the SM and its
couplings are induced by Higgs mixing, and gradually add additional production
and decay modes, as well as additional states that can be produced together
with dark Higgs bosons and give rise to richer experimental signatures. While
our primary focus is on collider and accelerator experiments, we also discuss
the resulting DM phenomenology and its implications for astrophysics and
cosmology.

##  2 Terminology and outline

A dark Higgs field is a complex scalar field, which is a singlet under the SM
gauge group but carries charge under a new $U(1)^{\prime}$ gauge
group.222Constructions with non-Abelian gauge groups also exist in the
literature [9], but we will focus on the simpler Abelian case here. This field
acquires a vacuum expectation value, thereby breaking the $U(1)^{\prime}$
gauge symmetry spontaneously and giving mass to the corresponding gauge boson.
In the process it may also give mass to other dark sector particles. We will
denote the vacuum expectation value of the dark Higgs field by $w$ and the
resulting physical dark Higgs boson by $\phi$.

Our definition of a dark Higgs boson has the central implication that $\phi$
will in general couple linearly to other fields in the theory. In particular,
it is generally expected to be unstable and decay into any pair of particles 1
and 2 with $m_{1}+m_{2}<m_{\phi}$. If the dark Higgs bosons decay dominantly
into stable (i.e. invisible) dark sector particles, it will be challenging
(but not impossible) to observe them experimentally. In the following, we will
therefore focus primarily on the case that the dark Higgs boson is the
lightest state in the dark sector. In this case, the only possible decay modes
are those involving SM particles.

Indeed, such decay modes are generally induced via Higgs mixing. Before
symmetry breaking the scalar potential contains a term of the form

$V(\Phi,H)=\lambda_{h\phi}|\Phi|^{2}|H|^{2}\;,$ (2.1)

where $\Phi$ ($H$) denotes the complex dark Higgs (SM Higgs) field. After
symmetry breaking and the replacements

$\displaystyle\Phi$ $\displaystyle\to\frac{\phi+w}{\sqrt{2}}$ $\displaystyle
H$ $\displaystyle\to\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ h+v\end{pmatrix}$

this term leads to mixing between the SM Higgs boson $h$ and the dark Higgs
boson $\phi$ with mixing angle $\theta$ given by333This expression assumes
$\theta\ll 1$, as required by the observed properties of the SM-like Higgs
boson. The general expression valid also for large mixing can be found in Ref.
[10].

$\theta\approx\frac{\lambda_{h\phi}\,v\,w}{m_{h}^{2}-m_{\phi}^{2}}\;.$ (2.2)

The mixing between the two Higgs bosons can be captured by the replacement

$\displaystyle h$ $\displaystyle\to\cos\theta\,h+\sin\theta\,\phi$ (2.3)
$\displaystyle\phi$ $\displaystyle\to-\sin\theta\,h+\cos\theta\,\phi\;,$ (2.4)

which leads to three main consequences:

  1. 1.

The dark Higgs boson obtains couplings to SM particles proportional to
$\sin\theta$. It will therefore have the same decay modes as a SM-like Higgs
boson with mass $m_{\phi}$, with each partial decay width suppressed by a
factor $\sin^{2}\theta$.

  2. 2.

The couplings of the SM-like Higgs boson to the other SM particles are
suppressed proportional to $\cos\theta$, which may be in conflict with the
observed agreement of the Higgs signal strength with SM expectations.

  3. 3.

The SM-like Higgs boson obtains couplings to dark sector particles
proportional to $\sin\theta$, which may induce new decay modes and thereby
shift the branching ratios away from the SM prediction.

Furthermore, for $m_{\phi}<m_{h}/2$ the SM Higgs boson can decay into a pair
of dark Higgs bosons, whereas the opposite decay is possible for
$m_{h}<m_{\phi}/2$.

In the simplest case, where no other dark sector states are kinematically
accessible, the phenomenology of a dark Higgs boson can be fully characterised
by its mass $m_{\phi}$ and the mixing angle $\sin\theta$. We review the
production and decay modes of this simple scenario in section 3. Constraints
from the observed properties of the SM-like Higgs boson and from direct
searches for dark Higgs bosons at accelerators are discussed in sections 4 and
5, respectively.

A straight-forward extension of this model that has received great interest in
recent years allows for additional interactions of the new gauge boson, which
is commonly called dark photon and denoted by $A^{\prime}$ (see Refs. [11, 12]
for recent reviews). Such couplings arise either directly, if SM fermions
carry charge under the $U(1)^{\prime}$ gauge group, or indirectly through
kinetic mixing. As a result, the dark photon provides a new way to produce
dark Higgs bosons via dark Higgs Strahlung: $\text{SM SM}\to A^{\prime\ast}\to
A^{\prime}+\phi$, as well as a new decay mode: $\phi/h\to 2A^{\prime}\to
4\,\text{SM}$. This makes it possible to search for dark Higgs bosons even if
the Higgs mixing angle $\theta$ is tiny. We discuss the corresponding
experimental strategies in section 6.

The main motivation to study dark Higgs bosons is to address the DM puzzle. To
do so, the model introduced above can be easily extended by a third dark
sector particle, namely a fermion that is a SM singlet and carries charge
under the $U(1)^{\prime}$ gauge group.444The case of a complex scalar DM
candidate is also possible and leads to largely the same phenomenology. The
gauge symmetry ensures the stability of the fermion, such that it becomes an
attractive DM candidate. The simplest version of this model introduces a Dirac
fermion $\chi$ and assumes that its mass is generated exclusively via the dark
Higgs field, such that $m_{\chi}=y_{\chi}\,w/\sqrt{2}$ in terms of the Yukawa
coupling $y_{\chi}$ between the dark Higgs boson and the DM particle. This
set-up opens up the possibility to search for dark Higgs bosons in events with
missing energy, arising from invisible decays of either the dark Higgs boson
or the dark photon, as discussed in sections 7 and 8.

As a final extension of the model, one can replace the Dirac fermion $\chi$ by
two Weyl fermions $\psi_{\mathrm{L,R}}$ with opposite $U(1)^{\prime}$ charge,
for which one can write down a gauge-invariant Dirac mass term
$m_{\mathrm{D}}\bar{\psi}\psi$ with $\psi=(\psi_{L},\psi^{c}_{R})$.
Spontaneous symmetry breaking will then generate additional Majorana mass
terms $m_{L}\overline{\psi^{c}_{L}}\psi_{L}$ and
$m_{R}\overline{\psi^{c}_{R}}\psi_{R}$. Assuming $m_{L}\approx m_{R}\ll
m_{D}$, the diagonalisation of the mass matrix then leads to so-called Pseudo-
Dirac [13] (or inelastic [14]) DM: two Majorana fermions $\chi_{1}$ and
$\chi_{2}$ with small mass splitting $\Delta=m_{2}-m_{1}\ll m_{1},m_{2}$ and
off-diagonal couplings to the dark photon. This type of model has received
great interest [15, 16, 17, 18] due to its ability to evade direct and
indirect detection constraints, while at the same time predicting exciting new
signatures at accelerators resulting from the production of long-lived
$\chi_{2}$ particles. We will discuss the experimental implications of such a
set-up in section 9.

To conclude our review of dark Higgs bosons, we consider in section 10
complementary constraints from astrophysics and cosmology, as well as from DM
direct and indirect detection experiments.

##  3 Phenomenology of dark Higgs bosons

(a) Gluon fusion production.

(b) Top-associated production.

(c) Decay of the SM-like Higgs boson.

(d) Dark Higgs Strahlung.

(e) Dark Higgs Strahlung and decay into two dark photons.

(f) Rare $K$ and $B$ decays.

Figure 3.1:  Feynman diagrams depicting the leading Dark Higgs search
channels. Depending on context, $f$ denotes either a SM fermion or a DM
particle, and $V$ denotes either a SM gauge boson or a dark photon.

###  3.1 Dark Higgs production

Dark Higgs bosons that mix with the SM Higgs boson can be produced in proton-
proton collisions in complete analogy to the SM Higgs boson. For heavy dark
Higgs bosons, the dominant production mode is gluon fusion (see figure 1(a)),
but other modes, such as vector boson fusion or production in association with
gauge bosons or heavy quarks (see figure 1(b)), may be of interest depending
on the final state under consideration. If the dark Higgs boson has a mass
$m_{\phi}<m_{h}/2$ it can also be produced in decays of the SM-like Higgs
boson (see figure 1(c)). The corresponding decay width is given by

$\Gamma_{h\to\phi\phi}=\frac{(m_{h}^{2}+2\,m_{\phi}^{2})^{2}\,\sin^{2}2\theta}{128\pi\,m_{h}}\left(1-\frac{4\,m_{\phi}^{2}}{m_{h}^{2}}\right)^{1/2}\left(\frac{1}{w}\cos\theta+\frac{1}{v}\sin\theta\right)^{2}\;.$
(3.1)

We note that for $\theta\ll 1$ this decay width is proportional to
$\theta^{2}/w^{2}$, which for $m_{\phi}\ll m_{h}$ is proportional to
$\lambda_{h\phi}^{2}$. This is a direct consequence of the fact that this
decay mode is allowed even in the limit $w\to 0$, i.e. for unbroken
$U(1)^{\prime}$ gauge symmetry. In order to compare constraints on this decay
mode to other constraints (which depend exclusively on $\sin\theta$) it is
therefore necessary to assume a specific value of $w$. In the following, we
will consider the benchmark choice $w=100\,\mathrm{GeV}$, keeping in mind that
smaller (larger) values of $w$ will enhance (suppress) constraints from
$h\to\phi\phi$ for fixed value of $\sin\theta$. We remind the reader that –
just as in the SM – it is perfectly possible to have $m_{\phi}\ll w$, while
the opposite case ($m_{\phi}\gg w$) generally leads to unitarity violation
[19].

If the dark Higgs field gives mass to an $A^{\prime}$ gauge boson that couples
to SM particles (either via direct charges or through kinetic mixing), it can
furthermore be produced in association with the $A^{\prime}$ in a process
analogous to Higgs Strahlung:

$\text{SM}+\text{SM}\to A^{\prime\ast}\to A^{\prime}+\phi\;,$ (3.2)

see figures 1(d) and 1(e) This process is of particular interest if the
$A^{\prime}$ subsequently decays invisibly (e.g. into a pair of DM particles),
such that the dark Higgs boson is produced in association with missing energy.
The same signature may also be obtained from any process that produces DM
particles through final-state radiation of a dark Higgs boson, provided that
the DM particles obtain (part of) their mass from the dark Higgs field as
depicted in Fig. 2(a).

Variations on the idea of dark Higgs Strahlung include the production of dark
Higgs bosons in $Z$ boson decays ($Z\to A^{\prime}+\phi$), which exploits the
mass mixing between the dark photon and the $Z$ boson [20], and the production
of dark Higgs bosons in $\rho$ meson decays [21], which exploits the
$\rho_{0}$–$\gamma$ mixing.

(a) Inelastic dark matter.

(b) Dark Higgs Strahlung and inelastic dark matter.

Figure 3.2:  Feynman diagrams depicting the leading Dark Higgs production
channels in association with DM particles $\chi$.

Finally, the strong experimental programme searching for rare decays of kaons
and $B$ mesons offers a unique opportunity to search for low-mass dark Higgs
bosons. Indeed, dark Higgs bosons can participate in flavour-changing decays
via the penguin diagram shown in figure 1(f). The corresponding effective
Lagrangian after electroweak symmetry breaking (and for $m_{\phi}<v$) can be
written as [22]

$\mathcal{L}_{bs}=h_{bs}\phi\bar{s}_{L}b_{R}+\text{h.c.}\;,$ (3.3)

where

$h_{bs}=\frac{\sin\theta\,m_{b}}{v}\frac{3m_{t}^{2}V_{ts}^{\ast}V_{tb}}{16\pi^{2}v^{2}}$
(3.4)

with $m_{b,t}$ being the quark masses and $V_{ij}$ denoting the CKM matrix
elements.555Ref. [23] identified a second contribution to $h_{bs}$ arising
from the dark Higgs boson coupling to the longitudinal (i.e. Goldstone) mode
of the $W$ boson in the loop. This contribution however turns out to be
numerically sub-dominant. Given this effective coupling, a simple estimate of
the inclusive decay rate $B\to X_{s}\phi$ can be obtained via [24]666Early
studies of the production of SM-like Higgs bosons in rare $B$ meson decays
provide a similar expression with $m_{B}$ replaced by the quark mass $m_{b}$
[25, 26]. Although this expression predicts the wrong kinematic limit for dark
Higgs boson production, it is still widely used in the community.

$\Gamma_{B\to X_{s}\phi}=|h_{bs}|^{2}\frac{(m_{B}^{2}-m_{\phi}^{2})^{2}}{32\pi
m_{B}^{3}}\;.$ (3.5)

Close to the kinematic threshold, i.e. $m_{\phi}\approx m_{B}-m_{K}$ a better
estimate is obtained by considering separately the exclusive decays $B\to
K\phi$ and $B\to K^{\ast}\phi$ with partial decay widths given by [27]

$\displaystyle\Gamma(B\rightarrow K\,\phi)=$
$\displaystyle\frac{|h_{sb}|^{2}}{64\pi\,m_{B}^{3}}\lambda^{1/2}(m_{B}^{2},m_{K}^{2},m_{\phi}^{2})\left|f_{0}^{B^{0}}(m_{\phi}^{2})\right|^{2}\left(\frac{m_{B}^{2}-m_{K}^{2}}{m_{b}-m_{s}}\right)^{2}\;,$
(3.6) $\displaystyle\Gamma(B\rightarrow K^{\ast}\,\phi)=$
$\displaystyle\frac{|h_{sb}|^{2}}{64\pi\,m_{B}^{3}}\lambda^{3/2}(m_{B}^{2},m_{K^{\ast}}^{2},m_{\phi}^{2})\left|A_{0}^{B^{0}}(m_{\phi}^{2})\right|^{2}\frac{1}{\left(m_{b}+m_{s}\right)^{2}}\;,$
(3.7)

with $\lambda(a,b,c)=(a-b-c)^{2}-4\,b\,c$. The hadronic form factors are often
parametrised as [28, 29, 30]

$\displaystyle f_{0}^{B^{0}}(q^{2})$
$\displaystyle=\frac{0.33}{1-\tfrac{q^{2}}{38\>\text{GeV}^{2}}}\;,$ (3.8)
$\displaystyle A_{0}^{B^{0}}(q^{2})$
$\displaystyle=\frac{1.36}{1-\tfrac{q^{2}}{28\>\text{GeV}^{2}}}-\frac{0.99}{1-\tfrac{q^{2}}{37\>\text{GeV}^{2}}}\;.$
(3.9)

More accurate numerical results for the hadronic form factors have recently
been provided in Ref. [31]. Even more exclusive final states have been
considered in Ref. [32].

The effective Lagrangian for the transition $s\to d\phi$ can be obtained by
replacing $b\to s$ and $s\to d$ in eqs. (3.3) and (3.4). The resuling decay
width of the kaon is given by [33, 34]

$\displaystyle\Gamma_{K^{\pm}\to\pi^{\pm}\phi}$
$\displaystyle=\frac{\lambda(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})^{1/2}}{m_{K}^{3}}\frac{|\mathcal{M}|^{2}}{16\pi}$
(3.10) $\displaystyle\Gamma_{K_{L}\to\pi^{0}\phi}$
$\displaystyle=\frac{\lambda(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})^{1/2}}{m_{K}^{3}}\frac{\text{Re}(\mathcal{M})^{2}}{16\pi}$
(3.11) $\displaystyle\Gamma_{K^{\pm}\to\pi^{\pm}\phi}$
$\displaystyle=\frac{\lambda(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})^{1/2}}{m_{K}^{3}}\frac{\text{Im}(\mathcal{M})^{2}}{16\pi}$
(3.12)

with

$\mathcal{M}\approx\frac{h_{sd}}{2}\frac{m_{K}^{2}-m_{\pi}^{2}}{m_{s}-m_{d}}+\frac{7\,\gamma_{1}\sin\theta}{18}\frac{m_{K}^{2}-m_{\phi}^{2}+m_{\pi}^{2}}{v}$
(3.13)

and $\gamma_{1}=3.1\times 10^{-7}$. The second term, arising from the
effective Higgs-meson coupling, is numerically subleading and often neglected.

In principle, dark Higgs bosons can also be produced in $D$ meson decays. The
corresponding effective coupling is however suppressed by small CKM matrix
elements and the ratio $m_{b}^{2}/m_{t}^{2}$, rendering it generally
irrelevant for phenomenology.

Finally, dark Higgs bosons may also be produced in proton bremsstrahlung
through their effective coupling to nucleons (see section 10.2). This
production mode may be of particular interest for forward experiments at the
LHC [35].

###  3.2 Dark Higgs decays

The decay modes of a real scalar with Higgs-like couplings to SM particles
have been studied extensively in the literature and are summarised in figure
3.3. For example, the leading-order partial decay widths into leptons are
given by

$\Gamma_{\phi\to\ell^{+}\ell^{-}}=\frac{\sin^{2}\theta\,m_{\phi}\,m_{\ell}^{2}}{8\pi\,v^{2}}\sqrt{1-\frac{4m_{\ell}^{2}}{m_{\phi}^{2}}}\;.$
(3.14)

The public tool HDECAY [36] provides partial decay widths for SM-like Higgs
bosons with a mass in the range $2m_{D}\lesssim m_{\phi}\lesssim
1\,\mathrm{TeV}$ including many relevant higher-order corrections.777We
emphasize that HDECAY does not account for the reduced phase space due to
confinement, i.e. the decay into charm (bottom) quarks opens up for
$m_{\phi}>2m_{c}$ ($m_{\phi}>2m_{b}$) rather than $m_{\phi}>2m_{D}$
($m_{\phi}>2m_{B}$). To first approximation, this effect can be included by
multiplying the decay widths from HDECAY with a correction factor of the form
$\sqrt{1-4m_{D,B}^{2}/m_{\phi}^{2}}/\sqrt{1-4m_{c,b}^{2}/m_{\phi}^{2}}$. For
even larger masses, the decay width becomes unphysical due to diverging next-
to-leading order electroweak corrections, and one should revert to the tree-
level decay widths instead [37].

For masses below the D meson threshold, it becomes essential to take into
account the confinement of the final-state particles. For $m_{\phi}\lesssim
1.3\,\mathrm{GeV}$ relatively accurate estimates for the decays into $\pi\pi$
and $KK$ can be obtained using dispersion relations [38], while the decay into
photons can be deduced from low-energy theorems [39]. For higher masses,
additional final states become important and an accurate prediction of the
various decay widths becomes very challenging. So far no fully satisfactory
agreement has been achieved in the literature. The most widely adopted
approach is the one from Ref. [24], which switches from a dispersive analysis
to a perturbative spectator model at 2 GeV (see the top panel of figure 3.3).
We note, however, that this approach has been questioned in Refs. [40, 41],
where it has been argued that for $m_{\phi}$ close to the $f_{0}(980)$
resonance the partial decay width into $2\pi$ is overestimated (see also Ref.
[42]).

(a) Light dark Higgs bosons.

(b) Heavy dark Higgs bosons.

Figure 3.3:  Decay widths for the dominant decay modes of (a) light and (b)
heavy dark Higgs bosons. The grey shaded region in the top panel indicates the
mass range where neither the dispersive analysis (following Ref. [24]) nor the
perturbative spectator model give reliable predictions, leading to a
discontinuity at $2\,\mathrm{GeV}$. We emphasize that the hadronic partial
decay widths in the GeV mass range are affected by large theoretical
uncertainties and various different approaches are used in the community. The
decay widths in the bottom panel are taken from HDECAY [36], except for
$\Gamma_{\phi\to hh}$, for which we use the leading-order result from eq.
(3.15) and approximate $\cos\theta\approx 1$.

A notable deviation from the expectations for a SM-like Higgs boson arises for
$m_{\phi}>2m_{h}$ (see the bottom panel of figure 3.3). In this case the dark
Higgs boson can decay into a pair of SM Higgs bosons with a decay width given
by

$\Gamma_{\phi\to
hh}=\frac{(m_{\phi}^{2}+2\,m_{h}^{2})^{2}\,\sin^{2}2\theta}{128\pi\,m_{\phi}}\left(1-\frac{4\,m_{h}^{2}}{m_{\phi}^{2}}\right)^{1/2}\left(\frac{1}{v}\cos\theta+\frac{1}{w}\sin\theta\right)^{2}\;.$
(3.15)

This decay mode is of particular interest in the context of searches for Higgs
boson pair production at the LHC.

In the presence of a dark photon, there are two additional decay modes of
interest for phenomenology. The first is the tree-level decay $\phi\to
A^{\prime}A^{\prime}$ followed by the decay of each dark photon into SM
fermions. If $m_{\phi}>2m_{A^{\prime}}$ the dark photons are on-shell and the
decay width is given by [7]

$\Gamma_{\phi\to A^{\prime}A^{\prime}}=\frac{g^{\prime 2}m_{\phi}^{3}}{32\pi
m_{A^{\prime}}^{2}}\sqrt{1-\frac{4m_{A^{\prime}}^{2}}{m_{\phi}^{2}}}\left(1-\frac{4m_{A^{\prime}}^{2}}{m_{\phi}^{2}}+\frac{12m_{A^{\prime}}^{4}}{m_{\phi}^{4}}\right)\;,$
(3.16)

where $g^{\prime}$ denotes the $U(1)^{\prime}$ gauge coupling. If all four SM
fermions are detected, it is possible to reconstruct both the dark photon and
the dark Higgs mass from the final state. If the decay into on-shell dark
photons is kinematically forbidden, it becomes interesting instead to consider
the loop-induced decay into two SM fermions [7]:

$\Gamma_{\phi\to f\overline{f}}=\frac{3\,g_{f}^{4}\,g^{\prime
2}\,m_{\phi}}{128\pi^{5}}\frac{m_{f}^{2}}{m_{A^{\prime}}^{2}}\left(1-\frac{4m_{f}^{2}}{m_{\phi}^{2}}\right)^{3/2}\left|I\left(\frac{m_{\phi}^{2}}{m_{A^{\prime}}^{2}},\frac{m_{f}}{m_{A^{\prime}}^{2}}\right)\right|^{2}\;,$
(3.17)

where $g_{f}$ denotes the coupling of the dark photon to SM fermions and

$I(x_{\phi},x_{f})\equiv\int_{0}^{1}\mathrm{d}y\int_{0}^{1-y}\mathrm{d}z\frac{2-(y+z)}{(y+z)+(1-y-z)^{2}x_{f}-yzx_{\phi}}\approx\frac{3}{2}$
(3.18)

for $m_{A^{\prime}}\gg m_{\phi},m_{f}$. This decay mode will typically only be
relevant if the mixing with the SM Higgs boson is extremely small [21], but
may give rise to interesting phenomenology, such as long-lived dark Higgs
bosons [43]. We also note that an interesting feature of the loop-induced
decays into SM fermions is that the branching ratios may differ from the ones
obtained through Higgs mixing. For example, for a dark photon with kinetic
mixing, the partial decay widths into the various quarks are proportional to
$q_{q}^{4}m_{q}^{2}$, where $q_{q}$ denotes the electromagnetic charge of
quark $q$. In most of the parameter space, decays into charm quarks would
therefore dominate over decays into bottom quarks.

###  3.3 Dark Higgs decay length

The distance $d$ travelled by a particle with lifetime $\tau_{\phi}$, mass
$m_{\phi}$ and momentum $p_{\phi}$ before decaying follows an exponential
distribution

$P(d)=\frac{1}{l}\exp(-d/l)$ (3.19)

with $l=c\tau_{\phi}p_{\phi}/m_{\phi}$. As long as $l$ is small compared to
the typical vertex resolution of a given experiment, the majority of decays
will appear prompt. Conversely, if $l$ is large compared to the size of the
detector, most particles will decay outside the detector and hence appear
invisible. In the intermediate regime, it may be possible to reconstruct a
displaced decay vertex, offering a powerful handle for background suppression.

For concreteness, let us consider two examples. A dark Higgs boson with
$m_{\phi}=200\,\mathrm{MeV}$ may be produced in $B$ meson decays with
$p_{\phi}\approx m_{B}/2$. In this scenario, we find

$l\approx\frac{7\,\mathrm{cm}}{\sin^{2}\theta}\;,$ (3.20)

such that for $\sin\theta\lesssim 0.1$ the dark Higgs boson is expected to
escape from the detector unnoticed in a large fraction of events.

As a second example, let us consider a dark Higgs boson with
$m_{\phi}=500\,\mathrm{MeV}$ produced in a 400 $\mathrm{\,Ge\kern-1.00006ptV}$
proton beam-dump experiment. Given the typical momentum $p_{\phi}\approx
10\,\mathrm{GeV}$ [34], we find

$l\approx 20\,\mathrm{cm}\left(\frac{10^{-3}}{\sin\theta}\right)^{2}\;.$
(3.21)

##  4 Constraints from observations of the SM-like Higgs boson

###  4.1 Signal strength

Both ATLAS and CMS search extensively for non-standard couplings of the SM-
like Higgs boson. The simplest extension is to consider a common signal
strength modifier $\mu$ affecting equally all production and decay modes of
the SM-like Higgs boson, keeping all branching fractions equal to the SM
predictions. In the case of mixing with a dark Higgs boson, the signal
strength of the SM-like Higgs boson is suppressed according to [44]

$\mu=\cos^{2}\theta\frac{\Gamma^{\text{SM}}_{h}\cos^{2}\theta}{\Gamma^{\text{SM}}_{h}\cos^{2}\theta+\Gamma_{h\to\text{dark}}}\;,$
(4.1)

where $\Gamma^{\text{SM}}_{h}$ denotes the decay width of the SM Higgs boson
and $\Gamma_{h\to\text{dark}}$ denotes the decay width for any decays of the
SM-like Higgs boson into dark sector states, irrespective of the experimental
signature. If all dark sector states are heavier than $m_{h}/2$, this
expression reduces to $\mu=\cos^{2}\theta$. The latest bounds on $\mu$ are
given by [2, 1]

$\mu=\begin{cases}1.05\pm 0.06&\;\text{(ATLAS)}\\\ 1.00\pm
0.06&\;\text{(CMS)}.\end{cases}$ (4.2)

Rather than attempting to combine these two measurements considering all
correlations, we will use the average of the two results and assume that the
dominant uncertainties are of systematic nature. The resulting signal strength
of $1.03\pm 0.06$ corresponds to

$\sin\theta<0.27$ (4.3)

at 95% confidence level (CL) in the case of no dark decays.

###  4.2 Invisible decays

Additional constraints arise in the case that the SM-like Higgs boson can
decay into fully invisible final states. In our set-up, such decays arise if
the SM-like Higgs boson can decay into a pair of dark Higgs bosons
($h\to\phi\phi$) and both dark Higgs bosons decay invisibly or escape from the
detector before decaying, or if the SM-like Higgs boson inherits the couplings
of the dark Higgs boson to other dark sector states. Experimental constraints
are quoted in terms of

$\mathcal{B}_{\text{inv}}=\frac{\sigma}{\sigma_{\text{SM}}}\frac{\Gamma_{h\to\text{inv}}}{\Gamma_{h}}\;,$
(4.4)

which in our case corresponds to

$\mathcal{B}_{\text{inv}}=\cos^{2}\theta\frac{\Gamma_{h\to\text{inv}}}{\Gamma^{\text{SM}}_{h}\cos^{2}\theta+\Gamma_{h\to\text{dark}}}\;.$
(4.5)

The leading constraints on $\mathcal{B}_{\text{inv}}$ stem from Higgs
production in vector boson fusion and yield [45, 46]

$\mathcal{B}_{\text{inv}}<\begin{cases}0.15&\;\text{(ATLAS)}\\\
0.18&\;\text{(CMS)}\end{cases}\;$ (4.6)

at 95% confidence level. An even stronger bound is obtained by combining
different production modes, giving $\mathcal{B}_{\text{inv}}<0.11$ [47]. For
$m_{\phi}\ll m_{h}/2$ and assuming invisible decays of the dark Higgs boson,
we find

$\mathcal{B}_{\text{inv}}\approx 0.05\left(\frac{\sin\theta}{0.01}\right)^{2}$
(4.7)

for $w=100\,\mathrm{GeV}$, and hence $\sin\theta\lesssim 0.02$. On the other
hand, using $\mu=\cos^{2}\theta-\mathcal{B}_{\text{inv}}$, the measurement of
the Higgs signal strength discussed above gives $\sin\theta\lesssim 0.01$. As
pointed out in Ref. [48], this bound is always stronger than the ones obtained
from invisible Higgs decays, so we will not consider the latter further in the
following.

###  4.3 Future projections and proposed experiments

The relative importance of the Higgs signal strength measurement and searches
for invisible Higgs decays is expected to change with HL-LHC. Indeed, Ref.
[49] finds that the bound on $\mu$ will only improve significantly if
systematic uncertainties can be reduced, and even under optimistic assumptions
will only reach an expected 95% CL lower bound of $\mu>0.96$, assuming that
the best-fit value agrees with the SM. The upper bound on the invisible
branching ratio, on the other hand, will continue to improve substantially,
with an expected upper bound of $\mathcal{B}_{\text{inv}}<0.025$ at 95% CL
with an integrated luminosity of $3000\,\mathrm{fb^{-1}}$.

Future electron-positron colliders will provide tremendous progress in
constraining the properties of the SM-like Higgs boson. In particular, they
will provide very accurate measurements of the $Zh$ coupling, which should
constrain $\mu$ at the level of $0.005$ [50], corresponding to an expected
upper bound $\sin\theta\lesssim 4\times 10^{-3}$ at 95% CL [51], while the
expected upper bound on the invisible branching ratio is
$\mathcal{B}_{\text{inv}}\lesssim 0.002$. As for HL-LHC, the latter will give
the dominant bound on light dark Higgs bosons, giving $\sin\theta\lesssim
2\times 10^{-3}$ [52].

##  5 Direct dark Higgs searches

In this section we consider searches for dark Higgs bosons that do not rely on
any other state in the dark sector, i.e. the dark Higgs boson is produced
directly from SM states and decays back into SM states. These searches are
hence reminiscent of the long search for a SM-like Higgs boson at colliders,
except that for the case of the dark Higgs boson the coupling strength is
unknown. Moreover, the SM-like Higgs boson can now appear as initial or final
state, participating in dark Higgs production or decay.

We note that in the absence of other dark sector states the dark Higgs boson
can be mapped onto the more general class of dark scalar models first
introduced in Ref. [53]. In these models the linear coupling $\mu\phi
H^{\dagger}H$ and the quadratic coupling $\lambda\phi^{2}H^{\dagger}H$ are
allowed to vary independently, whereas for the dark Higgs boson they are
related via $\mu=2w\lambda$ due to the underlying symmetry. Nevertheless, in
both settings the leading effect is parametrised by the effective Higgs mixing
angle $\sin\theta$, for which constraints can be obtained as a function of
$m_{\phi}$. We show a summary of these constraints in figures 5.1, 5.2 and 5.3
and discuss the individual searches in detail below. Unless explicitly stated
otherwise, the constraints discussed in sections 5.1 and 5.2 have been
translated to the parameter space of the dark Higgs model using HiggsBounds
[54] as implemented in HiggsTools [55] and are shown at 95% confidence level
(CL). We focus on the searches and results that give the strongest constraints
on dark Higgs bosons. We do not consider the case where the dark Higgs boson
is close in mass to the SM-like Higgs boson and therefore exclude the mass
range $100\mathrm{\,Ge\kern-1.00006ptV}\leq m_{\phi}\leq
150\mathrm{\,Ge\kern-1.00006ptV}$, which has recently been studied in great
detail in Ref. [44].

###  5.1 Collider limits for $m_{\phi}>m_{h}$

####  $pp\to\phi\to ZZ/WW$ at CMS

Using an integrated luminosity of up to $5.1\,\mathrm{fb}^{-1}$ collected in
2011 at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and up to
$19.7\,\mathrm{fb}^{-1}$ collected in 2012 at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, CMS performed a search for heavy
Higgs bosons decaying into a pair of $W$ or $Z$ bosons [56]. Several final
states of the $\phi\to WW$ and $\phi\to ZZ$ decays are analyzed. The combined
upper limit at the 95% CL on the product of the cross section and branching
fraction excludes a Higgs boson with SM-like couplings in the range
$145<m_{\phi}<1000\mathrm{\,Ge\kern-1.00006ptV}$. This can be translated into
limits on the mixing angle. With an upper bound of $\sin\theta\lesssim 0.26$,
the strongest limits are reached in the mass range below approximately
$260\mathrm{\,Ge\kern-1.00006ptV}$.

####  $pp\to\phi\to ZZ$ at CMS

Higgs masses up to $3\mathrm{\,Te\kern-1.00006ptV}$ are probed by CMS in an
analysis [57] based on $pp$ collisions at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ recorded in 2016, and corresponding
to an integrated luminosity of $35.9\,\mathrm{fb}^{-1}$. The search considers
both gluon fusion as well as electroweak production of a heavy Higgs boson
$\phi$ decaying into a pair of $Z$ bosons. Interference effects between signal
and background amplitudes for a resonance of arbitrary width are included.
$ZZ$ decays are reconstructed using the 4$\ell$, 2$\ell$2q, and 2$\ell$2$\nu$
final states, where $\ell$ = e or $\mu$ and $q$ a quark leading to a final
state jet. In order to categorize events according to their production
mechanism or to separate signal from the dominant background, all relevant
matrix element probabilities are calculated for each event and compared to
each other. The analysis provides great sensitivity in the mass range between
200 and $600\mathrm{\,Ge\kern-1.00006ptV}$ and the strongest upper bound of
$\sin\theta\lesssim 0.17$ is reached for resonance mass around
$550\mathrm{\,Ge\kern-1.00006ptV}$.

####  $pp\to\phi\to ZZ/WW/hh/\ell\ell/\ell\nu$ at ATLAS

A search for a heavy Higgs boson decaying into a pair of $Z$, $W$ or Higgs
bosons as well as directly into leptons has been published by ATLAS [58]. The
analysis is based on a data sample corresponding to $36.1\,\mathrm{fb}^{-1}$
of pp collisions at $\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ collected
during 2015 and 2016.

To separate the SM background from a potential signal, the invariant final
state mass distribution is examined. While the signal shape is extracted from
Monte Carlo (MC) simulations, the background shape and normalization are
estimated using a mixture of MC simulation and data from dedicated control
regions. The final result is based on the combination of 12 different final
states from the bosonic decay channels as well as the $\ell\nu$ and $\ell\ell$
final states.

Above about $600\mathrm{\,Ge\kern-1.00006ptV}$ this search provides the
strongest direct limits over a wide mass range with the best upper bound of
$\sin\theta\lesssim 0.2$ at $600\mathrm{\,Ge\kern-1.00006ptV}$.

####  $pp\to\phi\to WW/ZZ/WZ$ at ATLAS

The first search for heavy Higgs bosons based on the full LHC $pp$ run at
$13\mathrm{\,Te\kern-1.00006ptV}$ has been published by ATLAS [59]. The data
corresponding to a total integrated luminosity of $139\,\mathrm{fb}^{-1}$ were
recorded between 2015 and 2018. The search for $\phi\to WW/ZZ/WZ$ is performed
in final states in which one $W$ or $Z$ boson decays leptonically, and the
other $W$ boson or $Z$ boson decays hadronically. To discriminate signal from
background, the reconstructed (transverse) mass of the $VV$ system is used
when the leptonically decaying $V$ boson decays into a pair of leptons
(neutrinos). As the analysis adapts the reconstruction of the hadronically
decaying $V$ boson for high transverse momenta, the search is particularly
sensitive above around $800\mathrm{\,Ge\kern-1.00006ptV}$. However, with the
best upper limit of $\sin\theta\lesssim 0.35$, this search is less sensitive
than indirect limits from measurements of the Higgs signal strength.

####  $pp\to\phi\to ZZ$ at ATLAS

The strongest limits on the mixing angle are provided by the ATLAS experiment
[60]. Using the complete LHC $13\mathrm{\,Te\kern-1.00006ptV}$ $pp$ dataset
recorded betwenn 2015 and 2018, ATLAS published a search for heavy Higgs
bosons decaying via a pair of $Z$ bosons into $4\ell$ and $2\ell 2\nu$ final
states. The analysis looks for an excess in the invariant mass of the four
charged leptons respectively the transverse mass of the two charged leptons
and the two neutrinos. The results are interpreted separately for the gluon-
gluon and vector-boson fusion production modes. At about
$340\mathrm{\,Ge\kern-1.00006ptV}$ the best constraints are reached with an
upper bound of $\sin\theta\lesssim 0.13$.

####  $pp\to\phi\to hh$ at CMS

Using an integrated luminosity of $137\,\mathrm{fb}^{-1}$ of
$13\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions recorded between 2016 and
2018, CMS performed a search for heavy Higgs bosons from gluon-gluon fusion
decaying into the SM Higgs boson and another Higgs boson [61]. The mass of
both BSM Higgs bosons is a free parameter of the search.

To maximize the statistics while still having a clean signature, the analysis
explores final states with two $\tau$ leptons and two $b$ quarks. A fully
connected, feed-forward neural network is used to classify events into five
signal and background-enriched categories per year of data-taking and per
$\tau$ decay channel. The background-enriched categories are used to further
constrain systematic uncertainties.

For our re-interpretation, the heavy Higgs boson is associated with the dark
Higgs boson, while the second Higgs boson from the decay is assumed to be the
SM Higgs boson. With an upper bound of $\sin\theta\lesssim 0.23$, the best
sensitivity of this search is reached between 400 and
$500\mathrm{\,Ge\kern-1.00006ptV}$.

####  $pp\to\phi\to hh$ at ATLAS

Similar to the $pp\to\phi\to hh$ search of CMS, ATLAS searched for Higgs boson
pair production in events with two $b$-jets and two $\tau$-leptons using an
integrated luminosity of $139\,\mathrm{fb}^{-1}$ [62] of $pp$ collision data
recorded between 2015 and 2018. The analysis targets off-shell and scalar,
narrow-resonant production in the mass range from about $250$ to
$1600\mathrm{\,Ge\kern-1.00006ptV}$. Events in which at least one $\tau$
lepton decays hadronically are considered, and multivariate discriminants are
used to reject and constrain background events and to extract the $hh$ signal
yields in a binned maximum-likelihood fit.

Since this search is not yet available in the public version of HiggsBounds,
we perform the translation ourselves, using the production cross section of a
SM-like Higgs boson from Ref. [63]. As in the case of the CMS result, the best
sensitivity is reached between 400 and $500\mathrm{\,Ge\kern-1.00006ptV}$ with
a best upper bound of $\sin\theta\lesssim 0.26$.

Figure 5.1:  95 % CL upper limits on the mixing angle $\sin\theta$ as function
of scalar mass $m_{S}$ for heavy dark Higgs bosons from CMS [56, 57, 61] and
ATLAS [58, 59, 60, 62]. For limits from Higgs signal strength see Sec. 4. For
limits from direct detection experiments see Sec. 10.2. Constraints colored in
gray with dashed outline are reinterpretations not performed by the
experimental collaborations and without access to raw data.

###  5.2 Collider limits for $10~{}\mathrm{\,Ge\kern-1.00006ptV}\leq
m_{\phi}\leq m_{h}$

####  $e^{+}e^{-}\to Z^{*}\phi$ at L3

Using an integrated luminosity of $114\,\mathrm{pb}^{-1}$ the L3 experiment at
the $e^{+}e^{-}$ collider LEP has searched for $e^{+}e^{-}\to Z^{*}\phi$,
combining events with $Z^{*}$ decays into $\nu\bar{\nu}$, $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ and $\phi$ reconstructed as one, two and three jets depending
on the $\phi$ mass [64]. For $m_{\phi}>2m_{\mu}$, the $\phi$ decays promptly.
We reinterpret the 95 % CL limits derived by L3 using all $Z^{*}$ decay
channels to constrain scalar masses
$2m_{\mu}<m_{\phi}<60\,\mathrm{\,Ge\kern-1.00006ptV}$ at the level of
$\sin\theta=\sqrt{\Gamma(Z\to Z^{*}\phi)/\Gamma^{SM}(Z\to Z^{*}H)}\lesssim
0.1$. For scalar masses $m_{\phi}<2m_{\mu}$ the decay length of the dark Higgs
boson becomes macroscopic, such that the assumption of prompt $\phi$ decays
made in the L3 analysis is not satisfied for a large fraction of events. We do
not attempt a reinterpretation which would require assumptions on the L3
detector response in this mass range, since several stronger limits from beam-
dumps, fixed target experiments, and Belle II exist.

####  $e^{+}e^{-}\to Z^{*}\phi$ at LEP2

Similarly to the L3 result, the limits of the combined search for a SM Higgs
boson of the four LEP experiments [65] can be translated into limits on the
dark Higgs boson. Using an integrated luminosity of up to
$2461\,\mathrm{pb}^{-1}$ of $e^{+}e^{-}$ data at centre-of-mass energies
between 189 and $209\mathrm{\,Ge\kern-1.00006ptV}$, the LEP experiments
searched for $e^{+}e^{-}\to Z^{*}\phi$ in four-jet final state $(\phi\to
b\bar{b})(Z\to q\bar{q})$, the missing energy final state $(\phi\to
b\bar{b})(Z\to\nu\bar{\nu})$, the leptonic final state $(\phi\to
b\bar{b})(Z\to\ell^{+}\ell^{-})$, where $\ell$ denotes an electron or a muon,
and the $\tau$ lepton final states $(\phi\to b\bar{b})(Z\to\tau^{+}\tau^{-})$
and $(\phi\to\tau^{+}\tau^{-})(Z\to q\bar{q})$. In most channels the input is
binned in two variables: the reconstructed Higgs boson mass as well as a
variable that combines several event features such as $b$-tagging variables,
or neural network outputs, which allow to discriminate signal from background.
Between 10 and $80\mathrm{\,Ge\kern-1.00006ptV}$, the obtained exclusion
limits for $\sin\theta$ range from 0.15 to 0.2.

####  $pp\to h\to\phi\phi$ at CMS

Limits on the production cross section of the SM Higgs boson decaying into a
pair of light pseudoscalar bosons can directly be translated into limits on
the production of light dark Higgs bosons from SM Higgs decays. A first such
search in final states with two $b$ quarks and two $\tau$ leptons has been
performed by CMS in 2016 using a total integrated luminosity of
$35.9\,\mathrm{fb}^{-1}$ [66] of $13\mathrm{\,Te\kern-1.00006ptV}$ pp
collisions data.

Four categories are defined on the basis of the invariant mass of the
$b$-tagged jet with the highest transverse momentum and the visible decay
products of the $\tau$ leptons. For signal events this variable is bound from
above by the mass of the SM Higgs boson, while it is on average much larger
for background events, where the three objects do not originate from a
resonance decay. The final results of the search is extracted from a fit of
the visible $\tau\tau$ mass distributions in each category. With upper limits
of $\sin\theta\lesssim 0.02$ , this search provides the strongest direct
limits on low-mass dark Higgs bosons in a mass range between 30 and
$45\mathrm{\,Ge\kern-1.00006ptV}$.

####  $pp\to h\to\phi\phi$ at ATLAS

Using all 13 $\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collision data from the LHC
recorded between 2015 and 2018, ATLAS performed a search for decays of the SM
Higgs boson into a pair of new pseudoscalar particles in final states with two
muons and two $b$-quarks [67]. A narrow dimuon resonance is searched for in
the invariant mass spectrum between $16\mathrm{\,Ge\kern-1.00006ptV}$ and
$62\mathrm{\,Ge\kern-1.00006ptV}$. Additionally, boosted decision tree
techniques are used to improve the separation of the signal from the
backgrounds.

Compared to the $pp\to h\to\phi\phi$ CMS search discussed above, this analysis
achieves significantly higher sensitivities in the mass range between 20 and
30 $\mathrm{\,Ge\kern-1.00006ptV}$, with exclusions reaching down to
$\sin\theta\lesssim 0.015$.

Figure 5.2:  95 % CL upper limits on the mixing angle $\sin\theta$ as function
of scalar mass $m_{S}$ for light dark Higgs bosons from CMS [66], ATLAS [67],
LEP2 [65], and L3 [64]. For limits from Higgs signal strength see Sec. 4. For
limits from direct detection experiments see Sec. 10.2. Constraints colored in
gray with dashed outline are reinterpretations not performed by the
experimental collaborations and without access to raw data.

###  5.3 Collider limits for $50~{}\mathrm{\,Me\kern-1.00006ptV}\leq
m_{\phi}\leq 5~{}\mathrm{\,Ge\kern-1.00006ptV}$

####  $B\to K^{(*)}\phi(\to
e^{+}e^{-},\mu^{+}\mu^{-},\pi^{+}\pi^{-},K^{+}K^{-})$ at Belle II

Belle II at the asymmetric $e^{+}e^{-}$ collider SuperKEKB in Japan has
searched directly for long-lived spin-0 mediators produced in $b\to s$
transitions in $B\text{-meson}$ decays [68] using an integrated luminosity of
189 $\,\mathrm{fb}^{-1}$. The search has been conducted model-independently in
eight exclusive final states using $K$ and $K^{*}(892)^{0}$-mesons and
mediator decays into $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\pi^{+}\pi^{-}$, and
$K^{+}K^{-}$ for various lifetimes but excluding prompt decays. These are so
far the only exclusive limits for scalar decays into hadrons. Using a combined
fit, the results are also presented as 95 % CL limits in the
$m_{\phi}$–$\sin\theta$ plane and constrain the mixing angle for dark Higgs
boson masses $25\,\mathrm{\,Me\kern-1.00006ptV}<m_{\phi}\lesssim
2.5\,\mathrm{\,Ge\kern-1.00006ptV}$. For
$m_{\phi}>2.0\,\mathrm{\,Ge\kern-1.00006ptV}$ only decays into muons are used
in the combined fit to avoid large uncertainties in the branching fraction
calculations into two hadrons. Note that in contrast to searches for Kaon
decays, $K\to\pi\phi(\to\text{inv)}$, $B\text{-meson}$ decays $B\to
K\phi(\to\text{inv)}$ are not competitive with direct searches for long-lived
particles if the dark Higgs decays predominantely into SM particles.

####  $B\to X_{S}\phi(\to
e^{+}e^{-},\mu^{+}\mu^{-},\pi^{+}\pi^{-},K^{+}K^{-})$ at BaBar

BaBar at the asymmetric $e^{+}e^{-}$ collider PEP-II in the US has used an
integrated luminosity of 404 $\,\mathrm{fb}^{-1}$ to perform a search for
mediator decays into $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\pi^{+}\pi^{-}$, and
$K^{+}K^{-}$ for various lifetimes [69]. The results are presented as 90 % CL
limits on the product of branching fractions $\mathcal{B}(B\to
X_{s}\phi)\times\mathcal{B}(\phi\to x^{+}x^{-})$ for different $\phi$
lifetimes between 1 $\mathrm{\,cm}$ and 100 $\mathrm{\,cm}$. In contrast to
the Belle II analysis (see above), BaBar did not reconstruct any exclusive
$B\text{-meson}$ final states, but only reconstructs the displaced $\phi$
candidate. This leads to larger backgrounds, but also to a higher production
rate. The results have been reinterpreted as limits on the dark Higgs mixing
$\sin\theta$ for the muon and pion final state only [24, 70].

####  $B\to K^{(*)}\phi(\to\mu^{+}\mu^{-})$ at LHCb

Using $pp$ collisions at the LHC, the LHCb experiment has performed two
dedicated searches for scalar particles produced in $b\to s$ transitions in
$B\text{-meson}$ decays [71, 72] using an integrated luminosity of 3
$\,\mathrm{fb}^{-1}$. The scalar is searched for as a narrow di-muon resonance
as a function of lifetime and includes both prompt and displaced scalar decays
$\phi\to\mu^{+}\mu^{-}$. For all masses above $m_{\phi}\gtrsim
0.5\,\mathrm{\,Ge\kern-1.00006ptV}$, the search in the final state $B^{+}\to
K^{+}\phi$ is more sensitive than $B^{0}\to K^{*}(892)^{0}\phi$. The limits
derived by LHCb have been rederived in Ref. [24] using different scalar decay
rates. The limits at 95 % CL are constraining scalar masses
$2m_{\mu}<m_{\phi}\lesssim 4.8\mathrm{\,Ge\kern-1.00006ptV}$. Ref. [24] has
also recast a published spectrum of prompt $B^{+}\to K^{+}\mu\mu$ decays [73]
which is, however, significantly weaker than the dedicated searches.

####  $K^{\pm}\to\pi^{\pm}\phi(\to\mathrm{inv})$ at NA62

The fixed target experiment NA62 uses the CERN SPS beam to search for the very
rare decay $K^{\pm}\to\pi^{\pm}\nu\bar{\nu}$. The extracted
$400\,\mathrm{\,Ge\kern-1.00006ptV}$ proton beam is dumped on a 40
$\mathrm{\,cm}$ long beryllium rod to produce a secondary beam containing a
small fraction of about 6% of charged kaons. Using data collect until 2018,
NA62 has also searched directly for invisible scalar particles or long-lived
scalar particles that decay outside of the detector. The limits at 90 % CL are
constraining scalar masses $m_{\phi}<250$ $\mathrm{\,Me\kern-1.00006ptV}$ with
masses around $m_{\pi^{0}}$ vetoed from the search [74]; the region around
$m_{\pi^{0}}$ is included in a second search specifically targeting this mass
region [75].

####  $K^{\pm}\to\pi^{\pm}\phi(\to\mathrm{inv})$ at E949

The combined results of the experiments E787 and E949 of the main search for
the SM decay $K^{\pm}\to\pi^{\pm}\nu\bar{\nu}$ at Brookhaven National
Laboratory (BNL) have also been analysed by the collaboration as a search for
invisible $\phi$ decays [76]. Those experiments used a stopped kaon beam and
identified events with a charged pion while rejecting events with any
additional activity in the detector. For the case of an unstable scalar
particle the experiments assumed a 100 % efficiency for detecting and vetoing
such decays if the $\phi$ decayed inside the outer radius of the barrel veto.
The sensitivity of E787 and E949 hence increases with the lifetime of the
scalar. The limits constrain scalar masses $0<m_{\phi}<250$
$\mathrm{\,Me\kern-1.00006ptV}$ with masses around $m_{\pi^{0}}$ vetoed from
the search. The experiment has presented exclusions in the
$m_{\phi}$–$\mathcal{B}(B)$ plane for different lifetimes between 100 ps and
infinity. The corresponding exclusion in the $m_{\phi}$–$\sin\theta$ plane has
been calculated in Ref. [24] and constrains the mixing angle for $\phi$ masses
$m_{\phi}\lesssim 210\,\mathrm{\,Me\kern-1.00006ptV}$ with an insensitive
vetoed region around $m_{\pi^{0}}$.

####  $K^{\pm}\to\pi^{\pm}\phi(\to e^{+}e^{-},\mu^{+}\mu^{-})$ and
$K^{0}_{L}\to\pi^{0}\phi(\to e^{+}e^{-},\mu^{+}\mu^{-})$ at PS191

The fixed target experiment PS191 used 19.2 $\mathrm{\,Ge\kern-1.00006ptV}$
protons from the CERN PS to search for heavy neutrinos [77, 78]. The
experiment operated in the early 1980s and collected a dataset of about
$8.6\times 10^{18}$ protons on target. Protons hitting the 80 $\mathrm{\,cm}$
long beryllium target produced mostly pions and kaons, that subsequently
decayed in an approximately 50 $\mathrm{\,m}$ long, helium-filled decay
volume, followed by a 5 $\mathrm{\,m}$ iron absorber. The detector was placed
at a distance of about 128 $\mathrm{\,m}$ from the beryllium target, about 2.3
∘ off-axis. The distance between the iron absorber and the detector was filled
with dirt and soil. The detector consisted of eight $(6\times 3)$
$\mathrm{\,m^{2}}$ flash counter tracking planes perpendicular to the beam
axis equally spaced over 12 m, interspaced with helium bags. The tracking
system was followed by a 7.2 radiation length deep calorimeter. The detector
was triggered by a coincidence of the PS extraction signal and a hodoscope
embedded into the calorimeter. The trigger was efficient for pairs of
electrons or muons. Published limits on heavy neutrino decays were recast in
Ref. [79] as limits on a dark scalar produced in lepton-flavour violating
$s\to d$ transitions in $K$-meson decays taking into account $K^{0}_{L}$ and
$K^{+}$ production in the beryllium target, the decay volume, the surrounding
dirt and the iron absorber. The limits at 90 % CL constrain the mass range
$2m_{e}<m_{\phi}<250\,\mathrm{\,Me\kern-1.00006ptV}$. The recast is based on
simulations to calculate signal yields including an approximate trigger
simulation. It conservatively ignores the presence of a focusing magnet for
positively charged kaons that would increase the scalar yield.

####  $K^{+}\to\pi^{+}\phi(\to e^{+}e^{-})$ at MicroBooNE

The MicroBooNE experiment is a liquid Argon time-projection chamber primarily
designed for neutrino scattering measurements in the Booster Neutrino Beam
(BNB) at Fermilab. However, to search for a dark scalar the experiment used a
dataset with an exposure of $1.93\times 10^{20}$ protons on target using the
Fermilab NuMI neutrino beam [80]. The NuMi beam is produced by 120
$\mathrm{\,Ge\kern-1.00006ptV}$ protons hitting a graphite target producing
secondary hadrons that decay in a helium-filled volume downstream of the
target. Remaining hadrons are then stopped in a 5 $\mathrm{\,m}$ deep hadron
absorber, including charged kaons which will decay at rest and produce dark
scalars in flavour-changing $s\to d$ transitions. MicroBooNE then searches for
visible decays of long-lived dark scalars. The hadron absorber is located at a
distance of 100 $\mathrm{\,m}$ and an angle of 125 ∘ with respect to the BNB
direction, such that any particles coming from the absorber enter the detector
in almost the opposite direction than neutrinos from the on-axis BNB. With one
observed candidate and a background expectation of about 2 events, they set 95
% CL upper limits for long lifetimes in the $m_{\phi}$–$\sin\theta$ plane.
Using the published model-independent limits on the product of branching ratio
and lifetime, we derive 95 % CL upper limits for short lifetimes where
MicroBooNE looses acceptance to close the exclusion contour towards large
values of $\sin\theta$. The limits constrain scalar masses for
$3\,\mathrm{\,Me\kern-1.00006ptV}\lesssim m_{\phi}\lesssim
210\,\mathrm{\,Me\kern-1.00006ptV}$.

####  $B\to X_{s}\phi(\to e^{+}e^{-},\mu^{+}\mu^{-})$ and $K\to\pi\phi(\to
e^{+}e^{-},\mu^{+}\mu^{-})$ at CHARM

The CERN SPS beam has been used to directly search for visible axion-like
particle decays into $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ in an empty decay region
of 35 m length and 9 m2 cross-section [81]. This region was located parallel
to the neutrino beam line used at the CHARM detector, about 10 mrad off-axis.
The distance from the target to the detector was 480 m. The 400
$\mathrm{\,Ge\kern-1.00006ptV}$ proton beam was dumped on a copper target to
produce mostly kaons, and a very small fraction of $B\text{-mesons}$. The
experiment used a dataset of about $2.4\times 10^{18}$ protons on target and
did not observe any events with an efficiency of about 0.5. The corresponding
exclusion in the $m_{\phi}$–$\sin\theta$ plane has been calculated in Ref.
[24] taking into account kaon absorption in the copper target888Recent studies
in the context of future heavy neutral lepton searches at SHiP [82] suggest
that a detailed simulation of Kaon interactions in the beam dump targets may
yield a significantly smaller geometric acceptance and hence weaker limits
than that assumed in [24]., but neglecting kaon regeneration by secondary
interactions and assuming that the number of kaons escaping the target is
negligible. The limits constrain the mixing angle for dark Higgs boson masses
$m_{\phi}\lesssim 280\,\mathrm{\,Me\kern-1.00006ptV}$.

####  $K^{0}_{L}\to\pi^{0}\phi(\to\mu^{+}\mu^{-})$ at KTeV

With the main scientific goal of measuring the branching fraction
$\mathcal{B}(K^{0}_{L}\to\pi^{0}\mu^{+}\mu^{-})$, the KTeV experiment used the
Tevatron proton beam at Fermilab to search for two photons from a $\pi^{0}$
decay and two oppositely charged tracks, all coming from the same vertex. The
800 $\mathrm{\,Ge\kern-1.00006ptV}$ proton beam was directed on a 30
$\mathrm{\,cm}$ BeO target. A 65 $\mathrm{\,m}$ long instrumented vacuum decay
volume was located 94 $\mathrm{\,m}$ downstream of the target. KTeV observed
two events consistent with the SM background expectation and placed an upper
limit of $\mathcal{B}(K^{0}_{L}\to\pi^{0}\mu^{+}\mu^{-})<3.8\times 10^{-10}$
(90 % CL) [83]. Since KTeV did not perform an invariant mass selection for the
muon pair, the result can be reinterpreted as an upper bound on a promptly
decaying dark Higgs boson using an estimated vertex resolution of 4
$\mathrm{\,mm}$ [84]. The corresponding exclusion in the
$m_{\phi}$–$\sin\theta$ plane for $2m_{\mu}<m_{\phi}\lesssim
350\,\mathrm{\,Me\kern-1.00006ptV}$ has been calculated in [24].

####  $K^{0}_{L}\to\pi^{0}\phi(\to\mathrm{inv})$ at KOTO

The KOTO experiment at J-PARC is using a beam of $K^{0}_{L}$-mesons decaying
in flight to search for the rare decay $K^{0}_{L}\to\pi^{0}\nu\bar{\nu}$. The
$K^{0}_{L}$ beam is produced using 30 $\mathrm{\,Ge\kern-1.00006ptV}$ protons
from the J-PARC main ring. Protons hitting a gold target produce about
$1.2\times 10^{-7}$ $K^{0}_{L}$ per proton on target. The experiment used a
dataset of about $3.05\times 10^{19}$ protons on target. After an early
analysis had revealed an anomalous excess of events [85], also called the
‘KOTO-anomaly’, the final re-analysis yielded a result consistent with the
background expectation. The corresponding upper limit is
$\mathcal{B}(K^{0}_{L}\to\pi^{0}\nu\bar{\nu})<4.9\times 10^{-9}$ (90 % CL)
[86]. This result can be used to set limit on scalars that escape the KOTO
detector before decaying into visible SM particles. We derive an exclusion
region in the $m_{\phi}$–$\sin\theta$ plane for $m_{\phi}<2m_{\mu}$ following
the description in Ref. [87] and using an average scalar momentum of 1.5 GeV
and a conservative efficiency estimate of $\varepsilon=0.75$ for all scalar
masses to correct for the difference between the SM 3-body kinematics
$K^{0}_{L}\to\pi^{0}\nu\bar{\nu}$ and the BSM 2-body kinematics
$K^{0}_{L}\to\pi^{0}\phi(\to\mathrm{inv})$ [88].

Figure 5.3:  Upper limits on the mixing angle $\sin\theta$ as function of
scalar mass $m_{S}$ from PS191 (90 % CL [79]), E949 (90 % CL [76]), NA62 (90 %
CL [74, 75]), MicroBooNE (95 % CL [80] and this work), KOTO (90 % CL [86] and
this work), KTeV (90 % CL [83]), L3 (95 % CL [64] and this work), CHARM (95 %
CL [81, 24]), LHCb (95 % CL [71, 72, 24]), Belle II (95 % CL [68]), BaBar (90
% CL [69, 24]). For limits from Higgs signal strength see Sec. 4. For limits
from BBN see Sec. 10.1. For limits from direct detection experiments see Sec.
10.2. Constraints colored in gray with dashed outline are reinterpretations
not performed by the experimental collaborations and without access to raw
data.

###  5.4 Hints for dark Higgs bosons in hyperon decays

In analogy to the rare meson decays discussed above, dark Higgs bosons can
also be produced in hyperon decays, in particular $\Sigma^{+}\to p\phi$, which
is sensitive to the flavour-changing effective coupling $h_{sd}$. In 2005,
HyperCP searched for this decay in the $p\mu^{+}\mu^{-}$ final state and
observed three candidate events with an invariant di-muon mass just above the
di-muon threshold [89]. These events are consistent with a two-body decay
$\Sigma^{+}\to p+X$ with $m_{X}\approx 214\,\mathrm{MeV}$ and
$\text{BR}(\Sigma^{+}\to p+X)\approx 3\times 10^{-8}$.

While a translation of this result to fundamental model parameters is
complicated by the uncertain hadronic form factors, it was quickly pointed out
that the interpretation in terms of a light scalar Higgs boson was
incompatible with constraints from rare meson decays, while an interpretation
in terms of a light pseudoscalar remained viable [90, 91, 92, 93, 94]. In
2017, however, LHCb repeated the same measurement [95], finding a much broader
distribution of di-muon invariant masses at a level consistent with the SM
prediction. No evidence for a new particle produced in Hyperon decays was
found, and the upper bound clearly excludes the HyperCP anomaly.

###  5.5 Future projections and proposed experiments

#### Prospects for direct medium and high-mass dark Higgs searches

With $3\,\mathrm{ab}^{-1}$ of $14\mathrm{\,Te\kern-1.00006ptV}$ $pp$ data
collected at the future HL-LHC, the sensitivity to both medium and high-mass
dark Higgs boson can be significantly improved. In the high mass range, the
HL-LHC will allow to improve the sensitivity by about a factor of three,
probing mixing angles $\sin\theta$ e.g. down to 0.06 at
$400\mathrm{\,Ge\kern-1.00006ptV}$ and 0.17 at
$1\mathrm{\,Te\kern-1.00006ptV}$ [96]. Mixing angles even reaching down to
0.01 for dark higgs mass below $3\mathrm{\,Te\kern-1.00006ptV}$ could be
tested with $20\,\mathrm{ab}^{-1}$ of a future
$14\mathrm{\,Te\kern-1.00006ptV}$ muon collider [97].

In the intermediate mass range ($10\mathrm{\,Ge\kern-1.00006ptV}\lesssim
m_{h}\lesssim 100\mathrm{\,Ge\kern-1.00006ptV}$), the HL-LHC will be able to
probe mixing angles around 0.005 [98]. A future electron-positron collider
will provide further improvements on $\sin\theta$ using both associate
production $e^{+}e^{-}\to Z\phi$ [99] and exotic decays of the SM-like Higgs
boson $h\to\phi\phi$ [100]. The former will be particularly important in the
otherwise difficult to constrain mass range above $m_{h}/2$.

#### Future experiments probing low-mass dark scalars

Many experiments have been proposed to improve the sensitivity to dark Higgs
bosons in the GeV range. Among the most mature and promising proposals are
SHiP [101], SHADOWS [102], HIKE [103], FASER 2 [104], DUNE [105], CODEX-b
[106], MATHUSLA [107], FACET [108] and DarkQuest [109]. But also existing
experiments such as Belle II [23, 70] and LHCb [110] promise substantial gains
in sensitivity in coming years. A detailed review of all of these ideas is
beyond the scope of this review, and we therefore refer to the activities of
the Feebly Interacting Particle Physics Centre of the Physics Beyond Colliders
Initiative at CERN999See Ref. [111, 112, 113] and https://pbc.web.cern.ch/fpc-
mandate. instead.

##  6 Searches for decays into dark photons

While it is plausible to assume that the dark Higgs boson is the lightest
state in the dark sector, it is also possible to have
$m_{A^{\prime}}<m_{\phi}/2$. This case has two important phenomenological
consequences, as illustrated in figure 1(e). First, the dark photon may give a
relevant contribution to dark Higgs boson production via dark Higgs-strahlung.
And second, both dark and SM-like Higgs bosons can now decay into pairs of
dark photons, which would subsequently decay into SM particles (typically
pairs of charged leptons or mesons) through mixing with the SM
photon.101010For a detailed discussion of the dark photon branching ratios, we
refer to Ref. [114]. Several experiments have searched for the resulting
signatures. A summary of these searches is shown in figure 6.1.

Figure 6.1:  Constraints on dark Higgs models arising from searches for dark
photons. The upper limits from Belle (90 % CL [115]) and BaBar (90 % CL [116])
in the left half of the plot result from searches for $e^{+}e^{-}\to
A^{\prime}\phi(\to A^{\prime}A^{\prime})$, which constrains the product of the
kinetic mixing parameter $\epsilon$ and the dark gauge coupling $g^{\prime}$.
The upper limits from CMS (95 % CL [117]) and ATLAS (95 % CL [118]) in the
right half of the plot result from searches for $pp\to h\to
A^{\prime}A^{\prime}$, which constrains the Higgs mixing parameter
$\sin\theta$.

###  6.1 Decays of the dark Higgs boson

####  $e^{+}e^{-}\to A^{\prime}\phi,\phi\to A^{\prime}A^{\prime}$ at Belle

Belle at the asymmetric $e^{+}e^{-}$ collider KEKB in Japan has used the full
dataset of 977 $\,\mathrm{fb}^{-1}$ at center-of-mass energies corresponding
to the $\Upsilon(1S)$ to $\Upsilon(5S)$ resonances to search for dark Higgs-
strahlung in fully visible final states $A^{\prime}\to
e^{+}e^{-},\mu^{+}\mu^{-},\pi^{+}\pi^{-}$ assuming $m_{\phi}>2m_{A^{\prime}}$
with both promptly decaying dark Higgs and dark photons [115]. The process
involves the coupling of the dark photon to SM particles via kinematic mixing
$\epsilon$ with the SM photon, and the coupling of the dark photon to the dark
Higgs $\alpha_{\mathrm{D}}\equiv g^{\prime 2}/(4\pi)$. For dark photon masses
$0.1\,\mathrm{\,Ge\kern-1.00006ptV}<m_{A^{\prime}}<3.5\,\mathrm{\,Ge\kern-1.00006ptV}$
and dark Higgs masses
$0.2\,\mathrm{\,Ge\kern-1.00006ptV}<m_{\phi}<10.5\,\mathrm{\,Ge\kern-1.00006ptV}$,
Belle has set 90% CL upper limits on the product coupling
$\epsilon^{2}\alpha_{\mathrm{D}}$ of the level of $10^{-10}$–$10^{-8}$. The
Belle limits are the strongest limit to date in these mass ranges.

####  $e^{+}e^{-}\to A^{\prime}\phi,\phi\to A^{\prime}A^{\prime}$ at BaBar

BaBar has used a dataset of 516 $\,\mathrm{fb}^{-1}$ at center-of-mass
energies corresponding to the $\Upsilon(2S)$ to $\Upsilon(4S)$ resonances to
search for dark Higgs-strahlung with a technique comparable to the
aforementioned Belle analysis [116]. For dark photon masses
$0.25\,\mathrm{\,Ge\kern-1.00006ptV}<m_{A^{\prime}}<3.0\,\mathrm{\,Ge\kern-1.00006ptV}$
and dark Higgs masses
$0.8\,\mathrm{\,Ge\kern-1.00006ptV}<m_{A^{\prime}}<10.0\,\mathrm{\,Ge\kern-1.00006ptV}$,
BaBar has set 90% confidence level upper limits on the product coupling
$\epsilon^{2}\alpha_{\mathrm{D}}$ of the level of $10^{-9}$–$10^{-8}$.

###  6.2 Decays of the SM-like Higgs boson

If the dark Higgs boson mixes with the SM-like Higgs boson, the latter
inherits the coupling of the former to dark photons. For
$m_{A^{\prime}}<m_{h}/2$ it then becomes possible for SM-like Higgs bosons to
decay into pairs of dark photons [119]. According to the Goldstone equivalence
theorem, for $m_{\phi},m_{A^{\prime}}\ll m_{h}$ the corresponding decay width
$\Gamma_{h\to A^{\prime}A^{\prime}}$ is equal to $\Gamma_{h\to\phi\phi}$ as
given in eq. (3.1) [120]. In other words, constraints on $\Gamma_{h\to
A^{\prime}A^{\prime}}$ can be directly translated into bounds on the Higgs
mixing parameter $\sin\theta$ (for a given value of the dark vev $w$).

####  $pp\to h\to A^{\prime}A^{\prime}$ at CMS

Using an integrated luminosity of $137\,\mathrm{fb}^{-1}$ of
$13\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions collected between 2016 and
2018, CMS has carried out a search for SM-like Higgs bosons decaying into a
pair of dark photons, which subsequently decay into either $4\mu$, 2e2$\mu$,
or 4e [117]. The analysis covers the dark photon mass range from
$4.2\mathrm{\,Ge\kern-1.00006ptV}$ to $60\mathrm{\,Ge\kern-1.00006ptV}$
(except for a veto region around the $\Upsilon$ mass) and achieves a
sensitivity of approximately $\sin\theta\lesssim 10^{-4}$–$10^{-3}$, depending
on the assumed value of the dark photon mass but independent of the dark Higgs
boson mass as long as $m_{\phi}\ll m_{h}$.

####  $pp\to h\to A^{\prime}A^{\prime}$ at ATLAS

The corresponding ATLAS search [118] extends the mass range considered by CMS
by an additional window between $1\mathrm{\,Ge\kern-1.00006ptV}$ and
$2\mathrm{\,Ge\kern-1.00006ptV}$. While the search in the high mass range
between 15 and $60\mathrm{\,Ge\kern-1.00006ptV}$ explores $4\mu$, 2e2$\mu$, or
4e final states, the search in the low mass region up to
$15\mathrm{\,Ge\kern-1.00006ptV}$ focuses on $4\mu$ final states only. A
sensitivity comparable to CMS is achieved.

###  6.3 Future projections and proposed experiments

The ATLAS and CMS bounds on exotic Higgs decays discussed above are expected
to improve by an order of magnitude with HL-LHC data [98], implying
corresponding improvements for the bounds on $\sin\theta$ [52]. At future
electron-positron colliders, the most promising search strategy is analogous
to the Belle and BaBar searches discussed above, but with the dark photon
replaced by a $Z$ boson, i.e. $e^{+}e^{-}\to Z+\phi(\to
A^{\prime}A^{\prime})$.

Finally, we would like to mention that it may also be possible to search for
dark Higgs bosons decaying into dark photons in exotic $Z$ boson decays.
Indeed, Ref. [121] has pointed out that dark photons also mix with the SM $Z$
boson, and hence one can search for the exotic decay $Z\to A^{\prime}\phi(\to
A^{\prime}A^{\prime})$. The authors estimate that for dark Higgs and dark
photon masses above the $B$ meson threshold, LHC searches for this decay mode
may be competitive with direct bounds on the kinetic mixing parameter
$\epsilon$ from LHCb and CMS.

##  7 Searches for invisible dark Higgs decays

Figure 7.1:  Upper limits on the mixing angle $\theta$ as function of scalar
mass $m_{\phi}$ from NA62 (95 % CL [79]), E949 (90 % CL [76]), and BaBar (95 %
CL [122] and this work). Constraints colored in gray with dashed outline are
reinterpretations not performed by the experimental collaborations and without
access to raw data. The green dashed lines indicates the parameter
combinations that correspond to branching ratios between $5\times
10^{-7}\leq\mathcal{B}(B^{+}\to K^{+}\phi)\leq 1\times 10^{-5}$.

If the dark Higgs boson couples to a DM particle with $m_{\chi}<m_{\phi}/2$,
we expect invisible decays to dominate the branching ratios of the dark Higgs
boson. While this would obviously make it much harder to conclusively discover
dark Higgs bosons, we can exploit signatures with missing (transverse) energy
to probe and constrain such models. The specific search strategies depend on
the range of $m_{\phi}$.

###  7.1 Heavy dark Higgs bosons ($m_{\phi}>m_{h}$)

For heavy dark Higgs bosons, the case of an invisibly decaying dark Higgs
boson is experimentally very challenging, as it combines a moderate production
cross section with a rather unspecific final state. At the LHC, traditional DM
searches for jets in association with missing energy are generally not
sensitive to invisibly decaying dark Higgs bosons, because the production
cross section is either suppressed by a small Yukawa coupling, if the dark
Higgs is emitted from a light quark, or by a loop factor, if the dark Higgs is
emitted from a top-quark loop [123]. This makes it interesting to consider
searches for heavy-quark final states in association with missing energy. Due
to destructive interference effects in the production of a single top quark
and a Higgs boson [124], the strongest limits are provided by the
$t\bar{t}+\text{MET}$ channel.

####  $pp\to t\bar{t}\phi(\to\text{inv})$ at ATLAS

ATLAS has performed a statistical combination [125] of final states with 0
[126], 1 [127] or 2 [128] leptons, $\ell=e,\mu$, (all with an integrated
luminosity of $139\,\mathrm{fb}^{-1}$). The result is an observed upper bound
of approximately $\sin^{2}\theta\lesssim 0.2$ for $m_{\phi}\lesssim m_{h}$.

####  $pp\to t\bar{t}\phi(\to\text{inv})$ at CMS

CMS has released a search for large missing transverse momentum in association
with a $t\bar{t}$ pair using $137\,\mathrm{fb}^{-1}$ of data recorded at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ between 2016 and 2018 [129]. The
analysis combines previous searches in final states with 0 [130], 1 [131] or 2
[132] leptons, $\ell=e,\mu$. While the primary target of the analyses is stop-
quark pair production, the combined result is re-interpreted in a simplified
DM model with scalar mediators and can be translated into limits on the mixing
angle $\sin^{2}\theta\lesssim 0.2$ for $m_{\phi}\lesssim m_{h}$.

The result is thus comparable to that published by ATLAS and significantly
weaker than the bound from the SM-like Higgs signal strength. We therefore do
not show a figure for this case.

###  7.2 Light dark Higgs bosons ($m_{\phi}\ll m_{h}$)

For sufficiently light dark Higgs bosons, invisible decays lead to strong
constraints from searches for $K\to\pi\phi(\to\text{inv)}$, which we already
discussed in the context of long-lived dark Higgs bosons in section 5. In
addition, it now becomes interesting to also consider rare $B\text{-meson}$
decays involving invisible final states.

####  $B\to K\phi(\to\mathrm{inv})$ at BaBar

In Ref. [122] BaBar presents results in terms of $\mathcal{B}(B\to
K\nu\bar{\nu})$ for different bins of $s_{B}=m_{\nu\nu}^{2}/m_{B}^{2}$. Since
the detector resolution for $s_{B}$ is at the level of a few percent, the
signal from an invisibly decaying dark Higgs boson is expected to be strongly
peaked at $s_{B}=m_{\phi}^{2}/m_{B}^{2}$. We hence reinterpret the BaBar
result by assuming that the dark Higgs signal is fully contained in a single
bin, multiplying the quoted uncertainties by 1.96 to obtain the approximate
95% CL upper bound. We show the result of this reinterpretation together with
the bounds from NA62 and E949 and the constraint from the SM-like Higgs signal
strength in Fig. 7.1.

####  $e+e^{-}\to\mu^{+}\mu^{-}\phi(\to\text{inv})$ at KLOE-2

Additional search strategies open up if dark Higgs bosons can be produced in
association with a kinetically-mixed dark photon (see figure 1(d)) for dark
Higgs boson masses as large as 0.5$\mathrm{\,Ge\kern-1.00006ptV}$. In this
case the production cross section becomes proportional to
$\epsilon^{2}\alpha_{\mathrm{D}}$, where $\alpha_{\mathrm{D}}=g^{\prime
2}/(4\pi)$ governs the probability for dark Higgs-strahlung. The final state
is then a pair of SM fermions with invariant mass equal to the dark photon
mass together with missing energy from the invisible dark Higgs boson decay.
The first search for this signature was performed by the KLOE-2 experiment
[133], which used bins of $m_{\mu\mu}$ and $m_{\text{miss}}$ to reach a
sensitivity of $\epsilon^{2}\alpha_{\mathrm{D}}\lesssim
10^{-9}\text{--}10^{-8}$ for sub-GeV dark photons.

####  $e+e^{-}\to\mu^{+}\mu^{-}\phi(\to\text{inv})$ at Belle II

Belle II has searched for dark Higgs bosons produced in association with a
kinetically-mixed dark photon via $e^{+}e^{-}\to
A^{\prime}(\to\mu^{+}\mu^{-})\phi(\to\mathrm{inv.})$ for dark Higgs boson
masses as large as 4.5$\mathrm{\,Ge\kern-1.00006ptV}$ using an early data set
based on an integrated luminosity of $8.34\,\mathrm{fb}^{-1}$ [134]. The best
sensitivity ($\epsilon^{2}\alpha_{\mathrm{D}}\lesssim 10^{-7}$) is achieved
for heavy dark photons, where the energy in visible final states is maximised.

###  7.3 Future projections and proposed experiments

At the moment, the LHC only probes models of invisibly decaying scalars that
have an enhanced production cross section compared to Higgs mixing.
Nevertheless, in particular for the two-lepton final state, current searches
are still limited by statistics, and improvements can be expected from HL-LHC
[135]. Further improvements can be achieved with future lepton colliders,
where the dark Higgs boson would be produced via Higgs mixing in association
with a $Z$ boson. Ref. [136] showed that the ILC with
$\sqrt{s}=250\,\mathrm{GeV}$ and an integrated luminosity of
$2\,\mathrm{ab}^{-1}$ may be sensitive to mixing angles as small as
$\sin\theta\approx 0.02$. Even stronger bounds may be obtained from future $Z$
factories, using the exotic decay $Z\to Z^{\ast}\phi$ followed by
$Z^{\ast}\to\ell^{+}\ell^{-}$ and an invisible dark Higgs boson decay [137].

For dark Higgs boson masses below the $B\text{-meson}$ mass, significant
advances are expected from Belle II with larger data sets. With an integrated
luminosity of only 63 $\,\mathrm{fb}^{-1}$ and using an inclusive tagging
method, Belle II already excluded $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})>4.1\times 10^{-5}$ at the 90% CL [138]. This inclusive
method provides a higher signal efficiency and also sensitivity compared to
the hadronic tag method used by BaBar, but has almost no mass resolution. As a
result, this method is ideal to constrain the three-body decay $B^{+}\to
K^{+}\nu\bar{\nu}$, where the missing mass of the neutrino pair follows a
broad distribution, but is less sensitive than the hadronic tag method for the
two-body decay $B^{+}\to K^{+}\phi$, where the narrow peak in the missing mass
provides a key handle to distinguish signal and background (see Ref. [139] for
a detailed discussion in the context of a similar model). This also makes it
difficult to interpret the inclusive tag method in case of anomalously large
$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$ as a signal of dark Higgs
bosons.111111As pointed out in Ref. [140], it is possible to obtain a broad
distribution distribution of missing masses from the decay $B\to
K\chi\bar{\chi}$ via an off-shell dark Higgs boson. The branching ratio scales
parametrically as $\sin^{2}\theta y_{\chi}^{2}/m_{\phi}^{4}$, which is
unobservably small (given experimental bounds on $\sin\theta$) unless there is
a second contribution to the mass of the DM particle such that $y_{\chi}\gg
m_{\chi}/w$.

The Belle II sensitivity to measure the SM $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})$ for an integrated luminosity of 50 $\,\mathrm{ab}^{-1}$
using a hadronic tag method like BaBar is about 11% [141]. Assuming that Belle
II observes a SM-like branching fraction $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})=(5.67\pm 0.38)\times 10^{-6}$ [142] with that precision,
one can exclude dark Higgs mixing angles down to about $\sin\theta>10^{-3}$
(compare Fig. 7.1). For the inclusive tag a similar sensitivity of about 11%
is expected [143], while combining the different tags could yield a
sensitivities as low as 8%. A combined analysis of all available
$B\text{-meson}$ decays $B^{0}\to K^{0}_{S}\phi$, $B^{0}\to K^{*0}\phi$, and
$B^{+}\to K^{+}\phi$ in an optimized search for this two-body decay instead of
the SM three body decay may provide even better sensitivity.

Finally, Ref. [143] has considered a search at Belle II analogous to the ones
discussed in section 6.1, i.e. $e^{+}e^{-}\to A^{\prime}\phi$, but with the
dark Higgs boson decaying invisibly (rather than into a pair of dark photons).
The sensitivity that can be achieved is comparable to current bounds on
$\epsilon g^{\prime}$ for fully visible decays.

##  8 Searches for dark Higgs bosons in events with missing energy

In the previous section we considered dark Higgs models with an additional
dark fermion $\chi$ with $m_{\chi}<m_{\phi}/2$. If this inequality is not
satisfied, the dark Higgs boson cannot decay invisibly. Nevertheless, the dark
fermions may still plan an important role in the search for dark Higgs bosons
if both are produced together. This can happen for example if a dark photon is
produced at a collider and radiates off a dark Higgs boson before decaying
into a pair of dark fermions (see figures 1(d) and 1(e)), or if a dark Higgs
boson is radiated . Such events can be searched for at the LHC by looking for
large missing transverse momentum produced in association with a dark Higgs
boson.

As pointed out in Ref. [144], for sufficiently heavy dark photon masses, the
cross section for $pp\to\chi\chi\phi$ may be comparable to the one for
$pp\to\chi\chi j$, where the jet arises from initial state radiation. However,
the decays of the dark Higgs boson lead to much more distinctive experimental
signatures, such as boosted topologies, meaning that backgrounds can be more
easily suppressed and greater sensitivities can be achieved. While in
principle any decay mode of the dark Higgs boson may be of interest, only two
been investigated so far: the decay of the dark Higgs boson into two $b$-quark
jets as well as the decay into two vector bosons.

####  $pp\to\chi\bar{\chi}\phi(\to b\bar{b})$ at ATLAS

Decays into bottom quarks dominate if the dark Higgs boson has a mass below
about $135\,\mathrm{GeV}$. This decay mode was targeted by ATLAS in Ref. [145]
by reinterpreting a search for DM produced in association with a SM-like Higgs
boson based on an integrated luminosity of $79.8\,\mathrm{fb}^{-1}$ [146]. The
reinterpretation was made using the RECAST framework [147] and considers
different $E^{\text{miss}}_{T}$ bins, which correspond to either two small-
radius $b$ jets with radius parameter $R=0.4$ (resolved case) or a single
large-radius jet with $R=1.0$ (merged fat jet). Discrimination between signal
and background is achieved by considering the invariant mass of the jet pair
(or the fat jet), which for the signal is strongly peaked at $m_{\phi}$. Very
recently, the same strategy was applied in Ref. [148] to a DM search based on
the full 13 $\mathrm{\,Te\kern-1.00006ptV}$ $pp$ LHC dataset [149] using
active learning. This approach makes it possible to obtain approximate limits
for values of the dark photon couplings different from the ones usually
considered. The sensitivity of this re-interpretation exceeds that of Ref.
[145] and is thus the only one shown in Fig. 8.1 for the $E_{T}^{\rm miss}+bb$
final state.

####  $pp\to\chi\bar{\chi}\phi(\to VV\to q\bar{q}q\bar{q})$ at ATLAS

If the dark Higgs boson decays dominantly into gauge bosons, a number of
different search strategies become available. In Ref. [150] ATLAS considers
the fully hadronic final state using $139\,\mathrm{fb}^{-1}$ of data, which
offers the possibility to reconstruct the dark Higgs boson mass, in particular
if the dark Higgs is produced with sufficient boost that all its decay
products are merged into a so-called track-assisted reclustering (TAR) jet.
Nevertheless, this final state suffers from large backgrounds from
$V+\text{jets}$, which limit the achievable sensitivity.

####  $pp\to\chi\bar{\chi}\phi(\to WW\to q\bar{q}\ell\nu)$ at ATLAS

Substantially higher sensitivity can be achieved in the semi-leptonic final
state, where backgrounds are reduced at the expense of losing information
about the dark Higgs boson mass. In Ref. [151], ATLAS performs such a search
both for the case that the two jets are resolved or that they are merged into
a TAR jet. Special care is taken for highly boosted Higgs bosons where the
lepton potentially merges with the TAR jet so that only hadronic objects are
included and the reconstructed mass close to the $W$ boson is preserved. They
construct a kinematic variable $m_{s}^{\text{min}}$, which satisfies
$m_{s}^{\text{min}}<m_{\phi}$ and can be used to separate signal and
background.

####  $pp\to\chi\bar{\chi}\phi(\to WW\to q\bar{q}\ell\nu)$ and
$pp\to\chi\bar{\chi}\phi(\to WW\to\ell\nu\ell\nu)$ at CMS

In Ref. [152] CMS considered the same final state for the full run-2 dataset
using a BDT to separate signal from background. In addition, the analysis also
considers the fully leptonic final state, for which a kinematic variable
called $m_{T}^{\ell_{\text{min}},p_{T}^{\text{miss}}}$ can be constructed,
which peaks at the $W$ boson mass for the background but takes larger values
for the signal.

Figure 8.1:  95 % CL exclusion limits on a visibly decaying dark Higgs boson
produced in association with an invisibly decaying dark photon $A^{\prime}$
(or $Z^{\prime}$) as a function of the two masses $m_{A^{\prime}}$ and
$m_{\phi}$ from ATLAS [148, 150, 151] and CMS [152]. Here we assume that SM
quarks carry direct charge $g_{q}=0.25$ under the new $U(1)^{\prime}$ gauge
group, while the DM particles carry charge $g_{\chi}=1$ and have a mass
$m_{\chi}=200\,\mathrm{GeV}$ such that the decay $\phi\to\chi\bar{\chi}$ is
kinematically forbidden. The Higgs mixing angle has been fixed to
$\sin\theta=0.01$, but its precise value is inconsequential.

A comparison of existing results from the LHC is shown in figure 8.1. In this
plot the dimensionality of the parameter space has been reduced by fixing
$m_{\chi}=200\,\mathrm{GeV}$ and $\sin\theta=0.01$, with the precise value of
the Higgs mixing angle being inconsequential as long as it is large enough to
ensure prompt dark Higgs boson decays. Furthermore, it is assumed that the
dark photon has flavour-universal direct couplings to SM quarks given by
$g_{q}=0.25$ and a direct coupling to the dark fermion $g_{\chi}=1$. For a
detailed discussion of how these choices affect the exclusion limits, we refer
to Ref. [144].

##  9 Proposed searches for dark Higgs bosons in events with displaced
vertices

In this section we discuss future searches for dark Higgs bosons in events
with with displaced vertices, arising from the decay of a neutral long-lived
particle.121212We emphasize that neutral long-lived particles may also decay
exclusively into neutral final states, such as photons or neutral pions, which
would not allow for the reconstruction of the decay vertex. These decays are
much more difficult to identify and distinguish from background, and will
therefore not be considered further in this review. Such a signature may arise
from processes analogous to the ones considered in section 8, if one of the
particles produced in association with the dark Higgs boson is long-lived
rather than stable. This set-up is realised in models of Pseudo-Dirac dark
matter, where two states $\chi_{1,2}$ with mass splitting $\delta$ couple off-
diagonally to the dark photon, leading to the decay chain
$A^{\prime}\to\chi_{1}\chi_{2}(\to\chi_{1}+f\bar{f})$, where $f\bar{f}$
denotes a pair of charged SM states produced via an off-shell dark photon.

Since $\chi_{2}$ decays into a three-body final state, the invariant mass of
the visible final states follow a broad distribution, which makes it difficult
to distinguish signal from background even if a displaced vertex can be
reconstructed. Collider searches for Pseudo-Dirac dark matter are therefore
most promising if additional particles can be produced in association with the
$\chi_{1}$–$\chi_{2}$ pair. Such additional particles may either come from
initial-state radiation [153, 154] or if the dark photon emits dark Higgs-
strahlung before decaying [18]. In the latter case, we can expect another pair
of charged SM states, emitted either from a prompt or displaced vertex (see
the left diagram in Fig 2(b)). This second pair has an invariant mass equal to
$m_{\phi}$, leading to a striking signature that can be readily distinguished
from backgrounds. Ref. [18] studied the sensitivity of Belle II to this
signature, finding promising prospects already for early data sets.

Depending on the mass spectrum of the dark sector, there can be a number of
variations of the signature discussed above. For example, if $m_{\phi}<\delta$
and if the dark Higgs boson is responsible for generating the mass splitting,
it may also be possible to produce a dark Higgs boson in the decay
$\chi_{2}\to\chi_{1}+\phi$ [155]. If the excited dark matter state is long-
lived, this decay chain leads to a displaced vertex even if the dark Higgs
decays promptly (see the diagram in Fig 2(a)). In contrast to the case
discussed above, the invariant mass of the particles originating from the
displaced vertex would reconstruct to the dark Higgs boson mass. Conversely,
if $m_{\phi}>2m_{\chi_{2}}$, it may be possible to produce long-lived
particles in the decays of a dark Higgs boson, produced either via Higgs
mixing or dark Higgs-strahlung. This process has been studied in Ref. [156] in
order to estimate the sensitivity of FASER to inelastic dark matter. It is
however not possible in this set-up to reconstruct the properties of the dark
Higgs boson from the visible decay products.

To conclude this section, let us mention a search for long-lived dark photons
produced in dark Higgs boson decays carried out by the ATLAS experiment [157]
with an integrated luminosity of $36.1\,\mathrm{fb}^{-1}$. In this search, the
dark photons are assumed to be several orders of magnitude lighter than the
dark Higgs boson, such that they are produced with high boost and their decay
products resemble a displaced jet. ATLAS then searches for pairs of such
displaced dark photon jets. While the primary target are exotic decays of the
SM-like Higgs boson, the search is also interpreted in terms of heavier scalar
resonances.

##  10 Constraints from cosmology, astrophysics and non-collider experiments

To conclude our discussion of dark Higgs bosons, we discuss the role that they
may play in the early universe, in astrophysical systems and as mediator of
the interactions of dark matter.

###  10.1 Cosmological and indirect detection constraints

The first question of interest is whether dark Higgs bosons would thermalise
with the plasma of SM particles. While thermalisation can in principle happen
already in the unbroken phase (i.e. before the two Higgs bosons aquire their
vacuum expectation values), it is most efficient once the two Higgs bosons can
mix with each other. The various processes that contribute to the
thermalisation of dark Higgs bosons have been studied in detail in Ref. [158].
The dominant contributions are found to come from $h\to\phi\phi$ (if
kinematically allowed) and $qg\to q\phi$. Broadly speaking, the conclusion is
that for $\sin\theta\gtrsim 10^{-7}$ the dark Higgs boson will enter into
thermal equilibrium with the SM thermal bath, although this equilibrium may
not be maintained at low temperatures, when the number densities of the SM
Higgs bosons and heavy quarks become Boltzmann suppressed.

If dark Higgs bosons enter into thermal equilibrium with the SM thermal bath,
it is essential that they decay or annihilate away before the beginning of Big
Bang Nucleosynthesis (BBN) in order to avoid spoiling the successes of
standard cosmology. A commonly quoted [159] requirement on the lifetime of the
dark Higgs boson is $\tau<1\,\mathrm{s}$, but closer analysis reveals some
dependence of this bound on the dark Higgs mass and the decay products. For
example, for a dark Higgs boson mass of $100\,\mathrm{MeV}$, the bound is
$\tau<0.2\,\mathrm{s}$ [160], while even stronger bounds are expected if pions
are produced in the dark Higgs boson decays [161].

If, on the other hand, the mixing angle is so small that the dark Higgs bosons
do not enter into thermal equilibrium, it may still be possible to produce
them non-thermally via the freeze-in mechanism [162, 163]. In this case, the
constraints from BBN are relaxed considerably, and it is in fact possible for
dark Higgs bosons to be stable on cosmological scales, such that they may
constitute the dominant form of dark matter [164, 165, 166]. The leading
constraints on this scenario stem from searches for mono-energetic x-ray and
$\gamma$-ray lines produced in the decays of dark Higgs bosons [164], bounds
on dark matter self-interactions from the Bullet Cluster [167] and, for dark
Higgs boson masses in the keV range, from warm dark matter bounds [168, 169].
The viable mixing angles are so small that the dark Higgs boson would be
unobservable in laboratory experiments, and therefore we do not consider this
case further.131313We note that it is also possible for dark Higgs bosons to
decay between BBN and recombination, provided their abundance is small enough.
For a more detailed discussion, we refer to Ref. [170].

Even if the dark Higgs boson itself does not constitute a dark matter
candidate, it may help to explain the observed dark matter relic abundance via
the freeze-out mechanism either as a mediator for dark matter annihilations
(i.e. processes of the form $\text{DM DM}\to\phi^{(\ast)}\to\text{SM SM}$
[171]) or as final state in the secluded annihilation process $\text{DM
DM}\to\phi\phi$ [172]. The latter case is particularly predictive, as it fixes
the coupling $y_{\chi}$ between dark matter and dark Higgs bosons as a
function of the dark matter mass (with negligible dependence on $m_{\phi}$ for
$m_{\phi}\ll m_{\text{DM}})$). To first approximation, the observed relic
abundance is reproduced for $y_{\chi}\approx
0.04\times\sqrt{m_{\text{DM}}/\text{GeV}}$ [173].141414We emphasize that this
estimate assumes that the dark sector remains in thermal equilibrium with the
SM thermal bath during freeze-out. While this assumption is questionable if
the dark Higgs is the only mediator between the dark and the visible sector
[174], it is generally plausible in models that also contain a dark photon.

If annihilations into dark Higgs bosons are responsible for setting the dark
matter relic abundance, these processes may still be observable in the present
universe through indirect detection experiments. Indeed, it has been shown in
Refs. [175, 176] that this set-up can in principle fit the $\gamma$-ray
Galactic Centre Excess [177, 178] for dark matter masses around
$100\,\mathrm{GeV}$. However, in the standard set-up with a fermionic dark
matter particle, annihilation into a pair of Higgs bosons is a $p$-wave
process, meaning that the cross section today is suppressed proportional to
the dark matter velocity squared, $v_{\text{DM}}^{2}\sim 10^{-6}$. Ref. [179]
pointed out a promising alternative, namely the $s$-wave process
$\text{DM}+\text{DM}\to\phi+Z^{\prime}$. Indeed, if this process is
kinematically allowed, one finds strong constraints from Fermi-LAT
observations of dwarf spheroidal galaxies [180]. Furthermore, Ref. [181]
showed that bound-state formation may lift the $p$-wave suppression and lead
to strong indirect detection constraints (see also Ref. [182]). More complex
indirect detection signatures have been discussed in Ref. [183].

An entirely different avenue to probe dark Higgs bosons is to search for the
stochastic gravitational wave background emitted during the phase transition
that leads to spontaneous symmetry breaking, which is found to be first-order
in parts of the parameter space [184, 185]. Whether or not the resulting
signal may be observable with near-future gravitational wave observatories
depends on both the strength of the transition as well as on the temperature
of the dark sector relative to the SM thermal bath [186, 187]. In addition,
the phase transition may also lead to the production of dark matter particles
through the decay of dark Higgs bosons crossing the bubble walls that separate
the two phases, even if $m_{\phi}\ll m_{\text{DM}}$ [188].

Figure 10.1:  Upper limits on the spin-independent DM-nucleon cross section
$\sigma$ as function of DM mass $m_{\mathrm{DM}}$ for heavy dark Higgs bosons
from LZ [189]. For limits from direct dark Higgs searches see Sec. 5, and for
limits from Higgs signal strength see Sec. 4. Constraints colored in gray with
dashed outline are reinterpretations not performed by the experimental
collaborations and without access to raw data. Figure 10.2:  Upper limits on
the spin-independent DM-nucleon cross section $\sigma$ as function of DM mass
$m_{\mathrm{DM}}$ for light dark Higgs bosons from CRESST-III [190], DarkSide
[191], XENON1T [192], and LZ [189]. For limits from direct dark Higgs
searches, limits from flavour experiments and fixed-target experiments see
Sec. 5, and for limits from Higgs signal strength see Sec. 4. We also indicate
the approximate lower bound on $\sin\theta$ imposed by BBN [160] and the
parameter region excluded by the duration of the neutrino signal from SN1987a
[193]. Constraints colored in gray with dashed outline are reinterpretations
not performed by the experimental collaborations and without access to raw
data.

###  10.2 Astrophysical and direct detection constraints

Sub-GeV dark Higgs bosons can be produced in astrophysical systems and thereby
constitute an exotic cooling mechanism [194]. Of particular interest in this
context is SN1987a, which is commonly interpreted as a core-collapse supernova
explosion. Dark Higgs bosons can be produced in the hot and dense supernova
core via nucleon-nucleon bremsstrahlung ($NN\to NN+\phi$) [159]. If the dark
Higgs bosons escape from the supernova core without decaying or being
absorbed, they would reduce the luminosity in neutrinos, in conflict with
observations [195]. A careful recent analysis [193] showed that this
consideration excludes mixing angles in the range $\sin\theta\sim
10^{-6}\text{--}10^{-5}$ for $m_{\phi}\lesssim 100\,\mathrm{MeV}$. For sub-MeV
dark Higgs bosons, even stronger bounds arise from the cooling of horizontal
branch stars and red giants, which require $\sin\theta\lesssim 10^{-9}$ for
$m_{\phi}<10\,\mathrm{keV}$ [196].

Furthermore, dark Higgs bosons may mediate the scattering of dark matter
particles on nuclei, leading to potentially observable signals in direct
detection experiments.151515Due to Higgs mixing, DM-nucleon scattering can
also be mediated by the SM Higgs boson. In the limit $m_{\phi}\gg m_{h}$ one
recovers so-called Higgs portal DM models, which we will not discuss further
here. Instead, we refer to Ref. [197] for a recent analysis of these models.
The DM-nucleon scattering cross section at zero momentum transfer is given by

$\sigma_{N}=\frac{\mu_{N}^{2}\,f_{N}^{2}\,m_{p}^{2}\,y_{\chi}^{2}\,\cos^{2}\theta\,\sin^{2}\theta}{2\pi\,v^{2}}\left(\frac{1}{m_{\phi}^{2}}-\frac{1}{m_{h}^{2}}\right)^{2}\;,$
(10.1)

where $\mu_{N}$ is the reduced DM-nucleon mass and $f_{N}\approx 0.3$ is the
effective Higgs-proton coupling [198] with some uncertainty due to the
strange-quark content of the nucleon [199].

For $m_{\phi}\gg m_{\text{DM}}$ the DM annihilation cross section is also
proportional to $y_{\chi}^{2}\cos^{2}\theta\sin^{2}\theta/m_{\phi}^{4}$, such
that direct detection constraints can be directly compared to the relic
density requirement $\Omega_{\text{DM}}\approx 0.12$ [200]. The outcome of
this comparison is that, given the very strong constraints from liquid-xenon
based experiments like XENONnT [201], PandaX-4T [202] and LZ [189], it is very
difficult to reproduce the observed relic density unless the nuclear recoil
energy is below threshold, corresponding to sub-GeV DM masses. For such small
DM masses, however, the relic density requirement is typically incompatible
with constraints on the invisible Higgs decay $h\to\chi\chi$, see Ref. [197].

The more attractive option is therefore that $m_{\phi}<m_{\text{DM}}$, and the
relic density requirement simply fixes $y_{\chi}$ as a function of
$m_{\text{DM}}$ (see above). In this case, direct detection constraints place
an upper bound on the mixing angle $\theta$ for given values of
$m_{\text{DM}}$ and $m_{\phi}$. These bounds get stronger as the ratio
$m_{\text{DM}}/m_{\phi}$ increases. In the following, when showing direct
detection constraints, we will conservatively take
$m_{\text{DM}}/m_{\phi}=2$.161616We note that this choice is generally
compatible with the assumption that the DM particle obtains its mass from the
dark Higgs boson via a suitable choice of the dark vev $w$. In figures 10.1
and 10.2 we show the comparison of direct detection constraints with other
constraints on dark Higgs bosons. The leading constraints at 90% confidence
level stem from CRESST-III [190], DarkSide [191], XENON1T [192], and LZ [189].
These constraints are found to be stronger than the ones from invisible Higgs
decays.

No relevant constraints on $\sin\theta$ arise from searches for DM-electron
scattering due to the smallness of the corresponding Yukawa coupling. However,
these searches are very sensitive to the couplings $g^{\prime}$ and $\epsilon$
of the dark photon, offering an alternative avenue to constrain dark Higgs
boson models.

##  11 Outlook

Dark Higgs bosons arising from the spontaneous breaking of a $U(1)^{\prime}$
gauge extension of the SM provide a simple and well-motivated framework to
connect the key frontiers of modern particle physics:

  * •

detailed studies of the production and decay modes of the SM-like Higgs boson;

  * •

LHC searches for new particles at the TeV scale, for example extended Higgs
sectors;

  * •

precision measurements of rare meson decays;

  * •

exotic collider signatures with high-multiplicity final states and/or missing
energy;

  * •

searches for long-lived particles at high-intensity facilities.

At the same time, dark Higgs bosons may be directly connected to the DM
puzzle, providing a viable mechanism to reproduce the observed DM relic
abundance and predicting detectable signals in laboratory experiments and
astrophysical observations.

On all of these fronts, tremendous progress can be expected over the next two
decades, as the LHC and Belle II continue to collect data and new experiments
with high intensity or ultra-low background will be constructed. Among the
most exciting opportunities in the near future are the measurement of $B\to
K+\text{invisible}$ at Belle II and the construction of a new beam-dump and/or
a new forward-physics facility at CERN.

At the same time, progress in the theoretical description of dark Higgs bosons
is needed to match the experimental improvements. For example, there is still
considerable uncertainty in the lifetime and branching ratios of dark Higgs
bosons with mass at the GeV scale, and in the constraints imposed on dark
Higgs bosons from stellar cooling and supernova explosions. Finally, the
rapidly growing field of gravitational wave detection motivates more detailed
studies of the phase transition that triggers spontaneous symmetry breaking.

In this review, we have shown how results from many different experiments and
measurements can be presented or reinterpreted in terms of dark Higgs bosons,
both in minimal models, where the dark Higgs boson is the only kinematically
accessible state, and in simple extensions, where dark photons and DM
particles may give rise to distinctive experimental signatures. This
comparison makes the complementarity of different experimental strategies
apparent and highlights the need to either agree on benchmark models across
collaborations, or to make all relevant data publicly available. We hope that
the plots shown in this review will be populated with many more exclusion
contours and, eventually, closed confidence regions.

## Acknowledgements

It is a great pleasure to thank (in alphabetical order) Thomas Biekötter,
Pawel Guzowski, Belina von Krosigk, Gaia Lanfranchi, Ulrich Nierste, Maksym
Ovchynnikov and Martin W. Winkler for discussions, and Thomas Biekötter,
Sascha Dreyer, Maksym Ovchynnikov, Kai Schmidt-Hoberg, and Susanne Westhoff
for feedback and comments on earlier versions of the manuscript. This work was
supported by the Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF)
through the Young Investigators Group VH-NG-1303, and by the Deutsche
Forschungsgemeinschaft (DFG) through the Emmy Noether Grant No. KA 4662/1-2
and grant 396021762 – TRR 257. Moreover, the authors acknowledge funding by
the DFG under Germany‘s Excellence Strategy – EXC 2121 “Quantum Universe” –
390833306.

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