($m_{s}=200\,{\rm GeV}$). It is also evident from the figure that our recast
of the dijet Aad  _et al._ (2020b) and $t\bar{t}$ search Sirunyan  _et al._
(2019d) has a higher mass reach in $m_{Z^{\prime}}$ than the mono-$X$
searches. We note that the 2MDM model can also give rise to $hh+E_{T}^{\rm
miss}$ and $t\bar{t}+E_{T}^{\rm miss}$ signatures. The former signal can be
constrained for instance by using the results of Sirunyan  _et al._ (2018f);
Aaboud  _et al._ (2018e) that studies final states with at least three
$b$-jets and $E_{T}^{\rm miss}$ (see also Blanke  _et al._ (2019) for an
exploration of the $hh+E_{T}^{\rm miss}$ signature in simplified models of
hidden sectors). Estimating the sensitivity of multi $b$-jet plus $E_{T}^{\rm
miss}$ searches to the $hh+E_{T}^{\rm miss}$ signature in the 2MDM model is
however beyond the scope of this review. The latter signal can be targeted by
standard $t\bar{t}+E_{T}^{\rm miss}$ searches (cf. for example Aaboud  _et
al._ (2018a); Aad  _et al._ (2021c, e)). Employing Aad  _et al._ (2021e) we
expect that for (45) the HL-LHC with $3\,{\rm ab}^{-1}$ may be able to set the
bounds $M_{Z^{\prime}}\gtrsim 500\,{\rm GeV}$ for $m_{s}\simeq 750\,{\rm GeV}$
and $M_{Z^{\prime}}\gtrsim 1.5\,{\rm TeV}$ for $m_{s}\simeq 400\,{\rm GeV}$.
We finally add that the 2HDM model can lead to interesting LLP signatures if
the mixing between the SM and the dark Higgs is switched off, because in such
a case the dark Higgs can only decay through EW gauge boson loops. For details
see the recent publication Bernreuther  _et al._ (2021).

##  5 Models with exotic Higgs decays involving LLPs

BSM scenarios with hidden sectors that are connected to the SM sector through
the $125\,{\rm GeV}$ Higgs boson are being actively explored at colliders.
Such scenarios are often characterised by new electrically neutral LLPs. These
LLPs typically decay into SM particles, leaving a displaced vertex signature
in the detector. The identification of such signatures is non-trivial and
often requires dedicated triggering and reconstruction algorithms — see Lee
_et al._ (2019); Alimena  _et al._ (2020) for a detailed review of
experimental aspects of LLPs at the LHC. Generally speaking, LLP signatures
can arise in a multitude of BSM models ranging from SUSY, theories of neutral
naturalness, hidden valley models and composite Higgs theories, to just name a
few examples. Depending on the context, the appearance of BSM LLPs can be
theoretically motivated for instance by the naturalness problem of the Higgs
mass, the DM puzzle, baryogenesis or the smallness of neutrino masses — a
comprehensive collection of BSM theories with LLPs can be found in Curtin  _et
al._ (2019). Below we will consider three hidden sector models in which the
LLPs arise from decays of the $125\,{\rm GeV}$ Higgs boson. The collider
phenomenology of axion-like particles (see for example Bauer  _et al._ (2017b,
2019a) for recent comprehensive studies) that might also lead to LLP
signatures from Higgs decays will not be discussed.

###  5.1 Neutral naturalness

The discovery of a light, seemingly elementary Higgs boson has escalated the
seriousness of the EW hierarchy problem, while the steadily increasing LHC
limits on coloured BSM states exclude more and more of the natural parameter
space of the standard solution to the EW hierarchy problem such as SUSY or
compositeness. Models of neutral naturalness like twin Higgs Chacko  _et al._
(2006), folded SUSY Burdman  _et al._ (2007), quirky little Higgs Cai  _et
al._ (2009) and orbifold Higgs Craig  _et al._ (2015a) provide compelling
alternative solutions to the EW hierarchy problem. In these theories, the
large radiative corrections to the Higgs mass associated with the top quark,
are cancelled by top partners that carry no colour, thereby relaxing the most
stringent LHC constraints that follow from strong production. The cancellation
is achieved with discrete symmetries that must be nearly exact in the top
sector, but may be approximate for the other partner or mirror states Craig
_et al._ (2015b). In fact, since the QCD coupling drives the renormalisation
group (RG) running of the top-quark Yukawa coupling, for the near-exact
discrete symmetry in the top sector to be preserved, viable theories of
neutral naturalness contain at least one new QCD-like hidden gauge group with
a coupling $\hat{\alpha}_{s}$ that is comparable in strength to its SM
counterpart $\alpha_{s}$. Connecting DM with neutral naturalness is possible
(see for instance Garcia Garcia  _et al._ (2015a); Craig and Katz (2015);
Garcia Garcia  _et al._ (2015b); Farina (2015); Freytsis  _et al._ (2016);
Farina  _et al._ (2016); Barbieri  _et al._ (2016, 2017); Hochberg  _et al._
(2019); Cheng  _et al._ (2018); Terning  _et al._ (2019); Koren and McGehee
(2020)) but more model-dependent than the LLP phenomenology on which we will
focus in the following.

####  5.1.1 Theory

In models of neutral naturalness the coupling between the $125\,{\rm GeV}$
Higgs boson and the top partners gives rise to an effective coupling
$h\hat{g}\hat{g}$ between the $125\,{\rm GeV}$ Higgs $h$ and the hidden gluons
$\hat{g}$ at the one-loop level. This is in full analogy to the SM where top-
quark loops provide the dominant contribution to the effective $hgg$ coupling.
The effective $h\hat{g}\hat{g}$ interactions can be parameterised by

${\cal L}=\frac{\hat{\alpha}_{s}}{12\pi}\hskip 0.7113pt\frac{h}{v}\hskip
1.42262pt\hat{G}_{\mu\nu}^{a}\hat{G}^{a,\mu\nu}\hskip 0.7113pt\hat{\zeta}\,,$
(50)

where $\hat{G}_{\mu\nu}^{a}$ denotes the field strength tensor of the hidden
$SU(3)$ gauge group and $\hat{\zeta}$ is a model-dependent mixing angle. One
generically has $|\hat{\zeta}|={\cal O}(v^{2}/M^{2})$ where depending on the
model $M$ is either the scale of spontaneous global symmetry breaking or the
mass of the top partners. Fine-tuning arguments bound the scale $M$ to lie at
or below the TeV scale, implying that the relevant mixing angles fall into the
range $0.1\lesssim|\hat{\zeta}|\lesssim 1$.

The coupling (50) provides a portal for production of states in the hidden QCD
sector. Once produced, states in the hidden sector cascade down to the
lightest accessible mirror state, which is typically a bound state of hidden
QCD. This state can then decay back to SM particles with the associated
lifetime depending on the exact nature of the hidden sector. A canonical
signal that can appear in models of neutral naturalness without light mirror
matter, such as folded SUSY and quirky little Higgs and some realisations of
twin Higgs, is the formation of mirror glueballs Juknevich  _et al._ (2009);
Juknevich (2010).

The lightest mirror glueball is a scalar state with quantum numbers
$J^{CP}=0^{++}$ and its mass $\hat{m}_{0}$ is entirely determined by the RG
running of the strong coupling $\hat{\alpha}_{s}$ in the hidden sector. In
fact, $\hat{m}_{0}$ is related to the mass $m_{0}\simeq 1.7\,{\rm GeV}$ of the
$0^{++}$ glueball in QCD Morningstar and Peardon (1999); Chen  _et al._
(2006); Meyer (2009) by the following simple rescaling Curtin and Verhaaren
(2015)

$\hat{m}_{0}=\frac{\hat{\Lambda}_{\rm Landau}}{\Lambda_{\rm
Landau}}\,m_{0}\,,$ (51)

with $\Lambda_{\rm Landau}\simeq 150\,{\rm MeV}$ the Landau pole in QCD, i.e.
the scale where $1/\alpha_{s}(\Lambda_{\rm Landau})=0$. Without knowing the
exact mass spectrum of states in the hidden sector it is not possible to give
a precise value of $\hat{\Lambda}_{\rm Landau}$ and therefore $\hat{m}_{0}$.
One can however estimate the typical range of $\hat{m}_{0}$ values by studying
the following toy model. Let $M_{\ast}$ be the scale where the discrete
symmetry $\hat{\alpha}_{s}(M_{\ast})=\alpha_{s}(M_{\ast})$ between the two
strong couplings is broken and denote with $M$ the scale of the lightest top
partner. The one-loop beta function in the hidden sector takes the form

$\hat{\beta}=\left\\{\begin{matrix}11-\displaystyle\frac{2}{3}\hskip
0.35565pt\hat{N}_{f}\,,&M<\mu<M_{\ast}\,,\\\\[11.38109pt]
11\,,&\mu<M\,,\end{matrix}\right.$ (52)

where $\hat{N}_{f}$ denotes the number of flavours that are active for
renormalisation scales $\mu$ in the range $M<\mu<M_{\ast}$. Solving the one-
loop RG equation of $\hat{\alpha}_{s}$ it is easy to show that the Landau pole
in our toy model occurs at

$\hat{\Lambda}_{\rm
Landau}=M_{\ast}\,\left(\frac{M_{\ast}}{M}\right)^{-\frac{2\hskip
0.35565pt\hat{N}_{f}}{33}}e^{-\frac{2\pi}{11\hskip
0.35565pt\alpha_{s}(M_{\ast})}}\,,$ (53)

where

$\alpha_{s}(M_{\ast})=\alpha_{s}(m_{t})\hskip
1.42262pt\left[1-\frac{7\alpha_{s}(m_{t})}{2\pi}\ln\left(\frac{m_{t}}{M_{\ast}}\right)\right]^{-1}\,,$
(54)

with $\alpha_{s}(m_{t})\simeq 0.11$. Assuming now that there is only one light
hidden state of mass $M$, while all other mirror states have a mass
$M_{\ast}$, and taking $M=v$ and $M_{\ast}=1\,{\rm TeV}$ or $M=2\,{\rm TeV}$
and $M_{\ast}=20\,{\rm TeV}$, one finds using (51), (53) and (54) the
following approximate range of mirror glueball masses:

$15\,{\rm GeV}\lesssim\hat{m}_{0}\lesssim 50\,{\rm GeV}\,.$ (55)

While more sophisticated calculations (see for instance Curtin and Verhaaren
(2015)) lead to slightly larger mass ranges, they do not change the conclusion
that a representative mirror sector gives rise to glueballs that can be pair-
produced in the decays of the $125\,{\rm GeV}$ Higgs. Glueball production is
therefore a smoking gun signature in many theories of neutral naturalness.

Under the assumption that the $0^{++}$ mirror glueballs are dominantly
produced in symmetric two-body Higgs decays, one can estimate the
corresponding exclusive Higgs branching ratio. One finds Curtin and Verhaaren
(2015)

${\rm BR}\left(h\to 0^{++}0^{++}\right)=\left(1-\frac{4\hskip
0.35565pt\hat{m}_{0}^{2}}{m_{h}^{2}}\right)^{1/2}\left(\frac{\hat{\alpha}_{s}(m_{h})}{\alpha_{s}(m_{h})}\,\hat{\zeta}\right)^{2}\,\kappa(\hat{m}_{0})\,{\rm
BR}\left(h\to gg\right)_{\rm SM}\,,$ (56)

where ${\rm BR}\left(h\to gg\right)_{\rm SM}\simeq 8.2\%$ is the Higgs to
digluon branching ratio in the SM. The parameter $\kappa(\hat{m}_{0})$ encodes
our ignorance about the hadronisation of the lightest mirror glueball and the
mixing effects of excited hidden glueball states with the $125\,{\rm GeV}$
Higgs. In the article Curtin and Verhaaren (2015) it has been argued that
$\kappa(15\,{\rm GeV})\simeq 0.1$ and $\kappa(50\,{\rm GeV})\simeq 1$. Using
these numbers as well as the estimate $\hat{\zeta}\simeq v^{2}/M^{2}$, one
obtains for the two benchmarks that led to (55) the following range of
branching ratios:

$2\cdot 10^{-5}\lesssim{\rm BR}\left(h\to 0^{++}0^{++}\right)\lesssim 1\cdot
10^{-2}\,.$ (57)

In view of all the approximations and estimates that went into (57), the given
range should only be taken as an indication of the typical values of the $h\to
0^{++}0^{++}$ branching ratio that arise in theories of neutral naturalness.

Since the $0^{++}$ mirror glueball has the same quantum numbers as the SM
Higgs both states mix by virtue of (50). Once produced, the $0^{++}$ mirror
glueballs can hence decay to all kinematically available SM particles $Y$ via
an off-shell Higgs, i.e. through the process $0^{++}\to h^{\ast}\to YY$. The
corresponding partial decay widths have been calculated in the work Juknevich
(2010) and in the case of (50) take the form

$\Gamma\left(0^{++}\to
YY\right)=\left(\frac{\hat{\alpha}_{s}(\hat{m}_{0})\hskip
0.7113pt\hat{f}_{0}\hskip 1.42262pt\hat{\zeta}}{6\hskip 0.35565pt\pi\hskip
0.35565ptv\left(m_{h}^{2}-\hat{m}_{0}^{2}\right)}\right)^{2}\;\Gamma\left(h^{\ast}\to
YY\right)_{\rm SM}\,,$ (58)

where $\hat{f}_{0}$ is the annihilation matrix element of the lightest mirror
glueball through (50) and $\Gamma\left(h^{\ast}\to YY\right)_{\rm SM}$ denotes
the partial decay width for a SM Higgs boson with mass $\hat{m}_{0}$. It
follows that the decay pattern of the lightest scalar mirror glueball
resembles that of a SM Higgs boson of appropriate mass. For mirror glueball
masses in the range (55) the $0^{++}$ decays around 80%, 10% and 10% of the
time to $b\bar{b}$, $c\bar{c}$ and $\tau^{+}\tau^{-}$ final states,
respectively. Using again $\hat{\zeta}\simeq v^{2}/M^{2}$ as well as
$\hat{\alpha}_{s}(\hat{m}_{0})\hat{f}_{0}\hskip 1.42262pt\simeq 0.18\hskip
0.7113pt\hat{m}_{0}^{3}$ Meyer (2009); Curtin and Verhaaren (2015) together
with (58) the proper decay length of the lightest mirror glueball can be
approximated by

$c\hskip 0.35565pt\tau_{0^{++}}\simeq 2\,{\rm m}\,\cdot\,\left(\frac{15\,{\rm
GeV}}{\hat{m}_{0}}\right)^{7}\left(\frac{M}{1\,{\rm TeV}}\right)^{4}\,.$ (59)

For typical values of $\hat{m}_{0}$ and $M$ realised in theories neutral
naturalness, the proper decay length of the $0^{++}$ mirror glueball ranges
from microns to kilometers. In fact, the strong scaling of (59) with both
$\hat{m}_{0}$ and $M$ suggests that $c\hskip 0.35565pt\tau_{0^{++}}$ can in
practice be treated as an almost free parameter in the framework of neutral
naturalness.

Notice that hidden valley models Strassler and Zurek (2007, 2008); Han  _et
al._ (2008) share many of the phenomenological features discussed above. Like
in theories of neutral naturalness also in hidden valley models a new
confining gauge group is added to the SM. However, the confining gauge group
in hidden valley models makes, in full analogy to QCD, hidden hadrons out of
hidden quarks. If the hidden sector comprises two light flavours of quarks the
spectrum of hidden hadrons contains a hidden pion $\pi_{h}$. Given its
pseudoscalar nature the hidden pion preferentially decays to heavy SM flavours
for example $\pi_{h}\to b\bar{b}$. As argued in the articles Strassler and
Zurek (2007, 2008); Han  _et al._ (2008), the typical $\pi_{h}$ masses and
proper decay lengths fall into the ballpark of (55) and (59), respectively.
Hidden valley models may also contain hidden Higgses which partake in the mass
generation of the hidden quarks. If one of these hidden Higgs fields mixes
with the $125\,{\rm GeV}$ Higgs boson it is possible to obtain
$h\to\pi_{h}\pi_{h}$ branching ratios that are observable at the LHC Strassler
and Zurek (2007, 2008); Han  _et al._ (2008). The phenomenology of the
$\pi_{h}$ is therefore very similar to that of the $0^{++}$ with the most
obvious difference that the hidden pion is a pseudoscalar whereas the lightest
mirror glueball is a scalar.

While the above considerations broadly motivate searches for displaced Higgs
decays at the LHC, the models presently employed in the interpretation of such
searches by the experimental collaborations are more generic than the theories
of neutral naturalness or the hidden valley models discussed above. The used
simplified models assume SM production of a Higgs boson and its subsequent
decay to a pair of scalar ($s$) or pseudoscalar ($a)$ particles. The $s$ ($a$)
is assumed to decay like the SM Higgs, while its mass $m_{s}$ ($m_{a}$), the
relevant Higgs branching ratio ${\rm BR}\left(h\to ss\right)$ $\big{(}$${\rm
BR}\left(h\to aa\right)$$\big{)}$ and its proper decay length $c\tau_{s}$
($c\tau_{a}$) are treated as free parameters. For what concerns models of
neutral naturalness, this approach is motivated by (55), (57) and (59). The
scalar (pseudoscalar) case can be thought to cover theories of neutral
naturalness (hidden valley models) with the lightest mirror glueball (the
hidden pion) being the LLP. For definiteness we will hereafter refer to the
LLP produced in exotic Higgs decays as an $a$ particle.

####  5.1.2 Experimental constraints

The first searches for pair-produced neutral LLPs in the context of Higgs
portal models have been performed by the CDF Aaltonen  _et al._ (2012) and DØ
Abazov  _et al._ (2009) collaborations at the Tevatron. Both searches looked
for displaced vertices in their tracking system only, thereby setting limits
on LLP mean decay lengths of the order of a few centimetres. At the LHC,
searches for Higgs decays into LLP have been carried out by the ATLAS, CMS and
LHCb collaborations in different final states, covering a wide range of mean
decay lengths. The LLP mean decay length determines the search strategies and
reconstruction techniques that are employed. Below, we discuss the relevant
LHC searches, starting with the shortest mean decay lengths considered.

ATLAS has performed searches for the decay $h\to aa\to 4b$ optimised for
prompt decays or small proper decay lengths $c\tau_{a}\lesssim 6\cdot
10^{-3}\,{\rm m}$ Aaboud  _et al._ (2018f); Aad  _et al._ (2020e). The
searches select events corresponding to associated $Vh$ production with decays
of the EW bosons into leptons, as displayed on the left-hand side of Figure
19. Since the targeted LLP mean decay lengths were small, standard
track/vertex reconstruction and $b$-jet identification techniques were used.
LHCb has also performed a search for $gg\to h\to aa$ with $a$ decaying to
hadronic jets Aaij  _et al._ (2017), which is sensitive to small mean decay
lengths in the ballpark of a few millimetres. The corresponding signal process
is illustrated on the right in Figure 19. At least one displaced vertex was
required in the event due to the limited acceptance of the LHCb vertex
detector. The data were recorded by requiring the presence of an energetic
charged lepton or hadron in the event in the hardware trigger, and either one
track with high transverse momentum ($p_{T})$ and a large impact parameter, or
a displaced vertex of at least two tracks in the software trigger. Further
requirements on the displaced vertex properties were applied in the last stage
of the software trigger and in the data analysis.

Figure 19:  Representative diagrams that can give rise to LLP signatures at
the LHC. The left and right graph displays associated $Zh$ production followed
by $Z\to\ell^{+}\ell^{-}$ and $h\to aa$ and ggF Higgs production with $h\to
aa$, respectively. The $a$ dominantly decays to bottom-quark pairs leading in
both case to a four-bottom final state. Most existing searches have targeted
this final state but analyses that look for multi-jet events have also been
performed.

For mean decay lengths in the range of $10^{-3}\,{\rm m}$ to $1\,{\rm m}$, a
substantial fraction of the LLPs is expected to decay inside the inner
detector (ID) of the LHC experiments. This allows for a direct reconstruction
of the displaced decay vertex and hence dramatically reduces the SM background
rate, which becomes dominated by long-lived hadrons and instrumental
backgrounds. At the same time, the macroscopic mean decay lengths
significantly decrease the efficiency of standard track reconstruction
algorithms and thereby also the identification efficiency of $b$-jets from LLP
decays. Searches relying on displaced vertex signatures in the inner tracker
systems of ATLAS and CMS are described in the following.

In order to significantly increase the efficiency for reconstructing displaced
vertices, dedicated reconstruction techniques, so-called large-radius tracking
(LRT), were developed at ATLAS ATL (2017). Properties of displaced vertices
reconstructed from LRT tracks like the invariant mass or the number of
associated tracks can then be used to discriminate signal from background.
Since the LRT reconstruction algorithms are computationally intensive, they
were not employed in the ATLAS trigger. Hence, the ATLAS search employing LRT
techniques targeted the associated $Zh$ production channel with $h\to aa\to
4b$ and $Z\to\ell^{+}\ell^{-}$. The leptons from the $Z\to\ell^{+}\ell^{-}$
decays were used for triggering, and backgrounds were essentially eliminated
by requiring two displaced vertices in candidate events Aad  _et al._ (2021i).

CMS has recently searched for $h\to aa\to 4b$ and $Z\to\ell^{+}\ell^{-}$ using
the same process CMS (2021d). A trigger and selections based on dilepton
$Z$-boson decays provide sensitivity to light LLPs with masses of $15\,{\rm
GeV}$ or less. The decays of the LLPs are selected by requiring the presence
of displaced jets which are identified using information from the tracking
system. CMS in addition searched for the $h\to aa\to 4b$ process targeting
$c\tau_{a}$ between $10^{-3}\,{\rm m}$ and $3\,{\rm m}$ by requiring displaced
vertices consistent with LLP decays to be reconstructed in the inner tracker
Sirunyan  _et al._ (2019f, 2021b). Unlike the ATLAS analysis Aad  _et al._
(2021i), the CMS search Sirunyan  _et al._ (2021b) considered the ggF topology
and relied on jets for triggering. Two triggers were used, both requiring a
large scalar sum of transverse jet energies in the event and at least two
energetic jets consistent with a LLP decay, i.e. at most two associated prompt
tracks with $p_{T}>1\,{\rm GeV}$ and at least one track consistent with
originating from a displaced vertex. Benefiting from the large ggF Higgs
production cross section, the CMS search Sirunyan  _et al._ (2021b) provides
currently the best sensitivity for proper decay lengths between $10^{-3}\,{\rm
m}$ and $10^{-1}\,{\rm m}$. CMS has also searched for displaced leptons
arising from ggF Higgs production followed by $h\to aa\to 4\ell$ CMS (2021c).
Since this search assumes that the LLP has equal probability to decay to two
muons and two electrons, whereas (58) implies that ${\rm
BR}\left(a\to\mu^{+}\mu^{-}\right)={\cal O}(10^{-4})$ and ${\rm BR}\left(a\to
e^{+}e^{-}\right)={\cal O}(10^{-8})$, an interpretation of CMS (2021c) in the
context of neutral naturalness/hidden valleys leads to no meaningful
constraint on the $h\to aa$ branching ratio.

For mean decay lengths in the range of $1\,{\rm m}$ to $10^{2}\,{\rm m}$, a
significant fraction of the LLPs is expected to decay in the outermost layers
of the detector, i.e. inside the calorimeter (CM) or the muon spectrometer
(MS). Searches in this regime are described in the following.

ATLAS searched for $h\to aa\to 4f$ decays in the CM targeting event topologies
compatible with ggF Higgs production Aaboud  _et al._ (2019f). The decay of a
LLP inside the CM is typically reconstructed as a single jet with striking
characteristics, namely a narrow width in pseudorapidity-azimuthal-angle
($\eta\hskip 0.7113pt$–$\hskip 0.7113pt\phi$) space, a high ratio of energy
deposited in the hadronic CM to that registered in the electromagnetic CM, and
no or only a few low-momentum tracks associated with it. The search uses these
characteristics both in dedicated triggers Aad  _et al._ (2013) and when
performing the offline analysis employing an artificial neural network. ATLAS
recently performed a search for the same process using the MS ATL (2021d). The
tracking capabilities of the MS allow for an explicit reconstruction of a
displaced vertex, which dramatically reduces SM and instrumental backgrounds,
and allows for a dedicated triggering strategy Aad  _et al._ (2013) based on
the overall activity in the MS. The backgrounds are essentially eliminated by
requiring two displaced vertices from candidate $h\to aa\to 4b$ decays within
the MS. The sensitivity for even longer decay times can be extended by
requiring only one displaced vertex to be detected in the MS. This strategy
has been applied in a similar analysis Aaboud  _et al._ (2019g) and the
results were statistically combined with the CM search Aaboud  _et al._
(2019f). Finally, ATLAS carried out a search using the combination of the ID
and the MS, targeting short and long mean decay lengths Aad  _et al._ (2020f).

CMS performed a LLP search using the endcap of the MS Tumasyan  _et al._
(2021b), targeting $c\tau_{a}$ between $10^{-3}\,{\rm m}$ and $10^{2}\,{\rm
m}$ and decay chains $h\to aa$ with $a\to b\bar{b},c\bar{c},\tau^{+}\tau^{-}$.
While many features are similar to the corresponding ATLAS searches ATL
(2021d); Aaboud  _et al._ (2019g), the search philosophy differs in one key
aspect: the magnetic field return yoke, interleaved with the tracking layers
of the MS was used as a sampling CM. Only the endcap region of the CMS MS is
considered, as it provides a greater depth of up to 30 nuclear interaction
lengths. A dedicated algorithm was applied to cluster the hits in the muon
system, and the hit multiplicity in an azimuthal slice close to the
$E_{T}^{\rm miss}$ vector was used as the final discriminant.

ATLAS and CMS also performed other searches for neutral LLPs decaying to jets
targeting SUSY models Sirunyan  _et al._ (2018g, h, 2019g); Aad  _et al._
(2020g); Sirunyan  _et al._ (2021c) that are not explicitly optimised for
$h\to aa\to 4f$ signatures, for example due to very high trigger thresholds,
and therefore will not be discussed below.

Figure 20:  Observed exclusions at 95% CL on the branching ratio ${\rm
BR}\left(h\to aa\right)$ as a function of the proper decay length $c\tau_{a}$
of the LLP. The shown limits are taken from the ATLAS Aad  _et al._ (2021i);
Aaboud  _et al._ (2018f, 2019f), CMS Sirunyan  _et al._ (2021b); CMS (2021d)
and LHCb Aaij  _et al._ (2017) publications, respectively. For comparison the
95% CL limits on the branching ratio of undetected and invisible Higgs decays
(60) are also displayed. The boundary between the constraints from undetected
and invisible Higgs decays is taken to be $c\tau_{a}=3.5\,{\rm GeV}$
corresponding to $m_{a}=35\,{\rm GeV}$. For further explanations consult the
main text.

An indirect way to constrain models where the $125\,{\rm GeV}$ Higgs boson
decays into LLPs is provided by the combination of precision measurements of
the Higgs couplings in visible and invisible final states ATL (2020a, b); CMS
(2020b). In particular, under the assumption that the coupling modifiers
$\kappa_{V}=g_{hVV}/g_{hVV}^{\rm SM}$ of the $125\,{\rm GeV}$ Higgs boson to
EW gauge bosons satisfy $\kappa_{V}\leq 1$, which generically holds in models
with Higgs mixing, the following 95% CL constraints

${\rm BR}\left(h\to{\rm undet}\right)<0.19\,,\qquad{\rm BR}\left(h\to{\rm
inv}\right)<0.09\,,$ (60)

can be placed ATL (2020b). Here, the undetected category includes events with
undetected BSM particles that do not provide a significant $E_{T}^{\rm miss}$
contribution such as the ones typically selected by the aforementioned LLP
analyses with short mean decay lengths below ${\cal O}(1\,{\rm m})$. In
contrast, LLPs that decay outside the tracker and calorimeters, i.e. having
mean decay lengths larger than ${\cal O}(1\,{\rm m})$, are not registered by
standard reconstruction algorithms, resulting in a $E_{T}^{\rm miss}$
contribution. Hence, for moderate boosts the above bound on ${\rm
BR}\left(h\to{\rm undet}\right)$ and ${\rm BR}\left(h\to{\rm inv}\right)$ can
be interpreted as an indirect limit on ${\rm BR}\left(h\to aa\right)$ for
proper decays lengths of $c\tau_{a}\lesssim 1\,{\rm m}$ and $c\tau_{a}\gtrsim
1\,{\rm m}$, respectively. In the case of larger boosts, the boundary between
the constraints from undetected and invisible Higgs decays is instead
approximately given by $c\tau_{a}\simeq 0.1\,{\rm m}\hskip
1.42262pt(m_{a}/{\rm GeV})$ Curtin and Peskin (2018); ATL (2021e).

The 95% CL limits of the above searches on the branching ratio ${\rm
BR}\left(h\to aa\right)$ as a function of the proper decay length $c\tau_{a}$
of the LLP are shown in Figure 20. The displayed masses of the LLP $a$ all
fall into the central region of $0^{++}$ mirror glueball masses as predicted
by neutral naturalness (55). For proper decay lengths $c\tau_{a}$ of a few
meters the LHC searches are able to set a limit ${\rm BR}\left(h\to
aa\right)\lesssim 10^{-3}$. In view of (57) and (59) this is an interesting
finding, as it allows to test certain model realisations of neutral
naturalness. Notice that the limits from ${\rm BR}\left(h\to{\rm
undet}\right)$ and ${\rm BR}\left(h\to{\rm inv}\right)$ provide currently the
strongest LHC bound on ${\rm BR}\left(h\to aa\right)$ for $c\tau_{a}\lesssim
10^{-4}\,{\rm m}$ and $c\tau_{a}\gtrsim 10^{3}\,{\rm m}$, respectively.
Besides these two constraints, proper decay lengths $c\tau_{a}\gtrsim
10^{3}\,{\rm m}$ are currently unexplored by collider measurements. Dedicated
detectors like MATHUSLA Curtin  _et al._ (2019), CODEX-b Gligorov  _et al._
(2018) and ANUBIS Bauer  _et al._ (2019) may address such long mean decay
lengths at the HL-LHC.

###  5.2 Dark photons

In the case of neutral naturalness the LLP is a composite spin-0 particle.
However, hidden sector models with a spin-1 LLP also exist. The model
considered in this subsection is based on an extra $U(1)_{X}$ symmetry in the
hidden sector, where the associated vector field $X$ is given a mass via a
dark Higgs mechanism involving the singlet scalar field $S$. As explained
below, in certain regions of parameter space this model predicts displaced
dilepton vertex signatures that arise from the exotic decays of the $125\,{\rm
GeV}$ Higgs boson to a pair of dark photons $Z_{d}$ followed by
$Z_{d}\to\ell^{+}\ell^{-}$. This feature makes the discussed hidden sector
model experimentally distinct from theories of neutral naturalness where the
LLP decay products consist primarily of hadrons. We will not discuss the dark
photon searches Sirunyan  _et al._ (2019h, 2021d); Aad  _et al._ (2021j) which
are motivated by the theoretical works Gabrielli and Raidal (2014); Gabrielli
_et al._ (2014); Biswas  _et al._ (2016); Tsai  _et al._ (2017). We emphasise
that the dark photon models that are discussed in this subsection do not have
a DM candidate. For comprehensive discussions of dark photon models with an
additional DM candidate see for example the reviews Essig  _et al._ (2013);
Alexander  _et al._ (2016); Fabbrichesi  _et al._ (2020).

####  5.2.1 Theory

The interaction terms of the hidden sector model that we consider include both
a hypercharge and a Higgs portal. See Curtin  _et al._ (2014, 2015) for
details and further relevant literature. We write these terms in the following
way

${\cal L}=\frac{\epsilon}{2\cos\theta_{w}}\hskip 0.7113ptB_{\mu\nu}\hskip
0.7113ptX^{\mu\nu}-\kappa\hskip 0.7113pt(H^{\dagger}H)\hskip 0.7113ptS^{2}\,,$
(61)

where $B^{\mu\nu}=\partial^{\mu}B^{\nu}-\partial^{\nu}B^{\mu}$ and
$X^{\mu\nu}=\partial^{\mu}X^{\nu}-\partial^{\nu}X^{\mu}$ is the $U(1)_{Y}$ and
$U(1)_{X}$ gauge field strength tensor, respectively, $\epsilon$ denotes the
hypercharge mixing parameter, while $\kappa$ is the Higgs portal coupling.
After EW symmetry breaking $\langle H\rangle=(0,v/\sqrt{2})^{T}$ and
spontaneous symmetry breaking of the $U(1)_{X}$ symmetry by $\langle
S\rangle=v_{S}/\sqrt{2}$, the mass spectrum of the hidden sector model
contains two heavy neutral gauge bosons and two neutral Higgs bosons. We will
denote these states by $Z$, $Z_{d}$, $h$ and $h_{d}$. For small $\epsilon$
($\kappa$) the $Z$ ($h$) is essentially the SM $Z$ boson (Higgs boson), while
the dark photon $Z_{d}$ (dark Higgs $h_{d}$) is mostly $X$-like ($S$-like).

The hypercharge portal leads to a modification of the couplings of the neutral
gauge bosons to fermions. In the case of the $Z$-boson couplings the
corrections start at ${\cal O}(\epsilon^{2})$, while the dark photon vector
couplings receive corrections already at ${\cal O}(\epsilon)$. Explicitly, one
has

$g_{Z_{d}f\bar{f}}=\epsilon\hskip 0.35565pte\hskip
0.35565ptQ_{f}+\epsilon\hskip 0.35565pte\hskip
0.7113pt\left(Q_{f}-\frac{Y_{f}}{\cos^{2}\theta_{w}}\right)\hskip
0.7113pt\frac{m_{Z_{d}}^{2}}{m_{Z}^{2}}\,,$ (62)

where $e=\sqrt{4\pi\alpha}$ is the electromagnetic coupling and $Q_{f}$
($Y_{f}$) is the electric charge (hypercharge) of the relevant fermion. Notice
that for $m_{Z_{d}}\ll m_{Z}$ the $Z_{d}$-boson coupling to fermions is
photon-like, while for $m_{Z_{d}}\simeq m_{Z}$ the $Z_{d}$ boson couples to
fermions like the SM Z boson. Referring to the $Z_{d}$ boson as dark photon is
hence a bit of a misnomer, because a massive $Z_{d}$ boson always couples to
neutrinos. We will however follow this established naming convention. At
${\cal O}(\epsilon)$ the hypercharge portal also leads to a coupling between
the $125\,{\rm GeV}$ Higgs, a $Z$ and a $Z_{d}$ boson. To this order the
coupling takes the form

$g_{hZZ_{d}}=\frac{2\hskip
0.35565pt\epsilon\tan\theta_{w}}{v}\frac{m_{Z_{d}}^{2}m_{Z}^{2}}{m_{Z}^{2}-m_{Z_{d}}^{2}}\,.$
(63)

For $\epsilon\gtrsim{\cal O}(10^{-4})$ a dark photon with mass
$m_{Z_{d}}\gtrsim 1\,{\rm GeV}$ decays promptly Curtin  _et al._ (2014, 2015).
At the LHC, such dark photons can be searched for in Drell-Yan (DY) dimuon
production $pp\to Z_{d}\to\mu^{+}\mu^{-}$ and in four-lepton final states that
arise from the process $pp\to h\to ZZ_{d}\to 4\ell$. The corresponding Feynman
diagrams are shown on the left-hand side and in the middle of Figure 21.

The Higgs portal gives rise to a coupling between the $125\,{\rm GeV}$ Higgs
and two dark photons. To lowest order in $\kappa$ this coupling can be written
as

$g_{hZ_{d}Z_{d}}=\frac{2\hskip 0.35565pt\kappa\hskip 0.35565ptv\hskip
0.35565ptm_{Z_{d}}^{2}}{m_{h_{d}}^{2}-m_{h}^{2}}\,.$ (64)

For $m_{Z_{d}}<m_{h}/2$ this coupling allows for Higgs decays of the form
$h\to Z_{d}Z_{d}$ — a representative Feynman diagram is shown on the right in
Figure 21. At the same order in $\kappa$ also interactions between the
$125\,{\rm GeV}$ Higgs and two dark Higgses and a single dark Higgs and two
dark photons exist. In the following, we will assume that the dark Higgs is
heavy, i.e. $m_{h_{d}}\gg m_{h}/2$. In such a case the exotic Higgs decay
$h\to h_{d}h_{d}$ is kinematically forbidden and the production cross section
for $pp\to h_{d}\to Z_{d}Z_{d}$ is always smaller than $pp\to h\to
Z_{d}Z_{d}$. As a result, for sufficiently light dark photons the best probe
of the Higgs portal parameter $\kappa$ is the exotic Higgs decay $h\to
Z_{d}Z_{d}$. The corresponding partial decay width is given to leading order
in $\kappa$ by the following expression Curtin  _et al._ (2014, 2015)

$\Gamma\left(h\to Z_{d}Z_{d}\right)=\frac{\kappa^{2}\hskip
0.35565ptv^{2}}{32\hskip 0.35565pt\pi\hskip
0.35565ptm_{h}}\,\frac{(1-4x_{Z_{d}/h})^{1/2}}{\left(1-x_{h_{d}/h}\right)^{2}}\,\left(1-4x_{Z_{d}/h}+12x_{Z_{d}/h}^{2}\right)\,.$
(65)

Notice that given the small total width of the SM Higgs boson, the branching
ratio following from (65) can easily reach the few percent level for values of
$\kappa\ll 1$. For example, taking $\kappa=0.07$, $m_{Z_{d}}=30\,{\rm GeV}$
and $m_{h_{d}}=300\,{\rm GeV}$ leads to ${\rm BR}(h\to Z_{d}Z_{d})=0.15$,
which is close to the model-independent limit on ${\rm BR}(h\to{\rm undet})$
reported in (60).

Figure 21:  Example contributions in the dark photon model (61) to DY dimuon
production (left diagram) and four-lepton final state production arising from
the exotic Higgs decays $h\to ZZ_{d}$ (middle diagram) and $h\to Z_{d}Z_{d}$
(right diagram), respectively. In the latter two cases the production
mechanism of the $125\,{\rm GeV}$ Higgs boson is not shown.

The best way to constrain the Higgs portal coupling now depends on the size of
kinetic mixing. For $\epsilon\gtrsim{\cal O}(10^{-4})$ the dark photon decays
promptly and one can again study four-lepton final states Aaboud  _et al._
(2018g); CMS (2020c) to constrain the $h\to Z_{d}Z_{d}$ branching ratio and
thereby $\kappa$. For $\epsilon\lesssim{\cal O}(10^{-4})$ the decay length of
the dark photon starts to become macroscopic. In fact, for ${\cal
O}(10^{-8})\lesssim\epsilon\lesssim{\cal O}(10^{-4})$ the decays of the
$Z_{d}$ bosons are displaced with a large fraction of events ending up in the
LHC detectors Curtin  _et al._ (2014, 2015). This leads to the exciting
opportunity to probe very small values of $\epsilon$ that are inaccessible by
other means, provided some Higgs mixing is present in the dark photon model.
Below we summarise the LHC searches that have considered the case of LLPs in
dark photon models with both a hypercharge and a Higgs portal.

####  5.2.2 Experimental constraints

The ATLAS, CMS and LHCb collaborations have carried out searches for prompt
dark photon decays targeting hypercharge mixing parameters $\epsilon\gtrsim
10^{-4}$ in both the DY channel $pp\to Z_{d}\to\mu^{+}\mu^{-}$ Aaij  _et al._
(2018, 2020); Sirunyan  _et al._ (2020e) and in four-lepton production
associated with $h\to ZZ_{d}$ and $h\to Z_{d}Z_{d}$ Aaboud  _et al._ (2018g);
CMS (2020c). ATLAS has also searched for LLPs in the $h\to ZZ_{d}$ channel
Aaboud  _et al._ (2019h). In addition, smaller $\epsilon$ values have been
tested by dedicated searches for long-lived dark photons in final states with
displaced dimuon vertices arising from $h\to Z_{d}Z_{d}$. Building on earlier
LHC Run 1 analyses Khachatryan  _et al._ (2015); Aad  _et al._ (2015), the
latest ATLAS and CMS results of this kind can be found in Aaboud  _et al._
(2019h) and CMS (2021), respectively. In the following we review only LHC
searches for LLP signatures involving dark photons.

From all possible dark photon decay modes, the $Z_{d}\to\mu^{+}\mu^{-}$
process is experimentally the most accessible one. First, the LHC experiments
can trigger on muons with a high efficiency of up to 90%. Second, the rate of
SM background processes is moderate, allowing for relatively low $p_{T}$
thresholds in the trigger. Third, the sophisticated MSs have a large
acceptance for muon tracks with large impact parameters coming from displaced
vertices. As a result, a wide range of proper lifetimes can be covered by the
combination of the inner tracker employing LRT techniques and the MS.

Figure 22:  95% CL upper limits on the branching ratio $h\to Z_{d}Z_{d}$ as a
function of the decay length of the LLP. The displayed exclusions correspond
to the ATLAS Aaboud  _et al._ (2019h) and CMS Khachatryan  _et al._ (2015);
CMS (2021) results for different mass hypotheses of the dark photon. For
comparison the model-independent limits (60) found by ATLAS in ATL (2020b) are
also shown. The boundary between the constraints from undetected and invisible
Higgs decays is taken to be $c\tau_{Z_{d}}=1\,{\rm GeV}$ corresponding to
$m_{Z_{d}}=10\,{\rm GeV}$. See text for additional details.

ATLAS searched for dark photons in the $Z_{d}\to\mu^{+}\mu^{-}$ decay mode
targeting proper decay lengths of $10^{-3}\,{\rm m}<c\tau_{Z_{d}}<10^{3}\,{\rm
m}$ by requiring displaced vertices consistent with LLP decays to be
reconstructed using the muon system Aaboud  _et al._ (2019h). A combination of
single muon, dimuon and trimuon triggers with progressively lower thresholds
down to $p_{T}>6\,{\rm GeV}$, as well as a $E_{T}^{\rm miss}$ trigger have
been employed. The reconstruction efficiency for muon tracks from displaced
vertices ranges from 70% for small impact parameters to 10% at the acceptance
limit of $4\,{\rm m}$. Finally, displaced vertices of oppositely charged muons
with an invariant mass above $15\,{\rm GeV}$ are selected within the fiducial
volume.

CMS also performed a search for dark photons in $\mu^{+}\mu^{-}$ final states
CMS (2021). In contrast to the aforementioned ATLAS search, this search
analysed muon tracks reconstructed in the inner tracker, targeting small
proper decay lengths of $10^{-3}\,{\rm m}<c\tau_{Z_{d}}<10^{-1}\,{\rm m}$. The
novelty of this analysis is that it searches for pairs of oppositely charged
muons with an invariant mass down to the dimuon production threshold. This is
achieved using the so-called data scouting technique Khachatryan  _et al._
(2016a); Sirunyan  _et al._ (2020e), where only partial event information, as
reconstructed by the high-level trigger system, is recorded. This dramatically
reduces the event size, thereby allowing to store orders of magnitude more
candidate events for analysis, and lower the muon $p_{T}$ thresholds down to
$4\,{\rm GeV}$. Finally, candidate muon pairs from the
$Z_{d}\to\mu^{+}\mu^{-}$ decay are selected in events with two muons targeting
DY production, and four muons with an invariant mass consistent with the
$125\,{\rm GeV}$ Higgs boson aiming for $pp\to h\to ZZ_{d}$ and $pp\to h\to
Z_{d}Z_{d}$. A sliding window fit is then applied to the distributions of
dimuon invariant masses in several signal regions to look for a potential
excess from $Z_{d}\to\mu^{+}\mu^{-}$ decays away from the known SM resonances
like the $J\\!/\\!\psi$ and $\Upsilon$ mesons.

The region with $c\tau_{Z_{d}}<10^{-3}\,{\rm m}$ is covered by the CMS search
for displaced $Z_{d}\to e^{+}e^{-},\mu^{+}\mu^{-}$ decays performed at
$8\,{\rm TeV}$ Khachatryan  _et al._ (2015), which targets the proper decay
length range between $10^{-4}\,{\rm m}$ and $10^{2}\,{\rm m}$. This analysis
is similar to the $13\,{\rm TeV}$ CMS search Sirunyan  _et al._ (2020e),
except that it does not apply data scouting techniques and considers both the
dielectron and the dimuon channel. Both search channels rely on the dilepton
triggers.

In Figure 22 we summarise the existing 95% CL bounds on ${\rm BR}\left(h\to
Z_{d}Z_{d}\right)$ as a function of the proper decay length $c\tau_{Z_{d}}$ of
the dark photon. The shown ATLAS limits Aaboud  _et al._ (2019h) correspond to
the different dark photon masses $m_{Z_{d}}=20\,{\rm GeV},60\,{\rm GeV}$,
while in the case of CMS we display bounds for $m_{Z_{d}}=20\,{\rm
GeV},50\,{\rm GeV}$ from Khachatryan  _et al._ (2015) and for
$m_{Z_{d}}=0.65\,{\rm GeV},2\,{\rm GeV},15\,{\rm GeV},30\,{\rm GeV}$,
$50\,{\rm GeV}$ from CMS (2021). Notice that for dark photons with masses of a
few tens of GeV, the strongest bound on ${\rm BR}\left(h\to Z_{d}Z_{d}\right)$
typically arises for the lightest choice of the dark photon mass. For
$m_{Z_{d}}\lesssim 5\,{\rm GeV}$ the limits on ${\rm BR}\left(h\to
Z_{d}Z_{d}\right)$ however become significantly weaker with decreasing dark
photon mass even if data scouting techniques are employed. Specifically, for
dark photons with masses above $15\,{\rm GeV}$ the ATLAS (CMS) measurements
exclude ${\rm BR}\left(h\to Z_{d}Z_{d}\right)\gtrsim 10^{-4}$ ($10^{-5}$) for
proper decay lengths of around $10^{-1}\,{\rm m}$ ($10^{-2}\,{\rm m}$), while
for GeV-scale dark photons the corresponding CMS exclusions are weaker by
about two orders of magnitude. One also observes that for proper decay lengths
in the range of around $10^{-3}\,{\rm m}$ to $10^{2}\,{\rm m}$ the bound on
${\rm BR}\left(h\to Z_{d}Z_{d}\right)$ that derives from the LLP searches
Aaboud  _et al._ (2019h); CMS (2021) is stronger than the model-independent
limits quoted in (60). For shorter or longer proper decay lengths the ATLAS
Higgs coupling measurement ATL (2020b) however provides currently the best
constraint on the branching ratio of $h\to Z_{d}Z_{d}$.

Figure 23:  95% CL excluded regions in the $m_{Z_{d}}\hskip 0.7113pt$–$\hskip
1.42262pt\epsilon$ plane following from the ATLAS Aaboud  _et al._ (2019h) and
CMS CMS (2021) measurement. In the case of the ATLAS (CMS) results the shown
solid red (dashed blue) bound corresponds to ${\rm BR}\left(h\to
Z_{d}Z_{d}\right)=10^{-3}$. The 90% CL upper limit on the hypercharge mixing
parameter that derives from DY dimuon production by LHCb Aaij  _et al._ (2020)
and CMS Sirunyan  _et al._ (2020e) is also displayed for comparison as a solid
green and a dashed yellow line, respectively. The 95% CL limit on $\epsilon$
from the ATLAS search Aaboud  _et al._ (2018g) for $pp\to h\to ZZ_{d}\to
4\ell$ is finally indicated by a solid orange line. The latter three bounds
hold for any value of the $h\to Z_{d}Z_{d}$ branching ratio. Finally, the
indirect bound that follows from the limit (60) on invisible Higgs decays is
shown as a dashed black line. For further details consult the text.

The limits presented in Figure 22 can also be translated into 95% CL
exclusions in the $m_{Z_{d}}\hskip 0.7113pt$–$\hskip 1.42262pt\epsilon$ plane.
This is done in Figure 23 where we show the constraints that derive from the
ATLAS Aaboud  _et al._ (2019h) and CMS CMS (2021) measurement assuming ${\rm
BR}\left(h\to Z_{d}Z_{d}\right)=10^{-3}$. For the same $h\to Z_{d}Z_{d}$
branching ratio, the exclusion that follows from Khachatryan  _et al._ (2015)
almost exactly resembles that of CMS (2021) for dark photon masses between
$20\,{\rm GeV}$ and $50\,{\rm GeV}$. This limit is therefore not displayed in
the figure. One observes that $\epsilon$ values of order $10^{-8}$ and
$10^{-7}$ are excluded by these searches. Notice that the CMS search leads to
weaker limits in terms of $\epsilon$ than the ATLAS search, because CMS does
not consider decay lengths beyond $10^{-1}\,{\rm m}$. For comparison we also
depict in Figure 23 the 90% CL bounds on the hypercharge mixing parameter that
follow from the searches by LHCb and CMS in DY dimuon production Aaij  _et
al._ (2020); Sirunyan  _et al._ (2020e) as well as the ATLAS search Aaboud
_et al._ (2018g) that focuses on the $pp\to h\to ZZ_{d}\to 4\ell$ channel.
These prompt limits are a few orders of magnitude weaker than the bounds that
arise from the LLP searches, but they make no assumption about the amount of
Higgs mixing in the dark photon model. In addition, searches for $pp\to h\to
Z_{d}Z_{d}\to 4\ell$ Aaboud  _et al._ (2018g); CMS (2020c) also provide some
sensitivity to $\epsilon$, but the resulting upper limits are significantly
weaker than those that stem from the DY searches and hence not reported in the
figure.

An indirect constraint on the $m_{Z_{d}}\hskip 0.7113pt$–$\hskip
1.42262pt\epsilon$ plane can also be derived from the available limits on the
invisible Higgs branching ratio ATL (2020a, b). Imposing ${\rm
BR}\left(h\to{\rm inv}\right)=0.1$, we obtain the dashed black contour shown
in Figure 23 under the assumption that the dark photon is not registered by
standard reconstruction algorithms for $c\tau_{Z_{d}}>0.1m\hskip
1.42262pt(m_{Z_{d}}/{\rm GeV})$, resulting in a $E_{T}^{\rm miss}$
contribution. Another assumption is that the kinematic distributions such as
the $E_{T}^{\rm miss}$ spectrum are the same for $pp\to h+X\to Z_{d}Z_{d}+X$
and for the SM Higgs production channels that go into the ${\rm
BR}\left(h\to{\rm inv}\right)$ bounds given in (3) and (60). While ATL (2021e)
suggests that these are good assumptions, we cannot quantify the associated
systematic uncertainties. The shown indirect bound from invisible Higgs decays
has therefore only an indicative character. However, it should be
straightforward for ATLAS and CMS to directly reinterpret their Higgs to
invisible searches in the context of long-lived dark photons thereby improving
on our naive estimate.

###  5.3 Models with a vector and a fermion portal

Hidden sector models with both a vector and a fermion portal such as the
Falkowski-Ruderman-Volansky-Zupan (FRVZ) model Falkowski  _et al._ (2010a, b)
also include a hidden photon as a possible LLP and can lead to signatures with
displaced charged leptons. However, in models of this type, the hidden photons
are not directly produced in the exotic decays of the $125\,{\rm GeV}$ Higgs
boson, but through a cascade involving SUSY and hidden sector particles with
masses of ${\cal O}(10\,{\rm GeV})$ or below. This implies that the hidden
photons have to be even lighter than the other particles in the decay chain,
with masses of ${\cal O}(1\,{\rm GeV})$ in order to be kinematically
accessible. Due to their small mass, the hidden photons are preferentially
produced with large boosts at the LHC, resulting in so-called lepton-jets
Arkani-Hamed and Weiner (2008); Baumgart  _et al._ (2009); Cheung  _et al._
(2010), i.e. collimated groups of leptons in a jet-like structure. These
lepton-jet events are accompanied by varying amounts of $E_{T}^{\rm miss}$
depending on the precise pattern of the Higgs decay.

Figure 24:  Left diagram: Higgs decay to a pair of the lightest MSSM
neutralinos. Middle and right diagram: Possible decays of the MSSM bino to the
hidden sector, which follow from gaugino kinetic mixing (66). See text for
further details.

####  5.3.1 Theory

A minimal model that realises the general idea proposed in Falkowski  _et al._
(2010a); Arkani-Hamed and Weiner (2008); Baumgart  _et al._ (2009); Cheung
_et al._ (2010); Falkowski  _et al._ (2010b) contains a massive hidden photon
$\gamma_{d}$ that communicates with the visible sector through mixing with the
hypercharge field — see the first term in (61). In the FRVZ model, the
particle content of the visible sector is that of the minimal supersymmetric
SM (MSSM). Supersymmetrising the hypercharge portal leads to a mixing of the
visible bino ($\tilde{B}$) and the hidden gaugino ($\tilde{\gamma}_{d}$).
Removing the kinetic mixing between the $\tilde{B}$ and the
$\tilde{\gamma}_{d}$ then gives rise to interactions between all hidden fields
charged under $U(1)_{d}$ and the visible neutralinos that are proportional to
the hypercharge mixing parameter $\epsilon$. In particular, one obtains an
interaction term of the following form

${\cal L}=-\frac{\epsilon\hskip
0.7113ptg_{d}}{\cos\theta_{w}}\,\tilde{B}\hskip 0.7113pt\sum_{i}\hskip
0.7113ptq_{i}\hskip 0.7113pth_{d}^{i{\dagger}}\hskip
0.7113pt\tilde{h}_{d}^{i}\,,$ (66)

where $g_{d}$ is the $U(1)_{d}$ gauge coupling, $h_{d}^{i}$ are the hidden
scalar fields, $\tilde{h}_{d}^{i}$ are the hidden gauginos and $q_{i}$ is the
relevant $U(1)_{d}$ charge.

The interactions (66) lead to vertices involving a visible neutralino, a
hidden neutralino and a hidden photon or a hidden Higgs boson. An exotic Higgs
decay signal can therefore arise in the FRVZ model as follows. Initially the
Higgs decays into a pair of the lightest visible neutralinos
($\tilde{N}_{1}$), as indicated by the Feynman diagram on the left-hand side
of Figure 24. In the pure MSSM without the hypercharge portal, the
$\tilde{N}_{1}$ could be a DM candidate, i.e. the lightest SUSY particle or
LSP, but in the FRVZ model the presence of the term (66) allows the
$\tilde{N}_{1}$ to decay into hidden sector states. Example diagrams are shown
in the middle and on the right in Figure 24. In the first case, the
$\tilde{N}_{1}$ decays to the lightest hidden sector gaugino ($\tilde{N}_{d}$)
and a hidden photon, while in the second case the final-state $\tilde{N}_{d}$
is accompanied by a hidden Higgs boson ($h_{d}$). The lightest hidden sector
neutralino $\tilde{N}_{d}$ is stable and therefore a DM candidate, but the
$\gamma_{d}$ and the $h_{d}$ decay further. If the hidden photon is the
lightest hidden state then $\gamma_{d}\to\ell^{+}\ell^{-}$ and
$h_{d}\to\gamma_{d}\gamma_{d}\to 4\ell$ are possible decay chains that will
lead to the aforementioned lepton-jet signatures. Since both the hypercharge
portal in (61) as well as the gaugino kinetic mixing (66) are proportional to
$\epsilon$, depending on the magnitude of this parameter the lepton-jets can
be either prompt or displaced.

Figure 25:  Benchmark topologies that have been studied in the context of the
FRVZ model at the LHC. In the first process (left diagram) both neutralinos
produced in $h\to\tilde{N}_{1}\tilde{N}_{1}$ decay through
$\tilde{N}_{1}\to\tilde{N}_{d}\gamma_{d}$. The hidden photon further decays to
all kinematically accessible SM fermions, leading to four-fermion final
states. In the second process (right diagram) the neutralinos instead decay
via $\tilde{N}_{1}\to\tilde{N}_{d}h_{d}$ followed by the decay of the hidden
Higgs $h_{d}\to\gamma_{d}\gamma_{d}$. After the decay of the hidden photons
the associated final state contains eight SM fermions. The hidden neutralinos
$\tilde{N}_{d}$ escape the LHC detectors undetected and therefore appear as a
$E_{T}^{\rm miss}$ signature.

Notice that models of inelastic DM (see for instance Tucker-Smith and Weiner
(2001); Izaguirre  _et al._ (2016, 2017); Berlin and Kling (2019); Duerr  _et
al._ (2020, 2021)) also possess many of the features discussed above. Such
models typically contain two fermionic states $\chi_{1}$ and $\chi_{2}$ with a
small mass splitting $\Delta m=m_{\chi_{2}}-m_{\chi_{1}}>0$. The role of
$\tilde{N}_{1}$ ($\tilde{N}_{d}$) is played by $\chi_{2}$ ($\chi_{1}$) in
inelastic DM models, and the DM candidate $\chi_{1}$ can be excited to its
heavier twin $\chi_{2}$ by absorbing a massive dark photon $\gamma_{d}$. The
simplest realisation of such a scenario consists in assuming a hypercharge
portal and postulating a $U(1)_{X}$ symmetry that is spontaneously broken by a
dark Higgs $h_{d}$. This dark Higgs couples to a pair of dark photons. The
particle content and coupling structure of inelastic DM models therefore
resemble quite closely those of the simplest FRVZ models. The difference
between the two types of models is that, unlike the
$\gamma_{d}\tilde{N}_{1}\tilde{N}_{d}$ coupling (66), the
$\gamma_{d}\chi_{1}\chi_{2}$ coupling is not proportional to the hypercharge
mixing parameter $\epsilon$, but only involves the $U(1)_{X}$ coupling
constant. The simplest inelastic DM models are hence in some sense a
generalisation of the FRVZ idea. In fact, all the existing FRVZ
interpretations of LHC searches correspond to such a generalisation, where the
$\gamma_{d}\chi_{1}\chi_{2}$, $\gamma_{d}f\bar{f}$, $h_{d}\chi_{1}\chi_{2}$
and $h\chi_{2}\chi_{2}$ couplings are effectively treated as free parameters.
As a concrete example, the ATLAS collaboration has used the following
parameters in their interpretations

$\begin{split}&\hskip 34.1433ptg_{\gamma_{d}\chi_{1}\chi_{2}}=0.31\,,\quad
g_{h_{d}\chi_{1}\chi_{2}}=0.1\,,\\\\[5.69054pt] &m_{\chi_{1}}=2\,{\rm
GeV}\,,\quad m_{\chi_{2}}=5\,{\rm GeV}\,,\quad m_{h_{d}}=2\,{\rm
GeV}\,,\end{split}$ (67)

where the choice of $g_{\gamma_{d}\chi_{1}\chi_{2}}$ corresponds to
$g_{\gamma_{d}\chi_{1}\chi_{2}}=e=\sqrt{4\pi\alpha}$ with $\alpha$ the
electromagnetic fine structure constant at the EW scale. The hypercharge
mixing parameter $\epsilon$ and the coupling $g_{h\chi_{2}\chi_{2}}$ are not
directly used as external parameters, but expressed through
$c\tau_{\gamma_{d}}$ and ${\rm BR}\left(h\to\chi_{2}\chi_{2}\right)$,
respectively, which then serve as input. Notice that for the parameter choices
(67) and sufficiently large ${\rm BR}\left(h\to\chi_{2}\chi_{2}\right)$
values, the decays $h\to\chi_{2}\chi_{2}$ and $\chi_{2}\to\chi_{1}\gamma_{d}$
are necessarily prompt. From a theoretical point of view the generalised FRVZ
model is therefore quite similar to the dark photon model discussed in Section
5.2 if ${\rm BR}\left(h\to\chi_{2}\chi_{2}\right)$ is identified with ${\rm
BR}\left(h\to Z_{d}Z_{d}\right)$.

####  5.3.2 Experimental constraints

The ATLAS and CMS collaborations have searched for collimated groups of
charged leptons or light hadrons in a jet-like structure to constrain exotic
Higgs decays by exploring both prompt Aad  _et al._ (2016); Khachatryan  _et
al._ (2016b) and displaced Aad  _et al._ (2020g); Sirunyan  _et al._ (2019i)
signatures. The results were interpreted in the generalised FRVZ framework
described above, as well as in the context of other portal models. In the case
of the generalised FRVZ model, the published LHC searches have focused on the
two benchmark processes illustrated in Figure 25, with the Higgs boson
produced in the ggF topology.

The searches for prompt Higgs decays targeted the $h\to\gamma_{d}\gamma_{d}+X$
topology (cf. left diagram in Figure 25), where the hidden photon is assumed
to decay into charged leptons. The ATLAS search Aad  _et al._ (2016) included
$\gamma_{d}$ decays to $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$, while the CMS search
Khachatryan  _et al._ (2016b) only considered dimuons. Hadronic decays of the
hidden photon were not included, since they cannot be easily separated from
the QCD multijet background. These searches probe signal hypotheses with
hidden photon masses in the ranges of $0.1\,{\rm GeV}$ to $2\,{\rm GeV}$ for
ATLAS and $0.25\,{\rm GeV}$ and $2\,{\rm GeV}$ for CMS. In the ATLAS analysis,
events are required to have at least two lepton-jets, which are reconstructed
by clustering tracks from the primary vertex within a radius $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}=0.5$ of the highest $p_{T}$ track,
and subsequently matching them to electron or muon candidates. In the CMS
analysis, events are required to have at least two dimuon pairs, which are
reconstructed by combining muon-candidate tracks into a common vertex.

In the domain of searches targeting LLP signatures, the ATLAS analysis Aad
_et al._ (2020g) has probed the $h\to 4\gamma_{d}+X$ topology (cf. right
diagram in Figure 25) considering both leptonic and hadronic decays of the
hidden photons. The mass of the hidden photon was set to $0.4\,{\rm GeV}$,
resulting in a 10% branching ratio into pions, and the rest of the branching
ratio divided equally between electrons and muons Falkowski  _et al._ (2010a).
The dark photon decays into muon pairs were targeted by reconstructing jets of
muons that were reconstructed using the MS only, while the decays into
electron pairs and hadrons were reconstructed as calorimeter jets with a high
fraction of energy deposited in the hadronic calorimeters. In both channels,
the jets were required to be isolated from any ID activity, as expected for
LLPs, and multivariate analysis techniques using timing and topological
information of the jet were employed to discriminate signal from background.
The CMS search Sirunyan  _et al._ (2019i) explored the
$h\to\gamma_{d}\gamma_{d}+X$ topology (cf. left diagram in Figure 25),
considering only $\gamma_{d}\to\mu^{+}\mu^{-}$ decays. Hidden photon masses in
the range $0.25\,{\rm GeV}$ and $8.5\,{\rm GeV}$ were explored, with the upper
limit set by the requirement $m_{\mu\mu}<9\,{\rm GeV}$ in order to
sufficiently suppress the background from DY and $\Upsilon\to\mu^{+}\mu^{-}$
production. Proper decay lengths below $c\tau_{\gamma_{d}}<0.1\,{\rm m}$ were
targeted, explicitly including both prompt and displaced signatures. Candidate
events were required to have exactly two $\gamma_{d}\to\mu^{+}\mu^{-}$
candidates that are isolated from significant activity in the tracking system
and have an invariant mass consistent with each other.

Figure 26:  90% CL upper limits that follow from the ATLAS search Aad  _et
al._ (2020g) and the CMS search Sirunyan  _et al._ (2019i) for the exotic
Higgs boson decay $h\to\gamma_{d}\gamma_{d}+X$ in the generalised FRVZ model.
Both exclusions assume ${\rm BR}\left(h\to\gamma_{d}\gamma_{d}+X\right)=0.1$.
For comparison also constraints in the $m_{\gamma_{d}}\hskip 0.7113pt$–$\hskip
1.42262pt\epsilon$ plane from LHCb Aaij  _et al._ (2020), BaBar Lees  _et al._
(2014), KLOE-2 Anastasi  _et al._ (2018) and $\nu$-CAL I Blümlein  _et al._
(1992) are shown. The latter bounds hold for any value of the
$h\to\gamma_{d}\gamma_{d}+X$ branching ratio. Finally, the indirect upper
limit that follows from (60) is depicted. Consult the text for further
explanations.

Figure 26 contains a summary of 90% CL exclusions in the $m_{\gamma_{d}}\hskip
0.7113pt$–$\hskip 1.42262pt\epsilon$ plane in the generalised FRVZ model for
hidden photon masses in the range of $0.1\,{\rm GeV}$ to $10\,{\rm GeV}$. The
constraints following from the ATLAS Aad  _et al._ (2020g) and CMS Sirunyan
_et al._ (2019i) search for the exotic Higgs decay
$h\to\gamma_{d}\gamma_{d}+X$ apply in the generalised FRVZ model and assume
${\rm BR}\left(h\to\gamma_{d}\gamma_{d}+X\right)=0.1$. The ATLAS analysis
employs (67) and focuses on the mass range $0.2\,{\rm GeV}\lesssim
m_{\gamma_{d}}\lesssim 3.6\,{\rm GeV}$ and small kinetic mixings that lead to
LLP signatures. Depending on $m_{\gamma_{d}}$ it excludes hypercharge mixing
parameters within $1\cdot 10^{-5}\lesssim\epsilon\lesssim 3\cdot 10^{-7}$. The
CMS search probes hidden photon masses in the range $0.25\,{\rm GeV}\leq
m_{\gamma_{d}}\leq 8.5\,{\rm GeV}$ and targets proper decay lengths of
$c\tau_{\gamma_{d}}\leq 0.1\,{\rm m}$. This analysis is able to exclude
hypercharge mixing parameters $\epsilon\lesssim 3\cdot 10^{-6}$
($\epsilon\lesssim 7\cdot 10^{-8}$) at low (high) $m_{\gamma_{d}}$. The
displayed LHCb Aaij  _et al._ (2020), BaBar Lees  _et al._ (2014), KLOE-2
Anastasi  _et al._ (2018) and $\nu$-CAL I Blümlein  _et al._ (1992) bounds
have been taken from the Darkcast package developed in Ilten  _et al._ (2018).
They hold irrespectively of the value of the $h\to\gamma_{d}\gamma_{d}+X$
branching ratio. The exclusion that follows from the limit (60) is obtained
from our recast as described in Section 5.2 and assumes ${\rm
BR}\left(h\to{\rm inv}\right)=0.1$. The limit arising from the recent ATLAS
analysis ATL (2021e) of mono-jet signatures is not shown in the figure because
it is not sensitive to ${\rm BR}\left(h\to\gamma_{d}\gamma_{d}+X\right)=0.1$.
Figure 26 clearly shows that in the context of the generalised FRVZ model, LHC
searches for exotic Higgs decays provide an opportunity to probe values of
$\epsilon$ that are at present inaccessible by other means. Future LLP
experiments like MATHUSLA, CODEX-b, ANUBIS, FASER Ariga  _et al._ (2019) and
FASER2 Anchordoqui  _et al._ (2021) located at the LHC, in combination with
Belle II Duerr  _et al._ (2020, 2021), are expected to set additional
stringent constraints on the $m_{\gamma_{d}}\hskip 0.7113pt$–$\hskip
1.42262pt\epsilon$ plane in the generalised FRVZ or inelastic DM frameworks.

##  6 Outlook

In this article we have reviewed the status of the LHC constraints on DM
scenarios where the $125\,{\rm GeV}$ Higgs boson plays a prominent role.
Specifically, we have covered the case of SM Higgs portals (Section 2), models
with extended Higgs and gauge sectors with one mediator (Section 3) or two
mediators (Section 4) as well as theories that can lead to LLP signatures
induced by the decay of the $125\,{\rm GeV}$ Higgs (Section 5). In all cases,
we have collected the existing constraints from the different LHC experiments
to obtain state-of-the-art summary plots that show the current LHC
sensitivities to the relevant model parameters in various benchmark scenarios.
In some cases, we have also indicated in these summary plots the restrictions
that other measurements impose, including those of the properties of the
$125\,{\rm GeV}$ Higgs boson, flavour physics, EW precision measurements, DM
DD and ID experiments, and the relic density to just name a few.

While our review mostly focuses on “known knowns”, i.e. existing experimental
constraints and their interpretations, we have also tried to discuss aspects
of the considered models that are in principle known but that might be unknown
to at least some of the readers, i.e. “unknown knowns”. For instance, in the
case of the Higgs portal models we have emphasised in Section 2.4 that
$E_{T}^{\rm miss}$ searches for off-shell Higgs production in the VBF,
$t\bar{t}$ and $tW$, and potentially other channels provide sensitivity to the
parameter space where $m_{\rm DM}>m_{h}/2$. In fact, in the case of the
kinetic Higgs portal such searches are the only known way to test DM masses
above the Higgs threshold, making dedicated experimental searches and/or
interpretations of the relevant mono-$X$ signatures in our opinion an
important goal for future LHC runs. In the context of the Higgs portal models,
we have furthermore argued in Section 2.5 that interpreting LHC limits on the
invisible Higgs branching ratio in terms of a SI DM-nucleon cross section
leads to somewhat unintuitive results, in the sense that the indirectly
obtained $\sigma_{\rm SI}$ bounds do not become constant in the limit $m_{\rm
DM}\ll m_{h}/2$, as one would naively expect. We have explained the reason for
this unexpected feature. To avoid potential confusion, we suggest to present
the LHC limits on ${\rm BR}\left(h\to{\rm inv}\right)$ as well as the limits
on $\sigma_{\rm SI}$ obtained by DD experiments in terms of the effective
interaction strength of the different Higgs portal models, for example in
terms of $|c_{m}|$ in the case of the marginal Higgs portal. This is the
standard presentation in the theoretical literature on EFT Higgs portals, and
adding it to the interpretation of results from the LHC and/or DM DD
experiments would further promote a fruitful exchange between these
communities as well as the theory community.

We have stressed at the end of Section 3.1 that all existing collider studies
of the 2HDM+$a$ model have assumed a Yukawa sector of type-II. For this
choice, the bounds from FCNC processes and LHC searches for heavy Higgs bosons
are strong, pushing the masses of the additional 2HDM Higgs bosons above the
$500\,{\rm GeV}$ range. In fermiophobic 2HDM models of type-I the constraints
on the additional Higgs bosons can be significantly relaxed, thereby allowing
for new scalars and pseudoscalars with EW-scale masses. The fermiophobic
nature of the Higgs bosons can lead to unconventional production mechanisms
and decay patterns of the non-SM spin-0 particles. In our opinion, the
mono-$X$ phenomenology in a fermiophobic 2HDM+$a$ deserves dedicated studies.
The Yukawa sector of the model also plays an important role for the 2HDM+$s$
model considered in Section 3.2. In particular, we have shown that a suitable
choice of the Yukawa sector allows to tame excessive 2HDM+$s$ corrections to
the SI DM-nucleon cross section, opening up the possibility to test model
realisations at the LHC which lead to a viable DM phenomenology that is
consistent with the observed relic density.

The existing LHC interpretations of the 2HDM-$Z^{\prime}$ model and the 2MDM
model discussed in Section 4.1 and Section 4.2, respectively, have all focused
on the mono-Higgs channel with $h\to b\bar{b},\gamma\gamma$ in the former case
and on the mono-$s$ channel with $s\to b\bar{b},VV$ in the latter case. In
order to present a more global picture of the LHC sensitivity to these models,
we have reinterpreted existing LHC searches for heavy spin-1 resonances
decaying to visible particles. Our studies show that mono-$X$ searches are not
the only way to probe these two-mediator models at the LHC. In fact, both
discussed DM models are in general more tightly constrained by resonance
searches for the $Z^{\prime}$ boson in dijet or $t\bar{t}$ final states than
by mono-$X$ searches. We believe this to be a rather generic feature of DM
models with a heavy spin-1 mediator. Despite the dominant sensitivity of
resonance searches in the context of the 2HDM-$Z^{\prime}$ and 2MDM models,
searches for $X+E_{T}^{\rm miss}$ signals are important given their distinct
experimental signature. We have furthermore pointed out that the
2HDM-$Z^{\prime}$ model can also be probed by a search for $Z^{\prime}\to Zh$
resonances. Similarly, we advocate the exploration of the $hh+E_{T}^{\rm
miss}$ and $tt+E_{T}^{\rm miss}$ topologies in the context of the 2MDM model.
We are convinced that only through a combined exploration of the whole suite
of searches in various channels can the full LHC potential be exploited.

Searches for BSM LLPs have gained a lot of momentum in LHC Run 2. In Section
5.1, Section 5.2 and Section 5.3, we have presented state-of-the-art summary
plots for models of neutral naturalness/hidden valleys, dark photon models and
BSM theories with both a hypercharge and a fermion portal, respectively, that
can lead to LLP signatures in the decay of the $125\,{\rm GeV}$ Higgs boson.
We have stressed that an indirect way to constrain models of this type is
provided by the precision measurements of the $125\,{\rm GeV}$ Higgs
properties in visible and invisible final states. For instance, in the case of
neutral naturalness we find that the limits from ${\rm BR}\left(h\to{\rm
undet}\right)$ and ${\rm BR}\left(h\to{\rm inv}\right)$ provide currently the
strongest LHC bound on ${\rm BR}\left(h\to aa\right)$ for $c\tau_{a}\lesssim
10^{-4}\,{\rm m}$ and $c\tau_{a}\gtrsim 10^{3}\,{\rm m}$, respectively. To
further emphasise the possibility to test LLP scenarios via $E_{T}^{\rm miss}$
searches, we have in the case of the two models with a hypercharge portal
performed recasts of the limits imposed by the latest $h\to{\rm inv}$ results
to derive upper limits on the hypercharge mixing parameter $\epsilon$ as a
function of the dark photon mass.

The “known knowns” and the “unknown knowns” discussed in this review are
clearly only a snapshot of the broad landscape of collider searches for DM
through the Higgs lens. We tried our best to represent a complete picture of
experimental searches and the relevant theoretical works, and apologise for
any potential omission. We hope that our review can trigger the experimental
exploration of new search strategies and possibly even groundbreaking new
ideas or discoveries in the years leading to the HL-LHC era.

###### Acknowledgements.

We would like to thank Andreas Albert, Martino Borsato, Karri Folan Di
Petrillo and Felix Kahlhoefer for their useful comments on the manuscript.
This work is supported by the German Research Foundation (DFG) under grant No.
AR 1321/1-1 and the Isaac Newton Trust under grant No. G101121 References

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