# Review of LHC Dark Matter Searches

Felix Kahlhoefer  DESY, Notkestraße 85, D-22607 Hamburg, Germany
<EMAIL_ADDRESS>
###### Abstract

This review discusses both experimental and theoretical aspects of searches
for dark matter at the LHC. An overview of the various experimental search
channels is given, followed by a summary of the different theoretical
approaches for predicting dark matter signals. A special emphasis is placed on
the interplay between LHC dark matter searches and other kinds of dark matter
experiments, as well as among different types of LHC searches.

###### keywords:

Dark matter; Models beyond the standard model; Limits on production of
particles

PACS numbers: 95.35.+d, 12.60.-i, 13.85.Rm
Preprint number: DESY-17-024

###### Contents

  1. 1 Introduction
  2. 2 Experimental searches
  3. 3 Predicting dark matter signals
    1. 3.1 Effective theories
    2. 3.2 Simplified models
    3. 3.3 Towards complete models
    4. 3.4 Numerical tools
  4. 4 A question of complementarity
    1. 4.1 Relic density and indirect detection
    2. 4.2 Direct detection
    3. 4.3 Searches for mediators
  5. 5 Conclusions

##  1 Introduction

At first sight it may seem preposterous to search for dark matter (DM) at
hadron colliders. After all, we have so far only observed the gravitational
interactions of DM and it therefore remains unknown whether DM particles
interacts sufficiently strongly with ordinary matter to enable us to produce
them in collisions of Standard Model (SM) particles. Even if such a production
is possible in principle, the unknown DM mass means we do not know whether the
center-of-mass energy available in present experiments is sufficient for the
production to be kinematically allowed. Finally, should the production of DM
particles succeed against these odds, the resulting experimental signature
would be most unspectacular: The DM particles would be invisible to any
detector in the vicinity of the collision point and would hence reveal their
presence only via an apparent imbalance in the total transverse momentum (so-
called missing transverse momentum or, more colloquially, missing energy).

Nevertheless, searches for DM particles at the Large Hadron Collider (LHC) are
a thriving research field, which have attracted significant interest from both
the experimental and the theoretical community in recent years. The main
reason is the experimental feasibility of these searches. The SM backgrounds
and their differential distributions are now understood to sufficient accuracy
that even small distortions in the missing energy spectrum (in particular in
its high-energy tail) may be observable and can be used to constrain DM models
that predict these distortions.

This leads to the question of how to construct DM models for predicting
missing energy signals at the LHC. Clearly, any such model must contain a new
neutral stable particle consistent with the properties inferred from the
behavior of DM on astrophysical scales (regarding e.g. structure formation).
This requirement alone is however too general to provide useful guidance for
experiments and to yield instructive results. Instead, the fundamental
requirement should be that these models provide a mechanism for how DM was
produced in the early Universe, so that the predictions of the model can be
compared to the one well-measured property of DM, namely its cosmological
relic density $\Omega_{\text{DM}}\,h^{2}\approx 0.12$ [1].

One of the most successful paradigms for DM production in the early Universe
(in the sense of its predictivity and its ability to reproduce the observed
relic density) is the idea of thermal freeze-out [2]. This idea is based on
the assumption that DM particles interact sufficiently strongly with ordinary
matter that they enter into thermal equilibrium with the bath of SM states at
high temperatures. The relic abundance is then essentially set by the
temperature at which the DM annihilation rate drops below the expansion rate
of the Universe and therefore the DM interactions become insufficient to
maintain thermal equilibrium – the DM particles freeze out.

In principle, the freeze-out mechanism can work for a wide range of masses and
couplings.111In practice there is an upper bound on the DM mass of about
$m_{\text{DM}}\lesssim 100\>\text{TeV}$ from the requirement of perturbative
unitarity [3] and in many models one finds lower bounds of
$m_{\text{DM}}\gtrsim 10\>\text{GeV}$ from the requirement that DM thermal
freeze-out does not spoil the successful predictions of recombination [1].
The crucial point is however that in any case interactions between the DM
particle and SM states have to be sizable. Specifically, the velocity-averaged
DM annihilation cross section should roughly be given by

$\langle\sigma_{\text{DM DM}\to\text{SM SM}}\,v\rangle\approx 3\times
10^{-26}\mathrm{cm^{3}\,s^{-1}}\;.$ (1)

The generic expectation would then be that the inverse process also has a
sizable cross section and that therefore particle colliders can be used to
invert the annihilation processes that happened frequently in the early
Universe, see figure 1. This strong link to the idea of thermal freeze-out
justifies the excitement for DM searches at the LHC.

Figure 1: Illustration of the connection between thermal freeze-out and DM
searches at the LHC. The observed DM relic abundance can be readily reproduced
if the cross section for DM annihilation (left) is sizeable. It should then be
possible to invert this annihilation process by colliding SM particles. To
obtain an observable DM signal at the LHC, it is necessary to produce DM in
association with SM states (right).

Another key reason to search for DM at the LHC is that new stable particles at
the electroweak scale are a central ingredient of models that attempt to
address the gauge hierarchy problem, such as supersymmetry [4]. Intriguingly,
if the DM mass is comparable to the electroweak scale (i.e. of order of a few
hundred GeV) and the coupling strength is comparable to that of the weak
interactions, the required DM annihilation cross section can be obtained
rather naturally.222Such DM candidates are often referred to as weakly-
interacting massive particles (WIMPs) and the successful prediction of their
relic abundance is sometimes called the _WIMP miracle_.  In such a set-up the
LHC is expected to produce DM particles in abundance via the decays of heavier
(colored) states with large production cross sections, leading to
characteristic missing energy signatures. Many searches for these well-
motivated extensions of the SM are therefore ultimately also searches for DM.

Nevertheless, in models like supersymmetry, DM particles are typically only
produced together with a significant number of additional SM particles from
the decay chain, implying that there is no direct connection between the
annihilation and the production process.333On the other hand, the presence of
these additional states means that it may be much easier to infer the
properties of the DM particle, such as its mass, from kinematic distributions.
Furthermore, most constraints on these models are in fact unrelated to the
actual properties of the DM particle. In the context of these searches, the DM
particle is therefore often a mere tool rather than the actual object of
interest.

This review will therefore adopt a more narrow definition of LHC DM searches
and focus on _searches for the direct pair-production of DM making use of SM
particles emitted from an initial, intermediate or final state_.444For the
same reason we will also not discuss the (very interesting) signatures arising
if DM is produced in the decay of long-lived particles. The review is
structured as follows. First, in section 2 the relevant experimental
signatures are discussed and some recent results are presented. Section 3 then
provides an overview of different approaches for predicting the expected DM
signals. Finally, section 4 focuses on the important issue of how DM searches
at the LHC can be connected to other LHC searches, as well as to non-collider
DM searches, in order to obtain complementary information.

##  2 Experimental searches

In this section we review the experimental strategies relevant for DM searches
at the LHC and provide references to the most recent experimental results. The
main focus of this discussion is on so-called _mono- $X$ searches_, which are
concerned with the production of a single SM particle in association with
missing transverse momentum, but some more complicated DM signatures are also
mentioned. To avoid repetition, it is worth pointing out that at the time of
writing none of the DM searches discussed below have observed a significant
excess over expected backgrounds.

### Mono-jet

If it is at all possible to produce DM particles in proton-proton collisions,
it should also be possible to produce them in association with one or more QCD
jets from initial state radiation. Searches for events in which a jet with
high transverse momentum $p_{T}$ is produced in association with large missing
transverse momentum $E_{T,\rm miss}$ have therefore become emblematic for LHC
DM searches. The popular name _mono-jet search_ is however rather misleading,
because the probability to produce just one highly energetic jet is in fact
rather low [5]. Mono-jet searches therefore typically only impose a strict
veto on events containing leptons, but do include events with several high
$p_{T}$ jets.

In fact, with increasing center-of-mass energy these searches have become more
and more inclusive. For example, the most recent ATLAS analysis allows up to
four jets with $p_{T}>30\>\text{GeV}$ and pseudorapidity $|\eta|<2.8$, while
the leading (i.e. most energetic) jet is required to have
$p_{T}>250\>\text{GeV}$ and $|\eta|<2.4$ [6]. CMS does not constrain the total
number of jets at all and only requires that the leading jet satisfy
$p_{T}>100\>\text{GeV}$ and $|\eta|<2.5$ [7].

Clearly it becomes very challenging to model distributions of missing
transverse momentum in events with such a large number of jets, making it
necessary to use data-driven methods based on control regions for the
background estimation. For example, the pertinent background from $pp\to
Z(\to\nu\bar{\nu})+\text{jets}$ can be inferred from analogous events in which
the $Z$ boson decays leptonically. Similarly, backgrounds arising from $pp\to
W(\to\ell\nu)+\text{jets}$ and an unobserved charged lepton can be inferred
from events in which the lepton is correctly identified.

Detector effects, in particular jet mismeasurement, can also lead to events
that appear to have unbalanced transverse momentum. However, these multi-jet
backgrounds can be very efficiently suppressed by requiring that the missing
transverse momentum vector does not point into the (azimuthal) direction of
any of the leading jets. In combination, these methods allow to describe the
background distributions rather well. Nevertheless, the sensitivity of mono-
jet searches is often limited by systematic uncertainties and improving it
with increasing luminosity is rather challenging. To make progress it will be
essential to better understand the impact of electroweak corrections on the
$W+\text{jets}$ to $Z+\text{jets}$ ratio [8].

In some models the DM particles are produced preferentially in association
with one or more bottom quarks [9, 10]. Searches for such models are
conceptually similar to mono-jet searches, except that they require at least
one hard jet to pass $b$-tagging requirements [11, 12, 13].

### Mono-$V$

In a similar manner as in mono-jet events, DM may also be produced together
with a vector boson $V=\gamma,W,Z$, which is radiated off a quark in the
initial state. While the corresponding production cross section is
significantly smaller than for QCD radiation, the process is much cleaner and
can therefore be searched for with higher sensitivity. Moreover, if DM
particles couple directly to a pair of gauge bosons, mono-$V$ processes may in
fact be the dominant way in which DM is produced at the LHC [14].

Mono-photon searches are among the conceptually simplest searches for DM,
requiring typically only the presence of a high $p_{T}$ photon and no isolated
leptons [15, 16]. Although both detector effects (for example electron or jet
misidentification) and beam-induced events can potentially fake mono-photon
events, background levels are typically very low and the experimental
sensitivity is therefore limited only by statistics.

Leptonically decaying $Z$ bosons also yield a very clean signal [17, 18, 19].
By requiring that the transverse momentum of the di-lepton system is opposite
in azimuthal direction and similar in magnitude to the missing transverse
momentum vector and that the di-lepton invariant mass is close to the $Z$
boson mass, backgrounds can be very much suppressed and only the irreducible
backgrounds from di-boson production remain relevant.

If a $W$ boson produced in association with DM decays leptonically, the
neutrino adds to the missing transverse momentum and one obtains a so-called
mono-lepton event [20, 21]. Since the experimental signature is essentially
identical to the one from the leptonic decay of an off-shell $W$ boson,
background suppression is challenging and requires an accurate estimate of the
transverse mass distribution.

It is also possible to search for the production of a $W$ or $Z$ boson in
association with missing transverse momentum in the hadronic final state.
Searches for hadronically decaying $W$ or $Z$ bosons are similar to mono-jet
searches but use a larger distance parameter for the leading jet and employ
additional criteria such as requiring the mass of the fat jet to be consistent
with a $W$ or $Z$ boson [22, 7].

We note that mono-$W$ searches have received significant interest in the
context of so-called _isospin-violating DM_ , i.e. DM particles with different
couplings to up and down quarks [23]. This interest results from the
observation that a typical mono-$W$ signal arises from the interference of two
different diagrams, in which DM couples to up quarks and down quarks,
respectively. Mono-$W$ searches should therefore have a unique sensitivity to
the relative phase of the two couplings. However, it was subsequently pointed
out that it is very difficult to consistently study this set-up without
violating gauge invariance [24]. We will return to this issue in section 3.3.

### Mono-Higgs

Searches for a SM-like Higgs boson in association with missing transverse
momentum were first proposed in phenomenological studies [25, 26, 27] and have
since triggered a number of experimental searches both in the $\gamma\gamma$
[28, 29, 30] and in the $b\bar{b}$ [31, 32, 33] final state. In the former
case, backgrounds are very small and hence only a relatively loose cut on
$E_{T,\rm miss}$ is necessary. For example, the most recent CMS search [29]
requires $E_{T,\rm miss}>105\>\text{GeV}$. Moreover, these searches can make
use of the excellent resolution in the invariant mass of the photon pair,
$m_{\gamma\gamma}$, to suppress non-resonant backgrounds from SM processes
with mismeasured $E_{T,\rm miss}$. As a result, mono-Higgs searches in the di-
photon channel are currently only limited by statistics.

In the $b\bar{b}$ final state, on the other hand, background rejection is of
crucial importance. Fortunately, these searches can draw from a number of
techniques developed to identify Higgs bosons with high $p_{T}$. A
particularly interesting situation occurs in the case that the SM Higgs boson
is produced with sufficiently high transverse momentum that the two $b$-jets
from its decay merge into a single fat jet. Very roughly, this is the case if
$2m_{h}/p_{T}<R$, where $R$ is the distance parameter of the fat jet
clustering algorithm. For example, the ATLAS collaboration searches for events
with $E_{T,\rm miss}>500\>\text{GeV}$ in which there is a single fat jet with
$R=1.0$ and $p_{T}>250\>\text{GeV}$ that contains two $b$-tagged sub-jets with
$R=0.2$ [32]. This procedure achieves a Higgs tagging efficiency of up to
$40\%$ [34]. Again, the dominant backgrounds ($t\bar{t}$ and
$Z,W+\text{jets}$) are non-resonant, so that the invariant mass of the fat jet
can be used to discriminate signal from background. These searches are still
very much limited by statistics.

### DM + top quarks

If DM particles couple dominantly to heavy quark flavors, a promising way to
discover DM at the LHC is to search for a top-quark pair in association with
missing transverse momentum [9, 35]. Both ATLAS [11] and CMS [36] have
performed such searches in the channel where at least one of the $W$ bosons
from the decay $t\to bW$ decays into hadrons. Top quarks produced in
association with DM are typically not highly boosted, so their decay products
can be fully resolved. A number of cuts on the kinematics of the individual
jets as well as $b$-tagging techniques can then be used to efficiently
suppress backgrounds. In addition, CMS has also performed a search in the
fully leptonic channel [37], requiring at least two jets, out of which at
least one is $b$-tagged, as well as exactly two leptons.

While searches in the semi-leptonic channel provide the strongest constraints
at the moment [38], searches in the fully-leptonic channel are nevertheless
interesting for a number of reasons. First of all, these searches are
currently only limited by statistics and therefore promise significant gains
in sensitivity with luminosity [39]. Furthermore, the angular distribution of
the two leptons can be used not only to distinguish signal from background,
but also to determine the CP nature of the DM interactions [40, 39].

Finally, it is also conceivable that DM is produced in association with a
single top quark [41], and corresponding searches have been carried out by
both ATLAS [42] (in the semi-leptonic channel) and by CMS [43, 44] (in the
fully hadronic channel). It is worth noting, however, that (due to the absence
of top quarks in the initial state) this signature can only arise either from
sizable flavor-changing transitions, or from a bottom quark in the initial
state analogous to single-top production [45].

### Invisible Higgs decays

If the DM mass is less than half of the mass of the SM Higgs boson, it may be
possible to produce pairs of DM particles in Higgs decays. Such invisible
Higgs decays can be searched for in a number of different ways. First of all,
indirect constraints can be obtained by combining the visible decay modes in
order to construct an upper bound on all unobserved decay channels. This
approach, however, requires an assumption on the Higgs production cross
section, which is typically taken to be given by the SM prediction.

Alternatively, one can directly search for invisible Higgs decays by
triggering on the particles that signal the production of a Higgs boson. The
two most relevant production modes in this context are vector boson fusion
(VBF) and production in association with a massive vector boson (VH). However,
the gluon-fusion mode can also be interesting if the Higgs boson is produced
with an additional jet from initial state radiation.

Searches for the associated production of a Higgs boson together with a
massive vector boson followed by an invisible decay of the Higgs boson are
conceptually very similar to the mono-$V$ signatures discussed above. They can
be searched for both in the leptonic decays of a $Z$ boson or in the hadronic
decays of a $W$ or $Z$ boson [46, 47, 17]. The jet-associated gluon-fusion
production mode, on the other hand, essentially yields a mono-jet signature as
discussed in detail above [46].

A truly novel signature is obtained in the VBF case [48]. In this channel the
production of a Higgs boson is signaled by the presence of two jets with large
separation in pseudo-rapidity and large invariant mass [49, 46]. This
distinctive topology can be exploited to discriminate hypothetical invisible
Higgs decays from the large SM backgrounds. Indeed, searches for invisible
Higgs decays in the VBF production mode typically yield the strongest bounds
on the Higgs invisible branching ratio.

Combinations of the various direct searches for invisible Higgs decays have
been performed by both ATLAS [50] and CMS [46], the latter including first
results from data taken at 13 TeV. The resulting upper bounds at 95%
confidence level on the invisible branching ratio are
$\text{BR}_{\text{inv}}<0.25$ and $\text{BR}_{\text{inv}}<0.24$, respectively.
These bounds are comparable to the ones obtained indirectly from the visible
decay modes [50, 51].

##  3 Predicting dark matter signals

Having discussed the various ways in which one can search for the production
of DM at the LHC, we now turn to the essential question: How do we know what a
DM signal at the LHC will look like? Which of the search channels is the most
promising? And how do we calculate the details of the expected distributions?
It should be clear that (for the time being) there is no single correct answer
to these questions. Our ignorance of the particle physics nature of DM means
that a variety of different approaches need to be considered. Each approach
needs to find a compromise between two conflicting requirements: generality
and plausibility (or equivalently minimality and realism) [52].

The requirement of generality (or minimality) means that we want to make as
few assumptions as possible on the presence of new particles in the dark
sector. This approach is desirable both from the practical perspective
(because only a small number of new parameters are introduced) and from the
philosophical perspective (along the lines of Occam’s razor). The requirement
of plausibility (or realism), on the other hand, compels us to give preference
to models that are in agreement with well-established principles of particle
physics, for example the absence of large $\mathrm{CP}$ or flavor violation.
This requirement may also include aesthetic or practical arguments, for
example that the model under consideration should have a perturbative ultra-
violet (UV) completion that is consistent with the structure of the SM gauge
group before electroweak symmetry breaking.

Another way to characterize these two conflicting approaches is to describe
them as the bottom-up approach on the one side, in which we try to add the
minimum amount of additional structure to the SM, and the top-down approach on
the other side, in which we start from well-motivated UV completions in order
to gain intuition for the construction of DM models. We will begin with the
bottom-up approach below and then extend this approach further and further
until we connect to models obtained from the top-down approach.

###  3.1 Effective theories

The most minimal assumption possible is that the DM particle is the only new
state beyond the SM that is kinematically accessible at the LHC. In this case,
the interactions between DM and SM states can be described by an effective
field theory (EFT) containing operators of mass dimension larger than
four.555There are a small number of DM models that pursue an even more minimal
approach by coupling the DM particle to the SM via renormalizable
interactions, such as sterile neutrinos [53], hidden photons [54], scalar
singlets [55, 56, 57] or $SU(2)_{L}$ multiplets with a stable neutral
component [58]. With the exception of the scalar singlets, which we will
discuss below in the context of Higgs portal models, these models do however
not predict any observable signals at the LHC. This approach was first
suggested under the name _Maverick Dark Matter_ [59] and was subsequently
popularized as the _EFT approach_ by a number of detailed studies [60, 61, 62,
63, 64, 65].

An extensive classification of the lowest-dimension effective operators
describing the interactions between fermionic or scalar DM and quarks or
gluons has been performed in refs. 60, 62.666The relevance of higher-dimension
operators is discussed in ref. 66.  Each operator is characterized by only two
parameters: the effective suppression scale $\Lambda$ and the DM mass
$m_{\text{DM}}$. For example, a frequently studied operator is the so-called
axial-vector operator:

$\mathcal{O}=\frac{1}{\Lambda^{2}}(\bar{q}\gamma^{\mu}\gamma^{5}q)(\bar{\chi}\gamma_{\mu}\gamma^{5}\chi)\;,$
(2)

where $\chi$ denotes a spin-1/2 DM particle, which can be either a Dirac
fermion (in which case the operator is usually labeled D8 [62]) or a Majorana
fermion (labeled M6 [60]). This operator has been the subject of a number of
LHC studies [67, 68].

A crucial property of the EFT approach is that the shape of all kinematic
distributions is independent of the suppression scale $\Lambda$. For a
dimension-6 operator proportional to $\Lambda^{-2}$, for example, all cross
sections are simply proportional to $\Lambda^{-4}$. It is therefore
technically very easy to present experimental results in terms of lower bounds
on $\Lambda$ as a function of $m_{\text{DM}}$. For DM masses smaller than the
typical cut on missing transverse momentum, kinematic distributions and hence
the resulting bounds become independent of $m_{\text{DM}}$, implying that LHC
searches can be sensitive to arbitrarily small DM masses.

For hadron colliders, the strongest constraints on the suppression scale
$\Lambda$ are obtained for effective operators involving quarks and gluons.
Another interesting possibility however are contact interactions between DM
particles and SM gauge bosons [14, 69, 70] or Higgs bosons [26]. In such a
set-up, any gauge boson or Higgs boson produced at the LHC can radiate off a
pair of DM particles, potentially leading to mono-$V$ or mono-Higgs signals.

Effective interactions between DM particles and Higgs bosons have also been
studied in the context of so-called Higgs portal models [71, 72, 73, 74].
Indeed, one of the simplest ways to couple fermionic DM to the SM is via the
dimension-5 operator

$\mathcal{O}\sim\frac{1}{\Lambda}H^{\dagger}H\,\bar{\chi}\chi\;,$ (3)

where $H$ denotes the SM Higgs doublet. After electroweak symmetry breaking,
this operator gives rise to an $h\bar{\chi}\chi$ vertex, where $h$ denotes the
physical Higgs boson. For $m_{\text{DM}}<m_{h}/2$ this interaction leads to
invisible Higgs decays, which are strongly constrained by experimental data.
For $m_{\text{DM}}>m_{h}/2$, on the other hand, the DM production cross
section at the LHC is strongly suppressed [75]. We note that, if DM is a
scalar singlet, the corresponding Higgs portal operator is in fact
renormalizable, leading to the arguably simplest model for DM production at
the LHC.

#### EFT validity

The original appeal of the EFT approach was based on the idea that bounds on
effective operators are model-independent, in the sense that it is not
necessary at any point of the analysis to specify the details of the
underlying UV completion. However, this hope has been challenged by two
related observations. First, it has become clear that there are many
interesting models describing the production of DM at the LHC which are not
correctly captured by the EFT approach [61, 64, 76]. In other words, these
models predict kinematic distributions that differ significantly from the ones
obtained from contact interactions. And second, it was shown that – at least
for certain values of the suppression scale $\Lambda$ – the effective operator
approach makes unphysical predictions so that it becomes impossible to find a
plausible UV completion [77, 76, 78].

Both of these observations are connected to the way in which effective
operators are obtained from a more fundamental theory. For example, the axial-
vector operator from eq. (2) can be obtained from a theory containing a heavy
spin-1 particle $V^{\mu}$ with axial couplings to DM and quarks:

$\mathcal{L}\supset\frac{m_{V}^{2}}{2}V^{\mu}V_{\mu}+V^{\mu}g_{q}\bar{q}\gamma_{\mu}\gamma^{5}q+V^{\mu}g_{\text{DM}}\bar{\chi}\gamma_{\mu}\gamma^{5}\chi\;.$
(4)

In the context of DM searches at the LHC, such a new particle is often
referred to as the _mediator_ of the interactions between quarks and DM. If
the mediator is exchanged in the $s$-channel of a $2\rightarrow 2$ process
with center-of-mass energy $\sqrt{s}$, the resulting matrix element will
contain a propagator of the form

$\mathcal{M}\propto\frac{g_{q}\,g_{\text{DM}}}{m_{V}^{2}-s}\;.$ (5)

In the limit $m_{V}^{2}\gg s$ this propagator becomes
$g_{q}\,g_{\text{DM}}/m_{V}^{2}$ and one obtains the axial-vector operator
from above with

$\frac{1}{\Lambda^{2}}=\frac{g_{q}\,g_{\text{DM}}}{m_{V}^{2}}\;.$ (6)

If, on the other hand, the mass of the mediator is comparable to or smaller
than the momentum transfer in the process, the contact interaction does not
provide an accurate description of the kinematics, because terms that are of
higher order in $s/m_{V}^{2}$ cannot be neglected. In other words, if we are
interested in a theory where DM interacts with quarks via the exchange of a
mediator with mass at or below the TeV scale, the EFT approach will not
correctly capture this model at LHC energies.777We emphasize that it becomes
more complicated to apply this argument to effective operators for which the
physical interpretation of the suppression scale is less clear, such as the
one introduced in eq. (3).

As long as $m_{V}$ is sufficiently large, the EFT approach is fully justified
at LHC energies. However, while the validity of the EFT approach depends on
the value of $m_{V}$, the sensitivity of experimental searches depends only on
the suppression scale $\Lambda$. It turns out that the LHC has no sensitivity
to effective operators involving DM with a suppression scale larger than a few
TeV. In fact, to obtain sizable cross sections $\Lambda$ typically has to be
below the TeV scale. At first sight, this does not necessarily pose a problem:
If the couplings $g_{q}$ and $g_{\text{DM}}$ are very large, it follows from
eq. (6) that $m_{V}$ can be much larger than $\Lambda$ and hence the EFT
approach may still be valid.

The problem we then face is that the matrix element obtained from the
effective operator scales proportional to $s/\Lambda^{2}$ with increasing
center-of-mass energy (or equivalently that the cross section grows
proportional to $s/\Lambda^{4}$). At some point the EFT therefore violates the
requirement of perturbative unitarity, which essentially demands that the
matrix element (or, more precisely, all of its partial waves) is smaller than
unity [77, 76, 78]. Typically, this happens for
$\sqrt{s}\sim(2\text{--}3)\Lambda$.

The conclusion is that for large momentum transfer the EFT makes unphysical
predictions. In other words, the energy transfer for at least some fraction of
events at the LHC is sufficient to resolve the underlying micro-physics. For a
complete description of all processes at the LHC the effective operator must
then be replaced by a more complete theory, in which unitarity is restored
[79].

#### EFT truncation

A possible way to restore the validity of the EFT approach for suppression
scales $\Lambda$ comparable to LHC energies is to ensure that the EFT is only
applied to processes with sufficiently small momentum transfer that the
predictions of the effective operator can be trusted. This process, referred
to as _EFT truncation_ , allows to obtain a conservative but model-independent
bound.

The most straightforward way to perform such a truncation is to assume that
the EFT approach becomes invalid at energies $E\gtrsim g_{\ast}\Lambda$ [80,
81]. In the example discussed above, this scale would be given by $m_{V}$,
i.e. one can identify $g_{\ast}=\sqrt{g_{q}\,g_{\text{DM}}}$. For a specific
choice of $\Lambda$ and $g_{\ast}$ one then disregards all events that have a
momentum transfer larger than $g_{\ast}\Lambda$ and determines whether the
remaining number of events is sufficient to exclude the assumed value of
$\Lambda$.888Since the DM particles in the final state cannot be detected, the
actual momentum transfer in a given event is not observable, so this
requirement can only be implemented in Monte Carlo generators. Iterating this
procedure, one can construct a self-consistent bound on $\Lambda$ for an
assumed value of $g_{\ast}$ [67]. An alternative approach [82, 27, 83] is to
determine the fraction of events with $E>g_{\ast}\Lambda$ and to use this
number to rescale bounds obtained without truncation. Note that, since the
energy required to produce a pair of DM particles is at least
$E=2\,m_{\text{DM}}$, it is never possible to constrain suppression scales
with $g_{\ast}\Lambda<2\,m_{\text{DM}}$.

If the underlying model of DM is strongly interacting, i.e. if $g_{\ast}\gg
1$, the EFT approach together with an appropriate truncation procedure yields
the most suitable description of DM production at the LHC [81], avoiding the
problem that the EFT makes unphysical predictions. If, on the other hand, DM
is weakly coupled, so that $g_{\ast}\lesssim 1$, kinematic distributions are
typically so different from the EFT approach that it is not possible to obtain
relevant constraints using effective operators. Clearly, a different
description of the interactions of DM is necessary. This alternative approach
is provided by the framework of DM simplified models.

###  3.2 Simplified models

So far we have discussed possible ways to describe the interactions of DM
particles at the LHC under the assumption that they are the only particles
kinematically accessible. Now we will go one step further and consider
possible descriptions in which there is a second light particle, which is
responsible for mediating the interactions of quarks and DM. To limit the
number of possibilities, we now require all interactions to be renormalizable,
i.e. we will only consider interactions of dimension four or less. This
approach [84] has come to be known as _DM simplified models_ [85].

Clearly, including a light mediator in the model requires the introduction of
a number of new parameters, on which kinematic distributions will depend in a
non-trivial way. While this additional complexity is a challenge for
experimental searches, there are a number of good reasons to consider such a
set-up:

  1. 1.

There is no simple way to translate an exclusion limit obtained within the EFT
approach to models with a light mediator. A naive conversion of a bound on
$\Lambda$ into a bound on the mediator mass $m_{\text{med}}$ may both
overestimate or underestimate the actual strength of the constraint by orders
of magnitude [84]. Simplified models aim to fill this gap and provide bounds
for models that cannot be mapped onto effective operators.

  2. 2.

Contact interactions between DM particles and SM states predict rather
specific distributions, in particular very hard missing energy spectra. If
experiments relied only on these predictions for developing analysis
strategies, the searches would likely not have the optimum sensitivity for
models predicting softer spectra. Considering mediators of different masses
makes it possible to consider different kinematic distributions and optimize
the experimental sensitivity for each case.

  3. 3.

From a theoretical point of view, it is quite natural to assume that the DM
particle is comparable in mass to the particle responsible for its
interactions. In particular it turns out to be very difficult to obtain the
required DM relic abundance if the mediator is too heavy, challenging one of
the primary motivations for LHC DM searches. In the presence of a light
mediator, on the other hand, the relic abundance can be readily reproduced.
[86]

Note that, while the name suggests a top-down motivation, simplified models
are really most strongly motivated from a bottom-up perspective. We will
return to the question of how these models are connected to well-motivated UV
completions in section 3.3.

Following a joint effort of the LHC experiments and the theoretical community,
a number of particularly interesting simplified models have been identified
[87] and their LHC phenomenology has been investigated in the context of the
ATLAS/CMS DM Forum (DMF) [88]. These simplified models have subsequently been
the subject of a number of theoretical studies [89, 90, 91, 92, 93, 94] and
experimental analyses [6, 7].

####  $s$-channel mediators

We have already mentioned a first example for such a simplified model above,
namely the axial-vector mediator

$\mathcal{L}\supset
g_{q}\,V^{\mu}\sum_{q}\bar{q}\gamma_{\mu}\gamma^{5}q+g_{\text{DM}}\,V^{\mu}\bar{\chi}\gamma_{\mu}\gamma^{5}\chi\;.$
(7)

Analogously, one can consider the case of a vector mediator999Note that the
vector current vanishes for Majorana fermions and for real scalars.
Consequently, the interactions between DM and a vector mediator can only
sensibly be defined if the DM particle is a Dirac fermion or a complex scalar.

$\mathcal{L}\supset
g_{q}\,V^{\mu}\sum_{q}\bar{q}\gamma_{\mu}q+g_{\text{DM}}\,V^{\mu}\bar{\chi}\gamma_{\mu}\chi\;.$
(8)

Note that, to be consistent with the hypothesis of minimal flavor violation,
the mediators are assumed to couple to all quarks with equal strength [87].
Because of the way how DM is produced in these simplified models, they are
referred to as spin-1 $s$-channel models. In a similar way one can construct
spin-0 $s$-channel models. The case of a scalar mediator $\phi$ is given by

$\mathcal{L}\supset
g_{q}\,\phi\sum_{q}\frac{y_{q}}{\sqrt{2}}\bar{q}q+g_{\text{DM}}\,\phi\bar{\chi}\chi\;,$
(9)

while the pseudoscalar mediator $a$ is described by

$\mathcal{L}\supset
g_{q}\,a\sum_{q}\frac{y_{q}}{\sqrt{2}}\bar{q}\gamma^{5}q+g_{\text{DM}}\,a\bar{\chi}\gamma^{5}\chi\;.$
(10)

The fact that these interactions are taken to be proportional to the Yukawa
couplings $y_{q}$ is again a result of the hypothesis of minimal flavor
violation [87]. In the context of spin-0 $s$-channel mediators, DM is
therefore expected to couple most strongly to top quarks [40, 95, 39, 35, 45].
As we will discuss below, $g_{q}$ should be thought of as a numerical factor
(such as a mixing angle) rather than an independent coupling.

Each of the four simplified models described above is fully characterized by
five parameters: the masses of the two particles $m_{\text{DM}}$ and
$m_{\text{med}}$, the two couplings $g_{q}$ and $g_{\text{DM}}$ and the width
of the mediator $\Gamma_{\text{med}}$. In fact, the shapes of the various
kinematic distributions depend only on the two masses and the mediator width,
whereas a variation of the couplings simply rescales the spectra. Varying the
couplings while keeping the width fixed may however lead to unphysical results
if the total width is assumed to be smaller than the sum of the partial widths
for decays into DM and quarks. To avoid this problem, it is usually assumed
that the mediator does not couple to any other light particles, so that its
width can be calculated in terms of the remaining four parameters (often
referred to as the _minimal width assumption_):

$\Gamma=\Gamma_{\chi\bar{\chi}}+\sum_{q}\Gamma_{q\bar{q}}+\Gamma_{gg}\;,$ (11)

where the final term is relevant only for spin-0 $s$-channel mediators.

Replacing the mediator width by the minimal width eliminates one parameter,
but it has the disadvantage that now the shapes of the different spectra
depend on all four remaining parameters in a non-trivial way. Nevertheless, in
many cases the dependence on the couplings is trivial, whereas the two masses
turn out to be decisive for the experimental sensitivity. It has therefore
become common to study simplified models by considering specific (fixed)
choices of couplings and investigate the experimental sensitivity in the
parameter plane spanned by the two masses [96].

#### Presenting experimental results

A sketch of such a study is shown in figure 2. Roughly, the mass-mass plane
can be divided into three different regions. For very large mediator masses,
one recovers the EFT limit discussed above. LHC searches typically have no
sensitivity to this case unless the couplings are assumed to be close to the
perturbativity bound. For smaller mediator masses, the phenomenology depends
decisively on the ratio of the mediator mass and the DM mass. For
$m_{\text{med}}>2m_{\text{DM}}$, the mediator can be produced on-shell and
subsequently decays into a pair of DM particles. As a result, the DM
production cross section receives a resonant enhancement. Conversely, for
$m_{\text{med}}<2m_{\text{DM}}$ the production of DM can only proceed via an
off-shell mediator and is correspondingly suppressed. To first approximation,
LHC searches therefore aim to explore the on-shell region.

Figure 2: Sketch of the mass-mass plane used to present experimental results
for simplified DM models. See text for details.

Presenting experimental results in the mass-mass plane is very convenient for
comparing the sensitivity of different LHC searches. In addition, it is
possible to indicate in this plane the parameter combinations for which the
total DM annihilation cross section is equal to the thermal cross section, so
that the observed DM relic abundance can be reproduced (see section 4). The
drawback of this approach is that it is far from obvious how to reinterpret a
specific experimental result in terms of a model with couplings that differ
from the assumed values (or with additional contributions to the mediator
width).

One possible solution is to make use of the narrow-width approximation, which
states that in the on-shell region the cross section for any mono-$X$ process
factorizes into the production cross section of the mediator together with the
state $X$ and the branching ratio of the mediator to decay into DM particles:

$\mathrm{d}\sigma(pp\to\chi\bar{\chi}+X)=\mathrm{d}\sigma(pp\to
V+X)\cdot\text{BR}(V\to\chi\bar{\chi})\;.$ (12)

In particular, this approximation, which is valid as long as
$\Gamma_{\text{med}}\lesssim 0.3\,m_{\text{med}}$, implies that the shape of
kinematic distributions depends _only_ on $m_{\text{med}}$. It is then
possible to infer a bound on any model containing an $s$-channel mediator as
follows

  1. 1.

For the model of interest, calculate the branching ratio
$\text{BR}_{0}=\text{BR}(V\to\chi\bar{\chi})$.

  2. 2.

Identify the simplified model most similar to the model of interest, pick a
parameter point within the on-shell region with the same mediator mass and
read off the bound on the signal strength $\mu$ (i.e. the ratio of excluded
cross section to predicted cross section).

  3. 3.

For the chosen parameter point in the simplified model, calculate the
branching ratio $\text{BR}_{1}=\text{BR}(V\to\chi\bar{\chi})$.

  4. 4.

The bound on the signal strength for the model of interest is then given by
$\bar{\mu}\equiv\mu\frac{\text{BR}_{1}}{\text{BR}_{0}}$. For $\bar{\mu}<1$,
the model is excluded by the search under consideration.

This procedure provides a good approximation to more detailed analyses as long
as the parameter point under consideration is well within the on-shell region
and the mediator width is sufficiently narrow [97, 94].

####  $t$-channel mediators

Another well-motivated class of simplified models focuses on the case that the
mediator couples to one quark and one DM particle. Clearly, such an
interaction is only possible if the mediator carries color charge. A
frequently studied case is the one where the DM particle is a fermion and the
mediator is a colored scalar:

$\mathcal{L}\supset g\sum_{i=1,2,3}\phi_{i}^{\ast}\bar{\chi}P_{R}u_{i}\;.$
(13)

Interactions with right-handed down-type quarks and left-handed quarks can be
constructed in complete analogy.

In order to be consistent with minimal flavor violation, the mediator needs to
carry a flavor index $i=1,2,3$, implying that there are really three mediators
of equal mass ($m_{1}=m_{2}=m_{3}$) and equal coupling strength
$g_{1}=g_{2}=g_{3}\equiv g$. It is possible, however, to break this
universality and consider the case where the third-generation mediator has
different mass and couplings from the first two generations. Most studies then
focus on the two mediators coupling to the first two generations. These models
are referred to as _$t$ -channel flavored mediators_ [98, 99, 100, 101, 102,
103, 104, 105, 106]. Another interesting possibility, named _flavored dark
matter_ is that the DM particle (rather than the mediator) carries the flavor
index [107, 10]. Models in which the third-generation mediator dominates the
phenomenology are often called _top-flavored dark matter_ [108, 109, 110].

The mediator in this set-up resembles very much the squarks in the minimal
supersymmetric Standard Model and indeed the collider phenomenology is very
similar. Compared to the case of an $s$-channel mediator, there are additional
contributions to mono-jet signals, because the mediator can radiate off gluon
jets or decay into a jet and a DM particle. Moreover, it is possible to pair-
produce the mediator, leading to a distinctive signature with two jets and two
DM particles in the final state: $pp\to\phi\phi^{\ast}\to\chi\bar{\chi}jj$
[100, 101, 102, 103]. Searches for mono-jets and di-jets in association with
missing transverse momentum have comparable sensitivity, unless the mass
spectrum is highly degenerate (i.e. $m_{\text{med}}-m_{\text{DM}}\ll
m_{\text{med}}$), in which case mono-jet searches are more promising [103].

To conclude the discussion of simplified models, we point out that there are a
number of further possibilities, for example those involving scalar DM
particles or fermionic mediators. For a more complete overview (as well as
comprehensive lists of analytical results), we refer to more focused reviews
of DM simplified models [85, 52]. For a discussion of simplified models with
spin-2 mediators, we refer to ref. 111.

###  3.3 Towards complete models

The central idea of the simplified model framework is that, while it abandons
some of the model independence of the EFT approach, it still captures the
phenomenology of a wide range of possible theories of DM. The aim is therefore
that one can obtain relevant constraints for any such theory by mapping it
onto the appropriate simplified model. It should be clear that in some cases
the constraints obtained in this way may not be the dominant ones, meaning
that the simplified model may not capture all of the relevant phenomenology.
However, at the very least this approach should make it possible to
conclusively rule out certain regions of parameter space based on the
predictions of the corresponding simplified model.

While the technical details of such a mapping from complete theories to
simplified models remain challenging (see above for a discussion of how this
can be done in the narrow width approximation), this section will focus on a
more fundamental question, namely whether the simplified models introduced
above are the appropriate choice for this purpose. Indeed, it is perfectly
conceivable that the models introduced above are too simplified, in the sense
that we have neglected additional states or couplings that make an essential
contribution to the phenomenology. Conversely, it is also possible that we
have been too general. For example, it may turn out that there are hidden
relations between the different parameters, or that certain regions of
parameter space are disfavored for theoretical reasons.

In principle one could try to address these questions from a top-down approach
by considering a large collection of UV-complete models of DM and performing
the mapping onto simplified models. Clearly, if a specific simplified model is
never found to give a relevant constraint, its inclusion in LHC studies would
be questionable. However, it is possible to address the same issue also from
the bottom-up perspective, by investigating whether the simplified models
introduced above fulfill certain theoretical consistency requirements, such as
gauge invariance and that perturbative unitarity is guaranteed in the relevant
regions of parameter space [112, 113]. As we will see below, these principles
provide a useful guidance for the construction of more realistic simplified
models.

#### Implications of perturbative unitarity

One of the main motivation for the development of simplified models was to
address the issue of unitarity violation ubiquitous in the EFT approach.
Indeed, it has been shown that the simplified models introduced above satisfy
the requirement of perturbative unitarity in mono-jet and mono-$Z$ searches up
to very large energies [114], provided all couplings are sufficiently small.
Nevertheless, some cases have been identified in which (seemingly
renormalizable) simplified models are in fact not well-behaved up to
arbitrarily high energies.

A particularly interesting case are mono-$W$ searches. For simplified models
with an $s$-channel spin-1 mediator, these searches are found to violate
perturbative unitarity at high energies unless DM couples with equal strength
to left-handed up and down quarks [24, 113, 115]. This unphysical behavior
turns out to result from the emission of longitudinal $W$ bosons and can only
be tamed if the simplified model is extended by an additional interaction
between the mediator and $W$-bosons [115]. A similar solution is found in
models with a $t$-channel colored scalar mediator, where the emission of
$W$-bosons from the mediator itself restores unitarity at high energies [113].

The simplified model with an axial-vector mediator turns out to be troublesome
for yet another reason: In contrast to a spin-1 mediator with purely vectorial
couplings, the longitudinal mode of the axial-vector mediator does not
decouple. In fact, if the transverse mode couples to a fermion of mass $m_{f}$
with coupling strength $g_{f}$, the corresponding coupling of the longitudinal
mode is proportional to $g_{f}\,m_{f}/m_{\text{med}}$, i.e. it is enhanced for
heavy fermions. In other words, for $m_{f}\gg m_{\text{med}}$ the coupling of
the mediator will become non-perturbative.101010In practice, this is not an
issue for mono-jet searches, where the mediator always couples to a light
quark in the initial state [114]. This situation changes, however, when
considering searches for top quarks in association with missing transverse
momentum, where all fermions involved in the process can be heavy [116, 114,
112].  This consideration implies that the DM mass should satisfy the
requirement [112]

$m_{\text{DM}}\leq\sqrt{\frac{\pi}{2}}\frac{m_{\text{med}}}{g_{\text{DM}}}\;,$
(14)

giving an example for how theoretical considerations can constrain the
parameter space of simplified models.

But even if all couplings are perturbative, the simplified model with an
axial-vector mediator still violates unitarity at high energies (for example
in the process $\chi\bar{\chi}\rightarrow VV$, which yields a matrix element
proportional to $\sqrt{s}$ for large energies) [112]. This issue is
reminiscent of the well-known problems with unitarity that the SM would face
in the absence of a Higgs boson. Clearly, the inconsistencies of the axial-
vector simplified model arise from the fact that we have not specified a
mechanism to generate the mediator mass. The simplest way to address this
issue is to introduce an additional Higgs boson that is a singlet under the SM
gauge group [112, 117]. This so-called _dark Higgs boson_ then acquires a
vacuum expectation value that generates the mediator mass.111111For a spin-1
mediator with purely vectorial couplings it is possible to generate the
mediator mass via a Stueckelberg mechanism [118] without the need to introduce
additional degrees of freedom.

The observation that simplified models with an axial-vector mediator are
necessarily incomplete has led to an increasing interest in simplified models
with more than one mediator [119, 120, 121]. Although it is necessary to
introduce at least one additional parameter (the mass of the second mediator),
these models are highly attractive due to their rich phenomenology. For
example, if the spin-1 mediator decays visibly, one may obtain a
mono-$Z^{\prime}$ signature [122, 123], while visible decays of the spin-0
mediator may lead to a mono-dark-Higgs signal [124].

Another consistency requirement, which is relevant for any UV completion of
spin-1 simplified models that involves an extension of the SM gauge group, is
the absence of gauge anomalies. One possible way to implement this requirement
is to choose the couplings of the mediator to SM fermions in such a way that
no anomalies arise [125, 126, 127]. Doing so necessarily implies that the
mediator couples to leptons, leading to tight constraints from searches for
di-lepton resonances (see below). Insisting on a mediator that couples only to
quarks, on the other hand, requires new states that cancel the anomalies [128,
129, 130]. Nevertheless, in many cases there is no color anomaly [112] and
therefore no new colored states are required in order to achieve anomaly
freedom. The additional states are therefore expected not to affect the
phenomenology of the model significantly.121212The detailed particle content
of the model does however become important when considering loop-induced
processes such as DM annihilation into gamma-rays [131].

#### Implication of gauge invariance

The example of mono-$W$ searches discussed above illustrates the problems that
may arise if a simplified model does not respect the full gauge symmetry of
the SM before electroweak symmetry breaking. It may nevertheless be of
interest from a phenomenological perspective to study simplified models with
interactions that can only arise after electroweak symmetry breaking. An
instructive example are simplified models with an $s$-channel scalar or
pseudoscalar mediator. In these models the mediator is a SM singlet which
couples to $\bar{q}q=\bar{q}_{L}q_{R}+\bar{q}_{R}q_{L}$ and
$\bar{q}\gamma^{5}q=\bar{q}_{L}q_{R}-\bar{q}_{R}q_{L}$, respectively. Clearly,
neither of these interactions is invariant under the SM gauge group before
electroweak symmetry breaking, so they must vanish in the limit that the
electroweak vacuum expectation value tends to zero and electroweak symmetry is
restored [24].

A possible way to obtain these interactions is to assume that the spin-0
mediator does not actually couple directly to SM quarks, but that it only
obtains these couplings after electroweak symmetry breaking from mixing with
the SM Higgs [132].131313Note that for a pseudoscalar mediator, such a mixing
would violate $\mathrm{CP}$ symmetry [133, 134, 135].  In this case, the new
mediator couples to fermions in exactly the same way as the SM Higgs, i.e.
with coupling strength proportional to the fermion masses. This construction
therefore leads to the same coupling structure that was motivated above by
invoking minimal flavor violation.

At first sight, such a _simplified model with mixing_ gives rise to a very
interesting phenomenology. The fact that the mediator couples most strongly to
heavy quarks means that gluon fusion via top-quark loops will give the
dominant contribution to its production cross section, leading to promising
mono-jet signals [136, 91, 90, 38]. Moreover, the mediator may be produced in
association with heavy quarks [9, 137], which may potentially allow to
distinguish scalar from pseudoscalar mediators [95, 39]. Finally, the mediator
would also obtain couplings to SM gauge bosons, so that even mono-$V$ signals
can be expected.

In practice, however, all of these promising signatures are forced to be small
by the measurements of the branching ratios of the SM-like Higgs boson. Even
if the DM particle is sufficiently heavy that invisible Higgs decays are
kinematically forbidden, bounds on the Higgs signal strength require the
mixing angle between the SM Higgs and the new mediator to be rather small. In
other words, gauge invariance implies that the most relevant constraints on
the simplified model with mixing will likely come from Higgs physics rather
than from LHC DM searches.

A possible way to evade this conclusion is to extend the Higgs sector with a
second Higgs doublet. The mediator that couples to DM can then obtain its
couplings to SM quarks from mixing with the second Higgs doublet.141414An
alternative approach is to consider an inert second Higgs doublet, which does
not have any direct couplings to SM fermions. The lightest component of this
inert doublet then is a candidate for scalar DM, which obtains couplings to SM
states via the Higgs portal [138].  This way, rather than modifying the
branching ratios or the signal strength of the SM Higgs boson, the mixing will
modify the properties of the second Higgs doublet (and may thereby even
explain why no heavier Higgs boson has yet been observed). Since the second
Higgs doublet contains both a scalar and a pseudoscalar degree of freedom, it
is easily possible within this approach to generate couplings between a
pseudoscalar singlet mediator and SM quarks without violating $\mathrm{CP}$
[139, 140].

Simplified models based on two Higgs doublet models have received significant
interest recently [139, 140, 141, 142]. This interest stems from the
observation that the additional heavy Higgs bosons may lead to novel
signatures not captured by the simplified model with just a single mediator.
In particular, the heavy Higgs bosons may decay into the singlet mediator and
a SM $Z$ or Higgs boson. If the singlet mediator then decays into DM
particles, one may obtain large mono-$Z$ or mono-Higgs signals. Furthermore,
it may be possible to produce the mediator in the VBF mode and perform
searches analogous to the ones relevant for constraining invisible Higgs
decays (see section 2) [143].

In the discussion above we have moved from the largely model-independent EFT
approach towards simplified models that predict a rich phenomenology and
finally to various extensions of these models that address potential
theoretical inconsistencies. The resulting variety of approaches provides a
continuous spectrum from the most minimal constructions to highly complex
models. Indeed, some of the extended simplified models discussed above could
equally well have been obtained from a top-down approach. For example,
simplified models containing a spin-1 mediator and a dark Higgs are very
similar to models of $U(1)^{\prime}$ extensions of the SM gauge group [128,
144], whereas simplified models containing two Higgs doublets and a SM singlet
spin-0 mediator resemble the Higgs sector of the next-to-minimal
supersymmetric Standard Model. Clearly, it is desirable to pursue all of these
different approaches in parallel, so that the balance between generality and
plausibility can be adjusted for each situation as appropriate.

###  3.4 Numerical tools

The effective theories and simplified models discussed above provide a
framework for predicting the various signatures of DM production at the LHC
discussed in section 2 and can therefore be used to devise the corresponding
experimental searches and to present the resulting constraints. To achieve
this goal it is necessary to implement these models into numerical codes for
Monte Carlo event generation capable of making accurate predictions for
collider observables.

Generating DM signals at next-to-leading order (NLO) is essential in
particular for signatures where the DM particle is produced in association
with QCD radiation. In the context of the EFT approach these QCD NLO
corrections have been calculated [145] and implemented into the Monte Carlo
generator MCFM [146]. This approach has subsequently been extended to
$s$-channel simplified models[5] and implemented into the POWHEG BOX [147].
All DMF $s$-channel simplified models have also been implemented into
FeynRules [148] and MadGraph5_aMC@NLO [149] at the NLO level under the name
DMsimp [150, 151, 152]. Very recently, model files for some extended
simplified models have also become available [153, 142].

Finally, we note that there are dedicated numerical tools for calculating DM
observables beyond the LHC, such as micrOMEGAs [154], MadDM [155] and DarkSUSY
[156]. An ongoing effort to combine all of this information into global fits
of DM models is performed by the GAMBIT collaboration [157]. This way it will
be possible to fully explore the complementarity of different DM searches,
which we will discuss in the next section.

##  4 A question of complementarity

The different approaches for predicting DM signals at the LHC presented in the
previous section all have in common that they can be used not only to obtain
the rates for DM production in the collisions of SM particles, but also to
calculate the cross sections for other types of processes involving DM
particles. Of particular interest in this context are the DM annihilation
cross section, which is relevant for calculating the DM relic abundance and
for indirect detection experiments, and the cross section for DM-nucleon
scattering, which determines the event rates in direct detection experiments.
Figure 3 illustrates this observation for the case of a simplified model with
an $s$-channel mediator.

If the same model can be used to obtain predictions for several types of
experiments, we can use it to compare the sensitivity of the different
experimental strategies and to map bounds from one kind of search onto the
parameter space relevant for another approach. Such a comparison can be
important for a number of reasons. As long as there is no conclusive signal,
we may be interested in understanding which experimental strategy is the most
promising and where we would expect to first see a DM signal. Furthermore, we
may hope to learn how large the allowed parameter space is, telling us what
sensitivity we need to aim for and when to stop pursuing a specific search.

Figure 3: Illustration of the different processes involving a DM particle
$\chi$ and an $s$-channel mediator $R$. From left to right the diagrams
correspond to DM annihilation, DM-nucleon scattering, DM production together
with a jet from initial state radiation and the decay of the mediator into a
di-jet resonance.

The possibility to compare different search strategies will become even more
important once a conclusive mono-$X$ signal has been observed at the LHC.
Crucially, the LHC alone cannot establish the stability of invisible particles
it produces. To infer the DM nature of such particles necessarily involves the
connection to non-collider experiments and to cosmological observations. It is
therefore essential for any _DM search_ at the LHC (as opposed to a mere
search for invisible particles) to be constructed in a way that facilitates
such a connection.

A detailed discussion of how the comparison between different experimental
strategies is made for the different approaches discussed above is beyond the
scope of this review (some of the possible caveats are discussed in refs. 158,
159). Instead, we will discuss the general approach and some of the inherent
complications, and present some of the key results.

###  4.1 Relic density and indirect detection

As discussed in the introduction, one of the central motivations for DM
searches at the LHC stems from the idea of thermal freeze-out, which motivates
sizable interactions between DM particles and SM states. It is therefore a
pertinent question whether this link holds in practice, i.e. whether the LHC
achieves sufficient sensitivity to test the freeze-out paradigm.

Within a specific model, it is straightforward to calculate the DM
annihilation cross section and compare it to the thermal cross section
$\langle\sigma v\rangle=3\cdot 10^{-26}\>\mathrm{cm^{3}s^{-1}}$ $\approx
2.5\cdot 10^{-9}\>\text{GeV}^{-2}$. If, for example, DM interacts with quarks
via a dimension-6 effective operator with suppression scale $\Lambda$, we
expect from dimensional arguments an annihilation cross section of the order
of $m_{\text{DM}}^{2}/\Lambda^{4}$ (assuming $m_{\text{DM}}\gg m_{q}$). If the
DM particle is light (i.e. of order of a few tens of GeV or less) this
estimate implies that $\Lambda$ must be well below the TeV scale (and
therefore within the reach of colliders) in order to achieve a sufficiently
large annihilation cross section. Collider constraints therefore typically
severely constrain thermal production for low-mass DM [160, 161, 162, 86].

There are a number of caveats to this line of reasoning. First of all, LHC DM
searches primarily probe the interactions of DM with quarks and gluons. If DM
interacts primarily with color-neutral SM states such as leptons, it will be
very difficult for the LHC to constrain the coupling strengths implied by
thermal freeze-out. Furthermore, there are many ways in which the standard
thermal production mechanism can be altered. For example, there could be late-
time entropy injection (reducing the DM abundance) or a particle-antiparticle
asymmetry in the dark sector (increasing the DM abundance) [160]. Finally,
there could be additional particles in the dark sector, so that the DM
particle can experience co-annihilation [163, 164, 165]. Nevertheless, it
remains interesting to understand whether standard thermal production via the
couplings between DM and quarks can yield the observed relic abundance or
whether LHC bounds imply the presence of some additional ingredient (in the
form of additional couplings or modifications of standard cosmology) to match
observations.

The same processes that set the DM relic abundance can be searched for in
indirect detection experiments. If DM annihilates into quarks, the ensuing
hadronization produces a continuum of $\gamma$ rays as well as a
characteristic flux of anti-protons, which can potentially be observed with
satellites such as Fermi-LAT [166] or AMS-02 [167]. The bounds obtained from
these searches can then be mapped onto the parameter space relevant for LHC
searches [168, 169, 170, 171, 172]. Conversely, one can use the simplified
model framework to compare LHC constraints with astrophysical excesses to
infer whether a consistent DM interpretation is possible [173, 174].

It is important to keep in mind that in contrast to LHC searches, indirect
detection experiments rely on the presence of DM in astrophysical objects
(such as the Galactic Center or Milky Way dwarf spheroidals) and therefore
suffer from astrophysical uncertainties related to the DM distribution. We
will return to this issue in the context of direct detection experiments.

###  4.2 Direct detection

Direct detection experiments aim to detect the scattering of DM particles off
nuclei in low-background underground detectors. Since DM particles in the
Galactic halo move with velocities of order $v\sim 10^{-3}$, the momentum
transfer in a DM scattering event is typically $\mu v\lesssim 100\>\text{MeV}$
with $\mu$ being the DM-nucleus reduced mass [175]. DM particles therefore
interact not with individual quarks but coherently with the entire nucleus. As
a result there are two main types of scattering: spin-independent scattering
and spin-dependent scattering. In the former case the DM particle couples to
the mass $A$ (or charge $Z$) of the entire nucleus and scattering rates
therefore receive a coherent enhancement proportional to $A^{2}$ (or $Z^{2}$).
In the latter case, the DM particle couples dominantly to unpaired nucleons
within the nucleus, so that there is no large enhancement factor [96].

DM direct detection experiments thus place very strong bounds on any model of
DM that predicts spin-independent interactions [176, 177], in particular
simplified models with vector or scalar mediators as well as the corresponding
effective operators [89]. For spin-dependent interactions, on the other hand,
constraints are comparable in strength to the ones obtained from the LHC for a
wide range of DM masses. While direct detection is more sensitive to DM in the
TeV range (where the LHC runs out of energy), the LHC has a distinct advantage
for low-mass DM. The reason is that for DM masses below a few GeV, the
momentum transfer in direct detection experiments becomes too small to be
detectable for existing experimental strategies.

An alternative way to probe spin-dependent scattering is to consider the
capture of DM particles in the sun, followed by DM annihilation [178].
Typically, these two processes are in equilibrium, so that the annihilation
rate is determined by the capture rate and hence directly comparable to direct
detection experiments. If the DM annihilation process leads to the production
of neutrinos (for example as a result of hadronization), the capture rate can
be constrained by neutrino telescopes such as IceCube [179] or Super-
Kamiokande [180]. Indeed, these constraints are comparable to the ones
obtained by the most sensitive direct probes of spin-dependent scattering
[181, 182] and provide complementary constraints on DM simplified models [183,
127].

On closer inspection, it turns out that the simple division into spin-
independent and spin-dependent interactions does not capture all
possibilities. A general approach in terms of a non-relativistic effective
theory of DM scattering identifies twelve additional operators that predict
smaller but potentially relevant scattering rates [184, 185, 186]. It is
possible to construct a mapping from the various simplified models to this set
of operators and hence calculate event rates in direct detection experiments
for all of the models discussed in section 3 [187]. It turns out that some of
these models predict a strong suppression of event rates in the non-
relativistic limit and are therefore completely unconstrained by direct
detection experiments. This is for example the case for simplified models with
a pseudoscalar mediator, which have received significant interest for
precisely this reason [188, 189].

A further subtlety in the mapping from collider constraints to direct
detection experiments results from the large separation of scales between the
energies relevant for LHC collisions and the ones describing the scattering of
DM particles from the Galactic halo. Indeed, it has been shown that
renormalization group evolution can lead to important effects, in particular
for models where the tree-level calculation predicts a strong suppression of
the event rates [190, 162, 191, 192, 193, 194, 186].

Unfortunately, the comparison between LHC results and direct detection
experiments is complicated by a number of theoretical uncertainties. First of
all, calculating event rates in direct detection experiments requires
knowledge of the relevant nuclear form factors, which can carry uncertainties
of $10\%$ or more [195]. Even larger uncertainties result from our lack of
knowledge of the local DM density and the corresponding velocity distribution
[196]. There is consequently a great need for techniques to compare LHC and
direct detection experiments without the need to make strong assumptions on
the astrophysical parameters (so-called halo-independent methods) [197, 198].
Finally, it is important to keep in mind that direct detection experiments
rely on the assumption that all of DM consists of only one type of particles.
If there are instead several different sub-components, direct detection
experiments will be at a disadvantage compared to the LHC [116].

###  4.3 Searches for mediators

In addition to the complementarity between the various DM searches discussed
above, there is an important interplay between all these searches and
processes not involving any DM particles at all. For example, if DM can be
produced at the LHC via an $s$-channel mediator, this mediator may always
decay back into quarks or gluons (see panel (d) in figure 3) [199, 200, 201].
Consequently searches for di-jet resonances place strong bounds on models for
DM production at the LHC [202, 116, 203, 204, 205, 206]. A similar argument
can be made in the EFT approach. If for example DM particles interact with
quarks via an effective operator of the form
$\frac{1}{\Lambda^{2}}\bar{q}\gamma^{\mu}q\,\bar{\chi}\gamma_{\mu}\chi$, one
would in general also expect the presence of an analogous operator involving
only quarks: $\frac{1}{\Lambda^{\prime
2}}\bar{q}\gamma^{\mu}q\,\bar{q}\gamma_{\mu}q$ with
$\Lambda^{\prime}\approx\Lambda$ [207].

In the context of spin-1 $s$-channel simplified models, the strong constraints
from searches for di-jet resonances imply that sizable mono-$X$ signals can
only be expected if the mediator couples much more strongly to DM than to
quarks [116, 203]. In this case, there is an intriguing interplay between di-
jet searches and mono-$X$ searches: In the on-shell region, where
$m_{\text{DM}}<m_{\text{med}}/2$, the mediator decays dominantly invisibly and
di-jet searches are much less sensitive than mono-$X$ searches, whereas in the
off-shell region the mediator can only decay visibly and hence di-jet searches
give the best constraints.

It is worth noting that searches for di-jet resonances lose sensitivity for
low-mass resonances (because of the overwhelming QCD background) and for large
widths (because of the uncertainty in the background shape). The former
problem can be addressed by searching for di-jet resonances in association
with electroweak gauge bosons, which suppresses gluon-induced backgrounds
[201, 116, 204]. The latter problem can be addressed by studying not only the
invariant mass of the two jets but also their angular distribution [208, 209].

Of course, the mediator may also have further decay modes that can be
detectable. For example, a spin-0 $s$-channel mediator with
$m_{\text{med}}>2m_{t}$ would be expected to decay dominantly into top quarks
[85]. Searches for $t\bar{t}$ resonances are however complicated by the fact
that both the predicted new-physics signal and the SM background arise from
gluon initial states, so that interference effects become important [210, 211,
212, 213]. Because of the resulting peak-dip structure, searches for
$t\bar{t}$ resonances are typically less sensitive than searches for di-jet
resonances.

Finally, we note that many UV completions of simplified models with spin-1
mediators predict that the mediator couples not only to quarks, but also to
charged leptons [214, 215, 216, 217, 112]. Correspondingly, one could hope to
detect the mediator of the DM interactions also in searches for di-lepton
resonances [218, 219, 220]. Indeed, these searches are so sensitive, that
either the production cross section of the resonance or its branching fraction
into leptons must be tiny. In the context of LHC searches for DM it is
therefore typically assumed that the mediator does not couple to leptons at
high scales and couplings are only introduced at the loop level (if at all)
[121].

##  5 Conclusions

This review has demonstrated that searching for DM at the LHC is a thriving
research field – both in terms of experimental strategies and theoretical
developments. A large number of different mono-$X$ searches are now performed
by the experimental collaborations and a number of different theoretical
approaches are available for predicting DM signals and interpreting results. A
particularly exciting aspect is the complementarity between LHC searches and
alternative approaches to the search for DM, which promise to cover the full
parameter space of many well-motivated models of DM.

It is clear that a discovery of DM at the LHC is not guaranteed – many well-
motivated DM models like axions or sterile neutrinos predict production cross
sections that are much too small to be observable. Nevertheless, the LHC will
be able to comprehensively test the paradigm that DM was produced via thermal
freeze-out, provided DM couples to quarks or gluons. In other words, if DM
ever was in thermal equilibrium with SM states, the potential to discover DM
at the LHC is large. Conversely, a non-observation of DM at the LHC would have
important implications for our understanding of DM, in particular when
combined with the information from other types of DM searches.

While there have been many studies on the potential sensitivity of different
search strategies and many proposals for how to compare experimental bounds
from different experiments, up to now there is a surprising lack of studies
for how to interpret the observation of an excess at the LHC. One explanation
for this may be that such an interpretation is inherently difficult: The
missing energy spectrum predicted in most DM models is essentially featureless
and depends only mildly on the properties of DM. Reconstructing the mass, spin
and couplings of the DM particle will therefore likely require input from
several mono-$X$ searches, and possibly even the combination of LHC and non-
LHC searches for DM.

Simplified models may prove to be a useful tool for such a comparison, but it
is also conceivable that further extensions of this framework will be
necessary. In any case, it is certainly timely to explore these issues
further. The next few years promise spectacular advances in the sensitivity
for DM, and we may move very quickly from setting bounds to interpreting hints
and finally to mapping out the DM parameter space. No experiment or search
channel will achieve this goal in isolation – so it is imperative to maintain
a broad experimental program and an open mind about the models used to
describe the interactions of DM.

## Acknowledgments

F. Kahlhoefer would like to thank Sebastian Bruggisser, Alexander Grohsjean,
Ulrich Haisch, Enrico Morgante and Kai Schmidt-Hoberg for useful comments on
the manuscript.

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