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I review the construction of simplified models for dark matter searches. After discussing the philosophy and some simple examples, I turn the attention to the aspect of the theoretical consistency and to the implications of the necessary extensions of these models.

Producing and studying the properties of the dark matter (DM) particles at the LHC are an extremely exciting possibility that would open the door to a new understanding of the interplay between astrophysics, cosmology, and particle physics. Essentially all the naturalness-inspired scenarios can accommodate the presence of a good dark matter candidate: a neutral and very long-lived particle that was copiously produced in the early universe and then lost thermal contact with the SM (if it ever occurred) leaving a relic density

A key task in these studies is that of choosing a theoretical framework to compare with data and compare the results of different experiments. Given the plethora of particle physics models beyond the SM providing a WIMP candidate, it is highly desirable to study the signatures of this DM candidate in a model-independent way. In the early stages of the LHC, this was achieved by means of the effective field theory approach (EFT). In this framework, the Standard Model (SM) is complemented by a set of non-renormalizable operators that parametrize the interaction of the DM particle with SM fields in terms of one effective scale

The important drawback of the EFT description is its intrinsic energy limitation. At energies larger than some cutoff

Partly in response to the problems of EFTs, and partly inspired by their rich phenomenological implications, in more recent years the LHC community has turned its attention to the tool-kit of simplified models. Such models are characterized by the most important state mediating the interaction of the DM particle with the SM, as well as the DM particle itself (see, for example, [

This paper is structured as follows. In Section

Thorough discussions about simplified DM models may be found in [

As in the case of the EFT, the idea behind simplified models is to provide a good representation of possibly all realistic WIMP scenarios within the energy reach of the LHC, restricting to the smallest possible set of benchmark models, each with the minimal number of free parameters. Simplified models should be complete enough to give an accurate description of the physics at the scale probed by colliders, but at the same time they must have a limited number of new states and parameters.

The starting point is always the SM Lagrangian, complemented with a DM particle and a mediator that couples to it, through renormalizable operators, to quarks and gluons, which is necessary for the production of these states at a hadron collider. A coupling to other SM particles can be included as well and will add interesting experimental signatures to the model. In general, some simplifying assumptions can be made: for example, one can take all couplings to be equal, or the couplings to third generation's quarks to be dominant. Interactions that violate the accidental global symmetries of the SM must be handled with great care. Indeed, constraints on processes that violate these symmetries are typically very strong and may overcome those coming from DM searches or even rule out all of the interesting parameter space of the simplified model. For this reason, CP, lepton number, and baryon number conservation is typically assumed, together with minimal flavour violation (MFV). (Constraints on BSM models from CP and flavour violating observables are very strong, and the energy scale at which new physics may show up must be larger than tens of TeV in the best case, if the flavour structure of the model is generic. Minimal Flavour Violation is a way to reconcile these constraints with possible new physics at the TeV scale [

Most simplified models of interest may be understood as the limit of a more general new physics scenario, where all new states but a few are integrated out because they have a mass larger than the energy scale reachable at the LHC or because they have no role in DM interactions with the SM. Similarly, in the limit where the mass of the mediator is very large, the EFT framework may be recovered by integrating out the mediator. On the contrary, there are new physics models which cannot be recast in terms of vanilla simplified models, typically because more than just one operator is active at the same time, possibly interfering with each other. The situation is summarized in Figure

Even if this may sound obvious, we should stress that the correspondence between simplified models and EFT is not one to one. Simplified models that involve mediators of different spin nature may give rise to the same effective operator after a Fierz rotation, as pointed out in [

Even when a simple correspondence between the EFT and the simplified model is assumed, limits on the EFT cannot be readily translated onto the simplified model because of the possible resonant enhancement (that would make the limit stronger) or the typically softer missing energy spectrum (that would weaken the limit) [

From the point of view of LHC searches, the enlarged physical spectrum and parameter space of simplified models with respect to the EFT represents a challenge, and a greater variety of search channels is involved. While within the EFT approach the mono-X searches hold the stage, simplified models of DM can be constrained also with multi-jet + MET searches, with di-jet and di-leptons resonance searches and many others, depending on the degree of sophistication and on the ingredients of the model. Interestingly, many of these searches do not involve the DM particle, but only the mediator, and constraints are often stronger than the mono-X ones. On the other hand, the EFT is still a useful tool when dealing with strongly coupled theories, where a description in terms of a perturbative simplified model is not viable [

We are now going to list a few examples of the simplified models of relevance for LHC searches, starting with those that include a fermionic DM

Let us briefly point out, and we will come back to this in Section

Consistently with the MFV hypothesis, we force the couplings to be diagonal:

Feynman diagrams for the production of a DM pair in association with a quark or a gluon, leading to a mono-jet signature. Additional diagrams obtained by exchanging a fermion and a anti-fermion, as well as the ones obtained by permuting the gluon vertices in the loop in the (pseudo)scalar case, are neglected. The diagrams enclosed in the dashed box in the

In addition to the parameters of the Lagrangian, in the calculation of scattering amplitudes one must also include the decay width

The exclusion lines that LHC draws have typically a simple structure. In MET+X searches in which DM is pair produced from the mediator and recoils against a SM particle (a photon, a hadronic jet or other) that is necessary to tag the event, the best sensitivity is obtained for

Sometimes, it proves very useful to show the constraints on

Missing transverse energy searches are not the only handle that we have on simplified models. Searches for the mediator, for example, in the resonant di-jet channel, can lead to more stringent bounds in

Constraints on

A very interesting phenomenology arises in the case where the couplings to third-generation quarks is larger than the couplings to the first two, as it is the case in the spin-0 models with MFV introduced above. Strong constraints on these models come from searches for one or two

Another interesting possibility is that of a coloured fermionic mediator with an interaction vertex between quarks and the WIMP resulting in a

As it can be easily understood, also in the case of

Two interesting features of this model are worth listing that makes it qualitatively different from its low energy EFT limit. Firstly, being the squarks coloured, gluons may be emitted not only as initial state radiation but also from the mediator itself. This process is suppressed in the EFT limit by two powers of

An interesting phenomenology arises in the case in which the

In addition to the models listed above, many interesting ones may be constructed that cannot be addressed here. Those include spin-2 mediators [

The simplified models that we discussed so far can be viewed as an improvement of effective operators, where the effective scale

The reason why the theoretical consistency of the simplified models is important is twofold. On the one hand, violation of perturbative unitarity can lead to spuriously large predictions (e.g., in the process of

The scalar and pseudo-scalar models of (

As we will detail below, this issue can be fixed if the mediator is assumed to couple at tree level

In [

A model of this kind naturally replicates the SM Yukawa-like structure of (

The vector and axial-vector simplified models of (

In the axial-vector model (

Again referring to the axial-vector model of (

If the interaction of DM with SM fermions is due to an extended gauge symmetry, in order for the theory to be consistent at the quantum level the charge assignment under the new gauge group cannot be generic. If all the fermions of the dark sector are uncharged under

Alternatively, additional heavy fermions, charged under the SM, may be added to the model. The mass of these fermions cannot be arbitrarily large: in order to cancel the anomalies, they must be chiral at least under the dark gauge group, and their mass is given by the vev of the dark Higgs. Fortunately, imposing the invariance of the SM Yukawa terms as discussed above in Section

This model is one of the simplest possible extensions of the SM, in which the gauge group is enlarged by an additional

A number of studies have addressed DM

The dark Higgs becomes relevant when the annihilation rate is concerned. Indeed, the

Models in which the (pseudo-)scalar mediator that couples to DM obtains its coupling to SM quarks from mixing with a second Higgs doublet have received a significant attention recently. With respect to the scenario in which the singlet mixes directly with the SM Higgs discussed in Section

A thorough discussion of 2HDM (not related to DM) is given in [

In the context of DM, many search channels have been used to constrain this class of models. In particular, searches for mono-

The Higgs’ width to invisibles constrains is

Interestingly, since in such a model the coupling of DM to the heavy quarks is naturally enhanced, the Galactic Centre excess could be explained by a model which is testable at the LHC [

Let us just mention another related possibility, which is the one of a second inert Higgs doublet, which does not couple to SM fermions except for its mixing with the Higgs. The lightest component of this doublet is a perfect candidate for scalar DM [

The question of which DM models should be adopted in defining new search strategies and in presenting experimental results is a pressing one, primarily for LHC searches. Simplified DM models are a possible answer to this question, living in between the effective operators approach (with a limited applicability at the LHC) and the realm of well-motivated BSM theories.

From a bottom-up viewpoint, the idea of simplified models is to expand the effective operators including mediator particles in the description, thus avoiding the energy limitations of the EFT approach and adding a richer phenomenology, new search channels, etc. In a top-down framework, instead, simplified models can be seen as a way to simplify the phenomenology of complex new physics models in such a way to restrict to the phenomena related to DM.

In order not to deal with unphysical results, the vanilla simplified models have to be supplied with additional constraints, couplings, and states, in a kind of second-order improvement. The typical consequence is that the strongest LHC constraints on the dark sector come from many possible observables other than DM production processes (as the mono-X searches) and di-jet searches (e.g., di-lepton resonances, mixing with

Theoretically consistent simplified models tend to loose part of their generality and to mimic richer BSM theories. For example, models containing a vector mediator and a dark Higgs may descend from gauged

Simplified models cannot (or only partially) be viewed as an exhaustive toolbox to constrain all possible WIMP scenarios at once. For this reason, it is of extreme importance that the LHC collaborations publish their results on simple, search-specific, models in such a way that they are recastable for any other model (as it is for cut-and-count analyses). In turn, theoreticians should keep working in close contact with experimentalists in order to maximise the utility of the simplified models tool-kit. Finally, the use of (truncated) EFT should not be disregarded, since this is the most model-independent approach and it is economical from the point of view of the reduced dimensionality of its parameter space.

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author is grateful to Michael Duerr and Davide Racco for the many useful comments on this manuscript.