Decoding the Nature of Dark Matter at Current and Future Experiments

1. Introduction

Understanding the nature of Dark Matter (DM) is one of the greatest puzzles of modern particle physics and cosmology. Although overwhelming observational evidences from galactic to cosmological scales point to the existence of DM [1–3], after decades of experimental effort only its gravitational interaction has been experimentally confirmed. Currently, no information is available on the DM properties, such as its spin, mass, interactions other than gravitational, symmetry responsible for its stability, number of states associated to it, and possible particles that would mediate the interactions between DM and the standard model (SM) particles.

If DM is light enough and interacts with SM particles directly or via some mediators with a strength beyond the gravitational one, its elusive nature can be detected or constrained in different ways: (a) from direct production at colliders, resulting in a signature exhibiting an observed SM object, such as jet, Higgs, Z, W, photon, or top-quark(s) that recoils against the missing energy from the DM pair [4–8]; (b) via the relic density constraint obtained through the observations of cosmic microwave background (CMB) anisotropies, such as those of WMAP and PLANCK collaborations [1, 9, 10]; (c) from DM direct detection (DD) experiments, which are sensitive to elastic spin independent (SI) or spin dependent (SD) DM scattering off nuclei [11–14]; (d) from DM indirect detection searches, that look for SM particles produced in the decay or annihilation of DM present in the cosmos, both with high energy observables (gamma-rays, neutrinos, charge cosmic rays) produced in the local Universe [15–20], and by studying the effects of energy produced by DM annihilation in the early universe on the properties of the CMB spectrum [1, 21, 22].

It is clear that decoding the nature of DM requires the respective signal at least in one of the search experiments. We do not have one yet. However, even without having the signal at the moment we can already conclude on what kind of DM models are excluded. Moreover, by exploring different signatures of one particular model, their correlation and interplay we can prepare ourselves to discovery of DM and their identification.

2. Contact Interactions

Let us start our discussion with the three simplest scenarios for the DM particles: complex scalars (ϕ), Dirac fermions (χ), and complex vectors (V_μ) within the effective field theory (EFT) approach. In the EFT approach we parametrize the DM interactions with the SM quarks and gluons with the effective coupling and the scale describing operators of dimension six or five. In Table 1 we have summarized a minimal set of independent dimension-5 and dimension-6 operators for complex scalar, Dirac fermion, and complex vector DM coupling to quarks and gluons, adopting the widely used notations of Goodman et al. [23], Kumar et al. [24], and Belyaev et al. [25]. Since different operators have different energy behavior and respective different invariant mass distributions: typically softer for majority operators with scalar DM, intermediate for fermion DM and the hardest for vector DM and because of relation of M_inv(DM, DM) and $E_{\text{T}}^{\text{miss}}$ slope one can distinguish several operators and related underlying theories using the shape of the $E_{\text{T}}^{\text{miss}}$ signal: C1-C2,C5-C6,D9-D10,V1-V2,V3-V4,V5-V6, and V11-12 pairs among each other [25].

TABLE 1

Table 1. Minimal basis of dimension 5 and 6 operators for complex scalar DM (ϕ), Dirac fermion DM (χ) or complex vector DM (V^μ) interacting with SM quarks (q) or gluons.

Notice the presence of the coupling g_* in the definition of the effective operators, which we insert according to the Naive Dimensional Analysis [26]. Moreover, for the vector DM case we choose the parametrization suggested in Belyaev et al. [25] that takes into account the high energy behavior of the scattering amplitudes that are enhanced by an energy factor (E/m_DM) for every longitudinal vector DM polarization. These operators are gauge invariant and provides the minimal and simplistic description of underlying theory of DM. The scale Λ is related to the mass of the mediator, while coupling g^* is related to the product of DM and SM couplings to the mediator.

These operators provide monojet-signature, the shapes of $E_{\text{T}}^{\text{miss}}$ distributions for which is presented in Figure 1 from Belyaev et al. [25] for DM mass of 10 GeV. One can observe a big difference in $E_{\text{T}}^{\text{miss}}$ shapes of the groups of the operators, primarily split into groups of operators with scalar, femion and vector DM. The origin of the different $E_{\text{T}}^{\text{miss}}$ shapes from different operators can be related to a combination of effects. First, for a fixed Lorentz structure of the SM part of the EFT operators, the same invariant mass distribution of the DM pair, M_inv(DM, DM), uniquely defines the shape of the $E_{\text{T}}^{\text{miss}}$ distribution. Moreover, with the increase of M_inv(DM, DM), the $E_{\text{T}}^{\text{miss}}$ shape falls less and less steeply (again, for a given SM component of the EFT operator).

FIGURE 1

Figure 1. $E_{\text{T}}^{\text{miss}}$ parton level distributions for a representative subset of the EFT operators from Table 1 for 13 TeV LHC energy and M_DM = 10 GeV.

It was found in Belyaev et al. [25] that the reason why the bigger invariant mass of DM is correlated with flatter $E_{\text{T}}^{\text{miss}}$ behavior is related to phase space and parton density effects: when M_inv(DM, DM) is small, the radiation of a high p_T jet will “cost” a large relative shift in x, the transferred momentum of the parton, leading to a rapidly falling $E_{\text{T}}^{\text{miss}}$ distribution; on the contrary, when M_inv(DM, DM) is large, the radiation of a high p_T jet will “cost" a small relative shift in x, which will lead to a more slowly falling $E_{\text{T}}^{\text{miss}}$ distribution in comparison to the first case.

Therefore if one theory predicts higher values of the invariant mass of DM-pair, M(DM, DM), than the other theory, one expects the flatter $E_{\text{T}}^{\text{miss}}$ distribution for the first one. In Figure 2 we present M(DM, DM) distributions for all EFT operators in Table 1 where one clearly observes that the mean values of M(DM, DM) distributions for vector DM operators are larger than those for most of fermion DM operators which are in their turn have higher mean value of M(DM, DM) than most of scalar DM operators. Now we can see the connection of M(DM, DM) distributions shape and the slope of the $E_{\text{T}}^{\text{miss}}$ which was presented in Figure 1.

FIGURE 2

Figure 2. Invariant mass of DM pair distributions normalized to unity for EFT operators in Table 1 for 13 TeV LHC energy, M_DM = 10 GeV and p_{T, jet} ≥ 100 GeV cut applied.

One should stress that non-collider DM searches play an important complementary role in probing DM parameter space. As an example in Figure 3 (left) we present the non-collider constraints for the operators D2, which exhibit pseudo-scalar interactions of fermion Dirac DM with quarks.

FIGURE 3

Figure 3. (Left) Non-collider constraints on D2 operator with fermion DM: (i) SI DM DD searches (shaded blue region below the lowest blue contour), (ii) constraints from relic density (above the yellow dashed line), (iii) constraints from the CMB (shaded green area), and (iv) constraints from the validity of the EFT (Λ > 2m_DM). (Right) LHC monojet constraints on D2 EFT operator. The area inside the red, orange and blue solid curves is excluded by current LHC data at 95% CL for g_⋆ = 1, 6, and 4π, respectively. The projected LHC limits for 300 fb⁻¹ are indicated by dashed thin lines. The combined exclusion regions from CMB and DM DD searches for g_⋆ = 1 are given by the light-purple area. See details and complete set of plots in Belyaev et al. [27].

One can see that even for momentum-suppressed operator D2 (because of its pseudo-scalar nature) DM DD constraints from Xenon [28] play an important role which is comparable to collider constraints, presented in Figure 3 (right). It is important to stress that both LHC and DM DD searches set an upper limit on value of Λ. The LHC limit is of the order of 1 TeV for present LHC data while DM DD searches the limit strongly depend on the operator. For example for non-suppressed operators conserving parity the limit on Λ is about 3 orders of magnitude above the LHC one. On the other hand LHC limit is beyond DM DD searches for operators with suppressed elastic scattering cross sections on the nuclei (C2,C4,C6,D2,D3,D4,D6-10,V2,V4-V10). Moreover, for operators with pseudo-vector currents which have suppressed DM DD rates, one should take into account effect of their running from TeV energy scale at the LHC down to low energy scale at DM DD experiments, due to which an operator acquires non-negligible vector component [29–31].

3. Beyond EFT

The analysis of $E_{\text{T}}^{\text{miss}}$ shape presented here can be applied to different scenarios, beyond the EFT approach in general, where the DM mediator is not produced on-the-mass-shell, such as the case of t-channel mediator or mediators with mass below 2M_DM, where the M_inv(DM, DM) is not fixed. This case covers a wide range of theories.

As an example in Figure 4 (left) the normalized shape for $E_{\text{T}}^{\text{miss}}$ distribution from $pp\to {\chi }_1^{+}{\chi }_1^{-}/{\chi }_1^{\pm }{\chi }_1^0\to {\chi }_1^0{\chi }_1^0+\text{soft leptons}/\text{jets}$ Minimal Supersymmetric Model(MSSM) signal and its dominant irreducible background $Z+jet\to \nu \bar{\nu }+jet$ (Zj) is presented for LHC@13TeV [32]. The model parameter space in the compressed chargino-neutralino scenario driven by small μ parameter is essentially characterized only by DM (${\chi }_1^0$) and chargino (${\chi }_1^{+}$) masses and mildly depends on the value of tanβ.

FIGURE 4

Figure 4. (Left) Normalized signal (dotted blue and dashed red) and Zj background (solid black) parton-level $p_T^j$ distributions for the 13 TeV LHC for the NSUSY scenario: normalized signal and Zj background distributions. See details in Barducci et al. [32]. (Right) Normalized $E_{\text{T}}^{\text{miss}}$ from h₁h₂j i2HDM signal vs. background for $\sqrt{s}=$ 13 TeV. See details in Belyaev et al. [33].

In the Figure 4 (right) we present $E_{\text{T}}^{\text{miss}}$ from h₁h₂j inert two Higgs doublet model (i2HDM) signal alongside the estimated (by CMS) experimental background for $\sqrt{s}=$ 13 TeV. The parameter space of the model, details of which can be found for example in Belyaev et al. [33], is characterized by DM mass (h₁), the mass of the second neutral scalar (h₂), the mass of the charged scalar (h⁺) and the DM-Higgs boson coupling. However, the h₁h₂j production cross section depends only on two parameters–h₁ and h₂ masses in analogy to the above SUSY scenario. An important feature of the signal vs. background shapes in these completely different theory cases is that the background falls more rapidly with $E_{\text{T}}^{\text{miss}}$, and the difference in the slope with respect to the signal is bigger for higher DM mass. This behavior has the same explanation as for EFT study case above—it is related to the bigger invariant mass of the invisible system for the signal—M(DM, DM) than for the background—M_Z. This feature is very instrumental to increase signal-to-background ratio (S/B) (which is typically below 1% for low $E_{\text{T}}^{\text{miss}}$ cuts) by increasing the value of $E_{\text{T}}^{\text{miss}}$ or by performing the signal-background shape analysis [33].

The role of non-collider DM searches is also crucial in case of these two complete and consistent models. And an example in Figure 5 we present the projected LHC reach for MSSM monojet signal in the $\Delta M=m_{{\chi }_1^{+}}-m_{{\chi }_1^0}$, $M_{DM}=m_{{\chi }_1^0}$ parameter space together with LUX and Xenon1T DM DD exclusion [32]. One can see that LHC would be able cover neutralino DM mass only below 250 GeV (with the assumption that S/B of the order of 3% will be under control) even with 3 ab⁻¹ total integrated luminosity. But coverage of this region is important as well as LHC-Xenon1T complementarity: LHC will be able to cover the region inaccessible by Xenon1T in small ΔM region, while Xenon1T is able to cover m_DM well beyond the LHC reach for ΔM > 3 − 5 GeV. In case of i2HDM model collider sensitivity with mono-jet signature is even more limited because of the lower production rates of the scalar DM, h₁, or its inert partners (h₂ and h⁺) and expected LHC reach is below 100 GeV for M_h1.

FIGURE 5

Figure 5. Exclusion contour lines for the 13 TeV LHC at the end of the LHC Run2 (light red region) and of the HL-LHC (light blue region). The region excluded by LUX and Xenon1T are also shown, together with the LEP limit. See details in Barducci et al. [32].

4. Beyond Mono-X Signature

While mono-X (with X being jet, γ, Z, H, t etc.) DM signatures at colliders are the most general ones, their rates are typically very low (usually at the percent level or even lower). Besides several other interesting but model-specific DM signature studies, one should stress one signature which can be also considered as quite generic one. In case when DM, D⁰, is embedded into electroweak multiplet and its mass split from the charged odd particle(s), D⁺, is generated only radiatively (preserving gauge invariance), the one can find that the value of this mass split is of the order of 0.2 GeV. In this case D⁺ has a very small width and respectively large life-time. D⁺ being long lived particle (LLP) dominantly decays into DM and very soft pion: D⁺ → D⁰π⁰. Production of D⁺ in pairs or in association with DM leads then to the typical signature from charged LLP: disappearing charged track (DCT) as soon as the track from LLP is long enough (from few cm to a meter). In case of such signature the S/B ratio is much higher than in case of mono-jet signal and therefore, substantially bigger DM masses can be probed with charged LLPs from DM sector [34–36]. As an example, we would like to present here results for the minimal vector triplet DM (V⁰) model [36] which predicts the right amount of DM for M_DM in the 3-4 TeV range depending on DM coupling to the Higgs boson.

In this model the SM is supplemented by a new massive vector boson V_μ in the adjoint representation of SU(2)_L, e.g., by two new massive vector particles: V⁰ and V^±. If V_μ transforms homogeneously (i.e., $V_{\mu }\to g_L^{†}{V}_{\mu }{g}_L$ where g_L∈SU(2)_L) and Z₂ symmetry is imposed (which links the quartic V coupling to the gauge coupling constant and makes theory unitary is unitary before EW symmetry breaking and in the absence of the Higgs boson as found in Zerwekh [37]) then Lagrangian can be written as:

$\begin{array}{l}{L}={{L}}_{SM}-Tr\left\{D_{\mu }{V}_{\nu }{D}^{\mu }{V}^{\nu }\right\}+Tr\left\{D_{\mu }{V}_{\nu }{D}^{\nu }{V}^{\mu }\right\}\\ -\frac{g^2}{2}Tr\left\{\left[V_{\mu },V_{\nu }\right]\left[V^{\mu },V^{\nu }\right]\right\}\\ -igTr\left\{W_{\mu \nu }\left[V^{\mu },V^{\nu }\right]\right\}+{\tilde{M}}^{2}Tr\left\{V_{\nu }{V}^{\nu }\right\}\\ +a\left({\Phi }^{†}\Phi \right)Tr\left\{V_{\nu }{V}^{\nu }\right\}\end{array}$

where D_μ = ∂_μ − ig[W_μ, ] is the usual SU(2)_L covariant derivative in the adjoint representation and ${{L}}_{SM}$ represents the SM Lagrangian. The main difference with respect to the model in Zerwekh [37] is that the SU(2)_L symmetry is broken by the Higgs mechanism and the associated gauge bosons have mass. We thus allow for a coupling of V to the Higgs scalar field Φ. Due to the Z₂ symmetry the neutral new vector boson, V⁰ is stable and therefore is the perfect DM candidate. The mass split, ΔM, between V⁰ and V^± is generated radiatively and its value is just above the pion mass which makes V^± long lived. In Figure 6 we present ΔM as a function of M_V, which was calculated in Belyaev et al. [36].

FIGURE 6

Figure 6. The one-loop radiatively induced mass splitting between the charged and neutral components of the vector DM. The solid lines represent ΔM computed at fixed values of the renormalization scale Q. The shaded green band indicates the range of values obtained by varying Q continuously between min{M_V/2, M_Z/2} and max{2M_V, 2M_Z} and thus constitutes an estimate of the uncertainty on ΔM. The solid black line is the one-loop mass splitting, with all higher order terms truncated [36].

In Figure 7 we present results for spin-independent cross-section for V⁰-nucleon elastic scattering as a function of M_V and for representative values of a. It is very important to note that Xenon1T experiment combined with DM relic density constraints excludes DM mass above 4 TeV.

FIGURE 7

Figure 7. Spin-independent cross-section for V⁰-nucleon elastic scattering as a function of M_V and for representative values of a. The continuous black curve represents the elastic cross-section computed with the values of M_V and a that saturate the measured DM relic density. The gray dashing highlights the parameter space where perturbative unitarity loss occurs at too low scale. See details in Belyaev et al. [36].

At the same time the production rate σ_eff for σ(pp → V^±V⁰) + 2σ(pp → V⁺V⁻) process which leads to the disappearing charge track signatures (that is why pp → V⁺V⁻ process comes with coefficient 2 which is equal to the number of disappearing charged tracks it provides) are high enough to probe this process at colliders. In particular, the current LHC@13TeV limit on M_V is about 1.4 TeV as one can see from Figure 8. Moreover, one can also see from this figure that 100 TeV collider will be able to exclude DM mass below 4 TeV, thus allowing to probe the entire parameter space of the model. One should also note that in case of i2HDM, DCT signature also allows to substantially enhance LHC potential and probe DM mass upto about 500 GeV [34] which is much higher than 100 GeV—the maximum DM mass which can be probed via mono-jet signature.

FIGURE 8

Figure 8. The effective cross-sections ${\sigma }_{eff}=\sigma (pp\to V^{\pm }{V}^0)+2\sigma (pp\to V^{+}{V}^{-})$ at leading order for the vector isotriplet model for 13 and 27 TeV LHC energies and for a 100 TeV future collider. The dashed lines corresponds to collider sensitivity. See details in Belyaev et al. [36].

5. Toward Decoding Framework

There is no framework at the moment which can solve the reverse engineering task—the task of decoding the nature of DM. It is not surprising why—we are all eagerly looking for the signal first of all and busy with the interpreting and exploring our own models. The huge amount of work has been done on the model building, phenomenology and experimental searches as well as on building different tools, examples of which has been given above. And there is really huge potential of combining different methods and signatures to probe different models. What is missing is the framework which joins all these pieces in one tool which would help us to decode underlying theory, in particular its part related to DM. The task of decoding of the whole underlying theory sounds probably too ambitious to the author, while decoding of its DM part sound more realistic since it contain specific and possibly much smaller piece of the theory. This framework requires the database of models, database of various signatures and set of tools which will be able to effectively explore not only parameter space of each particular model, but also the model parameter space and compare predicted signatures with the observed ones. Such a framework would allow objectively judge about preferred model or set of models which would fit signal best of all. An example of the prototype of such a framework actually already exists in the form of High Energy Physics Model Data Base (HEPMDB) (https://hepmdb.soton.ac.uk) [38], created at Southampton University in 2011. At the moment HEPMDB is created as a web-server accessible to everybody and is able to:

1. collect HEP models for all multipurpose Matrix Element (ME) generators in the form of Feynman rules and parameters written in the format specific for a given package;

2. collect models' sources which can be used to generate HEP models for various ME generators using FeynRules [39] or LanHEP [40];

3. allow users to perform simulations for their own models or models available at HEPMDB using the full power of the High Performance Computing (HPC) IRIDIS cluster standing behind the HEPMDB itself. Connection to HPC cluster is one of the key features of the HEPMDB: it provides a web interface to various ME generators (CalcHEP[41], Madgraph [42], and Whizard [43]) which can then also be run directly on the HPC cluster avoiding problems related to installing the actual software, which can sometimes be quite cumbersome;

4. collect simulated events and plot distributions using web interface.

Though the signature database at HEPMDB is at the development stage, users can indicate some essential features of the signatures which model can provide, such as presence of resonance, $E_{\text{T}}^{\text{miss}}$ etc. The next step of development of HEPMDB will include an addition of various packages to event analysis. Probably the most important feature of HEPMDB is that it can be developed by the whole HEP community—any registered user can add his/her own model and signature which would be used for identification of underlying theory when the experimental signal will be observed.

6. Conclusions

In the absence of DM signal we can still do a lot—we can prepare ourselves for its discovery and identification. $E_{\text{T}}^{\text{miss}}$ shape is quite instrumental in understanding the underlying theory at colliders, while direct and indirect DM searches are very powerful in complementing collider searches especially in the parameter space with large DM mass. We also advocate the usage of new DM signatures such as disappearing charge tracks which allows to substantially extend collider exploration of large DM mass. Moreover, we would like to stress the crucial role of 100 TeV pp collider which is likely to exclude substantial amount of thermal DM models. We show that collider and non-collider DM searches have a unique power to decode the nature of Dark Matter on the examples of several appealing DM theories. Such complementarity and usage of different signatures would allow us to decode the nature of DM, signals from which we are expecting in the near future. Finally, we advocate the importance of the framework which would combine the experience of HEP community and would allow to identify effectively the underlying theory of DM from the experimental signal which we will hopefully observe in the near future.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Conflict of Interest Statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

AB acknowledges partial support from the STFC grant ST/L000296/1 and Soton-FAPESP grant. AB also thanks the NExT Institute and Royal Society International Exchange grant IE150682.

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