^{1,2}

^{3}

^{4}

^{2,5}

^{1,2}

^{3}.

The inert two-Higgs-doublet model (i2HDM) is a theoretically well-motivated example of a minimal consistent dark matter (DM) model which provides monojet, mono-

The evidence for dark matter (DM) is well established from several independent cosmological observations, including galactic rotation curves, cosmic microwave background fits of the WMAP and PLANCK data, gravitational lensing, large scale structure of the Universe, as well as interacting galaxy clusters such as the Bullet Cluster. Despite these large-scale evidences, the microscopic nature of the DM particles remains unknown, since no experiment so far has been able to claim their detection in the laboratory and probe their properties. Potentially, DM can be produced at the LHC and probed in the DM direct detection (DD) underground experiments. The fundamental importance and vast experimental opportunities make the search for and investigation of DM one of the key goals in astroparticle physics and high energy physics (HEP), worthy of the intense efforts undertaken by the physics community.

At the other end of the length scale, the Standard Model (SM) of particle physics recently demonstrated its vitality once again. The scalar boson with mass

One way the particle theory community can respond to this situation is to propose simple, fully calculable, renormalizable BSM models with viable DM candidates. We do not know yet which of these models (if any) corresponds to reality, but all models of this kind offer an excellent opportunity to gain insight into the intricate interplay among various astrophysical and collider constraints. We call here these models minimal consistent dark matter (MCDM) models. MCDM models which can be viewed as toy models, are self-consistent and can easily be incorporated into a bigger BSM model. Because of these attractive features, MCDM models can be considered as the next step beyond DM effective field theory (EFT) (see e.g.

In this paper, we explore, in the light of the recent collider, astroparticle, and DD DM experimental data, the inert two-Higgs-doublet model (i2HDM), also known as the inert doublet model. This model is easily doable with analytic calculations, its parameter space is relatively small, and it can be strongly constrained by the present and future data. The model leads to a variety of collider signatures, and, in spite of many years of investigation, not all of them have yet been fully and properly explored. It is the goal of the present paper to investigate in fine detail the present constraints and the impact of the future LHC and DD DM data on the parameter space of this model.

The i2HDM

Even if we started with a complex

The idea that the symmetry-protected second Higgs doublet naturally produces a scalar dark matter candidate was first mentioned more that 30 years ago

Assuming that the lightest inert scalar is the only DM candidate, one typically finds that the low-mass region, below about 50 GeV, is excluded by the relic density constraints coupled with the LHC constraints on the invisible Higgs decay

The i2HDM can also have interesting cosmological consequences. Being an example of 2HDM, it possesses a rich vacuum structure, which evolves at high temperatures

There has also been a number of studies on collider signatures of the i2HDM. They focus on specific processes such as SM-like Higgs decays to

In the present work, to these many studies on the i2HDM, we add the following:

detailed combined analysis of the i2HDM model in its full five-dimensional (5D) parameter space, taking into account perturbativity and unitarity, LEP and electroweak precision data, Higgs data from the LHC, DM relic density, direct/indirect DM detection complemented by realistic (beyond-the-parton-level) LHC monojet analysis at the LHC;

quantitative exploration of the surviving regions of parameters, including very fine details and a qualitatively new region not seen in previous studies, which is enabled by our extensive numerical scans;

a combination of different processes giving the LHC monojet signatures: those with direct DM pair production and those with associate production of DM with another scalar with a close mass from the inert multiplet;

implication of experimental LHC studies on disappearing charged tracks relevant to a high (

separate, equally detailed analyses for the assumptions of the DM relic density being fitted to the PLANCK results or underabundant, allowing thus for additional allowed regions of the parameter space.

The paper is organized as follows. In Sec.

In order to represent a viable model, the potential

Once these restrictions are applied, the vacuum is neutral, and one can calculate the masses of the physical Higgs bosons. In addition to the SM-like scalar

One also notices that the sign of

With all these conventions, we describe the five-dimensional parameter space of i2HDM with the following phenomenologically relevant variables:

Another set of theoretical constraints comes from the symmetry breaking patterns in i2HDM

Restrictions on the

Rewriting conditions

We have implemented the i2HDM into the

To explore the phenomenology of the i2HDM we need to consider other constraints on its parameter space in addition to those coming from vacuum stability which we discussed above.

The first requirement we impose on the quartic couplings in

One should also stress that the vacuum stability condition given by Eq.

In Fig.

The part of the (

Very strong constraints on the i2HDM arise from precision data and searches from LEP experiments. First of all, the model should respect the precise measurements of the

While studying the phenomenology of the i2HDM, we should also make sure that Electroweak Precision Test (EWPT) data are respected. As we know, EWPT can be expressed in terms of three measurable quantities, called

Effect of the

We also excluded the region defined by the intersection of the conditions below:

The LHC Higgs data further restrict the i2HDM parameters space in the form of constraints on the couplings of the SM-like Higgs boson. A collection of combined fits from the Run I data, for both ATLAS and CMS, can be found in

A simpler possibility is, instead, to consider the best possible bound from the available fits on the two parameters. We follow this simpler procedure, confident that it will lead to a somewhat more conservative estimation of the bounds. For the invisible Higgs branching ratio, we consider the bound coming from the dedicated ATLAS search

One could also limit

For the second observable, the diphoton decay rate, we consider the result from the combined fit on the signal strength in the diphoton channel

It should be noted that we would expect a proper two-parameter fit to lead to stronger constraints than the ones we use; however, the qualitative impact of the constraints should be unchanged. For example, the partial decay width of the Higgs into DM which is defined by

The results from PLANCK

We have evaluated

Figure

The red-shaded region in Fig.

In the case of larger

The sharpest dip in the

At higher masses, we observe a wider and more shallow dip at around 80–90 GeV from

Finally, the last dip around 125 GeV corresponds to the reduction of the DM relic density due to the opening of the

The pattern of these last three dips is the same for the larger mass split scenario presented in Fig.

One can also observe qualitative differences in the asymptotic behavior of the DM relic density for small and large

For

The relic density,

We have also checked whether the i2HDM parameter space is consistent with the limits from DM DD experiments. We have evaluated the spin-independent cross section of DM scattering off the proton,

Rescaled spin independent direct detection rates

The flat asymptotic of

A related question is whether the model can be better probed by indirect detection (ID) experiments, i.e. the detection of energetic cosmic rays like

To have a complete picture of the properties of i2HDM in the whole parameter space, we have performed a five-dimensional random scan of the model parameter space with about

When performing the scan, we took into account the constraints mentioned above in the following succession. First, we applied only theoretical constraints from vacuum stability, perturbativity, and unitarity; second, we applied the collider constraints (LEP, EWPT, LHC Higgs data); last, we placed the upper bound on the DM relic density at

The salient features of the results of this scan, with all three groups of constraints applied successively, are presented in Fig.

The lower bound of

The LEP and LHC data also place constraints on the other inert scalars. Charged scalars lighter than 70 GeV as well as

The narrow strip at

The masses

Finally, we remark that above 200 GeV, EWPT forces

Color maps of DM relic abundance projected on the plane

Color maps of DM relic abundance projected on the planes

In summary, after all constraints mentioned here and exposed in more detail in Appendix, we found that the parameter space with

In our analysis, we generically allow the DM relic density to be equal or below the PLANCK constraints, Eq.

However, it is also instructive to explore the parameter space where both the upper and the lower PLANCK limits are satisfied. This parameter space region is presented in Fig.

Projection on two planes of the scan points passing all constraints, and fitting the PLANCK relic abundance within two sigmas. We show both a wide scan with masses between 10 and 1000 GeV, and a zoom on the low mass region.

Many interesting features of the i2HDM parameter space arise once the “correct” amount of DM relic density is required. One observes two very distinct

In the low mass region we clearly distinguish three regimes with specific physical properties:

A thin horizontal line with very small values of

The width of this strip is defined by the maximum allowed value of

For

In the region

The relic density can also be “just right” at large masses

Remarkably, the mass split is required to be

This small

Feynman diagram representing effective

From the Lagrangian above one can find the following formula for the

The decay width of

The next step is to check which

The

At the same time the figure presents the

After applying efficiency for the disappearing charged track signatures provided by CMS

One should also note that with increasing DM mass, the required split among

The i2HDM exhibits various signatures that are potentially accessible at the LHC. They can be generically described as “mono-object production,” that is, production of several final states in association with large missing transverse momentum. In this section, we undertake a detailed exploration of such processes which goes beyond the previously published state-of-the-art. We will first list the relevant processes, and then produce a cumulative plot which helps us compare their rates. With this knowledge, we will formulate convenient benchmark points which represent various qualitatively distinct regimes of i2HDM, and finally go into a more detailed calculation of monojet production.

The monojet signature originates from the

Feynman diagrams for the

There is one more process, namely

Feynman diagrams for the

Besides monojets, the i2HDM gives rise to a mono-

For the second diagram, the

Feynman diagrams for the

One should also note that for values of

The i2HDM could also provide a mono-Higgs signature via

Feynman diagrams for the

Feynman diagrams for the

Finally, one should mention the production of DM via vector boson fusion,

Diagrams for

When calculating the cross sections of mono-object production at the LHC, we used the following setup for the process evaluation:

the QCD renormalization and factorization scales

the PDF and the strong coupling constant are as provided by the NNPDF23LO (

for all processes a cut on the minimal value of missing transverse momentum of 100 GeV was applied;

the VBF cross section has been evaluated with the following additional cuts:

In Fig.

Cross sections versus dark matter mass,

It is also worth focusing on the new

The plot also shows that the mono-Higgs channels does not yield detectable rates.

The process

The experience we have gained so far, in both relic density and mono-object cross section calculations, allows us to discern several qualitatively distinct regimes of i2HDM and find their representative benchmark points. In Table

Benchmarks (BM) from the i2HDM parameter space together with corresponding observables: DM relic density (

The first two benchmarks have small and medium values of

BM2 differs from BM1 only by the value of

BM3 and BM4 correspond to the scenarios where

Finally, BM5 and BM6 represent the cases with a small (5 GeV) mass split and

From Table

In the previous subsections, we calculated the mono-object production cross sections at the LHC as a function of DM mass

In order to calculate the limits from the LHC at 8 TeV, we used the

The signature of both processes that we consider,

For both processes considered, we found that the lowest cross section limits for each benchmark point considered were provided by one of the ATLAS

In order to project these limits for increased luminosity and to 13 TeV, we use Monte Carlo events to estimate the efficiencies for the signal and background at 13 TeV, which is a function of

The results for the process

Cross sections and 95% C.L.s for

We should note that a similar projection of CMS monojet limits

In Fig.

Projected limit on

For

Cross sections and 95% C.L.s for

Taking into consideration these collider limits and also adding the projections of the direct detection XENON1T experiment, we are able to impose the complete set of constraints on the i2HDM parameter space. It is worth stressing that, as before, we present the limits using the rescaled DD cross section

The results of the constraints are presented in Fig.

Projection of the 5D random scan of the i2HDM into the (

Besides the AA region it is informative to find and analyze the region with

Projection of the 5D random scan of the i2HDM into the (

When comparing Figs.

From Figs.

Projection of the 5D random scan of the i2HDM into (a),(b) (

One should stress again the importance of the

We have also found the projected limits from colliders of monojet signatures and the XENON1T DD experiment for the i2HDM points which satisfy both the upper and the lower PLANCK limits, Eq.

The 2D projections of the 5D random scan of the i2HDM satisfying all constraints (Cut-1 to Cut-4) considered above for Figs.

As we can see, the incorporation of the DD constraint sets important restrictions on the parameter space. Still, in Fig.

The

The i2HDM is a clear example of a minimal consistent DM model which is very well motivated by theoretical considerations. At the same time this model could provide monojet, mono-

The model is implemented into the

In this paper we have performed detailed analysis of the constraints in the full 5D parameter space of the i2HDM from perturbativity, unitarity, electroweak precision data, Higgs data from the LHC, DM relic density, direct/indirect DM detection, and the LHC monojet analysis as well as implications of experimental LHC studies on disappearing charged tracks relevant to the high DM mass region. The LHC monojet analysis for the i2HDM model has been performed at the fast detector simulation level and provides new results together with limits from disappearing charged tracks at the LHC. Our results on non-LHC constraints are summarized in Figs.

Though in general the parameter space of the i2HDM is five-dimensional, the parameter space relevant to the LHC monojet signature is only one or two dimensional, so the model can be easily explored at the LHC. There are two qualitatively different and complementary channels in monojet searches:

Talking about quantitative results, the LHC has rather limited potential to probe

We have also explored the projected potential of XENON1T to probe the i2HDM parameter space and have found that it is quite impressive, confirming results of previous studies. In our study we have presented “absolutely allowed” and “absolutely excluded” points in different projections of the i2HDM 5D space demonstrating different features of the models and the potential of current and future experiments. In general, DM DD experiments and collider searches complement each other: the

A. B. thanks Tania Robens and Alexander Pukhov for useful discussions. A. B. and M. T. acknowledge partial support from the STFC Grant No. ST/L000296/1 and the NExT Institute, FAPESP Grant No. 2011/11973-4 for funding their visit to ICTP-SAIFR as well as SOTON-FAPESP collaboration grant. A. B. thanks the Royal Society Leverhulme Trust Senior Research Fellowship No. LT140094. M. T. acknowledges support from an STFC STEP award. G. C. acknowledges partial support from the Labex-LIO (Lyon Institute of Origins) under Grant No. ANR-10-LABX-66 and FRAMA (FR3127, Fédération de Recherche “André Marie Ampère”). I. P. I. acknowledges funding from

To have a complete picture of the properties of i2HDM in the whole parameter space, we have performed a five-dimensional random scan of the model parameter space with about

To better delineate the impact of each constraint, we have imposed different cuts on the parameter space sequentially, following the classification below:

Cut-1: theoretical constraints on the potential from vacuum stability [Eqs.

Cut-2: constraints from LEP [Eqs.

Cut-3: constraint on the relic density [

Cut-4: constraints from DM DD searches from LUX.

The results of the scan are presented in Fig.

Color maps of DM relic density for 2D projections of the 5D random scan of the i2HDM: each row demonstrates the effect of consequent application of the experimental and theoretical constraints in the

Color maps of DM relic density for 2D projections of the 5D random scan of the i2HDM for the parameter space restricted to (10 GeV–200 GeV) for

One can see from Figs.

One can also observe that the effect of Cut-1 plus Cut-2 is quite dramatic: (a)

The additional constraint from DM DD from LUX (Cut-4) removes a substantial portion of the parameter space for large and intermediate

We would also like to point to some features of the scan for the region of

Finally, for the case when the relic density is required to fit the PLANCK result within two sigmas, 2D projections on the

The 2D projections of the random scan of the i2HDM for points satisfying all constraints and providing the correct relic abundance within two sigmas of the PLANCK result. The top row corresponds to the “full” parameter space