Constraints to Dark Matter from Inert Higgs Doublet Model

We study the Inert Higgs Doublet Model and its inert scalar Higgs $H$ as the only source for dark matter. It is found that three mass regions of the inert scalar Higgs can give the correct dark matter relic density. The low mass region (between 3 and 50 GeV) is ruled out. New direct dark matter detection experiments will probe the intermediate (between 60 and 100 GeV) and high (heavier than 550 GeV) mass regions. Collider experiments are advised to search for $D^\pm \to HW^\pm$ decay in the two jets plus missing energy channel.


I. INTRODUCTION
Astrophysical observations provide strong evidence for the existence of Dark Matter (DM) [1], and its abundance in the current phase of the Universe [2]. According to the newest results from Planck collaboration, there is a 74% approximately of matter which is not directly visible but is observed due to its gravitational effects on visible matter. Additional evidence for the existence of DM comes from study the rotation curves of spiral galaxies [3], the analysis of the bullet cluster [4], and the study of baryon acoustic oscillations [5].
One of the most common hypothesis used to explain these phenomena is to postulate the existence of weakly interacting massive particles (WIMPs) [6].
In order to explain DM, we study a simple extension of the Standard Model (SM), called the Inert Higgs Doublet Model (IHDM). This model, which was originally proposed for studies on electroweak (EW) symmetry breaking [10], introduces an additional doublet and a discrete symmetry. These two characteristics of the model modify the SM phenomenology, but there are some regions of the parameter space which predict only small deviations from the SM. Nevertheless, one of the most attractive characteristic of the IHDM is the presence of an stable neutral particle which can be a DM candidate.
In this work the IHDM is revisited, considering various restrictions, focusing our analysis on the zones of the parameter space which reproduce the correct DM relic density according to the newest measurements [7,8]. In this context, three not connected mass regimes for the lightest inert particle are found. These regimes are also analyzed using LHC observables like branching ratios to invisible particles and an specific SM-like arXiv:1511.04429v3 [hep-ph] 23 Jan 2016 Higgs boson decay mode. Additionally, we study some inert decays modes. Finally, we use a direct detection approach to rule out one of the mass regimes. It is further shown that this regime can also be ruled out by constraints from collider physics [9]. This paper is organized as follows. After a short introduction in section I the model is introduced in section II by formulating the associated potential and constraints of the model, and by exploring the parameter space and its characteristics in section III. In section IV the behavior of the model is presented from a collider physics perspective, studying the modifications of the SM and its implications for the IHDM. In section V the results of this study are complemented by an analysis from the Dark Matter perspective. Finally, in section VI we remark on the most important conclusions of our work.

II. THE INERT HIGGS DOUBLET MODEL
Consider an extension of Standard Model (SM), which contains two Higgs doublets Φ S,D and a discrete . All fields of the SM are invariants under the discrete symmetry, and Φ S is completely analogous to the SM Higgs doublet.
The most general renormalizable SU (2) × U (1) invariant Higgs potential, that also preserves the discrete symmetry is, where A, B, C are given by, The parameters µ 2 i (i = 1, 2) and λ j (j = 1 . . . 4) are intrinsically real, and λ 5 will be assumed to be real [11]. After Spontaneous Symmetry Breaking, the vacuum expectation values of the Higgs doublets are, where Φ D is forced by the discrete symmetry, and v = 246 GeV. Expanding the fields around those vacua, we define, where G 0 and G ± are the neutral and charged Goldstone bosons, and h is the SM-like Higgs boson. The fields in the second doublet belong to the so called "dark" or "inert" sector. They are the scalar H and pseudoscalar A, both neutral, and the charged scalar D ± . As in the SM, the parameter µ 2 1 is related to λ 1 by the tree-level tadpole condition µ 2 1 = −λ 1 v 2 . The masses of physical states [12,13] are, with As independent and free parameters we take the masses m H , m A , and m D at tree-level, and the couplings λ 2 and λ 345 . The SM-like Higgs boson mass is fixed now thanks to the measurement m h = 125 GeV [14,15].
The constraints we initially impose includes vacuum stability at tree-level, where constraints on the λ i couplings and µ i mass terms appear [16][17][18]; perturbativity (|λ i | < 8π) [19,20] and unitarity [21], where we impose that the scalar potential is unitary and that several scattering processes between scalar and gauge bosons are bounded; electroweak precision tests through the S, T , and U parameters [22] applied to the IHDM [21,23], with 3σ values given by with a correlation coefficient of +0.91 [24] and collider constraints [16,[25][26][27], where we satisfy lower bonds on the Higgs boson masses.
The DM particle must be neutral. In our analysis we assume it is the H boson, thus m H < m A and m H < m D , which due to (5) translates to We do not consider A as the DM candidate because it is analogous to consider H as the DM candidate We randomly scan the parameter space of the IHDM, taking into account all the constraints mentioned in the previous section. Additionally, we compute some astrophysical properties of the model using the micrOMEGAs software [28]. We consider masses satisfying 1 GeV < m i < 1 TeV, where i = H, A, D.
In addition, we consider cosmological measurements: the DM relic density Ω DM h 2 is a property related to its abundance in the current phase of the Universe. This quantity is well measured by WMAP [29] and Planck [30] experiments. Following [31] to combine both measurements we obtain, In Figure 1 the coupling λ 345 , is shown as a function of the Higgs boson mass m H varying (µ i , λ i , m A , m D and m H ). We work with he hypothesis that the light inert Higgs boson H is providing the complete DM density Ω DM h 2 given in (9). The color code is as follows: red points (dark gray) produce a relic density above the 3σ limit given in eq. (9); blue points (black) produce a relic density within the 3σ region; green points (light gray) produce a relic density below the 3σ limit. Regarding the points that satisfy the relic density we see three clear regions [12,32], one for low m H (3 < m H < 50 GeV), another one for medium m H (60 < m H < 100 GeV) and finally one for high values of m H (m H > 550 GeV). The explanation for the gap is related to annihilation processes and it will be given later. At m H < 3 GeV the IHDM can no longer be compatible with vacuum existence and stability [33,34]. A, and D are close to each other, and that this feature is more pronounced when the masses of these particles are near the TeV scale (due to the logarithmic scale). Finally we notice that Ω DM h 2 is more sensible to the parameters λ 345 and m H (Fig. 1) than the masses of the other inert particles (Fig. 2).

IV. COLLIDER PHYSICS
As we mentioned before, the Higgs boson discovered at CERN in 2012 is the SM-like Higgs boson h of our model from the non-inert Higgs doublet field Φ S . This particle h couples to the charged Higgs pair (D ± ), which contributes to the diphoton decay width 1 Γ(h → γγ) [35]. For the same reason D ± also contribute to It is convenient to work with the parameter [21,34,38] The value we use for the SM is Γ(h SM → γγ) = 4.1 MeV [39]. ATLAS [40] and CMS [41] collaborations have studied this decay mode, and if we combined both results [31] obtain R exp γγ = 1.14 ± 0.18. the decay h → γγ such that R γγ is close to unity. Interestingly, the right panel of Fig. 3 shows that perturbative values (|λ 345 | < 1) are preferred.
In addition, if the inert particles are light enough, there are two other two-body decays which are: There is no phase space for a two-body decay h → D ± D ∓ . It is possible to define the parameter R Zγ , in analogy to R γγ defined in eq. (10). It is interesting that even though the decay h → Zγ is not well measured, it still can give additional insight to the model [43,44]. As Fig. 4 shows, there appears a very narrow correlation between R γγ and R Zγ , which is a common feature for R γγ v/s R Zγ plots [17,18,45].  [47] and CMS [48], gives B(h → inv) < 0.43 for the SM-like Higgs boson. In Fig. 5 we show the branching To finalize this section we discuss the branching ratios for the two observable inert Higgs bosons A and D ± . In Fig. 6 we have the branching ratios for the pseudoscalar Higgs boson (left panel) as a function of its mass, and the branching ratios for the charged Higgs boson (right panel) as a function of its mass. In branching ratio for the decay D ± → HW ± and in dash we have D ± → AW ± . In the case of D ± , there is no crossing point, thus B(D ± → HW ± ) is always larger than B(D ± → AW ± ). We remind the reader that the presence of a Higgs boson H in a final state is seen as missing momentum at the LHC.

V. COSMOLOGY AND DARK MATTER
The existence of Dark Matter seems to be well established now [49]. There are several candidates for DM, among them the previously mentioned WIMPs. A good particle candidate for DM must be neutral and stable (or quasi-stable). The Z 2 discrete symmetry in the model studied in this article ensures that the lightest of the inert Higgs bosons is stable. Observation implies it is either H or A (or in a fine tuned scenario both).
In this article we study the former case. An important restriction this candidate must satisfy is that its mass density must agree with experimental observations. We calculate the relic density of our DM candidate using micrOMEGAs software [28]. To better understand the results on the relic density, we calculate also the thermal averaged annihilation cross section times the relative velocity σv , or annihilation cross section for short. In Fig. 7 we plot the H relic density as a function of the DM candidate mass m H (left panel), and the annihilation cross section also as a function of the DM candidate mass m H (right panel), calculated with [28].
For the relic density case, we also show, as an horizontal dashed line, the experimentally measured value for Ω DM h 2 as given in eq. (9). The scan shows a large distribution with differences that can reach more than where HH → W W (ZZ) become available, and continue later with the threshold m H ≈ m h where the channel HH → hh opens up. The annihilation channel HH → tt also helps. All these new annihilation channels make the DM annihilation very efficient, and it is not possible to obtain a relic density according to observations. On the other hand, for larger m H , it is possible to get a correct relic density if the difference between the three inert scalar masses is not so large and λ 345 remains small enough [50] (see Fig. 1).
We finally study the direct detection prospects of our DM candidate. We do that through the tree-level spin-independent DM-nucleon interaction cross section [51], which applied to our case is, Here m h is the mass of the SM-like Higgs boson, m H is the mass of the DM candidate, m N is the nucleon mass, taken here to be m N = 0.939 GeV as the average of the proton and neutron masses, λ 345 is the combined coupling defined in eq. (6), and f N is a form factor that depends on hadronic matrix elements [52,53]. In Fig. 8 [56][57][58][59]. Notice that the dispersion of points for high m H can be understood from analogous dispersion seen in Fig. 1, and the same situation occurs with the line-like distribution for light m H . We also show the coherent neutrino scattering upper limit [60].
This curve represents the threshold below which the detector sensitivity is such that not only the possible DM scattering effects can be observed, but also the indistinguishable scattering effects associated with neutrinos.
Thus, this indicates a region where the neutrino background becomes dominant and and little information can be obtained on DM effects. Current direct detection of DM exclude all the low DM mass points, and most of the medium DM mass points. Allowed are a narrow region near 60 GeV, and all the high mass region. Note, that the absence of points in the range of ∼ 100 − 550 GeV in this plot is due to the fact that the plotted points are only those that give the right dark matter density ("blue points"). Future experiments will be able to test large parts of these two regions, but will not be able to rule them out entirely if there is no signal.

VI. CONCLUSIONS
In this article the Inert Higgs Doublet Model is studied, with the inert Higgs boson H as a DM candidate, using the latest results for DM relic density, annihilation cross section, and collider searches. As a summary we highlight, • The branching ratios for the charged Higgs D ± and for the pseudoscalar Higgs A are studied and shown in Fig. 6. The Z 2 symmetry strongly reduces the number of different decay channels. Considering the Higgs boson on-shell and allowing the gauge boson to be off-shell (a different choice would produce different decay channels, but with smaller branching ratios), we find that B(D ± → HW ± ) > B(D ± → AW ± ) as opposed to the A decays, where there is a crossing point not far from the threshold. For this reason, in collider searches we recommend to look for a signal for a D ± : two jets (consistent with a W ) and missing energy (from the DM candidate H).
• Three distinct H mass regions are found that produced the correct relic density (i.e. H is the only source for DM) as it can be seen in Fig. 7. The low mass region (between 3 and 50 GeV approximately) is already ruled out because it produces a too small value for R γγ (Fig. 3), because it produces a too high value for B(h → HH) (Fig. 5), and because of direct DM searches (Fig. 8). The intermediate mass region (between 60 and 100 GeV approximately) and the high mass region (heavier than 550 GeV approximately) are allowed.
• In Fig. 8 we study the DM candidate direct detection. The low mass region is also ruled out by present experiments. In addition, future experiments will probe intermediate and high mass regions. Nevertheless, in absence of signals, it will not be possible to rule out these two regions. Notice the proximity of the region where the coherent neutrino scattering is an irreducible background.
With this one sees that the Inert Higgs Doublet Model gives a still viable DM candidate, which will most likely be tested by direct DM detection experiments.