Constraining Inert Dark Matter by R_{\gamma\gamma} and WMAP data

We discuss the constraints on the Dark Matter coming from the LHC Higgs data and WMAP relic density measurements for the Inert Doublet Model (IDM), which is one of the simplest extensions of the Standard Model providing a Dark Matter (DM) candidate. We found that combining the diphoton rate R_{\gamma\gamma} and the \Omega_{DM}h^2 data one can set strong limits on the parameter space of the IDM, stronger or comparable to the constraints provided by XENON100 experiment for low and medium DM mass.


Introduction
Dark Matter (DM) is thought to constitute around 25% of the Universe's mass-energy density, but its precise nature is yet unknown. The DM relic density Ω DM h 2 is well measured by WMAP and the current value of Ω DM h 2 is [1]: Various direct and indirect experiments have reported signals that can be interpreted as the observation of the DM particles. Low DM mass 10 GeV is favoured by DAMA/LIBRA [2], CoGeNT [3,4] and recently by CDMS-II [5] experiments, while the medium mass of 25 − 60 GeV by CRESST-II [6]. There have also been reports of the observation of the γ-lines due to the DM annihilation pointing to the DM particles with mass equal to 130 − 150 GeV [7,8]. All those events lie in the regions excluded by the XENON10 and XENON100 experiments, which set the strongest limits on the DM-nucleon scattering cross-section [9].
There have been many attempts to explain those contradictory results either by assuming some experimental inaccuracies coming from the incorrectly determined physical quantities in astrophysics or nuclear physics, or by interpreting the results in modified astrophysical models of DM [see e.g. [10][11][12][13][14]]. However, so far no agreement has been reached, and the situation in direct and indirect detection experiments is not yet clear [13,[15][16][17].
In this paper we attempt to set constraints on the scalar DM particle from the Inert Doublet Model (IDM), using solely the LHC Higgs data and relic density measurements. The IDM provides an example of a Higgs portal DM. In a vast region of the allowed DM masses, particularly in the range that the LHC can directly test, the main annihilation channel of DM particles and their interaction with nucleons, relevant for the direct DM detection, are processed by exchange of the Higgs particle. We found that the h → γγ data for the SM-like Higgs particle with mass M h ≈ (125 − 126) GeV can set strong constraints on the allowed masses and couplings of DM in the IDM. Combining them with the WMAP results leads to an exclusion of a large part of the IDM parameter space, setting stronger limits on DM than XENON100.

Inert Doublet Model
The Inert Doublet Model is defined as a 2HDM with an exact D (Z 2 type) symmetry: φ S → φ S , φ D → −φ D [18,19], i.e. a 2HDM with a D-symmetric potential, vacuum state and Yukawa interaction (Model I). In the IDM only one doublet, φ S , is involved in the Spontaneous Symmetry Breaking, while the D-odd doublet, φ D , is inert, having vev = 0 and no couplings to fermions. The D-symmetric potential of the IDM has the following form: with all parameters real (see eg. [20]). The vacuum state in IDM is given by: The first doublet, φ S , gives the SM-like Higgs boson h with mass M h equal to The second doublet, φ D , contains four dark (inert) scalars H, A, H ± , which do not interact with fermions. Due to an exact Z 2 symmetry the lightest neutral 1 scalar H (or A) is stable and can play a role of the DM. The masses of the dark particles read: We take H to be the DM candidate and so M H < M A , M H ± (λ 5 < 0, λ 45 < 0). The properties of the IDM can be described by the parameters of the potential m 2 ii , λ i or by the masses of the scalar particles and their physical couplings. λ 345 (5) is related to a triple and quartic coupling between the SM-like Higgs h and the DM candidate H, while λ 3 describes the Higgs particle interaction with charged scalars H ± . λ 2 gives the quartic self-couplings of dark particles. Physical parameters are limited by various theoretical and experimental constraints (see e.g. [18,[21][22][23][24][25][26][27][28][29][30][31][32][33][34]). We take the following conditions into account: Vacuum stability A stable vacuum exists iff: Existence of inert vacuum In the IDM two minima of the different symmetry properties can coexist [20,35,36]. For the state (2) to be not just a local, but the global minimum, the following condition has to be fulfilled [20]: Perturbative unitarity Parameters of the potential are constrained by the eigenvalues of the high-energy scattering matrix of the scalar sector: |Λ i | < 8π [30][31][32], which leads to the limit for the DM quartic self coupling: The Higgs boson mass (3) and conditions (5,6,7) provide the following constraints [32]: EWPT Values of the S and T parameters should lie within 2σ ellipses of the (S, T ) plane with the following central values [37]: S = 0.03 ± 0.09, T = 0.07 ± 0.08, with correlation equal to 87%.

LEP limits
The LEP II analysis excludes the region of masses in IDM where simultaneously: For δ A < 8 GeV the LEP I limit applies [33,34]: The standard limits for the charged scalar do not apply, as H ± has no couplings to fermions. Its mass is indirectly constrained by the supersymmetry studies at LEP to be: Relic density constraints In most regions of the parameter space the value of Ω DM h 2 predicted by the IDM is too low meaning that H does not constitute 100% of DM in the Universe. However, there are three regions of M H in agreement with WMAP data (1): (i) light DM particles with mass 10 GeV, (ii) medium DM mass of 40 − 150 GeV and (iii) heavy DM with mass 500 GeV. Proper relic density (1) can be obtained by tuning the λ 345 coupling, and in some cases also by the coannihilation between H and other dark scalars and interference processes with virtual EW gauge bosons [18, 19, 22-28, 35, 36, 38]. The IDM provides, apart from the DM candidate, also a good framework for the studies of thermal evolution of the Universe and strong electroweak phase transition, see e.g. Ref. [39][40][41].
3 R γγ constraints for the DM sector A SM-like Higgs particle was discovered at the LHC in 2012. The measurement of R γγ , the ratio of the diphoton decay rate of the observed h to the SM prediction, is sensitive to the "new physics". The current measured values of R γγ provided by the ATLAS and CMS collaborations are respectively [42,43]: Both of them are in 2σ agreement with the SM value R γγ = 1, however a deviation from that value is still possible and would be an indication of physics beyond the Standard Model. The ratio R γγ in the IDM is given by: where the fact that the main production channel is gluon fusion and that the Higgs particle from IDM is SM-like, so σ(gg → h) IDM = σ(gg → h) SM , was used. In the IDM two sources of deviation from R γγ = 1 are possible. First is a charged scalar contribution to the partial decay width Γ(h → γγ) IDM [44][45][46][47][48]: where M SM is the SM amplitude and δM IDM is the H ± contribution 2 . The interference between M SM and δM IDM can be either constructive or destructive, leading to an increase or a decrease of the decay rate (15). The second source of the modifications to R γγ are the possible invisible decays h → HH and h → AA, which can augment strongly the total decay width Γ IDM (h) with respect to the SM case. Partial widths for these decays are given by: with M H exchanged to M A and λ 345 to λ − 345 (λ − 345 = λ 3 + λ 4 − λ 5 ), for the h → AA decay. Using formulas from Eq. (4)  If M H < M h /2 then the h → HH invisible channel is open and it is not possible to obtain R γγ > 1, as shown in [48,49]. If an enhancement (12) in the diphoton channel is confirmed, this DM mass region is already excluded. However, if the final value of R γγ is below 1, as suggested by the CMS data (13), then it limits the parameters of the IDM on the basis of the following reasoning. For any given values of the dark scalars' masses R γγ is a function of one parameter: λ 345 , the behaviour of which is presented in Fig. 1 for M H = 55 GeV, M A = 60 GeV, M H ± = 120 GeV (the same shape of the curve is preserved for different values of masses). It can be observed, that setting a lower bound on R γγ leads to upper and lower bounds on λ 345 . We will explore those bounds, as functions of M H and δ A in Sections 3.1 and 3.2 for three cases that are in 1σ region of the CMS value: R γγ > 0.7, 0.8, 0.9, respectively.

HH, AA decay channels open
If both M H , M A < M h /2 then the LEP constraint (9) enforces δ A < 8 GeV and so Eq. (10) limits the allowed values of the DM particle mass as follows: In this region, the invisible decay channels have stronger influence on the R γγ value than the contribution from the charged scalar loop [49], and so the exact value of M H ± influences the results less than the other scalar masses. In the following examples we use M H ± = 120 GeV, which is a good benchmark value of the charged scalar mass in the DM analysis for the low and medium DM mass regions, discussed later in section 4. Due to dependence of the partial width Γ(h → AA) on |λ − 345 | the obtained lower and upper bounds are not symmetric with respect to λ 345 = 0.
Diphoton rate constraints Fig. 2a show the upper and lower limits for the λ 345 coupling if R γγ > 0.7. The allowed values of λ 345 are small, typically between (−0.04, 0.04), depending on the difference between masses of H and A. In general, for R γγ > 0.8 the allowed values of λ 345 are smaller than for R γγ > 0.7. Also, region of larger δ A is excluded (Fig. 2b). In contrast to the previous cases condition R γγ > 0.9 strongly limits the allowed parameter space of IDM, as shown in Fig. 2c, where a large portion of the parameter space is excluded. The allowed A, H mass difference is δ A 4 GeV, and values of λ 345 are smaller than in the previous cases. Requesting larger R γγ leads to the exclusion of the whole region of masses, apart from M H ≈ M A ≈ M h /2.
In principle, while discussing the M H < M h /2 region, one should also include the constraints from existing LHC data on the invisible channels branching ratio [50]. However, constraints on λ 345 obtained by requesting Br(h → inv) < 40 %, as estimated in [51], are up to 50% weaker than those coming from R γγ , compare Fig. 2 and Fig. 3. The recent value of Br(h → inv) < 65% [42] weakens the limits obtained for λ 345 even more.

AA decay channel closed
When the AA decay channel is closed, very light DM particle can exist. Of course, if AA channel is closed the results do not depend on the value of M A while the charged scalar contribution becomes more relevant. A clear dependence on the H ± mass appears, especially for M H ± 120 GeV. Fig. 4 shows the limits for λ 345 coupling that allow the values of R γγ higher than 0.7, 0.8 and 0.9 for M H ± = 70, 120 and 500 GeV, respectively. Larger value of R γγ leads to smaller allowed values of λ 345 . In case of R γγ > 0.9 large region of DM masses is excluded, as it is not possible to obtain the requested value of R γγ for any value of λ 345 .

DM-nucleon cross section
In the IDM the DM-nucleon scattering cross-section σ DM,N is given by: where we take M h = 125 GeV, m N = 0.939 GeV and f N = 0.326 as the universal Higgs-nucleon coupling 3 . Value of the λ 345 coupling is essential for the value of σ DM,N in the IDM and so we translate the limits for λ 345 obtained from R γγ measurements to (M H , σ DM,N ) plane, used in direct detection experiments. Exclusion bounds for cases R γγ > 0.7, 0.8 are shown in Fig. 6, along with the XENON10/100 limits [9]. If H should constitute 100% of DM in the Universe, then the limits set by R γγ measurements are much stronger than the ones provided by XENON10/100 experiments for M H 20 GeV. Even for R γγ > 0.7 it provides stronger or comparable limits for σ DM,N for M H 60 GeV.

Invisible decay channels closed
If M H > M h /2, and consequently M A > M h /2, the invisible channels are closed and the only modification to R γγ may come from the charged scalar loop (15), so the most important parameters are M H ± , M H and λ 3 (or equivalently λ 345 ). The contribution from the SM (M SM ) is real and negative, while δM IDM is also real with sign correlated with the sign of λ 3 . Enhancement is possible when λ 3 < 0 [48,49,53,54], with a maximal R γγ for λ 3 = −1.47, i.e. the smallest value of this parameter allowed by model constraints (8).
The contribution to the amplitude from the charged scalar loop (δM IDM ) is a decreasing function of M H ± , and so in general the larger R γγ is, the smaller M H ± should be. For example, R γγ > 1.2 gives 70 GeV < M H ± < 154 GeV [49].   (Fig. 7). Bound in Fig. 8 was obtained by assumption, that H and H ± have the degenerated masses and shows a relation between R max γγ > 1 and M H ± allowed by the model constraints. In general, as the previous studies have shown, the very heavy mass region is consistent with the very small deviations from R γγ = 1, but one cannot reconcile the ATLAS results (12) with this region of masses. R γγ < 1 is possible if the invisible channels are closed and λ 3 > 0. Looking for points allowed by the R γγ smaller than the assumed value one can also limit the allowed parameter space. For example, if R γγ < 0.8 with channels closed then M H < 200 GeV [49].
Comparison with XENON100 results If the dark scalars H constitute 100% of dark matter in the Universe, then the σ DM,N measurements done by the direct detection experiments bound the λ 345 parameter, which is also constrained by the R γγ value (Fig. 7). For given scalar masses one can test the compability between the two limits, and Fig. 7 shows that R γγ > 1 and agreement with XENON-100 need almost degenerated masses of H and H ± .
One can get the relation between maximal R γγ and M H ± allowed by XENON experiments, see Fig. 8. For M H ≈ M H ± = 70 GeV, the R γγ is bounded by (1.09, 1.04, 1.02) for f N = (0.14, 0.326, 0.66) respectively. If δ H ± 0 then R γγ requiers larger λ 345 , and that violates the XENON bounds (Fig. 7). It is also not possible to have R γγ > 1.09 with agreement with XENON, unless the dark scalar H constitutes only a part of the dark matter relic density.

Combining R γγ and relic density constraints on DM
In this section we compare the limits on the λ 345 parameter obtained from R γγ in the previous section with those coming from the requirement that the DM relic density is in agreement with the WMAP measurements (1). We use the micrOMEGAs package [56] to calculate Ω DM h 2 for chosen values of DM masses. We demand the obtained value to lie in the 3σ WMAP limit:

Low DM mass
In the IDM the low DM mass region corresponds to the masses of H below 10 GeV, while the other dark scalars are heavier, M A ≈ M H + ≈ 100 GeV. In this region the main annihilation channel is HH → h →bb and to In the low mass region the invisible channel h → HH is open, meaning that R γγ > 1 is not possible, so we can conclude that the ATLAS measurement (12) excludes the low DM mass region in the IDM. If R γγ < 1, as suggested by the CMS data (13), the low DM mass could be in principle allowed. However, our results, described in the previous section, show, that it is not possible, as the allowed coupling is of an order of magnitude smaller than needed, |λ 345 | ∼ 0.02. So we can conclude that the low DM mass region cannot be accommodated in IDM with recent LHC results, regardless if H is the only, or just a subdominant, DM candidate.

Medium DM mass
Invisible decay channels open Let us first consider the case with AA invisible channel closed, where we chose M A = M H ± = 120 GeV. In this case the main annihilation channels are HH → h →f f , when HHh coupling is large enough and HH → W + W − , when HHh coupling is suppressed, typically leading to Ω DM h 2 above the WMAP limit. Lower values of M H require rather large λ 345 -in this sense this region resembles the low DM mass region. As M H grows towards M H = M h /2, λ 345 required to obtain the proper relic density gets smaller, leading eventually to the Ω DM h 2 below WMAP limit, apart from extremely tunned and small values of λ 345 .
Those results are presented in Fig. 9, where WMAP allowed range of Ω DM h 2 is denoted by the dark gray bound. White excluded region between the WMAP bounds corresponds to the Ω DM h 2 too large leading to the overclosing of the Universe. If we consider H as a subdominant DM candidate with Ω H h 2 < Ω DM h 2 then also the regions below and above the grey bounds in Fig. 9 is allowed. This usually corresponds to the larger values of λ 345 . Horizontal lines correspond to the maximal λ 345 allowed by R γγ analysis from section 3.2. It can be clearly seen that for a large portion of the parameter space limits for λ 345 from R γγ , even for the least stringent case R γγ > 0.7, cannot be reconciled with the WMAP-allowed region.
Invisible decay channels closed In this analysis we choose δ H ± = δ A = 50 GeV and M H varying between M h /2 and 83 GeV. The main annihilation channels are as in the previous case, with the gauge channels getting more important as the mass of the DM particle grows. This, and the three body final states with virtual W ± , are the main reason why the WMAP-allowed region (the red bound) presented in Fig. 10 is not symmetric around zero, eventually leaving no positive values of λ 345 allowed. The absolute values of λ 345 that lead to the proper relic density are in general larger than in the case of M H < M h /2. Fig. 10 presents the values of R γγ for chosen masses and couplings compared to the WMAP-allowed/excluded region. It can be seen that this region is consistent with R γγ < 1. It is in agreement with results obtained before (Fig. 7), as mass difference δ H ± = 50 GeV and R γγ > 1 requires λ 345 ≈ (0.3 − 0.4), a value larger than the one obtained from the relic density limits.
We can conclude, that R γγ > 1 and relic density constraints (18) cannot be fulfilled for the middle DM mass region. If IDM is the source of all DM in the Universe and M H ≈ (63 − 83) GeV then the maximal value of R γγ is around 0.98. A subdominant DM candidate, which corresponds to larger λ 345 , is consistent with R γγ > 1.

Summary
The IDM is a simple extension of the Standard Model that can provide a scalar DM candidate. This candidate is consistent with the WMAP results on the DM relic density and in three regions of masses it can explain 100 % of the DM in the Universe. In a large part of the parameter space it can also be considered as a subdominant DM candidate. Measurements of the diphoton ratio, R γγ , recently done by the ATLAS and CMS experiments at the LHC can set strong limits on masses of the DM and other dark scalars, as well as the self-couplings, especially λ 345 . In this paper we discuss the obtained constraints for various possible values of R γγ , that are in agreement with the recent LHC measurements, and combine them with WMAP constraints.
The main results of the present paper are as follows: • If invisible Higgs decays channels are open (M H < M h /2) then R γγ measurements can constrain the maximal value of |λ 345 |. This sets strong limits especially on the low DM mass region in IDM. Values of |λ 345 | that lead to the proper relic density in the 3σ WMAP range are of order of magnitude larger, than the ones allowed by assuming that R γγ > 0.7. We conclude, we can exclude the low DM mass region in IDM, i.e. M H 10 GeV.
• R γγ provides also strong limits for the larger values of M H . First, demanding that R γγ > 0.9 leaves only a small part of the allowed parameter space, excluding M A − M H 2 GeV if both invisible decay channels are open or M H 43 GeV if AA channel is closed. Second, comparing R γγ limits with the WMAP allowed region, we found that masses M H 53 GeV, which require larger values of λ 345 to be in agreement with WMAP, cannot be reconciled with R γγ > 0.7.
• The R γγ sets limits on the DM-nucleon scattering cross-section that are stronger, or comparable, with the XENON100 experiment in the low and medium DM mass region.
• If the invisible decay channels are closed, then R γγ > 1 is possible. This however leads to the constraints on masses and couplings. In general, R γγ > 1 favours the degenerated H and H ± . When the mass difference is large, δ H ± ≈ (50 − 100) GeV, then the required values of |λ 345 | that provide R γγ > 1 are bigger than the ones allowed by WMAP measurements. We conclude it is not possible to have all DM in the Universe explained by the IDM and R γγ > 1. If R γγ > 1 then H may be a subdominant DM candidate. If R γγ < 1 then M H ≈ (63 − 80) GeV can explain 100 % of DM in the Universe.