Constraining Inert Dark Matter by Rγγ and WMAP data

Maria Krawczyk, Dorota Soko lowska, Pawe l Swaczyna, Bogumi la Swiezewska

Univeristy of Warsaw, Faculty of Physics

Hoza 69, 00-681 Warszawa, POLAND

April 1, 2021

Abstract

We discuss the constraints on the Dark Matter coming from the LHC Higgs data and WMAP relic densitymeasurements for the Inert Doublet Model (IDM), which is one of the simplest extensions of the StandardModel providing a Dark Matter (DM) candidate. We found that combining the diphoton rate Rγγ and theΩDMh2 data one can set strong limits on the parameter space of the IDM, stronger or comparable to theconstraints provided by XENON100 experiment for low and medium DM mass.

1 Introduction

Dark Matter (DM) is thought to constitute around 25% of the Universe’s mass-energy density, but its precisenature is yet unknown. The DM relic density ΩDMh

2 is well measured by WMAP and the current value ofΩDMh

2 is [1]:ΩDMh

2 = 0.1126± 0.0036. (1)

Various direct and indirect experiments have reported signals that can be interpreted as the observation ofthe DM particles. Low DM mass . 10 GeV is favoured by DAMA/LIBRA [2], CoGeNT [3, 4] and recently byCDMS-II [5] experiments, while the medium mass of 25 − 60 GeV by CRESST-II [6]. There have also beenreports of the observation of the γ-lines due to the DM annihilation pointing to the DM particles with massequal to 130 − 150 GeV [7, 8]. All those events lie in the regions excluded by the XENON10 and XENON100experiments, 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 experimentalinaccuracies coming from the incorrectly determined physical quantities in astrophysics or nuclear physics, or byinterpreting the results in modified astrophysical models of DM [see e.g. [10–14]]. However, so far no agreementhas been reached, and the situation in direct and indirect detection experiments is not yet clear [13,15–17].

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 Higgsportal 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 DMdetection, are processed by exchange of the Higgs particle. We found that the h→ γγ data for the SM-like Higgsparticle with mass Mh ≈ (125− 126) GeV can set strong constraints on the allowed masses and couplings of DMin the IDM. Combining them with the WMAP results leads to an exclusion of a large part of the IDM parameterspace, setting stronger limits on DM than XENON100.

2 Inert Doublet Model

The Inert Doublet Model is defined as a 2HDM with an exact D (Z2 type) symmetry: φS → φS , φD → −φD[18,19], i.e. a 2HDM with a D-symmetric potential, vacuum state and Yukawa interaction (Model I). In the IDMonly one doublet, φS , is involved in the Spontaneous Symmetry Breaking, while the D-odd doublet, φD, is inert,having vev = 0 and no couplings to fermions. The D-symmetric potential of the IDM has the following form:

V = − 12

[m2

11(φ†SφS)+m222(φ†DφD)

]+ λ1

2 (φ†SφS)2+ λ2

2 (φ†DφD)2

+λ3(φ†SφS)(φ†DφD)+λ4(φ†SφD)(φ†DφS) + λ5

2

[(φ†SφD)2+(φ†DφS)2

],

1

arX

iv:1

305.

6266

v1 [

hep-

ph]

27

May

201

3

with all parameters real (see eg. [20]). The vacuum state in IDM is given by:

〈φS〉 =1√2

(0v

), 〈φD〉 =

1√2

(00

), v = 246 GeV. (2)

The first doublet, φS , gives the SM-like Higgs boson h with mass Mh equal to

M2h = λ1v

2 = m211 = (125 GeV)

2. (3)

The second doublet, φD, contains four dark (inert) scalars H, A,H±, which do not interact with fermions. Dueto an exact Z2 symmetry the lightest neutral1 scalar H (or A) is stable and can play a role of the DM. Themasses of the dark particles read:

M2H± =

1

2

(λ3v

2 −m222

), M2

A = M2H± +

1

2(λ4 − λ5) v2 , M2

H = M2H± +

1

2(λ4 + λ5) v2 . (4)

We take H to be the DM candidate and so MH < MA,MH± (λ5 < 0, λ45 < 0).The properties of the IDM can be described by the parameters of the potential m2

ii, λi or by the masses ofthe scalar particles and their physical couplings. λ345 (5) is related to a triple and quartic coupling between theSM-like Higgs h and the DM candidate H, while λ3 describes the Higgs particle interaction with charged scalarsH±. λ2 gives the quartic self-couplings of dark particles. Physical parameters are limited by various theoreticaland experimental constraints (see e.g. [18, 21–34]). We take the following conditions into account:

Vacuum stability A stable vacuum exists iff:

λ1 > 0, λ2 > 0, λ3 +√λ1λ2 > 0, λ345 +

√λ1λ2 > 0 (λ345 = λ3 + λ4 + λ5). (5)

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]:

m211/√λ1 > m2

22/√λ2. (6)

Perturbative unitarity Parameters of the potential are constrained by the eigenvalues of the high-energyscattering matrix of the scalar sector: |Λi| < 8π [30–32], which leads to the limit for the DM quartic selfcoupling:

λmax2 = 8.38. (7)

The Higgs boson mass (3) and conditions (5,6,7) provide the following constraints [32]:

λ1 = 0.258, m222 . 9 · 104 GeV2, λ3, λ345 > −

√λ1λ2 = −1.47. (8)

EWPT Values of the S and T parameters should lie within 2σ ellipses of the (S, T ) plane with the followingcentral 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:

MH < 80 GeV,MA < 100 GeV and δA = MA −MH > 8 GeV. (9)

For δA < 8 GeV the LEP I limit applies [33,34]:

MH +MA > MZ . (10)

The standard limits for the charged scalar do not apply, as H± has no couplings to fermions. Its mass is indirectlyconstrained by the supersymmetry studies at LEP to be:

MH± & 70− 90 GeV. (11)

1Charged DM in IDM is excluded by the interplay between perturbativity and positivity constraints [20].

2

Relic density constraints In most regions of the parameter space the value of ΩDMh2 predicted by the IDM

is too low meaning that H does not constitute 100% of DM in the Universe. However, there are three regions ofMH in agreement with WMAP data (1): (i) light DM particles with mass . 10 GeV, (ii) medium DM mass of40− 150 GeV and (iii) heavy DM with mass & 500 GeV. Proper relic density (1) can be obtained by tuning theλ345 coupling, and in some cases also by the coannihilation between H and other dark scalars and interferenceprocesses 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 evolutionof the Universe and strong electroweak phase transition, see e.g. Ref. [39–41].

3 Rγγ constraints for the DM sector

A SM-like Higgs particle was discovered at the LHC in 2012. The measurement of Rγγ , the ratio of the diphotondecay rate of the observed h to the SM prediction, is sensitive to the “new physics”. The current measured valuesof Rγγ provided by the ATLAS and CMS collaborations are respectively [42,43]:

ATLAS : Rγγ = 1.65± 0.24(stat)+0.25−0.18(syst), (12)

CMS : Rγγ = 0.79+0.28−0.26. (13)

Both of them are in 2σ agreement with the SM value Rγγ = 1, however a deviation from that value is still possibleand would be an indication of physics beyond the Standard Model.

The ratio Rγγ in the IDM is given by:

Rγγ :=σ(pp→ h→ γγ)IDM

σ(pp→ h→ γγ)SM≈ Γ(h→ γγ)IDM

Γ(h→ γγ)SMΓ(h)SM

Γ(h)IDM, (14)

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 Rγγ = 1 are possible.First is a charged scalar contribution to the partial decay width Γ(h→ γγ)IDM [44–48]:

Γ(h→ γγ)IDM =GFα

2M3h

128√

2π3

∣∣∣∣ 4

3A1/2

(4M2

t

M2h

)+A1

(4M2

W

M2h

)︸ ︷︷ ︸

MSM

+λ3v

2

2M2H±

A0

(4M2

H±

M2h

)︸ ︷︷ ︸

δMIDM

∣∣∣∣2 , (15)

where MSM is the SM amplitude and δMIDM is the H± contribution2. The interference between MSM andδMIDM 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 Rγγ are the possible invisible decays h→ HH and h→ AA, whichcan augment strongly the total decay width ΓIDM(h) with respect to the SM case. Partial widths for these decaysare given by:

Γ(h→ HH) =λ2345v

2

32πMh

√1−

4M2H

M2h

, (16)

with MH exchanged to MA and λ345 to λ−345 (λ−345 = λ3 + λ4 − λ5), for the h → AA decay. Using formulasfrom Eq. (4) the couplings λ3 and λ−345 can be reexpressed in terms of MH , MA, MH± , λ345. So from Eq. (14)and (16) Rγγ depends only on the masses of the dark scalars and λ345.

For MH > Mh/2 (and MA > Mh/2) the invisible channels are closed, and Rγγ > 1 is possible, with themaximal value of Rγγ equal to 3.69 for MH = MH± = 70 GeV.

If MH < Mh/2 then the h → HH invisible channel is open and it is not possible to obtain Rγγ > 1,as shown in [48, 49]. If an enhancement (12) in the diphoton channel is confirmed, this DM mass region isalready excluded. However, if the final value of Rγγ is below 1, as suggested by the CMS data (13), then itlimits the parameters of the IDM on the basis of the following reasoning. For any given values of the darkscalars’ masses Rγγ is a function of one parameter: λ345, the behaviour of which is presented in Fig. 1 forMH = 55 GeV, MA = 60 GeV, MH± = 120 GeV (the same shape of the curve is preserved for different values ofmasses). It can be observed, that setting a lower bound on Rγγ leads to upper and lower bounds on λ345. Wewill explore those bounds, as functions of MH and δA in Sections 3.1 and 3.2 for three cases that are in 1σ regionof the CMS value: Rγγ > 0.7, 0.8, 0.9, respectively.

2The definition of the functions Ai can be found in Ref. [44, 45].

3

-0.10 -0.05 0.00 0.05 0.100.0

0.2

0.4

0.6

0.8

1.0

Λ345

RΓΓ

Λ345,m

in=-0.023

Λ345,m

ax=0.009

RΓΓHΛ345L

RΓΓ=0.7

Figure 1: Rγγ as a function of λ345 for the following dark scalars’ masses: MH = 55 GeV, MA = 60 GeV, MH± =120 GeV. The bounds on λ345 coming from the requirement that Rγγ > 0.7 are shown.

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

MH±=120 GeV

Λ345

0

0.01

0.02

0.03

0.04

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

MH±=120 GeV

Λ345

-0.04

-0.03

-0.02

-0.01

0

(a) Rγγ > 0.7

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2MH

±=120 GeVΛ345

0

0.01

0.02

0.03

0.04

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

MH±=120 GeV

Λ345

-0.04

-0.03

-0.02

-0.01

0

(b) Rγγ > 0.8

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

MH±=120 GeV

Λ345

0

0.01

0.02

0.03

0.04

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

MH±=120 GeV

Λ345

-0.04

-0.03

-0.02

-0.01

0

(c) Rγγ > 0.9

Figure 2: Upper (upper panel) and lower (lower panel) limits on λ345 coming from the requirement that (a)Rγγ > 0.7, (b) Rγγ > 0.8, (c) Rγγ > 0.9, expressed as functions of MH and δA for the case when the h→ HH,AAchannels are open. MH± is set to 120 GeV. The lower left corner is excluded by LEP.

3.1 HH,AA decay channels open

If both MH ,MA < Mh/2 then the LEP constraint (9) enforces δA < 8 GeV and so Eq. (10) limits the allowedvalues of the DM particle mass as follows: MA + MH ≈ 2MH & 80 GeV, so MH & 40 GeV. In this region, theinvisible decay channels have stronger influence on the Rγγ value than the contribution from the charged scalarloop [49], and so the exact value of MH± influences the results less than the other scalar masses. In the followingexamples we use MH± = 120 GeV, which is a good benchmark value of the charged scalar mass in the DManalysis for the low and medium DM mass regions, discussed later in section 4. Due to dependence of the partialwidth Γ(h→ AA) on |λ−345| the obtained lower and upper bounds are not symmetric with respect to λ345 = 0.

Diphoton rate constraints Fig. 2a show the upper and lower limits for the λ345 coupling if Rγγ > 0.7. Theallowed values of λ345 are small, typically between (−0.04, 0.04), depending on the difference between masses ofH and A. In general, for Rγγ > 0.8 the allowed values of λ345 are smaller than for Rγγ > 0.7. Also, region oflarger δA is excluded (Fig. 2b). In contrast to the previous cases condition Rγγ > 0.9 strongly limits the allowedparameter space of IDM, as shown in Fig. 2c, where a large portion of the parameter space is excluded. Theallowed A,H mass difference is δA . 4 GeV, and values of λ345 are smaller than in the previous cases. Requestinglarger Rγγ leads to the exclusion of the whole region of masses, apart from MH ≈MA ≈Mh/2.

Br(h→ inv)Br(h→ inv)Br(h→ inv) In principle, while discussing the MH < Mh/2 region, one should also include the constraintsfrom existing LHC data on the invisible channels branching ratio [50]. However, constraints on λ345 obtained by

4

requesting Br(h→ inv) < 40 %, as estimated in [51], are up to 50% weaker than those coming from Rγγ , compareFig. 2 and Fig. 3. The recent value of Br(h→ inv) < 65% [42] weakens the limits obtained for λ345 even more.

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

0

0.02

0.04

0.06

45 50 55 60

0

2

4

6

8

MH@GeVD

∆A@G

eVD

LEPexcl.

MA =M

h 2

-0.06

-0.04

-0.02

0

Figure 3: Upper (upper panel) and lower (lower panel) limits on λ345 coming from the requirement that Br(h→inv) < 40%, expressed as functions of MH and δA for the case when the h → HH,AA channels are open. Thelower left corner is excluded by LEP.

3.2 AA decay channel closed

When the AA decay channel is closed, very light DM particle can exist. Of course, if AA channel is closed theresults do not depend on the value of MA while the charged scalar contribution becomes more relevant. A cleardependence on the H± mass appears, especially for MH± . 120 GeV. Fig. 4 shows the limits for λ345 couplingthat allow the values of Rγγ higher than 0.7, 0.8 and 0.9 for MH± = 70, 120 and 500 GeV, respectively. Largervalue of Rγγ leads to smaller allowed values of λ345. In case of Rγγ > 0.9 large region of DM masses is excluded,as it is not possible to obtain the requested value of Rγγ for any value of λ345.

MH± @GeVD

70 120 500

10 20 30 40 50 60MH@GeVD

-0.04

-0.02

0.02

0.04

Λ345 MA Mh2

(a) Rγγ > 0.7

MH± @GeVD

70 120 500

10 20 30 40 50 60MH@GeVD

-0.04

-0.02

0.02

0.04

Λ345 MA Mh2

(b) Rγγ > 0.8

MH± @GeVD

70 120 500

10 20 30 40 50 60MH@GeVD

-0.04

-0.02

0.02

0.04

Λ345 MA Mh2

(c) Rγγ > 0.9

Figure 4: Upper and lower limits on λ345 coming from the requirement that (a) Rγγ > 0.7, (b) Rγγ > 0.8, (c)Rγγ > 0.9, expressed as functions of MH , for the case when the h→ AA channel is closed. Three values of MH±

are considered MH± = 70 GeV, MH± = 120 GeV, MH± = 500 GeV.

If Rγγ > 0.7 then an exact value of MH± is not crucial for the obtained limits for λ345, and allowed valuesof |λ345| are ∼ 0.02. For Rγγ > 0.8 the results are different for MH± = 70 GeV and 120 GeV. Smaller H± massleads to the stronger limits, for λ345 to be . 0.005, while larger masses of H± allow |λ345| . 0.015.

Condition Rγγ > 0.9 limits the IDM parameter space strongly. It is not possible to have Rγγ > 0.9 if MH . 45GeV. For the larger masses only relatively small values of λ345 ( below 0.02) are allowed. It is interesting to note,that in this case not smaller, but larger MH± leads to more stringent limits on λ345.

Br(h→ inv)Br(h→ inv)Br(h→ inv) Similarly to the MA < Mh/2 case, the constraint from the invisible decay branching ratio doesnot further limit the values of λ345, compare Fig. 4 and Fig. 5.

5

10 20 30 40 50 60MH@GeVD

-0.10

-0.05

0.05

0.10

Λ345

MA>Mh2

Figure 5: Upper and lower limits on λ345 coming from the requirement that Br(h→ inv) < 40%, expressed as afunction of MH , for the case when the h→ AA channel is closed.

DM-nucleon cross section In the IDM the DM-nucleon scattering cross-section σDM,N is given by:

σDM,N =λ2345

4πM4h

m4N

(mN +MH)2f2N , (17)

where we take Mh = 125 GeV, mN = 0.939 GeV and fN = 0.326 as the universal Higgs-nucleon coupling3.Value of the λ345 coupling is essential for the value of σDM,N in the IDM and so we translate the limits for λ345obtained from Rγγ measurements to (MH , σDM,N ) plane, used in direct detection experiments.

Exclusion bounds for cases Rγγ > 0.7, 0.8 are shown in Fig. 6, along with the XENON10/100 limits [9]. If Hshould constitute 100% of DM in the Universe, then the limits set by Rγγ measurements are much stronger thanthe ones provided by XENON10/100 experiments for MH . 20 GeV. Even for Rγγ > 0.7 it provides stronger orcomparable limits for σDM,N for MH . 60 GeV.

MH± @GeVD

70

120

500

0 10 20 30 40 50 6010-10

10-9

10-8

10-7

10-6

10-5

MH@GeVD

ΣDM,N@pbD

XENON10

XENON100

(a) Rγγ > 0.7

MH± @GeVD

70

120

500

0 10 20 30 40 50 6010-10

10-9

10-8

10-7

10-6

10-5

MH@GeVD

ΣDM,N@pbD

XENON10

XENON100

(b) Rγγ > 0.8

Figure 6: Upper limits on σDM,N coming from the requirement that (a) Rγγ > 0.7, (b) Rγγ > 0.8, expressedas functions of MH , for the case when the h → AA channel is closed. Three values of MH± are considered:MH± = 70 GeV, MH± = 120 GeV, MH± = 500 GeV and fN = 0.326. For comparison also the upper bounds setby XENON10 and XENON100 are shown.

3.3 Invisible decay channels closed

If MH > Mh/2, and consequently MA > Mh/2, the invisible channels are closed and the only modification toRγγ may come from the charged scalar loop (15), so the most important parameters are MH± ,MH and λ3 (orequivalently λ345). The contribution from the SM (MSM) is real and negative, while δMIDM is also real withsign correlated with the sign of λ3. Enhancement is possible when λ3 < 0 [48,49,53,54], with a maximal Rγγ forλ3 = −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 (δMIDM) is a decreasing function of MH± ,and so in general the larger Rγγ is, the smaller MH± should be. For example, Rγγ > 1.2 gives 70 GeV < MH± <154 GeV [49].

3There is no agreement to the value of the fN coupling and various estimations exit in the literature. Here we consider the themiddle value of 0.14 < fN < 0.66 [52], and comment on the other possible values later in the text. See also discussion in [50]

6

100 200 300 400 500 600 700

0

20

40

60

80

100

MH@GeVD

∆H±@G

eVD

RΓΓ=1.01

Λ345

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

100 200 300 400 500 600 700

0

20

40

60

80

100

MH@GeVD

∆H±@G

eVD

RΓΓ=1.02

Λ345

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

Figure 7: Allowed regions in (MH , δH±) plane for two values of Rγγ : 1.01 (left panel), 1.02 (right panel). Darkgray region is excluded due to LEP bounds (left lower corner) and the vacuum stability constraint (right uppercorner). Red lines show bound from XENON100 (solid for fN = 0.326, dashed for fN = 0.14 and fN = 0.66) –region above this line is excluded if we assume that the dark scalar H constitutes all dark matter relict density.

Fig. 7 shows the ranges of λ345 in the (MH± , δH±) plane for two values of Rγγ close to 1, Rγγ = 1.01 and 1.02.One can see that even the small deviation from Rγγ = 1 requires a relatively large λ345, if the mass differenceδH± is of the order (50− 100) GeV. Small values of |λ345| are preferred if mass difference is small.

Unitarity limits on λ3,345 constrain the allowed value of MH± (and thus also MH) for a given Rmaxγγ . For

δH± ≈ 0 and Rmaxγγ = 1.01 masses MH± & 700 GeV are excluded, and if Rmax

γγ > 1.02 this bound is stronger,forbidding MH± & 480 GeV (Fig. 7). Bound in Fig. 8 was obtained by assumption, that H and H± havethe degenerated masses and shows a relation between Rmax

γγ > 1 and MH± allowed by the model constraints.In general, as the previous studies have shown, the very heavy mass region is consistent with the very smalldeviations from Rγγ = 1, but one cannot reconcile the ATLAS results (12) with this region of masses.

Rγγ < 1 is possible if the invisible channels are closed and λ3 > 0. Looking for points allowed by the Rγγsmaller than the assumed value one can also limit the allowed parameter space. For example, if Rγγ < 0.8 withchannels closed then MH < 200 GeV [49].

Comparison with XENON100 results If the dark scalars H constitute 100% of dark matter in the Universe,then the σDM,N measurements done by the direct detection experiments bound the λ345 parameter, which is alsoconstrained by the Rγγ value (Fig. 7). For given scalar masses one can test the compability between the twolimits, and Fig. 7 shows that Rγγ > 1 and agreement with XENON-100 need almost degenerated masses of Hand H±.

One can get the relation between maximal Rγγ and MH± allowed by XENON experiments, see Fig. 8. ForMH ≈ MH± = 70 GeV, the Rγγ is bounded by (1.09, 1.04, 1.02) for fN = (0.14, 0.326, 0.66) respectively. IfδH± 0 then Rγγ requiers larger λ345, and that violates the XENON bounds (Fig. 7). It is also not possibleto have Rγγ > 1.09 with agreement with XENON, unless the dark scalar H constitutes only a part of the darkmatter relic density.

4 Combining Rγγ and relic density constraints on DM

In this section we compare the limits on the λ345 parameter obtained from Rγγ in the previous section withthose 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 ΩDMh

2 for chosen values of DM masses. We demandthe obtained value to lie in the 3σ WMAP limit:

0.1018 < ΩDMh2 < 0.1234 . (18)

If this condition is fulfilled, then H constitutes 100% of DM in the Universe. Values of ΩHh2 > 0.1234 are

excluded, while ΩHh2 < 0.1018 is still allowed if H is a subdominant DM candidate.

4.1 Low DM mass

In the IDM the low DM mass region corresponds to the masses of H below 10 GeV, while the other dark scalarsare heavier, MA ≈ MH+ ≈ 100 GeV. In this region the main annihilation channel is HH → h → bb and to

7

200 400 600 800 10000.95

1.00

1.05

1.10

1.15

1.20

1.25

MH±@GeVD

Maxim

alRΓΓ

Unitarity

XENON100

Figure 8: Maximal value of Rγγ , allowed by the condition λ345 > −1.47 (Eq. (8)) as a function of MH (blueline). Maximal value of Rγγ , allowed by the XENON100 constraints on λ345 (derived using the assumption thatH constitutes 100% of DM) as a function of MH (red lines). Solid red line corresponds to the bounds obtainedfor fN = 0.326 [55], while upper dashed for fN = 0.14 and lower dashed for fN = 0.66.

have the proper relic density, the HHh coupling (λ345) has to be large, above O(0.1). For example, for CDMS-IIfavoured mass M = 8.6 GeV [5] one gets relic density in agreement with bound (18) for |λ345| = (0.35 − 0.41),while |λ345| . 0.35 are excluded.

In the low mass region the invisible channel h → HH is open, meaning that Rγγ > 1 is not possible, so wecan conclude that the ATLAS measurement (12) excludes the low DM mass region in the IDM. If Rγγ < 1, assuggested by the CMS data (13), the low DM mass could be in principle allowed. However, our results, describedin the previous section, show, that it is not possible, as the allowed coupling is of an order of magnitude smallerthan needed, |λ345| ∼ 0.02. So we can conclude that the low DM mass region cannot be accommodated in IDMwith recent LHC results, regardless if H is the only, or just a subdominant, DM candidate.

4.2 Medium DM mass

Invisible decay channels open Let us first consider the case with AA invisible channel closed, where wechose MA = MH± = 120 GeV. In this case the main annihilation channels are HH → h → ff , when HHhcoupling is large enough and HH → W+W−, when HHh coupling is suppressed, typically leading to ΩDMh

2

above the WMAP limit. Lower values of MH require rather large λ345 – in this sense this region resembles thelow DM mass region. As MH grows towards MH = Mh/2, λ345 required to obtain the proper relic density getssmaller, leading eventually to the ΩDMh

2 below WMAP limit, apart from extremely tunned and small values ofλ345.

Those results are presented in Fig. 9, where WMAP allowed range of ΩDMh2 is denoted by the dark gray

bound. White excluded region between the WMAP bounds corresponds to the ΩDMh2 too large leading to the

overclosing of the Universe. If we consider H as a subdominant DM candidate with ΩHh2 < ΩDMh

2 then alsothe regions below and above the grey bounds in Fig. 9 is allowed. This usually corresponds to the larger valuesof λ345. Horizontal lines correspond to the maximal λ345 allowed by Rγγ analysis from section 3.2. It can beclearly seen that for a large portion of the parameter space limits for λ345 from Rγγ , even for the least stringentcase Rγγ > 0.7, cannot be reconciled with the WMAP-allowed region.

Invisible decay channels closed In this analysis we choose δH± = δA = 50 GeV and MH varying betweenMh/2 and 83 GeV. The main annihilation channels are as in the previous case, with the gauge channels gettingmore important as the mass of the DM particle grows. This, and the three body final states with virtual W±, arethe main reason why the WMAP-allowed region (the red bound) presented in Fig. 10 is not symmetric aroundzero, eventually leaving no positive values of λ345 allowed. The absolute values of λ345 that lead to the properrelic density are in general larger than in the case of MH < Mh/2.

Fig. 10 presents the values of Rγγ for chosen masses and couplings compared to the WMAP-allowed/excludedregion. It can be seen that this region is consistent with Rγγ < 1. It is in agreement with results obtained before(Fig. 7), as mass difference δH± = 50 GeV and Rγγ > 1 requires λ345 ≈ (0.3− 0.4), a value larger than the oneobtained from the relic density limits.

We can conclude, that Rγγ > 1 and relic density constraints (18) cannot be fulfilled for the middle DM mass

8

50 52 54 56 58 60

-0.10

-0.05

0.00

0.05

0.10

MH@GeVD

Λ345

WMAP excluded

RΓΓ

0.1

0.3

0.5

0.7

0.9

Figure 9: Comparison of the values of Rγγ and region allowed by the relic density measurements for the middleDM mass region with HH invisible channel open and MA = MH± = 120 GeV. Red bound: region in agreementwith WMAP (18). Grey area: excluded by WMAP. Rγγ > 0.7 limits the allowed values of masses to MH > 53GeV.

region. If IDM is the source of all DM in the Universe and MH ≈ (63− 83) GeV then the maximal value of Rγγis around 0.98. A subdominant DM candidate, which corresponds to larger λ345, is consistent with Rγγ > 1.

5 Summary

The IDM is a simple extension of the Standard Model that can provide a scalar DM candidate. This candidate isconsistent with the WMAP results on the DM relic density and in three regions of masses it can explain 100 % ofthe DM in the Universe. In a large part of the parameter space it can also be considered as a subdominant DMcandidate. Measurements of the diphoton ratio, Rγγ , recently done by the ATLAS and CMS experiments at theLHC can set strong limits on masses of the DM and other dark scalars, as well as the self-couplings, especiallyλ345. In this paper we discuss the obtained constraints for various possible values of Rγγ , that are in agreementwith 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 (MH < Mh/2) then Rγγ measurements can constrain themaximal value of |λ345|. This sets strong limits especially on the low DM mass region in IDM. Values of|λ345| that lead to the proper relic density in the 3σ WMAP range are of order of magnitude larger, thanthe ones allowed by assuming that Rγγ > 0.7. We conclude, we can exclude the low DM mass region inIDM, i.e. MH . 10 GeV.

• Rγγ provides also strong limits for the larger values of MH . First, demanding that Rγγ > 0.9 leaves onlya small part of the allowed parameter space, excluding MA−MH . 2 GeV if both invisible decay channelsare open or MH . 43 GeV if AA channel is closed. Second, comparing Rγγ limits with the WMAP allowedregion, we found that masses MH . 53 GeV, which require larger values of λ345 to be in agreement withWMAP, cannot be reconciled with Rγγ > 0.7.

• The Rγγ sets limits on the DM-nucleon scattering cross-section that are stronger, or comparable, with theXENON100 experiment in the low and medium DM mass region.

• If the invisible decay channels are closed, then Rγγ > 1 is possible. This however leads to the constraints onmasses and couplings. In general, Rγγ > 1 favours the degenerated H and H±. When the mass differenceis large, δH± ≈ (50 − 100) GeV, then the required values of |λ345| that provide Rγγ > 1 are bigger thanthe ones allowed by WMAP measurements. We conclude it is not possible to have all DM in the Universeexplained by the IDM and Rγγ > 1. If Rγγ > 1 then H may be a subdominant DM candidate. If Rγγ < 1then MH ≈ (63− 80) GeV can explain 100 % of DM in the Universe.

9

65 70 75 80

-0.2

-0.1

0.0

0.1

0.2

MH@GeVD

Λ345

∆A=∆H±=50 GeV

WMAP

excluded

RΓΓ

0.86

0.90

0.94

0.98

Figure 10: Comparison of the values of Rγγ and region allowed by the relic density measurements for the middleDM mass region with HH invisible channel closed and δA = δH± = 50 GeV. Red bound: region in agreementwith WMAP (18). Grey area: excluded by WMAP. Rγγ > 1 is not possible, unless H is a subdominant DMcandidate.

Acknowledgments This work was supported in part by a grant NCN OPUS 2012/05/B/ST2/03306 (2012-2016).

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