Inert Doublet Dark Matter with an additional scalar singlet and 125 GeV Higgs Boson
IInert Doublet Dark Matter with an additional scalar singlet and 125GeV Higgs Boson
Amit Dutta Banik , Debasish Majumdar Astroparticle Physics and Cosmology Division,Saha Institute of Nuclear Physics,1/AF Bidhannagar, Kolkata 700064, India
Abstract
In this work we consider a model for particle dark matter where an extra inert Higgs doubletand an additional scalar singlet is added to the Standard Model (SM) Lagrangian. The dark mattercandidate is obtained from only the inert doublet. The stability of this one component dark matteris ensured by imposing a Z symmetry on this additional inert doublet. The additional singlet scalarhas a vacuum expectation value (VEV) and mixes with the Standard Model Higgs doublet resultingin two CP even scalars h and h . We treat one of these scalars, h , to be consistent with the SMHiggs like boson of mass around 125 GeV reported by the LHC experiment. These two CP evenscalars affect the annihilation cross-section of this inert doublet dark matter resulting in a largerdark matter mass region that satisfies the observed relic density. We also investigate the h → γγ and h → γZ processes and compared these with LHC results. This is also used to constrain thedark matter parameter space in the present model. We find that the dark matter candidate in themass region m < m H < m W GeV ( m = 125 GeV, mass of h ) satisfies the recent bound from LUXdirect detection experiment. email: [email protected] email: [email protected] a r X i v : . [ h e p - ph ] A p r Introduction
Existence of a newly found Higgs-like scalar boson of mass about 125 GeV has been reported by recentLHC results. ATLAS [1] and CMS [2] independently confirmed the discovery of a new scalars andmeasured signal strengths of the Higgs-like scalar to various decay channels separately. ATLAS hasreported a Higgs to di-photon signal strength ( R γγ ) about 1 . +0 . − . [3]. On the other hand Higgs todi-photon signal strength evaluated by CMS experiment is found to be about 0 . +0 . − . [4]. Despite thesuccess of Standard Model (SM) of particle physics, it fails to produce a plausible explanation of darkmatter (DM) in modern cosmology. Existence of dark matter is firmly established by the observationsof galaxy rotation curves and analysis of cosmic microwave background (CMB) etc. DM relic densitypredicted by the PLANCK [5] and WMAP [6] results suggest that about 26 .
5% of our Universe isconstituted by DM. The particle constituent of dark matter is still unknown and SM of particle physicsappears inadequate to address the issues regarding dark matter. The observed dark matter relic densityreported by CMB anisotropy probes suggests that weakly interacting massive particle or WIMP [7, 8].or WIMP can be assumed to serve as a feasible candidate for dark matter. Thus, in order to proposea feasible candidate for dark matter one sould invoke a theory beyond SM and in this regard simpleextension of SM scalar or fermion sector or both could be of interest in respect of addressing the problemof a viable candidate of dark matter and dark matter physics. There are other theories though beyondStandard Model (BSM) such as the elegant theory of Supersymmetry (SUSY) in which the dark mattercandidate is supposedly the LSP or lightest SUSY particle formed by the superposition of neutralgauge bosons and Higgs boson [9]. Extra dimension models [10] providing Kaluza-Klein dark mattercandidates are also explored at length in literature. Comprehensive studies on simplest extension of SMwith additional scalar singlet where a discrete Z symmetry stabilizes the scalar is studied elaborately inearlier works such as [11]-[22]. It is also demonstrated by previous authors that singlet fermion extensionof SM can also be a viable candidate of dark matter [23]-[25]. SM extensions with two Higgs doubletsand a singlet are also addressed earlier where the additional singlet is the proposed of dark mattercandidate [26, 27]. Among various extensions of SM, a simplest model is to introduce an additionalSU(2) scalar doublet which produces no VEV. The resulting model namely Inert Doublet Model (IDM)provides a viable explanation for DM. Satbility of this inert doublet ensured by a discrete Z symmetryand the lightest inert particle (LIP) in this theory can be assumed to be a plausible DM candidate.Phenomenology of IDM has been elaborately studied in literatures [28]-[37]. In the present work, weconsider a two Higgs doublet model (THDM) with an additional scalar singlet, where one of the Higgsdoublet is identical to the inert doublet, i.e., one of the doublet assumes no VEV and all the SM sectorincluding the newly added singlet are even under an imposed discrete symmetry ( Z ) while the inertdoublet is odd uner that Z symmetry. Since inert scalars do not interact with SM particles, are stableand LIP is considered as a potential DM candidate. Presence of an additional singlet scalar enrichesthe phenomenology of Higgs sector and DM sector.Various ongoing direct detection experiments such as XENON100 [38], LUX [39], CDMS [40] etc.2rovide upper limits on dark matter-nucleon scattering cross sections for different possible dark mattermass. The CDMS [40] experiment also claimed to have observed three potential signals of dark matterat low mass region ( ∼ R γγ for h → γγ signal and comparingthe same with those given by LHC experiment.In this work, we consider an Inert Doublet Model (IDM) along with an additional singlet scalarfield S . A discrete Z symmetry, under which all SM particles along with the singlet scalar S areeven while the inert doublet considered is odd, allows the LIP ( H ) to remain stable and serve as aviable dark matter candidate. Additional scalar singlet having a non zero VEV mixes with the SMHiggs, provides two CP even Higgs states. We consider one of the scalars, h , to be the SM-like Higgs.Then h should be compatible with SM Higgs and one can compare the relevant calculations for h with that obtained in LHC experiment. The model parameter space is first constrained by theoreticalconditions such as vacuum stability, perturbativity, unitarity and then by the relic density bound givenby PLANCK/WMAP experiments. We evaluate the direct detection scattering cross-section σ SI withthe resulting constrained parameters for different LIP masses m H and investigate the regions in σ SI − m H plane that satisfy the bounds from experiments like LUX, XENON etc. We also calculate the signalstrength R γγ for h → γγ channel in the present framework and compare them with the experimentallyobtained limits for this quantity from CMS and ATLAS experiments. This will further constrain themodel parameter space. We thus obtain regions in σ SI − m H plane in the present framework thatsatisfy not only the experimental results for dark matter relic density and scattering cross-sections butcompatible with LHC results too.The paper is organised as follows. In Sec. 2 we present a description of the model and model param-eters with relevant bounds from theory (vacuum stability, pertubativity and unitarity) and experiments(PLANCK/WMAP, direct detection experiments, LHC etc.). In Sec. 3 we describe the relic density andannihilation cross section measurements for dark matter and modified R γγ and R γZ processes due toinert charged scalars. We constrain the model parameter space satisfying the relic density requirementsof dark matter and present the correlation between R γγ and R γZ processes in Section 4. In Sec. 5, wefurther constrain the results by direct detection bounds on dark matter. Finally, in Sec. 6 we summarizethe work briefly with concluding remarks. In our model we add an additional SU(2) scalar doublet and a a real scalar singlet S . Similar to thewidely studied inert doublet model or IDM where the added SU(2) scalar doublet to the SM Lagrangian3s made “inert” (by imposing a Z symmetry that ensures no interaction with SM fermions and the inertdoublet does not generate any vev), here too the extra doublet is assumed to be odd under a discrete Z symmetry (IDM). Under this Z symmetry however, all SM particles as also the added singlet S remain unchanged. The potential is expressed as V = m Φ † Φ + m Φ † Φ + 12 m s S + λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ λ (Φ † Φ )(Φ † Φ ) + 12 λ [(Φ † Φ ) + (Φ † Φ ) ] + ρ (Φ † Φ ) S + ρ (cid:48) (Φ † Φ ) S + ρ S (Φ † Φ ) + ρ (cid:48) S (Φ † Φ ) + 13 ρ S + 14 ρ S , (1)where m k ( k = 11 , , s ) etc. and all the coupling parameters ( λ i , ρ i , ρ (cid:48) i , i = 1 , , , ... etc.) are assumedto be real. In Eq. 1, Φ is the ordinary SM Higgs doublet and Φ is the inert Higgs doublet. Afterspontaneous symmetry breaking Φ and S acquires VEV and expressed asΦ = (cid:32) √ ( v + h ) (cid:33) , Φ = (cid:32) H +1 √ ( H + iA ) (cid:33) , S = v s + s . (2)In the above v s denotes the VEV of the field S and s is the real singlet scalar. Relation among modelparameters can be obtained from the extremum conditions of the potenial expressed in Eq. 1 and aregiven as m + λ v + ρ v s + ρ v s = 0 ,m s + ρ v s + ρ v s + ρ v v s + ρ v = 0 . Mass terms of various scalar particles as derived from the potential are µ h = 2 λ v µ s = ρ v s + 2 ρ v s − ρ v v s µ hs = ( ρ + 2 ρ v s ) vm H ± = m + λ v ρ (cid:48) v s + ρ (cid:48) v s m H = m + ( λ + λ + λ ) v ρ (cid:48) v s + ρ (cid:48) v s m A = m + ( λ + λ − λ ) v ρ (cid:48) v s + ρ (cid:48) v s . (3)The mass eigenstates h and h are linear combinations of h and s and can be written as h = s sin α + h cos α ,h = s cos α − h sin α , (4)4 being the mixing angle between h and h , is given bytan α ≡ x √ x , (5)where x = µ hs ( µ h − µ s ) . Masses of the physical neutral scalars h and h are m , = µ h + µ s ± µ h − µ s (cid:112) x . (6)We consider h with mass m = 125 GeV as the SM-like Higgs boson and the mass of the other scalar h in the model is denoted as m with m > m . Couplings of the physical scalars h and h with SMparticles are modified by the factors cos α and sin α respectively. In the present framework H and A arestable as long as the Z symmetry is unbroken and hence these neutral scalars can be viable candidatesfor dark matter. Here, the coupling λ serves as a mass splitting factor between H and A . We consider H to be the lightest inert particle (LIP) which is stable and is DM candidate in this work. We take λ < H to be the lightest stable inert particle. In the present framework, both thescalars h and h couple with the lightest inert particle H . Couplings of the scalar bosons ( h and h )with the inert dark matter H are given by λ h HH v = (cid:18) λ c α − λ s s α (cid:19) v ,λ h HH v = (cid:18) λ s α + λ s c α (cid:19) v (7)where λ = λ + λ + λ , λ s = ρ (cid:48) +2 ρ (cid:48) v s v and s α ( c α ) denotes sin α (cos α ). Couplings of scalar bosonswith charged scalars H ± are λ h H + H − v = ( λ c α − λ s s α ) v ,λ h H + H − v = ( λ s α + λ s c α ) v. (8) The model parameters are bounded by theoretical and experimental constraints. • Vacuum Stability - Vacuum stability constraints requires the potential to remain bounded frombelow. Conditions for the stability of the vacuum are λ , λ , ρ > ,λ + 2 (cid:112) λ λ > ,λ + λ + λ + 2 (cid:112) λ λ > ,ρ + (cid:112) λ ρ > ,ρ (cid:48) + (cid:112) λ ρ > . (9)5 Pertubativity - For a theory to be acceptable in perturbative limits, we have to constrain highenergy quartic interactions at tree level. The eigenvalues | Λ i | of quartic couplings (scattering)matrix must be smaller than 4 π . • LEP
LEP[44] results constrains the Z boson decay width and masses of scalar particles m H + m A > m Z ,m H ± > . . (10) • Relic Density - Parameter space is also constrained by the measurement of relic density of darkmatter candidate. Relic density of the lightest inert particle (LIP) serving as a viable candidatefor dark matter in the present model must satisfy PLANCK/WMAP results,Ω DM h = 0 . ± . . (11) • Higgs to Diphoton Rate R γγ Bound on Higgs to two photon channel has been obtained fromexperiments performed by LHC. The reported singal strength for the Higgs to diphoton channelfrom ATLAS and CMS are given as R γγ | ATLAS = 1 . +0 . − . , R γγ | CMS = 0 . +0 . − . . • Direct Detection Experiments - The bounds on dark matter from direct detection experi-ments are based on the elastic scattering of the dark matter particle off a scattering nucleus.Dark matter direct detection experiments set constraints on the dark matter - nucleus (nucleon)elastic scattering cross section. Limits on scattering cross sections for different dark matter masscause further restrictions on the model parameters. Experiments like CDMS, DAMA, CoGeNT,CRESST etc. provide effective bounds on low mass dark matter. Stringent bounds on midddlemass and high mass dark matter are obtained from XENON100 and LUX experiments.
Relic density of dark matter is constrained by the results of PLANCK and WMAP. Dark matter relicabundance for the model is evaluated by solving the evolution of Boltzmann equation given as [45]d n H d t + 3H n H = −(cid:104) σ v (cid:105) ( n H − n H eq ) . (12)In Eq. 12, n H ( n Heq ) denotes the number density (equilibrium number density) of dark matter H andH is the Hubble constant. In Eq. 12, (cid:104) σ v (cid:105) denotes the thermal averaged annihilation cross section of6ark matter particle to SM species. The dark matter relic density can be obtained by solving Eq. 12and is written as Ω DM h = 1 . × x F √ g ∗ M Pl (cid:104) σ v (cid:105) . (13)In the above, M Pl = 1 . × GeV, is the Planck scale mass whereas g ∗ is the effective number ofdegrees of freedom in thermal equilibrium and h is the Hubble parameter in unit of 100 km s − Mpc − .In Eq. 13, x F = M/T F , where T F is the freeze out temperature of the annihilating particle and M isthe mass of the dark matter ( m H for the present scenario). Freeze out temperature T F for the darkmatter is obtained from the iterative solution to the equation x F = ln M π (cid:115) M g ∗ x F (cid:104) σ v (cid:105) . (14) Annihilation of inert dark matter H to SM particles is governed by processes involving scalar ( h , h )mediated s( (cid:39) m H ) channels. Thermal averaged annihilation cross section (cid:104) σ v (cid:105) of dark matter H toSM fermions are given as (cid:104) σ v HH → f ¯ f (cid:105) = n c m f π β f (cid:12)(cid:12)(cid:12)(cid:12) λ h HH cos α m H − m + i Γ m + λ h HH sin α m H − m + i Γ m (cid:12)(cid:12)(cid:12)(cid:12) . (15)In the above, m x represents mass of the particle x ( ≡ f, H etc.), n c is the colour quantum number (3for quarks and 1 for leptons) with β a = (cid:114) − m a m H and Γ i ( i = 1 ,
2) denotes the total decay width ofeach of the two scalars h and h . For DM mass m H > ( m W , m Z ), annihilation of DM to gauge boson( W or Z ) channels will yield high annihilation cross-section. Since Ω DM ∼ (cid:104) σ v (cid:105) − (Eq. 13), the relicdensity for the dark matter with mass m H > m W or m Z in the present model in fact falls below therelic density given by WMAP or PLANCK as the four point interaction channel HH → W + W − or ZZ will be accessible and as a result increase in total annihilation cross-section will be observed. Thus thepossibility of a single component DM in the present framework is excluded for mass m H > m W , m Z .Higgs like boson h and the scalar h may also decay to dark matter candidate H when the condition m H < m i / i = 1 ,
2) is satisfied. Contributions of invisible decay widths of h and h are taken intoaccount when the condition m H < m i / i = 1 ,
2) is satisfied. Invisible decay width is represented bythe relation Γ inv ( h i → H ) = λ h i HH v πm i (cid:115) − m H m i . (16) Similar results for IDM are also obtained in previous work (Ref. [51]) where a two component dark matter wasconsidered in order to circumvent this problem. .3 Modification of R γγ and R γZ Recent studies of IDM [46, 47, 48] and two Higgs doublet models [49, 50] have reported that a low masscharged scalar could possibly enhance the h → γγ signal strength R γγ . Correlation of R γγ with R γZ is also accounted for as well [47, 50]. The quantities R γγ and R γZ are expressed as R γγ = σ ( pp → h ) σ ( pp → h ) SM Br ( h → γγ ) Br ( h → γγ ) SM (17) R γZ = σ ( pp → h ) σ ( pp → h ) SM Br ( h → γZ ) Br ( h → γZ ) SM , (18)where σ is the Higgs production cross section and Br represents the branching ratio of Higgs to finalstates. Branching ratio to any final state is given by the ratio of partial decay width for the particularchannel to the total decay width of decaying particle. For IDM with additional singlet scalar, theratio σ ( pp → h ) σ ( pp → h ) SM in Eqs. 17-18 is represented by a factor cos α . Standard Model branching ratios Br ( h → γγ ) SM and Br ( h → γZ ) SM for a 125 GeV Higgs boson is 2 . × − and 1 . × − respectively [52]. To evaluate the branching ratios Br ( h → γγ ) and Br ( h → γZ ), we compute thetotal decay width of h . Invisible decay of h to dark matter particle H is also taken into account andevaluated using Eq. 16 when the condition m H < m / h → γγ )and Γ( h → γZ ) according to the model are given asΓ( h → γγ ) = G F α s m √ π (cid:12)(cid:12)(cid:12)(cid:12) cos α (cid:18) F / (cid:18) m t m (cid:19) + F (cid:18) m W m (cid:19)(cid:19) + λ h H + H − v m H ± F (cid:18) m H ± m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , Γ( h → γZ ) = G F α s π m W m (cid:18) − m Z m (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − α − s W c W F (cid:48) / (cid:18) m t m , m t m Z (cid:19) − cos αF (cid:48) (cid:18) m W m , m W m Z (cid:19) + λ h H + H − v m H ± (1 − s W ) c W I (cid:18) m H ± m , m H ± m Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (19)where G F is the Fermi constant, m x denotes the mass of particle x ( x ≡ , W, Z, t, H ± ) etc. and s W ( c W ) represents sin θ W (cos θ W ), θ W being the weak mixing angle. Expressions for various loopfactors ( F / , F , F , F (cid:48) / , F (cid:48) and I ) appeared in Eq. 19 are given in Appendix A. It is to be notedthat a similar derivation of decay widths and signal strengths ( R (cid:48) γγ or R (cid:48) γ Z ) for the other scalar h can be obtained by replacing m , cos α, λ h H + H − with m , sin α, λ h H + H − respectively and this isaddressed in Sec. 5. R γγ and R γZ In this section we compute the quantities R γγ and R γZ in the framework of the present model. Werestrict the allowed model parameter space for our analysis using the vacuum stability, perturbativeunitarity, LEP bounds along with the relic density constraints described in Section 2.2. Dark matter8elic density is evaluated by solving the Boltzmann equation presented in Section 3.1 with the expressionfor annihilation cross section given in Eq. 15. Model parameters ( λ i , ρ i ), should remain small in orderto satisfy perturbative bounds and relic density constraints. Calculations are made for the modelparameter limits given below, m = 125 GeV ,
80 GeV ≤ m H ± ≤
400 GeV , < m H < m H ± , m A , < α < π/ , − ≤ λ ≤ , − ≤ λ ≤ , − ≤ λ s ≤ . (20)The couplings λ h HH and λ h HH (Eq. 7) are required to calculate the scattering cross-section of thedark matter off a target nucleon. Dark matter direct detection experiments are based on this scatteringprocesses whereby the recoil energy of the scattered nucleon is measured. Thus the couplings λ h HH and λ h HH can be constrained by comparing the computed values of the scattering cross-section for differentdark matter masses with those given by different dark matter direct detection experiments. In thepresent work, | λ h HH , λ h H + H − | ≤ λ h HH and λ h H + H − (Eqs. 7-8). Using Eqs. 12-16 we scan over the parameter spacementioned in Eq. 20 where we also impose the conditions | λ h H + H − , λ h HH | ≤ h ) mass namely m = 140 ,
150 and 160 GeV. Scanning ofthe full parameter space yields that for all the cases considered, the limits − . ≤ λ h HH ≤ . | λ h HH | ≤ . | λ h H + H − | ≤ λ h H + H − . Our calculation reveals that | λ h H + H − | ≤ R γγ and R γZ (Eqs. 17-18) by evaluating the corresponding decay widthsgiven in Eq. 19.In Fig. 1(a-c), shown are the regions in the R γγ − m H plane for the parameter values that satisfyDM relic abundance. As mentioned earler, results are presented for three values of h mass namely140 ,
150 and 160 GeV. Since for low mass DM region, invisible decay channel of h to DM pair remainsopen, enhancement of R γγ is not possible in this regime. R γγ becomes greater than unity near theregion where m H (cid:38) m /
2. The region that describe the R γγ enhancement is reduced with increasing h mass and thus enhacement is not favoured for higher values of h mass. For the rest of the allowedDM mass parameter space, R γγ remains less than 1. The results presented in Fig. 1 indicate that9 a) (b)(c) Figure 1: Variation of R γγ with DM mass m H satifying DM relic density for m = 140 ,
150 and 160GeV.observed enhancement of the h → γγ signal could be a possible indication of the presence of h since R γγ (cid:38) h . The R γγ value depends on the coupling λ h H + H − and becomesgreater than unity only for λ h H + H − < R γγ depends on the values of h mass, charged scalar mass m H ± , coupling λ h H + H − andthe decay width of invisible decay channel (Γ inv ( h → HH )). A similar variation for the h → γZ channel (computed using Eqs. 18-19 and Eq. 20) yields lesser enhancement for R γZ in comparison with R γγ . This phenomenon can also be verified from the correlation between R γγ and R γZ . The correlationbetween the signals R γγ and R γZ is shown in Fig. 2a - Fig. 2c for m = 140 , ,
160 GeV respectively.10 a) (b)(c)
Figure 2: Correlation plots between R γγ and R γZ for three choices of h mass (140 ,
150 and 160 GeV).Variations of R γγ and R γZ satisfy all necessary parameter constraints taken into account inclusive ofthe relic requirements for DM. In this case (Fig. 2), we further constrain the parameter space of α mentioned in Eq. 20 by imposing the condition 0 < α < π/
4. This condition ensures that h is the SM-like Higgs boson [23, 25]. Fig. 2 also indicates that, with increase in the mass ( m ) of h , enhancementof R γγ and R γZ are likely to reduce. For m = 140 GeV, R γγ enhances up to four times whereas R γZ increases nearly by a factor 2 with respect to corresponding values predicted SM. On the other hand,for m = 160 GeV, R γγ varies linearly with R γZ ( R γγ (cid:39) R γZ ) without any significant enhacement. Inaddition Fig. 2 suggests that for cos α (cid:38) / √
2, a considerable portion of allowed parameter space withlower values of R γγ will disappear. For | λ h HH | < .
05, variation of R γγ with R γZ is almost linear (with11lope ≈
1) which is presented by the line passing through origin shown in plots of Fig. 2. The scatteredplots represent the correlation for other values of λ h HH . For low mass dark matter ( m H (cid:46) m / h remains open and the processes h → γγ and h → γZ suffer considerablesuppressions. These result in the correlation between the channels h → γγ and h → γZ to becomestronger and R γγ vs R γZ plot shows more linearity with increase in h mass. For larger h masses, thecorresponding charged scalar ( H ± ) masses for which R γγ,γZ >
1, tends to increase. Since any increasein H ± mass will affect the contribution from charged scalar loop, the decay widths Γ( h → γγ, γZ ) orsignal strengths R γγ,γZ are likely to reduce. Our numerical results exhibit a positive correlation betweenthe signal strengths R γγ and R γZ . This is an important feature of the model. Since signal strengthstend to increase with relatively smaller values of m , possibility of having a light singlet like scalar is notexcluded. The coupling of h with SM sector is suppressed by a factor sin α which results in a decreasein the signal strengths from h and makes their observations difficult. Within the framework of our model and allowed values of parameter region obtained in Sec. 4, wecalculate spin independent (SI) elastic scattering cross-section for the dark matter candidate in ourmodel off a nucleon in the detector material. We then compare our results with those given by variousdirect detection experiments and examine the plausibility of our model in explaining the direct detectionexperimental results. The DM candidate in the present model, interacts with SM via processes led byHiggs exchange. The spin-independent elastic scattering cross section σ SI is of the form σ SI (cid:39) m r π (cid:18) m N m H (cid:19) f (cid:18) λ h HH cos αm + λ h HH sin αm (cid:19) , (21)where m N and m H are the masses of scattered nucleon and DM respectively, f represents the scatteringfactor that depends on pion-nucleon cross-section and quarks involved in the process and m r = m N m H m N + m H is the reduced mass. In the present framework f = 0 . σ SI for the dark matter candidate in the present model are carried out with those values of the couplingsrestricted by the experimental value of relic density.In Fig. 3(a-c), we present the variation of elastic scattering cross section calculated using Eq. 21,with LIP dark matter mass ( m H ) for three values of h masses m = 140 ,
150 and 160 GeV. We assume h to be SM-like Higgs and restrict the mixing angle α such that the conditon cos α (cid:38) / √ σ SI − DM massobtained from DM direct search experiments such as XENON100, LUX, CDMS, CoGeNT, CRESST.From Fig. 3 one notes that in the low mass region, the DM candidate in our model satisfies boundsobtained from experiments like CoGeNT, CDMS, CRESST. We further restrict the σ SI − m H spaceby identifying in Fig. 3(a-c) the region for which the CMS limit of R γγ ( R γγ = 0 . +0 . − . ) is satisfied.In each of the σ SI − m H plots of Fig. 3(a-c) the light blue region satisfies CMS limit of R γγ for threechosen values of m . Also marked in black are the specific zones that correspond to the central value of12 a) (b)(c) Figure 3: Allowed regions in m H − σ SI plane for m = 140 ,
150 and 160 GeV. R γγ | CMS = 0 .
78. It is therfore evident from Fig. 3(a-c) that imposition of signal strength ( R γγ ) resultsobtained from LHC, further constraints the allowed scattering cross-section limits obtained from directdetection experimental results for the DM candidate in our model. Investigating the region allowed byLUX and XENON experiments along with other direct dark matter experiments such as CDMS etc., itis evident from Fig. 3(a-c) that our model suggests a DM candidate within the range m / < m H < m W GeV with scattering cross-section values ∼ − − − cm with m = 125 GeV, i.e., SM-like scalar.There are however few negligibly small allowed parameter space with σ SI below ∼ − cm . It mayalso be noticed from Fig. 3 that the present model with all the constraints including R γγ condition alsopartly agrees with allowed contour given by DAMA experiment. However, DAMA contour is ruled out13 a) Figure 4: The m H vs σ SI parameter space for R γγ (cid:38) m = 140 −
160 GeV.by recent results from experiments like LUX and XENON100. Similar procedure has been adopted forrestricting the σ SI − m H space with R γγ limits from ATLAS experiment. In Fig. 4, the region shownin red corresponds to the region satisfying R γγ (cid:38) h varied from 140 GeV to 160 GeV.Also shown in Fig. 4, the scattered region in green (blue) represents the signal strength R γγ = 1 . +0 . − . ( R γγ = 1 .
65) as obtained from ATLAS experiment respectively. Fig. 4 shows that the part of theregion constrained by ATLAS result is more stringent than that for CMS case and appears to satisfyonly a part of DAMA allowed contour. There is however a negiligibly small allowed region satisfyingthe domain constrained by LUX or XENON100 expreiments. Similar to the case for R γγ limit fromCMS, here too, the allowed zone lies in the range around m H = 70 GeV. Hence, in the present model H can serve as a potential dark matter candidate and future experiments with higher sensitivity likeXENON1T [54], SuperCDMS [55] etc. are expected to constrain or rule out the viability of this model.In the present model we so far adopt the consideration that h plays the role of SM Higgs and hencein our discussion we consider h → γγ for constraining our parameter space. The model considered inthis work also provides us with a second scalar namely h . Since LHC has not yet observed a secondscalar, it is likely that the other scalar h is very weakly coupled to SM sector so that the correspondingbranching ratios (signal strengths) are small. This may be justified in the present scenario if in case the h → γγ branching ratio or signal strength ( R (cid:48) γγ ) is very small compared to that for h . Needless tomention that the couplings required to compute R γγ and R (cid:48) γγ are restricted by dark matter constraints.One also has to verify whether the process R (cid:48) γγ can play significant role in restricting the dark mattermodel parameter space in the present framework. We address these issues by computing R (cid:48) γγ values andcomparing them with R γγ . The computation of R γγ and R (cid:48) γγ initially involves the dark matter model Since R (cid:48) γγ and R (cid:48) γZ are correlated, any suppression in h → γγ will be followed by similar effects in h → γZ . a) (b)(c) Figure 5: Allowed regions in R γγ − R (cid:48) γγ plane for m = 140 ,
150 and 160 GeV.parameter space that yields the dark matter relic density in agreement with PLANCK data as also thestringent direct detection cross-section bound obtained from LUX. R γγ values thus obtained are notfound to satisfy the experimental range given by ATLAS experiment. The resulting R γγ − R (cid:48) γγ is furtherrestricted for those values of R γγ which are within the limit of R γγ | CMS given by CMS experiment. Theregion with black scattered points in Fig. 5(a-c) corresponds to the R γγ − R (cid:48) γγ space consistent withthe model parameters that are allowed by DM relic density obtained from PLANCK, direct detectionexperiment bound from LUX and R γγ | CMS for three different values of m (140 , , and 160 GeV).Fig. 5(a-c) reveals that R (cid:48) γγ ≤ . m = 140 ,
150 GeV whereas for m = 160 GeV it is even less( R (cid:48) γγ < .
25) for R γγ values compatible with CMS results. In Table 1 we further demonstrate that15 m H m H ± α R γγ R (cid:48) γγ Br ( h → γγ ) σ SI in GeV in GeV in GeV in cm h mass.within the framework of our proposed model for LIP dark matter, R (cid:48) γγ is indeed small compared to R γγ . We tabulate the values of both R γγ and R (cid:48) γγ for some chosen values of LIP dark matter mass m H .These numerical values are obtained from the computational results consistent with LUX direct DMsearch bound. Also given in Table 1 the corresponding mixing angles α between h and h , the scalarmasses m ± H , h to di-photon branching ratio and the scattering cross-section σ SI for three differentvalues of m considered in the work. It is also evident from Table 1 that R γγ >> R (cid:48) γγ and mixingangles corresponding to respective values are small. In fact for some cases such as for m H = 70 .
13 GeV( m = 140 GeV) R γγ = 0 .
889 whereas R (cid:48) γγ ∼ − and α is as small as 2. This demonstrates thatthe scalar h in Eq. 4 is mostly dominated by SM-like Higgs component and the major component inthe other scalar is the real scalar singlet s of the proposed model. Table 1 also exhibits that the signalstrength R (cid:48) γγ for h → γγ channel is negligibly weak. In this paper we have proposed a model for dark matter where we consider an extended two Higgsdoublet model with an additional singlet scalar. The DM candidate follows by setting one of theHiggs doublet to be identical with an inert Higgs doublet imposing a Z symmetry on the potential.This ensures the DM candidate that follows from the added inert doublet is stable. The inert doubletdoes not generate any VEV and hence cannot couple to Standard Model fermions directly. The scalarsinglet, having no such discrete symmetry aquires a non zero VEV and mixes up with SM Higgs. Theunknown couplings appearing in the model, which are basically the model parameters, are restricted16ith theoretical and experimental bounds. The mixing of the SM and the singlet scalar gives rise totwo sclar states namely h and h . For small mixing h behaves as the SM Higgs and h as the addedscalar. We extensively explored the scalar sector of the model and studied the signal streghts R γγ and R γZ for the SM- like Higgs ( h ) in the model. The range and region of enhancement depens onthe mass of the singlet like scalar h . Appreciable enhancement of signals depends on h mass whichoccurs near the Higgs resonance. Increase in signal strengths is not allowed for heavier values of h mass. Enhancement of signals are forbidden when the invisible decay channel remains open. The extentof enhancement depends on the charged scalar mass and occurs only when the Higgs-charged scalarcoupling λ h H + H − <
0. We first restrict our parameter space by calculating the relic density of LIPdark matter in the framework of our model. Using the resultant paramter space obtained from relicdensity bounds we evalute the signal strengths R γγ and R γZ for different dark matter mass. We thenrestrict the parameter space by calculating the spin independent scattering cross-section and comparingit with the existing limits from ongoing direct detection experiments like CDMS, CoGeNT, DAMA,XENON100, LUX etc. Employing additional constraints by requiring that R γγ and R γZ will satisfythe CMS bounds and ATLAS bounds, we see that the present model not only provides a good DMcandidate in middle mass region consistent with LUX and XENON100 bounds. The possibility that R γγ ( > .
0) in the present framework does not seem to be favoured by LUX and XENON100 data.However, DAMA results appear to favour R γγ ≥
1. Therefore, we conclude that under the presentframework, Inert Doublet Model with additional scalar singlet provide a viable DM candidate withmass range m / < m H < m W GeV that not only is consistent with the direct detection experimentalbounds and PLANCK results for relic density but also in agreement with the Higgs search results ofLHC. A singlet like scalar that couples weakly with SM Higgs may also exist that could enrich the Higgssector and may be probed in future collider experiments.
Acknowledgments : A.D.B. would like to thank A. Biswas, D. Das and K.P. Modak for usefuldiscussions.
Appendix A
In Section 3.3 we have derived the decay widths h → γγ and h → γZ in terms of the loop factors F / , F , F , F (cid:48) / , F (cid:48) and I respectively. Factors F / , F , F , for the measurement of h → γγ decay width can be written as [56, 57, 58] F / ( τ ) = 2 τ [1 + (1 − τ ) f ( τ )] ,F ( τ ) = − [2 + 3 τ + 3 τ (2 − τ ) f ( τ )] ,F ( τ ) = − τ [1 − τ f ( τ )] , and f ( τ ) = arcsin (cid:16) √ τ (cid:17) for τ ≥ , − (cid:104) log (cid:16) √ − τ −√ − τ (cid:17) − iπ (cid:105) for τ < . h → γZ are expressed following Refs. [56, 57, 58] F (cid:48) / ( τ, λ ) = I ( τ, λ ) − I ( τ, λ ) ,F (cid:48) ( τ, λ ) = c W (cid:26) (cid:18) − s W c W (cid:19) I ( τ, λ ) + (cid:20)(cid:18) τ (cid:19) s W c W − (cid:18) τ (cid:19)(cid:21) I ( τ, λ ) (cid:27) , where I ( a, b ) = ab a − b ) + a b a − b ) [ f ( a ) − f ( b )] + a b ( a − b ) [ g ( a ) − g ( b )] ,I ( a, b ) = − ab a − b ) [ f ( a ) − f ( b )] . 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