ULB-TH/06-27 The Inert Doublet Model: an Ar hetype for Dark Matter Laura Lopez Honorez, Emmanuel Nezri, Josep F. Oliver, Mi hel H.G. Tytgat Servi e de Physique Théorique, Université Libre de Bruxelles, CP225, Bld du Triomphe, 1050 Brussels, Belgium 7 0 0 2 Abstra t n The Inert Doublet Model (IDM), a two Higgs extension of the Standard Model with an unbroken Z a 2 symmetry, is a simple and yet ri h model of dark matter. We present a systemati analysis of the J darkmatterabundan e and investigate the potentialities fordire t and gamma indire t dete tion. We 9 showthat the modelshouldbe within the rangeoffuture experiments, likeGLAST and ZEPLIN. The lightest stable s alar in the IDM is a perfe t example, or ar hetype of a weakly intera ting massive 2 v parti le. 5 7 2 1 Introdu tion 2 1 Contemporary osmologi alobservations on urtoindi atethatthemajorityofmatterintheuniverse 6 not only does not shine but is not even made of ordinary atoms [1, 2℄. De iphering the nature of this 0 so- alled Dark Matter has be ome one of the most important issue at the frontier of parti le physi s, / h astrophysi s and osmology. A profusion of dark parti les have been proposed over the years and it p is mu h hoped that present and forth oming experiments will throw some light on the matter. For a - p review, see for instan e [3, 4℄. e In the present arti le weinvestigate further arather mundane, although in our opinion quite inter- h esting, form of dark matter. The parti le andidate is a weakly intera ting massive s alar and it has : H1 H2 v been advo ated re ently in [5℄ and [6, 7℄. The framework is that of a two Higgs doublets, and , Z Xi version of the Standard Model with a 2 symmetry su h that H H H H . r 1 → 1 and 2 →− 2 a Z 2 All the (cid:28)elds of the Standard Model are even under . This model has been (cid:28)rst dis ussed by Z 2 Deshpande and Ma in [8℄. Following [6℄, we will assume that is not spontaneously broken, i.e. H 2 does not develop an expe tation value. Among other things, the dis rete symmetry prevents the H 0 2 appearan e of (cid:29)avour hanging neutral urrents. The model is de(cid:28)nitely not the h i → limit of a generi two Higgs model like, for instan e, the Higgs se tor of the MSSM. The most general, albeit renormalizable, potential of the model an be written as λ V =µ2 H 2+µ2 H 2+λ H 4+λ H 4+λ H 2 H 2+λ H†H 2+ 5 (H†H )2+h.c. . 1| 1| 2| 2| 1| 1| 2| 2| 3| 1| | 2| 4| 1 2| 2 h 1 2 i (1) U(1) λ = 0 5 There is an Pe ei-Quinn global symmetry if . This limit is however not favored by dark matter dire t dete tion experiments (se tion 2.3). SU(2) U(1) H 1 The × symmetry is broken by the va uum expe tation value of , v H = 1 h i √2 v = µ2/λ =248 µ2 >0 with − 1 1 GeV while, assuming 2 , H =0. 2 h i 1 h The mass of the Brout-Englert-Higgs parti le ( or Higgs for short) is M2 = 2µ2 2λ v2 h − 1 ≡ 1 (2) H+ H A H 0 0 2 while the mass of the harged, , and two neutral, and , omponents of the (cid:28)eld aregiven by M2 = µ2+λ v2/2 H+ 2 3 M2 = µ2+(λ +λ +λ )v2/2 H0 2 3 4 5 M2 = µ2+(λ +λ λ )v2/2. A0 2 3 4− 5 (3) H A H 0 0 2 Forappropriatequarti ouplings, or isthelightest omponentofthe doublet. Intheabsen e Z 2 of any other lighter -odd (cid:28)eld, either one is a andidate for dark matter. For de(cid:28)niteness we hoose H A 0 0 . All our on lusions are un hanged if the dark matter andidate is instead. Following [6℄ we H λ = (λ +λ +λ )/2 0 L 3 4 5 parameterize the ontribution from symmetry breaking to the mass of by , h H 0 whi h is also the oupling onstant between the Higgs (cid:28)eld and our dark matter andidate . Of v the seven parameters of the potential (1), one is known, , and four an be related to the mass of the µ λ 2 2 s alar parti les. In the sequel, we take and as the last two independent parameters. The later a tually plays little role for the question of dark matter. H 0 The abundan e of as the dark matter of the universe as well as the onstraints from dire t dete tion experiments have been tou hed upon in the re ent literature [6, 7℄ (see also [9℄). In the H 0 presentarti leweinvestigatethe potentialitiesforindire tdete tionof usinggammarayteles opes and dete tors. Forthe sakeof ompleteness, wealsopresenta systemati (albeit tree-level)analysisof H 0 the abundan e of the in the light of WMAP data, as well as the prospe t for dire t dete tion by underground dete tors. Before doing so, we would like to brie(cid:29)y emphasize some of the virtues of this, so- alled, Inert Doublet Model (IDM), both in general and for the issue of dark matter in parti ular. True, the model is ad ho . Also, it does not address any deep issue, say the hierar hy problem. Yet it is a very simple extension of the Standard Model and its phenomenology is nevertheless very ri h [5, 6, 10, 11℄. In referen e [6℄, Barbieri et al onsidered this model as a way of pushing the mass of the Higgs parti le towardtheTeVs ale,without ontradi tingLEPpre isionmeasurements(seealso[10℄,andse tion3.1 ofthepresentpaper). Also,inthesimplestversiondis ussedheretherearenoYukawa ouplingstothe H 2 doublet, but it is somewhat natural to imagine oupling it to (odd) right-handed neutrinos. This opensthepossibilityofgeneratingthemassoftheSMneutrinosthroughloop orre tions,ame hanism introdu ed by Ma in [5℄ and further addressed in a series of papers [12, 13, 14℄. H 0 As a dark matter andidate, is not without interest either. It belongs to the family of Weakly Intera tingMassiveParti les(WIMP) andhasbeen proposedasadarkmatter andidatebyMa. The most a laimed member of this family is ertainly the neutralino, the lightest supersymmetri parti le (LSP).Thesupersymmetri extensionsoftheSMarewithoutdoubtveryinterestingandwellmotivated but they involve many new parameters and their phenomenology is, to say the least, omplex. It is perhaps far fet hed to ompare the respe tive advantages and disadvantages of the IDM and of the H 0 MSSM but it is interesting that the shares many similarities with the LSP (weak intera tions, similar mass s ale, et ) while being onsiderably mu h simpler to analyze. Interestingly, if we take H 0 seriouslythe ideathat isthedominantformofdarkmatter, theparameterspa eoftheIDMmodel is quite onstrained. It is also, to some extent, omplementary to that of the MSSM, as we show in H 0 the on lusions. And, herry on the top, the phenomenologyof the as a andidate for dark matter isneatly intertwined with thatof the Higgsparti le. In ouropinion,it is this onjun tion ofsimpli ity and ri hness that makes the lightest stable s alar a perfe t example, or ar hetype, of dark matter. The plan of the paper is as follow. In the next se tion (2) we give a summary of our methodology forthe al ulationsofthedarkmatterabundan e,thephoton(cid:29)uxforindire tdete tionbygammaray teles opesandthe ross-se tionsfordire tdete tion. Wethendis ussinsomedetailstheresultsofour analysis, in the light of WMAP data, and study the prospe ts for both indire t and dire t dete tion by existing and forth oming experiments. We give our on lusions in the last se tion, together with omparison of the Model with that of the MSSM, and the prospe ts for future analysis. 2 2 Dark matter aspe ts 2.1 Dark Matter abundan e H 0 The abundan e of has been estimated in [6℄ and in [7℄ for two di(cid:27)erent mass ranges. As in these papers, we onsider only thHe standard freeze-out me hanismW. ThereZare then esseMntially>tw80oGreeVgimes, depending on the mass of 0 with respe t to that of the and bosons. If H0 , the H M >M ∼ 0 annihilate essentially into Z and W boson pairs (see Figure (1)). For H0 h, the annihilation h H 0 hannel into pairs opens (see Figure (2)). Otherwise, the annihilate essentially through an H 0 intermediate Higgs, provided the Higgs itself is not too heavy. Some amount of oannihilation of A H+ Z W+ 0 with the next-to-lightest s alar parti le (resp. ) through a (resp. ) boson an be present [15℄, a feature that substantially ompli ates the determination of the dark matter abundan e (see Figure (3)). H 0 We have omputed the reli abundan e of using mi rOMEGAs2.0, a new and versatilepa kage for the numeri al al ulation of Dark Matter abundan e from thermal freeze-out [16℄. The latest implementation of this ode, whi h was originally developed to study supersymmetri models, allows onetoenteranymodel ontainingadis retesymmetrythatguaranteesthestabilityofthedarkmatter parti le. The ode takesadvantageofthe fa tthatea hodd(i.e. nonstandard)parti lesofthemodel H 0 will eventually de ay into the lightest odd parti le, in our ase the . One an then sum the system of Boltzmann equations of all odd spe ies and the reli density of dark matter is al ulated by solving the Boltzmann equation for the lightest parti le only [17℄: dY πg (T) = ∗ M σv (Y2(T) Y2(T)), dT r 45 Plh i − eq (4) Y n /s DM where ≡ is the omovingdensity of dark matter. All the pro essesof the model enterin the σv T = T =T 0 thermally averaged ross-se tion h i. Integrating Eq.(4) from ∞ to the reli abundan e is given by M Ω h2 =2.72 108 DMY(T ). DM 0 × GeV (5) Mi rOMEGAs2.0 itself is build upon CALCHEP, a ode for omputing tree-level ross-se tions. We have used both Mi rOMEGAs2.0 and CALCHEP to ompare with our analyti al al ulations and to performanumber∗of ross- he ksthathavestrengthenedourfaith in thenumeri alresultsreportedin the present paper. 2.2 Indire t dete tion Themeasurementofse ondaryparti les omingfromdarkmatterannihilationinthehaloofthegalaxy isapromisingwayofde ipheringthenatureofdarkmatter. Thispossibilitydependshowevernotonly on the properties of the dark matter parti le, through its annihilation ross-se tions, but also on the astrophysi alassumptionsmade on erningthedistributionofdarkmatterinthehalothatsupposedly surroundsourgalaxy. Thegala ti enter(GC)regionispotentiallyaveryattra tivetargetforindire t dete tion of dark matter, in parti ular through gamma rays. The produ ed gamma ray (cid:29)ux from the annihilation of dark matter parti les an be expressed as Φ dNi 1 γ = γ σ v ρ2 dl, dΩdE dE h i i4πm2 Z (6) Xi γ DM l.o.s. m ρ σ v dNi/dE where DM is the darkmatter parti le mass, is the dark matterdensity pro(cid:28)le, h i i and γ γ v are,respe tively,thethermallyaveragedannihilation rossse tiontimestherelativevelo ity andthe di(cid:27)erential gamma spe trum per annihilation oming from the de ay of annihilation produ ts of (cid:28)nal i state . The integral is taken along the line of sight. The pro esses involved are shown in the Figure (3), (cid:28)rst diagram and Figures (1) and (2). ∗ The Feynman rules and de(cid:28)nition (cid:28)les of the IDM to be used with mi rOMEGAs an be obtained upon request to the authors of thepresentarti le and will be in ludedin thenextavailable version of the ode. 3 10−8 The mcmod−e2ls.sw−1ill be onstrainedbythe existingEGRET [18℄ experimentallim10i−t1o0n th1e0(cid:29)−u11x,∼ pcmho−t2o.nss−1 ∆Ω,=a1n0d−3the for1t0h− 5oming GLAST [19℄ expe ted sensitivity, ∼ − photons for and srad respe tively. Our purpose for the time being is merely to prospe t the IDM with regard to gamma ray indire t dete tion. In this paper, we fo us on the parti le physi s parameter dependen e of the model and work withinRa=(cid:28)x8e.5d astrophysi al framework, the popular Navarro-Frankρ-W=h0it.e3(GNeFVW.cm) −ha3lo pro(cid:28)le 0 0 [20℄. With kp equaltothedistan efromtoSuntotheGCand thedark matter density in our neighborhood, the NFW pro(cid:28)le density an be parameterized as [1+(R /a)α](β−γ)/α R γ 0 0 ρ(r)=ρ , 0 [1+(r/a)α](β−γ)/α (cid:18) r (cid:19) (7) (a,α,β,γ) = (20,1,3,1) with . It should be emphasized that the (dark) matter distribution in the γ innermost region of the galaxy is poorly known so that is not very onstrained, a freedom that an give rise to very di(cid:27)erent values of the gamma ray (cid:29)ux. Typi ally a suppression or an enhan ement of two orders of magnitude an be obtained if one onsiders respe tively a halo with a (cid:29)at ore (e.g. γ = 0 γ 1.5 isothermal, ) or a deeper usp ( ∼ ) [21℄ oming for instan e from baryoni infall. Hen e, dependingontheastrophysi sassumptions,theestimationofgammaraysignal anvaryquitestrongly (see e.g. [23, 22℄). 1GeV To a∆l Ωul=ate10t−h3e (cid:29)ux we have integrat∆edΩ(=6)10fo−r5a NFW pro(cid:28)le above and around a solid angleof sradforEGRET and sradforGLAST.Thedi(cid:27)erentialspe traofea h hannel are given by mi rOMEGAs (we have he ked the agreement with Pythia simulations of [23℄) σv as well as h i at rest and we have integrated the square of the dark matter density along the line of sight. 2.3 Dire t dete tion A lo al distribution of weakly intera ting dark matter ould be dete ted [24℄ by measuring the energy deposited in a low ba kground dete tor by the s attering of a dark matter parti le with a nu lei of H 0 the dete tor. One distinguishes spin dependent and spin independent intera tions. For intera ting Z H q A q 0 0 with the quarks of the nu lei, there are two spin independent pro esses at tree level, −→ h H q H q 0 0 and −→ (see Figure (4)). The experiments have rea hed su h a level of sensitivity that the Z Z ex hange ontribution is ex luded by the urrent experimental limits. Consequently, to forbid A H 0 0 ex hange by kinemati s, the mass of the parti le must be higher than the mass of by a few 100keV λ 0 5 (we will thus disregard the → limit). We assume that the main pro ess here is the one h with Higgs parti le ex hange . The ross se tion at tree-level is [6℄ m2 λ 2 σh = r L f2m2, H0−p 4π (cid:18)M M2(cid:19) p (8) H0 h f f 0.3 m r where is a form fa tor estimated in the literature to be ∼ and is the redu ed mass of the M < M system. [25℄M. In a>ddMition, for H0 W, we have negle ted the oσne-loop ex4 .h6a1n0g−e13opfbtwo gauge bosons. For H0 W, we in lude this pro ess with, following [7℄, 1H00−−6p ≃ . We take into a ount the onstraints given by the CDMS [26℄ experiment (∼ pb) and the model is also ompared to the next generation experiments like EDELWEISS II [2σ7℄ or10a−t8on-sizσe ex1p0e−ri1m0ent like ZEPLIN [28℄ with the valley of their sensitivities respe tively around ∼ and ∼ pb. 3 Analysis σv The pro esses driving the ( o)annihilation ross se tion h i relevant for the reli density from freeze- H 0 out and indire t dete tion of gamma rays are shown in the Figures (1), (2) and (3). The -proton intera tions relevant for dire t dete tion are shown in Figure (4). H W 0 There are two qualitatively distin t regimes, depending on whether the is lighter than the Z and and/or the Higgs boson, in whi h ase annihilation pro eeds through the diagrams of Figure 4 (3). At low mass, the se ond diagram of (3) may ontribute to the dark matter abundan e if there is A 0 a substantial number of at the time of freeze-out (i.e. oannihilation). H 0 In the plots of Figure (5), the reli abundan e of dark matter parti les, the gamma ray (cid:29)ux H 0 due to annihilation at the gala ti enter (indire t dete tion) and, (cid:28)nally, their ross-se tion for σ elasti s attering o(cid:27) a proton through Higgs boson H0−p (dire t dete tion) ex hange are shown, for M =120 200 M ,µ twoparti ularHiggsmasses( h GeV and GeV) in the ( H0 2) plane. We refer to this as H H M =120 0 0 h thelow massregime. Forhigher masses,similarplotsaredisplayedin Figure(6)(for GeV), respe tively for the darkmatterabundan e, the gamma ray (cid:29)ux and the ross-se tionfor dire t dloegt(eΩ tionh.2)Inloge(aΦ h(of t−h2es−e1)t)hree laosge(sσ, the ( olo)r) gradients orrespond respe tively to gradients in H0 , γ m s and H0−p pb . M ,µ λ =0 Sin e weplot ourresultsin the ( H0 2)plane,thediagonalline orrespondsto L , i.e. to no H λ λ <0 λ >0 0 L L L ouplingbetween andtheHiggsboson. Awayfromthisline, in reases,with (resp. ) ∆MA =M M ∆MH =M M above (resp. below) the diagonal. Also, we write 0 A0 − H0 and c H+ − H0. In the plots of the dark matter abundan e, the areas between the two dark lines orrespond to 0.094 < Ω h2 < 0.129 DM regions of the parameter spa e su h that , the range of dark matter energy densities onsistentwithWMAPdata. Inthe rossse tionandgammaray(cid:29)uxplots,thelinesindi ate the areasof the parameter spa e within rea h of the various experiments we take in onsideration. 3.1 Constraints The shaded areas in the plots of Figures (5) and (6) orrespond to regions that are ex luded by the following onstraints: • Va uum stability: Va uum stability (at tree level) demands that λ > 0, 1,2 λ ,λ +λ λ > 2 λ λ . 3 3 4 5 1 2 −| | − (9) p λ < 0 L As a result, negative ouplings, and among others, are largely ex luded. This is the µ >M shaded area in the domains 2 H0 of Figures (5) and (6). • Perturbativity: λ > 4π 1 < λ < 4π i i Strong ouplings | | are ex luded but intermediate ouplings, | | , might be tolerated. The latter orrespond to the regions with horizontal lines. Noti e that the mass µ 2 splitting relative to the s ale is smaller in the high mass regime (see Eq.(3)). For this reason, goingawayfrom the diagonalline in the plotsof Figure(6), werunfasterinto the large oupling regime than in those of Figure (5). The regions ex luded by the strong oupling onstraint is of µ =M ourse symmetri with respe t to the 2 H0 axis. • Charged Higgs s alar: H+ The mass of the harged s alar is onstrained by LEP data to be larger than 79.3 GeV [M30℄. AH0M, re- (cid:29)e ting the dependen e of the annihilation ross se tion on | L|. For H0 W (resp. H0 h), H H ¯bb H H hh ∼ M < M <∼ M 0 0 → (resp. 0 0 → ) is the dominant pro ess. Otherwise, for W H0 h, H H WW,ZZ ∼ ∼ 0 0 → dominate. µ >InMregio,nM3, the ouplings areratherlarge,but nevertheless onsistentwiHthvHa uumWs+taWbi−lity. Sin e 2 H0 W, the dominant ontribution to the ross se tion is given by 0 0 → pro ess, large enough to bring the reli abundan e far below WMAP data. W Region4isbelowthe threshold. Itistheonlyregion onsistentwiththedarkmatterabundan e predi tedby WMAP datain the lowmassregime. The pro ess thatdetermines the reli abundan eof H 0 is again the annihilation through the Higgs. Coannihilation pro esses be ome dominant near the M +M M H M 40 resonan eat A0 H0 ≈ Z. Thisistheoriginofthedipinthe 0reli abundan earound H0 ≈ M 50 GeV ( orresponding to A0 ≈ GeV for our hoi e of mass splittings). Similarly, annihilation near †H H+ ∆MH 0 c oannihilation is suppressedfor our hoi e of . 7 M M /2 the Higgs resonan e generates a se ond dip around H0 ≈ h . There is an island onsistent with M 80 WW WMAPdata,whi hextendsupto H0 ∼ GeV,atwhi hpoint annihilationbe omesimportant and suppresses the reli abundan e. M > M Region 5, (cid:28)nally, orresponds to H0 W. Annihilation into a gauge boson pair is dominant, M M 2 H0 while Higgs pair produ tion is dominantMif 2 <60H00. In all instan es, the reli abundan e is too suppressed to be onsistent with WMAP for H0 GeV. ∼ As expe ted, and as revealedby visualinspe tion, the photon (cid:29)ux plots shares some of the hara - teristi softheabundan eplots. Theonlynewsalientfeatureistheabsen eofadipattheZresonan e. A 0 This is of ourse be ause today, in ontrast with the early universe, all are gone. Existing experiments are not very onstraining but future gamma ray dete tion experiments, su h as GLAST, might severely hallenge the model. As usual we should bear in mind the astrophysi al un ertainties: by hanging the gala ti dark matter density pro(cid:28)le, we an get higher (or smaller) (cid:29)uxes. Theplotofthe rossse tionfordire tdete tionisprettytransparent. Su(cid:30) estonoti e(seese tion σ λ2 2.3)that H0−p at treelevelisproportionalto L. The dominant ontributiontotheelasti s attering µ =M µ M ross se tion is thus zero on the 2 H0 axis while it in reases for larger values of | 2− H0|. σ The dependen e on the Higgs mass (Eq. 8) learly appears when omparing the H0−p plots for M = 120 M = 200 M > M h GeV and h GeV. For H0 W, the one-loop ontribution to the ross se tion W from10−13ex hangeistakenintoa ount. HoweveritdHoesnotaA(cid:27)e tmu htheresultsasit onlyamounts 0 0 for pb. Unless we suppose that the mass of and are nearly degenerate, existing dire t dete tion experiments are not very onstraining. Forth oming experiments however might put a dent on some of the solutions onsistent with WMAP. M M H W 3.3 The high mass regime, ≫ The results of this se tion are shown in Figure (6), left olumn. No new annihilation hannel opens if M H0 is heavierthan the Higgs orgaugebosons. There arethen essentiallytwosort of pro esseswhi h ontrolboththeabundan eandthegammaray(cid:29)ux: theannihilationintotwogaugebosons,dominant µ M if 2 H0, and the annihilation into twoHiggs,whi hdominatesif 2 H0. Coannihilationplays little role. It a(cid:27)e ts a bit the reli abundan e alongthe diagonalbut, even so, it is not the key pro ess to get the WMAP abundan e. The abundan e of dark matter is suppressed over most of the area of the plot be ause of large quarti oupling e(cid:27)e ts on the ross se tions. Strong ouplings are ex luded on a physi al basis, but we found this limit nevertheless useful to understand the interplay between the various pro esses. In thepresentsubse tion,wewill arguethatit ispossibletorea hagreementwithWMAP data,but only to the pri e of some (cid:28)ne tuning between the di(cid:27)erent annihilation hannels. Consider(cid:28)rst theannihilationinto twoHiggsbosons(the dashedline in thegraphsFigure(7)). Its λ =0 µ 0 ross se tion is vanishing for L . For 2 H0 ( orresponding to L ), there is a destru tive µ > M interferen e between the diagrams of Figure (2), whi h is absent if 2 H0. The annihilation into gauge bosons depends on the quarti ouplings between the s alars (see Figure (1)). Indeed, the λ L annihilation through an intermediate Higgs is ontrolled by . Also, the t and u hannels ex hange H 2 diagramsaresensitivetothemassdi(cid:27)eren esbetweenthe omponentsofthe doublet,whi hdepend λ Z λ +λ W 5 4 5 respe tivelyon fortheannihilationinto bosonsandon fortheannihilationinto bosons. λ 0 L If ≃ , thereisno orlittleannihilationinto aHiggs. The relevantdiagramsarethenthequarti A H+ λ =0 0 5 vertex with two gauge bosonsand the t and u hannels with (resp. ) ex hange. If (resp. λ +λ = 0 Z W α2 /M2 4 5 ) the ross se tion into a (resp. ) boson pair is minimal and s ales like W H0. λ = 0 Z 5 If, for instan e, 6 the annihilation into a boson pair re eives a ontribution whi h grows like α2 (M M )2/M4 W W A0 − H0 Z. A similar result holds for the annihilation into a boson pair provided λ +λ = 0 α2 (M M )2/M4 λ2/M2 4 5 6 . Noti e that W A0 − H0 Z ∼ 5 H0. The e(cid:27)e ts of weak isospin breaking H λ 2 1 betweenthe omponentsof isreminis entofwhat happensfortheHiggsin the regimeof strong oupling (the so- alled Goldstone boson equivalen e theorem [31℄). 8 To have an abundan e of dark matter in agreement with WMAP, the mass splittings between the H 2 omponents of must be kept relatively small. First be ause large mass splittings orrespond to large ouplings and se ond be ause the di(cid:27)erent ontributions to the annihilation ross se tion must λ = 0 M µ M M be suppressed at the same lo ation, around L (ie H0 ≃ ≃ A0 ≃ H+ in this ase). This is illustrated in Figure (7) for two di(cid:27)erent mass splittings. The (cid:28)rst plot is for small mass di(cid:27)eren es ∆MA =5 ∆MH =10 ∆MA =10 0 c 0 ( GeVand GeV), these ondonefor(relatively)largersplittings ( ∆MH =50 c GeV and GeV). In the se ond ase, the ross se tion is too large to obtain an abundan e σv pb WMAP onsistent with WMAP, h i ∼ . λ =0 L In theMlimit>of60sm0 all masssplittings and vanishing wehavetherightamountof darkmatter provided H0 GeV.Thisregime orrespondstothenarrowregionaroundthediagonalin Figure ∼ M (6). The abundan e in reases for in reasing H0, but this an be somewhat ompensated by playing with∆tMheAm=ass5splittings,∆wMhi Hh, =as1d0is ussed above, tend to in rease the ross sMe tio>n.8F0o0r instan e, for 0 GeV and c GeV, we get the right reli abundan e for H0 GeV. ∼ There is however a limit to this trend. Indeed, the total annihilation ross se tion of a s alar H 0 parti le, like our , is onstrained by unitarity to be smaller than 4π x σv unit f h iv→0 ≈ M2 r π (10) H0 aresultwhi h,inthe ontextofdarkmatterreli sfromfreeze-out,hasbeen(cid:28)rstputforwardbyGriest andKamionkowski[32℄. Inthepresentmodel,in reasingthemasssplittingdrivestheannihilation ross se tion toward the strong quarti oupling regime. A similar result holds, for instan e, for a weakly intera ting massiveneutrino andidate of dark matter. In this ase, both the neutrino and its harged 1 partner should be kept lighter than, say TeV, sin e their mass omes from Yukawa ouplings to the Higgs (cid:28)eld. (See [33℄ and the dis ussion in [32℄). In our ase, the bulk of the mass of the omponents H µ 2 2 of the doublet omes from the mass s ale , whi h is a prioriMarbit

The Inert Doublet Model: an Archetype for Dark Matter

Similar The Inert Doublet Model: an Archetype for Dark Matter