MMCTP-11-33SLAC-PUB-14584
Singlet–Doublet Dark Matter
Timothy Cohen,
1, 2
John Kearney and Aaron Pierce, and David Tucker-Smith SLAC National Accelerator Laboratory,2575 Sand Hill Rd, Menlo Park, CA 94025 Michigan Center for Theoretical Physics (MCTP)Department of Physics, University of Michigan, Ann Arbor, MI 48109 Williams College Department of Physics,Williams College, Williamstown, MA 01267 (Dated: November 10, 2011)
Abstract
In light of recent data from direct detection experiments and the Large Hadron Collider, weexplore models of dark matter in which an SU (2) L doublet is mixed with a Standard Model singlet.We impose a thermal history. If the new particles are fermions, this model is already constraineddue to null results from XENON100. We comment on remaining regions of parameter space andassess prospects for future discovery. We do the same for the model where the new particles arescalars, which at present is less constrained. Much of the remaining parameter space for bothmodels will be probed by the next generation of direct detection experiments. For the fermionmodel, DeepCore may also play an important role. PACS numbers: 95.25.+d,98.80.Cq,12.60.-i a r X i v : . [ h e p - ph ] N ov . INTRODUCTION A weakly interacting massive particle (WIMP) remains an attractive candidate to explaindark matter. But what exactly is meant by “weakly?” Often, all that is implied is thatannihilation cross sections are parametrically suppressed by the weak mass scale, σ ann ∼ m − W ;the precise mechanism of annihilation may or may not involve the bosons of the electroweaktheory. As an example consider supersymmetry, where annihilations may be mediated byparticles of the supersymmetric sector.In this paper we address the following question: does a strictly weakly interacting particle,i.e., one whose annihilation is controlled by the W , Z and Higgs bosons, remain an attractivedark matter candidate? Such a dark matter candidate would not require the introductionof new mediators, and would thus provide a well-motivated, economical scenario. A particlepossessing full-strength interactions with the Z boson, e.g. a heavy Dirac neutrino, wouldhave a direct detection cross section many orders of magnitude in excess of present limits.A simple remedy is to mix a sterile state with this active state. This mixing yields twoeffects: it reduces the size of the coupling to the gauge bosons and, in the case of fermions,can transform the dark matter from a Dirac particle into a Majorana particle. Together,these variations enable the dark matter to have both an annihilation cross section consistentwith a thermal history and a direct detection cross section that is not yet excluded. Insupersymmetry, the bino may play the role of this sterile state, and can be mixed withthe Higgsinos to achieve a well-tempered neutralino, a possibility emphasized in [1]. For adifferent approach to strictly weakly interacting Dark Matter, see [2].Here, we do not confine ourselves to supersymmetric models, but instead explore moregenerically the consequences of mixing a Standard Model singlet with an active particle.The particular case where the charged state has the quantum numbers of a doublet isworthy of special attention. In this case, the mixing can naturally be provided by arenormalizable coupling to the Higgs field. This fermionic singlet–doublet model has beenpreviously explored in the literature [3–6], and serves to inform us about the viability ofstrictly weakly interacting dark matter in light of recent improvement in direct detectionbounds and the negative searches for the Higgs boson at the Large Hadron Collider (LHC).We also consider the scalar analog of this model, in which a scalar doublet is mixed with areal scalar singlet[7, 8].After imposing a thermal history, much of the parameter space for the fermionic model hasbeen excluded. To avoid tension with direct detection bounds, we find one of the followingexceptional cases must apply:1. The dark matter mass could be close to half the mass of either the Higgs or Z boson.2. Masses in the dark matter sector could be arranged such that co-annihilation isimportant.3. The couplings to the Higgs boson could be small. This does not necessarily implythat the couplings that induce the mixing are small, as there is room for non-trivialcancellations.4. The Higgs boson could be heavy. This can be made consistent with precisionelectroweak constraints without the need for any additional physics, since this modelcan give a large positive contribution to the T parameter in a straight-forward way[5, 6].We explore these possibilities in detail in Sec. II. Recent data from the LHC have had animpact on the fourth possibility. ATLAS [9] and CMS [10] have greatly constrained the rangeof allowed values for the Higgs boson mass, m h . A naive combination of the results fromthese experiments disfavors Higgs boson masses in the range 150 GeV (cid:46) m h (cid:46)
450 GeV.Consequently, to avoid direct detection bounds by making the Higgs boson heavy, i.e., heavierthan ∼
150 GeV, now requires a significant increase in the Higgs boson mass. Motivatedby these findings, we mainly consider two scenarios: a light Higgs boson ( m h = 140 GeV),and a heavy Higgs boson ( m h = 500 GeV). We also comment on an intermediate case( m h = 200 GeV) in which the dark sector could conceivably contribute significantly tothe invisible width of the Higgs boson such that the recent experimental bounds are evaded.Both spin-independent and spin-dependent direct detection searches will be important futureprobes of this model.The physics of the scalar model can be quite different. For instance, because of the possiblepresence of a singlet–Higgs boson mixed quartic, no mixing is necessary to achieve a darkmatter-Higgs boson coupling. While at present this scalar model is less constrained, spin-independent direct detection experiments will probe much of its parameter space in the nearfuture. We examine this model in Sec. III. II. THE SINGLET DOUBLET FERMION MODEL
We consider an extension of the Standard Model consisting of a gauge singlet fermion anda pair of fermionic electroweak doublets. The doublets have a vector-like mass term, andthe neutral components of the doublets mix with the gauge singlet through renormalizablecouplings to the Higgs boson. These fields are odd under a Z symmetry, ensuring the3tability of the lightest state. We denote the singlet as S and the doublets as D and D c : D = νE D c = − E c ν c , (1)with hypercharges − and + respectively, implying that the ν states are electrically neutral.Mass terms and interactions for this model are given by:∆ L = − λDHS − λ (cid:48) D c ˜ HS − M D DD c − M S S + h.c. , (2)where SU (2) doublets are contracted with the Levi-Civita symbol (cid:15) ij and ˜ H ≡ iσ H ∗ .Field re-definitions leave one physical phase for the set of parameters { M S , M D , λ, λ (cid:48) } . Forsimplicity we take them to be real. Discussions of the consequences of introducing a non-zero phase may be found in [4, 5]. As alluded to in the introduction, in addition to beingan interesting candidate for dark matter in its own right, this model is similar to neutralinodark matter in the MSSM (or Split Supersymmetry), in which the sterile Bino mixes withthe electroweak doublet Higgsinos (in the limit where the Wino decouples, M → ∞ ).Consequently, it provides a laboratory where one can potentially gain insight into the physicsof MSSM dark matter. Expanding the Higgs field around its vacuum expectation value, v = 246 GeV, we can writethe neutral mass terms in the basis ψ = ( S, ν, ν c ) as:∆ L ⊃ −
12 ( ψ ) T M ψ + h.c. = −
12 ( ψ ) T M S λ √ v λ (cid:48) √ v λ √ v M Dλ (cid:48) √ v M D ψ + h.c. (3)It can also be instructive to write this in terms of the rotated basis ψ r = ( S, ν c + ν √ , ν c − ν √ ):∆ L ⊃ −
12 ( ψ r ) T M S λ + v λ − v λ + v M D λ − v − M D ψ r + h.c. (4)where λ ± = λ (cid:48) ± λ . The three physical mass eigenstates for the neutral particles are a linear In fact, [11], where a singlet-doublet model was considered (but without a majorana mass for S ), was animportant historical step on the road towards supersymmetric electroweak theories [12]. ν i = ϑ i S + α i ν + β i ν c , ( i = 1 , , . (5)We let ν denote the lightest (Majorana) neutral state — this is our dark matter candidate.The spectrum also contains a Dirac fermion ψ E composed of the fields E and E c with mass M D .As a linear combination of singlet and doublet states, ν generically has a coupling to theHiggs boson and a coupling to the Z . These couplings can provide channels for dark matterannihilation in the early universe through s -channel Higgs and Z boson exchange. If the ν ν h coupling is considerable, this coupling may also yield a large spin-independent cross section.Rotating the Feynman diagram for annihilation of dark matter to quarks via an s -channelHiggs boson produces a diagram that contributes to spin-independent direct detection, asillustrated in Fig. 1. Similarly, a large ν ν Z coupling may yield a large spin-dependentcross section. This is a salient feature of strictly WIMP dark matter — generically, themediators responsible for annihilation ( h and Z , in particular) also couple to protons, whichcan result in observable direct detection signals. However, there do exist additional processesby which the dark matter can annihilate in the early universe, including annihilation directlyto gauge bosons via t -channel exchange of various beyond the Standard Model particles (for m ν > m W ), and co-annihilation [13]. These processes are also illustrated in Fig. 1, and unlikethe s -channel processes have no tree-level direct detection analog. That said, the couplingsinvolved depend on the mixing angles, so there can still be non-trivial correlations betweendark matter annihilation in the early universe and direct detection cross sections.Recent data from direct detection experiments, notably XENON100 [14] and SIMPLE[15], have substantially improved the sensitivity to both spin-independent and -dependentscattering with no evidence for the detection of dark matter. Although DAMA [16], CoGeNT[17], and CRESST [18] have reported possible evidence for dark matter scattering, thisinterpretation seems to be in serious tension with null results from XENON100, CDMS andEDELWEISS [19], and other direct detection experiments, and a coherent explanation forthese possible signals is lacking at present. It is conceivable that a consistent picture mayone day emerge, but in this paper we operate under the assumption that existing data donot indicate signals, and dark matter detection cross sections should lie beneath currentbounds. We agree with the expressions for the masses and mixing angles given in [6] with the caveat that the thirdmass eigenvalue given in their Eq. (A.1) corresponds to the mass of the lightest particle, and the first tothe mass of the heaviest. hν ν qq → (cid:2) hqν qν ( σ SI ) (cid:3) Zν ν qq → (cid:4) Zqν qν ( σ SD ) (cid:5) Eν ν WW → No tree-level direct detection analog (cid:6) ν E W → No tree-level direct detection analog
FIG. 1: Relevant diagrams for annihilation and corresponding direct detection diagrams, whereapplicable. Achieving sufficient dark matter annihilation in the early universe in order to obtainthe measured relic density requires at least one of these diagrams to be significant. In the case of s -channel Higgs or Z boson exchange, this may imply correspondingly large σ SI or σ SD respectively.In the case of t -channel annihilation or co-annihilation, there is not a clear direct detection analog,but the processes will be related through couplings and mixing angles. . Relic density and cross section calculations Given the above discussion it is interesting to ask whether this simple WIMP model alwayshas large direct detection signals, or whether it is possible to have highly suppressed spin-independent cross sections σ SI and/or spin-dependent cross sections σ SD3 . To address thisand related questions we calculate relic densities and direct detection cross sections in micrOMEGAs micrOMEGAs employsthe following values for the scalar nuclear matrix elements: f ( p ) T u = 0 . f ( p ) T d = 0 . f ( p ) T s = 0 . f ( n ) T u = 0 . f ( n ) T d = 0 . f ( n ) T s = 0 . , although recent lattice measurements suggest that smaller values may be more accurate,which would weaken direct detection bounds [21]. The above choices correspond to aneffective Higgs boson–proton coupling of f =0.467, whereas the lattice evaluation correspondsto a f = 0 . ± .
015 [21]. The difference is a decrease of the quoted spin-independent crosssections by roughly a factor of 2.5.It is worth mentioning two approximations employed by micrOMEGAs . First, micrOMEGAs does not include loops effects or the (velocity suppressed) contribution to the spin-independent cross section due to Z exchange (the ( ¯ ψ ν γ µ γ ψ ν )(¯ qγ µ q ) effective operator).While these contributions are generally sub-dominant to those due to Higgs boson exchange,if the ν ν h coupling were to be suppressed, these effects would play a significant role indetermining σ SI . Since the spin-independent cross sections produced by such effects tendbe well below the current bounds [2, 22, 23], we neglect these effects throughout our paper.Rather, spin-independent cross sections (cid:46) − pb should be taken as illustrative of thevery small direct detection cross sections at these points, and not as precise values. Asimilar caveat holds for tiny spin-dependent cross sections. Second, it should be noted that micrOMEGAs accounts only for two-to-two scattering when computing the relic abundance.Three-body processes can be relevant near the opening of a new channel, see e.g. [24]. Forinstance, as m ν → m W , the ν ν → W W ∗ annihilation channel can become particularlyrelevant, but will be neglected in our calculations. Similarly, as m ν → m t , the ν ν → tt ∗ final state can become relevant. This is especially important for dark matter that annihilatesthrough an s -channel Z boson, as the ν ν → Z → tt process does not suffer from p -wavesuppression. Throughout this paper, σ SI is strictly the cross section off of the proton, but for the class of modelsconsidered the spin-independent cross sections off the proton and neutron are equal to an excellentapproximation . Suppression of σ SI and σ SD For certain values of the parameters, it is indeed possible to cancel the tree-level couplingof the dark matter to the Higgs or Z bosons, thereby realizing suppressed σ SI and σ SD respectively. The case of the Z is straightforward: the ν ν Z coupling goes as ( α − β ) inthe notation of Eq. (5). Thus, whenever ν contains approximately equal amounts of ν and ν c the coupling to the Z boson will be small. This occurs for either λ + = 0 or λ − = 0.From Eq. (4), we see that in either case mixing occurs between the S and only one of therotated doublet states, ν c ± ν √ . Consequently all neutral states mix with either ν c + ν √ or ν c − ν √ ,meaning they will contain equal amounts of ν and ν c , and thus the ν ν Z coupling willvanish. λ ± = 0 ⇒ λ (cid:48) = ± λ corresponds to the maintenance of a custodial SU (2) symmetryin the new sector.We now derive the condition for eliminating the coupling between the Higgs boson and ν .For M S < M D , the mass of the lightest neutral particle can be written as: m ν = M S + v f ( M S , M D , λ v, λ (cid:48) v ) . (6)By gauge invariance, the ν ν h coupling is also proportional to f . Thus, a choice ofparameters parameters that satisfies m ν = M S for M S < M D also eliminates the coupling tothe Higgs boson. The following relationship, derived from the characteristic mass eigenvalueequation, cancels the ν ν h coupling: λ (cid:48) crit = − λ M S M D ± (cid:115) − (cid:18) M S M D (cid:19) − . (7)Note, for M S < M D , it is not possible to simultaneously satisfy this condition and one of theconditions λ + = 0 or λ − = 0. In other words, it is impossible in this case to simultaneouslycancel the ν ν h and ν ν Z couplings.An example of these cancellations for M S < M D is shown in Fig. 2. There, we fix M S , M D , and λ , and vary λ (cid:48) . With M S = 200 GeV, M D = 300 GeV and λ = 0 .
36, for mostvalues of λ (cid:48) the relic density is set by annihilation through an s -channel Z . Consequently,for λ (cid:48) ≈ − .
36 = − λ (where the ν ν Z coupling cancels) the annihilation cross sectiondecreases and there is a dramatic increase in the relic density. Meanwhile, aside from thisspecial point, s -channel Higgs boson exchange does not contribute significantly to the darkmatter annihilation. Correspondingly, at the point λ (cid:48) ≈ − .
138 = λ (cid:48) crit where the ν ν h coupling vanishes, the relic density is essentially unaffected. Since σ SI ∼ ( λ (cid:48) − λ (cid:48) crit ) , evena 10% “accident” where λ (cid:48) takes on values close to this critical value can have importantimplications for spin-independent direct detection.8 IG. 2: An example of the suppression of σ SI and σ SD as a function of λ (cid:48) for M S = 200 GeV, M D = 300 GeV and λ = 0 .
36. The critical value for ν ν h cancellation is λ (cid:48) = − . ν ν Z cancellation is λ (cid:48) = ± .
36. The lines shown are σ ( p )SD [green], σ SI [red] and Ω h [blue]. For the alternative case where M D < M S , the analogous analysis reveals the condition for ν ν h cancellation to be λ (cid:48) crit = − λ ⇒ λ + = 0 ( m ν = M D ). The resultant WIMP is ν = √ ( ν c + ν ), and has suppressed coupling to both the Higgs and Z boson. However, thedark matter particle retains a full-strength coupling to the charged dark sector fermion andthe W boson. Because the E fermion also has mass M D , there is significant contributionto dark matter annihilation from co-annihilation with the charged state. As this couplingstrength is fixed, to achieve the correct relic density the value of M D is constrained to M D (cid:38) M D ∼ . h . So, there is thepossibility that m ν (cid:38) M S = 2 TeV and λ = − λ (cid:48) = 0 .
2, the correctrelic density is achieved for M D = 1 . σ SI and σ SD are heavily suppressedas the ν ν h and ν ν Z couplings are small, and m ν − m ν ∼ Note that, in fact, the E will be slightly heavier than the WIMP due to Coulombic radiative corrections. λ coupling allows a wider range of (all heavy) M D values.In models that have built-in relations between λ and λ (cid:48) , such as the MSSM, there is a questionas to whether these cancellations are still possible. In the MSSM, we find cancellations andan appropriate relic density are indeed simultaneously realizable, but only for small valuesof tan β . In particular, the λ + = 0 condition just discussed is achieved for tan β = 1 (itis impossible to achieve λ − = 0 due to the relative signs between off-diagonal couplings inthe MSSM), and for M < µ (analogous to M S < M D ) we find the cancellation of the darkmatter coupling to the Higgs boson and the correct relic density only for values of tan β (cid:46) M < µ, M , the highdegree of symmetry between the off-diagonal entries in the neutralino mass matrix resultsin the condition for canceling the dark matter-Higgs boson coupling being the identical forany M > M .Returning now to the singlet-doublet model, for a small ν ν h coupling (and σ SI ) a sizeable ν ν Z coupling (and σ SD ) might still be required to achieve sufficient dark matter annihilationin the early universe, or vice-versa. We now investigate the general size of the direct detectioncross sections for a dark matter relic density of 0 . ≤ Ω h ≤ . σ range determinedby the combination of the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) andother data on large scale structure [25]. In what follows we will investigate the differencesin the dark matter phenomenology associated with a light versus a heavy Higgs boson. Thiswill provide us with a sense of the likelihood of discovery of this particular model as directdetection experiments increase in sensitivity in the coming years, and of the fate of fermionicWIMP dark matter in general. C. Light Higgs boson m h = 140 GeV
Some previous studies of this model have focused on the possibility of new dark statescharged under SU (2) L generating a large contribution to the oblique T parameter [6]. Fora relatively light Higgs boson with m h = 140 GeV, such a large contribution is undesirable.We require the contribution to the T parameter from the dark sector lie in the range: − . ≤ ∆ T ≤ .
21 (8)Exact expressions for ∆ T can be found in [5]. As in [6], we neglect the new physicscontributions to S and U , which are significantly smaller than the contributions to T . The10ange given above represents the shift in ∆ T required by the new physics to ensure thatthe oblique parameter values for the model remain within the 68% ellipse in the ( S, T )plane. We perform a random scan of the parameter space with 0 GeV ≤ M S ≤
800 GeV, 80 GeV ≤ M D ≤ − ≤ λ ≤ ≤ λ (cid:48) ≤
2. We permit relatively large values of λ and λ (cid:48) to avoid imposing any theory bias. However, we note that restricting to smaller couplings − ≤ λ ≤ ≤ λ (cid:48) ≤ . ≤ Ω h ≤ . ≤ m ν ≤
500 GeV. Points with m ν much less than 40 GeV wouldtypically lead to an excessive contribution to the invisible width of the Z . This contributioncan be turned off by setting λ (cid:48) = ± λ . However, doing so leaves Higgs boson exchange as theonly annihilation process in the early universe, and for these small values of m ν it turns outthat Higgs boson exchange alone cannot yield a realistic relic density.Plots of σ SI and σ ( p )SD against m ν are shown in Fig. 3, along with exclusion limits fromXENON100 [14] and SIMPLE/Super-K/IceCube [15, 27, 28]. The exclusion curves shownassume a local dark matter density of ρ = 0 . . A recent evaluation suggests asomewhat higher density [29], which would give rise to proportionally stronger bounds. Itshould be noted that the spin-dependent limits shown for m ν (cid:38) m W are model-dependentindirect detection limits, which assume certain dark matter annihilation channels in theSun and Earth. The limits shown for m W (cid:46) m ν (cid:46) m t are taken directly from Super-K’spaper [27]. They assume dark matter annihilations to τ τ (presumably neglecting neutrinooscillations). For the points in Fig. 3 near these limits, the dark matter has sizable ν ν Z coupling, and will exhibit dominant annihilation in the Sun and Earth via an s -channel Z to bb , which dominates in the v → τ τ as well,but annihilation to W boson pairs is tiny due to velocity suppression. So, while the limitsare representative, they are not precise. If m ν (cid:38) m t , the hard limits shown from IceCubeassume annihilation to W W . In fact, in this region, the points nearest the limits will be onceagain be characterized by Dark Matter that annihilates predominantly via an s -channel Z ,although in this case to tt , in the Sun and Earth. The tops will decay to produce fairly hard W bosons, so in this case the limits shown are representative of the actual model-dependentlimits but the actual limits will be slightly weaker.In addition, the lack of signal events in XENON100 also implies new direct detection limitson σ ( p,n )SD . Based on the fact that the σ SI limits have improved by approximately a factor of10 between XENON10 [30] and XENON100, we use the XENON10 spin-dependent limits toproject that the σ SD limits will be O (10 − pb) and O (10 − pb) for scattering off of neutrons This ellipse is larger than the restrictive 39.35% ellipse shown in [26]. IG. 3: Plots of spin-independent [top] and spin-dependent [bottom] cross sections against darkmatter mass for m h = 140 GeV. Points satisfy the thermal relic density constraint. Shaded regionsrepresent σ SI exclusion limits from XENON100 [14] [top] and combined σ ( p )SD exclusion limits fromSIMPLE, Super-K and IceCube (hard) [15, 27, 28] [bottom]. Exclusion curves assume a local darkmatter density of ρ = 0 . . σ ( p )SD /σ ( n )SD (cid:39) . Z to protons and neutrons). Thus, we expect XENON100 limitson σ ( n )SD to be competitive with those from SIMPLE and Super-K on σ ( p )SD for m ν (cid:46)
200 GeV(for higher masses, the significantly stronger limits from IceCube become relevant). Whileat first glance it may appear that much of the parameter space is out of the reach of bothpresent or near future direct detection, it is important to consider the correlation between σ SI and σ SD . This is represented in Fig. 4, which depicts the allowed points in the σ ( p )SD vs. σ SI plane.We see that in a large portion of the parameter space permitted by constraints on Ω h , pointshave either a significant spin-independent or spin-dependent cross section. For heavier darkmatter (with m ν ≥
85 GeV), the majority of points lie in either a horizontal band at the topof the plot or a vertical band to the right. The horizontal band consists of points for whichthe relic density is predominantly set by annihilation via s -channel Z exchange, and thesepoints correspondingly have the largest spin-dependent cross sections. The vertical bandcontains points for which the dark matter annihilates predominantly via s -channel Higgsboson exchange, resulting in larger spin-independent cross sections. The horizontal band isat lower values of σ ( p )SD for 175 GeV ≤ m ν ≤
500 GeV than for 85 GeV ≤ m ν ≤
160 GeVdue to the opening of the ν ν → tt channel. The ν ν → Z → tt channel is significant, soits opening permits a smaller ν ν Z coupling, yielding smaller spin-dependent cross sections.The location of the vertical band is largely unchanged as the top threshold is crossed becausethe ν ν → h → V V (where V is W or Z ) channel dominantes the ν ν → h → tt channelfor m ν ≥ m t . Notably, both spin-independent and spin-dependent searches are vital forprobing this parameter space, as while many points have small σ SI or σ SD , relatively fewexhibit suppression of both.Points that do have relatively small σ SI and σ SD (those that do not clearly fall into aband) are those for which co-annihilation and t -channel annihilation to gauge bosons areparticularly significant in the early universe. This permits smaller couplings of the darkmatter to the Higgs and Z bosons, producing smaller spin-independent and -dependentcross sections. In general points outside of, but near to, the bands are those for which t -channel processes are significant. The masses of other dark sector particles are close enoughto m ν that t -channel exchange is not heavily suppressed, but sufficiently separated that co-annihilation is not relevant in the early universe. As the masses of the dark sector particlesbecome increasingly degenerate, t -channel annihilation processes increase in significance,and eventually co-annihilation becomes relevant. The points further from both bands arethose for which t -channel annihilation and co-annihilation are the dominant processes insetting the relic density, so σ SI and σ SD can be small (and in general must be to avoidover-annihilation). 13 IG. 4: Scatter plots of σ ( p )SD against σ SI depicting points with the correct relic density. Shown are m ν ≤
70 GeV [top] and m ν ≥
85 GeV [bottom]. At bottom, the blue/light gray points represent85 GeV ≤ m ν ≤
160 GeV and green/darker gray represent 175 GeV ≤ m ν ≤
500 GeV; thesemass ranges are chosen to avoid regions where
W W ∗ and tt ∗ final states are expected to becomeimportant (see text for discussion). In both plots, gray indicates points excluded by current directdetection limits. ≤ m ν ≤
70 GeV (the upper plot in Fig. 4), there is no clear banding structure.In this mass regime, lower spin-independent and spin-dependent cross sections can beachieved due to the presence of the Higgs and Z boson poles. This allows the relic densityto still be set by s -channel Higgs or Z boson exchange but with significantly smaller ν ν Z or ν ν h couplings to compensate for the enhancement in the annihilation cross section dueto the small propagator. The contribution to the cross section from the propagator in theearly universe goes as ( s − m h/Z ) − (cid:39) (4 m ν − m h/Z ) − , whereas for direct detection thepropagator contribution goes as m − h/Z . As a result, enhancement of the annihilation crosssection near a pole does not imply a similar enhancement of direct detection cross sections.Points exhibiting this enhancement are numerous; the dark matter need not be exactly onresonance to take advantage of a reduced s -channeled propagator. Furthermore, the energiesof the dark matter particles follow a Boltzmann distribution, so for m ν (cid:46) m Z / , m h / Z boson that is enhanceddue to the presence of an s -channel pole in the early universe. This allows smallercouplings to the Higgs and Z bosons, and suppressed spin-independent and spin-dependent cross sections respectively.2. The dark sector masses are sufficiently close that dark matter annihilation in the earlyuniverse is predominantly due to t -channel processes or co-annihilation. For many suchmodels, direct detection is unobservable.3. The dark matter coupling to the Higgs boson is small, suppressing σ SI . The relic densityis set by Z exchange, which generically leads to large spin-dependent cross sections.Many of these models may be ruled out within the coming years by direct detectionexperiments. In particular, models with suppressed σ SI and 85 GeV ≤ m ν ≤
160 GeVin which relic density is set by s -channel Z exchange are already beginning to beexcluded by spin-dependent direct detection experiments.In each of these scenarios, some tuning of the parameters is required. In the first case, it isnecessary to have m ν (cid:46) m Z / m h /
2. For case 2, the masses of the dark sector particlesmust be nearly degenerate, ∆ m (cid:46) T fo (cid:39) m/
20, and σ SI and σ ( p )SD must also be fairly small.This usually requires M S (cid:39) M D , and small λ and λ (cid:48) . In the final case, for a given value of λ , λ (cid:48) must be tuned to be approximately λ (cid:48) crit . At present, the required a tuning is mild, atthe level of approximately ten percent; setting λ (cid:48) to within ∼
10% of λ (cid:48) crit will suppress σ SI by a factor of O (10 ).The allowed parameter space will become even more restricted with imminent developments15n dark matter detection experiments. A one-ton Xe experiment could potentially improvebounds on spin-independent cross section by orders of magnitude. For points with suppressed σ SI , improvements in experiments that probe σ SD will be very important. Projected limitsfrom the COUPP experiment [31] are on the order of σ SD ∼ − − − pb for dark mattermasses between 10 and 500 GeV. In addition, experiments other than those that focus ondirect detection of dark matter may begin to play a role. For instance, recent work hasshown that bounds on monojet events the LHC on σ SD are rapidly becoming comparable todirect detection bounds [32]; however, these currently only apply if the operator mediatingdirect detection is effectively parameterized by a contact operator at the LHC. Here, whereZ boson exchange is relevant, a preliminary investigation indicates that the collider boundsare significantly degraded. A more promising probe is the DeepCore extension to IceCube,which should also provide stringent limits on σ SD for dark matter in this mass range [33].A recent study [34] has found that the expected atmospheric background rate for muonevents DeepCore is approximately 2.3 events per year. This informs the estimate thatthe dark matter annihilations in the Sun must yield approximately 10 muon events peryear for discovery. We can thus approximate the capture and annihilation rates in the sunnecessary to produce this required number of events, and consequently the spin-dependentcross sections that we expect to be probed by DeepCore. We rescale points A and D from[34], accounting for the dominant mass dependent effects. Doing so, we find that for a darkmatter candidate annihilating primarily to τ τ and bb (for m W (cid:46) m DM (cid:46) m t ) or tt (for m DM (cid:38) m t ), the approximate σ ( p )SD required for discovery rises from ∼ × − pb for a 100GeV dark matter candidate to around 10 − pb for a 500 GeV dark matter candidate. Thisis comparable to, although slightly less optimistic than, the projected limits given in [35],which assume a lower energy threshold will be attainable. For points with these relativelyhigh spin-dependent cross sections, annihilation rates are sufficiently high that the WIMPsin the sun are in equilibrium.If no hint of dark matter is seen at DeepCore, we expect the experiment will severely limit theavailable parameter space for the fermionic singlet-doublet model in the case of m ν ≥ m W .For m W ≤ m ν < m t , points with suppressed σ SI , and relic density and neutrino spectrumset by annihilation via an s -channel Z (to W W in the early universe and to τ τ , bb in theSun and Earth - those in the horizontal blue band of Fig. 4) could soon be readily excludedby a combination of direct detection experiments sensitive to spin-dependent couplings andDeepCore. In the case of m ν (cid:38) m t , points with suppressed σ SI , with correct relic densityand neutrino spectrum set by annihilation to tt via s -channel Z exchange (the horizontalgreen band of Fig. 4) generally exhibit spin-dependent cross sections that are comparableto (if not slightly greater than) the expected DeepCore limits after one year of running.Consequently, for m ν ≥ m W , it may soon be the case that scenario 2 is the only viableoption for avoiding experimental constraints. For m ν < m W , the situation is less clear:16here are a number of points with lower σ SD , and the annihilation of lighter dark matterwill yield a softer neutrino spectrum, so the prospects for detection will depend significantlyon the precise muon detection energy threshold achieved by DeepCore. Direct detectionexperiments will still be important in this range.One clear take-away from this analysis is that a combination of spin-independent and spin-dependent experiments will be necessary to effectively probe the variety of dark mattermodels; neither one will be sufficient on its own to eliminate the majority of the parameterspace for this model of dark matter. Furthermore, given the correspondence between directdetection and annihilation in the early universe, measurements from both types of experimentmay be vital to determine the properties of a dark matter particle. D. Heavier Higgs bosons
We now consider how the situation changes when we increase the mass of the Higgs boson.Within the Standard Model, recent ATLAS [9] and CMS [10] results disfavor most of therange 150 (cid:46) m h (cid:46)
450 GeV. For moderate values of the Higgs boson mass, however, LHCproduction cross sections not much below the Standard Model rate are allowed. In the modelwith mixed singlet-doublet fermion dark matter, there is the possibility that the Higgs bosondecays invisibly into pairs of neutral Z -odd fermions with an appreciable branching ratioallowing evasion of the ATLAS and CMS 95% CL limits. However, for a Higgs boson inthis mass range, invisible decays compete with decays to W W , so achieving even an (cid:39) T ) with the additional requirement that the Higgs boson has a ≥
10% branching ratio todark sector particles, we find no allowed points. This is true for Higgs bosons in the entireATLAS/CMS exclusion range as well.A Higgs boson heavy enough to evade LHC searches, m h ∼ >
450 GeV, requires a large positivecontribution to the T parameter from new physics in order to be consistent with precisionelectroweak data. As has been pointed out in [5, 6], it is possible for this correction to arisefrom the effects of the dark sector itself. To explore the viable parameter space for a heavyHiggs, we repeat the scans that produced Figs. 3 and 4, this time with m h = 500 GeV, andwith the dark sector’s contribution to the T parameter constrained to be in the range0 . < ∆ T < . . (9)We scan over the same parameter ranges as for the m h =140 GeV case. While we assume17 IG. 5: Plots of spin-independent [top] and spin-dependent [bottom] cross section against darkmatter mass for m h = 500 GeV. Exclusion contours are as in Fig. 3. the new ∆ T contribution arises from the dark sector itself, it is possible to imagine a morebaroque model where the additional new physics contributes to ∆ T . In this case, an increasein m h can generically be used to suppress σ SI . We do not focus on this case here as it isphenomenologically straightforward.In Fig. 5 we show the results for spin-independent and spin-dependent cross sections versus18ark matter mass. At tree-level the spin-independent cross section depends on the ν ν h coupling, which can be arbitrarily small given the potential cancellations discussed inSec. II B. For m ν < m W , Z exchange regulates the relic abundance. For dark matter massesabove m W , Higgs boson mediated annihilations to W W can instead set the abundance,but the possibility of using the Z coupling alone to do so persists in this regime as well.Regardless of whether m ν lies below or above m W , it is therefore possible to tune the ν ν h coupling away and still achieve a realistic relic abundance. Although the great majorityof points have spin-independent cross sections within roughly two orders of magnitude ofcurrent limits, points with tiny spin-independent cross sections consequently show up in thefull mass range from ∼ −
170 GeV.An important feature of both plots in Fig. 5 is that no points show up for m ν > m t . Ourrequirement that the dark sector produces a large ∆ T (which goes parametrically as ( λ − λ (cid:48) ) ) forces λ and λ (cid:48) to have very different magnitudes, which in turn means that the ν ν Z coupling will generally be significant. Since Z boson mediated annihilations to t ¯ t do not sufferfrom p -wave suppression, achieving the correct relic density when the ν ν → Z → tt channelis open requires a small ν ν Z coupling. Hence, it is impossible to simultaneously satisfy therequirement of large ∆ T and the constraint on the relic density, thereby prohibiting pointswith m ν > m t . If we were to relax our requirement that ∆ T come from the dark sector,smaller values of the ν ν Z coupling would be possible and the m ν > m t region would open.Next we turn our attention to the second plot in Fig. 5. For m ν < m W , where annihilationthrough an s -channel Z sets the abundance, the ν ν Z coupling required to obtain the correctrelic density gets smaller as m ν approaches m Z / s -channel propagator. This results in smaller spin-dependent cross sections. When m ν gets sufficiently close to m Z /
2, the propagator enhancement becomes so large that itbecomes impossible to find λ and λ (cid:48) values such that ∆ T is large enough enough while ν ν Z is simultaneously small enough to acheive a realistic relic abundance. This explainswhy no points are realized for m ν (cid:46)
50 GeV for both plots in Fig. 5. Analogously to the m t cutoff discussed in the previous paragraph, the cutoff at around 50 GeV is tied to our ∆ T requirement.At larger values of m ν the spin-dependent cross section is rather large, σ ( p )SD ∼ > × − pb,for points where the abundance is set by the coupling to the Z . Note that for these largermasses, the W W and ZZ final states are also available. Therefore, a non-trivial contributionto annihilation from Higgs boson exchange is possible, and a realistic abundance may befound for smaller Z couplings. This yields points with smaller spin-dependent cross sections,although these cross sections are non-vanishing because the ∆ T requirement prevents the ν ν Z coupling from being extremely suppressed.19 IG. 6: Scatter plots of σ ( p )SD against σ SI depicting points with the correct relic density, for m h = 500GeV. Shown are m ν ≤
70 GeV [top] and m ν ≥
85 GeV [bottom]. In both plots, gray representspoints already excluded by direct detection experiments. Z exchange necessarilyregulates the abundance for masses below m W — this places a minimum value on the spin-dependent cross section of σ ( p )SD ∼ × − pb. Even if there is a delicately canceled ν ν h coupling, the spin-dependent cross section will be large enough to be seen at upcomingexperiments. The second plot in Fig. 6 shows that this is also true for larger m ν values. Forthis higher mass region the effect is more pronounced, with spin-independent cross sectionssmaller than 10 − pb requiring spin-dependent cross sections (cid:38) × − pb. Consequently,many of these points are excluded by current experimental bounds. In this high mass region,if the Higgs boson coupling is suppressed, there is no pole enhancement for Z -mediatedannihilation so we must regulate the abundance with a “full-strength” Z coupling, producinglarger spin-dependent cross sections for points with suppressed σ SI than in the low massregion.As for the case of a light Higgs boson, DeepCore and direct detection experiments shouldbe sufficiently sensitive to probe the points with m W ≤ m ν < m t and suppressed σ SI .Furthermore, since in this case there is a floor on σ SD for m ν < m W , these experimentscould also have interesting implications for lighter dark matter. Consequently, in this regimethe most difficult points to probe may be those for which m ν (cid:39) m t . As the ν ν → Z → tt ∗ annihilation channel begins to turn on, a smaller ν ν Z coupling can be allowed (and thus asmaller σ SD ), implying that these points are more difficult to probe.In summary, we see that for m h = 500 GeV, the vast majority of points will be probedthrough their spin-independent cross sections once the experiments improve their reach byabout two orders of magnitude. Even points with unusually small spin-independent crosssections should be probed through their spin-dependent cross sections in the near future.These conclusions are sensitive to our assumption that the dark sector produces a large∆ T . III. THE SCALAR MODEL
We now consider the analogous model where the fermions are replaced with scalars. Asimple candidate model of dark matter, it displays a broader range of phenomenology thanthe simplest model of scalar WIMP dark matter where the abundance of a real singlet scalaris set via a quartic coupling to the Higgs field [36–38]. While scalar singlet dark matter is notyet ruled out, future direct detection experiments may soon begin to eliminate this simplestmodel for lighter Higgs boson masses. Consequently, it is worthwhile to consider whetherextending such a model to include an additional doublet can potentially allow for evasion of21uture direct detection bounds.We introduce a real scalar singlet S and a complex doublet Φ (with hypercharge 1 /
2) andthe Lagrangian∆ L = D µ Φ † D µ Φ − m D Φ † Φ + 12 ( ∂ µ S ) − m S S − g ( S Φ † H + h.c.) − λ S S H † H − λ ( H † H )(Φ † Φ) − λ (cid:0) (Φ † H ) + h.c. (cid:1) − λ (Φ † H )( H † Φ) , (10)where SU (2) indices are contracted within parentheses, and the doublet isΦ ≡ φ +1 √ ( φ + iA ) . (11)We neglect other possible allowed couplings containing only dark sector particles that are notrelevant to the dark matter phenomenology, e.g. S (Φ † Φ). For non-zero trilinear coupling g ,the singlet and the doublet mix when the Higgs boson takes on its vev. The resulting darkmatter is: X = cos θ S + sin θ φ . (12)We denote the orthogonal neutral scalar as X .In contrast to the fermion case, annihilations through the Higgs boson can be present withoutinducing mixing, for instance due to the presence of the S ( H † H ) coupling. In the presenceof non-zero mixing, the coupling to the Higgs boson is given by: L ⊃ − ( λ S v cos θ + λ v sin θ − g sin θ cos θ ) X h ≡ − A eff X h, (13)where we have introduced the effective coupling of the neutral doublet scalar to the Higgsboson λ ≡ λ + 2 λ + λ .The dominant processes that contribute to early universe annihilation in this model (for m X > m W ) are shown in Fig. 7. For masses beneath the W -boson mass, the relic abundanceis essentially determined by the s -channel Higgs boson exchange diagram, with coupling A eff and a b ¯ b final state.It is instructive to examine the region of correct thermal relic density in the sin θ − A eff plane (the upper panel of Fig. 8). In this figure, we have shown the allowed region for threechoices of Higgs boson mass, m h = 115 , ,
250 GeV with m X = 95 GeV and m D = 125GeV. The λ i = 0, for i = 1 , ,
3. Setting these couplings to zero ensure the absence ofany co-annihilation, a possibility we will revisit below. Scalars contribute less to the T parameter than fermions with similar strength couplings, so we do not require internal ∆ T to compensate for heavier Higgs boson masses. Moreover, for λ = 2 λ a custodial SU (2)22 hX X W + W − (cid:2) X X W + W − (cid:3) φ + X X W + W − FIG. 7: The dominant annihilation processes for singlet–doublet scalar dark matter in the regime m X > m W . is maintained in the new sector, such that ∆ T vanishes for all of the points shown in theseplots (as λ = 2 λ = 0). At sin θ = 0, for m S < m D , we recover the “pure singlet model” (remove all terms withΦ from Eq. (10)) and its attendant value of | A eff | . Moving away from sin θ = 0, otherprocesses begin to contribute to X X → W + W − . The dominant effect is due to the direct4-point vertex (the middle diagram in Fig. 7); the t -channel exchange is usually smaller. Thepresence of these additional diagrams requires a new value of A eff to maintain the correctrelic abundance. Notably, there exists a value of sin θ for which the correct relic densityis maintained only via the gauge interactions, and the contribution from the Higgs bosonvanishes ( A eff =0). At this point, the spin-independent detection cross section plummets.This explains the deep trough in the lower panel of Fig. 8. Once again, it should be notedthat where tiny cross sections appear here (and elsewhere in this section), loop inducedeffects which we have neglected in our numerical studies would be relevant.We now discuss the interplay between the contributions from Higgs boson exchange and the4-point diagram to X X → W + W − in more detail. The interference between these twodiagrams can be constructive or destructive. This depends on two factors: the sign of A eff and the size of the Higgs boson mass. The latter (in combination with the dark mattermass) sets the sign of the s -channel propagator. Examining the lower panel of Fig. 8, thereis a plateau of relatively large σ SI values. There the relic density is set dominantly via s -channel Higgs boson exchange. The 4-point diagram makes a subdominant contribution thatinterferes destructively with the Higgs diagram. Consequently, the | A eff | must be increasedto maintain the correct relic abundance. In the top panel, this can be seen for the lower(upper) branches of the curve for m h = 115 ,
140 (250) GeV. Due to the increased size of | A eff | , direct detection cross sections are greater than those found in the model with nodoublet at all. For sufficiently large values of sin θ , there is another possibility exhibiting This custodial symmetry can be made manifest as follows. Write Ω H = ( ˜ HH ) which transforms under SU (2) L × SU (2) R as Ω H → L Ω H R † . The Φ doublet has the same quantum numbers as the Higgsdoublet, so we can have an analogous Ω Φ that transforms identically. Then, for λ = 2 λ , we can write∆ L ⊃ − gS tr(Ω † Φ Ω H ) − λ tr(Ω † Φ Ω Φ )tr(Ω † H Ω H ) − λ [tr(Ω † Φ Ω H )] , and the custodial symmetry is explicit. IG. 8: In the top panel, we show the coupling to the Higgs boson, A eff (see Eq. (13)), neededto achieve the correct relic density as a function of the mixing angle sin θ . Regions are shown forthree different Higgs boson masses: m h = 115 GeV, m h = 140 GeV, and m h = 250 GeV. Thedark matter mass is fixed, m X = 95 GeV, and all λ i = 0, i=1,2,3. In the bottom panel, we plot σ SI vs. sin θ , from top to bottom, m h = 115 , ,
250 GeV. The shaded region corresponds to theXENON100 exclusion for this mass [14]. s -channel Higgs boson contribution that interferes destructively. In the toppanel, this corresponds to the segment that extends from | A eff | = 0 up to the tip of thecurve. In the lower panel, this segment extends from the trough up to values of peak crosssection at large sin θ . The tip of the curve is characterized by points at which the destructiveinterference between the four-point and the s -channel Higgs boson diagrams is most severe.In this region, other processes such as t -channel charged scalar exchange, annihilation viaan s -channel Higgs boson to heavy quarks (for instance, tt for m ν > m t ) or annihilationto Higgs boson pairs (for m ν > m h ) can play significant roles. Finally, there is a regionwhere the interference is constructive . In the upper plot, this segment runs from sin θ = 0(where only Higgs boson exchange contributes) out to (sin θ, A eff ) = (0.35, 0), where onlythe four-point diagram contributes. In the lower plot, this explains the lower left portion ofthe triangular region.To summarize, the presence of additional contributions to the X X → W + W − annihilationchannel can either increase or decrease the direct detection cross section with respect to adark matter candidate that relies on annihilation via a Higgs boson alone. XENON100 hasalready begun to probe this model for lower values of the Higgs boson mass. To explorethe achievable direct detection cross sections in this model, we performed a scan over allparameters with the ranges: 10 GeV ≤ m X ≤
500 GeV, 80 GeV ≤ m D ≤ | λ i | ≤ ≤ g ≤ v . We imposed the same ∆ T requirements as in fermion case with a light Higgsboson, and required that the sum of each scalar mass and the pseudoscalar mass be greaterthan m Z (to avoid Z -width constraints). Note, there is a possibility that the dark mattermight be quite light, (cid:46) few GeV, consistent with current direct detection bounds. In thiscase, the phenomenology is essentially that of the pure singlet, coupled to a Higgs boson.This window was studied recently in [39], see also [40].The result is shown in Fig. 9. Superimposed on this plot is a scan over the pure singletmodel. In the singlet model, all dynamics are controlled by the Higgs–dark matter coupling.The precise measurement of the dark matter relic abundance determines λ S , which in turndetermines σ SI , resulting in the thin band in the figure. The addition of the doublet allowsdeviations from this curve. Points approximately along the curve are those whose relicabundance is set by the coupling to the Higgs boson, A eff , of Eq. (13). For m X > m W otherchannels can now contribute to annihilation in the early universe, and the firm connectionbetween (Higgs boson mediated) direct detection and cosmology is broken. Nevertheless,many of the points in the plot will be probed by a future generation of direct detection As alluded to previously, this prohibits very few points due to the difficulty of achieving large ∆ T contributions from scalars. However, we include this requirement for consistency. tt and hh thresholds, wherenew final states open up. Some of the points with the lowest σ SI are due to the minimumexhibited in Fig. 8 (where four point diagram X X → W + W − sets the relic density). FIG. 9: σ SI vs. m X for the scalar singlet doublet model. The Higgs boson mass is m h = 140GeV. Superimposed is the narrower band that corresponds to the pure singlet model. Also shownis the exclusion region from XENON100 [14]. σ SI can also be suppressed if co-annihilation is relevant. Since the dark matter is a real scalar,it does not possess diagonal couplings with the Z boson. Any mass splitting between A and X which is (cid:38)
100 keV avoids an enormous (lethal) Z -boson mediated spin-independentcross section. If the splitting is close to this value, the scattering is inelastic [41]. Sincewe are considering g ∼ > O (GeV), it is unlikely that such a small splitting will be realized.However, it is possible that the pseudo-scalar may have mass sufficiently close to the scalarso that this off-diagonal coupling is relevant for setting the relic density in the early universevia co-annihilation. Similarly, the charged scalar, φ + may co-annihilate with the dark mattervia the W boson.To demonstrate the possible relevance of co-annihilation, we again examine the A eff − sin θ λ i = 0. For concreteness, we choose acombination of λ i to allow the possibility that m φ + ≈ m X , but we leave the pseudo-scalarmass fixed at m A = m D = 125 GeV. The dark matter mass is again fixed at 95 GeVand m h = 140 GeV. With respect to the analogous upper plot in Fig. 8, we notice thepossibility of points within the interior of the curve. These are precisely the points whereco-annihilation and t -channel exchange are relevant, and a smaller coupling to the Higgsboson may be accommodated. For direct detection, the lower panel of Fig. 10, there is thepossibility of points with reduced detection cross-sections and small sin θ .Finally, we note that a (nearly) pure doublet scalar can yield the correct relic density. Allthat is needed is a tiny splitting ( (cid:38)
100 keV) between the scalar and pseudo scalar state toavoid the enormous Z -boson mediated direct detection signals. This can be accomplished viaa tiny mixing with the singlet. In this case, the right relic density is achieved for m D (cid:39) IV. CONCLUSIONS
We have explored models of strictly weakly interacting dark matter; specifically, dark matterwhose annihilation, spin-independent and spin-dependent cross sections are controlled bythe W , Z and Higgs bosons. Since the neutral component of a pure electroweak doubletwith full-strength coupling to the Z -boson has a fatally high direct detection cross section,we have considered the case in which these electroweak doublet couplings are diluted bymixing with a sterile state. This singlet–doublet model serves as a proxy for strictly weaklyinteracting dark matter. Other similar models are possible, such as mixing active darkmatter in other representations of SU (2) L with a Standard Model singlet. However, thesinglet-doublet model is particularly appealing since it allows mixing between the active andsterile states to arise from renormalizable couplings to the Higgs field. We have analyzedthis type of model for the case where the new dark sector particles are fermions, and wherethey are scalars. These models subsume other models of weakly-interacting mixed singlet-doublet dark matter, such as a mixed Bino-Higgsino state in supersymmetric extensions ofthe Standard Model.We find that, for the case of both the fermion and scalar, current direct and indirect detectionexperiments are already beginning to probe the parameter space consistent with the requiredthermal relic density of 0 . ≤ Ω h ≤ . IG. 10: The coupling to the Higgs boson, A eff (see Eq. (13)), needed to achieve the corrrect relicdensity as a function of the mixing angle sin θ (top). The Higgs boson and dark matter massesare fixed: m h = 140 GeV and m X = 95 GeV. Unlike Fig. 8 we allow λ i (cid:54) = 0, i = 1 , , m A = 125 GeV (see text for further discussion). Points interior to the curve illustrate the possiblerelevance of co-annihilation. At bottom, we plot σ SI vs. sin θ . When compared to Fig. 8, there arepoints with reduced σ SI . The shaded region corresponds to the XENON100 exclusion for this mass[14]. Z bosons required to achieve sufficient dark matter annihilation in theearly universe imply correspondingly large spin-independent and/or spin-dependent crosssections, respectively.For a fermionic singlet–doublet WIMP, the prospects for discovery or exclusion are veryoptimistic. While it is possible to suppress either σ SI or σ SD in the context of this model, therequirement of sufficient dark matter annihilation in the early universe makes suppressingboth cross sections extremely difficult. Notably, this means that both σ SI - and σ SD -based darkmatter detection experiments will be vital for discovering or excluding this class of models.As spin-independent and spin-dependent limits improve, for instance with the advent ofa one-ton XENON experiment and the DeepCore extension to IceCube, the most viableoptions for evading direct detection bounds are limited if the Higgs boson is light: either theannihilation in the early universe is enhanced by a small s -channel propagator (due to theHiggs or Z boson poles) or coannihilation occurs.A heavy Higgs boson is also an option for avoiding σ SI limits. However, recent ATLASand CMS limits have constrained “heavy” to imply m h (cid:38)
450 GeV for a Standard Model-like Higgs boson. In this case, the large contribution to the T parameter from the Higgsboson will require cancellation for consistency with electroweak precision constraints. Sucha contribution could come from the dark sector. In the case of the fermionic singlet–doublet model this implies spin-dependent cross sections well within the reach of futureexperiments.The scalar model also exhibits sizable spin-independent cross sections in much of theparameter space. If the model is not discovered in the near future, coannihilation or enhanced s -channel propagators again provide options for avoiding direct detection limits. For scalars,however, the is another option: σ SI can be heavily suppressed while the correct relic densityis achieved by a sizable four-point XXV V (with V as W or Z ) coupling. There is noappreciable σ SD in this case. So, direct detection will be very difficult, but indirect detectionsignals (such as neutrino flux from dark matter annihilations to gauge bosons) may beobservable.For a strict WIMP, the possibilities for avoiding direct and indirect detection are beginningto be constrained. Furthermore, these possibilities tend to involve some fine tuning. Hence,if the dark matter is strictly weakly interacting, the prospects for detection or exclusion inthe near future are extremely promising. 29 cknowledgments The work of T.C. was supported in part by DOE Grants [1] N. Arkani-Hamed, A. Delgado, and G. F. Giudice, Nucl. Phys.
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