Simplified Models of Mixed Dark Matter
CCALT 68-2870
Simplified Models of Mixed Dark Matter
Clifford Cheung ∗ and David Sanford † California Institute of Technology, Pasadena, CA 91125, USA
We explore simplified models of mixed dark matter (DM), defined here to be astable relic composed of a singlet and an electroweak charged state. Our setup de-scribes a broad spectrum of thermal DM candidates that can naturally accommodatethe observed DM abundance but are subject to substantial constraints from currentand upcoming direct detection experiments. We identify “blind spots” at which theDM-Higgs coupling is identically zero, thus nullifying direct detection constraints onspin independent scattering. Furthermore, we characterize the fine-tuning in mix-ing angles, i.e. well-tempering, required for thermal freeze-out to accommodate theobserved abundance. Present and projected limits from LUX and XENON1T forcemany thermal relic models into blind spot tuning, well-tempering, or both. Thissimplified model framework generalizes bino-Higgsino DM in the MSSM, singlino-Higgsino DM in the NMSSM, and scalar DM candidates that appear in models ofextended Higgs sectors.
PACS numbers: 95.35.+d, 12.60.Jv ∗ cliff[email protected] † [email protected] a r X i v : . [ h e p - ph ] N ov CONTENTS
I. Introduction 3II. Model Definitions 5A. Singlet-Doublet Fermion 7B. Singlet-Doublet Scalar 10C. Singlet-Triplet Scalar 13III. Model A: Singlet-Doublet Fermion DM 15A. Exclusion Plots (General) 15B. Exclusion Plots (Thermal Relic) 19C. Exclusion Plots (Thermal Relic and Marginal Exclusion) 23IV. Model B: Singlet-Doublet Scalar DM 26A. Exclusion Plots (General) 27B. Exclusion Plots (Thermal Relic) 28V. Model C: Singlet-Triplet Scalar DM 32A. Exclusion Plots (General) 32B. Exclusion Plots (Thermal Relic) 35VI. Conclusions and Future Directions 37Acknowledgments 38References 38
I. INTRODUCTION
The gravitational evidence for dark matter (DM) is very strong, but its precise particleproperties remain elusive. Long ago, laboratory experiments excluded the simplest modelsof weakly interacting massive particle (WIMP) DM which predicted DM-nucleon scatteringvia spin independent Z boson exchange. Today, direct detection experiments, particularlyXENON100 [1, 2] and LUX [3], have become sensitive to the large class of theories thatpredict spin independent (SI) DM-nucleon scattering mediated by the Higgs boson. Futureexperiments such as XENON1T [4] and LZ [5] will have improved sensitivities and theirresults will have even stronger implications. In light of the discovery of the Higgs bosonat the LHC [6, 7], present and future limits on Higgs-mediated scattering can be recast interms of the effective parameter space defined by the DM mass, M χ , and its coupling to theHiggs boson, c hχχ .What are the natural values for M χ and c hχχ ? In the absence of additional theory input,these parameters are arbitrary – there is simply no reason why DM should be accessiblethrough direct detection. For example, DM could be completely inert and thus imperviousto non-gravitational probes. While many theories offer a DM candidate as part of a newphysics framework, the only general impetus for couplings between DM and the StandardModel (SM) is cosmological in nature: if DM is a thermal relic, then it is reasonable for itbe thermalized with the SM in the early universe. In minimal extensions of the SM, DMcouples via electroweak gauge interactions and/or via the Higgs portal. Going beyond thissetup requires more elaborate models that entail richer structures like dark force carriers[8–15] or other mediators [16, 17]. Thus, an important question for present and upcomingexperiments is the status of thermal relic DM, broadly defined.The literature provides a litany of well-motivated theories of DM, both within and outsideof broader new physics frameworks, though by far the most popular is neutralino DM insupersymmetry (SUSY). While supersymmetric theories are a useful benchmark for models,analyses of SUSY DM are often colored by theory biases and disparate connections tounrelated experimental data. For instance, under specific model assumptions, issues ofnaturalness are still taken as a hard constraint on the parameter space of SUSY DM models.Another example is the discrepancy in g − µ parameter, influencing the perceived viability of SUSY DM. However, given the currentsensitivity of experiments, overarching theory assumptions like SUSY are not required topare down the parameter space – experiments will do so.Instead it can be fruitful to take the approach of simplified models: effective theoriesthat describe a broad class of theories but are tailored to extract maximal information fromexperimental results. Simplified models of DM have appeared in the literature in a numberof guises. In the case of minimal DM [21, 22], pure gauge representations were considered.Others have studied simplified models of a singlet and colored particle, a.k.a. the effective“bino-squark” system [23–25]. For thermal relics, it was found that many of these modelsare bowing under the weight of present experimental constraints from direct detection andthe LHC. Recently, there has also been growing interest in effective operator descriptions ofDM [26–31]. Modulo the well-known limits of their validity [27, 29, 32–34], these effectivetheories have been used to determine quite general bounds on DM from colliders.In the present work, we consider simplified models of mixed DM, defined here as renormal-izable theories of fermion or scalar DM comprised of a singlet and an additional electroweakcharged state. Generically, the singlet and charged states will mix after electroweak symme-try breaking. As a consequence, the DM possesses annihilation channels inherited from itselectroweak charged component, and thermal relic DM can be achieved with an appropriatedegree of mixing. In a sense, this simplified model is a generalization of the “well-tempered”neutralino [35–37] found most commonly in focus point SUSY scenarios [38–41] to a more di-verse set of DM charges and spins. By enumeration, there exist three renormalizable, gaugeinvariant simplified models of mixed dark matter: fermion singlet-doublet ( Model A ), scalarsinglet-doublet (
Model B ), and scalar singlet-triplet (
Model C ). More complicated modelsnecessarily include additional degrees of freedom or higher dimension operators to inducemixing. We evaluate the viability of models based upon current limits at LUX [3] and theexpected reach of XENON1T [4]. Our main conclusions are: • In light of current LUX limits and the projected reach of XENON1T, we have de-termined the viable parameter space of thermal relic DM in Fig. 9 (singlet-doubletfermion), Fig. 15 (singlet-doublet scalar), and Fig. 19 (singlet-triplet scalar). We havecast our results in terms of the parameter space of physical quantities: the DM mass, M χ , and the DM-Higgs coupling, c hχχ . • Model A: Singlet-Doublet Fermion . LUX stringently constrains this model except inregions with relative signs among the DM-Higgs Yukawa couplings. Given the overallYukawa coupling strength y defined in Sec. II A, XENON1T will eliminate all of theviable parameter space with y > ∼ . < ∼ y < ∼ .
1, but they require < ∼
10% tuning in order to accommodate the observed relic density. • Model B: Singlet-Doublet Scalar . LUX places modest limits on this model but leavesmuch of the parameter space still open. Given the overall DM-Higgs quartic couplingstrength λ defined in Sec. II B, XENON1T will eliminate essentially all of the param-eter space for λ ≤
0. For λ >
0, blind spots appear but a typical tuning of < ∼ • Model C : Singlet-Triplet Scalar . LUX places relatively weak limits on the thermal relicparameter space because triplets annihilate very efficiently in the early universe. Giventhe overall DM-Higgs quartic coupling strength λ defined in Sec. II C, XENON1T willstrongly constrain models with λ ≤
0, requiring a tuning of < ∼
1% to match theobserved DM abundance except for nearly pure triplet DM. For λ >
0, the couplingis suppressed and there are regions with all fine-tunings alleviated to > ∼ Models A,B, and C , and discuss general aspectsof the thermal relic abundance and the DM-nucleon scattering cross-section. For the latter,we derive analytic formulas indicating when the DM-Higgs coupling identically vanishes. InSec. III, Sec. IV, and Sec. V, we examine the parameter spaces of Models A,B, and C , bothin generality and for the case of thermal relic DM which saturates the observed abundance.Current bounds from LUX and projected reach of XENON1T bounds are shown throughout.Finally, in Sec. VI we present a discussion of our results and concluding thoughts. II. MODEL DEFINITIONS
In this section we explicitly define our simplified models. Throughout, we focus on thecase of a singlet mixed with a non-singlet, which is a natural generalization of many modelsof theoretical interest. Of course, mixing among non-singlet states is also viable, but inthis case the preferred mass range for DM is typically in the multi-TeV range, with a lowerbound of several hundred GeV.Furthermore, we restrict our discussion to models with renormalizable interactions. Whilenon-renormalizable interactions are of course allowed, they are competitive with renormal-izable operators only when the cutoff is so low that the effective theory is invalid. Indeed,mixing induced by higher dimension operators is highly suppressed by the electroweak sym-metry breaking scale divided by the cutoff scale. Even in scenarios where such a theory isvalid and produces the appropriate relic density, a large degree of well-tempering is requiredfor even marginal mixing, disfavoring it for this study. Restricting to renormalizable modelslimits us to three simplified models of mixed DM: • Model A:
Majorana fermion DM composed of a Majorana fermion singlet and Diracfermion doublet with hypercharge Y = 1 / • Model B:
Real scalar DM composed of a real scalar singlet and complex scalar doubletwith hypercharge Y = 1 / • Model C:
Real scalar DM composed of a real scalar singlet and real scalar triplet withhypercharge Y = 0.In principle, one can consider singlets which are Dirac fermions or complex scalars, but thesetheories have more degrees of freedom and the analysis does not qualitatively change.Throughout, we take Ω χ to be the relic abundance for the DM predicted by a thermal his-tory. In all cases, the relic abundance and direct detection cross-section are calculated with micrOMEGAs 2.4.5 [42–44] using model files generated using FeynRules 1.6.11 [45].We are predominantly interested in the parameter space that saturates the DM abundanceobserved by Planck [46], Ω DM h (cid:39) . ± . . (1)Obviously, the DM relic abundance will drastically vary if the cosmological history is non-thermal, and in such cases there is no requirement that DM couples to the SM at all. Ourinitial analysis for each model will highlight the location of the Ω χ = Ω DM line in parameterspace, along with regions of Ω χ < Ω DM and Ω χ > Ω DM . Our more detailed analysis will berestricted to the Ω χ = Ω DM region, examining the behavior of other observables within thethermal relic context.In all of our models, the DM particle is either a real scalar or Majorana fermion. Con-sequently, SI scattering through the Z boson is inelastic and can be ignored. On the otherhand, SI scattering through the Higgs boson is mediated via mixing between the singletand and non-singlet components, though it is suppressed at direct detection “blind spots”:regions of parameter space at which the coupling of DM to the Higgs boson vanishes iden-tically. As noted in [47], the existence of blind spots depend sensitively on relative signsamong the DM parameters. In the blind spot parameter space, the SI scattering DM-nucleoncross-section is zero at tree-level. Radiative corrections are typically sub-dominant in theparameter space except very close to the blind spot cancellation points. However, a properevaluation of these higher order effects may become important for the status of DM if directdetection experiments do not observe SI scattering. We also neglect radiative correctionsto the masses, which are important for the phenomenology of minimal DM [21] but aresub-dominant when large mixing effects are introduced.In the spirit of low energy Higgs theorems, we can straightforwardly compute the couplingof DM to the Higgs via a Taylor expansion of the DM mass term with respect to the Higgsvacuum expectation value (VEV), v . For Majorana fermion DM, we obtain − L = 12 M χ ( v + h ) χχ (2)= 12 M χ ( v ) χχ + 12 ∂M χ ( v ) ∂v hχχ + O ( h ) , (3)where v = 246 GeV. Eq. (3) implies a dimensionless DM-Higgs boson coupling given by c hχχ = ∂M χ ( v ) /∂v . For real scalar DM the same formula applies except with the replacement M χ ( v ) → M χ ( v ). In this case we define a dimensionful coupling a hχχ = ∂ (cid:2) M χ ( v ) (cid:3) /∂v whichis proportional to the DM mass, though for ease of discussion we will sometimes use theeffective dimensionless coupling, c hχχ = a hχχ /M χ instead. As discussed in Ref. [47], the blindspot is defined by c hχχ = 0, computed by taking the ∂/∂v derivative of the characteristiceigenvalue equation for the DM mass.The DM-Higgs coupling maps straightforwardly onto limits from direct detection. Thespin independent DM-nucleon cross-section is mediated by Higgs exchange and scales as σ SI ∝ µ m h × c hχχ , (4)where we use m h = 125 . µ and c hχχ varywith M χ . In our region of interest, though, we require M χ > ∼
100 GeV to avoid LEPconstraints on additional charged states which accompany the DM particle. Thus, µ isapproximately equal to the nucleon mass. Meanwhile, we can compare the σ SI computedfrom theory with the limits from LUX and XENON1T. These limits have complicated massdependence at low mass due to reduced efficiency in observing low energy events. However,for M χ > ∼
100 GeV, the cross-section bounds σ SILUX , σ
SIX1T rise linearly with M χ because theevent rates are proportional to the DM density, which falls with 1 /M χ . Throughout, we usethe lattice values for the quark content of the nucleon from [48].We will be interested in models which evade present and projected limits from direct de-tection while accommodating a thermal relic abundance consistent with observation. How-ever, these theories may require tuning for either or both of these aspects. For directdetection, the DM-Higgs coupling is of the schematic form c hχχ = a + b , where a and b depend on different sets of model parameters. We can characterize the degree of tuningrequired for a blind spot cancellation by ξ BS = | a + b || a | + | b | . (5)Blind spot tuning grows more severe as ξ BS →
0. This effectively captures the tuninginherent in c hχχ → a and b have thesame sign, no cancellation in c hχχ is possible, and ξ BS = 1.Meanwhile, achieving the correct thermal relic abundance may require fine-tuning ofmixing angles, or “well-tempering”, since the DM must be the appropriate admixture ofsinglet and non-singlet state. Heuristically, well-tempering is correlated with the existenceof small mass splittings in the DM multiplet relative to the dimensionful input parameters.Concretely, if the mass squared matrix is an N × N matrix M , then the severity of well-tempering is linked to the relative size of the traceless component of M (the mass splittings)relative to the trace of M (the overall mass scale). Hence, we define a parameter thatdescribes the well-tempering in the mixing angle, ξ WT = (cid:18) N Tr[ M ]Tr[ M ] − (cid:19) , (6)which is related to the variance of the M matrix. Indeed, ξ WT is precisely the fractionalstandard deviation of the eigenvalues of M . In the limit that the entire DM multiplet isexactly degenerate, the mixing angle is very fine-tuned, M ∝ and ξ WT → A. Singlet-Doublet Fermion
In this section we define a simplified model for fermionic mixed DM comprised of a Majo-rana singlet and Dirac doublet of SU (2) W × U (1) Y . Here we have introduced a Dirac doubletin order to cancel anomalies and allow for a bare mass term. A priori, the Dirac doubletcan have arbitrary hypercharge, but to induce mixing via renormalizable interactions, wedemand that Y = ± / Model A
Field Charges Spin S ( ,
0) 1/2 D ( , − /
2) 1/2 D ( , /
2) 1/2 ,and the general renormalizable Lagrangian is − L
Model A = 12 M S S + M D D D + y D SHD + y D SH † D + H.c. , (7)where we have dropped the kinetic terms for simplicity. In our notation, D and D havethe same quantum numbers as ˜ H d and ˜ H u of the MSSM, respectively. We will sometimesparametrize the Yukawa couplings in polar coordinates, y D = y cos θ (8) y D = y sin θ . (9)Throughout, we work in a convention where M S and M D are positive but y D and y D have indefinite sign. Note, however, that using a parity transformation S → − S , we cansimultaneously flip the signs of y D and y D , so only their relative sign is physical. Likewise,a parity transformation D → − D or D → − D flips the signs of M D and either y D or y D , respectively, so only a singlet sign among the three parameters is physical. Afterelectroweak symmetry breaking, S mixes with D and D , and this mixing simultaneouslycontrols the thermal relic density of the DM as well as its coupling to the Higgs boson.The lightest neutral state, χ , is stable DM, and is defined as a linear combination of the y = 0.3tan θ = 2 M ( GeV ) χ N S y = 0.3tan θ = 2 M ( GeV ) χ N S20.010.001 (TeV) D M ( T e V ) S M (a) . . y = 0.3tan θ = -2 M ( GeV ) χ N S0.001 2 (TeV) D M ( T e V ) S M (b) y = 0.3tan θ = 10 M ( GeV ) χ N S20.010.001 (TeV) D M ( T e V ) S M (c) y = 0.3tan θ = -10 M ( GeV ) χ N S20.010.001 (TeV) D M ( T e V ) S M (d) FIG. 1.
Mass and mixing angle contours in singlet-doublet fermion DM for y = 0 . and tan θ = ± , ± . Shown are contours of the DM mass M χ in GeV (red dot-dashed) and the DM-singletmixing angle squared N S (blue dotted). interaction eigenstates, χ = N S S + N D D + N D D , where N S + N D + N D = 1. Thereis also an electrically charged state in the spectrum which we denote by χ ± . Two heavierneutral states are also present, but they have little effect except when coannihilation controlsthe thermal relic density.Eq. (7) parametrizes a broad class of models which have been discussed in the literature.Most prominently, it describes bino-Higgsino DM in the MSSM, with y = g (cid:48) / √ θ = β .It also describes singlino-Higgsino mixing in the NMSSM, where mixing is controlled by thesuperpotential term W = λSH u H d , so y = λ and θ and β are offset by π/
2. Singlet-doubletMajorana DM has also been discussed in great detail in Ref. [49]. y = 1.5tan θ = 2 M ( GeV ) χ N S20.90.1 (TeV) D M ( T e V ) S M (a) y = 1.5tan θ = -2 M ( GeV ) χ N S2 (TeV) D M ( T e V ) S M (b) y = 1.5tan θ = 10 M ( GeV ) χ N S2 (TeV) D M ( T e V ) S M (c) y = 1.5tan θ = -10 M ( GeV ) χ N S20.01 (TeV) D M ( T e V ) S M (d) FIG. 2.
Mass and mixing angle contours in singlet-doublet fermion DM for y = 1 . and tan θ = ± , ± . Contours shown are the same as in Fig. 1.
Next, we present an analytic derivation of the relevant properties of χ . In the basis( S, D , D ), the neutral mass matrix is M = M S √ y D v √ y D v √ y D v M D √ y D v M D . (10)The characteristic equation of the mass matrix is0 = (cid:0) M χ − M D (cid:1) ( M S − M χ ) + M D y D y D v + 12 M χ (cid:0) y D + y D (cid:1) v = (cid:0) M χ − M D (cid:1) ( M S − M χ ) + 12 y v ( M χ + M D sin 2 θ ) , (11)0where we are interested in the smallest eigenvalue, M χ . Since y labels the overall magnitudeof the Yukawa couplings, its sign is unphysical and the characteristic equation depends onlyon y . On the other hand, the sign of y D /y D is physical, and is represented by the sign oftan θ (or equivalently sin 2 θ ). The DM-Higgs coupling, c hχχ = ∂M χ ( v ) /∂v , can be computedexactly by differentiating the characteristic equation in Eq. (11) with respect to ∂/∂v andsolving, yielding c hχχ = − y v ( M χ + M D sin 2 θ ) M D + 2 M S M χ − M χ + y v / . (12)We define “blind spot” for spin independent direct detection by all parameter points whichsatisfy c hχχ = 0, so M χ + M D sin 2 θ = 0 , (13)as discussed in [47]. Because M S and M D are positive definite, a blind spot can only occurswhen sin 2 θ < ξ WT = (cid:113) M D − M S ) + y v + 2 y v [ M D + 2 M S + 3 M D M S sin 2 θ ]2 M D + M S + y v (14) ξ BS = (cid:12)(cid:12)(cid:12)(cid:12) M χ + M D sin 2 θM χ + M D | sin 2 θ | (cid:12)(cid:12)(cid:12)(cid:12) , (15)corresponding to the amount of tuning required for a properly well-tempered thermal relic,and to the amount of tuning required to reside sufficiently close to a blind spot cancellation,respectively.Finally, let us consider the DM mass and mixing angles in the parameter space of singlet-doublet DM. Figs. 1 and 2 show contours of M χ and DM-singlet mixing angle squared, N S ,in the ( M D , M S ) plane for tan θ = ± , ±
10 for y = 0 . y = 1 .
5, respectively. For y = 0 . M χ is equal to the lower of M S or M D in most of the parameter space, only deviating near M S ≈ M D . Significant mixing occurs when M S ≈ M D , and away from this region the DMquickly approaches a pure singlet or pure doublet. For y = 1 . M χ is significantly offsetfrom both M S and M D throughout the plotted range for tan θ >
0, with a larger offsetaround M S ≈ M D . A sizable region even exists with M χ <
100 GeV for low values of M S and/or M D . For tan θ <
0, however, the offset is more modest. As indicated, the degree ofmixing typically much greater for y = 1 . y = 0 . B. Singlet-Doublet Scalar
Next, we define another simplified DM model with singlet-doublet mixing, only withscalars rather than fermions. As in the case of fermionic DM, singlet-doublet mixing forscalars requires that the doublet have hypercharge Y = 1 /
2. However, in this case a seconddoublet is not required to either accommodate a bare doublet mass or cancel anomalies.The field content is1 M ( GeV ) χ N S λ = 0A = 10 GeV (TeV) D M ( T e V ) S M (a) M ( GeV ) χ N S λ = 0A = 100 GeV (TeV) D M ( T e V ) S M (b) . . . M ( GeV ) χ N S λ = 0A = 1 TeV . N o So l u ti on (TeV) D M ( T e V ) S M (c) FIG. 3.
Mass and mixing angle contours for singlet-doublet scalar DM for λ = 0 and A =10 , , GeV.
Contours shown are the same as in Fig. 1.
Model B
Field Charges Spin S ( ,
0) 0 D ( , /
2) 0 ,with a corresponding Lagrangian − L
Model B = 12 M S S + M D | D | + 12 λ S S | H | + λ D | D | | H | + λ (cid:48) D (cid:12)(cid:12) HD † (cid:12)(cid:12) + 12 λ (cid:48)(cid:48) D (cid:104)(cid:0) HD † (cid:1) + h.c. (cid:105) + A (cid:2) SHD † + h.c. (cid:3) . (16)Here we drop kinetic terms, along with all interactions involving only S and D , which areDM self-interactions which do not effect DM annihilation or direct detection. For simplicity,we assume positive M S and M D , although strictly speaking this is not necessary becausethe singlet and doublet acquire masses after electroweak symmetry breaking. Note that byapplying a parity transformation S → − S , we can flip the sign of A ; consequently this signis unphysical.Scalar dark matter theories have a long pedigree. Scalar singlet DM is often considered themost minimal DM candidate, and has been studied extensively [50–53]. Current bounds onscalar singlet models with the correct relic density require M S > ∼
100 GeV except for a smallregion of viable parameter space for 50 GeV < ∼ M S < ∼
65 GeV [54], but XENON1T reachexpected to cover M S > ∼
10 TeV. Scalar doublet DM also has been studied extensively, mostoften in the case of a two-Higgs doublet model where only one Higgs receives a VEV [55],often called the “inert doublet model” [56]. Mixed singlet-doublet scalar models have alsobee considered previously [49, 57, 58], though most such studies have considered a sub-setof the possible phenomenology motivated by grand unification.Paralleling the fermion case, singlet-doublet mixing for scalars produces three real neutralscalars and one charged scalar. Mixing among states is induced by the A term after elec-troweak symmetry breaking. We can work in the basis of real neutral scalars, ( S, D R , D I ),where D R and D I are the real and imaginary components of the neutral component of D .2The mass squared matrix is M = M S + v λ S Av Av M D + v ( λ D + λ (cid:48) D + λ (cid:48)(cid:48) D ) 00 0 M D + v ( λ D + λ (cid:48) D − λ (cid:48)(cid:48) D ) . (17)We assume the absence of CP violating couplings, so D I cannot mix with either D R or S .Focusing on mixed DM, we choose a very tiny but negative value of λ (cid:48)(cid:48) D to ensure that thedoublet which mixes has a smaller mass term. With this restriction we define the lightestmixed state as the DM particle χ = N S S + N D D R , where N S + N D = 1.DM mixing is induced by the upper left 2 × (cid:16) M χ − ˜ M S (cid:17) (cid:16) M χ − ˜ M D (cid:17) − v A , (18)where ˜ M S = M S + v λ S and ˜ M D = M D + v ( λ D + λ (cid:48) D + λ (cid:48)(cid:48) D ). The two eigenvalues of themass-squared matrix are12 (cid:34) ˜ M S + ˜ M D ± (cid:114)(cid:16) ˜ M S − ˜ M D (cid:17) + 4 v A (cid:35) , (19)with the smaller eigenvalue corresponding to M χ . As shown earlier, the associated DM-Higgscoupling is given by the derivative of M χ with respect to v , a hχχ = 12 v ( λ S + λ D + λ (cid:48) D + λ (cid:48)(cid:48) D ) − vA + v (cid:16) ˜ M S − ˜ M D (cid:17) ( λ S − λ D − λ (cid:48) D − λ (cid:48)(cid:48) D ) (cid:114)(cid:16) ˜ M S − ˜ M D (cid:17) + 4 v A . (20)In our analysis we will make use of the simplifying limit λ S = λ D = λ and λ (cid:48) D = λ (cid:48)(cid:48) D = 0. Inthis limit, a hχχ simplifies to a hχχ = λv − vA (cid:113) ( M S − M D ) + 4 v A . (21)In analogy with the case of fermion DM, we define the blind spot region by the condition a hχχ = 0. However, in the scalar case it is complicated by the presence of Higgs couplingsto the pure states. For fermionic DM, mixing is induced by Yukawa terms, leading to acorrelation between mixing strength and the Higgs coupling. In the scalar case, however, forany degree of mixing, the direct quartic couplings can be modified to create a blind spot.The enhancement or suppression of a hχχ depends on the sign of λ . For positive λ , a blindspot can occur, and interestingly, this is the sign preferred in general by considerations oftree-level vacuum stability. In particular, if λ is too negative then the potential may containunbounded from below directions at large field values.As in the case of singlet-doublet fermion DM, we can characterize the amount of tuningrequired to accommodate a thermal relic abundance and evade direct detection. Again using3 M ( GeV ) χ N S λ = 0 κ = 0.3 (TeV) T M ( T e V ) S M (a) M ( GeV ) χ N S λ = 0 κ = 3 (TeV) T M ( T e V ) S M (b) FIG. 4.
Mass and mixing angle contours for singlet-triplet scalar DM for λ = 0 and κ = 0 . , . Contours shown are the same as in Fig. 1.
Eq. (5) and Eq. (6), we define (using only the 2 × ξ WT = (cid:113) ( M S − M D ) + 4 v A ( M S + M D + λv ) (22) ξ BS = (cid:12)(cid:12)(cid:12)(cid:12) λv (cid:113) ( M S − M D ) + 4 v A − vA (cid:12)(cid:12)(cid:12)(cid:12) | λv | (cid:113) ( M S − M D ) + 4 v A + 2 vA , (23)to be the tuning measures for well-tempering and blind spot cancellations, respectively.Last of all, we consider the DM mass and mixing angles in singlet-doublet scalar DM.Fig. 3 shows contours of M χ and N S in the ( M D , M S ) plane, with A = 10 , , λ S = λ D = λ (cid:48) D = λ (cid:48)(cid:48) D = 0. For sufficiently small masses in the A = 100 GeV and A = 1 TeV cases, one of the eigenstates becomes tachyonic and thus the region is excluded.The well-mixed scenario occurs near M S ≈ M D , with the degree of mixing dropping rapidlyexcept for the very large value of A = 1 TeV. Likewise, except for A = 1 TeV the masscontours are restricted to M χ ≈ M S or M χ ≈ M D except for very close to the M S ≈ M D line. Allowing for non-zero quartic DM-Higgs couplings does not qualitatively change themass and mixing contours, but simply shifts their positions. C. Singlet-Triplet Scalar
Lastly, we consider scalar DM comprised of a mixed singlet and triplet. For the sake ofsimplicity, we consider a real triplet, which necessarily carries zero hypercharge. The fieldcontent of this model is4
Model C
Field Charges Spin S ( ,
0) 0 T ( ,
0) 0 ,with a corresponding Lagrangian, − L
Model C = 12 M S S + M T tr (cid:0) T (cid:1) + 12 λ S S | H | + λ T tr (cid:0) T (cid:1) | H | + κSH † T H , where once again we have dropped kinetic terms and interactions involving only S and T . After electroweak symmetry breaking, the singlet and triplet mix via the dimensionlessquartic interaction κ . Note that by applying the parity transformation, S → − S , we canfreely flip the sign of κ , so its sign is unphysical.While models involving triplet scalar DM have not received the same level of attentionas those with singlet or doublet scalar DM, both pure triplet [21, 59] and mixed singlet-triplet [60, 61]. However, as for mixed singlet-doublet scalar DM, previous studies haveprimarily focused on a sub-set of parameter space motivated by grand unification.The mass matrix for the singlet-triplet scalar case resembles the mixed sub-matrix forthe singlet-doublet scalar, with the substitution A → − κv/ S, T ) basis, M = (cid:18) M S + v λ S − κv − κv M T + v λ T (cid:19) . (24)After diagonalization, the DM particle is χ = N S S + N T T , where N S + N T = 1. Likewise,the mass-squared eigenvalues become12 (cid:34) ˜ M S + ˜ M T ± (cid:114)(cid:16) ˜ M S − ˜ M T (cid:17) + 14 κ v (cid:35) , (25)where ˜ M S = M S + λ S v / M T = M T + λ T v /
2. However, due to the v -dependence ofthe mixing term for the singlet-triplet case, the same substitution does not apply to a hχχ ,which has the form a hχχ = 12 v ( λ S + λ T ) − κ v + v (cid:16) ˜ M S − ˜ M T (cid:17) ( λ S − λ T ) (cid:114)(cid:16) ˜ M S − ˜ M T (cid:17) + κ v . (26)The contribution to the Higgs coupling from the mixing term in the singlet-triplet case isroughly twice as large relative to the singlet-doublet case for models with an equivalent massspectrum and mixing. For λ S = λ D = λ , this reduces to a hχχ = λv − κ v (cid:113) ( M S − M T ) + κ v . (27)As in the singlet-doublet case, cancellations in the DM-Higgs coupling only occur if λ is positive, with the same implication that vacuum stability favors positive values of λ and5thus cancellation. From Eq. (5) and Eq. (6), we define tuning measures for blind spotcancellations and well-tempering, ξ WT = (cid:113) ( M T − M D ) + κ v ( M S + M T + λv ) (28) ξ BS = (cid:12)(cid:12)(cid:12)(cid:12) λv (cid:113) ( M S − M T ) + κ v − κ v (cid:12)(cid:12)(cid:12)(cid:12) | λv | (cid:113) ( M S − M T ) + κ v + κ v . (29)As before, we plot the DM mass, M χ , and singlet mixing angle squared, N S , as a functionof the parameter space in Fig. 4. Qualitatively, our results are similar to the singlet-doubletcase. However, there is a major quantitative difference, which is that mixing effects areminimal away from M S ≈ M D even at very large values of the coupling, κ = 3. As we willsee in Sec. V, this implies substantially different experimental constraints on singlet-doubletversus singlet-triplet scalar DM. III. MODEL A: SINGLET-DOUBLET FERMION DM
To begin, we will analyze the full four dimensional parameter space of singlet-doubletfermion DM, ( M S , M D , y D , y D ), imposing no constraints beyond the definition of the theoryin Sec. II A. We will display experimental constraints and regions consistent with Ω χ = Ω DM as a function of the bare masses M S and M D , fixing ( y, θ ) to several characteristic values.We will then focus on the subspace of thermal relic DM, fixing one of the model parameters( M D , M S , or θ , depending on the plot) to accommodate Ω χ = Ω DM . Lastly, we will studythe thermal relic scenario further restricted to the parameter space residing exactly at thepresent (and future) limits of direct detection experiments. In particular, for this analysiswe will fix y to σ SI = σ SILUX or σ SI = σ SIX1T , corresponding to the space of models which are“marginally excluded” by LUX or XENON1T, respectively.
A. Exclusion Plots (General)
First, we consider the unconstrained parameter space, focusing on the position of theΩ χ = Ω DM line. A pure singlet Majorana fermion does not couple to the SM at the renor-malizable level, so its thermal relic abundance is typically very large. Meanwhile, a puredoublet has Ω χ = Ω DM for M D (cid:39) χ > Ω DM for M D > ∼ χ < Ω DM for M D < ∼ N D and N D . In principle, anni-hilation can also occur via Higgs exchange, and this will be enhanced at large y . However,annihilation via the Higgs is a p-wave suppressed process for Majorana fermions, and thussub-dominant to gauge boson processes unless y is very large.In Fig. 5, we have plotted constraints in the ( M D , M S ) plane, fixing y = 0 . θ = ± ±
10 (bottom). For M S < ∼ χ = Ω DM for all parameter combinations. This is the casebecause yv (cid:28) M S,D , requiring M S ≈ M D for any significant level of mixing. For M S > ∼ Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded y = 0.3tan θ = 2 σ ( zb ) SI L U X E xc l uded (TeV) D M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ y = 0.3tan θ = -2 σ ( zb ) SI XEN O N E xc l uded (TeV) D M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded L U X E xc l uded y = 0.3tan θ = 10 σ ( zb ) SI (TeV) D M ( T e V ) S M (c) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded L U X E xc l uded y = 0.3tanθ = -10 σ ( zb ) SI (TeV) D M ( T e V ) S M (d) FIG. 5.
Direct detection prospects for y = 0 . and tan θ = ± , ± . Shown are contours of σ SI in zb (green, various styles) together with current bounds from LUX (blue shaded) and projectedreach at XENON1T (gold shaded). Away from the line consistent with the observed relic density(teal dashed), we compute LUX and XENON1T bounds assuming that the DM relic density isequal to the observed one due to a non-thermal cosmology. the thermal relic line asymptotes to M D (cid:39) χ = Ω DM is not possible for M D > y = 0 . σ SI depends sensitively on the sign of tan θ .For tan θ > .
01 zb < ∼ σ SI < ∼
10 zb throughout most of the regionshown, and exhibits no blind spot behavior. Conversely, for tan θ <
0, the maximum value7of σ SI is reduced, and a blind spot occurs where σ SI vanishes (the minimum value shown is0 .
001 zb). The position of the blind spot changes for different values of tan θ , and is locatedat M S + M D sin 2 θ ≈
0, roughly consistent with the blind spot condition of Eq. (13).For positive tan θ , the thermal line is constrained by LUX for M S > ∼ . θ = 2and M S > ∼ θ = 10. For negative tan θ , however, LUX provides no boundfor tan θ = − θ = − θ , XENON1T constrains the thermal scenario well intothe nearly pure doublet region, as well as large swaths of non-thermal scenarios. Even afterXENON1T, large swaths of parameter space will still be allowed at small and negative tan θ ,and for tan θ = − M S ∼
200 GeV remains viable.The relatively low values of σ SI and weak exclusion for tan θ = − θ = − | tan θ | (corresponding to largetan β in the MSSM), the sign of tan θ becomes unphysical [47], as shown by the relativesimilarity of the contours for tan β = ±
10 as opposed to tan θ = ± y = 0 .
3. XENON1T will further constrainthis region, excluding at least up to the point at which DM is nearly a pure doublet exceptfor a very finely tuned region of parameter space. Second, the sensitivity of limits dependsgreatly on the sign of tan θ , due to blind spot cancellation points, and in small regions ofparameter space even relatively light masses of a few hundred GeV remain viable.As shown in Fig. 6, the situation changes drastically when the Higgs-DM coupling isincreased to y = 1 .
5. There is a significant shift in the position of the thermal relic line,primarily toward larger values of M D for a given value of M S . This behavior stems from theincrease in the size of the off-diagonal terms in the mass matrix, resulting in yv ∼ M S,D fora larger portion of the space scanned in the figures. As a result, well-tempering is no longerrequired to produce significant mixing, and so M D must be increased significantly to suppressDM annihilation mediated through the doublet component. Moreover, a significant portionof the Ω χ = Ω DM line is located at M D > y issubstantially stronger than for a pure doublet. The dominant effect is enhanced annihilationmediated by a Z -boson due to a modification of the χ − Z coupling, ic Zχχ χγ µ γ χZ µ . Formixed scenarios c Zχχ = g θ W (cid:0) N D − N D (cid:1) , (30)where g is the weak gauge coupling and θ W is the Weinberg angle. In the y → N D = N D → M D > M S or | N D | = | N D | → / √ M D < M S . In either case, c Zχχ →
0, so long as the verysmall splitting necessary to break the doublet into two Majorana states remains. However,for larger values of y and tan θ (cid:54) = ±
1, the degree of S − D mixing and S − D mixing isdifferent, resulting in | N D | (cid:54) = | N D | and thus c Zχχ (cid:54) = 0. The size of c Zχχ grows with thedegree of mixing, and thus with y , though it of course can never exceed g/ θ W . For large y , enhanced Higgs-mediated annihilation also becomes important, though such processesremain p-wave suppressed and are somewhat smaller than the Z -mediated diagrams.While stronger annihilation channels shifts the Ω χ = Ω DM line to larger values of M D > M S ∼ − M D ≈ θ < c hχχ is suppressed, so Z -mediated processes dominateannihilation. As a result, annihilation becomes markedly stronger approaching the M S ≈ M D line, and the Ω χ = Ω DM line moves to larger M D with increasing M S . The coupling8 Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ Ω = Ω D M χ
100 100300300 L UX / XENON E xc l uded y = 1.5tan θ = 2 σ ( zb ) SI (TeV) D M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
100 10 1101 . XEN O N E xc l uded y = 1.5tan θ = -2 σ ( zb ) SI (TeV) D M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ y = 1.5tan θ = 10 σ ( zb ) SI L UX / XENON E xc l uded (TeV) D M ( T e V ) S M (c) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ . XEN O N E xc l udedL UX E xc l uded y = 1.5tan θ = -10 σ ( zb ) SI (TeV) D M ( T e V ) S M (d) FIG. 6.
Direct detection prospects for y = 1 . and tan θ = ± , ± . Contours shown are the sameas in Fig. 5. c Zχχ is marginally smaller for tan θ = − θ = −
10, but not sufficiently so toproduce significantly different results. For tan θ >
0, however, the Ω χ = Ω DM line showsdistinctly different behavior for tan θ = 2. Because Z -mediated processes vanish identicallyin the limit of c Zχχ → θ → ±
1, this coupling is suppressed in the neighborhood oftan θ ≈
1. As a result, the Ω χ = Ω DM contour is located at a lower value of M D for tan θ = 2.Higgs-mediated annihilation compensates somewhat at lower M S , and indeed the tan θ = 2case has the largest Higgs coupling of the cases shown, but Higgs-mediated annihilation doesnot exhibit the same degree of enhancement for M D ≈ M S . As a result, the Ω χ = Ω DM lineis located at large M D for small M S and shifts to smaller values of M D as M S increases. Thetan θ = 10 case, however, is far enough from tan θ = 1 that the behavior seen in tan θ < χ = Ω DM line is nearly horizontal over a range of severalhundred GeV for M D in multiple plots for M S ∼ m t . The increase of M D for nearly fixed M S in this region effectively reduces the mixing angle to compensate for an enhancement inannihilation from the opening of the χχ → t ¯ t annihilation channel. Likewise, the singlet-doublet mixing terms are so large that M χ → θ > M S,D of order a fewhundred GeV. For tan θ < y = 1 . , tan θ = 2 and the Ω χ = Ω DM line is roughly parallel tothe low mass DM contours. This occurs because the relic abundance is strongly controlledby the opening of annihilation channels when the DM mass crosses the bottom quark, W boson, and top quark thresholds.Regarding direct detection, for y = 1 .
5, the raw value of σ SI is increased by more thanan order of magnitude relative to y = 0 .
3. LUX excludes the entire region shown fortan θ >
0, except for a small region at low M S,D for which M χ < ∼ m b is below the experimentalthreshold. However, this low mass region retains a large Higgs coupling and will be excludedby constraints on the invisible decay width of the Higgs [63]. Blind spot cancellations occurfor tan θ <
0, however, with large portions of the thermal relic line remaining viable givenLUX limits. XENON1T still has much greater reach for tan θ <
0, with the increased Higgscoupling resulting in only the blind spot and an associated small portion of the thermal relicline evading XENON1T sensitivity.
B. Exclusion Plots (Thermal Relic)
Next, we consider singlet-doublet fermion DM in a reduced parameter space. We focuson thermal relic DM by restricting to regions that saturate the observed DM abundance,defined by Ω χ = Ω DM . Concretely, we fix one of parameters ( M S , M D , y, θ ) to saturate thethermal relic constraint.Fig. 7 depicts constraints in the ( y, M S ) plane, fixing tan θ to various values and setting M D so that Ω χ = Ω DM . As before, the sign of tan θ has a significant effect on DM properties.For tan θ > y < ∼ .
5, the cross-section σ SI grows almostmonotonically with y and decreases with M S ∼ M χ . In this region, DM annihilation resultsprimarily from gauge interactions typical to nearly-pure doublet DM. For y > ∼ .
5, however,Higgs-mediated and Z -mediated diagrams become important, with their relative contribu-tions increasing with y . As a result, σ SI actually decreases for increasing y for y > ∼
1, dueto a reduction in the mixing to limit the Higgs- and Z -mediated annihilation processes thatincrease with y . A small region of larger σ SI >
100 zb is also present for M S < ∼
200 GeV inthe tan θ = 2 case, associated with large splitting which drives M χ < m W and thus requiresa larger Higgs coupling to produce appropriate levels of annihilation. Fig. 7 also shows thatdirect detection limits from LUX bound y < ∼ . M S < ∼ θ > y for any M S < ∼ θ , and even up to M S = 2 TeV XENON1T limits y < ∼ . θ <
0, both the general suppression in the c hχχ and the existence of blind spotssignificantly affect the results. The shape of the σ SI contours for small y is similar to thetan θ > θ = − y < ∼ . M S , andonly bounds M S > ∼ y = 2. Even with this suppression, however, XENON1Tstill has strong constraining power down to small values of y for M S < ∼ . - - L UX E xc l uded tan θ = 2 ξ σ SI (zb) WT y ( T e V ) S M (a) - - .
10 1 . . . . XENON1TExclusion L UX E xc l uded tan θ = -2 ξ σ SI (zb) WT y ( T e V ) S M (b) . - - . XENON1TExclusion L UX E xc l uded tan θ = 10 ξ σ SI (zb) WT y ( T e V ) S M (c) - - . . - . . XENON1TExclusion L UX E xc l uded tan θ = -10 ξ σ SI (zb) WT y ( T e V ) S M (d) FIG. 7.
Well-tempering and direct detection prospects for fixed Ω χ = Ω DM as a function of y and M S . Shown are the well-tempering measure ξ WT (red dashed) and direct detection cross-section σ SI in zb (green, various styles). Also shown are the regions currently excluded by LUX (blueshaded) and the projected reach of XENON1T (gold shaded). of blind spots which also avoid XENON1T projected bounds is set by M D , and they occurprimarily at larger values of y since such values are required to produce sufficient splittingbetween M S and M D for the values of tan θ shown. These blind spots are more difficult toaccommodate at large tan θ , where they require a larger hierarchy between M χ and M D .These plots also depict the well-tempering parameter, ξ WT , which characterizes the levelof degeneracy required among the mixed neutral states in order to yield the correct relicabundance. The existing LUX bound still allows for thermal relics with moderate well-tempering, ξ WT ∼ .
1, especially for tan θ <
0. It also places moderate limits on y < ∼ . θ >
0, and allows larger values of y for tan θ <
0. However, XENON1T will strongly alter1 . . XEN O N E xc l uded L UX E xc l uded N o S o l u ti on M = 500 D ξ σ SI (zb) WT MSSM D y -1.5 -1 -0.5 0 0.5 1 1.5 D y (a) . . XEN O N E xc l uded L U X E xc l uded N o S o l u ti on M = 800 D ξ σ SI (zb) WT MSSM D y -1.5 -1 -0.5 0 0.5 1 1.5 D y (b) FIG. 8.
Well-tempering and direct detection prospects for fixed Ω χ = Ω DM as a function of y D and y D . Contours shown are the same as in Fig. 7. the available parameter space of this simplified model. Projected limits from XENON1Ttypically constrain ξ WT < ∼ . M χ < ∼ ξ WT < ∼ − for 700 GeV < ∼ M χ < ∼ y D , y D ) plane at fixed M D , and fixing M S to satisfy the relic constraint. Forcertain regions within this plane there are multiple solutions for Ω χ = Ω DM ; in such casesthe solution with the largest value of M χ is displayed . Here tan θ = y D /y D is positive(negative) in the right (left) quadrant. As in Fig. 7, ξ WT is small for small y and grows as y becomes large. However, in contrast with Fig. 7 where M D was allowed to grow to offsetthe increasing value of yv , in Fig. 8 M D < M S is never even approximately decoupled.For tan θ > y D >
0) the large y region induces a significant mass splitting, and is thusdominated by small values of M χ < ∼ m t and correspondingly large σ SI , closing annihilationchannels and reducing coannihilation to compensate for enhanced annihilation strength.For tan θ < y D <
0) the induced mass splitting is significantly smaller, particularly fortan θ ≈ −
1, and so no such low mass solutions exist. This produces a region for whichΩ χ = Ω DM is unachievable by varying M S alone. The size of both the small mass and nosolution regions decreases with increasing M D .At present, LUX strongly constrains this simplified model for y D >
0, only allowing asmall region at small y ; however, once again a blind spot exists for y D < y (limited, of course, by perturbativity), though the trajectory varies with mass Secondary solutions for tan θ > θ < σ SI to the primary solutions. M = T e V D M = G e V D M = G e V D M = G e V D M = G e V D θ < π Ω = Ω DM χ XE N O N XENON1T L U X (GeV) χ M
100 200 300 400 500 600 700 800 900 1000 χχ h C Singlet-Doublet Fermion (a) M = T e V D M = G e V D M = G e V D M = G e V D M = G e V D Viable Regions< θ < 0 π Ω = Ω DM χ XE N O N XENON1T L U X (GeV) χ M
100 200 300 400 500 600 700 800 900 1000 χχ h C Singlet-Doublet Fermion (b)
FIG. 9.
Viable regions with Ω χ = Ω DM as a function of M χ and c hχχ . Regions are shown forvarious values of M D and for the ranges 0 < θ < π/ − π/ < θ < | c hχχ | allowed by XENON100 (teal dashed), LUX (blue dot-dashed) andprojected upper limits from XENON1T (gold dashed). at large y . However, after XENON1T, only models very close to this blind spot will still beviable. Regarding well-tempering, for y D >
0, only a very small value of y will be allowedafter XENON1T, corresponding to a significant degree of well-tempering, ξ WT < ∼ .
03. For y D <
0, well-tempering is still substantial, but is alleviated when residing precisely on theblind spot.Fig. 8 also contains a line corresponding to bino-Higgsino DM in the MSSM. In thiscase, y = g (cid:48) / √
2, and θ = β is restricted to be in the range − π/ π/ β correspond to a negative µ parameter in the MSSM.Lastly, Fig. 9 shows the theoretically available parameter space with Ω χ = Ω DM in termsof more physical quantities: the DM mass, M χ , and the DM-Higgs coupling, ∂M χ /∂v = c hχχ .By definition, the blind spot is defined at the bottom of the plot where | c hχχ | = 0. We cansubstitute the model inputs M S and y for these more physical parameters using the relations M S = M χ − c hχχ v ∆ M B ∆ M B + c hχχ v (∆ M + 2 M χ B ) / y = − c hχχ vB (cid:0) ∆ M ( M S − M χ ) + B (cid:0) ∆ M + 2 M χ ( M S − M χ ) (cid:1)(cid:1) (32)∆ M = M D − M χ (33) B = M χ + M D sin 2 θ . (34)We have marginalized over all values of tan θ positive (left) and negative (right) to fixΩ χ = Ω DM for various values of M χ ± = M D . In keeping with comparison to direct detectionthe sign of c hχχ is also left undetermined. This issue will be discussed further in Sec. III C.3For tan θ > M χ . Theposition of the upper edge depends strongly on M D , and increases more quickly for small M D ,while the position of the lower edge is almost independent of M D . This behavior is modifiedas M χ and M D become degenerate, with the upper edge peaking at M χ ≈ M D −
100 GeVand dropping steadily thereafter. The lower edge also drops below the general trend linein the same mass range. Both effects are due to coannihilation, which requires a spectrumwith relatively suppressed Higgs couplings. The presence of a general lower bound on c hχχ is due to the lack of a blind spot for tan θ >
0. However, a small region with c hχχ → θ > M χ → M D limit as small cross-terms are required for mixing. Note thatthe absence of this region in Fig. 9(a) is due to a binning artifact.For tan θ <
0, the behavior of the upper edge is similar to that for tan θ >
0, thoughfound at somewhat different values of | c hχχ | . There is no lower bound, however, due tothe possibility of blind spot cancellations for tan θ <
0. In terms of constraints, LUXcurrently excludes most of the tan θ > θ < θ <
0, XENON1T will not be able toeliminate the model.
C. Exclusion Plots (Thermal Relic and Marginal Exclusion)
So far, we have reduced the dimensionality of the parameter space by imposing Ω χ = Ω DM .However, we can further reduce the parameter space by fixing σ SI , which is controlled by c hχχ coupling. In particular, we will now focus on the parameter space of thermal relic DM thatexactly saturates present LUX limits or projected XENON1T reach (in addition to fixingΩ χ = Ω DM ). This defines a space of marginally excluded, thermal relic DM models whichcan accommodate the observed abundance today. This space of marginally excluded thermalrelics represents the class of models at the edge of direct detection limits. Thus, comparingthe space fixed to XENON1T projected constraints against that fixed to current LUX limitsunambiguously demonstrates the effect of improved direct detection sensitivity. This is aidedby the additional benefit of reducing the parameter space to only two dimensions; hence theentire surviving parameter space can be described in a single plane. Moreover, because thedirect detection limits bound σ SI from above, the values y shown can be identified as themaximal DM-Higgs coupling allowed by a given experiment. The corresponding M χ shouldbe interpreted as the minimal DM mass allowed by direct detection, at least for M χ > ∝ M χ .First we consider the parameter space of marginally excluded thermal relics in the( M D , M χ ) plane for tan θ <
0. For every pair of ( M χ , M D ), there are a maximum of fourviable solutions – each with c hχχ either positive or negative and tan θ either positive ornegative. From Eq. (12), c hχχ ∝ − ( M χ + M D sin 2 θ ) , (35)so it is negative definite for tan θ positive, leaving a maximum of three viable solutions.Furthermore, from the discussion in Sec. III B it is clear that the viable region for tan θ > y and significant well-tempering, and thus would occupyonly a small sliver along the M χ = M D line in this plane; an explicit scan confirms thisassumption.4 < . < 0.1 c > 0 h χχ Ω = Ω DM χ σ = σ SI SILUX - π/2 < θ < 0 N o S o l u t i on y χ M < M χ + ξ WT ξ BS (TeV) D M ( T e V ) χ M (a) < . < . < . c < 0 h χχ Ω = Ω DM χ σ = σ SI SILUX - π/2 < θ < 0 y χ M < M χ + ξ WT ξ BS (TeV) D M ( T e V ) χ M (b) < . < . <0.01 < . c > 0 h χχ Ω = Ω DM χ σ = σ SI SIX1T - π/2 < θ < 0 y χ M < M χ + ξ WT ξ BS (TeV) D M ( T e V ) χ M (c) < . < . < . <0.03 c < 0 h χχ Ω = Ω DM χ σ = σ SI SIX1T - π/2 < θ < 0 y χ M < M χ + ξ WT ξ BS (TeV) D M ( T e V ) χ M (d) FIG. 10.
Tuning for models with fixed Ω χ and σ SI as a function of M D and M χ . Shown arepoints which satisfy the relic density constraint and exactly saturate σ SI limits from LUX (top) orprojected reach at XENON1T (bottom). Regions with significant tuning are shown for ξ WT (redshaded) and ξ BS (green shaded). The white regions correspond to parameters with lesser tuningthan the shaded regions: ξ WT , ξ BS > ∼ . ξ WT > ∼ . . > ∼ ξ BS > ∼ . c hχχ > c hχχ <
0; inboth cases tan θ < The two remaining interesting cases are shown in Fig. 10 for tan θ < c hχχ > c hχχ < y is shown, along withregions showing various degrees of well-tempering and blind spot tuning. Models marginallyexcluded by LUX (top) have minimal tuning, with most regions subject to ξ WT , ξ BS > ∼ . c hχχ . For models within the marginalsensitivity of XENON1T, however, ξ BS < . < . < 0.1 No Solution y ξ WT ξ BS y > π c > 0 h χχ DM χ Ω = Ω
LUXSI σ = SI σ π / θ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ( G e V ) χ M (a) No Solutiony ξ WT ξ BS c < 0 h χχ DM χ Ω = Ω
LUXSI σ = SI σ π / θ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ( G e V ) χ M (b) No Solution y ξ WT ξ BS y > π c > 0 h χχ DM χ Ω = Ω
X1TSI σ = SI σ π / θ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ( G e V ) χ M (c) < . NoSolution y ξ WT ξ BS c < 0 h χχ DM χ Ω = Ω
X1TSI σ = SI σ π / θ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ( G e V ) χ M (d) FIG. 11.
Tuning for models with fixed Ω χ and σ SI as a function of θ and M χ . Contours andregions shown are the same as in Fig. 10. c hχχ , and ξ BS < .
03 throughout most of the plane. While the parameter space remainsviable for DM, it is clear that some degree of fine-tuning is necessary for consistency withXENON1T projected limits.Fig. 11 show the parameter space of marginally excluded thermal relic DM in the ( M χ , θ )plane, with M D set to fix Ω χ = Ω DM . As discussed above, no solution exists for c hχχ > θ >
0. In principle, solutions exist throughout the plane for both signs of c hχχ andtan θ <
0. However, at sufficiently large values of y , the potential solutions for c hχχ < θ = − π/
4. For c hχχ > ξ WT , ξ BS > ∼ . ξ BS < . ξ BS < .
03 throughout most ofthe region for XENON1T. In contrast, for c hχχ < θ >
0, we find that ξ BS = 1 by definition because no cancellationsare possible to eliminate the DM-Higgs coupling. However, to evade direct detection, westill need tiny values of y , requiring coannihilation to produce Ω χ = Ω DM . This produces ξ WT < . < ∼ M χ < ∼
900 GeV for LUX, while for XENON1T, ξ WT < . ξ WT < .
01 for 750 GeV < ∼ M χ < ∼
900 GeV.
IV. MODEL B: SINGLET-DOUBLET SCALAR DM
Next, we will analyze the case of singlet-doublet scalar DM. This simplified model hasqualitative similarities to the fermionic version discussed in Sec. III. However, there is animportant physical difference: while a pure fermion singlet is inert, a pure scalar singletcan have renormalizable interactions to the SM through the Higgs boson. In particular,for scalar DM the quartic couplings between the DM and the Higgs can have a substantialeffect on the thermal relic abundance and direct detection constraints. The addition ofa quartic Higgs coupling to the singlet produces a line in the M χ vs. σ SI plane consistentwith the thermal relic density for pure singlet [54] DM, while a quartic Higgs coupling to thedoublet shift the value of M D which gives Ω χ = Ω DM independent of M S for pure doublet [64]DM. However, we are specifically interested in the case of mixed DM, whereby the dominantannihilation channels relevant to freeze-out derive from mixing of the singlet and non-singletstates. Thus, we will focus on the parameter space where the cubic term A [ SHD ∗ + h.c. ]is large enough to induce significant mixing.From the practical standpoint of analyzing constraints, scalar DM models tend to havemore parameters than fermionic DM – in the case of singlet-doublet DM, the scalar modelhas seven while the fermionic has only four. The large number of parameters renders a com-prehensive parameter scan like that used for fermions in Sec. III impractical. Fortunately,four of the free parameters for scalar DM are quartic couplings of the Higgs directly to thesinglet or doublet states, and thus do not induce singlet-doublet mixing. We will thereforefocus on the case of λ S = λ D = λ (cid:48) D = λ (cid:48)(cid:48) D = 0 as the “minimal case” for mixed singlet-doubletscalar DM. As we move away from this simplifying limit, mixing becomes less important forthe relic abundance and direct detection constraints, and the DM properties approach thatof a pure singlet or doublet.In principle, there are three distinct Higgs couplings to the doublet component of DM.However, only one combination of these enters into the couplings of the neutral mixeddoublet state. Thus, many combinations of the three couplings will result in the samedynamics for the DM state alone. On the other hand, these different combinations willresult in modified dynamics for processes involving DM and another doublet state, whichhere is limited to coannihilation for nearly pure doublet states. While these effects may beimportant in certain regions of parameter space, they are not the primary focus here.The dominant effect of including Higgs couplings to pure states is a modification of a hχχ and the coupling associated with the χχhh operator. If re-expressed in terms of the singlet-doublet mixing angles, a hχχ becomes [49] a hχχ = v (cid:0) λ S N S + [ λ D + λ (cid:48) D + λ (cid:48)(cid:48) D ] N D (cid:1) − AN S N D . (36)As shown in Fig. 3, for the majority of the parameter space either N S (cid:28) N D (cid:28)
1, soeither the singlet or doublet Higgs coupling contribution to a hχχ and the associated χχhh operator will be sub-dominant. Using this feature, we further simplify the parameter space7by taking λ S = λ D = λ (37) λ (cid:48) D = λ (cid:48)(cid:48) D = 0 (38)This simplified parameter space carries most of the qualitative features of the full theory,diverging primarily for M S ≈ M D . A. Exclusion Plots (General)
To begin, we analyze the unconstrained parameter space of scalar singlet-doublet DM.In Fig. 12 we restrict to the case λ = 0, fixing A = 10 , , A = 10 GeV, the associated cubic coupling is too weak to produce significant splitting orcontribute to the relic density directly, thus requiring large mixing for M S < ∼ −
550 GeV.For M S > ∼
600 GeV, DM with Ω χ = Ω DM becomes a nearly pure doublet with M D ≈
550 GeV [21, 64]. In this case direct detection prospects are minimal, with XENON1Tsensitivity only for M S,D < ∼
200 GeV.For A = 100 GeV, the position of the Ω χ = Ω DM line changes substantially for M S < ∼
800 GeV, shifting to larger values of M D as large as ≈
750 GeV. However, it eventuallyasymptotes to pure doublet behavior for larger M S . This shift is qualitatively similar tothe enhanced Higgs-mediated annihilation for fermions in the case of large couplings, butthe quantitative results diverge substantially. As discussed in Sec. III, while Majoranafermion DM can annihilate through the Higgs, this process appears at p-wave and onlybecomes important for large couplings. For scalars, however, annihilation via the Higgsis s-wave and interferes strongly with t-channel annihilation diagrams involving the othercharged and neutral scalars. Qualitatively, the relative strength of Higgs annihilation canbe seen in the direct detection coverage – the sensitivity of LUX and XENON1T shown inFig. 12(b) is much weaker for equivalent M χ than the full-plane coverage present for fermionsin Figs. 5(a) and 5(c), but despite this achieving Ω χ = Ω DM requires less well-tempering, atleast for M S < ∼
800 GeV.The relevance of annihilation through the Higgs is even more pronounced for A = 1 TeV.In this case, Ω χ = Ω DM cannot be achieved for small DM masses due to Higgs-mediatedannihilation. The Ω χ = Ω DM line occurs at M S > ∼ . ∼
550 GeV required for a pure doublet thermal relic. The Ω χ = Ω DM line occurs at somewhat smaller DM mass for M D < M S than for M D > M S due to the moreefficient annihilation from the typical doublet annihilation processes, but in both cases therelatively small mixing angle still produces large direct detection cross-sections. As a result,a portion of the Ω χ = Ω DM line remains outside of LUX bounds despite the relatively largevalues of σ SI , though it can be probed at XENON1T.The situation changes drastically if there are quartic couplings between the Higgs andthe singlet and doublet components of the DM, shown in Fig. 13 for λ = ± .
25. Themost distinctive feature is the possibility of Ω χ = Ω DM for pure singlet DM, occurring at M S ≈
800 GeV for a small mixing term A = 10 GeV. Furthermore, a Higgs coupling to thedoublet also shifts the mass of the Ω χ = Ω DM for pure doublet DM to M D ≈
650 GeV. For A (cid:28) λv the Higgs couplings to pure states dominates, and the sign of λ is irrelevant exceptfor M S ≈ M D . Direct detection sensitivity is strong even for the modest value of λ = ± . M S or M D of just under 400 GeV, and XENON1T sensitivityreaching out to M S , M D < ∼ . χ = Ω DM line.8 Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ - - . .
01 0 . . .
11 0 . XENON E xc l uded λ = 0A = 10 GeV σ ( zb ) SI (TeV) D M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ . . . . . . XE NON TE xc l udedL U XE xc l uded λ = 0A = 100 GeV σ ( zb ) SI (TeV) D M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ XE NON TE xc l udedL U X E xc l uded λ = 0A = 1 TeV σ ( zb ) SI N o So l u ti on (TeV) D M ( T e V ) S M (c) FIG. 12.
Direct detection prospects for λ = 0 and A = 10 , , GeV.
Contours shown arethe same as in Fig. 5.
The sign of λ becomes important for A ∼ λv . As can be seen from Eq. (21), the contribu-tion of A (cid:54) = 0 to the DM-Higgs coupling is always negative, leading to an enhancement of thecouplings for λ < λ >
0. The corresponding blind spot cancellationregions are located near but slightly offset above and below the M S = M D line in Fig. 13(b)for λ = 0 .
25, with no corresponding behavior present in Fig. 13(e) for λ = − .
25. For λ = 0 . , A = 100 GeV, certain points in the blind spot region are actually consistent withΩ χ = Ω DM , while for λ = − . , A = 100 GeV the entire Ω χ = Ω DM contour is within thesensitivity range of XENON1T. A small region also exists for λ = 0 . , A = 100 GeV withΩ χ = Ω DM for M S , M D < ∼
200 GeV, resulting from the cancellation in the Higgs coupling;by varying values of A for appropriate suppression this low mass contour can be shiftedanywhere with M S < M D .For A = 1 TeV (cid:29) λv , the mixing term dominates for both signs of λ , so no blind spot ispresent; however, σ SI is enhanced for λ = − .
25 relative to λ = 0 .
25 throughout the planeand constraints are stronger, as shown in Figs. 13(c) and 13(f). As in Fig. 12(c), when A = 1 TeV, the Higgs coupling is so large that the majority of parameter space is alreadyexcluded by LUX. The relative sign of λ also has a significant effect on the location of theΩ χ = Ω DM line, with the enhanced coupling for λ = − .
25 pushing the contour to largermass, while the suppressed coupling for λ = 0 .
25 shifts the line to lower mass. As for λ = 0,the portions of the Ω χ = Ω DM lines in the region shown are not excluded by LUX, but arewithin XENON1T sensitivity for λ = ± . B. Exclusion Plots (Thermal Relic)
Next, we consider the singlet-doublet scalar model for Ω χ = Ω DM . In keeping with ourinterest in mixed DM, we fix to various values of λ , leaving only three variables in the re-maining parameter space, ( M S , M D , A ). Within this sub-space, for fixed ( M S , A ) or ( M D , A )a varying number of solutions exist for Ω χ = Ω DM ,as evidenced by Figs. 12 and 13. In partic-ular, for λ (cid:54) = 0 sufficiently large, a solution always exists for which Ω χ = Ω DM is independentof M D and M S , respectively, above a certain critical value. This limits the usefulness of ananalog to Fig. 7 or Fig. 8 where either M S or M D is used to fix the relic density.As such, we consider the ( M D , M S ) plane in Fig. 14 with λ = 0 , ± .
25 and A fixed to9 Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ XE NON TE xc l uded λ = 0.25A = 10 GeV σ ( zb ) SI L U X E xc l uded (TeV) D M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ . XE NON TE xc l uded λ = 0.25A = 100 GeV σ ( zb ) SI L U X E xc l uded (TeV) D M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ XE NON TE xc l uded λ = 0.25A = 1 TeV σ ( zb ) SI L U X E xc l uded N o So l u ti on (TeV) D M ( T e V ) S M (c) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
10 1100 XE NON TE xc l udedL U X E xc l uded λ = -0.25A = 10 GeV σ ( zb ) SI (TeV) D M ( T e V ) S M (d) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
10 1100 XE NON TE xc l uded λ = -0.25A = 100 GeV σ ( zb ) SI L U X E xc l uded (TeV) D M ( T e V ) S M (e) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ XE NON TE xc l uded λ = -0.25A = 1 TeV σ ( zb ) SI L U X E xc l uded N o So l u ti on (TeV) D M ( T e V ) S M (f) FIG. 13.
Direct detection prospects for λ = 0 and A = 10 , , GeV.
Contours shown arethe same as in Fig. 5. produce Ω χ = Ω DM . For λ = 0, no solution is present for M D < M S and M D < ∼
550 GeV, as anearly pure doublet is under-dense in this region. The same effect is present for λ = ± .
25 for M D < ∼
650 GeV. Throughout the rest of the plane in Fig. 14(a), direct detection sensitivityto the Ω χ = Ω DM scenario is strong, with the best sensitivity present for M S ≈ M D anddropping off as the masses become less degenerate. In fact, direct detection sensitivitygrows at large DM mass because the mixing required for the thermal relic abundance requires( vA ) ∼ M χ at large mass, resulting in c hχχ ∼ M χ /v . The DM-nucleon cross-section thusscales as σ SI ∝ c hχχ ∝ ( M χ /v ) , which grows faster than the direct detection limits weaken: σ SILUX , σ
SIX1T ∝ M χ . For λ = 0, the blind spot is present for M S > M D and M D near the “nosolution” region.For λ (cid:54) = 0 the interplay of mixing and non-mixing Higgs couplings modifies the locationof the “no solution” and blind spot regions. For λ = − .
25, no solution exists for M S < ∼
800 GeV except for M S ≈ M D , as the Higgs-mediated annihilation cross-section is too largein this region. However, despite an enhanced Higgs coupling, the LUX exclusion is marginally weaker for λ = − .
25 – the direct coupling alone is insufficient to saturate the LUX boundfor M S (cid:54)≈ M D , but it does reduce the required degree of mixing necessary to produceΩ χ = Ω DM . However, the entire viable region for λ = − .
25 falls within project XENON1Tsensitivity. For λ = 0 .
25, the LUX exclusion is correspondingly stronger than for λ = 0;however, the blind spot is shifted away from the “no solution” region to M D ≈
725 GeVand M S > ∼
750 GeV. A region with multiple solutions for Ω χ = Ω DM also exists for small0 . . . .
01 0 . XE NON TE xc l uded L U X E xc l uded N o So l u ti on λ = 0 ξ σ SI (zb) WT (TeV) D M ( T e V ) S M (a) . . .
01 0 . XE NON TE xc l uded L U X E xc l uded M u lti p l e So l u ti on s N o So l u ti on λ = 0.25 ξ σ SI (zb) WT ξ < 0.1 BS (TeV) D M ( T e V ) S M (b) . . . . XE NON TE xc l uded L U X E xc l uded N o So l u ti on λ = -0.25 ξ σ SI (zb) WT (TeV) D M ( T e V ) S M (c) FIG. 14.
Well-tempering and direct detection prospects for fixed Ω χ = Ω DM as a function of M D and M S . Contours shown are the same as in Fig. 7, with the addition of a region correspondingto ξ BS < . M S and λ = 0 .
25, resulting from the interplay of Higgs coupling cancellation with varying A . In this region XENON1T constrains most solutions, but a set exists for which the singletand doublet states are highly mixed but a hχχ vanishes due to a cancellation between thecontributions, producing a blind spot.The blind spot tuning measure, ξ BS , only has meaningful implications for the λ > ξ BS = 1 identically for λ ≤
0, with partialcancellation in the top term possible only for λ >
0. This gives ξ BS < ∼ . λ > M χ ≈
700 GeV.The experimentally viable region for singlet-doublet scalar DM is depicted in Fig. 15 inthe plane of physical variables, ( M χ , c hχχ ). To produce this figure we have replaced themodel parameters ( M S , A ) with ( M χ , c hχχ ), and set M D to a value in order to accommodateΩ χ = Ω DM . As noted before, this produces multiple solutions at most points in each region,particularly for | c hχχ | > ∼ .
1. For | c hχχ | > ∼ . χ < Ω DM both for M D ≈ M χ due to coannihilation and for M D → ∞ due to pure Higgs coupling. However, Ω χ > Ω DM for an intermediate range of M D due to destructive interference between pure gauge andHiggs-mediated annihilation diagrams, producing at least two solutions for Ω χ = Ω DM .According to Eq. (21), c hχχ ≤ λ = 0. For M χ < ∼
550 GeV, arbitrarily small values of c hχχ are viable, since a sufficiently small splitting | M D − M S | can always be chosen to producethe correct degree of mixing through pure gauge diagrams. For M χ > ∼
550 GeV, some degreeof Higgs-mediated annihilation is required to produce Ω χ = Ω DM , and the region boundariesare set by the relative size of Higgs-mediated versus pure gauge annihilation diagrams. Theupper boundary along c hχχ ≈ . M χ > ∼
550 GeV has the relic density set entirely byHiggs-mediated diagrams, with gauge diagrams suppressed by M D (cid:29) M χ . In this case A /M D ∼ M χ /v , so the underlying Higgs coupling must increase dramatically with M χ andwill eventually become non-perturbative. For larger values of | c hχχ | the degree of mixingis increased, with the lower boundary set by near-maximal mixing. The largest values of c hχχ are achieved when significant interference is present between pure gauge and Higgs-mediated diagrams, resulting in BR ( χχ → W + W − , ZZ ) → χχ → t ¯ t ) →
1. The1 λ = 0.25 λ = 0 λ = -0.25 XE N O N XE N O N T L U X (TeV) χ M h xx c -0.5-0.4-0.3-0.2-0.100.1 Viable Regions Singlet-Doublet Scalar
FIG. 15.
Viable regions with Ω χ = Ω DM as a function of M χ and c hχχ . Regions are shown for λ = 0 (red shaded), λ = 0 .
25 (green shaded) and λ = − .
25 (gray shaded) with M D fixed toproduce the correct relic density. Also shown are upper limits on | c hχχ | from XENON100 (tealdashed), LUX (blue dot-dashed), and projected upper limits from XENON1T (gold dashed). nearly pure doublet case for M χ > ∼
550 GeV is found at intermediate points with M S (cid:29) M χ and Higgs-mediated diagrams compensating for insufficient pure gauge annihilation.For λ (cid:54) = 0, the viable region is modified significantly at low mass but remains similar inshape at high mass. This is a result of the scaling of the terms in Eq. (21), with the unmixedcoupling term λv becoming sub-dominant for increasing M χ , while the term proportional to A remains constant or increases with increasing M χ . For λ = 0 .
25, positive c hχχ is possiblefor M χ < ∼
700 GeV, and significantly larger values of | c hχχ | are viable for M χ < ∼
200 GeV dueto interference effects in annihilation to gauge bosons for masses insufficiently large to allowfor annihilation to t ¯ t . For large M χ , however, the upper boundary asymptotes to c hχχ ≈ . λ = 0 case, while the lower boundary is offset to more negative c hχχ by a smallvalue. For λ = − .
25, no solution is present for M χ < ∼
600 GeV – both contributions to c hχχ are negative, and the resulting value always produces Ω χ < Ω DM . While relatively largevalues of | c hχχ | were feasible at low mass for λ = 0 .
25, in such cases A and | M D − M S | couldbe relatively large due to cancellation between the contributions, resulting in a suppressionof coannihilation contributions, while for λ = − .
25 the contributions are always additive.For large M χ the upper boundary of the λ = − .
25 region also asymptotes to c hχχ ≈ . c hχχ by an amount somewhat larger thanthe offset for λ = 0 . c hχχ /M χ , while the region with a viable thermal relic abundanceis bounded by constant c hχχ and constant c hχχ /M χ . The blind spot for small M χ remainsconstrained to low M χ , though larger values of M χ would become viable for increased λ .A significant region of parameter space remains unconstrained by LUX limits, particularlyat large mass, but XENON1T projected sensitivity covers all viable regions except for thelow mass blind spot. If XENON1T yields null results, then theories with negative λ will be2strongly constrained. V. MODEL C: SINGLET-TRIPLET SCALAR DM
Finally, we examine the experimental limits on mixed singlet-triplet scalar DM. Thephenomenology of the singlet-triplet scalar model is similar to that of the singlet-doubletscalar model – the relic density is set primarily by DM-gauge interactions and Higgs-mediateddiagrams. Moreover, these contributions can interference substantially in a way that stronglyaffects the final relic density determination. However, several features significantly alter thedetailed phenomenology. First, the cubic DM-gauge interaction vanishes because of the SU (2) L symmetry. The leading DM-gauge interaction is quartic, and has a coupling whichis effectively four times stronger than in the singlet-doublet scalar case simply due to grouptheory factors. Thus, the model can accommodate Ω χ = Ω DM primarily through gaugeinteractions with sufficient well-tempering up to larger values of M χ . Secondly, for a givenmixing angle, the trilinear DM-Higgs coupling is larger by a factor of 2 for singlet-triplet DMas compared to singlet-doublet DM. This modifies the relative strength of contributions toannihilation and scattering. More significantly, however, is the DM-Higgs quartic interactioninduced by the κ mixing term. While a similar interaction is induced at tree level by themixing term in the singlet-doublet scalar case, the contribution from the direct coupling forthe singlet-triplet is larger for a spectrum with similar mass splittings.Singlet-triplet scalar DM involves the same basic processes as singlet-doublet scalar DM,so in the following analysis we focus primarily on the differences between these theories.Once again either N S (cid:28) N T (cid:28) λ S and λ T both play an important role in the dynamics is limited to M S ≈ M T ; thus forsimplicity we set λ S = λ T = λ (39)In the singlet-triplet case this is a somewhat better approximation, as the absence of addi-tional quartic couplings reduces the range of possible divergent results for M T (cid:54)≈ M S . A. Exclusion Plots (General)
To analyze the unconstrained singlet-triplet DM parameter space, we fix λ = 0 to elimi-nate the effects of the Higgs-DM quartic couplings. In the case of singlet-triplet DM, mixingis controlled by the dimensionless coefficient κ , which we set to κ = 0 . , M T , M S ) plane. For the case of κ = 0 . M S and M T must be verydegenerate to accommodate a thermal relic consistent with observations, and σ SI is smallthroughout the plane. For M S (cid:54)≈ M T , LUX only constrains M S , M T < ∼
200 GeV, and evenXENON1T only has sensitivity for M S , M T < ∼
400 GeV. In Fig. 16(b), the relatively largevalue of κ = 3 shifts the Ω χ = Ω DM line substantially, primarily due to an increase in the an-nihilation process χχ → hh with κ . This depletes the DM abundance and implies that verylarge DM masses – upwards of multi-TeV – may be necessary to accommodate the observedDM relic abundance. While the both LUX and XENON1T strongly constrain the plane,with generic sensitivity to M S , M T < ∼
800 GeV and M S , M T < ∼ . Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded λ = 0 κ = 0.3 σ ( zb ) SI (TeV) T M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded λ = 0 κ = 3 σ ( zb ) SI L UX E xc l uded (TeV) T M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ XEN O N E xc l uded λ = 0 κ = 0.3 σ ( zb ) SI L U X E xc l uded LEP Bound (GeV) S - M T M -200 -150 -100 -50 0 50 100 150 200 ( T e V ) S M (c) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ Ω = Ω D M χ Ω > Ω
DMχ
XEN O N E xc l uded λ = 0 κ = 3 σ ( zb ) SI L UX E xc l uded LEP Bound (GeV) S - M T M -200 -150 -100 -50 0 50 100 150 200 ( T e V ) S M (d) FIG. 16.
Direct detection prospects for λ = 0 and κ = 0 . , . Contours shown are the same as inFig. 5. in the Ω χ = Ω DM line places some regions with a viable thermal relic beyond XENON1Treach.The lower panels of Fig. 16 show the same information as the top panels, only plottedin the ( M T − M S , M S ) plane so as to focus on the diagonal region in which the singlet andtriplet are well mixed. Sensitivity at direct detection experiments to the M S ≈ M T region issignificantly improved relative to other parts of the plane. For κ = 0 . κ = 0 . χ = Ω DM contour up to M S < ∼
500 GeV and bound M S ≈ M T < ∼
900 GeV. While experimentalsensitivity improves for κ = 3, with reach for M S ≈ M T > χ = Ω DM contour is only within XENON1T reach for M S < ∼ . λ = ± .
25 are shown in Fig. 17, which is otherwise identical to the upper4 Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded λ = 0.25 κ = 0.3 σ ( zb ) SI L UX E xc l uded (TeV) T M ( T e V ) S M (a) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
Ω > Ω
DMχ XEN O N E xc l uded λ = 0.25 κ = 3 σ ( zb ) SI L UX E xc l uded (TeV) T M ( T e V ) S M (b) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded λ = -0.25 κ = 0.3 σ ( zb ) SI L UX E xc l uded (TeV) T M ( T e V ) S M (c) Ω = Ω D M χ Ω < Ω
DMχ
Ω > Ω
DMχ
Ω > Ω
DMχ
XEN O N E xc l uded λ = -0.25 κ = 3 σ ( zb ) SI L UX E xc l uded (TeV) T M ( T e V ) S M (d) FIG. 17.
Direct detection prospects for λ (cid:54) = 0 and κ = 0 . , . Contours shown are the same as inFig. 5. panels of Fig. 16. As can be seen from Eq. (27), a blind spot cancellation can occur if λ ispositive, in which case there will be destructive interference against Higgs exchange arisingfrom the κ induced mixing. This blind spot is visible in Fig. 16(b) for points near (but notdirectly on) the M S ≈ M T line. Similarly to the behavior in Fig. 13(b) for singlet-doubletscalars, the DM-Higgs coupling is controlled by mixing induced by κ near M S ≈ M T . Awayfrom this line, the DM-Higgs coupling is controlled by the direct quartic DM-Higgs couplingsproportional to λ . Partial cancellation is also present in Fig. 16(a), though κ = 0 . λ = 0 .
25. For λ = − .
25 the direct detection cross-sections is increased relative to λ = 0, with a corresponding LUX constraint of M S , M T > ∼ M S , M T < ∼ . M S ≈ M T .5 . . . . XE NON TE xc l udedL U X E xc l uded No Solution λ = 0 ξ σ SI (zb) WT (TeV) T M ( T e V ) S M (a) . . . . . . XE NON TE xc l uded No Solution λ = 0.25 ξ σ SI (zb) WT ξ < 0.1 BS (TeV) T M ( T e V ) S M (b) . . . . . XE NON TE xc l uded No Solution λ = -0.25 ξ σ SI (zb) WT (TeV) T M ( T e V ) S M (c) FIG. 18.
Well-tempering and direct detection prospects for fixed Ω χ = Ω DM as a function of M T and M S . Contours shown are the same as in Fig. 7.
B. Exclusion Plots (Thermal Relic)
For the singlet-triplet scalar model, we set Ω χ = Ω DM by varying κ in the ( M T , M S ) planein order to produce the observed relic density. Fig. 18 depicts the viable parameter space ofthermal relic singlet-triplet DM subject to current constraints from LUX and projected reachfrom XENON1T. For λ = 0, there is no viable thermal relic for M T < M S and M T < ∼ . χ < Ω DM for all such models. For M T > M S and 1 . < ∼ M T < M S , Ω χ = Ω DM canbe achieved throughout for sufficiently large κ , with M T ≈ . , M S (cid:29) M S correspondingto pure triplet DM with the correct thermal relic density [21, 64]. Moreover, σ SI varies byjust over an order of magnitude over most of the viable range shown. This occurs becausethe dominant annihilation channel, χχ → hh , scales with mixing terms in the same way asthe direct detection cross-section when κ is sufficiently large. Hence, the direct detectionand relic abundance are correlated. For M S ≈ M T annihilation is enhanced for identicalvalues of κ reducing σ SI at large masses by roughly an order of magnitude. Because anequivalent value of a hχχ implies stronger annihilation than in the singlet-doublet scalar case,the current LUX bounds only constrain a few points at small mass. However, because σ SI isonly weakly dependent on DM mass, XENON1T projected bounds constrain the majority ofthe parameter space shown. The only blind spot occurs along the edge of the “no solution”region for M T ≈ . λ = 0 .
25 in Fig. 18(b) the “no solution”region extends into the low mass M S < M T region. For M S < ∼
400 GeV, the direct singlet-Higgs quartic coupling results in Ω χ < Ω DM , and increasing κ sufficiently to cancel the directcoupling contributions induces sufficient annihilation in other channels that χ is remainsunder-abundant, resulting in no viable solution for any value of κ . The allowed region stillextends to lower M S than the Ω χ = Ω DM contour in Fig. 17(a), however, and a blind spotoccurs at M S ≈
650 GeV consistent with this cancellation. This blind spot extends tolarge masses below and roughly parallel to the M S = M T line. A second blind spot liesabove this line and extends to large M S for M T ≈ a hχχ required toaccommodate the observed relic density is significantly smaller. In the singlet-doublet case,gauge interactions are insufficient to set Ω χ = Ω DM for any mixing angle for M χ > ∼
550 GeV,and the induced DM-Higgs quartic coupling is relatively small. For singlet-triplet scalar6 λ = 0.25 λ = 0 λ = -0.25 XE N O N XE N O N T L U X (GeV) X M h xx c -0.2-0.15-0.1-0.0500.050.1 Viable Regions Singlet-Triplet Scalar
FIG. 19.
Viable regions with Ω χ = Ω DM as a function of M χ and c hχχ . Contours and regionsshown are the same as in Fig. 15.
DM, however, gauge interactions are strong throughout the entire range shown, and theDM-Higgs quartic coupling is sufficiently large that small values of c hχχ are viable and evenpreferred. LUX has no constraining power for λ = 0 .
25, and XENON1T sensitivity has onlymoderate coverage of the parameter space.For λ = − .
25 in Fig. 18(c), the “no solution” region covers M S < ∼
850 GeV for M S < M T .No blind spot regions exist for λ = − .
25, but the relative strength of the gauge andfour point interactions produces regions with small σ SI along the upper portion of the “nosolution” boundary and along the high mass M S ≈ M T line. LUX has no constrainingpower in the plane, and the regions with small σ SI avoid even XENON1T reach despite thelack of true blind spot behavior. For both λ = 0 and λ = − .
25 well-tempering of at least ξ WT < ∼ . ξ WT < ∼ . λ ≤ ξ BS = 1,and ξ BS is only physically meaningful when λ >
0. For λ >
0, however, the degree of fine-tuning needed to produce cancellation is significantly smaller than in the singlet-doubletscalar case. The pink region in Fig. 18(b) has tuning of ξ BS < ∼ . ξ BS > ∼ .
1. The lower portion of this allowedarea also has minimal tuning from well-tempering, ξ WT > .
1. Hence, for λ >
0, singlet-triplet DM can accommodate viable thermal relic DM with minimal tuning.The viable parameter space of singlet-triplet DM is depicted in the physical ( M χ , c hχχ )plane in Fig. 19. For λ = 0, the behavior is similar to the singlet-doublet case, except theregion is “stretched” horizontally and “squeezed” vertically – c hχχ can be close to zero up to M χ < ∼ . | c hχχ | along the lower boundary is more gradual. A smallregion extending to c hχχ ≈ − .
14 is allowed for very low mass, where M χ is below the Higgsproduction threshold and thus κ can be significantly larger. For λ = 0 .
25, most of the viableregion is restricted to M χ > ∼
500 GeV, while for λ = − . M χ > ∼
700 GeV is required. For M χ > ∼
500 GeV the behavior for λ = ± .
25 is similar to the singlet-doublet scalar case. For7all of the choices λ = 0 , ± .
25, there are parameter regions which are beyond the projectedreach of XENON1T. In contrast to the singlet-doublet scalar case, the sensitivity of directdetection experiments weakens as M χ increases. VI. CONCLUSIONS AND FUTURE DIRECTIONS
Simplified models are a powerful tool for studying the generic behavior of WIMP DM.Theories in which DM couples to the SM via the Higgs are of particular interest becauseHiggs-mediated DM-nucleon scattering is just now being probed by the current generationof direct detection experiments. In this paper we have constructed and analyzed simplifiedmodels of mixed DM describing a stable particle composed of a mixture of a singlet and anelectroweak doublet or triplet. In these models DM undergoes thermal freeze-out throughelectroweak interactions to accommodate the observed DM relic abundance. Mixing betweenthe singlet and non-singlet states is induced via DM-Higgs couplings, and is in generalcorrelated with signals in direct detection.We have determined the viable parameter space of these models subject to current LUXlimits and the projected reach of XENON1T. Present experimental constraints from LUXplace stringent limits on mixed DM models, with a DM mass of at least a few hundred GeVin most cases. The projected reach of XENON1T is significantly stronger, extending tomasses of at least 1 TeV, and in many cases larger. Using simplified models of mixed DM,we have identified direct detection blind spots, which are parameter regions at which σ SI vanishes identically, nullifying experimental limits on spin independent DM-nucleon scatter-ing. Finally, we have quantified the degree of fine-tuning required for mixing angles ( ξ WT )and for blind spot cancellations ( ξ BS ) required for thermal relic DM which is experimentallyviable. Our results for each of our simplified models are summarized in the discussion below.First, we studied singlet-doublet Majorana fermion DM, which is a generalization of mixedbino-Higgsino DM in the MSSM or singlino-Higgsino DM in the NMSSM. In these models theobserved thermal relic density can be produced for M χ < M D < ∼ χ = Ω DM with large Higgs couplings, particularly with M D > ∼ ξ WT , ξ BS > .
1, afterXENON1T nearly all models either exhibit a significant degree of blind spot tuning, ξ BS < .
1, or must have mixing angles which are sensitively well-tempered to produce Ω χ = Ω DM through coannihilation with small Higgs couplings with ξ WT < .
1. Thus, XENON1Tstrongly constrains the parameter space of singlet-doublet DM.The constraints placed by direct detection on singlet-doublet scalar models are alsosubstantial. In such models Ω χ = Ω DM can be achieved at any mass through mixing for M χ < M D < ∼
550 GeV with small DM-Higgs couplings, requiring significant well-temperingas in the fermionic case. For M χ > ∼
550 GeV, however, annihilation through Higgs-mediatedprocesses is required to accommodate Ω χ = Ω DM . Hence, these models require larger Higgscouplings and are subject to stronger direct detection bounds. Current limits from LUXplace strong bounds up to large M χ and XENON1T constrains almost the entire parameterspace studied. In the examined parameter space, the few allowed regions needed significantcoannihilation, with fine-tuning of ξ BS < . M χ < M T < ∼ . hχχ and quartic hhχχ interactionsare stronger than in the singlet-doublet scalar case, enhancing annihilation even for smallermixing angles and thus weakening direct detection bounds. LUX has little constrainingpower on the singlet-triplet parameter space. XENON1T constraints are strong for vanish-ing or negative quartic couplings, allowing for minimal well-tempering only for nearly puretriplet models with 1 . < ∼ M χ < ∼ . ξ WT , ξ BS > .
1, remains viable.The present work has focused exclusively on experimental constraints from spin indepen-dent direct detection. However, many complementary probes exist. For example, even atcancellation points with vanishing DM-Higgs coupling, there will generically be Z -mediatedspin dependent DM-nucleon scattering. Future direct detection probes [65–67] and both cur-rent and future constraints from neutrino telescopes [68–70] will place significant constraintson many of these models. Moreover, these models will also be constrained by astrophysicalprobes such as FERMI [71] and HESS [72]. We leave these analyses for future work. ACKNOWLEDGMENTS
We are grateful to the Kavli Institute for Theoretical Physics at Santa Barbara and theAspen Center for Physics, where part of this work was conducted. DS is supported in partby U.S. Department of Energy grant DE–FG02–92ER40701 and by the Gordon and BettyMoore Foundation through Grant No. 776 to the Caltech Moore Center for TheoreticalCosmology and Physics. CC is supported by a DOE Early Career Award DE-SC0010255. [1]
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